Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 945030 12 pageshttpdxdoiorg1011552013945030

Research ArticleMinimum Fuel Low-Thrust Transfers for Satellites Usinga Permanent Magnet Hall Thruster

Thais Carneiro Oliveira1 Evandro Marconi Rocco1

Joseacute Leonardo Ferreira2 and Antonio F B A Prado1

1 DMC Space Mechanics and Control Division National Institute for Space Research (INPE)Avenida dos Astronautas 1758 12227-010 Sao Jose dos Campos SP Brazil

2 Universidade de Brasılia Campus Darcy Ribeiro Plano Piloto 70910-900 Asa Norte DF Brazil

Correspondence should be addressed to Evandro Marconi Rocco evandrodeminpebr

Received 12 October 2012 Revised 9 November 2012 Accepted 14 November 2012

Academic Editor Vivian Gomes

Copyright copy 2013 Thais Carneiro Oliveira et alThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Most of the satellite missions require orbital maneuvers to accomplish its goals An orbital maneuver is an operation where theorbit of a satellite is changed usually applying a type of propulsionThe maneuvers may have several purposes such as the transferof a satellite to its final orbit the interception of another spacecraft or the adjustment of the orbit to compensate the shifts causedby external forces In this situation it is essential to minimize the fuel consumption to allow a greater number of maneuvers tobe performed and thus the lifetime of the satellite can be extended There are several papers and studies which aim at the fuelminimization in maneuvers performed by space vehicles In this context this paper has two goals (i) to develop an algorithmcapable of finding optimal trajectories with continuous thrust that can fit different types of missions and constraints at the sametime and (ii) to study the performance of two propulsion devices for orbital maneuvers under development at the Universidade deBrasilia including a study of the effects of the errors in magnitude of these new devices

1 Introduction

The present paper is concerned with the optimization ofspacecraft trajectories with minimum fuel consumption thatuses a low thrust as a propulsion system This type ofpropulsion system is the most economical type available inaerospace technology It assumes that a force with a lowmagnitude is applied during a certain time to change thetrajectory of the spacecraft

This type ofmaneuver has been studied in the literature byseveral researches with different goals Lawden [1 2] was oneof the first to use this idea and showed a maneuver betweentwo points spending minimum fuel He defined the conceptof ldquoprimer vectorrdquo the Lagrangemultiplier associatedwith thevelocity which is used to provide the conditions for optimaltrajectories

After that several researches were performed on thistopic A series of analytical methods considering the power-limited (PL) low-thrust transfer optimization using the two-body dynamics started with Beletsky and Egorov [3] They

solved the point-to-point transfer using linearization arounda reference orbit After that this method was changedand new versions appeared in the literature to solve theoptimal transfer orbit between two given state vectors like inSukhanov [4 5] and Sukhanov and Prado [6] In particularSukhanov [4] shows an interesting case where some orbitalelements of the final orbit are specified while some others arefree A generalization of that method to cover the three-bodyproblem and considering a generic force field is also availablein Sukhanov and Prado [7] Situations where there areconstraints imposed on the thrust direction are consideredin Sukhanov and Prado [8 9] A related study using the samePL system can be found in Fernandes and Carvalho [10] whoconsider analytical first-order solutions for transfers usinglimited power between arbitrary elliptical coplanar orbits

Another important application that uses the low-thrustpropulsion system is the problem of station keeping ofsatellites Any satellite moves away from its position due toperturbations or to the launch errors So station-keeping

2 Mathematical Problems in Engineering

maneuvers are required to keep a satellite in its desired posi-tion Geostationary satellites and constellations of satellitesare good examples where this type of maneuver is requiredPapers studying this problem are found in Boucher [11]using electric propulsion Bowditch [12] using magneticplasma thrusters Oleson et al [13] performing studies forgeostationary satellites Ely and Howell [14] studying howto compensate the effects of the resonant tesseral harmonicsCirci [15] using solar sails for the maneuvers Romero etal [16] using feedback control techniques and Gomes andPrado [17] who studied how to control perturbations thatchange the plane of the orbit of the spacecraft

Applications of transfers from the Earth to the Moon canalso be studied under this model like done by Pierson andKluever [18] which consider the existence of three stages forthe mission Song et al [19] who allow the flexibility of a lowthrust with variable magnitude Fazelzadeh and Varzandian[20] who use an approach involving the time-domain finiteelementmethod orMingotti et al [21] whouse an interestingapproach involving dynamical system theory Objects thatare at larger distances from the Earth can also be reachedwith low thrust Brophy and Noca [22] Santos et al [23]Santos et al [24] and Santos and Prado [25] show someoptions including asteroids and planets with emphasis onthe asteroid Apophis which is in a trajectory with a closeapproach with the Earth

Thefirst contribution of the present paper is a formulationand description of an algorithm developed to solve thisproblem which has many important features to increase thenumber of maneuvers that get convergence and to increasethe velocity of convergence in those situations This problemis very sensible to small variation of parameters so severalactions are required to increase the efficiency of the conver-gence This algorithm is based in Biggs [26 27] Prado [28]and Oliveira [29] It has several important features combinedin a single algorithm which makes it very interesting for usein real missions

The conditions imposed on the final orbit must beat least in one of its Keplerian elements but can be inmore elements depending on the goals of the mission Thealgorithm also allows the consideration of restrictions likeforbidding thrusting in certain parts of the orbits specifiedby the interval between the true longitudes This restrictionimplies that both the start and the stop of the thrust have tooccur inside this interval This is an important characteristicof the algorithm because it is possible to avoid turningon the propulsion system when the satellite is not underdirect observation from the ground In addition restrictionsin the direction of the thrust can also be considered Thisis important in order to accommodate propulsion systemsthat have limits in the pointing of the thrust The presentalgorithm can also perform maneuvers with several arcs ofpropulsion so it is possible to accommodate constraints offorbidden regions of burn that repeat along the orbit likethe restriction of turning off the propulsion system whenthe satellite is not under tracking from the ground Thisis one of the most important and desired characteristicsof the algorithm because the operation of turning on theengine when the satellite is not under tracking is very risk

in particular when a new propulsion system is used Theseconstraints and considerations make the algorithm morecomplex but capable of finding optimal trajectories withrealistic constraints Another important point is the use ofnonsingular variables so equatorial and circular orbits canbe used for the satellite

Regarding the propulsion system it is assumed that thethrust has a constant magnitude of any level from very lowto very high its direction can be fully controlled and it can beturned on and off at any time In this way to find the optimalmaneuver it is necessary to find the time to start and to stopthe engine and the direction of the thrust at every instantof time specified by the values of the pitch and yaw anglesSo it is applicable to transfers of large amplitudes or station-keepingmaneuvers It is also possible to include coasting arcswhich gives even more flexibility to the method

The second task of this work is to simulate the optimalmaneuver found by the algorithm described previously in arealistic system that can consider errors in the propulsiondevice with embedded closed loop PID (proportional inte-gral and derivative) controller This task is very importantbecause one of the goals of the present paper is to test theperformance of a new propulsion system under developmentat the Universidade of Brasilia and the effects of its errorshave to be considered

The propulsion system developed at the Plasma Labora-tory of the Universidade de Brasilia (Ferreira et al [30]) isan electric propulsion called permanentmagnet hall thrusterThe hall thruster is an electromagnetic propulsion whichaccelerates an ionized propellant gas by the application ofboth electric andmagnetic fields as defined by Stuhlinger [31]and Jahn [32] The advantage of choosing electric propulsionis to reduce the fuel consumption of the maneuver since ithas a high exhaust velocity

The electric propulsion is already recognized as a success-ful technology for long-duration space missions (Ferreira etal [30]) It has been used as primary propulsion system onEarth-Moonorbit transfermissions for trajectories to cometsand asteroids and on commercially geosynchronous satelliteattitude control systems

In this way the present paper has the intention of showingan algorithm that is very flexible and realistic regardingconstraints and other specific conditions and then using thisalgorithm to verify the performance of a new propulsionsystem that is under development taking into account errorsin magnitude of the thrust

2 Mathematical Model

First of all it is necessary to introduce the notation for theorbital elements used in this paper and other importantdefinitions The notation is 119886 = semimajor axis of the orbit ofthe spacecraft 119890 = eccentricity of the orbit of the spacecraft119894 = inclination of the orbit of the spacecraft Ω = argumentof the ascending node of the orbit of the spacecraft 120596 =argument of the perigee of the orbit of the spacecraft 119907 = trueanomaly of the spacecraft 119904 = range angle of the spacecraftand Φ = 119907 + 120596 minus 119904

Mathematical Problems in Engineering 3

Reference lineS

Figure 1 Definition of the range angle and the arbitrary referenceline

The range angle ldquo119904rdquo is the independent variable of thiswork and it is the angle between the radius position vectorof the satellite and a reference arbitrary line which lies in theplane of the orbit as shown in Figure 1 It replaces the time inthe equations of motion

There are also two important angles pitch and yaw whichare used to define the direction of the thrust The angle ofpitch is designated by119860 while the angle yaw is designated by119861 The pitch is the angle between the thrust direction and theperpendicular to the radius of the spacecraft while yaw is theangle between the thrust direction and the orbital plane Thestate variables used here are defined as follows [26ndash29]

1198831=

119886 (1 minus 1198902)

120583

1198832= 119890 cos (120596 minus Φ)

1198833= 119890 sin (120596 minus Φ)

1198834=

fuel consumed1198980

1198835= time

1198836= cos( 119894

2

) cos(Ω + Φ

2

)

1198837= sin( 119894

2

) cos(Ω minus Φ

2

)

1198838= sin( 119894

2

) sin(Ω minus Φ

2

)

1198839= cos( 119894

2

) sin(Ω + Φ

2

)

(1)

where 120583 is the gravitational constant of the Earth and 1198980is

the initial mass of spacecraftThe advantage of using these state equations is to avoid

singularities if the inclination or the eccentricity is zero Theequations of motion and the control variables used here areshown in [26ndash29]The true longitude which can be used as aconstraint to define the prohibit region to apply the thrust isgiven by

120582 = 120596 + 119907 (2)

The constraints on the final Keplerian orbital elements aregiven by (3) All the orbital elements are scaled The desiredvalues are given by the index ldquolowastrdquo while the suffix ldquo

0rdquo denotes

the initial value The element without suffix or index is thecurrent value of the element The constraints on the finalorbital elements are given by

(119886 minus 119886lowast)

10038161003816100381610038161198860minus 119886lowast1003816100381610038161003816

= 0

(119890 minus 119890lowast)

10038161003816100381610038161198900minus 119890lowast1003816100381610038161003816

= 0

(119894 minus 119894lowast)

10038161003816100381610038161198940minus 119894lowast1003816100381610038161003816

= 0

(Ω minus Ωlowast)

1003816100381610038161003816Ω0minus Ωlowast1003816100381610038161003816

= 0

(120596 minus 120596lowast)

10038161003816100381610038161205960minus 120596lowast1003816100381610038161003816

= 0

(3)

The Pontryaginrsquos maximum principle is used to find theoptimal control that determinates the optimal trajectory Inthis case it is desirable to find the maximum final mass ofthe spacecraft given by the functional 119869(sdot) = minus119883

4(119904119891) This is

equivalent of minimizing the fuel consumption because theexpenditure of fuel is the only way that the satellite has to losemass More details of this process can be found in [26ndash29]The formulation of the problem is given by (4) to (6)

Maximize minus1198834(119904119891) (4)

with respect to 1199040 119904119891 1198600 1198610 1198601015840

0 1198611015840

0 119875119894(1199040) (5)

subjected to 119878119895(119883 (119904119891)) = 0 119895 = 1 119898 (6)

where 119878119895(119883(119904119891)) are the constraints stated in (2) and (3) or

any other typeSome of the initial variables that should be guessed are

the initial adjoint multipliers 119875119894(1199040) A good initial guess

for those variables is very important to assist convergenceand to reduce the time the algorithm needs to obtain thesolution Unfortunately the Lagrangemultipliers do not haveany physical meaning so it is hard to find a good initial guessfor them To solve this problem the adjoint-control trans-formation ([26ndash29]) is used to attribute a good estimativefor 119875119894(1199040) from variables that have physical meaning Those

variables are the angles 1198600and 119861

0and the values of 1198601015840

0and

1198611015840

0(Biggs [26] Prado [28])

3 Steps of the Algorithm

The final complete algorithm has the following steps [28 29]

(i) It starts from an initial guess for the set of variables(1198600 1198610 11986010158400 11986110158400 1199040 119904119891) With these initial values the

initial guess for the control is obtained

4 Mathematical Problems in Engineering

(ii) The ldquoadjoint-controlrdquo transformation is used to obtainthe initial values of the Lagrange multipliers requiredfor the numerical integrations

(iii) Now with all the initial values needed to solve theproblem the equations of motion and the adjointequations are simultaneously numerically integratedduring the propulsion arcThe values of the pitch andyaw angles are obtained at every step of the inte-gration by the principle of maximum of PontryaginAt the end the value of the final state 119883 after thepropulsion arc is obtained

(iv) The Keplerian elements of the orbit achieved are thenobtained from the final state119883 using (1)

(v) At this point the satisfaction of the constraints isverified If themagnitude of the vector that representsthe constraints satisfaction is smaller than a specifiedtolerance provided by the user (the numerical zero)the algorithm proceeds to step (vii) On the otherhand if it is bigger the algorithmproceeds to step (vi)

(vi) At this step the gradient of the constraints equations(nablaS) is obtained by perturbing the elements of thecontrol vector u and performing a new set of numer-ical integrations in order to obtain the numericalderivatives of the constraints equations with respectto the vector uThen a new set of values for the vectoru is found by (7) (Luemberger [33]) as follows

u119894+1

= u119894minus nablaS119879[nablaSnablaS119879]

minus1

nablaS (7)

and then the algorithm goes back to step (i) with thenew value of the vector u

(vii) Once the constraints satisfaction is achieved thenext step is to search for the minimum of the fuelconsumed The direction of the search d using thegradient method (Bazaraa et al [34]) is

d = minusRnabla119869 (u) (8)

where the vector R is

R = I minus nablaS119879[nablaSnablaS]minus1nablaS (9)

(viii) At this step the magnitude of the vector d is verifiedIf it is smaller than a value specified by the useras a direction search tolerance (numerical zero) thealgorithm goes to step (x) and if it is bigger thealgorithm proceeds to step (ix)

(ix) In order to get better results the initial data ldquoratio ofcontractionrdquo RC and the direction search toleranceare reduced as u gets closer to the minimum Themagnitude for the direction search is then given by

PB =

RC119869 (u)nabla119869 (u) d

(10)

So the complete step is given by

u119894+1

= u119894+ PB d

|d| (11)

and then the algorithm goes back to step (i)

(x) At this step the possibility of the control u to be aKuhn-Tucker point is verified To check this condi-tion the vector VT is built formed by the 119898 firstelements of the vector given by (Bazarra et al [34])as follows

W = minus(SS119879)minus1

Snabla119869 (u) (12)

where119898 is the number of active constraints If VT isnot positive definite the line 119895 which causes VT

119895to

be negative on the matrix S is deleted And then thealgorithm returns to step (v)

(xi) The current step is verified and if it is not the last onethe algorithm goes to a new search for the minimumfuel consumption with a smaller direction searchtolerance for the objective function If this step is thelast one the algorithm is finalized

4 Extensions of the Algorithm

There are several extensions of the algorithm that can beconsidered in order to make it more flexible The first oneis the possibility to perform maneuvers with more than onearc of propulsion Increasing the number of propulsion arcsmeans that the number of optimization variables is increasedFor example for one thrust arc there are 119906

1to 1199066control

variables For two thrust arcs similarly there is a set of 1199061to

11990612control variables and so on In addition by adding more

than one thrust arc the inclusion of constraints to prevent theoverlap of thrust arcs becomes necessary

Another extension is the possibility to consider forbiddenregions to apply the propulsion as a constraintThese regionsare defined by the true longitude as shown in (2) Both valuesof the initial and final true longitudes of the prohibit regionhave to be specified

In some situations the initial range angles 1199040can be less

than zero It means that when the thrust is started the time isnegative To avoid negative time it is added a constraint thatimposes to 119904

0 the time to be equal or bigger than zero This

constraint is only active if 1199040lt 0

The last extension is the possibility to consider constraintsrestricting the pitch and yaw angles In this case the boundsconsidered on 119860 and 119861 are the ones given by Biggs [27]119860119871le 119860 le 119860

119906and 119861

119871le 119861 le 119861

119906 where the suffix ldquo119906rdquo

represents the maximum angle and the suffix ldquo119871rdquo representstheminimumangle If these constraint are active after findingthe optimal angles 119860lowast and 119861lowast and this problem is solved asfollows [26ndash29] 119860 = 119860

119906if 119860lowast ge 119860

119906 119860 = 119860

119871if 119860lowast le 119860

119871

119860 = 119860lowast if 119860

119871le 119860 le 119860

119906 Those equations are similarly

applied to the optimal yaw angles 119861lowast

5 The Permanent Magnet Hall Thruster

As defined by Jahn [32] the electric propulsion is the gasacceleration to generate thrust via electric heating electricbody forces or both electric and magnetic body forces Thegreatest advantage of the electric propulsion is the veryhigh exhaust velocities resulting in a reduction of the fuel

Mathematical Problems in Engineering 5

consumption Although electric propulsion has low thrustthe high specific impulse guarantees a reduced burned fuelalong the maneuver

The hall thruster used in this work as the propulsionsystem was fundamentally envisaged in Russia in 1960 andthe first successful space mission with this thrust propulsionwith 60mN was used in the Meteor satellite series in 1972(Zhurin et al [35]) After that this kind of propulsion systemhas been studied and developed in several countries such asFrance USA Japan and Brazil

The working principle of the hall thrusters is the useof an electromagnet to produce the main magnetic fieldresponsible for the plasma acceleration and generation (Fer-reira et al [30]) This configuration requires the use ofenergy to produce the magnetic field Nevertheless thepermanent magnet hall thruster (PMHT) developed at thePlasma Laboratory of the Universidade de Brasilia uses anarray of permanent magnets instead of an electromagnet toproduce a radial magnetic field inside the cylindrical plasmadrift channel of the thruster (Ferreira et al [30]) The useof permanent magnets instead of electromagnet guaranteeslow-power consumption and it can be used in small- andmedium-size satellites (Ferreira et al [30])

In the Plasma Laboratory of the Universidade de Brasıliait has been developed two kinds of hall thruster called Phall Iand Phall II The first one to be constructed was the Phall I Ithas a stainless steel chamber with 30 centimeters of diameterIts magnetic field is produced by two concentric cylindricalarrays of ferrite permanent magnets 32 bars in the outer shelland 10 bars in the inner shellThemagnetic field in themiddleline of the plasma sourcersquos channel is 200 Gauss (Ferreiraet al [30]) On the other hand the second thruster PhallII has an aluminum plasma chamber and it is smaller with15 cm diameter and contains neodymium-iron magnets toproduce the magnetic field (Ferreira et al [30]) Table 1 givesthe parameters of the performance of both Phall I and PhallII thrusters (Ferreira et al [30])

In this paper it was found first the optimal maneuverswith the thrust parameters shown in Table 1 Then theseoptimal maneuvers were simulated with a PID control for along period of time based on the continuous monitoring ofthe thrust in a real-time performance correcting any possiblefailure

6 Spacecraft Trajectory Simulator

Themaneuver simulator used to simulate the optimalmaneu-vers in a more realistic way is the spacecraft trajectorysimulator (STRS) The STRS can consider constructionalfeatures and operation such as nonlinearities failures errorsand external and internal disturbances in order to determinethe deviation in the reproduction of the optimal solutionpreviously found

Furthermore maneuvers with continuous thrust are usedduring a long period of timeThis kind ofmaneuvers requiresa PID control system to guarantee the achievement of the finalconditions of the maneuver The operating structure of theSTRS can be described as follows (Rocco [36])

(a) The simulation occurs in a discrete way that is atevery simulation step the state of the vehicle (positionand velocity) must be computed For all purposes thedisturbances nonlinearities errors and failures mustbe considered

(b) Since a closed loop system was used to controlthe trajectory the reference state is determined byapplying the optimal maneuver Then the currentstate follows the reference but considers the effectsof disturbances and the constructive characteristics ofthe vehicle

(c) The difference between the reference state and thecurrent state creates an error signal that is insertedinto a PID control

(d) The controller produces a control signal according tothe PID control law and the gains that have been set

(e) The control signal is inserted into the actuator model(propellant thrusters) where the nonlinearities inher-ent to the construction of the actuator can be consid-ered Thus the behavior of the propellant thrusterscan be reproduced by the appropriate adjustmentof the model parameters which are supplied by themanufacturer of the thrust actuator

(f) Finally an actuation signal (Δ119881) is sent to the dynam-ical model of the orbital motion At this signal thevelocities increments due to the disturbances can beadded

(g) With the actuation signal added with the possible dis-turbances themodel of the orbital dynamics providesthe state (position and velocity) after the applicationof the propulsion

(h) Through the use of sensors that were modeled con-sidering its constructive aspects the current state isdetermined and compared with the reference state inorder to close the control loop

7 Results

Before performing tests with the propulsion system PhallsI and II that is one of the goals of the present paper it isimportant to test some of the particular characteristics ofthe algorithm developed here Those tests are performed inmaneuvers that have similar results available in the literatureso a comparison to validate the algorithm and the softwaredeveloped can be made After that the results of the optimalmaneuvers using the new propulsion devices as well as thesimulations of them in the STRS are presented

71 Optimal Maneuvers to Validate the Algorithm

711 Maneuver 1 This first maneuver described has a con-straint at the pitch angle in order to test this capability of theproposed algorithm

Initial orbital elements

119886 = 99000 km 119890 = 07 119894 = 10∘ Ω = 55∘ 120596 = 105∘ 119907 =minus105∘

6 Mathematical Problems in Engineering

Table 1 Comparative performance parameters measured in PHALL I and PHALL II

PHALL1 PHALL2Maximum expected thrust (mN) 126 126Average measured thrust (mN) 849 1200Thrust density (Nm2) 468 lt60Maximum specific impulse (s) 1607 1607Measured specific impulse (s) 1083 1600Ionized mass ratio () 33 sim30Propellant consumption (Kgs) 6 0 times 10

minus61 0 times 10

minus6

Energy consumption (W) 350 250ndash350Electrical efficiency () 339 60Total efficiency () 1012 500

Range angle (degrees)

Pitc

h (d

egre

es)

1101009080

012345

120 130 140

minus1minus2minus3minus4minus5

Figure 2 Optimal pitch angles of the maneuver 1

Vehicle mass = 300 kg thrust = 10 Newton initialposition 119904 = 0∘Condition on the final orbit 119886 = 104000 kmOther constraints 5∘ ge 119860 ge minus5∘ (pitch constraint)

Final orbital elements obtained after the maneuver

119886 = 10399921 km 119890 = 07143 119894 = 10∘Ω = 55∘ 120596 = 105∘119907 = 3024∘

Fuel consumption 24235 kg fuel consumption in Biggs[27] 244 kg fuel consumption in Prado [28] 245 kg

Figure 2 shows the direction of the thrustNow with the constraint in the direction of propulsion

the fuel consumption was increased when compared witha maneuver without this constraint [27 28] The extra fuelneeded due to this constraint is 38 times 10minus3 kg It is possible toview clearly that the pitch angle is confined to 5∘ at the endof the maneuver In any case the algorithm developed hereobtains better results when compared to Biggs [27] and Prado[28]

712 Maneuver 2 This maneuver has the constraint of aregion prohibited for thrusting This is one of the mostimportant capabilities of the algorithm developed here

Initial orbital elements

119886 = 99000 km 119890 = 07 119894 = 10∘ Ω = 55∘ 120596 = 105∘ 119907 =minus105∘

Range angle (degrees)

Pitc

h (d

egre

es)

55504540353025 60 65 70

minus20minus21minus22minus23minus24minus25minus26minus27minus28minus29minus30

Figure 3 Optimal pitch angles of the maneuver 2

Vehicle mass = 300 kg thrust = 10 Newton initialposition 119904 = 0∘Condition on the final orbit 119886 = 104000 kmOther constraints no thrust between the true longi-tudes 120∘ and 180∘

Final orbital elements

119886 = 10399985 km 119890 = 07134 119894 = 10∘Ω = 55∘120596 = 105∘119907 = 32080∘

Fuel consumption 27852 kg fuel consumption in Biggs[27] 281 kg fuel consumption in Prado [28] 281 kg

Figure 3 shows the direction of the thrustIn Figure 3 it is possible to note that the solution has the

thrust arc ending in 120582 = 120∘ The true longitude is definedhere as 120582 = 120596 + 119907 = 42580

∘ that is similar to 6580∘ that isthe result shown in Figure 3 In this way the region prohibitof thrust was obeyed and compared with the maneuver thatis free of this constraint the extra fuel needed to reach thefinal semimajor axis required was 03655 kg Once againthe algorithm proposed here obtained better results whencompared to Biggs [27] and Prado [28]

713 Maneuver 3 This maneuver has the requirements ofchanging three orbital elements of the orbit at the same time

Initial orbital elements

119886 = 419041 km 119890 = 0018 119894= 0688∘ Ω = minus298∘ 120596 =7∘ 119907 = minus972∘

Mathematical Problems in Engineering 7

Range angle (degrees)

Pitc

h (d

egre

es)

7570656055 80 85

200

150

100

50

0

minus50

minus100

minus150

minus200

Figure 4 Optimal pitch angles of the maneuver 3 for the first arc

Yaw

(deg

rees

)

Range angle (degrees)

7570656055 80 85

minus612

minus614

minus616

minus618

minus62

minus622

minus624

minus626

Figure 5 Optimal yaw angles of the maneuver 3 for the first arc

Vehicle mass = 300 kg thrust = 10 Newton initialposition 119904 = 0∘Condition on the final orbit 119886 = 421642 km 119890 = 0119894 = 0∘

Final orbital elements

119886 = 4216455 km 119890 = 000130 119894 = 00419∘Ω = 30060∘120596 = 10047∘ 119907 = 14364∘

Fuel consumption 5565 kg fuel consumption in Biggs[27] 5621 kg fuel consumption in Prado [28] 5579 kg

Figures 4ndash7 show the direction of the thrustIn this maneuver there are three Keplerian elements

fixed on the final orbit semimajor axis eccentricity andinclination Since the maneuver includes a change in theorbital plane the yaw angle is not zero The final conditionswere satisfied and the fuel consumption was reduced by0014 kg when compared to Prado [28]

72 Optimal Maneuvers Using the Phalls I and II In thissection we present the optimal maneuvers found by thealgorithm presented in the present paper when using PhallsI and II The optimal thrust angles are presented and thespecific impulse considered for each maneuver is given inTable 1 Figures 8ndash23 present the optimal angles for theproposedmaneuvers given by the initial and final range angleand the pitch angle at each arc of propulsionThe yaw angle isalways zero because all the maneuvers simulated are planar

Range angle (degrees)

Pitc

h (d

egre

es)

215205 210200190 195185 220 225

30

25

20

15

10

5

0

minus5

Figure 6 Optimal pitch angles of themaneuver 3 for the second arc

Range angle (degrees)

Yaw

(deg

rees

)

215205 210200190 195185

60

55

50

45

40

35

30220 225

Figure 7 Optimal yaw angles of the maneuver 3 for the second arc

All the maneuvers considered here have the same initialorbital elements and final condition imposed for the finalorbit Only the number of arcs and the propulsion system isdifferent at each maneuver The parameters of the maneuverproposed are as follows

Initial orbital elements 119886 = 7130865 km 119890 = 00035 119894= 985054∘ Ω = 0∘ 120596 = 0 119907 = 220∘Vehicle mass = 300 kgCondition imposed on the final orbit 119886 = 7200 km 119890= 0004

721 Maneuver 4 For this maneuver it was considered theexistence of three Phall I thrusters The propulsion force ofeach one is assumed to be the average measured value Theparameters considered in this maneuver are as follows

Total thrust = 0252 Newton specific impulse 1083 sSolution

Final orbital elements 119886 = 720000 km 119890 = 0004Fuel consumption 1046346 kgDuration of the maneuver 4406696 s

722 Maneuver 5 Thismaneuver has the same conditions ofmaneuver 4 but now two thrust arcs are used instead of oneThe parameters considered in this maneuver are as follows

8 Mathematical Problems in Engineering

Range angle (degrees)

30002500200015001000500

Pitc

h an

gle

(deg

rees

)

0

0

5

10

15

minus10

minus15

minus5

Figure 8 Optimal pitch angles for maneuver 4

Range angle (degrees)

Pitc

h an

gle

(deg

rees

) 03

010

02

0405

1400120010008006004002000

minus01minus02

minus03minus04minus05

Figure 9 Optimal pitch angles for the first arc of the maneuver 5

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

1

0

05

15

2

40003800360034003200300028002600

minus05

minus1

minus15

minus2

Figure 10 Optimal pitch angles for the second arc of maneuver 5

Total thrust = 0252 Newton specific impulse 1083 s

Solution

Final orbital elements after the first arc 119886 =716530 km 119890 = 00039Final orbital elements after the second arc 119886 =720000 km 119890 = 00040Fuel consumption 1013402 kgDuration of the maneuver 4267954 s

Note that the use of two arcs generates savings in the fuelconsumption as expected

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

0

5

10

15

22001800 2000160014001000 1200800600400200

minus5

minus10

minus15

Figure 11 Optimal pitch angle for maneuver 6

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

02

0

0604

081

1000900600 700 8005004003002001000

minus02

minus06

minus1

minus04

minus08

Figure 12 Optimal pitch angles for the first arc of maneuver 7

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

0

05

1

15

240023002100 2200200019001800170016001500

minus05

minus1

minus15

Figure 13 Optimal pitch angles for the second arc of maneuver 7

723 Maneuver 6 For this maneuver it was considered theexistence of three Phall II thrusters The propulsion force ofeach one is assumed to be the average measured value Theparameters considered in this maneuver are as follows

Total thrust = 0360 Newton specific impulse 1600 s

Solution

Final orbital elements 119886 = 720000 km 119890 = 0004Fuel consumption 0697930 kgDuration of the maneuver 3039873 s

Mathematical Problems in Engineering 9

Time (s)

minus5

minus10

0

0

5

05 1 15 2 25 3 35 4 45times104

Sem

imaj

or a

xis

dev

iati

on(m

)

Figure 14 Semimajor axis deviation in maneuver 4

02468

1012

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus2minus4minus6minus8

Ecc

entr

icit

y d

evia

tion

times10minus7

Figure 15 Eccentricity deviation in maneuver 4

0

0005

001

0015

Posi

tion

dev

iati

on (m

)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus0005

minus001

Figure 16 Position deviation in maneuver 4

Note that the use of the Phall II generates savings inthe fuel consumption since it has better parameters whencompared to the Phall I device

724 Maneuver 7 Thismaneuver has the same conditions ofmaneuver 6 but now with two thrust arcs instead of oneTheparameters considered in this maneuver are as follows

Total thrust = 0360 Newton specific impulse 1600 s

Solution

Final orbital elements after the first arc 119886 =716777 km 119890 = 00039

0

1

2

3

4

5

Vel

ocit

y d

evia

tion

(m

s)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus1

minus2

minus3

times10minus3

Figure 17 Velocity deviation in maneuver 4

Final orbital elements after the second arc 119886 =720000 km 119890 = 00040Fuel consumption 0686314 kgDuration of the maneuver 2989285 s

Note that the use of two arcs generates savings in the fuelconsumption one more time as expected

73 Optimal Maneuvers Simulations Using the STRS Nowwith the help of the optimal maneuvers found previously weshall simulate thosemaneuvers in the STRS using the optimalangles Some different aspects of the propulsion system wereconsidered to simulate amore realistic environment and alsothe PID control was considered

In this section for the propulsion system it was consid-ered that at each inertial direction 119909 119910 119911 has an error on theactuator signal (ms) with mean equal to zero and varianceequal to 10minus10 that is it was inserted an error in the velocity ofthe satelliteThis error has amplitude approximately close to 5times 10minus3N for each thrust which corresponds to the error of thePMHT (Ferreira et al [30]) There is a PID control to correctthe errors due the actuator but it is still possible to analyzethe deviations and the increase of the fuel consumption dueto the errors

731 Maneuver 4 in STRS In this case it is consideredmaneuver 4 including the solution found there

Final orbital elements 119886 = 719985 km 119890 = 0004Fuel consumption 10470 kgPropellant mass difference from the optimal maneu-ver 666 times 10minus4 kg

The error inserted in the propulsion system resulted in adeviation of the orbital elements as we can see in Figures 14and 15 for the semimajor axis and eccentricity If there wasnot any error at the propulsion system the deviation wouldbe zero all the time for the orbital elements In Figures 16 and17 we can see also the effect of the error in the velocity andposition as a function of these parameters Finally the thrusterror is shown in Figure 18

10 Mathematical Problems in EngineeringA

pplie

d th

rust

(N)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

0504

03

02

01

0

minus01

minus02

minus03

minus04

x axisy axisz axis

ThrustAverage

Figure 18 Thrust applied in maneuver 4

0

5

10

Time (s)

0 05 1 15 2 25 3 35times104

minus5

minus10

minus15

Sem

imaj

or a

xis

dev

iati

on(m

)

Figure 19 Semimajor axis deviation in maneuver 6

732 Maneuver 5 in STRS In this case maneuver 5 isconsidered including the solution found

Final orbital elements after the first arc 119886 =716523 km 119890 = 00039

Final orbital elements after the second arc 119886 =719980 km 119890 = 00040

Fuel consumption 1013362 kg

Propellant mass difference from the optimal maneu-ver 6384 times 10minus4 kg

733 Maneuver 6 in STRS Now maneuver 6 is considered

Final orbital elements 119886 = 719982 km 119890 = 0004

Fuel consumption 06979 kg

Propellant mass difference from the optimal maneu-ver 219 times 10minus4 kg

The same analysis for the figures made in Section 731 fitshere Figures 19ndash23 show the deviations and the error thatoccurred on the propulsion system

Time (s)

0 05 1 15 2 25 3 35times104

Ecc

entr

icit

y d

evia

tion

times10minus6

2

15

1

05

0

minus05

minus1

minus15

Figure 20 Eccentricity deviation in maneuver 6

734 Maneuver 7 in STRS Maneuver 7 is now studiedFinal orbital elements after the first arc 119886 =716770 km 119890 = 00039Final orbital elements after the second arc 119886 =719983 km 119890 = 00040Fuel consumption 0686279 kgPropellant mass difference from the optimal maneu-ver 21469 times 10minus4 kg

735 Maneuver 8 in STRS This maneuver uses the optimalangles given in maneuver 4 but includes the errors proposedfor the propulsion system In addition it was consideredother nonlinearities for the actuator two seconds delay torespond the control signal a rate limiter for the variationof the satelite velocity which the absolute limit of the rateis 000025ms dead zone where the propulsion system isnot turned on if the required velocity increment is lowerthan 00002ms Considering these nonlinearities for thepropulsion system the PID control was able to achievethe maneuver although the fuel consumption was slightlyincreased The results for this simulation are as follows

Final orbital elements 119886 = 719986 km 119890 = 0004Fuel consumption 11276322 kgPropellant mass difference from the optimal maneu-ver 0081429 kg

736 Maneuver 9 in STRS The maneuver simulated hereuses the optimal angles given in maneuver 6 with the errorsproposed for the propulsion system In addition it wasproposed the samenonlinearities (rate limiter dead zone andtime delay) of maneuver 8 Considering these nonlinearitiesfor the propulsion system the PID control was able to achievethe maneuver although the fuel consumption was increasedThe results for this maneuver are as follows

Final orbital elements 119886 = 719984 km 119890 = 00040Fuel consumption 0721921 kgThe propellant mass difference from the optimalmaneuver 002402 kg

Mathematical Problems in Engineering 11

Time (s)

0 05 1 15 2 25 3 35times104

002

0015

001

0005

0

minus0005

minus001

minus0015Po

siti

on d

evia

tion

(m)

Figure 21 Position deviation in maneuver 6

Vel

ocit

y d

evia

tion

(m

s)

Time (s)

0 05 1 15 2 25 3 35times104

8

6

4

2

0

minus2

minus4

times10minus3

Figure 22 Velocity deviation in maneuver 6

x axisy axisz axis

ThrustAverage

Time (s)

0 05 1 15 2 25 3 35times104

06

04

02

0

minus02

minus04

minus06

minus08

App

lied

thru

st (N

)

Figure 23 Thrust applied in maneuver 6

8 Conclusion

Primarily this paper established an algorithm to solve theproblem of orbital maneuvers using a low-thrust control fora spacecraft that is travelling around the Earth Then allthe maneuvers studied in this paper were successfully solvedwith the final conditions and constraints proposed and withreasonable fuel consumptions

It was possible to verify that maneuvers with morethrusting arcs consume less fuel as expected This occursbecause the algorithm has more variables to optimize so itis possible to reduce the fuel consumption

The second part of this work was to simulate the optimalmaneuvers found by the algorithm proposed here in a morerealistic environment which can consider errors on theactuator that was not considered in the search for the optimalmaneuver

Themaneuvers simulated in the STRSwere coherent withthe optimal maneuvers validating the integration of bothsoftwares

The error considered in the propulsion system opensthe possibility to study how the system would react tothose nonlinearities Although errors were inserted in thepropulsion system the PID control was able to correct andaccomplish the maneuver

It is possible to analyze the shifts caused by the error aswell as the deviations in semimajor axis and eccentricity It isalso possible to see that the propulsion thrust was not linear

For themaneuvers shown in Sections 735 and 736 somenonlinearities of the propulsion system such as a responsetime a rate limiter and a dead zone for the actuator signalwere considered All these errors occur in real systems Withthe help of the STRS it was possible to simulate these errorsand analyze the maneuvers studied with more accuracyThe STRS has shown to be an efficient simulator capableof simulating and studying optimal maneuvers in a morerealistic environment and to analyze many nonlinearities notconsidered in the calculation of the optimal maneuver

Evidently the phall II was more efficient than the phall Ias noticed by comparing maneuver 4 with maneuver 6 andmaneuver 5 with maneuver 7 This was expected since thephall II has a higher specific impulse and a higher averagepropulsion thrust as well Nevertheless the PID control wasable to control all the maneuvers and reaches the final orbitalconstraints

References

[1] D F Lawden ldquoMinimal rocket trajectoriesrdquo ARS Journal vol23 no 6 pp 360ndash382 1953

[2] D F Lawden ldquoFundamentals of space navigationrdquo Journal of theBritish Interplanetary Society vol 13 pp 87ndash101 1954

12 Mathematical Problems in Engineering

[3] V V Beletsky and V A Egorov ldquoInterplanetary flights withconstant output enginesrdquoCosmic Research vol 2 no 3 pp 303ndash330 1964

[4] A A Sukhanov ldquoOptimization of flights with low thrustrdquoCosmic Research vol 37 no 2 pp 182ndash191 1999

[5] A A Sukhanov ldquoOptimization of low-thrust interplanetarytransfersrdquo Cosmic Research vol 38 no 6 pp 584ndash587 2000

[6] A A Sukhanov and A F B A Prado ldquoA modification of themethod of transporting trajectoryrdquoCosmic Research vol 42 no1 pp 107ndash112 2004

[7] A A Sukhanov and A F B A Prado ldquoOptimization of lowthrust transfers in the three body problemrdquo Cosmic Researchvol 46 no 5 pp 440ndash451 2008

[8] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction Irdquo Cosmic Researchvol 45 no 5 pp 417ndash423 2007

[9] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction IIrdquo Cosmic Researchvol 46 no 1 pp 49ndash59 2008

[10] S S Fernandes and F D C Carvalho ldquoA first-order analyticaltheory for optimal low-thrust limited-power transfers betweenarbitrary elliptical coplanar orbitsrdquo Mathematical Problems inEngineering vol 2008 Article ID 525930 30 pages 2008

[11] R A Boucher ldquoElectrical propulsion for control of stationarysatellitesrdquo Journal of Spacecraft and Rockets vol 1 no 2 pp 164ndash169 1964

[12] D N Bowditch ldquoApplication of magnetic-expansion plasmathrusters to satellite station keeping and attitude control mis-sionsrdquo in Proceedings of the 4th Electric Propulsion Confer-ence AIAA-1964-677 pp 64ndash677 The American Institute ofAeronautics and Astronautics Philadelphia Pa USA Augest-September 1964

[13] S R Oleson R M Myers C A Kluever J P Riehl and F MCurran ldquoAdvanced propulsion for geostationary orbit insertionand north-south station keepingrdquo Journal of Spacecraft andRockets vol 34 no 1 pp 22ndash28 1997

[14] T A Ely and K C Howell ldquoEast-west stationkeeping of satelliteorbits with resonant tesseral harmonicsrdquoActa Astronautica vol46 no 1 pp 1ndash15 2000

[15] C Circi ldquoSimple strategy for geostationary stationkeepingmaneuvers using solar sailrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 249ndash253 2005

[16] P Romero J M Gambi and E Patino ldquoStationkeepingmanoeuvres for geostationary satellites using feedback controltechniquesrdquo Aerospace Science and Technology vol 11 no 2-3pp 229ndash237 2007

[17] V Gomes and A F B A Prado ldquoLow-thrust out-of-planeorbital station-keeping maneuvers for satellitesrdquo MathematicalProblems in Engineering vol 2012 Article ID 532708 14 pages2012

[18] B L Pierson and C A Kluever ldquoThree-stage approach to opti-mal low-thrust earth-moon trajectoriesrdquo Journal of GuidanceControl and Dynamics vol 17 no 6 pp 1275ndash1282 1994

[19] Y J Song S Y Park K H Choi and E S Sim ldquoA lunar cargomission design strategy using variable low thrustrdquo Advances inSpace Research vol 43 no 9 pp 1391ndash1406 2009

[20] S A Fazelzadeh and G A Varzandian ldquoMinimum-time earth-moon and moon-earth orbital maneuvers using time-domainfinite element methodrdquo Acta Astronautica vol 66 no 3-4 pp528ndash538 2010

[21] G Mingotti F Topputo and F Bernelli-Zazzera ldquoEfficientinvariant-manifold low-thrust planar trajectories to themoonrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 p 817 2012

[22] J R Brophy and M Noca ldquoElectric propulsion for solar systemexplorationrdquo Journal of Propulsion and Power vol 14 no 5 pp700ndash707 1998

[23] D P S Santos A F B A Prado L Casalino and G ColasurdoldquoOptimal trajectories towards near-earth-objects using solarelectric propulsion (sep) and gravity assistedmaneuverrdquo Journalof Aerospace Engineering Sciences and Applications vol 1 no 2pp 51ndash64 2008

[24] D P S Santos L Casalino G Colasurdo and A F B APrado ldquoOptimal trajectories towards near-earth-objects usingsolar electric propulsion (sep) and gravity assisted maneuverrdquoWSEAS Transactions on Applied andTheoretical Mechanics vol4 no 3 pp 125ndash135 2009

[25] D P S Santos and A F B A Prado ldquoOptimal low-thrusttrajectories to reach the asteroid apophisrdquo WSEAS MechanicalEngineering Series vol 7 pp 241ndash251 2012

[26] M C B BiggsThe Optimization of Spacecraft Orbital Manoeu-vres Part I Linearly Varying Thrust Angles The HatfieldPolytechnic Numerical Optimization Centre 1978

[27] M C B BiggsThe Optimisation of Spacecraft Orbital Manoeu-vres Part II Using Pontryaginrsquos Maximum Principle The Hat-field Polytechnic Numerical Optimization Centre 1979

[28] A F B A Prado Analise Selecao e Implementacao de Procedi-mentos que visemManobras Otimas de Satelites Artificiais [MSthesis] Instituto Nacional de Pesquisas Espaciais Sao Jose dosCampos 1988

[29] T C Oliveira Estrategias otimas para manobras orbitais uti-lizando propulsao contınua [MS thesis] Instituto Nacional dePesquisas Espaciais Sao Jose dos Campos 2012

[30] J L Ferreira G C Possa B SMoraes et al ldquoPermanentmagnethall thruster for satellite orbit maneuvering with low powerrdquoAdvances in Space Research vol 01 pp 166ndash182 2009

[31] E Stuhlinger Ion Propulsion for Space FlightMcGraw-Hill NewYork NY USA 1st edition 1964

[32] R G Jahn Physics of Electric Propulsion McGraw-Hill NewYork NY USA 1st edition 1968

[33] D G Luemberger Linear and Non-Linear ProgrammingSpringer Science amp Business Media New York NY USA 3rdedition 2008

[34] M S Bazaraa and C M ShettyNonlinear Programming-Theoryand Algorithms JohnWilley amp Sons New York NY USA 1979

[35] V V Zhurin H R Kaufman and R S Robinson ldquoPhysics ofclosed drift thrustersrdquo Plasma Sources Science and Technologyvol 8 no 1 pp R1ndashR20 1999

[36] E M Rocco ldquoPerturbed orbital motion with a PID controlsystem for the trajectoryrdquo in Proceedings of the 14th ColoquioBrasileiro De Dinamica Orbital Aguas de Lindoia AnaisUNESP Guaratingueta Brazil November 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

maneuvers are required to keep a satellite in its desired posi-tion Geostationary satellites and constellations of satellitesare good examples where this type of maneuver is requiredPapers studying this problem are found in Boucher [11]using electric propulsion Bowditch [12] using magneticplasma thrusters Oleson et al [13] performing studies forgeostationary satellites Ely and Howell [14] studying howto compensate the effects of the resonant tesseral harmonicsCirci [15] using solar sails for the maneuvers Romero etal [16] using feedback control techniques and Gomes andPrado [17] who studied how to control perturbations thatchange the plane of the orbit of the spacecraft

Applications of transfers from the Earth to the Moon canalso be studied under this model like done by Pierson andKluever [18] which consider the existence of three stages forthe mission Song et al [19] who allow the flexibility of a lowthrust with variable magnitude Fazelzadeh and Varzandian[20] who use an approach involving the time-domain finiteelementmethod orMingotti et al [21] whouse an interestingapproach involving dynamical system theory Objects thatare at larger distances from the Earth can also be reachedwith low thrust Brophy and Noca [22] Santos et al [23]Santos et al [24] and Santos and Prado [25] show someoptions including asteroids and planets with emphasis onthe asteroid Apophis which is in a trajectory with a closeapproach with the Earth

Thefirst contribution of the present paper is a formulationand description of an algorithm developed to solve thisproblem which has many important features to increase thenumber of maneuvers that get convergence and to increasethe velocity of convergence in those situations This problemis very sensible to small variation of parameters so severalactions are required to increase the efficiency of the conver-gence This algorithm is based in Biggs [26 27] Prado [28]and Oliveira [29] It has several important features combinedin a single algorithm which makes it very interesting for usein real missions

The conditions imposed on the final orbit must beat least in one of its Keplerian elements but can be inmore elements depending on the goals of the mission Thealgorithm also allows the consideration of restrictions likeforbidding thrusting in certain parts of the orbits specifiedby the interval between the true longitudes This restrictionimplies that both the start and the stop of the thrust have tooccur inside this interval This is an important characteristicof the algorithm because it is possible to avoid turningon the propulsion system when the satellite is not underdirect observation from the ground In addition restrictionsin the direction of the thrust can also be considered Thisis important in order to accommodate propulsion systemsthat have limits in the pointing of the thrust The presentalgorithm can also perform maneuvers with several arcs ofpropulsion so it is possible to accommodate constraints offorbidden regions of burn that repeat along the orbit likethe restriction of turning off the propulsion system whenthe satellite is not under tracking from the ground Thisis one of the most important and desired characteristicsof the algorithm because the operation of turning on theengine when the satellite is not under tracking is very risk

in particular when a new propulsion system is used Theseconstraints and considerations make the algorithm morecomplex but capable of finding optimal trajectories withrealistic constraints Another important point is the use ofnonsingular variables so equatorial and circular orbits canbe used for the satellite

Regarding the propulsion system it is assumed that thethrust has a constant magnitude of any level from very lowto very high its direction can be fully controlled and it can beturned on and off at any time In this way to find the optimalmaneuver it is necessary to find the time to start and to stopthe engine and the direction of the thrust at every instantof time specified by the values of the pitch and yaw anglesSo it is applicable to transfers of large amplitudes or station-keepingmaneuvers It is also possible to include coasting arcswhich gives even more flexibility to the method

The second task of this work is to simulate the optimalmaneuver found by the algorithm described previously in arealistic system that can consider errors in the propulsiondevice with embedded closed loop PID (proportional inte-gral and derivative) controller This task is very importantbecause one of the goals of the present paper is to test theperformance of a new propulsion system under developmentat the Universidade of Brasilia and the effects of its errorshave to be considered

The propulsion system developed at the Plasma Labora-tory of the Universidade de Brasilia (Ferreira et al [30]) isan electric propulsion called permanentmagnet hall thrusterThe hall thruster is an electromagnetic propulsion whichaccelerates an ionized propellant gas by the application ofboth electric andmagnetic fields as defined by Stuhlinger [31]and Jahn [32] The advantage of choosing electric propulsionis to reduce the fuel consumption of the maneuver since ithas a high exhaust velocity

The electric propulsion is already recognized as a success-ful technology for long-duration space missions (Ferreira etal [30]) It has been used as primary propulsion system onEarth-Moonorbit transfermissions for trajectories to cometsand asteroids and on commercially geosynchronous satelliteattitude control systems

In this way the present paper has the intention of showingan algorithm that is very flexible and realistic regardingconstraints and other specific conditions and then using thisalgorithm to verify the performance of a new propulsionsystem that is under development taking into account errorsin magnitude of the thrust

2 Mathematical Model

First of all it is necessary to introduce the notation for theorbital elements used in this paper and other importantdefinitions The notation is 119886 = semimajor axis of the orbit ofthe spacecraft 119890 = eccentricity of the orbit of the spacecraft119894 = inclination of the orbit of the spacecraft Ω = argumentof the ascending node of the orbit of the spacecraft 120596 =argument of the perigee of the orbit of the spacecraft 119907 = trueanomaly of the spacecraft 119904 = range angle of the spacecraftand Φ = 119907 + 120596 minus 119904

Mathematical Problems in Engineering 3

Reference lineS

Figure 1 Definition of the range angle and the arbitrary referenceline

The range angle ldquo119904rdquo is the independent variable of thiswork and it is the angle between the radius position vectorof the satellite and a reference arbitrary line which lies in theplane of the orbit as shown in Figure 1 It replaces the time inthe equations of motion

There are also two important angles pitch and yaw whichare used to define the direction of the thrust The angle ofpitch is designated by119860 while the angle yaw is designated by119861 The pitch is the angle between the thrust direction and theperpendicular to the radius of the spacecraft while yaw is theangle between the thrust direction and the orbital plane Thestate variables used here are defined as follows [26ndash29]

1198831=

119886 (1 minus 1198902)

120583

1198832= 119890 cos (120596 minus Φ)

1198833= 119890 sin (120596 minus Φ)

1198834=

fuel consumed1198980

1198835= time

1198836= cos( 119894

2

) cos(Ω + Φ

2

)

1198837= sin( 119894

2

) cos(Ω minus Φ

2

)

1198838= sin( 119894

2

) sin(Ω minus Φ

2

)

1198839= cos( 119894

2

) sin(Ω + Φ

2

)

(1)

where 120583 is the gravitational constant of the Earth and 1198980is

the initial mass of spacecraftThe advantage of using these state equations is to avoid

singularities if the inclination or the eccentricity is zero Theequations of motion and the control variables used here areshown in [26ndash29]The true longitude which can be used as aconstraint to define the prohibit region to apply the thrust isgiven by

120582 = 120596 + 119907 (2)

The constraints on the final Keplerian orbital elements aregiven by (3) All the orbital elements are scaled The desiredvalues are given by the index ldquolowastrdquo while the suffix ldquo

0rdquo denotes

the initial value The element without suffix or index is thecurrent value of the element The constraints on the finalorbital elements are given by

(119886 minus 119886lowast)

10038161003816100381610038161198860minus 119886lowast1003816100381610038161003816

= 0

(119890 minus 119890lowast)

10038161003816100381610038161198900minus 119890lowast1003816100381610038161003816

= 0

(119894 minus 119894lowast)

10038161003816100381610038161198940minus 119894lowast1003816100381610038161003816

= 0

(Ω minus Ωlowast)

1003816100381610038161003816Ω0minus Ωlowast1003816100381610038161003816

= 0

(120596 minus 120596lowast)

10038161003816100381610038161205960minus 120596lowast1003816100381610038161003816

= 0

(3)

The Pontryaginrsquos maximum principle is used to find theoptimal control that determinates the optimal trajectory Inthis case it is desirable to find the maximum final mass ofthe spacecraft given by the functional 119869(sdot) = minus119883

4(119904119891) This is

equivalent of minimizing the fuel consumption because theexpenditure of fuel is the only way that the satellite has to losemass More details of this process can be found in [26ndash29]The formulation of the problem is given by (4) to (6)

Maximize minus1198834(119904119891) (4)

with respect to 1199040 119904119891 1198600 1198610 1198601015840

0 1198611015840

0 119875119894(1199040) (5)

subjected to 119878119895(119883 (119904119891)) = 0 119895 = 1 119898 (6)

where 119878119895(119883(119904119891)) are the constraints stated in (2) and (3) or

any other typeSome of the initial variables that should be guessed are

the initial adjoint multipliers 119875119894(1199040) A good initial guess

for those variables is very important to assist convergenceand to reduce the time the algorithm needs to obtain thesolution Unfortunately the Lagrangemultipliers do not haveany physical meaning so it is hard to find a good initial guessfor them To solve this problem the adjoint-control trans-formation ([26ndash29]) is used to attribute a good estimativefor 119875119894(1199040) from variables that have physical meaning Those

variables are the angles 1198600and 119861

0and the values of 1198601015840

0and

1198611015840

0(Biggs [26] Prado [28])

3 Steps of the Algorithm

The final complete algorithm has the following steps [28 29]

(i) It starts from an initial guess for the set of variables(1198600 1198610 11986010158400 11986110158400 1199040 119904119891) With these initial values the

initial guess for the control is obtained

4 Mathematical Problems in Engineering

(ii) The ldquoadjoint-controlrdquo transformation is used to obtainthe initial values of the Lagrange multipliers requiredfor the numerical integrations

(iii) Now with all the initial values needed to solve theproblem the equations of motion and the adjointequations are simultaneously numerically integratedduring the propulsion arcThe values of the pitch andyaw angles are obtained at every step of the inte-gration by the principle of maximum of PontryaginAt the end the value of the final state 119883 after thepropulsion arc is obtained

(iv) The Keplerian elements of the orbit achieved are thenobtained from the final state119883 using (1)

(v) At this point the satisfaction of the constraints isverified If themagnitude of the vector that representsthe constraints satisfaction is smaller than a specifiedtolerance provided by the user (the numerical zero)the algorithm proceeds to step (vii) On the otherhand if it is bigger the algorithmproceeds to step (vi)

(vi) At this step the gradient of the constraints equations(nablaS) is obtained by perturbing the elements of thecontrol vector u and performing a new set of numer-ical integrations in order to obtain the numericalderivatives of the constraints equations with respectto the vector uThen a new set of values for the vectoru is found by (7) (Luemberger [33]) as follows

u119894+1

= u119894minus nablaS119879[nablaSnablaS119879]

minus1

nablaS (7)

and then the algorithm goes back to step (i) with thenew value of the vector u

(vii) Once the constraints satisfaction is achieved thenext step is to search for the minimum of the fuelconsumed The direction of the search d using thegradient method (Bazaraa et al [34]) is

d = minusRnabla119869 (u) (8)

where the vector R is

R = I minus nablaS119879[nablaSnablaS]minus1nablaS (9)

(viii) At this step the magnitude of the vector d is verifiedIf it is smaller than a value specified by the useras a direction search tolerance (numerical zero) thealgorithm goes to step (x) and if it is bigger thealgorithm proceeds to step (ix)

(ix) In order to get better results the initial data ldquoratio ofcontractionrdquo RC and the direction search toleranceare reduced as u gets closer to the minimum Themagnitude for the direction search is then given by

PB =

RC119869 (u)nabla119869 (u) d

(10)

So the complete step is given by

u119894+1

= u119894+ PB d

|d| (11)

and then the algorithm goes back to step (i)

(x) At this step the possibility of the control u to be aKuhn-Tucker point is verified To check this condi-tion the vector VT is built formed by the 119898 firstelements of the vector given by (Bazarra et al [34])as follows

W = minus(SS119879)minus1

Snabla119869 (u) (12)

where119898 is the number of active constraints If VT isnot positive definite the line 119895 which causes VT

119895to

be negative on the matrix S is deleted And then thealgorithm returns to step (v)

(xi) The current step is verified and if it is not the last onethe algorithm goes to a new search for the minimumfuel consumption with a smaller direction searchtolerance for the objective function If this step is thelast one the algorithm is finalized

4 Extensions of the Algorithm

There are several extensions of the algorithm that can beconsidered in order to make it more flexible The first oneis the possibility to perform maneuvers with more than onearc of propulsion Increasing the number of propulsion arcsmeans that the number of optimization variables is increasedFor example for one thrust arc there are 119906

1to 1199066control

variables For two thrust arcs similarly there is a set of 1199061to

11990612control variables and so on In addition by adding more

than one thrust arc the inclusion of constraints to prevent theoverlap of thrust arcs becomes necessary

Another extension is the possibility to consider forbiddenregions to apply the propulsion as a constraintThese regionsare defined by the true longitude as shown in (2) Both valuesof the initial and final true longitudes of the prohibit regionhave to be specified

In some situations the initial range angles 1199040can be less

than zero It means that when the thrust is started the time isnegative To avoid negative time it is added a constraint thatimposes to 119904

0 the time to be equal or bigger than zero This

constraint is only active if 1199040lt 0

The last extension is the possibility to consider constraintsrestricting the pitch and yaw angles In this case the boundsconsidered on 119860 and 119861 are the ones given by Biggs [27]119860119871le 119860 le 119860

119906and 119861

119871le 119861 le 119861

119906 where the suffix ldquo119906rdquo

represents the maximum angle and the suffix ldquo119871rdquo representstheminimumangle If these constraint are active after findingthe optimal angles 119860lowast and 119861lowast and this problem is solved asfollows [26ndash29] 119860 = 119860

119906if 119860lowast ge 119860

119906 119860 = 119860

119871if 119860lowast le 119860

119871

119860 = 119860lowast if 119860

119871le 119860 le 119860

119906 Those equations are similarly

applied to the optimal yaw angles 119861lowast

5 The Permanent Magnet Hall Thruster

As defined by Jahn [32] the electric propulsion is the gasacceleration to generate thrust via electric heating electricbody forces or both electric and magnetic body forces Thegreatest advantage of the electric propulsion is the veryhigh exhaust velocities resulting in a reduction of the fuel

Mathematical Problems in Engineering 5

consumption Although electric propulsion has low thrustthe high specific impulse guarantees a reduced burned fuelalong the maneuver

The hall thruster used in this work as the propulsionsystem was fundamentally envisaged in Russia in 1960 andthe first successful space mission with this thrust propulsionwith 60mN was used in the Meteor satellite series in 1972(Zhurin et al [35]) After that this kind of propulsion systemhas been studied and developed in several countries such asFrance USA Japan and Brazil

The working principle of the hall thrusters is the useof an electromagnet to produce the main magnetic fieldresponsible for the plasma acceleration and generation (Fer-reira et al [30]) This configuration requires the use ofenergy to produce the magnetic field Nevertheless thepermanent magnet hall thruster (PMHT) developed at thePlasma Laboratory of the Universidade de Brasilia uses anarray of permanent magnets instead of an electromagnet toproduce a radial magnetic field inside the cylindrical plasmadrift channel of the thruster (Ferreira et al [30]) The useof permanent magnets instead of electromagnet guaranteeslow-power consumption and it can be used in small- andmedium-size satellites (Ferreira et al [30])

In the Plasma Laboratory of the Universidade de Brasıliait has been developed two kinds of hall thruster called Phall Iand Phall II The first one to be constructed was the Phall I Ithas a stainless steel chamber with 30 centimeters of diameterIts magnetic field is produced by two concentric cylindricalarrays of ferrite permanent magnets 32 bars in the outer shelland 10 bars in the inner shellThemagnetic field in themiddleline of the plasma sourcersquos channel is 200 Gauss (Ferreiraet al [30]) On the other hand the second thruster PhallII has an aluminum plasma chamber and it is smaller with15 cm diameter and contains neodymium-iron magnets toproduce the magnetic field (Ferreira et al [30]) Table 1 givesthe parameters of the performance of both Phall I and PhallII thrusters (Ferreira et al [30])

In this paper it was found first the optimal maneuverswith the thrust parameters shown in Table 1 Then theseoptimal maneuvers were simulated with a PID control for along period of time based on the continuous monitoring ofthe thrust in a real-time performance correcting any possiblefailure

6 Spacecraft Trajectory Simulator

Themaneuver simulator used to simulate the optimalmaneu-vers in a more realistic way is the spacecraft trajectorysimulator (STRS) The STRS can consider constructionalfeatures and operation such as nonlinearities failures errorsand external and internal disturbances in order to determinethe deviation in the reproduction of the optimal solutionpreviously found

Furthermore maneuvers with continuous thrust are usedduring a long period of timeThis kind ofmaneuvers requiresa PID control system to guarantee the achievement of the finalconditions of the maneuver The operating structure of theSTRS can be described as follows (Rocco [36])

(a) The simulation occurs in a discrete way that is atevery simulation step the state of the vehicle (positionand velocity) must be computed For all purposes thedisturbances nonlinearities errors and failures mustbe considered

(b) Since a closed loop system was used to controlthe trajectory the reference state is determined byapplying the optimal maneuver Then the currentstate follows the reference but considers the effectsof disturbances and the constructive characteristics ofthe vehicle

(c) The difference between the reference state and thecurrent state creates an error signal that is insertedinto a PID control

(d) The controller produces a control signal according tothe PID control law and the gains that have been set

(e) The control signal is inserted into the actuator model(propellant thrusters) where the nonlinearities inher-ent to the construction of the actuator can be consid-ered Thus the behavior of the propellant thrusterscan be reproduced by the appropriate adjustmentof the model parameters which are supplied by themanufacturer of the thrust actuator

(f) Finally an actuation signal (Δ119881) is sent to the dynam-ical model of the orbital motion At this signal thevelocities increments due to the disturbances can beadded

(g) With the actuation signal added with the possible dis-turbances themodel of the orbital dynamics providesthe state (position and velocity) after the applicationof the propulsion

(h) Through the use of sensors that were modeled con-sidering its constructive aspects the current state isdetermined and compared with the reference state inorder to close the control loop

7 Results

Before performing tests with the propulsion system PhallsI and II that is one of the goals of the present paper it isimportant to test some of the particular characteristics ofthe algorithm developed here Those tests are performed inmaneuvers that have similar results available in the literatureso a comparison to validate the algorithm and the softwaredeveloped can be made After that the results of the optimalmaneuvers using the new propulsion devices as well as thesimulations of them in the STRS are presented

71 Optimal Maneuvers to Validate the Algorithm

711 Maneuver 1 This first maneuver described has a con-straint at the pitch angle in order to test this capability of theproposed algorithm

Initial orbital elements

119886 = 99000 km 119890 = 07 119894 = 10∘ Ω = 55∘ 120596 = 105∘ 119907 =minus105∘

6 Mathematical Problems in Engineering

Table 1 Comparative performance parameters measured in PHALL I and PHALL II

PHALL1 PHALL2Maximum expected thrust (mN) 126 126Average measured thrust (mN) 849 1200Thrust density (Nm2) 468 lt60Maximum specific impulse (s) 1607 1607Measured specific impulse (s) 1083 1600Ionized mass ratio () 33 sim30Propellant consumption (Kgs) 6 0 times 10

minus61 0 times 10

minus6

Energy consumption (W) 350 250ndash350Electrical efficiency () 339 60Total efficiency () 1012 500

Range angle (degrees)

Pitc

h (d

egre

es)

1101009080

012345

120 130 140

minus1minus2minus3minus4minus5

Figure 2 Optimal pitch angles of the maneuver 1

Vehicle mass = 300 kg thrust = 10 Newton initialposition 119904 = 0∘Condition on the final orbit 119886 = 104000 kmOther constraints 5∘ ge 119860 ge minus5∘ (pitch constraint)

Final orbital elements obtained after the maneuver

119886 = 10399921 km 119890 = 07143 119894 = 10∘Ω = 55∘ 120596 = 105∘119907 = 3024∘

Fuel consumption 24235 kg fuel consumption in Biggs[27] 244 kg fuel consumption in Prado [28] 245 kg

Figure 2 shows the direction of the thrustNow with the constraint in the direction of propulsion

the fuel consumption was increased when compared witha maneuver without this constraint [27 28] The extra fuelneeded due to this constraint is 38 times 10minus3 kg It is possible toview clearly that the pitch angle is confined to 5∘ at the endof the maneuver In any case the algorithm developed hereobtains better results when compared to Biggs [27] and Prado[28]

712 Maneuver 2 This maneuver has the constraint of aregion prohibited for thrusting This is one of the mostimportant capabilities of the algorithm developed here

Initial orbital elements

119886 = 99000 km 119890 = 07 119894 = 10∘ Ω = 55∘ 120596 = 105∘ 119907 =minus105∘

Range angle (degrees)

Pitc

h (d

egre

es)

55504540353025 60 65 70

minus20minus21minus22minus23minus24minus25minus26minus27minus28minus29minus30

Figure 3 Optimal pitch angles of the maneuver 2

Vehicle mass = 300 kg thrust = 10 Newton initialposition 119904 = 0∘Condition on the final orbit 119886 = 104000 kmOther constraints no thrust between the true longi-tudes 120∘ and 180∘

Final orbital elements

119886 = 10399985 km 119890 = 07134 119894 = 10∘Ω = 55∘120596 = 105∘119907 = 32080∘

Fuel consumption 27852 kg fuel consumption in Biggs[27] 281 kg fuel consumption in Prado [28] 281 kg

Figure 3 shows the direction of the thrustIn Figure 3 it is possible to note that the solution has the

thrust arc ending in 120582 = 120∘ The true longitude is definedhere as 120582 = 120596 + 119907 = 42580

∘ that is similar to 6580∘ that isthe result shown in Figure 3 In this way the region prohibitof thrust was obeyed and compared with the maneuver thatis free of this constraint the extra fuel needed to reach thefinal semimajor axis required was 03655 kg Once againthe algorithm proposed here obtained better results whencompared to Biggs [27] and Prado [28]

713 Maneuver 3 This maneuver has the requirements ofchanging three orbital elements of the orbit at the same time

Initial orbital elements

119886 = 419041 km 119890 = 0018 119894= 0688∘ Ω = minus298∘ 120596 =7∘ 119907 = minus972∘

Mathematical Problems in Engineering 7

Range angle (degrees)

Pitc

h (d

egre

es)

7570656055 80 85

200

150

100

50

0

minus50

minus100

minus150

minus200

Figure 4 Optimal pitch angles of the maneuver 3 for the first arc

Yaw

(deg

rees

)

Range angle (degrees)

7570656055 80 85

minus612

minus614

minus616

minus618

minus62

minus622

minus624

minus626

Figure 5 Optimal yaw angles of the maneuver 3 for the first arc

Vehicle mass = 300 kg thrust = 10 Newton initialposition 119904 = 0∘Condition on the final orbit 119886 = 421642 km 119890 = 0119894 = 0∘

Final orbital elements

119886 = 4216455 km 119890 = 000130 119894 = 00419∘Ω = 30060∘120596 = 10047∘ 119907 = 14364∘

Fuel consumption 5565 kg fuel consumption in Biggs[27] 5621 kg fuel consumption in Prado [28] 5579 kg

Figures 4ndash7 show the direction of the thrustIn this maneuver there are three Keplerian elements

fixed on the final orbit semimajor axis eccentricity andinclination Since the maneuver includes a change in theorbital plane the yaw angle is not zero The final conditionswere satisfied and the fuel consumption was reduced by0014 kg when compared to Prado [28]

72 Optimal Maneuvers Using the Phalls I and II In thissection we present the optimal maneuvers found by thealgorithm presented in the present paper when using PhallsI and II The optimal thrust angles are presented and thespecific impulse considered for each maneuver is given inTable 1 Figures 8ndash23 present the optimal angles for theproposedmaneuvers given by the initial and final range angleand the pitch angle at each arc of propulsionThe yaw angle isalways zero because all the maneuvers simulated are planar

Range angle (degrees)

Pitc

h (d

egre

es)

215205 210200190 195185 220 225

30

25

20

15

10

5

0

minus5

Figure 6 Optimal pitch angles of themaneuver 3 for the second arc

Range angle (degrees)

Yaw

(deg

rees

)

215205 210200190 195185

60

55

50

45

40

35

30220 225

Figure 7 Optimal yaw angles of the maneuver 3 for the second arc

All the maneuvers considered here have the same initialorbital elements and final condition imposed for the finalorbit Only the number of arcs and the propulsion system isdifferent at each maneuver The parameters of the maneuverproposed are as follows

Initial orbital elements 119886 = 7130865 km 119890 = 00035 119894= 985054∘ Ω = 0∘ 120596 = 0 119907 = 220∘Vehicle mass = 300 kgCondition imposed on the final orbit 119886 = 7200 km 119890= 0004

721 Maneuver 4 For this maneuver it was considered theexistence of three Phall I thrusters The propulsion force ofeach one is assumed to be the average measured value Theparameters considered in this maneuver are as follows

Total thrust = 0252 Newton specific impulse 1083 sSolution

Final orbital elements 119886 = 720000 km 119890 = 0004Fuel consumption 1046346 kgDuration of the maneuver 4406696 s

722 Maneuver 5 Thismaneuver has the same conditions ofmaneuver 4 but now two thrust arcs are used instead of oneThe parameters considered in this maneuver are as follows

8 Mathematical Problems in Engineering

Range angle (degrees)

30002500200015001000500

Pitc

h an

gle

(deg

rees

)

0

0

5

10

15

minus10

minus15

minus5

Figure 8 Optimal pitch angles for maneuver 4

Range angle (degrees)

Pitc

h an

gle

(deg

rees

) 03

010

02

0405

1400120010008006004002000

minus01minus02

minus03minus04minus05

Figure 9 Optimal pitch angles for the first arc of the maneuver 5

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

1

0

05

15

2

40003800360034003200300028002600

minus05

minus1

minus15

minus2

Figure 10 Optimal pitch angles for the second arc of maneuver 5

Total thrust = 0252 Newton specific impulse 1083 s

Solution

Final orbital elements after the first arc 119886 =716530 km 119890 = 00039Final orbital elements after the second arc 119886 =720000 km 119890 = 00040Fuel consumption 1013402 kgDuration of the maneuver 4267954 s

Note that the use of two arcs generates savings in the fuelconsumption as expected

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

0

5

10

15

22001800 2000160014001000 1200800600400200

minus5

minus10

minus15

Figure 11 Optimal pitch angle for maneuver 6

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

02

0

0604

081

1000900600 700 8005004003002001000

minus02

minus06

minus1

minus04

minus08

Figure 12 Optimal pitch angles for the first arc of maneuver 7

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

0

05

1

15

240023002100 2200200019001800170016001500

minus05

minus1

minus15

Figure 13 Optimal pitch angles for the second arc of maneuver 7

723 Maneuver 6 For this maneuver it was considered theexistence of three Phall II thrusters The propulsion force ofeach one is assumed to be the average measured value Theparameters considered in this maneuver are as follows

Total thrust = 0360 Newton specific impulse 1600 s

Solution

Final orbital elements 119886 = 720000 km 119890 = 0004Fuel consumption 0697930 kgDuration of the maneuver 3039873 s

Mathematical Problems in Engineering 9

Time (s)

minus5

minus10

0

0

5

05 1 15 2 25 3 35 4 45times104

Sem

imaj

or a

xis

dev

iati

on(m

)

Figure 14 Semimajor axis deviation in maneuver 4

02468

1012

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus2minus4minus6minus8

Ecc

entr

icit

y d

evia

tion

times10minus7

Figure 15 Eccentricity deviation in maneuver 4

0

0005

001

0015

Posi

tion

dev

iati

on (m

)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus0005

minus001

Figure 16 Position deviation in maneuver 4

Note that the use of the Phall II generates savings inthe fuel consumption since it has better parameters whencompared to the Phall I device

724 Maneuver 7 Thismaneuver has the same conditions ofmaneuver 6 but now with two thrust arcs instead of oneTheparameters considered in this maneuver are as follows

Total thrust = 0360 Newton specific impulse 1600 s

Solution

Final orbital elements after the first arc 119886 =716777 km 119890 = 00039

0

1

2

3

4

5

Vel

ocit

y d

evia

tion

(m

s)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus1

minus2

minus3

times10minus3

Figure 17 Velocity deviation in maneuver 4

Final orbital elements after the second arc 119886 =720000 km 119890 = 00040Fuel consumption 0686314 kgDuration of the maneuver 2989285 s

Note that the use of two arcs generates savings in the fuelconsumption one more time as expected

73 Optimal Maneuvers Simulations Using the STRS Nowwith the help of the optimal maneuvers found previously weshall simulate thosemaneuvers in the STRS using the optimalangles Some different aspects of the propulsion system wereconsidered to simulate amore realistic environment and alsothe PID control was considered

In this section for the propulsion system it was consid-ered that at each inertial direction 119909 119910 119911 has an error on theactuator signal (ms) with mean equal to zero and varianceequal to 10minus10 that is it was inserted an error in the velocity ofthe satelliteThis error has amplitude approximately close to 5times 10minus3N for each thrust which corresponds to the error of thePMHT (Ferreira et al [30]) There is a PID control to correctthe errors due the actuator but it is still possible to analyzethe deviations and the increase of the fuel consumption dueto the errors

731 Maneuver 4 in STRS In this case it is consideredmaneuver 4 including the solution found there

Final orbital elements 119886 = 719985 km 119890 = 0004Fuel consumption 10470 kgPropellant mass difference from the optimal maneu-ver 666 times 10minus4 kg

The error inserted in the propulsion system resulted in adeviation of the orbital elements as we can see in Figures 14and 15 for the semimajor axis and eccentricity If there wasnot any error at the propulsion system the deviation wouldbe zero all the time for the orbital elements In Figures 16 and17 we can see also the effect of the error in the velocity andposition as a function of these parameters Finally the thrusterror is shown in Figure 18

10 Mathematical Problems in EngineeringA

pplie

d th

rust

(N)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

0504

03

02

01

0

minus01

minus02

minus03

minus04

x axisy axisz axis

ThrustAverage

Figure 18 Thrust applied in maneuver 4

0

5

10

Time (s)

0 05 1 15 2 25 3 35times104

minus5

minus10

minus15

Sem

imaj

or a

xis

dev

iati

on(m

)

Figure 19 Semimajor axis deviation in maneuver 6

732 Maneuver 5 in STRS In this case maneuver 5 isconsidered including the solution found

Final orbital elements after the first arc 119886 =716523 km 119890 = 00039

Final orbital elements after the second arc 119886 =719980 km 119890 = 00040

Fuel consumption 1013362 kg

Propellant mass difference from the optimal maneu-ver 6384 times 10minus4 kg

733 Maneuver 6 in STRS Now maneuver 6 is considered

Final orbital elements 119886 = 719982 km 119890 = 0004

Fuel consumption 06979 kg

Propellant mass difference from the optimal maneu-ver 219 times 10minus4 kg

The same analysis for the figures made in Section 731 fitshere Figures 19ndash23 show the deviations and the error thatoccurred on the propulsion system

Time (s)

0 05 1 15 2 25 3 35times104

Ecc

entr

icit

y d

evia

tion

times10minus6

2

15

1

05

0

minus05

minus1

minus15

Figure 20 Eccentricity deviation in maneuver 6

734 Maneuver 7 in STRS Maneuver 7 is now studiedFinal orbital elements after the first arc 119886 =716770 km 119890 = 00039Final orbital elements after the second arc 119886 =719983 km 119890 = 00040Fuel consumption 0686279 kgPropellant mass difference from the optimal maneu-ver 21469 times 10minus4 kg

735 Maneuver 8 in STRS This maneuver uses the optimalangles given in maneuver 4 but includes the errors proposedfor the propulsion system In addition it was consideredother nonlinearities for the actuator two seconds delay torespond the control signal a rate limiter for the variationof the satelite velocity which the absolute limit of the rateis 000025ms dead zone where the propulsion system isnot turned on if the required velocity increment is lowerthan 00002ms Considering these nonlinearities for thepropulsion system the PID control was able to achievethe maneuver although the fuel consumption was slightlyincreased The results for this simulation are as follows

Final orbital elements 119886 = 719986 km 119890 = 0004Fuel consumption 11276322 kgPropellant mass difference from the optimal maneu-ver 0081429 kg

736 Maneuver 9 in STRS The maneuver simulated hereuses the optimal angles given in maneuver 6 with the errorsproposed for the propulsion system In addition it wasproposed the samenonlinearities (rate limiter dead zone andtime delay) of maneuver 8 Considering these nonlinearitiesfor the propulsion system the PID control was able to achievethe maneuver although the fuel consumption was increasedThe results for this maneuver are as follows

Final orbital elements 119886 = 719984 km 119890 = 00040Fuel consumption 0721921 kgThe propellant mass difference from the optimalmaneuver 002402 kg

Mathematical Problems in Engineering 11

Time (s)

0 05 1 15 2 25 3 35times104

002

0015

001

0005

0

minus0005

minus001

minus0015Po

siti

on d

evia

tion

(m)

Figure 21 Position deviation in maneuver 6

Vel

ocit

y d

evia

tion

(m

s)

Time (s)

0 05 1 15 2 25 3 35times104

8

6

4

2

0

minus2

minus4

times10minus3

Figure 22 Velocity deviation in maneuver 6

x axisy axisz axis

ThrustAverage

Time (s)

0 05 1 15 2 25 3 35times104

06

04

02

0

minus02

minus04

minus06

minus08

App

lied

thru

st (N

)

Figure 23 Thrust applied in maneuver 6

8 Conclusion

Primarily this paper established an algorithm to solve theproblem of orbital maneuvers using a low-thrust control fora spacecraft that is travelling around the Earth Then allthe maneuvers studied in this paper were successfully solvedwith the final conditions and constraints proposed and withreasonable fuel consumptions

It was possible to verify that maneuvers with morethrusting arcs consume less fuel as expected This occursbecause the algorithm has more variables to optimize so itis possible to reduce the fuel consumption

The second part of this work was to simulate the optimalmaneuvers found by the algorithm proposed here in a morerealistic environment which can consider errors on theactuator that was not considered in the search for the optimalmaneuver

Themaneuvers simulated in the STRSwere coherent withthe optimal maneuvers validating the integration of bothsoftwares

The error considered in the propulsion system opensthe possibility to study how the system would react tothose nonlinearities Although errors were inserted in thepropulsion system the PID control was able to correct andaccomplish the maneuver

It is possible to analyze the shifts caused by the error aswell as the deviations in semimajor axis and eccentricity It isalso possible to see that the propulsion thrust was not linear

For themaneuvers shown in Sections 735 and 736 somenonlinearities of the propulsion system such as a responsetime a rate limiter and a dead zone for the actuator signalwere considered All these errors occur in real systems Withthe help of the STRS it was possible to simulate these errorsand analyze the maneuvers studied with more accuracyThe STRS has shown to be an efficient simulator capableof simulating and studying optimal maneuvers in a morerealistic environment and to analyze many nonlinearities notconsidered in the calculation of the optimal maneuver

Evidently the phall II was more efficient than the phall Ias noticed by comparing maneuver 4 with maneuver 6 andmaneuver 5 with maneuver 7 This was expected since thephall II has a higher specific impulse and a higher averagepropulsion thrust as well Nevertheless the PID control wasable to control all the maneuvers and reaches the final orbitalconstraints

References

[1] D F Lawden ldquoMinimal rocket trajectoriesrdquo ARS Journal vol23 no 6 pp 360ndash382 1953

[2] D F Lawden ldquoFundamentals of space navigationrdquo Journal of theBritish Interplanetary Society vol 13 pp 87ndash101 1954

12 Mathematical Problems in Engineering

[3] V V Beletsky and V A Egorov ldquoInterplanetary flights withconstant output enginesrdquoCosmic Research vol 2 no 3 pp 303ndash330 1964

[4] A A Sukhanov ldquoOptimization of flights with low thrustrdquoCosmic Research vol 37 no 2 pp 182ndash191 1999

[5] A A Sukhanov ldquoOptimization of low-thrust interplanetarytransfersrdquo Cosmic Research vol 38 no 6 pp 584ndash587 2000

[6] A A Sukhanov and A F B A Prado ldquoA modification of themethod of transporting trajectoryrdquoCosmic Research vol 42 no1 pp 107ndash112 2004

[7] A A Sukhanov and A F B A Prado ldquoOptimization of lowthrust transfers in the three body problemrdquo Cosmic Researchvol 46 no 5 pp 440ndash451 2008

[8] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction Irdquo Cosmic Researchvol 45 no 5 pp 417ndash423 2007

[9] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction IIrdquo Cosmic Researchvol 46 no 1 pp 49ndash59 2008

[10] S S Fernandes and F D C Carvalho ldquoA first-order analyticaltheory for optimal low-thrust limited-power transfers betweenarbitrary elliptical coplanar orbitsrdquo Mathematical Problems inEngineering vol 2008 Article ID 525930 30 pages 2008

[11] R A Boucher ldquoElectrical propulsion for control of stationarysatellitesrdquo Journal of Spacecraft and Rockets vol 1 no 2 pp 164ndash169 1964

[12] D N Bowditch ldquoApplication of magnetic-expansion plasmathrusters to satellite station keeping and attitude control mis-sionsrdquo in Proceedings of the 4th Electric Propulsion Confer-ence AIAA-1964-677 pp 64ndash677 The American Institute ofAeronautics and Astronautics Philadelphia Pa USA Augest-September 1964

[13] S R Oleson R M Myers C A Kluever J P Riehl and F MCurran ldquoAdvanced propulsion for geostationary orbit insertionand north-south station keepingrdquo Journal of Spacecraft andRockets vol 34 no 1 pp 22ndash28 1997

[14] T A Ely and K C Howell ldquoEast-west stationkeeping of satelliteorbits with resonant tesseral harmonicsrdquoActa Astronautica vol46 no 1 pp 1ndash15 2000

[15] C Circi ldquoSimple strategy for geostationary stationkeepingmaneuvers using solar sailrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 249ndash253 2005

[16] P Romero J M Gambi and E Patino ldquoStationkeepingmanoeuvres for geostationary satellites using feedback controltechniquesrdquo Aerospace Science and Technology vol 11 no 2-3pp 229ndash237 2007

[17] V Gomes and A F B A Prado ldquoLow-thrust out-of-planeorbital station-keeping maneuvers for satellitesrdquo MathematicalProblems in Engineering vol 2012 Article ID 532708 14 pages2012

[18] B L Pierson and C A Kluever ldquoThree-stage approach to opti-mal low-thrust earth-moon trajectoriesrdquo Journal of GuidanceControl and Dynamics vol 17 no 6 pp 1275ndash1282 1994

[19] Y J Song S Y Park K H Choi and E S Sim ldquoA lunar cargomission design strategy using variable low thrustrdquo Advances inSpace Research vol 43 no 9 pp 1391ndash1406 2009

[20] S A Fazelzadeh and G A Varzandian ldquoMinimum-time earth-moon and moon-earth orbital maneuvers using time-domainfinite element methodrdquo Acta Astronautica vol 66 no 3-4 pp528ndash538 2010

[21] G Mingotti F Topputo and F Bernelli-Zazzera ldquoEfficientinvariant-manifold low-thrust planar trajectories to themoonrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 p 817 2012

[22] J R Brophy and M Noca ldquoElectric propulsion for solar systemexplorationrdquo Journal of Propulsion and Power vol 14 no 5 pp700ndash707 1998

[23] D P S Santos A F B A Prado L Casalino and G ColasurdoldquoOptimal trajectories towards near-earth-objects using solarelectric propulsion (sep) and gravity assistedmaneuverrdquo Journalof Aerospace Engineering Sciences and Applications vol 1 no 2pp 51ndash64 2008

[24] D P S Santos L Casalino G Colasurdo and A F B APrado ldquoOptimal trajectories towards near-earth-objects usingsolar electric propulsion (sep) and gravity assisted maneuverrdquoWSEAS Transactions on Applied andTheoretical Mechanics vol4 no 3 pp 125ndash135 2009

[25] D P S Santos and A F B A Prado ldquoOptimal low-thrusttrajectories to reach the asteroid apophisrdquo WSEAS MechanicalEngineering Series vol 7 pp 241ndash251 2012

[26] M C B BiggsThe Optimization of Spacecraft Orbital Manoeu-vres Part I Linearly Varying Thrust Angles The HatfieldPolytechnic Numerical Optimization Centre 1978

[27] M C B BiggsThe Optimisation of Spacecraft Orbital Manoeu-vres Part II Using Pontryaginrsquos Maximum Principle The Hat-field Polytechnic Numerical Optimization Centre 1979

[28] A F B A Prado Analise Selecao e Implementacao de Procedi-mentos que visemManobras Otimas de Satelites Artificiais [MSthesis] Instituto Nacional de Pesquisas Espaciais Sao Jose dosCampos 1988

[29] T C Oliveira Estrategias otimas para manobras orbitais uti-lizando propulsao contınua [MS thesis] Instituto Nacional dePesquisas Espaciais Sao Jose dos Campos 2012

[30] J L Ferreira G C Possa B SMoraes et al ldquoPermanentmagnethall thruster for satellite orbit maneuvering with low powerrdquoAdvances in Space Research vol 01 pp 166ndash182 2009

[31] E Stuhlinger Ion Propulsion for Space FlightMcGraw-Hill NewYork NY USA 1st edition 1964

[32] R G Jahn Physics of Electric Propulsion McGraw-Hill NewYork NY USA 1st edition 1968

[33] D G Luemberger Linear and Non-Linear ProgrammingSpringer Science amp Business Media New York NY USA 3rdedition 2008

[34] M S Bazaraa and C M ShettyNonlinear Programming-Theoryand Algorithms JohnWilley amp Sons New York NY USA 1979

[35] V V Zhurin H R Kaufman and R S Robinson ldquoPhysics ofclosed drift thrustersrdquo Plasma Sources Science and Technologyvol 8 no 1 pp R1ndashR20 1999

[36] E M Rocco ldquoPerturbed orbital motion with a PID controlsystem for the trajectoryrdquo in Proceedings of the 14th ColoquioBrasileiro De Dinamica Orbital Aguas de Lindoia AnaisUNESP Guaratingueta Brazil November 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Reference lineS

Figure 1 Definition of the range angle and the arbitrary referenceline

The range angle ldquo119904rdquo is the independent variable of thiswork and it is the angle between the radius position vectorof the satellite and a reference arbitrary line which lies in theplane of the orbit as shown in Figure 1 It replaces the time inthe equations of motion

There are also two important angles pitch and yaw whichare used to define the direction of the thrust The angle ofpitch is designated by119860 while the angle yaw is designated by119861 The pitch is the angle between the thrust direction and theperpendicular to the radius of the spacecraft while yaw is theangle between the thrust direction and the orbital plane Thestate variables used here are defined as follows [26ndash29]

1198831=

119886 (1 minus 1198902)

120583

1198832= 119890 cos (120596 minus Φ)

1198833= 119890 sin (120596 minus Φ)

1198834=

fuel consumed1198980

1198835= time

1198836= cos( 119894

2

) cos(Ω + Φ

2

)

1198837= sin( 119894

2

) cos(Ω minus Φ

2

)

1198838= sin( 119894

2

) sin(Ω minus Φ

2

)

1198839= cos( 119894

2

) sin(Ω + Φ

2

)

(1)

where 120583 is the gravitational constant of the Earth and 1198980is

the initial mass of spacecraftThe advantage of using these state equations is to avoid

singularities if the inclination or the eccentricity is zero Theequations of motion and the control variables used here areshown in [26ndash29]The true longitude which can be used as aconstraint to define the prohibit region to apply the thrust isgiven by

120582 = 120596 + 119907 (2)

The constraints on the final Keplerian orbital elements aregiven by (3) All the orbital elements are scaled The desiredvalues are given by the index ldquolowastrdquo while the suffix ldquo

0rdquo denotes

the initial value The element without suffix or index is thecurrent value of the element The constraints on the finalorbital elements are given by

(119886 minus 119886lowast)

10038161003816100381610038161198860minus 119886lowast1003816100381610038161003816

= 0

(119890 minus 119890lowast)

10038161003816100381610038161198900minus 119890lowast1003816100381610038161003816

= 0

(119894 minus 119894lowast)

10038161003816100381610038161198940minus 119894lowast1003816100381610038161003816

= 0

(Ω minus Ωlowast)

1003816100381610038161003816Ω0minus Ωlowast1003816100381610038161003816

= 0

(120596 minus 120596lowast)

10038161003816100381610038161205960minus 120596lowast1003816100381610038161003816

= 0

(3)

The Pontryaginrsquos maximum principle is used to find theoptimal control that determinates the optimal trajectory Inthis case it is desirable to find the maximum final mass ofthe spacecraft given by the functional 119869(sdot) = minus119883

4(119904119891) This is

equivalent of minimizing the fuel consumption because theexpenditure of fuel is the only way that the satellite has to losemass More details of this process can be found in [26ndash29]The formulation of the problem is given by (4) to (6)

Maximize minus1198834(119904119891) (4)

with respect to 1199040 119904119891 1198600 1198610 1198601015840

0 1198611015840

0 119875119894(1199040) (5)

subjected to 119878119895(119883 (119904119891)) = 0 119895 = 1 119898 (6)

where 119878119895(119883(119904119891)) are the constraints stated in (2) and (3) or

any other typeSome of the initial variables that should be guessed are

the initial adjoint multipliers 119875119894(1199040) A good initial guess

for those variables is very important to assist convergenceand to reduce the time the algorithm needs to obtain thesolution Unfortunately the Lagrangemultipliers do not haveany physical meaning so it is hard to find a good initial guessfor them To solve this problem the adjoint-control trans-formation ([26ndash29]) is used to attribute a good estimativefor 119875119894(1199040) from variables that have physical meaning Those

variables are the angles 1198600and 119861

0and the values of 1198601015840

0and

1198611015840

0(Biggs [26] Prado [28])

3 Steps of the Algorithm

The final complete algorithm has the following steps [28 29]

(i) It starts from an initial guess for the set of variables(1198600 1198610 11986010158400 11986110158400 1199040 119904119891) With these initial values the

initial guess for the control is obtained

4 Mathematical Problems in Engineering

(ii) The ldquoadjoint-controlrdquo transformation is used to obtainthe initial values of the Lagrange multipliers requiredfor the numerical integrations

(iii) Now with all the initial values needed to solve theproblem the equations of motion and the adjointequations are simultaneously numerically integratedduring the propulsion arcThe values of the pitch andyaw angles are obtained at every step of the inte-gration by the principle of maximum of PontryaginAt the end the value of the final state 119883 after thepropulsion arc is obtained

(iv) The Keplerian elements of the orbit achieved are thenobtained from the final state119883 using (1)

(v) At this point the satisfaction of the constraints isverified If themagnitude of the vector that representsthe constraints satisfaction is smaller than a specifiedtolerance provided by the user (the numerical zero)the algorithm proceeds to step (vii) On the otherhand if it is bigger the algorithmproceeds to step (vi)

(vi) At this step the gradient of the constraints equations(nablaS) is obtained by perturbing the elements of thecontrol vector u and performing a new set of numer-ical integrations in order to obtain the numericalderivatives of the constraints equations with respectto the vector uThen a new set of values for the vectoru is found by (7) (Luemberger [33]) as follows

u119894+1

= u119894minus nablaS119879[nablaSnablaS119879]

minus1

nablaS (7)

and then the algorithm goes back to step (i) with thenew value of the vector u

(vii) Once the constraints satisfaction is achieved thenext step is to search for the minimum of the fuelconsumed The direction of the search d using thegradient method (Bazaraa et al [34]) is

d = minusRnabla119869 (u) (8)

where the vector R is

R = I minus nablaS119879[nablaSnablaS]minus1nablaS (9)

(viii) At this step the magnitude of the vector d is verifiedIf it is smaller than a value specified by the useras a direction search tolerance (numerical zero) thealgorithm goes to step (x) and if it is bigger thealgorithm proceeds to step (ix)

(ix) In order to get better results the initial data ldquoratio ofcontractionrdquo RC and the direction search toleranceare reduced as u gets closer to the minimum Themagnitude for the direction search is then given by

PB =

RC119869 (u)nabla119869 (u) d

(10)

So the complete step is given by

u119894+1

= u119894+ PB d

|d| (11)

and then the algorithm goes back to step (i)

(x) At this step the possibility of the control u to be aKuhn-Tucker point is verified To check this condi-tion the vector VT is built formed by the 119898 firstelements of the vector given by (Bazarra et al [34])as follows

W = minus(SS119879)minus1

Snabla119869 (u) (12)

where119898 is the number of active constraints If VT isnot positive definite the line 119895 which causes VT

119895to

be negative on the matrix S is deleted And then thealgorithm returns to step (v)

(xi) The current step is verified and if it is not the last onethe algorithm goes to a new search for the minimumfuel consumption with a smaller direction searchtolerance for the objective function If this step is thelast one the algorithm is finalized

4 Extensions of the Algorithm

There are several extensions of the algorithm that can beconsidered in order to make it more flexible The first oneis the possibility to perform maneuvers with more than onearc of propulsion Increasing the number of propulsion arcsmeans that the number of optimization variables is increasedFor example for one thrust arc there are 119906

1to 1199066control

variables For two thrust arcs similarly there is a set of 1199061to

11990612control variables and so on In addition by adding more

than one thrust arc the inclusion of constraints to prevent theoverlap of thrust arcs becomes necessary

Another extension is the possibility to consider forbiddenregions to apply the propulsion as a constraintThese regionsare defined by the true longitude as shown in (2) Both valuesof the initial and final true longitudes of the prohibit regionhave to be specified

In some situations the initial range angles 1199040can be less

than zero It means that when the thrust is started the time isnegative To avoid negative time it is added a constraint thatimposes to 119904

0 the time to be equal or bigger than zero This

constraint is only active if 1199040lt 0

The last extension is the possibility to consider constraintsrestricting the pitch and yaw angles In this case the boundsconsidered on 119860 and 119861 are the ones given by Biggs [27]119860119871le 119860 le 119860

119906and 119861

119871le 119861 le 119861

119906 where the suffix ldquo119906rdquo

represents the maximum angle and the suffix ldquo119871rdquo representstheminimumangle If these constraint are active after findingthe optimal angles 119860lowast and 119861lowast and this problem is solved asfollows [26ndash29] 119860 = 119860

119906if 119860lowast ge 119860

119906 119860 = 119860

119871if 119860lowast le 119860

119871

119860 = 119860lowast if 119860

119871le 119860 le 119860

119906 Those equations are similarly

applied to the optimal yaw angles 119861lowast

5 The Permanent Magnet Hall Thruster

As defined by Jahn [32] the electric propulsion is the gasacceleration to generate thrust via electric heating electricbody forces or both electric and magnetic body forces Thegreatest advantage of the electric propulsion is the veryhigh exhaust velocities resulting in a reduction of the fuel

Mathematical Problems in Engineering 5

consumption Although electric propulsion has low thrustthe high specific impulse guarantees a reduced burned fuelalong the maneuver

The hall thruster used in this work as the propulsionsystem was fundamentally envisaged in Russia in 1960 andthe first successful space mission with this thrust propulsionwith 60mN was used in the Meteor satellite series in 1972(Zhurin et al [35]) After that this kind of propulsion systemhas been studied and developed in several countries such asFrance USA Japan and Brazil

The working principle of the hall thrusters is the useof an electromagnet to produce the main magnetic fieldresponsible for the plasma acceleration and generation (Fer-reira et al [30]) This configuration requires the use ofenergy to produce the magnetic field Nevertheless thepermanent magnet hall thruster (PMHT) developed at thePlasma Laboratory of the Universidade de Brasilia uses anarray of permanent magnets instead of an electromagnet toproduce a radial magnetic field inside the cylindrical plasmadrift channel of the thruster (Ferreira et al [30]) The useof permanent magnets instead of electromagnet guaranteeslow-power consumption and it can be used in small- andmedium-size satellites (Ferreira et al [30])

In the Plasma Laboratory of the Universidade de Brasıliait has been developed two kinds of hall thruster called Phall Iand Phall II The first one to be constructed was the Phall I Ithas a stainless steel chamber with 30 centimeters of diameterIts magnetic field is produced by two concentric cylindricalarrays of ferrite permanent magnets 32 bars in the outer shelland 10 bars in the inner shellThemagnetic field in themiddleline of the plasma sourcersquos channel is 200 Gauss (Ferreiraet al [30]) On the other hand the second thruster PhallII has an aluminum plasma chamber and it is smaller with15 cm diameter and contains neodymium-iron magnets toproduce the magnetic field (Ferreira et al [30]) Table 1 givesthe parameters of the performance of both Phall I and PhallII thrusters (Ferreira et al [30])

In this paper it was found first the optimal maneuverswith the thrust parameters shown in Table 1 Then theseoptimal maneuvers were simulated with a PID control for along period of time based on the continuous monitoring ofthe thrust in a real-time performance correcting any possiblefailure

6 Spacecraft Trajectory Simulator

Themaneuver simulator used to simulate the optimalmaneu-vers in a more realistic way is the spacecraft trajectorysimulator (STRS) The STRS can consider constructionalfeatures and operation such as nonlinearities failures errorsand external and internal disturbances in order to determinethe deviation in the reproduction of the optimal solutionpreviously found

Furthermore maneuvers with continuous thrust are usedduring a long period of timeThis kind ofmaneuvers requiresa PID control system to guarantee the achievement of the finalconditions of the maneuver The operating structure of theSTRS can be described as follows (Rocco [36])

(a) The simulation occurs in a discrete way that is atevery simulation step the state of the vehicle (positionand velocity) must be computed For all purposes thedisturbances nonlinearities errors and failures mustbe considered

(b) Since a closed loop system was used to controlthe trajectory the reference state is determined byapplying the optimal maneuver Then the currentstate follows the reference but considers the effectsof disturbances and the constructive characteristics ofthe vehicle

(c) The difference between the reference state and thecurrent state creates an error signal that is insertedinto a PID control

(d) The controller produces a control signal according tothe PID control law and the gains that have been set

(e) The control signal is inserted into the actuator model(propellant thrusters) where the nonlinearities inher-ent to the construction of the actuator can be consid-ered Thus the behavior of the propellant thrusterscan be reproduced by the appropriate adjustmentof the model parameters which are supplied by themanufacturer of the thrust actuator

(f) Finally an actuation signal (Δ119881) is sent to the dynam-ical model of the orbital motion At this signal thevelocities increments due to the disturbances can beadded

(g) With the actuation signal added with the possible dis-turbances themodel of the orbital dynamics providesthe state (position and velocity) after the applicationof the propulsion

(h) Through the use of sensors that were modeled con-sidering its constructive aspects the current state isdetermined and compared with the reference state inorder to close the control loop

7 Results

Before performing tests with the propulsion system PhallsI and II that is one of the goals of the present paper it isimportant to test some of the particular characteristics ofthe algorithm developed here Those tests are performed inmaneuvers that have similar results available in the literatureso a comparison to validate the algorithm and the softwaredeveloped can be made After that the results of the optimalmaneuvers using the new propulsion devices as well as thesimulations of them in the STRS are presented

71 Optimal Maneuvers to Validate the Algorithm

711 Maneuver 1 This first maneuver described has a con-straint at the pitch angle in order to test this capability of theproposed algorithm

Initial orbital elements

119886 = 99000 km 119890 = 07 119894 = 10∘ Ω = 55∘ 120596 = 105∘ 119907 =minus105∘

6 Mathematical Problems in Engineering

Table 1 Comparative performance parameters measured in PHALL I and PHALL II

PHALL1 PHALL2Maximum expected thrust (mN) 126 126Average measured thrust (mN) 849 1200Thrust density (Nm2) 468 lt60Maximum specific impulse (s) 1607 1607Measured specific impulse (s) 1083 1600Ionized mass ratio () 33 sim30Propellant consumption (Kgs) 6 0 times 10

minus61 0 times 10

minus6

Energy consumption (W) 350 250ndash350Electrical efficiency () 339 60Total efficiency () 1012 500

Range angle (degrees)

Pitc

h (d

egre

es)

1101009080

012345

120 130 140

minus1minus2minus3minus4minus5

Figure 2 Optimal pitch angles of the maneuver 1

Vehicle mass = 300 kg thrust = 10 Newton initialposition 119904 = 0∘Condition on the final orbit 119886 = 104000 kmOther constraints 5∘ ge 119860 ge minus5∘ (pitch constraint)

Final orbital elements obtained after the maneuver

119886 = 10399921 km 119890 = 07143 119894 = 10∘Ω = 55∘ 120596 = 105∘119907 = 3024∘

Fuel consumption 24235 kg fuel consumption in Biggs[27] 244 kg fuel consumption in Prado [28] 245 kg

Figure 2 shows the direction of the thrustNow with the constraint in the direction of propulsion

the fuel consumption was increased when compared witha maneuver without this constraint [27 28] The extra fuelneeded due to this constraint is 38 times 10minus3 kg It is possible toview clearly that the pitch angle is confined to 5∘ at the endof the maneuver In any case the algorithm developed hereobtains better results when compared to Biggs [27] and Prado[28]

712 Maneuver 2 This maneuver has the constraint of aregion prohibited for thrusting This is one of the mostimportant capabilities of the algorithm developed here

Initial orbital elements

119886 = 99000 km 119890 = 07 119894 = 10∘ Ω = 55∘ 120596 = 105∘ 119907 =minus105∘

Range angle (degrees)

Pitc

h (d

egre

es)

55504540353025 60 65 70

minus20minus21minus22minus23minus24minus25minus26minus27minus28minus29minus30

Figure 3 Optimal pitch angles of the maneuver 2

Vehicle mass = 300 kg thrust = 10 Newton initialposition 119904 = 0∘Condition on the final orbit 119886 = 104000 kmOther constraints no thrust between the true longi-tudes 120∘ and 180∘

Final orbital elements

119886 = 10399985 km 119890 = 07134 119894 = 10∘Ω = 55∘120596 = 105∘119907 = 32080∘

Fuel consumption 27852 kg fuel consumption in Biggs[27] 281 kg fuel consumption in Prado [28] 281 kg

Figure 3 shows the direction of the thrustIn Figure 3 it is possible to note that the solution has the

thrust arc ending in 120582 = 120∘ The true longitude is definedhere as 120582 = 120596 + 119907 = 42580

∘ that is similar to 6580∘ that isthe result shown in Figure 3 In this way the region prohibitof thrust was obeyed and compared with the maneuver thatis free of this constraint the extra fuel needed to reach thefinal semimajor axis required was 03655 kg Once againthe algorithm proposed here obtained better results whencompared to Biggs [27] and Prado [28]

713 Maneuver 3 This maneuver has the requirements ofchanging three orbital elements of the orbit at the same time

Initial orbital elements

119886 = 419041 km 119890 = 0018 119894= 0688∘ Ω = minus298∘ 120596 =7∘ 119907 = minus972∘

Mathematical Problems in Engineering 7

Range angle (degrees)

Pitc

h (d

egre

es)

7570656055 80 85

200

150

100

50

0

minus50

minus100

minus150

minus200

Figure 4 Optimal pitch angles of the maneuver 3 for the first arc

Yaw

(deg

rees

)

Range angle (degrees)

7570656055 80 85

minus612

minus614

minus616

minus618

minus62

minus622

minus624

minus626

Figure 5 Optimal yaw angles of the maneuver 3 for the first arc

Vehicle mass = 300 kg thrust = 10 Newton initialposition 119904 = 0∘Condition on the final orbit 119886 = 421642 km 119890 = 0119894 = 0∘

Final orbital elements

119886 = 4216455 km 119890 = 000130 119894 = 00419∘Ω = 30060∘120596 = 10047∘ 119907 = 14364∘

Fuel consumption 5565 kg fuel consumption in Biggs[27] 5621 kg fuel consumption in Prado [28] 5579 kg

Figures 4ndash7 show the direction of the thrustIn this maneuver there are three Keplerian elements

fixed on the final orbit semimajor axis eccentricity andinclination Since the maneuver includes a change in theorbital plane the yaw angle is not zero The final conditionswere satisfied and the fuel consumption was reduced by0014 kg when compared to Prado [28]

72 Optimal Maneuvers Using the Phalls I and II In thissection we present the optimal maneuvers found by thealgorithm presented in the present paper when using PhallsI and II The optimal thrust angles are presented and thespecific impulse considered for each maneuver is given inTable 1 Figures 8ndash23 present the optimal angles for theproposedmaneuvers given by the initial and final range angleand the pitch angle at each arc of propulsionThe yaw angle isalways zero because all the maneuvers simulated are planar

Range angle (degrees)

Pitc

h (d

egre

es)

215205 210200190 195185 220 225

30

25

20

15

10

5

0

minus5

Figure 6 Optimal pitch angles of themaneuver 3 for the second arc

Range angle (degrees)

Yaw

(deg

rees

)

215205 210200190 195185

60

55

50

45

40

35

30220 225

Figure 7 Optimal yaw angles of the maneuver 3 for the second arc

All the maneuvers considered here have the same initialorbital elements and final condition imposed for the finalorbit Only the number of arcs and the propulsion system isdifferent at each maneuver The parameters of the maneuverproposed are as follows

Initial orbital elements 119886 = 7130865 km 119890 = 00035 119894= 985054∘ Ω = 0∘ 120596 = 0 119907 = 220∘Vehicle mass = 300 kgCondition imposed on the final orbit 119886 = 7200 km 119890= 0004

721 Maneuver 4 For this maneuver it was considered theexistence of three Phall I thrusters The propulsion force ofeach one is assumed to be the average measured value Theparameters considered in this maneuver are as follows

Total thrust = 0252 Newton specific impulse 1083 sSolution

Final orbital elements 119886 = 720000 km 119890 = 0004Fuel consumption 1046346 kgDuration of the maneuver 4406696 s

722 Maneuver 5 Thismaneuver has the same conditions ofmaneuver 4 but now two thrust arcs are used instead of oneThe parameters considered in this maneuver are as follows

8 Mathematical Problems in Engineering

Range angle (degrees)

30002500200015001000500

Pitc

h an

gle

(deg

rees

)

0

0

5

10

15

minus10

minus15

minus5

Figure 8 Optimal pitch angles for maneuver 4

Range angle (degrees)

Pitc

h an

gle

(deg

rees

) 03

010

02

0405

1400120010008006004002000

minus01minus02

minus03minus04minus05

Figure 9 Optimal pitch angles for the first arc of the maneuver 5

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

1

0

05

15

2

40003800360034003200300028002600

minus05

minus1

minus15

minus2

Figure 10 Optimal pitch angles for the second arc of maneuver 5

Total thrust = 0252 Newton specific impulse 1083 s

Solution

Final orbital elements after the first arc 119886 =716530 km 119890 = 00039Final orbital elements after the second arc 119886 =720000 km 119890 = 00040Fuel consumption 1013402 kgDuration of the maneuver 4267954 s

Note that the use of two arcs generates savings in the fuelconsumption as expected

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

0

5

10

15

22001800 2000160014001000 1200800600400200

minus5

minus10

minus15

Figure 11 Optimal pitch angle for maneuver 6

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

02

0

0604

081

1000900600 700 8005004003002001000

minus02

minus06

minus1

minus04

minus08

Figure 12 Optimal pitch angles for the first arc of maneuver 7

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

0

05

1

15

240023002100 2200200019001800170016001500

minus05

minus1

minus15

Figure 13 Optimal pitch angles for the second arc of maneuver 7

723 Maneuver 6 For this maneuver it was considered theexistence of three Phall II thrusters The propulsion force ofeach one is assumed to be the average measured value Theparameters considered in this maneuver are as follows

Total thrust = 0360 Newton specific impulse 1600 s

Solution

Final orbital elements 119886 = 720000 km 119890 = 0004Fuel consumption 0697930 kgDuration of the maneuver 3039873 s

Mathematical Problems in Engineering 9

Time (s)

minus5

minus10

0

0

5

05 1 15 2 25 3 35 4 45times104

Sem

imaj

or a

xis

dev

iati

on(m

)

Figure 14 Semimajor axis deviation in maneuver 4

02468

1012

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus2minus4minus6minus8

Ecc

entr

icit

y d

evia

tion

times10minus7

Figure 15 Eccentricity deviation in maneuver 4

0

0005

001

0015

Posi

tion

dev

iati

on (m

)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus0005

minus001

Figure 16 Position deviation in maneuver 4

Note that the use of the Phall II generates savings inthe fuel consumption since it has better parameters whencompared to the Phall I device

724 Maneuver 7 Thismaneuver has the same conditions ofmaneuver 6 but now with two thrust arcs instead of oneTheparameters considered in this maneuver are as follows

Total thrust = 0360 Newton specific impulse 1600 s

Solution

Final orbital elements after the first arc 119886 =716777 km 119890 = 00039

0

1

2

3

4

5

Vel

ocit

y d

evia

tion

(m

s)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus1

minus2

minus3

times10minus3

Figure 17 Velocity deviation in maneuver 4

Final orbital elements after the second arc 119886 =720000 km 119890 = 00040Fuel consumption 0686314 kgDuration of the maneuver 2989285 s

Note that the use of two arcs generates savings in the fuelconsumption one more time as expected

73 Optimal Maneuvers Simulations Using the STRS Nowwith the help of the optimal maneuvers found previously weshall simulate thosemaneuvers in the STRS using the optimalangles Some different aspects of the propulsion system wereconsidered to simulate amore realistic environment and alsothe PID control was considered

In this section for the propulsion system it was consid-ered that at each inertial direction 119909 119910 119911 has an error on theactuator signal (ms) with mean equal to zero and varianceequal to 10minus10 that is it was inserted an error in the velocity ofthe satelliteThis error has amplitude approximately close to 5times 10minus3N for each thrust which corresponds to the error of thePMHT (Ferreira et al [30]) There is a PID control to correctthe errors due the actuator but it is still possible to analyzethe deviations and the increase of the fuel consumption dueto the errors

731 Maneuver 4 in STRS In this case it is consideredmaneuver 4 including the solution found there

Final orbital elements 119886 = 719985 km 119890 = 0004Fuel consumption 10470 kgPropellant mass difference from the optimal maneu-ver 666 times 10minus4 kg

The error inserted in the propulsion system resulted in adeviation of the orbital elements as we can see in Figures 14and 15 for the semimajor axis and eccentricity If there wasnot any error at the propulsion system the deviation wouldbe zero all the time for the orbital elements In Figures 16 and17 we can see also the effect of the error in the velocity andposition as a function of these parameters Finally the thrusterror is shown in Figure 18

10 Mathematical Problems in EngineeringA

pplie

d th

rust

(N)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

0504

03

02

01

0

minus01

minus02

minus03

minus04

x axisy axisz axis

ThrustAverage

Figure 18 Thrust applied in maneuver 4

0

5

10

Time (s)

0 05 1 15 2 25 3 35times104

minus5

minus10

minus15

Sem

imaj

or a

xis

dev

iati

on(m

)

Figure 19 Semimajor axis deviation in maneuver 6

732 Maneuver 5 in STRS In this case maneuver 5 isconsidered including the solution found

Final orbital elements after the first arc 119886 =716523 km 119890 = 00039

Final orbital elements after the second arc 119886 =719980 km 119890 = 00040

Fuel consumption 1013362 kg

Propellant mass difference from the optimal maneu-ver 6384 times 10minus4 kg

733 Maneuver 6 in STRS Now maneuver 6 is considered

Final orbital elements 119886 = 719982 km 119890 = 0004

Fuel consumption 06979 kg

Propellant mass difference from the optimal maneu-ver 219 times 10minus4 kg

The same analysis for the figures made in Section 731 fitshere Figures 19ndash23 show the deviations and the error thatoccurred on the propulsion system

Time (s)

0 05 1 15 2 25 3 35times104

Ecc

entr

icit

y d

evia

tion

times10minus6

2

15

1

05

0

minus05

minus1

minus15

Figure 20 Eccentricity deviation in maneuver 6

734 Maneuver 7 in STRS Maneuver 7 is now studiedFinal orbital elements after the first arc 119886 =716770 km 119890 = 00039Final orbital elements after the second arc 119886 =719983 km 119890 = 00040Fuel consumption 0686279 kgPropellant mass difference from the optimal maneu-ver 21469 times 10minus4 kg

735 Maneuver 8 in STRS This maneuver uses the optimalangles given in maneuver 4 but includes the errors proposedfor the propulsion system In addition it was consideredother nonlinearities for the actuator two seconds delay torespond the control signal a rate limiter for the variationof the satelite velocity which the absolute limit of the rateis 000025ms dead zone where the propulsion system isnot turned on if the required velocity increment is lowerthan 00002ms Considering these nonlinearities for thepropulsion system the PID control was able to achievethe maneuver although the fuel consumption was slightlyincreased The results for this simulation are as follows

Final orbital elements 119886 = 719986 km 119890 = 0004Fuel consumption 11276322 kgPropellant mass difference from the optimal maneu-ver 0081429 kg

736 Maneuver 9 in STRS The maneuver simulated hereuses the optimal angles given in maneuver 6 with the errorsproposed for the propulsion system In addition it wasproposed the samenonlinearities (rate limiter dead zone andtime delay) of maneuver 8 Considering these nonlinearitiesfor the propulsion system the PID control was able to achievethe maneuver although the fuel consumption was increasedThe results for this maneuver are as follows

Final orbital elements 119886 = 719984 km 119890 = 00040Fuel consumption 0721921 kgThe propellant mass difference from the optimalmaneuver 002402 kg

Mathematical Problems in Engineering 11

Time (s)

0 05 1 15 2 25 3 35times104

002

0015

001

0005

0

minus0005

minus001

minus0015Po

siti

on d

evia

tion

(m)

Figure 21 Position deviation in maneuver 6

Vel

ocit

y d

evia

tion

(m

s)

Time (s)

0 05 1 15 2 25 3 35times104

8

6

4

2

0

minus2

minus4

times10minus3

Figure 22 Velocity deviation in maneuver 6

x axisy axisz axis

ThrustAverage

Time (s)

0 05 1 15 2 25 3 35times104

06

04

02

0

minus02

minus04

minus06

minus08

App

lied

thru

st (N

)

Figure 23 Thrust applied in maneuver 6

8 Conclusion

Primarily this paper established an algorithm to solve theproblem of orbital maneuvers using a low-thrust control fora spacecraft that is travelling around the Earth Then allthe maneuvers studied in this paper were successfully solvedwith the final conditions and constraints proposed and withreasonable fuel consumptions

It was possible to verify that maneuvers with morethrusting arcs consume less fuel as expected This occursbecause the algorithm has more variables to optimize so itis possible to reduce the fuel consumption

The second part of this work was to simulate the optimalmaneuvers found by the algorithm proposed here in a morerealistic environment which can consider errors on theactuator that was not considered in the search for the optimalmaneuver

Themaneuvers simulated in the STRSwere coherent withthe optimal maneuvers validating the integration of bothsoftwares

The error considered in the propulsion system opensthe possibility to study how the system would react tothose nonlinearities Although errors were inserted in thepropulsion system the PID control was able to correct andaccomplish the maneuver

It is possible to analyze the shifts caused by the error aswell as the deviations in semimajor axis and eccentricity It isalso possible to see that the propulsion thrust was not linear

For themaneuvers shown in Sections 735 and 736 somenonlinearities of the propulsion system such as a responsetime a rate limiter and a dead zone for the actuator signalwere considered All these errors occur in real systems Withthe help of the STRS it was possible to simulate these errorsand analyze the maneuvers studied with more accuracyThe STRS has shown to be an efficient simulator capableof simulating and studying optimal maneuvers in a morerealistic environment and to analyze many nonlinearities notconsidered in the calculation of the optimal maneuver

Evidently the phall II was more efficient than the phall Ias noticed by comparing maneuver 4 with maneuver 6 andmaneuver 5 with maneuver 7 This was expected since thephall II has a higher specific impulse and a higher averagepropulsion thrust as well Nevertheless the PID control wasable to control all the maneuvers and reaches the final orbitalconstraints

References

[1] D F Lawden ldquoMinimal rocket trajectoriesrdquo ARS Journal vol23 no 6 pp 360ndash382 1953

[2] D F Lawden ldquoFundamentals of space navigationrdquo Journal of theBritish Interplanetary Society vol 13 pp 87ndash101 1954

12 Mathematical Problems in Engineering

[3] V V Beletsky and V A Egorov ldquoInterplanetary flights withconstant output enginesrdquoCosmic Research vol 2 no 3 pp 303ndash330 1964

[4] A A Sukhanov ldquoOptimization of flights with low thrustrdquoCosmic Research vol 37 no 2 pp 182ndash191 1999

[5] A A Sukhanov ldquoOptimization of low-thrust interplanetarytransfersrdquo Cosmic Research vol 38 no 6 pp 584ndash587 2000

[6] A A Sukhanov and A F B A Prado ldquoA modification of themethod of transporting trajectoryrdquoCosmic Research vol 42 no1 pp 107ndash112 2004

[7] A A Sukhanov and A F B A Prado ldquoOptimization of lowthrust transfers in the three body problemrdquo Cosmic Researchvol 46 no 5 pp 440ndash451 2008

[8] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction Irdquo Cosmic Researchvol 45 no 5 pp 417ndash423 2007

[9] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction IIrdquo Cosmic Researchvol 46 no 1 pp 49ndash59 2008

[10] S S Fernandes and F D C Carvalho ldquoA first-order analyticaltheory for optimal low-thrust limited-power transfers betweenarbitrary elliptical coplanar orbitsrdquo Mathematical Problems inEngineering vol 2008 Article ID 525930 30 pages 2008

[11] R A Boucher ldquoElectrical propulsion for control of stationarysatellitesrdquo Journal of Spacecraft and Rockets vol 1 no 2 pp 164ndash169 1964

[12] D N Bowditch ldquoApplication of magnetic-expansion plasmathrusters to satellite station keeping and attitude control mis-sionsrdquo in Proceedings of the 4th Electric Propulsion Confer-ence AIAA-1964-677 pp 64ndash677 The American Institute ofAeronautics and Astronautics Philadelphia Pa USA Augest-September 1964

[13] S R Oleson R M Myers C A Kluever J P Riehl and F MCurran ldquoAdvanced propulsion for geostationary orbit insertionand north-south station keepingrdquo Journal of Spacecraft andRockets vol 34 no 1 pp 22ndash28 1997

[14] T A Ely and K C Howell ldquoEast-west stationkeeping of satelliteorbits with resonant tesseral harmonicsrdquoActa Astronautica vol46 no 1 pp 1ndash15 2000

[15] C Circi ldquoSimple strategy for geostationary stationkeepingmaneuvers using solar sailrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 249ndash253 2005

[16] P Romero J M Gambi and E Patino ldquoStationkeepingmanoeuvres for geostationary satellites using feedback controltechniquesrdquo Aerospace Science and Technology vol 11 no 2-3pp 229ndash237 2007

[17] V Gomes and A F B A Prado ldquoLow-thrust out-of-planeorbital station-keeping maneuvers for satellitesrdquo MathematicalProblems in Engineering vol 2012 Article ID 532708 14 pages2012

[18] B L Pierson and C A Kluever ldquoThree-stage approach to opti-mal low-thrust earth-moon trajectoriesrdquo Journal of GuidanceControl and Dynamics vol 17 no 6 pp 1275ndash1282 1994

[19] Y J Song S Y Park K H Choi and E S Sim ldquoA lunar cargomission design strategy using variable low thrustrdquo Advances inSpace Research vol 43 no 9 pp 1391ndash1406 2009

[20] S A Fazelzadeh and G A Varzandian ldquoMinimum-time earth-moon and moon-earth orbital maneuvers using time-domainfinite element methodrdquo Acta Astronautica vol 66 no 3-4 pp528ndash538 2010

[21] G Mingotti F Topputo and F Bernelli-Zazzera ldquoEfficientinvariant-manifold low-thrust planar trajectories to themoonrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 p 817 2012

[22] J R Brophy and M Noca ldquoElectric propulsion for solar systemexplorationrdquo Journal of Propulsion and Power vol 14 no 5 pp700ndash707 1998

[23] D P S Santos A F B A Prado L Casalino and G ColasurdoldquoOptimal trajectories towards near-earth-objects using solarelectric propulsion (sep) and gravity assistedmaneuverrdquo Journalof Aerospace Engineering Sciences and Applications vol 1 no 2pp 51ndash64 2008

[24] D P S Santos L Casalino G Colasurdo and A F B APrado ldquoOptimal trajectories towards near-earth-objects usingsolar electric propulsion (sep) and gravity assisted maneuverrdquoWSEAS Transactions on Applied andTheoretical Mechanics vol4 no 3 pp 125ndash135 2009

[25] D P S Santos and A F B A Prado ldquoOptimal low-thrusttrajectories to reach the asteroid apophisrdquo WSEAS MechanicalEngineering Series vol 7 pp 241ndash251 2012

[26] M C B BiggsThe Optimization of Spacecraft Orbital Manoeu-vres Part I Linearly Varying Thrust Angles The HatfieldPolytechnic Numerical Optimization Centre 1978

[27] M C B BiggsThe Optimisation of Spacecraft Orbital Manoeu-vres Part II Using Pontryaginrsquos Maximum Principle The Hat-field Polytechnic Numerical Optimization Centre 1979

[28] A F B A Prado Analise Selecao e Implementacao de Procedi-mentos que visemManobras Otimas de Satelites Artificiais [MSthesis] Instituto Nacional de Pesquisas Espaciais Sao Jose dosCampos 1988

[29] T C Oliveira Estrategias otimas para manobras orbitais uti-lizando propulsao contınua [MS thesis] Instituto Nacional dePesquisas Espaciais Sao Jose dos Campos 2012

[30] J L Ferreira G C Possa B SMoraes et al ldquoPermanentmagnethall thruster for satellite orbit maneuvering with low powerrdquoAdvances in Space Research vol 01 pp 166ndash182 2009

[31] E Stuhlinger Ion Propulsion for Space FlightMcGraw-Hill NewYork NY USA 1st edition 1964

[32] R G Jahn Physics of Electric Propulsion McGraw-Hill NewYork NY USA 1st edition 1968

[33] D G Luemberger Linear and Non-Linear ProgrammingSpringer Science amp Business Media New York NY USA 3rdedition 2008

[34] M S Bazaraa and C M ShettyNonlinear Programming-Theoryand Algorithms JohnWilley amp Sons New York NY USA 1979

[35] V V Zhurin H R Kaufman and R S Robinson ldquoPhysics ofclosed drift thrustersrdquo Plasma Sources Science and Technologyvol 8 no 1 pp R1ndashR20 1999

[36] E M Rocco ldquoPerturbed orbital motion with a PID controlsystem for the trajectoryrdquo in Proceedings of the 14th ColoquioBrasileiro De Dinamica Orbital Aguas de Lindoia AnaisUNESP Guaratingueta Brazil November 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

(ii) The ldquoadjoint-controlrdquo transformation is used to obtainthe initial values of the Lagrange multipliers requiredfor the numerical integrations

(iii) Now with all the initial values needed to solve theproblem the equations of motion and the adjointequations are simultaneously numerically integratedduring the propulsion arcThe values of the pitch andyaw angles are obtained at every step of the inte-gration by the principle of maximum of PontryaginAt the end the value of the final state 119883 after thepropulsion arc is obtained

(iv) The Keplerian elements of the orbit achieved are thenobtained from the final state119883 using (1)

(v) At this point the satisfaction of the constraints isverified If themagnitude of the vector that representsthe constraints satisfaction is smaller than a specifiedtolerance provided by the user (the numerical zero)the algorithm proceeds to step (vii) On the otherhand if it is bigger the algorithmproceeds to step (vi)

(vi) At this step the gradient of the constraints equations(nablaS) is obtained by perturbing the elements of thecontrol vector u and performing a new set of numer-ical integrations in order to obtain the numericalderivatives of the constraints equations with respectto the vector uThen a new set of values for the vectoru is found by (7) (Luemberger [33]) as follows

u119894+1

= u119894minus nablaS119879[nablaSnablaS119879]

minus1

nablaS (7)

and then the algorithm goes back to step (i) with thenew value of the vector u

(vii) Once the constraints satisfaction is achieved thenext step is to search for the minimum of the fuelconsumed The direction of the search d using thegradient method (Bazaraa et al [34]) is

d = minusRnabla119869 (u) (8)

where the vector R is

R = I minus nablaS119879[nablaSnablaS]minus1nablaS (9)

(viii) At this step the magnitude of the vector d is verifiedIf it is smaller than a value specified by the useras a direction search tolerance (numerical zero) thealgorithm goes to step (x) and if it is bigger thealgorithm proceeds to step (ix)

(ix) In order to get better results the initial data ldquoratio ofcontractionrdquo RC and the direction search toleranceare reduced as u gets closer to the minimum Themagnitude for the direction search is then given by

PB =

RC119869 (u)nabla119869 (u) d

(10)

So the complete step is given by

u119894+1

= u119894+ PB d

|d| (11)

and then the algorithm goes back to step (i)

(x) At this step the possibility of the control u to be aKuhn-Tucker point is verified To check this condi-tion the vector VT is built formed by the 119898 firstelements of the vector given by (Bazarra et al [34])as follows

W = minus(SS119879)minus1

Snabla119869 (u) (12)

where119898 is the number of active constraints If VT isnot positive definite the line 119895 which causes VT

119895to

be negative on the matrix S is deleted And then thealgorithm returns to step (v)

(xi) The current step is verified and if it is not the last onethe algorithm goes to a new search for the minimumfuel consumption with a smaller direction searchtolerance for the objective function If this step is thelast one the algorithm is finalized

4 Extensions of the Algorithm

There are several extensions of the algorithm that can beconsidered in order to make it more flexible The first oneis the possibility to perform maneuvers with more than onearc of propulsion Increasing the number of propulsion arcsmeans that the number of optimization variables is increasedFor example for one thrust arc there are 119906

1to 1199066control

variables For two thrust arcs similarly there is a set of 1199061to

11990612control variables and so on In addition by adding more

than one thrust arc the inclusion of constraints to prevent theoverlap of thrust arcs becomes necessary

Another extension is the possibility to consider forbiddenregions to apply the propulsion as a constraintThese regionsare defined by the true longitude as shown in (2) Both valuesof the initial and final true longitudes of the prohibit regionhave to be specified

In some situations the initial range angles 1199040can be less

than zero It means that when the thrust is started the time isnegative To avoid negative time it is added a constraint thatimposes to 119904

0 the time to be equal or bigger than zero This

constraint is only active if 1199040lt 0

The last extension is the possibility to consider constraintsrestricting the pitch and yaw angles In this case the boundsconsidered on 119860 and 119861 are the ones given by Biggs [27]119860119871le 119860 le 119860

119906and 119861

119871le 119861 le 119861

119906 where the suffix ldquo119906rdquo

represents the maximum angle and the suffix ldquo119871rdquo representstheminimumangle If these constraint are active after findingthe optimal angles 119860lowast and 119861lowast and this problem is solved asfollows [26ndash29] 119860 = 119860

119906if 119860lowast ge 119860

119906 119860 = 119860

119871if 119860lowast le 119860

119871

119860 = 119860lowast if 119860

119871le 119860 le 119860

119906 Those equations are similarly

applied to the optimal yaw angles 119861lowast

5 The Permanent Magnet Hall Thruster

As defined by Jahn [32] the electric propulsion is the gasacceleration to generate thrust via electric heating electricbody forces or both electric and magnetic body forces Thegreatest advantage of the electric propulsion is the veryhigh exhaust velocities resulting in a reduction of the fuel

Mathematical Problems in Engineering 5

consumption Although electric propulsion has low thrustthe high specific impulse guarantees a reduced burned fuelalong the maneuver

The hall thruster used in this work as the propulsionsystem was fundamentally envisaged in Russia in 1960 andthe first successful space mission with this thrust propulsionwith 60mN was used in the Meteor satellite series in 1972(Zhurin et al [35]) After that this kind of propulsion systemhas been studied and developed in several countries such asFrance USA Japan and Brazil

The working principle of the hall thrusters is the useof an electromagnet to produce the main magnetic fieldresponsible for the plasma acceleration and generation (Fer-reira et al [30]) This configuration requires the use ofenergy to produce the magnetic field Nevertheless thepermanent magnet hall thruster (PMHT) developed at thePlasma Laboratory of the Universidade de Brasilia uses anarray of permanent magnets instead of an electromagnet toproduce a radial magnetic field inside the cylindrical plasmadrift channel of the thruster (Ferreira et al [30]) The useof permanent magnets instead of electromagnet guaranteeslow-power consumption and it can be used in small- andmedium-size satellites (Ferreira et al [30])

In the Plasma Laboratory of the Universidade de Brasıliait has been developed two kinds of hall thruster called Phall Iand Phall II The first one to be constructed was the Phall I Ithas a stainless steel chamber with 30 centimeters of diameterIts magnetic field is produced by two concentric cylindricalarrays of ferrite permanent magnets 32 bars in the outer shelland 10 bars in the inner shellThemagnetic field in themiddleline of the plasma sourcersquos channel is 200 Gauss (Ferreiraet al [30]) On the other hand the second thruster PhallII has an aluminum plasma chamber and it is smaller with15 cm diameter and contains neodymium-iron magnets toproduce the magnetic field (Ferreira et al [30]) Table 1 givesthe parameters of the performance of both Phall I and PhallII thrusters (Ferreira et al [30])

In this paper it was found first the optimal maneuverswith the thrust parameters shown in Table 1 Then theseoptimal maneuvers were simulated with a PID control for along period of time based on the continuous monitoring ofthe thrust in a real-time performance correcting any possiblefailure

6 Spacecraft Trajectory Simulator

Themaneuver simulator used to simulate the optimalmaneu-vers in a more realistic way is the spacecraft trajectorysimulator (STRS) The STRS can consider constructionalfeatures and operation such as nonlinearities failures errorsand external and internal disturbances in order to determinethe deviation in the reproduction of the optimal solutionpreviously found

Furthermore maneuvers with continuous thrust are usedduring a long period of timeThis kind ofmaneuvers requiresa PID control system to guarantee the achievement of the finalconditions of the maneuver The operating structure of theSTRS can be described as follows (Rocco [36])

(a) The simulation occurs in a discrete way that is atevery simulation step the state of the vehicle (positionand velocity) must be computed For all purposes thedisturbances nonlinearities errors and failures mustbe considered

(b) Since a closed loop system was used to controlthe trajectory the reference state is determined byapplying the optimal maneuver Then the currentstate follows the reference but considers the effectsof disturbances and the constructive characteristics ofthe vehicle

(c) The difference between the reference state and thecurrent state creates an error signal that is insertedinto a PID control

(d) The controller produces a control signal according tothe PID control law and the gains that have been set

(e) The control signal is inserted into the actuator model(propellant thrusters) where the nonlinearities inher-ent to the construction of the actuator can be consid-ered Thus the behavior of the propellant thrusterscan be reproduced by the appropriate adjustmentof the model parameters which are supplied by themanufacturer of the thrust actuator

(f) Finally an actuation signal (Δ119881) is sent to the dynam-ical model of the orbital motion At this signal thevelocities increments due to the disturbances can beadded

(g) With the actuation signal added with the possible dis-turbances themodel of the orbital dynamics providesthe state (position and velocity) after the applicationof the propulsion

(h) Through the use of sensors that were modeled con-sidering its constructive aspects the current state isdetermined and compared with the reference state inorder to close the control loop

7 Results

Before performing tests with the propulsion system PhallsI and II that is one of the goals of the present paper it isimportant to test some of the particular characteristics ofthe algorithm developed here Those tests are performed inmaneuvers that have similar results available in the literatureso a comparison to validate the algorithm and the softwaredeveloped can be made After that the results of the optimalmaneuvers using the new propulsion devices as well as thesimulations of them in the STRS are presented

71 Optimal Maneuvers to Validate the Algorithm

711 Maneuver 1 This first maneuver described has a con-straint at the pitch angle in order to test this capability of theproposed algorithm

Initial orbital elements

119886 = 99000 km 119890 = 07 119894 = 10∘ Ω = 55∘ 120596 = 105∘ 119907 =minus105∘

6 Mathematical Problems in Engineering

Table 1 Comparative performance parameters measured in PHALL I and PHALL II

PHALL1 PHALL2Maximum expected thrust (mN) 126 126Average measured thrust (mN) 849 1200Thrust density (Nm2) 468 lt60Maximum specific impulse (s) 1607 1607Measured specific impulse (s) 1083 1600Ionized mass ratio () 33 sim30Propellant consumption (Kgs) 6 0 times 10

minus61 0 times 10

minus6

Energy consumption (W) 350 250ndash350Electrical efficiency () 339 60Total efficiency () 1012 500

Range angle (degrees)

Pitc

h (d

egre

es)

1101009080

012345

120 130 140

minus1minus2minus3minus4minus5

Figure 2 Optimal pitch angles of the maneuver 1

Vehicle mass = 300 kg thrust = 10 Newton initialposition 119904 = 0∘Condition on the final orbit 119886 = 104000 kmOther constraints 5∘ ge 119860 ge minus5∘ (pitch constraint)

Final orbital elements obtained after the maneuver

119886 = 10399921 km 119890 = 07143 119894 = 10∘Ω = 55∘ 120596 = 105∘119907 = 3024∘

Fuel consumption 24235 kg fuel consumption in Biggs[27] 244 kg fuel consumption in Prado [28] 245 kg

Figure 2 shows the direction of the thrustNow with the constraint in the direction of propulsion

the fuel consumption was increased when compared witha maneuver without this constraint [27 28] The extra fuelneeded due to this constraint is 38 times 10minus3 kg It is possible toview clearly that the pitch angle is confined to 5∘ at the endof the maneuver In any case the algorithm developed hereobtains better results when compared to Biggs [27] and Prado[28]

712 Maneuver 2 This maneuver has the constraint of aregion prohibited for thrusting This is one of the mostimportant capabilities of the algorithm developed here

Initial orbital elements

119886 = 99000 km 119890 = 07 119894 = 10∘ Ω = 55∘ 120596 = 105∘ 119907 =minus105∘

Range angle (degrees)

Pitc

h (d

egre

es)

55504540353025 60 65 70

minus20minus21minus22minus23minus24minus25minus26minus27minus28minus29minus30

Figure 3 Optimal pitch angles of the maneuver 2

Vehicle mass = 300 kg thrust = 10 Newton initialposition 119904 = 0∘Condition on the final orbit 119886 = 104000 kmOther constraints no thrust between the true longi-tudes 120∘ and 180∘

Final orbital elements

119886 = 10399985 km 119890 = 07134 119894 = 10∘Ω = 55∘120596 = 105∘119907 = 32080∘

Fuel consumption 27852 kg fuel consumption in Biggs[27] 281 kg fuel consumption in Prado [28] 281 kg

Figure 3 shows the direction of the thrustIn Figure 3 it is possible to note that the solution has the

thrust arc ending in 120582 = 120∘ The true longitude is definedhere as 120582 = 120596 + 119907 = 42580

∘ that is similar to 6580∘ that isthe result shown in Figure 3 In this way the region prohibitof thrust was obeyed and compared with the maneuver thatis free of this constraint the extra fuel needed to reach thefinal semimajor axis required was 03655 kg Once againthe algorithm proposed here obtained better results whencompared to Biggs [27] and Prado [28]

713 Maneuver 3 This maneuver has the requirements ofchanging three orbital elements of the orbit at the same time

Initial orbital elements

119886 = 419041 km 119890 = 0018 119894= 0688∘ Ω = minus298∘ 120596 =7∘ 119907 = minus972∘

Mathematical Problems in Engineering 7

Range angle (degrees)

Pitc

h (d

egre

es)

7570656055 80 85

200

150

100

50

0

minus50

minus100

minus150

minus200

Figure 4 Optimal pitch angles of the maneuver 3 for the first arc

Yaw

(deg

rees

)

Range angle (degrees)

7570656055 80 85

minus612

minus614

minus616

minus618

minus62

minus622

minus624

minus626

Figure 5 Optimal yaw angles of the maneuver 3 for the first arc

Vehicle mass = 300 kg thrust = 10 Newton initialposition 119904 = 0∘Condition on the final orbit 119886 = 421642 km 119890 = 0119894 = 0∘

Final orbital elements

119886 = 4216455 km 119890 = 000130 119894 = 00419∘Ω = 30060∘120596 = 10047∘ 119907 = 14364∘

Fuel consumption 5565 kg fuel consumption in Biggs[27] 5621 kg fuel consumption in Prado [28] 5579 kg

Figures 4ndash7 show the direction of the thrustIn this maneuver there are three Keplerian elements

fixed on the final orbit semimajor axis eccentricity andinclination Since the maneuver includes a change in theorbital plane the yaw angle is not zero The final conditionswere satisfied and the fuel consumption was reduced by0014 kg when compared to Prado [28]

72 Optimal Maneuvers Using the Phalls I and II In thissection we present the optimal maneuvers found by thealgorithm presented in the present paper when using PhallsI and II The optimal thrust angles are presented and thespecific impulse considered for each maneuver is given inTable 1 Figures 8ndash23 present the optimal angles for theproposedmaneuvers given by the initial and final range angleand the pitch angle at each arc of propulsionThe yaw angle isalways zero because all the maneuvers simulated are planar

Range angle (degrees)

Pitc

h (d

egre

es)

215205 210200190 195185 220 225

30

25

20

15

10

5

0

minus5

Figure 6 Optimal pitch angles of themaneuver 3 for the second arc

Range angle (degrees)

Yaw

(deg

rees

)

215205 210200190 195185

60

55

50

45

40

35

30220 225

Figure 7 Optimal yaw angles of the maneuver 3 for the second arc

All the maneuvers considered here have the same initialorbital elements and final condition imposed for the finalorbit Only the number of arcs and the propulsion system isdifferent at each maneuver The parameters of the maneuverproposed are as follows

Initial orbital elements 119886 = 7130865 km 119890 = 00035 119894= 985054∘ Ω = 0∘ 120596 = 0 119907 = 220∘Vehicle mass = 300 kgCondition imposed on the final orbit 119886 = 7200 km 119890= 0004

721 Maneuver 4 For this maneuver it was considered theexistence of three Phall I thrusters The propulsion force ofeach one is assumed to be the average measured value Theparameters considered in this maneuver are as follows

Total thrust = 0252 Newton specific impulse 1083 sSolution

Final orbital elements 119886 = 720000 km 119890 = 0004Fuel consumption 1046346 kgDuration of the maneuver 4406696 s

722 Maneuver 5 Thismaneuver has the same conditions ofmaneuver 4 but now two thrust arcs are used instead of oneThe parameters considered in this maneuver are as follows

8 Mathematical Problems in Engineering

Range angle (degrees)

30002500200015001000500

Pitc

h an

gle

(deg

rees

)

0

0

5

10

15

minus10

minus15

minus5

Figure 8 Optimal pitch angles for maneuver 4

Range angle (degrees)

Pitc

h an

gle

(deg

rees

) 03

010

02

0405

1400120010008006004002000

minus01minus02

minus03minus04minus05

Figure 9 Optimal pitch angles for the first arc of the maneuver 5

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

1

0

05

15

2

40003800360034003200300028002600

minus05

minus1

minus15

minus2

Figure 10 Optimal pitch angles for the second arc of maneuver 5

Total thrust = 0252 Newton specific impulse 1083 s

Solution

Final orbital elements after the first arc 119886 =716530 km 119890 = 00039Final orbital elements after the second arc 119886 =720000 km 119890 = 00040Fuel consumption 1013402 kgDuration of the maneuver 4267954 s

Note that the use of two arcs generates savings in the fuelconsumption as expected

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

0

5

10

15

22001800 2000160014001000 1200800600400200

minus5

minus10

minus15

Figure 11 Optimal pitch angle for maneuver 6

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

02

0

0604

081

1000900600 700 8005004003002001000

minus02

minus06

minus1

minus04

minus08

Figure 12 Optimal pitch angles for the first arc of maneuver 7

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

0

05

1

15

240023002100 2200200019001800170016001500

minus05

minus1

minus15

Figure 13 Optimal pitch angles for the second arc of maneuver 7

723 Maneuver 6 For this maneuver it was considered theexistence of three Phall II thrusters The propulsion force ofeach one is assumed to be the average measured value Theparameters considered in this maneuver are as follows

Total thrust = 0360 Newton specific impulse 1600 s

Solution

Final orbital elements 119886 = 720000 km 119890 = 0004Fuel consumption 0697930 kgDuration of the maneuver 3039873 s

Mathematical Problems in Engineering 9

Time (s)

minus5

minus10

0

0

5

05 1 15 2 25 3 35 4 45times104

Sem

imaj

or a

xis

dev

iati

on(m

)

Figure 14 Semimajor axis deviation in maneuver 4

02468

1012

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus2minus4minus6minus8

Ecc

entr

icit

y d

evia

tion

times10minus7

Figure 15 Eccentricity deviation in maneuver 4

0

0005

001

0015

Posi

tion

dev

iati

on (m

)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus0005

minus001

Figure 16 Position deviation in maneuver 4

Note that the use of the Phall II generates savings inthe fuel consumption since it has better parameters whencompared to the Phall I device

724 Maneuver 7 Thismaneuver has the same conditions ofmaneuver 6 but now with two thrust arcs instead of oneTheparameters considered in this maneuver are as follows

Total thrust = 0360 Newton specific impulse 1600 s

Solution

Final orbital elements after the first arc 119886 =716777 km 119890 = 00039

0

1

2

3

4

5

Vel

ocit

y d

evia

tion

(m

s)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus1

minus2

minus3

times10minus3

Figure 17 Velocity deviation in maneuver 4

Final orbital elements after the second arc 119886 =720000 km 119890 = 00040Fuel consumption 0686314 kgDuration of the maneuver 2989285 s

Note that the use of two arcs generates savings in the fuelconsumption one more time as expected

73 Optimal Maneuvers Simulations Using the STRS Nowwith the help of the optimal maneuvers found previously weshall simulate thosemaneuvers in the STRS using the optimalangles Some different aspects of the propulsion system wereconsidered to simulate amore realistic environment and alsothe PID control was considered

In this section for the propulsion system it was consid-ered that at each inertial direction 119909 119910 119911 has an error on theactuator signal (ms) with mean equal to zero and varianceequal to 10minus10 that is it was inserted an error in the velocity ofthe satelliteThis error has amplitude approximately close to 5times 10minus3N for each thrust which corresponds to the error of thePMHT (Ferreira et al [30]) There is a PID control to correctthe errors due the actuator but it is still possible to analyzethe deviations and the increase of the fuel consumption dueto the errors

731 Maneuver 4 in STRS In this case it is consideredmaneuver 4 including the solution found there

Final orbital elements 119886 = 719985 km 119890 = 0004Fuel consumption 10470 kgPropellant mass difference from the optimal maneu-ver 666 times 10minus4 kg

The error inserted in the propulsion system resulted in adeviation of the orbital elements as we can see in Figures 14and 15 for the semimajor axis and eccentricity If there wasnot any error at the propulsion system the deviation wouldbe zero all the time for the orbital elements In Figures 16 and17 we can see also the effect of the error in the velocity andposition as a function of these parameters Finally the thrusterror is shown in Figure 18

10 Mathematical Problems in EngineeringA

pplie

d th

rust

(N)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

0504

03

02

01

0

minus01

minus02

minus03

minus04

x axisy axisz axis

ThrustAverage

Figure 18 Thrust applied in maneuver 4

0

5

10

Time (s)

0 05 1 15 2 25 3 35times104

minus5

minus10

minus15

Sem

imaj

or a

xis

dev

iati

on(m

)

Figure 19 Semimajor axis deviation in maneuver 6

732 Maneuver 5 in STRS In this case maneuver 5 isconsidered including the solution found

Final orbital elements after the first arc 119886 =716523 km 119890 = 00039

Final orbital elements after the second arc 119886 =719980 km 119890 = 00040

Fuel consumption 1013362 kg

Propellant mass difference from the optimal maneu-ver 6384 times 10minus4 kg

733 Maneuver 6 in STRS Now maneuver 6 is considered

Final orbital elements 119886 = 719982 km 119890 = 0004

Fuel consumption 06979 kg

Propellant mass difference from the optimal maneu-ver 219 times 10minus4 kg

The same analysis for the figures made in Section 731 fitshere Figures 19ndash23 show the deviations and the error thatoccurred on the propulsion system

Time (s)

0 05 1 15 2 25 3 35times104

Ecc

entr

icit

y d

evia

tion

times10minus6

2

15

1

05

0

minus05

minus1

minus15

Figure 20 Eccentricity deviation in maneuver 6

734 Maneuver 7 in STRS Maneuver 7 is now studiedFinal orbital elements after the first arc 119886 =716770 km 119890 = 00039Final orbital elements after the second arc 119886 =719983 km 119890 = 00040Fuel consumption 0686279 kgPropellant mass difference from the optimal maneu-ver 21469 times 10minus4 kg

735 Maneuver 8 in STRS This maneuver uses the optimalangles given in maneuver 4 but includes the errors proposedfor the propulsion system In addition it was consideredother nonlinearities for the actuator two seconds delay torespond the control signal a rate limiter for the variationof the satelite velocity which the absolute limit of the rateis 000025ms dead zone where the propulsion system isnot turned on if the required velocity increment is lowerthan 00002ms Considering these nonlinearities for thepropulsion system the PID control was able to achievethe maneuver although the fuel consumption was slightlyincreased The results for this simulation are as follows

Final orbital elements 119886 = 719986 km 119890 = 0004Fuel consumption 11276322 kgPropellant mass difference from the optimal maneu-ver 0081429 kg

736 Maneuver 9 in STRS The maneuver simulated hereuses the optimal angles given in maneuver 6 with the errorsproposed for the propulsion system In addition it wasproposed the samenonlinearities (rate limiter dead zone andtime delay) of maneuver 8 Considering these nonlinearitiesfor the propulsion system the PID control was able to achievethe maneuver although the fuel consumption was increasedThe results for this maneuver are as follows

Final orbital elements 119886 = 719984 km 119890 = 00040Fuel consumption 0721921 kgThe propellant mass difference from the optimalmaneuver 002402 kg

Mathematical Problems in Engineering 11

Time (s)

0 05 1 15 2 25 3 35times104

002

0015

001

0005

0

minus0005

minus001

minus0015Po

siti

on d

evia

tion

(m)

Figure 21 Position deviation in maneuver 6

Vel

ocit

y d

evia

tion

(m

s)

Time (s)

0 05 1 15 2 25 3 35times104

8

6

4

2

0

minus2

minus4

times10minus3

Figure 22 Velocity deviation in maneuver 6

x axisy axisz axis

ThrustAverage

Time (s)

0 05 1 15 2 25 3 35times104

06

04

02

0

minus02

minus04

minus06

minus08

App

lied

thru

st (N

)

Figure 23 Thrust applied in maneuver 6

8 Conclusion

Primarily this paper established an algorithm to solve theproblem of orbital maneuvers using a low-thrust control fora spacecraft that is travelling around the Earth Then allthe maneuvers studied in this paper were successfully solvedwith the final conditions and constraints proposed and withreasonable fuel consumptions

It was possible to verify that maneuvers with morethrusting arcs consume less fuel as expected This occursbecause the algorithm has more variables to optimize so itis possible to reduce the fuel consumption

The second part of this work was to simulate the optimalmaneuvers found by the algorithm proposed here in a morerealistic environment which can consider errors on theactuator that was not considered in the search for the optimalmaneuver

Themaneuvers simulated in the STRSwere coherent withthe optimal maneuvers validating the integration of bothsoftwares

The error considered in the propulsion system opensthe possibility to study how the system would react tothose nonlinearities Although errors were inserted in thepropulsion system the PID control was able to correct andaccomplish the maneuver

It is possible to analyze the shifts caused by the error aswell as the deviations in semimajor axis and eccentricity It isalso possible to see that the propulsion thrust was not linear

For themaneuvers shown in Sections 735 and 736 somenonlinearities of the propulsion system such as a responsetime a rate limiter and a dead zone for the actuator signalwere considered All these errors occur in real systems Withthe help of the STRS it was possible to simulate these errorsand analyze the maneuvers studied with more accuracyThe STRS has shown to be an efficient simulator capableof simulating and studying optimal maneuvers in a morerealistic environment and to analyze many nonlinearities notconsidered in the calculation of the optimal maneuver

Evidently the phall II was more efficient than the phall Ias noticed by comparing maneuver 4 with maneuver 6 andmaneuver 5 with maneuver 7 This was expected since thephall II has a higher specific impulse and a higher averagepropulsion thrust as well Nevertheless the PID control wasable to control all the maneuvers and reaches the final orbitalconstraints

References

[1] D F Lawden ldquoMinimal rocket trajectoriesrdquo ARS Journal vol23 no 6 pp 360ndash382 1953

[2] D F Lawden ldquoFundamentals of space navigationrdquo Journal of theBritish Interplanetary Society vol 13 pp 87ndash101 1954

12 Mathematical Problems in Engineering

[3] V V Beletsky and V A Egorov ldquoInterplanetary flights withconstant output enginesrdquoCosmic Research vol 2 no 3 pp 303ndash330 1964

[4] A A Sukhanov ldquoOptimization of flights with low thrustrdquoCosmic Research vol 37 no 2 pp 182ndash191 1999

[5] A A Sukhanov ldquoOptimization of low-thrust interplanetarytransfersrdquo Cosmic Research vol 38 no 6 pp 584ndash587 2000

[6] A A Sukhanov and A F B A Prado ldquoA modification of themethod of transporting trajectoryrdquoCosmic Research vol 42 no1 pp 107ndash112 2004

[7] A A Sukhanov and A F B A Prado ldquoOptimization of lowthrust transfers in the three body problemrdquo Cosmic Researchvol 46 no 5 pp 440ndash451 2008

[8] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction Irdquo Cosmic Researchvol 45 no 5 pp 417ndash423 2007

[9] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction IIrdquo Cosmic Researchvol 46 no 1 pp 49ndash59 2008

[10] S S Fernandes and F D C Carvalho ldquoA first-order analyticaltheory for optimal low-thrust limited-power transfers betweenarbitrary elliptical coplanar orbitsrdquo Mathematical Problems inEngineering vol 2008 Article ID 525930 30 pages 2008

[11] R A Boucher ldquoElectrical propulsion for control of stationarysatellitesrdquo Journal of Spacecraft and Rockets vol 1 no 2 pp 164ndash169 1964

[12] D N Bowditch ldquoApplication of magnetic-expansion plasmathrusters to satellite station keeping and attitude control mis-sionsrdquo in Proceedings of the 4th Electric Propulsion Confer-ence AIAA-1964-677 pp 64ndash677 The American Institute ofAeronautics and Astronautics Philadelphia Pa USA Augest-September 1964

[13] S R Oleson R M Myers C A Kluever J P Riehl and F MCurran ldquoAdvanced propulsion for geostationary orbit insertionand north-south station keepingrdquo Journal of Spacecraft andRockets vol 34 no 1 pp 22ndash28 1997

[14] T A Ely and K C Howell ldquoEast-west stationkeeping of satelliteorbits with resonant tesseral harmonicsrdquoActa Astronautica vol46 no 1 pp 1ndash15 2000

[15] C Circi ldquoSimple strategy for geostationary stationkeepingmaneuvers using solar sailrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 249ndash253 2005

[16] P Romero J M Gambi and E Patino ldquoStationkeepingmanoeuvres for geostationary satellites using feedback controltechniquesrdquo Aerospace Science and Technology vol 11 no 2-3pp 229ndash237 2007

[17] V Gomes and A F B A Prado ldquoLow-thrust out-of-planeorbital station-keeping maneuvers for satellitesrdquo MathematicalProblems in Engineering vol 2012 Article ID 532708 14 pages2012

[18] B L Pierson and C A Kluever ldquoThree-stage approach to opti-mal low-thrust earth-moon trajectoriesrdquo Journal of GuidanceControl and Dynamics vol 17 no 6 pp 1275ndash1282 1994

[19] Y J Song S Y Park K H Choi and E S Sim ldquoA lunar cargomission design strategy using variable low thrustrdquo Advances inSpace Research vol 43 no 9 pp 1391ndash1406 2009

[20] S A Fazelzadeh and G A Varzandian ldquoMinimum-time earth-moon and moon-earth orbital maneuvers using time-domainfinite element methodrdquo Acta Astronautica vol 66 no 3-4 pp528ndash538 2010

[21] G Mingotti F Topputo and F Bernelli-Zazzera ldquoEfficientinvariant-manifold low-thrust planar trajectories to themoonrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 p 817 2012

[22] J R Brophy and M Noca ldquoElectric propulsion for solar systemexplorationrdquo Journal of Propulsion and Power vol 14 no 5 pp700ndash707 1998

[23] D P S Santos A F B A Prado L Casalino and G ColasurdoldquoOptimal trajectories towards near-earth-objects using solarelectric propulsion (sep) and gravity assistedmaneuverrdquo Journalof Aerospace Engineering Sciences and Applications vol 1 no 2pp 51ndash64 2008

[24] D P S Santos L Casalino G Colasurdo and A F B APrado ldquoOptimal trajectories towards near-earth-objects usingsolar electric propulsion (sep) and gravity assisted maneuverrdquoWSEAS Transactions on Applied andTheoretical Mechanics vol4 no 3 pp 125ndash135 2009

[25] D P S Santos and A F B A Prado ldquoOptimal low-thrusttrajectories to reach the asteroid apophisrdquo WSEAS MechanicalEngineering Series vol 7 pp 241ndash251 2012

[26] M C B BiggsThe Optimization of Spacecraft Orbital Manoeu-vres Part I Linearly Varying Thrust Angles The HatfieldPolytechnic Numerical Optimization Centre 1978

[27] M C B BiggsThe Optimisation of Spacecraft Orbital Manoeu-vres Part II Using Pontryaginrsquos Maximum Principle The Hat-field Polytechnic Numerical Optimization Centre 1979

[28] A F B A Prado Analise Selecao e Implementacao de Procedi-mentos que visemManobras Otimas de Satelites Artificiais [MSthesis] Instituto Nacional de Pesquisas Espaciais Sao Jose dosCampos 1988

[29] T C Oliveira Estrategias otimas para manobras orbitais uti-lizando propulsao contınua [MS thesis] Instituto Nacional dePesquisas Espaciais Sao Jose dos Campos 2012

[30] J L Ferreira G C Possa B SMoraes et al ldquoPermanentmagnethall thruster for satellite orbit maneuvering with low powerrdquoAdvances in Space Research vol 01 pp 166ndash182 2009

[31] E Stuhlinger Ion Propulsion for Space FlightMcGraw-Hill NewYork NY USA 1st edition 1964

[32] R G Jahn Physics of Electric Propulsion McGraw-Hill NewYork NY USA 1st edition 1968

[33] D G Luemberger Linear and Non-Linear ProgrammingSpringer Science amp Business Media New York NY USA 3rdedition 2008

[34] M S Bazaraa and C M ShettyNonlinear Programming-Theoryand Algorithms JohnWilley amp Sons New York NY USA 1979

[35] V V Zhurin H R Kaufman and R S Robinson ldquoPhysics ofclosed drift thrustersrdquo Plasma Sources Science and Technologyvol 8 no 1 pp R1ndashR20 1999

[36] E M Rocco ldquoPerturbed orbital motion with a PID controlsystem for the trajectoryrdquo in Proceedings of the 14th ColoquioBrasileiro De Dinamica Orbital Aguas de Lindoia AnaisUNESP Guaratingueta Brazil November 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

consumption Although electric propulsion has low thrustthe high specific impulse guarantees a reduced burned fuelalong the maneuver

The hall thruster used in this work as the propulsionsystem was fundamentally envisaged in Russia in 1960 andthe first successful space mission with this thrust propulsionwith 60mN was used in the Meteor satellite series in 1972(Zhurin et al [35]) After that this kind of propulsion systemhas been studied and developed in several countries such asFrance USA Japan and Brazil

The working principle of the hall thrusters is the useof an electromagnet to produce the main magnetic fieldresponsible for the plasma acceleration and generation (Fer-reira et al [30]) This configuration requires the use ofenergy to produce the magnetic field Nevertheless thepermanent magnet hall thruster (PMHT) developed at thePlasma Laboratory of the Universidade de Brasilia uses anarray of permanent magnets instead of an electromagnet toproduce a radial magnetic field inside the cylindrical plasmadrift channel of the thruster (Ferreira et al [30]) The useof permanent magnets instead of electromagnet guaranteeslow-power consumption and it can be used in small- andmedium-size satellites (Ferreira et al [30])

In the Plasma Laboratory of the Universidade de Brasıliait has been developed two kinds of hall thruster called Phall Iand Phall II The first one to be constructed was the Phall I Ithas a stainless steel chamber with 30 centimeters of diameterIts magnetic field is produced by two concentric cylindricalarrays of ferrite permanent magnets 32 bars in the outer shelland 10 bars in the inner shellThemagnetic field in themiddleline of the plasma sourcersquos channel is 200 Gauss (Ferreiraet al [30]) On the other hand the second thruster PhallII has an aluminum plasma chamber and it is smaller with15 cm diameter and contains neodymium-iron magnets toproduce the magnetic field (Ferreira et al [30]) Table 1 givesthe parameters of the performance of both Phall I and PhallII thrusters (Ferreira et al [30])

In this paper it was found first the optimal maneuverswith the thrust parameters shown in Table 1 Then theseoptimal maneuvers were simulated with a PID control for along period of time based on the continuous monitoring ofthe thrust in a real-time performance correcting any possiblefailure

6 Spacecraft Trajectory Simulator

Themaneuver simulator used to simulate the optimalmaneu-vers in a more realistic way is the spacecraft trajectorysimulator (STRS) The STRS can consider constructionalfeatures and operation such as nonlinearities failures errorsand external and internal disturbances in order to determinethe deviation in the reproduction of the optimal solutionpreviously found

Furthermore maneuvers with continuous thrust are usedduring a long period of timeThis kind ofmaneuvers requiresa PID control system to guarantee the achievement of the finalconditions of the maneuver The operating structure of theSTRS can be described as follows (Rocco [36])

(a) The simulation occurs in a discrete way that is atevery simulation step the state of the vehicle (positionand velocity) must be computed For all purposes thedisturbances nonlinearities errors and failures mustbe considered

(b) Since a closed loop system was used to controlthe trajectory the reference state is determined byapplying the optimal maneuver Then the currentstate follows the reference but considers the effectsof disturbances and the constructive characteristics ofthe vehicle

(c) The difference between the reference state and thecurrent state creates an error signal that is insertedinto a PID control

(d) The controller produces a control signal according tothe PID control law and the gains that have been set

(e) The control signal is inserted into the actuator model(propellant thrusters) where the nonlinearities inher-ent to the construction of the actuator can be consid-ered Thus the behavior of the propellant thrusterscan be reproduced by the appropriate adjustmentof the model parameters which are supplied by themanufacturer of the thrust actuator

(f) Finally an actuation signal (Δ119881) is sent to the dynam-ical model of the orbital motion At this signal thevelocities increments due to the disturbances can beadded

(g) With the actuation signal added with the possible dis-turbances themodel of the orbital dynamics providesthe state (position and velocity) after the applicationof the propulsion

(h) Through the use of sensors that were modeled con-sidering its constructive aspects the current state isdetermined and compared with the reference state inorder to close the control loop

7 Results

Before performing tests with the propulsion system PhallsI and II that is one of the goals of the present paper it isimportant to test some of the particular characteristics ofthe algorithm developed here Those tests are performed inmaneuvers that have similar results available in the literatureso a comparison to validate the algorithm and the softwaredeveloped can be made After that the results of the optimalmaneuvers using the new propulsion devices as well as thesimulations of them in the STRS are presented

71 Optimal Maneuvers to Validate the Algorithm

711 Maneuver 1 This first maneuver described has a con-straint at the pitch angle in order to test this capability of theproposed algorithm

Initial orbital elements

119886 = 99000 km 119890 = 07 119894 = 10∘ Ω = 55∘ 120596 = 105∘ 119907 =minus105∘

6 Mathematical Problems in Engineering

Table 1 Comparative performance parameters measured in PHALL I and PHALL II

PHALL1 PHALL2Maximum expected thrust (mN) 126 126Average measured thrust (mN) 849 1200Thrust density (Nm2) 468 lt60Maximum specific impulse (s) 1607 1607Measured specific impulse (s) 1083 1600Ionized mass ratio () 33 sim30Propellant consumption (Kgs) 6 0 times 10

minus61 0 times 10

minus6

Energy consumption (W) 350 250ndash350Electrical efficiency () 339 60Total efficiency () 1012 500

Range angle (degrees)

Pitc

h (d

egre

es)

1101009080

012345

120 130 140

minus1minus2minus3minus4minus5

Figure 2 Optimal pitch angles of the maneuver 1

Vehicle mass = 300 kg thrust = 10 Newton initialposition 119904 = 0∘Condition on the final orbit 119886 = 104000 kmOther constraints 5∘ ge 119860 ge minus5∘ (pitch constraint)

Final orbital elements obtained after the maneuver

119886 = 10399921 km 119890 = 07143 119894 = 10∘Ω = 55∘ 120596 = 105∘119907 = 3024∘

Fuel consumption 24235 kg fuel consumption in Biggs[27] 244 kg fuel consumption in Prado [28] 245 kg

Figure 2 shows the direction of the thrustNow with the constraint in the direction of propulsion

the fuel consumption was increased when compared witha maneuver without this constraint [27 28] The extra fuelneeded due to this constraint is 38 times 10minus3 kg It is possible toview clearly that the pitch angle is confined to 5∘ at the endof the maneuver In any case the algorithm developed hereobtains better results when compared to Biggs [27] and Prado[28]

712 Maneuver 2 This maneuver has the constraint of aregion prohibited for thrusting This is one of the mostimportant capabilities of the algorithm developed here

Initial orbital elements

119886 = 99000 km 119890 = 07 119894 = 10∘ Ω = 55∘ 120596 = 105∘ 119907 =minus105∘

Range angle (degrees)

Pitc

h (d

egre

es)

55504540353025 60 65 70

minus20minus21minus22minus23minus24minus25minus26minus27minus28minus29minus30

Figure 3 Optimal pitch angles of the maneuver 2

Vehicle mass = 300 kg thrust = 10 Newton initialposition 119904 = 0∘Condition on the final orbit 119886 = 104000 kmOther constraints no thrust between the true longi-tudes 120∘ and 180∘

Final orbital elements

119886 = 10399985 km 119890 = 07134 119894 = 10∘Ω = 55∘120596 = 105∘119907 = 32080∘

Fuel consumption 27852 kg fuel consumption in Biggs[27] 281 kg fuel consumption in Prado [28] 281 kg

Figure 3 shows the direction of the thrustIn Figure 3 it is possible to note that the solution has the

thrust arc ending in 120582 = 120∘ The true longitude is definedhere as 120582 = 120596 + 119907 = 42580

∘ that is similar to 6580∘ that isthe result shown in Figure 3 In this way the region prohibitof thrust was obeyed and compared with the maneuver thatis free of this constraint the extra fuel needed to reach thefinal semimajor axis required was 03655 kg Once againthe algorithm proposed here obtained better results whencompared to Biggs [27] and Prado [28]

713 Maneuver 3 This maneuver has the requirements ofchanging three orbital elements of the orbit at the same time

Initial orbital elements

119886 = 419041 km 119890 = 0018 119894= 0688∘ Ω = minus298∘ 120596 =7∘ 119907 = minus972∘

Mathematical Problems in Engineering 7

Range angle (degrees)

Pitc

h (d

egre

es)

7570656055 80 85

200

150

100

50

0

minus50

minus100

minus150

minus200

Figure 4 Optimal pitch angles of the maneuver 3 for the first arc

Yaw

(deg

rees

)

Range angle (degrees)

7570656055 80 85

minus612

minus614

minus616

minus618

minus62

minus622

minus624

minus626

Figure 5 Optimal yaw angles of the maneuver 3 for the first arc

Vehicle mass = 300 kg thrust = 10 Newton initialposition 119904 = 0∘Condition on the final orbit 119886 = 421642 km 119890 = 0119894 = 0∘

Final orbital elements

119886 = 4216455 km 119890 = 000130 119894 = 00419∘Ω = 30060∘120596 = 10047∘ 119907 = 14364∘

Fuel consumption 5565 kg fuel consumption in Biggs[27] 5621 kg fuel consumption in Prado [28] 5579 kg

Figures 4ndash7 show the direction of the thrustIn this maneuver there are three Keplerian elements

fixed on the final orbit semimajor axis eccentricity andinclination Since the maneuver includes a change in theorbital plane the yaw angle is not zero The final conditionswere satisfied and the fuel consumption was reduced by0014 kg when compared to Prado [28]

72 Optimal Maneuvers Using the Phalls I and II In thissection we present the optimal maneuvers found by thealgorithm presented in the present paper when using PhallsI and II The optimal thrust angles are presented and thespecific impulse considered for each maneuver is given inTable 1 Figures 8ndash23 present the optimal angles for theproposedmaneuvers given by the initial and final range angleand the pitch angle at each arc of propulsionThe yaw angle isalways zero because all the maneuvers simulated are planar

Range angle (degrees)

Pitc

h (d

egre

es)

215205 210200190 195185 220 225

30

25

20

15

10

5

0

minus5

Figure 6 Optimal pitch angles of themaneuver 3 for the second arc

Range angle (degrees)

Yaw

(deg

rees

)

215205 210200190 195185

60

55

50

45

40

35

30220 225

Figure 7 Optimal yaw angles of the maneuver 3 for the second arc

All the maneuvers considered here have the same initialorbital elements and final condition imposed for the finalorbit Only the number of arcs and the propulsion system isdifferent at each maneuver The parameters of the maneuverproposed are as follows

Initial orbital elements 119886 = 7130865 km 119890 = 00035 119894= 985054∘ Ω = 0∘ 120596 = 0 119907 = 220∘Vehicle mass = 300 kgCondition imposed on the final orbit 119886 = 7200 km 119890= 0004

721 Maneuver 4 For this maneuver it was considered theexistence of three Phall I thrusters The propulsion force ofeach one is assumed to be the average measured value Theparameters considered in this maneuver are as follows

Total thrust = 0252 Newton specific impulse 1083 sSolution

Final orbital elements 119886 = 720000 km 119890 = 0004Fuel consumption 1046346 kgDuration of the maneuver 4406696 s

722 Maneuver 5 Thismaneuver has the same conditions ofmaneuver 4 but now two thrust arcs are used instead of oneThe parameters considered in this maneuver are as follows

8 Mathematical Problems in Engineering

Range angle (degrees)

30002500200015001000500

Pitc

h an

gle

(deg

rees

)

0

0

5

10

15

minus10

minus15

minus5

Figure 8 Optimal pitch angles for maneuver 4

Range angle (degrees)

Pitc

h an

gle

(deg

rees

) 03

010

02

0405

1400120010008006004002000

minus01minus02

minus03minus04minus05

Figure 9 Optimal pitch angles for the first arc of the maneuver 5

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

1

0

05

15

2

40003800360034003200300028002600

minus05

minus1

minus15

minus2

Figure 10 Optimal pitch angles for the second arc of maneuver 5

Total thrust = 0252 Newton specific impulse 1083 s

Solution

Final orbital elements after the first arc 119886 =716530 km 119890 = 00039Final orbital elements after the second arc 119886 =720000 km 119890 = 00040Fuel consumption 1013402 kgDuration of the maneuver 4267954 s

Note that the use of two arcs generates savings in the fuelconsumption as expected

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

0

5

10

15

22001800 2000160014001000 1200800600400200

minus5

minus10

minus15

Figure 11 Optimal pitch angle for maneuver 6

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

02

0

0604

081

1000900600 700 8005004003002001000

minus02

minus06

minus1

minus04

minus08

Figure 12 Optimal pitch angles for the first arc of maneuver 7

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

0

05

1

15

240023002100 2200200019001800170016001500

minus05

minus1

minus15

Figure 13 Optimal pitch angles for the second arc of maneuver 7

723 Maneuver 6 For this maneuver it was considered theexistence of three Phall II thrusters The propulsion force ofeach one is assumed to be the average measured value Theparameters considered in this maneuver are as follows

Total thrust = 0360 Newton specific impulse 1600 s

Solution

Final orbital elements 119886 = 720000 km 119890 = 0004Fuel consumption 0697930 kgDuration of the maneuver 3039873 s

Mathematical Problems in Engineering 9

Time (s)

minus5

minus10

0

0

5

05 1 15 2 25 3 35 4 45times104

Sem

imaj

or a

xis

dev

iati

on(m

)

Figure 14 Semimajor axis deviation in maneuver 4

02468

1012

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus2minus4minus6minus8

Ecc

entr

icit

y d

evia

tion

times10minus7

Figure 15 Eccentricity deviation in maneuver 4

0

0005

001

0015

Posi

tion

dev

iati

on (m

)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus0005

minus001

Figure 16 Position deviation in maneuver 4

Note that the use of the Phall II generates savings inthe fuel consumption since it has better parameters whencompared to the Phall I device

724 Maneuver 7 Thismaneuver has the same conditions ofmaneuver 6 but now with two thrust arcs instead of oneTheparameters considered in this maneuver are as follows

Total thrust = 0360 Newton specific impulse 1600 s

Solution

Final orbital elements after the first arc 119886 =716777 km 119890 = 00039

0

1

2

3

4

5

Vel

ocit

y d

evia

tion

(m

s)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus1

minus2

minus3

times10minus3

Figure 17 Velocity deviation in maneuver 4

Final orbital elements after the second arc 119886 =720000 km 119890 = 00040Fuel consumption 0686314 kgDuration of the maneuver 2989285 s

Note that the use of two arcs generates savings in the fuelconsumption one more time as expected

73 Optimal Maneuvers Simulations Using the STRS Nowwith the help of the optimal maneuvers found previously weshall simulate thosemaneuvers in the STRS using the optimalangles Some different aspects of the propulsion system wereconsidered to simulate amore realistic environment and alsothe PID control was considered

In this section for the propulsion system it was consid-ered that at each inertial direction 119909 119910 119911 has an error on theactuator signal (ms) with mean equal to zero and varianceequal to 10minus10 that is it was inserted an error in the velocity ofthe satelliteThis error has amplitude approximately close to 5times 10minus3N for each thrust which corresponds to the error of thePMHT (Ferreira et al [30]) There is a PID control to correctthe errors due the actuator but it is still possible to analyzethe deviations and the increase of the fuel consumption dueto the errors

731 Maneuver 4 in STRS In this case it is consideredmaneuver 4 including the solution found there

Final orbital elements 119886 = 719985 km 119890 = 0004Fuel consumption 10470 kgPropellant mass difference from the optimal maneu-ver 666 times 10minus4 kg

The error inserted in the propulsion system resulted in adeviation of the orbital elements as we can see in Figures 14and 15 for the semimajor axis and eccentricity If there wasnot any error at the propulsion system the deviation wouldbe zero all the time for the orbital elements In Figures 16 and17 we can see also the effect of the error in the velocity andposition as a function of these parameters Finally the thrusterror is shown in Figure 18

10 Mathematical Problems in EngineeringA

pplie

d th

rust

(N)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

0504

03

02

01

0

minus01

minus02

minus03

minus04

x axisy axisz axis

ThrustAverage

Figure 18 Thrust applied in maneuver 4

0

5

10

Time (s)

0 05 1 15 2 25 3 35times104

minus5

minus10

minus15

Sem

imaj

or a

xis

dev

iati

on(m

)

Figure 19 Semimajor axis deviation in maneuver 6

732 Maneuver 5 in STRS In this case maneuver 5 isconsidered including the solution found

Final orbital elements after the first arc 119886 =716523 km 119890 = 00039

Final orbital elements after the second arc 119886 =719980 km 119890 = 00040

Fuel consumption 1013362 kg

Propellant mass difference from the optimal maneu-ver 6384 times 10minus4 kg

733 Maneuver 6 in STRS Now maneuver 6 is considered

Final orbital elements 119886 = 719982 km 119890 = 0004

Fuel consumption 06979 kg

Propellant mass difference from the optimal maneu-ver 219 times 10minus4 kg

The same analysis for the figures made in Section 731 fitshere Figures 19ndash23 show the deviations and the error thatoccurred on the propulsion system

Time (s)

0 05 1 15 2 25 3 35times104

Ecc

entr

icit

y d

evia

tion

times10minus6

2

15

1

05

0

minus05

minus1

minus15

Figure 20 Eccentricity deviation in maneuver 6

734 Maneuver 7 in STRS Maneuver 7 is now studiedFinal orbital elements after the first arc 119886 =716770 km 119890 = 00039Final orbital elements after the second arc 119886 =719983 km 119890 = 00040Fuel consumption 0686279 kgPropellant mass difference from the optimal maneu-ver 21469 times 10minus4 kg

735 Maneuver 8 in STRS This maneuver uses the optimalangles given in maneuver 4 but includes the errors proposedfor the propulsion system In addition it was consideredother nonlinearities for the actuator two seconds delay torespond the control signal a rate limiter for the variationof the satelite velocity which the absolute limit of the rateis 000025ms dead zone where the propulsion system isnot turned on if the required velocity increment is lowerthan 00002ms Considering these nonlinearities for thepropulsion system the PID control was able to achievethe maneuver although the fuel consumption was slightlyincreased The results for this simulation are as follows

Final orbital elements 119886 = 719986 km 119890 = 0004Fuel consumption 11276322 kgPropellant mass difference from the optimal maneu-ver 0081429 kg

736 Maneuver 9 in STRS The maneuver simulated hereuses the optimal angles given in maneuver 6 with the errorsproposed for the propulsion system In addition it wasproposed the samenonlinearities (rate limiter dead zone andtime delay) of maneuver 8 Considering these nonlinearitiesfor the propulsion system the PID control was able to achievethe maneuver although the fuel consumption was increasedThe results for this maneuver are as follows

Final orbital elements 119886 = 719984 km 119890 = 00040Fuel consumption 0721921 kgThe propellant mass difference from the optimalmaneuver 002402 kg

Mathematical Problems in Engineering 11

Time (s)

0 05 1 15 2 25 3 35times104

002

0015

001

0005

0

minus0005

minus001

minus0015Po

siti

on d

evia

tion

(m)

Figure 21 Position deviation in maneuver 6

Vel

ocit

y d

evia

tion

(m

s)

Time (s)

0 05 1 15 2 25 3 35times104

8

6

4

2

0

minus2

minus4

times10minus3

Figure 22 Velocity deviation in maneuver 6

x axisy axisz axis

ThrustAverage

Time (s)

0 05 1 15 2 25 3 35times104

06

04

02

0

minus02

minus04

minus06

minus08

App

lied

thru

st (N

)

Figure 23 Thrust applied in maneuver 6

8 Conclusion

Primarily this paper established an algorithm to solve theproblem of orbital maneuvers using a low-thrust control fora spacecraft that is travelling around the Earth Then allthe maneuvers studied in this paper were successfully solvedwith the final conditions and constraints proposed and withreasonable fuel consumptions

It was possible to verify that maneuvers with morethrusting arcs consume less fuel as expected This occursbecause the algorithm has more variables to optimize so itis possible to reduce the fuel consumption

The second part of this work was to simulate the optimalmaneuvers found by the algorithm proposed here in a morerealistic environment which can consider errors on theactuator that was not considered in the search for the optimalmaneuver

Themaneuvers simulated in the STRSwere coherent withthe optimal maneuvers validating the integration of bothsoftwares

The error considered in the propulsion system opensthe possibility to study how the system would react tothose nonlinearities Although errors were inserted in thepropulsion system the PID control was able to correct andaccomplish the maneuver

It is possible to analyze the shifts caused by the error aswell as the deviations in semimajor axis and eccentricity It isalso possible to see that the propulsion thrust was not linear

For themaneuvers shown in Sections 735 and 736 somenonlinearities of the propulsion system such as a responsetime a rate limiter and a dead zone for the actuator signalwere considered All these errors occur in real systems Withthe help of the STRS it was possible to simulate these errorsand analyze the maneuvers studied with more accuracyThe STRS has shown to be an efficient simulator capableof simulating and studying optimal maneuvers in a morerealistic environment and to analyze many nonlinearities notconsidered in the calculation of the optimal maneuver

Evidently the phall II was more efficient than the phall Ias noticed by comparing maneuver 4 with maneuver 6 andmaneuver 5 with maneuver 7 This was expected since thephall II has a higher specific impulse and a higher averagepropulsion thrust as well Nevertheless the PID control wasable to control all the maneuvers and reaches the final orbitalconstraints

References

[1] D F Lawden ldquoMinimal rocket trajectoriesrdquo ARS Journal vol23 no 6 pp 360ndash382 1953

[2] D F Lawden ldquoFundamentals of space navigationrdquo Journal of theBritish Interplanetary Society vol 13 pp 87ndash101 1954

12 Mathematical Problems in Engineering

[3] V V Beletsky and V A Egorov ldquoInterplanetary flights withconstant output enginesrdquoCosmic Research vol 2 no 3 pp 303ndash330 1964

[4] A A Sukhanov ldquoOptimization of flights with low thrustrdquoCosmic Research vol 37 no 2 pp 182ndash191 1999

[5] A A Sukhanov ldquoOptimization of low-thrust interplanetarytransfersrdquo Cosmic Research vol 38 no 6 pp 584ndash587 2000

[6] A A Sukhanov and A F B A Prado ldquoA modification of themethod of transporting trajectoryrdquoCosmic Research vol 42 no1 pp 107ndash112 2004

[7] A A Sukhanov and A F B A Prado ldquoOptimization of lowthrust transfers in the three body problemrdquo Cosmic Researchvol 46 no 5 pp 440ndash451 2008

[8] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction Irdquo Cosmic Researchvol 45 no 5 pp 417ndash423 2007

[9] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction IIrdquo Cosmic Researchvol 46 no 1 pp 49ndash59 2008

[10] S S Fernandes and F D C Carvalho ldquoA first-order analyticaltheory for optimal low-thrust limited-power transfers betweenarbitrary elliptical coplanar orbitsrdquo Mathematical Problems inEngineering vol 2008 Article ID 525930 30 pages 2008

[11] R A Boucher ldquoElectrical propulsion for control of stationarysatellitesrdquo Journal of Spacecraft and Rockets vol 1 no 2 pp 164ndash169 1964

[12] D N Bowditch ldquoApplication of magnetic-expansion plasmathrusters to satellite station keeping and attitude control mis-sionsrdquo in Proceedings of the 4th Electric Propulsion Confer-ence AIAA-1964-677 pp 64ndash677 The American Institute ofAeronautics and Astronautics Philadelphia Pa USA Augest-September 1964

[13] S R Oleson R M Myers C A Kluever J P Riehl and F MCurran ldquoAdvanced propulsion for geostationary orbit insertionand north-south station keepingrdquo Journal of Spacecraft andRockets vol 34 no 1 pp 22ndash28 1997

[14] T A Ely and K C Howell ldquoEast-west stationkeeping of satelliteorbits with resonant tesseral harmonicsrdquoActa Astronautica vol46 no 1 pp 1ndash15 2000

[15] C Circi ldquoSimple strategy for geostationary stationkeepingmaneuvers using solar sailrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 249ndash253 2005

[16] P Romero J M Gambi and E Patino ldquoStationkeepingmanoeuvres for geostationary satellites using feedback controltechniquesrdquo Aerospace Science and Technology vol 11 no 2-3pp 229ndash237 2007

[17] V Gomes and A F B A Prado ldquoLow-thrust out-of-planeorbital station-keeping maneuvers for satellitesrdquo MathematicalProblems in Engineering vol 2012 Article ID 532708 14 pages2012

[18] B L Pierson and C A Kluever ldquoThree-stage approach to opti-mal low-thrust earth-moon trajectoriesrdquo Journal of GuidanceControl and Dynamics vol 17 no 6 pp 1275ndash1282 1994

[19] Y J Song S Y Park K H Choi and E S Sim ldquoA lunar cargomission design strategy using variable low thrustrdquo Advances inSpace Research vol 43 no 9 pp 1391ndash1406 2009

[20] S A Fazelzadeh and G A Varzandian ldquoMinimum-time earth-moon and moon-earth orbital maneuvers using time-domainfinite element methodrdquo Acta Astronautica vol 66 no 3-4 pp528ndash538 2010

[21] G Mingotti F Topputo and F Bernelli-Zazzera ldquoEfficientinvariant-manifold low-thrust planar trajectories to themoonrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 p 817 2012

[22] J R Brophy and M Noca ldquoElectric propulsion for solar systemexplorationrdquo Journal of Propulsion and Power vol 14 no 5 pp700ndash707 1998

[23] D P S Santos A F B A Prado L Casalino and G ColasurdoldquoOptimal trajectories towards near-earth-objects using solarelectric propulsion (sep) and gravity assistedmaneuverrdquo Journalof Aerospace Engineering Sciences and Applications vol 1 no 2pp 51ndash64 2008

[24] D P S Santos L Casalino G Colasurdo and A F B APrado ldquoOptimal trajectories towards near-earth-objects usingsolar electric propulsion (sep) and gravity assisted maneuverrdquoWSEAS Transactions on Applied andTheoretical Mechanics vol4 no 3 pp 125ndash135 2009

[25] D P S Santos and A F B A Prado ldquoOptimal low-thrusttrajectories to reach the asteroid apophisrdquo WSEAS MechanicalEngineering Series vol 7 pp 241ndash251 2012

[26] M C B BiggsThe Optimization of Spacecraft Orbital Manoeu-vres Part I Linearly Varying Thrust Angles The HatfieldPolytechnic Numerical Optimization Centre 1978

[27] M C B BiggsThe Optimisation of Spacecraft Orbital Manoeu-vres Part II Using Pontryaginrsquos Maximum Principle The Hat-field Polytechnic Numerical Optimization Centre 1979

[28] A F B A Prado Analise Selecao e Implementacao de Procedi-mentos que visemManobras Otimas de Satelites Artificiais [MSthesis] Instituto Nacional de Pesquisas Espaciais Sao Jose dosCampos 1988

[29] T C Oliveira Estrategias otimas para manobras orbitais uti-lizando propulsao contınua [MS thesis] Instituto Nacional dePesquisas Espaciais Sao Jose dos Campos 2012

[30] J L Ferreira G C Possa B SMoraes et al ldquoPermanentmagnethall thruster for satellite orbit maneuvering with low powerrdquoAdvances in Space Research vol 01 pp 166ndash182 2009

[31] E Stuhlinger Ion Propulsion for Space FlightMcGraw-Hill NewYork NY USA 1st edition 1964

[32] R G Jahn Physics of Electric Propulsion McGraw-Hill NewYork NY USA 1st edition 1968

[33] D G Luemberger Linear and Non-Linear ProgrammingSpringer Science amp Business Media New York NY USA 3rdedition 2008

[34] M S Bazaraa and C M ShettyNonlinear Programming-Theoryand Algorithms JohnWilley amp Sons New York NY USA 1979

[35] V V Zhurin H R Kaufman and R S Robinson ldquoPhysics ofclosed drift thrustersrdquo Plasma Sources Science and Technologyvol 8 no 1 pp R1ndashR20 1999

[36] E M Rocco ldquoPerturbed orbital motion with a PID controlsystem for the trajectoryrdquo in Proceedings of the 14th ColoquioBrasileiro De Dinamica Orbital Aguas de Lindoia AnaisUNESP Guaratingueta Brazil November 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Table 1 Comparative performance parameters measured in PHALL I and PHALL II

PHALL1 PHALL2Maximum expected thrust (mN) 126 126Average measured thrust (mN) 849 1200Thrust density (Nm2) 468 lt60Maximum specific impulse (s) 1607 1607Measured specific impulse (s) 1083 1600Ionized mass ratio () 33 sim30Propellant consumption (Kgs) 6 0 times 10

minus61 0 times 10

minus6

Energy consumption (W) 350 250ndash350Electrical efficiency () 339 60Total efficiency () 1012 500

Range angle (degrees)

Pitc

h (d

egre

es)

1101009080

012345

120 130 140

minus1minus2minus3minus4minus5

Figure 2 Optimal pitch angles of the maneuver 1

Vehicle mass = 300 kg thrust = 10 Newton initialposition 119904 = 0∘Condition on the final orbit 119886 = 104000 kmOther constraints 5∘ ge 119860 ge minus5∘ (pitch constraint)

Final orbital elements obtained after the maneuver

119886 = 10399921 km 119890 = 07143 119894 = 10∘Ω = 55∘ 120596 = 105∘119907 = 3024∘

Fuel consumption 24235 kg fuel consumption in Biggs[27] 244 kg fuel consumption in Prado [28] 245 kg

Figure 2 shows the direction of the thrustNow with the constraint in the direction of propulsion

the fuel consumption was increased when compared witha maneuver without this constraint [27 28] The extra fuelneeded due to this constraint is 38 times 10minus3 kg It is possible toview clearly that the pitch angle is confined to 5∘ at the endof the maneuver In any case the algorithm developed hereobtains better results when compared to Biggs [27] and Prado[28]

712 Maneuver 2 This maneuver has the constraint of aregion prohibited for thrusting This is one of the mostimportant capabilities of the algorithm developed here

Initial orbital elements

119886 = 99000 km 119890 = 07 119894 = 10∘ Ω = 55∘ 120596 = 105∘ 119907 =minus105∘

Range angle (degrees)

Pitc

h (d

egre

es)

55504540353025 60 65 70

minus20minus21minus22minus23minus24minus25minus26minus27minus28minus29minus30

Figure 3 Optimal pitch angles of the maneuver 2

Vehicle mass = 300 kg thrust = 10 Newton initialposition 119904 = 0∘Condition on the final orbit 119886 = 104000 kmOther constraints no thrust between the true longi-tudes 120∘ and 180∘

Final orbital elements

119886 = 10399985 km 119890 = 07134 119894 = 10∘Ω = 55∘120596 = 105∘119907 = 32080∘

Fuel consumption 27852 kg fuel consumption in Biggs[27] 281 kg fuel consumption in Prado [28] 281 kg

Figure 3 shows the direction of the thrustIn Figure 3 it is possible to note that the solution has the

thrust arc ending in 120582 = 120∘ The true longitude is definedhere as 120582 = 120596 + 119907 = 42580

∘ that is similar to 6580∘ that isthe result shown in Figure 3 In this way the region prohibitof thrust was obeyed and compared with the maneuver thatis free of this constraint the extra fuel needed to reach thefinal semimajor axis required was 03655 kg Once againthe algorithm proposed here obtained better results whencompared to Biggs [27] and Prado [28]

713 Maneuver 3 This maneuver has the requirements ofchanging three orbital elements of the orbit at the same time

Initial orbital elements

119886 = 419041 km 119890 = 0018 119894= 0688∘ Ω = minus298∘ 120596 =7∘ 119907 = minus972∘

Mathematical Problems in Engineering 7

Range angle (degrees)

Pitc

h (d

egre

es)

7570656055 80 85

200

150

100

50

0

minus50

minus100

minus150

minus200

Figure 4 Optimal pitch angles of the maneuver 3 for the first arc

Yaw

(deg

rees

)

Range angle (degrees)

7570656055 80 85

minus612

minus614

minus616

minus618

minus62

minus622

minus624

minus626

Figure 5 Optimal yaw angles of the maneuver 3 for the first arc

Vehicle mass = 300 kg thrust = 10 Newton initialposition 119904 = 0∘Condition on the final orbit 119886 = 421642 km 119890 = 0119894 = 0∘

Final orbital elements

119886 = 4216455 km 119890 = 000130 119894 = 00419∘Ω = 30060∘120596 = 10047∘ 119907 = 14364∘

Fuel consumption 5565 kg fuel consumption in Biggs[27] 5621 kg fuel consumption in Prado [28] 5579 kg

Figures 4ndash7 show the direction of the thrustIn this maneuver there are three Keplerian elements

fixed on the final orbit semimajor axis eccentricity andinclination Since the maneuver includes a change in theorbital plane the yaw angle is not zero The final conditionswere satisfied and the fuel consumption was reduced by0014 kg when compared to Prado [28]

72 Optimal Maneuvers Using the Phalls I and II In thissection we present the optimal maneuvers found by thealgorithm presented in the present paper when using PhallsI and II The optimal thrust angles are presented and thespecific impulse considered for each maneuver is given inTable 1 Figures 8ndash23 present the optimal angles for theproposedmaneuvers given by the initial and final range angleand the pitch angle at each arc of propulsionThe yaw angle isalways zero because all the maneuvers simulated are planar

Range angle (degrees)

Pitc

h (d

egre

es)

215205 210200190 195185 220 225

30

25

20

15

10

5

0

minus5

Figure 6 Optimal pitch angles of themaneuver 3 for the second arc

Range angle (degrees)

Yaw

(deg

rees

)

215205 210200190 195185

60

55

50

45

40

35

30220 225

Figure 7 Optimal yaw angles of the maneuver 3 for the second arc

All the maneuvers considered here have the same initialorbital elements and final condition imposed for the finalorbit Only the number of arcs and the propulsion system isdifferent at each maneuver The parameters of the maneuverproposed are as follows

Initial orbital elements 119886 = 7130865 km 119890 = 00035 119894= 985054∘ Ω = 0∘ 120596 = 0 119907 = 220∘Vehicle mass = 300 kgCondition imposed on the final orbit 119886 = 7200 km 119890= 0004

721 Maneuver 4 For this maneuver it was considered theexistence of three Phall I thrusters The propulsion force ofeach one is assumed to be the average measured value Theparameters considered in this maneuver are as follows

Total thrust = 0252 Newton specific impulse 1083 sSolution

Final orbital elements 119886 = 720000 km 119890 = 0004Fuel consumption 1046346 kgDuration of the maneuver 4406696 s

722 Maneuver 5 Thismaneuver has the same conditions ofmaneuver 4 but now two thrust arcs are used instead of oneThe parameters considered in this maneuver are as follows

8 Mathematical Problems in Engineering

Range angle (degrees)

30002500200015001000500

Pitc

h an

gle

(deg

rees

)

0

0

5

10

15

minus10

minus15

minus5

Figure 8 Optimal pitch angles for maneuver 4

Range angle (degrees)

Pitc

h an

gle

(deg

rees

) 03

010

02

0405

1400120010008006004002000

minus01minus02

minus03minus04minus05

Figure 9 Optimal pitch angles for the first arc of the maneuver 5

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

1

0

05

15

2

40003800360034003200300028002600

minus05

minus1

minus15

minus2

Figure 10 Optimal pitch angles for the second arc of maneuver 5

Total thrust = 0252 Newton specific impulse 1083 s

Solution

Final orbital elements after the first arc 119886 =716530 km 119890 = 00039Final orbital elements after the second arc 119886 =720000 km 119890 = 00040Fuel consumption 1013402 kgDuration of the maneuver 4267954 s

Note that the use of two arcs generates savings in the fuelconsumption as expected

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

0

5

10

15

22001800 2000160014001000 1200800600400200

minus5

minus10

minus15

Figure 11 Optimal pitch angle for maneuver 6

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

02

0

0604

081

1000900600 700 8005004003002001000

minus02

minus06

minus1

minus04

minus08

Figure 12 Optimal pitch angles for the first arc of maneuver 7

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

0

05

1

15

240023002100 2200200019001800170016001500

minus05

minus1

minus15

Figure 13 Optimal pitch angles for the second arc of maneuver 7

723 Maneuver 6 For this maneuver it was considered theexistence of three Phall II thrusters The propulsion force ofeach one is assumed to be the average measured value Theparameters considered in this maneuver are as follows

Total thrust = 0360 Newton specific impulse 1600 s

Solution

Final orbital elements 119886 = 720000 km 119890 = 0004Fuel consumption 0697930 kgDuration of the maneuver 3039873 s

Mathematical Problems in Engineering 9

Time (s)

minus5

minus10

0

0

5

05 1 15 2 25 3 35 4 45times104

Sem

imaj

or a

xis

dev

iati

on(m

)

Figure 14 Semimajor axis deviation in maneuver 4

02468

1012

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus2minus4minus6minus8

Ecc

entr

icit

y d

evia

tion

times10minus7

Figure 15 Eccentricity deviation in maneuver 4

0

0005

001

0015

Posi

tion

dev

iati

on (m

)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus0005

minus001

Figure 16 Position deviation in maneuver 4

Note that the use of the Phall II generates savings inthe fuel consumption since it has better parameters whencompared to the Phall I device

724 Maneuver 7 Thismaneuver has the same conditions ofmaneuver 6 but now with two thrust arcs instead of oneTheparameters considered in this maneuver are as follows

Total thrust = 0360 Newton specific impulse 1600 s

Solution

Final orbital elements after the first arc 119886 =716777 km 119890 = 00039

0

1

2

3

4

5

Vel

ocit

y d

evia

tion

(m

s)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus1

minus2

minus3

times10minus3

Figure 17 Velocity deviation in maneuver 4

Final orbital elements after the second arc 119886 =720000 km 119890 = 00040Fuel consumption 0686314 kgDuration of the maneuver 2989285 s

Note that the use of two arcs generates savings in the fuelconsumption one more time as expected

73 Optimal Maneuvers Simulations Using the STRS Nowwith the help of the optimal maneuvers found previously weshall simulate thosemaneuvers in the STRS using the optimalangles Some different aspects of the propulsion system wereconsidered to simulate amore realistic environment and alsothe PID control was considered

In this section for the propulsion system it was consid-ered that at each inertial direction 119909 119910 119911 has an error on theactuator signal (ms) with mean equal to zero and varianceequal to 10minus10 that is it was inserted an error in the velocity ofthe satelliteThis error has amplitude approximately close to 5times 10minus3N for each thrust which corresponds to the error of thePMHT (Ferreira et al [30]) There is a PID control to correctthe errors due the actuator but it is still possible to analyzethe deviations and the increase of the fuel consumption dueto the errors

731 Maneuver 4 in STRS In this case it is consideredmaneuver 4 including the solution found there

Final orbital elements 119886 = 719985 km 119890 = 0004Fuel consumption 10470 kgPropellant mass difference from the optimal maneu-ver 666 times 10minus4 kg

The error inserted in the propulsion system resulted in adeviation of the orbital elements as we can see in Figures 14and 15 for the semimajor axis and eccentricity If there wasnot any error at the propulsion system the deviation wouldbe zero all the time for the orbital elements In Figures 16 and17 we can see also the effect of the error in the velocity andposition as a function of these parameters Finally the thrusterror is shown in Figure 18

10 Mathematical Problems in EngineeringA

pplie

d th

rust

(N)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

0504

03

02

01

0

minus01

minus02

minus03

minus04

x axisy axisz axis

ThrustAverage

Figure 18 Thrust applied in maneuver 4

0

5

10

Time (s)

0 05 1 15 2 25 3 35times104

minus5

minus10

minus15

Sem

imaj

or a

xis

dev

iati

on(m

)

Figure 19 Semimajor axis deviation in maneuver 6

732 Maneuver 5 in STRS In this case maneuver 5 isconsidered including the solution found

Final orbital elements after the first arc 119886 =716523 km 119890 = 00039

Final orbital elements after the second arc 119886 =719980 km 119890 = 00040

Fuel consumption 1013362 kg

Propellant mass difference from the optimal maneu-ver 6384 times 10minus4 kg

733 Maneuver 6 in STRS Now maneuver 6 is considered

Final orbital elements 119886 = 719982 km 119890 = 0004

Fuel consumption 06979 kg

Propellant mass difference from the optimal maneu-ver 219 times 10minus4 kg

The same analysis for the figures made in Section 731 fitshere Figures 19ndash23 show the deviations and the error thatoccurred on the propulsion system

Time (s)

0 05 1 15 2 25 3 35times104

Ecc

entr

icit

y d

evia

tion

times10minus6

2

15

1

05

0

minus05

minus1

minus15

Figure 20 Eccentricity deviation in maneuver 6

734 Maneuver 7 in STRS Maneuver 7 is now studiedFinal orbital elements after the first arc 119886 =716770 km 119890 = 00039Final orbital elements after the second arc 119886 =719983 km 119890 = 00040Fuel consumption 0686279 kgPropellant mass difference from the optimal maneu-ver 21469 times 10minus4 kg

735 Maneuver 8 in STRS This maneuver uses the optimalangles given in maneuver 4 but includes the errors proposedfor the propulsion system In addition it was consideredother nonlinearities for the actuator two seconds delay torespond the control signal a rate limiter for the variationof the satelite velocity which the absolute limit of the rateis 000025ms dead zone where the propulsion system isnot turned on if the required velocity increment is lowerthan 00002ms Considering these nonlinearities for thepropulsion system the PID control was able to achievethe maneuver although the fuel consumption was slightlyincreased The results for this simulation are as follows

Final orbital elements 119886 = 719986 km 119890 = 0004Fuel consumption 11276322 kgPropellant mass difference from the optimal maneu-ver 0081429 kg

736 Maneuver 9 in STRS The maneuver simulated hereuses the optimal angles given in maneuver 6 with the errorsproposed for the propulsion system In addition it wasproposed the samenonlinearities (rate limiter dead zone andtime delay) of maneuver 8 Considering these nonlinearitiesfor the propulsion system the PID control was able to achievethe maneuver although the fuel consumption was increasedThe results for this maneuver are as follows

Final orbital elements 119886 = 719984 km 119890 = 00040Fuel consumption 0721921 kgThe propellant mass difference from the optimalmaneuver 002402 kg

Mathematical Problems in Engineering 11

Time (s)

0 05 1 15 2 25 3 35times104

002

0015

001

0005

0

minus0005

minus001

minus0015Po

siti

on d

evia

tion

(m)

Figure 21 Position deviation in maneuver 6

Vel

ocit

y d

evia

tion

(m

s)

Time (s)

0 05 1 15 2 25 3 35times104

8

6

4

2

0

minus2

minus4

times10minus3

Figure 22 Velocity deviation in maneuver 6

x axisy axisz axis

ThrustAverage

Time (s)

0 05 1 15 2 25 3 35times104

06

04

02

0

minus02

minus04

minus06

minus08

App

lied

thru

st (N

)

Figure 23 Thrust applied in maneuver 6

8 Conclusion

Primarily this paper established an algorithm to solve theproblem of orbital maneuvers using a low-thrust control fora spacecraft that is travelling around the Earth Then allthe maneuvers studied in this paper were successfully solvedwith the final conditions and constraints proposed and withreasonable fuel consumptions

It was possible to verify that maneuvers with morethrusting arcs consume less fuel as expected This occursbecause the algorithm has more variables to optimize so itis possible to reduce the fuel consumption

The second part of this work was to simulate the optimalmaneuvers found by the algorithm proposed here in a morerealistic environment which can consider errors on theactuator that was not considered in the search for the optimalmaneuver

Themaneuvers simulated in the STRSwere coherent withthe optimal maneuvers validating the integration of bothsoftwares

The error considered in the propulsion system opensthe possibility to study how the system would react tothose nonlinearities Although errors were inserted in thepropulsion system the PID control was able to correct andaccomplish the maneuver

It is possible to analyze the shifts caused by the error aswell as the deviations in semimajor axis and eccentricity It isalso possible to see that the propulsion thrust was not linear

For themaneuvers shown in Sections 735 and 736 somenonlinearities of the propulsion system such as a responsetime a rate limiter and a dead zone for the actuator signalwere considered All these errors occur in real systems Withthe help of the STRS it was possible to simulate these errorsand analyze the maneuvers studied with more accuracyThe STRS has shown to be an efficient simulator capableof simulating and studying optimal maneuvers in a morerealistic environment and to analyze many nonlinearities notconsidered in the calculation of the optimal maneuver

Evidently the phall II was more efficient than the phall Ias noticed by comparing maneuver 4 with maneuver 6 andmaneuver 5 with maneuver 7 This was expected since thephall II has a higher specific impulse and a higher averagepropulsion thrust as well Nevertheless the PID control wasable to control all the maneuvers and reaches the final orbitalconstraints

References

[1] D F Lawden ldquoMinimal rocket trajectoriesrdquo ARS Journal vol23 no 6 pp 360ndash382 1953

[2] D F Lawden ldquoFundamentals of space navigationrdquo Journal of theBritish Interplanetary Society vol 13 pp 87ndash101 1954

12 Mathematical Problems in Engineering

[3] V V Beletsky and V A Egorov ldquoInterplanetary flights withconstant output enginesrdquoCosmic Research vol 2 no 3 pp 303ndash330 1964

[4] A A Sukhanov ldquoOptimization of flights with low thrustrdquoCosmic Research vol 37 no 2 pp 182ndash191 1999

[5] A A Sukhanov ldquoOptimization of low-thrust interplanetarytransfersrdquo Cosmic Research vol 38 no 6 pp 584ndash587 2000

[6] A A Sukhanov and A F B A Prado ldquoA modification of themethod of transporting trajectoryrdquoCosmic Research vol 42 no1 pp 107ndash112 2004

[7] A A Sukhanov and A F B A Prado ldquoOptimization of lowthrust transfers in the three body problemrdquo Cosmic Researchvol 46 no 5 pp 440ndash451 2008

[8] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction Irdquo Cosmic Researchvol 45 no 5 pp 417ndash423 2007

[9] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction IIrdquo Cosmic Researchvol 46 no 1 pp 49ndash59 2008

[10] S S Fernandes and F D C Carvalho ldquoA first-order analyticaltheory for optimal low-thrust limited-power transfers betweenarbitrary elliptical coplanar orbitsrdquo Mathematical Problems inEngineering vol 2008 Article ID 525930 30 pages 2008

[11] R A Boucher ldquoElectrical propulsion for control of stationarysatellitesrdquo Journal of Spacecraft and Rockets vol 1 no 2 pp 164ndash169 1964

[12] D N Bowditch ldquoApplication of magnetic-expansion plasmathrusters to satellite station keeping and attitude control mis-sionsrdquo in Proceedings of the 4th Electric Propulsion Confer-ence AIAA-1964-677 pp 64ndash677 The American Institute ofAeronautics and Astronautics Philadelphia Pa USA Augest-September 1964

[13] S R Oleson R M Myers C A Kluever J P Riehl and F MCurran ldquoAdvanced propulsion for geostationary orbit insertionand north-south station keepingrdquo Journal of Spacecraft andRockets vol 34 no 1 pp 22ndash28 1997

[14] T A Ely and K C Howell ldquoEast-west stationkeeping of satelliteorbits with resonant tesseral harmonicsrdquoActa Astronautica vol46 no 1 pp 1ndash15 2000

[15] C Circi ldquoSimple strategy for geostationary stationkeepingmaneuvers using solar sailrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 249ndash253 2005

[16] P Romero J M Gambi and E Patino ldquoStationkeepingmanoeuvres for geostationary satellites using feedback controltechniquesrdquo Aerospace Science and Technology vol 11 no 2-3pp 229ndash237 2007

[17] V Gomes and A F B A Prado ldquoLow-thrust out-of-planeorbital station-keeping maneuvers for satellitesrdquo MathematicalProblems in Engineering vol 2012 Article ID 532708 14 pages2012

[18] B L Pierson and C A Kluever ldquoThree-stage approach to opti-mal low-thrust earth-moon trajectoriesrdquo Journal of GuidanceControl and Dynamics vol 17 no 6 pp 1275ndash1282 1994

[19] Y J Song S Y Park K H Choi and E S Sim ldquoA lunar cargomission design strategy using variable low thrustrdquo Advances inSpace Research vol 43 no 9 pp 1391ndash1406 2009

[20] S A Fazelzadeh and G A Varzandian ldquoMinimum-time earth-moon and moon-earth orbital maneuvers using time-domainfinite element methodrdquo Acta Astronautica vol 66 no 3-4 pp528ndash538 2010

[21] G Mingotti F Topputo and F Bernelli-Zazzera ldquoEfficientinvariant-manifold low-thrust planar trajectories to themoonrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 p 817 2012

[22] J R Brophy and M Noca ldquoElectric propulsion for solar systemexplorationrdquo Journal of Propulsion and Power vol 14 no 5 pp700ndash707 1998

[23] D P S Santos A F B A Prado L Casalino and G ColasurdoldquoOptimal trajectories towards near-earth-objects using solarelectric propulsion (sep) and gravity assistedmaneuverrdquo Journalof Aerospace Engineering Sciences and Applications vol 1 no 2pp 51ndash64 2008

[24] D P S Santos L Casalino G Colasurdo and A F B APrado ldquoOptimal trajectories towards near-earth-objects usingsolar electric propulsion (sep) and gravity assisted maneuverrdquoWSEAS Transactions on Applied andTheoretical Mechanics vol4 no 3 pp 125ndash135 2009

[25] D P S Santos and A F B A Prado ldquoOptimal low-thrusttrajectories to reach the asteroid apophisrdquo WSEAS MechanicalEngineering Series vol 7 pp 241ndash251 2012

[26] M C B BiggsThe Optimization of Spacecraft Orbital Manoeu-vres Part I Linearly Varying Thrust Angles The HatfieldPolytechnic Numerical Optimization Centre 1978

[27] M C B BiggsThe Optimisation of Spacecraft Orbital Manoeu-vres Part II Using Pontryaginrsquos Maximum Principle The Hat-field Polytechnic Numerical Optimization Centre 1979

[28] A F B A Prado Analise Selecao e Implementacao de Procedi-mentos que visemManobras Otimas de Satelites Artificiais [MSthesis] Instituto Nacional de Pesquisas Espaciais Sao Jose dosCampos 1988

[29] T C Oliveira Estrategias otimas para manobras orbitais uti-lizando propulsao contınua [MS thesis] Instituto Nacional dePesquisas Espaciais Sao Jose dos Campos 2012

[30] J L Ferreira G C Possa B SMoraes et al ldquoPermanentmagnethall thruster for satellite orbit maneuvering with low powerrdquoAdvances in Space Research vol 01 pp 166ndash182 2009

[31] E Stuhlinger Ion Propulsion for Space FlightMcGraw-Hill NewYork NY USA 1st edition 1964

[32] R G Jahn Physics of Electric Propulsion McGraw-Hill NewYork NY USA 1st edition 1968

[33] D G Luemberger Linear and Non-Linear ProgrammingSpringer Science amp Business Media New York NY USA 3rdedition 2008

[34] M S Bazaraa and C M ShettyNonlinear Programming-Theoryand Algorithms JohnWilley amp Sons New York NY USA 1979

[35] V V Zhurin H R Kaufman and R S Robinson ldquoPhysics ofclosed drift thrustersrdquo Plasma Sources Science and Technologyvol 8 no 1 pp R1ndashR20 1999

[36] E M Rocco ldquoPerturbed orbital motion with a PID controlsystem for the trajectoryrdquo in Proceedings of the 14th ColoquioBrasileiro De Dinamica Orbital Aguas de Lindoia AnaisUNESP Guaratingueta Brazil November 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Range angle (degrees)

Pitc

h (d

egre

es)

7570656055 80 85

200

150

100

50

0

minus50

minus100

minus150

minus200

Figure 4 Optimal pitch angles of the maneuver 3 for the first arc

Yaw

(deg

rees

)

Range angle (degrees)

7570656055 80 85

minus612

minus614

minus616

minus618

minus62

minus622

minus624

minus626

Figure 5 Optimal yaw angles of the maneuver 3 for the first arc

Vehicle mass = 300 kg thrust = 10 Newton initialposition 119904 = 0∘Condition on the final orbit 119886 = 421642 km 119890 = 0119894 = 0∘

Final orbital elements

119886 = 4216455 km 119890 = 000130 119894 = 00419∘Ω = 30060∘120596 = 10047∘ 119907 = 14364∘

Fuel consumption 5565 kg fuel consumption in Biggs[27] 5621 kg fuel consumption in Prado [28] 5579 kg

Figures 4ndash7 show the direction of the thrustIn this maneuver there are three Keplerian elements

fixed on the final orbit semimajor axis eccentricity andinclination Since the maneuver includes a change in theorbital plane the yaw angle is not zero The final conditionswere satisfied and the fuel consumption was reduced by0014 kg when compared to Prado [28]

72 Optimal Maneuvers Using the Phalls I and II In thissection we present the optimal maneuvers found by thealgorithm presented in the present paper when using PhallsI and II The optimal thrust angles are presented and thespecific impulse considered for each maneuver is given inTable 1 Figures 8ndash23 present the optimal angles for theproposedmaneuvers given by the initial and final range angleand the pitch angle at each arc of propulsionThe yaw angle isalways zero because all the maneuvers simulated are planar

Range angle (degrees)

Pitc

h (d

egre

es)

215205 210200190 195185 220 225

30

25

20

15

10

5

0

minus5

Figure 6 Optimal pitch angles of themaneuver 3 for the second arc

Range angle (degrees)

Yaw

(deg

rees

)

215205 210200190 195185

60

55

50

45

40

35

30220 225

Figure 7 Optimal yaw angles of the maneuver 3 for the second arc

All the maneuvers considered here have the same initialorbital elements and final condition imposed for the finalorbit Only the number of arcs and the propulsion system isdifferent at each maneuver The parameters of the maneuverproposed are as follows

Initial orbital elements 119886 = 7130865 km 119890 = 00035 119894= 985054∘ Ω = 0∘ 120596 = 0 119907 = 220∘Vehicle mass = 300 kgCondition imposed on the final orbit 119886 = 7200 km 119890= 0004

721 Maneuver 4 For this maneuver it was considered theexistence of three Phall I thrusters The propulsion force ofeach one is assumed to be the average measured value Theparameters considered in this maneuver are as follows

Total thrust = 0252 Newton specific impulse 1083 sSolution

Final orbital elements 119886 = 720000 km 119890 = 0004Fuel consumption 1046346 kgDuration of the maneuver 4406696 s

722 Maneuver 5 Thismaneuver has the same conditions ofmaneuver 4 but now two thrust arcs are used instead of oneThe parameters considered in this maneuver are as follows

8 Mathematical Problems in Engineering

Range angle (degrees)

30002500200015001000500

Pitc

h an

gle

(deg

rees

)

0

0

5

10

15

minus10

minus15

minus5

Figure 8 Optimal pitch angles for maneuver 4

Range angle (degrees)

Pitc

h an

gle

(deg

rees

) 03

010

02

0405

1400120010008006004002000

minus01minus02

minus03minus04minus05

Figure 9 Optimal pitch angles for the first arc of the maneuver 5

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

1

0

05

15

2

40003800360034003200300028002600

minus05

minus1

minus15

minus2

Figure 10 Optimal pitch angles for the second arc of maneuver 5

Total thrust = 0252 Newton specific impulse 1083 s

Solution

Final orbital elements after the first arc 119886 =716530 km 119890 = 00039Final orbital elements after the second arc 119886 =720000 km 119890 = 00040Fuel consumption 1013402 kgDuration of the maneuver 4267954 s

Note that the use of two arcs generates savings in the fuelconsumption as expected

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

0

5

10

15

22001800 2000160014001000 1200800600400200

minus5

minus10

minus15

Figure 11 Optimal pitch angle for maneuver 6

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

02

0

0604

081

1000900600 700 8005004003002001000

minus02

minus06

minus1

minus04

minus08

Figure 12 Optimal pitch angles for the first arc of maneuver 7

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

0

05

1

15

240023002100 2200200019001800170016001500

minus05

minus1

minus15

Figure 13 Optimal pitch angles for the second arc of maneuver 7

723 Maneuver 6 For this maneuver it was considered theexistence of three Phall II thrusters The propulsion force ofeach one is assumed to be the average measured value Theparameters considered in this maneuver are as follows

Total thrust = 0360 Newton specific impulse 1600 s

Solution

Final orbital elements 119886 = 720000 km 119890 = 0004Fuel consumption 0697930 kgDuration of the maneuver 3039873 s

Mathematical Problems in Engineering 9

Time (s)

minus5

minus10

0

0

5

05 1 15 2 25 3 35 4 45times104

Sem

imaj

or a

xis

dev

iati

on(m

)

Figure 14 Semimajor axis deviation in maneuver 4

02468

1012

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus2minus4minus6minus8

Ecc

entr

icit

y d

evia

tion

times10minus7

Figure 15 Eccentricity deviation in maneuver 4

0

0005

001

0015

Posi

tion

dev

iati

on (m

)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus0005

minus001

Figure 16 Position deviation in maneuver 4

Note that the use of the Phall II generates savings inthe fuel consumption since it has better parameters whencompared to the Phall I device

724 Maneuver 7 Thismaneuver has the same conditions ofmaneuver 6 but now with two thrust arcs instead of oneTheparameters considered in this maneuver are as follows

Total thrust = 0360 Newton specific impulse 1600 s

Solution

Final orbital elements after the first arc 119886 =716777 km 119890 = 00039

0

1

2

3

4

5

Vel

ocit

y d

evia

tion

(m

s)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus1

minus2

minus3

times10minus3

Figure 17 Velocity deviation in maneuver 4

Final orbital elements after the second arc 119886 =720000 km 119890 = 00040Fuel consumption 0686314 kgDuration of the maneuver 2989285 s

Note that the use of two arcs generates savings in the fuelconsumption one more time as expected

73 Optimal Maneuvers Simulations Using the STRS Nowwith the help of the optimal maneuvers found previously weshall simulate thosemaneuvers in the STRS using the optimalangles Some different aspects of the propulsion system wereconsidered to simulate amore realistic environment and alsothe PID control was considered

In this section for the propulsion system it was consid-ered that at each inertial direction 119909 119910 119911 has an error on theactuator signal (ms) with mean equal to zero and varianceequal to 10minus10 that is it was inserted an error in the velocity ofthe satelliteThis error has amplitude approximately close to 5times 10minus3N for each thrust which corresponds to the error of thePMHT (Ferreira et al [30]) There is a PID control to correctthe errors due the actuator but it is still possible to analyzethe deviations and the increase of the fuel consumption dueto the errors

731 Maneuver 4 in STRS In this case it is consideredmaneuver 4 including the solution found there

Final orbital elements 119886 = 719985 km 119890 = 0004Fuel consumption 10470 kgPropellant mass difference from the optimal maneu-ver 666 times 10minus4 kg

The error inserted in the propulsion system resulted in adeviation of the orbital elements as we can see in Figures 14and 15 for the semimajor axis and eccentricity If there wasnot any error at the propulsion system the deviation wouldbe zero all the time for the orbital elements In Figures 16 and17 we can see also the effect of the error in the velocity andposition as a function of these parameters Finally the thrusterror is shown in Figure 18

10 Mathematical Problems in EngineeringA

pplie

d th

rust

(N)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

0504

03

02

01

0

minus01

minus02

minus03

minus04

x axisy axisz axis

ThrustAverage

Figure 18 Thrust applied in maneuver 4

0

5

10

Time (s)

0 05 1 15 2 25 3 35times104

minus5

minus10

minus15

Sem

imaj

or a

xis

dev

iati

on(m

)

Figure 19 Semimajor axis deviation in maneuver 6

732 Maneuver 5 in STRS In this case maneuver 5 isconsidered including the solution found

Final orbital elements after the first arc 119886 =716523 km 119890 = 00039

Final orbital elements after the second arc 119886 =719980 km 119890 = 00040

Fuel consumption 1013362 kg

Propellant mass difference from the optimal maneu-ver 6384 times 10minus4 kg

733 Maneuver 6 in STRS Now maneuver 6 is considered

Final orbital elements 119886 = 719982 km 119890 = 0004

Fuel consumption 06979 kg

Propellant mass difference from the optimal maneu-ver 219 times 10minus4 kg

The same analysis for the figures made in Section 731 fitshere Figures 19ndash23 show the deviations and the error thatoccurred on the propulsion system

Time (s)

0 05 1 15 2 25 3 35times104

Ecc

entr

icit

y d

evia

tion

times10minus6

2

15

1

05

0

minus05

minus1

minus15

Figure 20 Eccentricity deviation in maneuver 6

734 Maneuver 7 in STRS Maneuver 7 is now studiedFinal orbital elements after the first arc 119886 =716770 km 119890 = 00039Final orbital elements after the second arc 119886 =719983 km 119890 = 00040Fuel consumption 0686279 kgPropellant mass difference from the optimal maneu-ver 21469 times 10minus4 kg

735 Maneuver 8 in STRS This maneuver uses the optimalangles given in maneuver 4 but includes the errors proposedfor the propulsion system In addition it was consideredother nonlinearities for the actuator two seconds delay torespond the control signal a rate limiter for the variationof the satelite velocity which the absolute limit of the rateis 000025ms dead zone where the propulsion system isnot turned on if the required velocity increment is lowerthan 00002ms Considering these nonlinearities for thepropulsion system the PID control was able to achievethe maneuver although the fuel consumption was slightlyincreased The results for this simulation are as follows

Final orbital elements 119886 = 719986 km 119890 = 0004Fuel consumption 11276322 kgPropellant mass difference from the optimal maneu-ver 0081429 kg

736 Maneuver 9 in STRS The maneuver simulated hereuses the optimal angles given in maneuver 6 with the errorsproposed for the propulsion system In addition it wasproposed the samenonlinearities (rate limiter dead zone andtime delay) of maneuver 8 Considering these nonlinearitiesfor the propulsion system the PID control was able to achievethe maneuver although the fuel consumption was increasedThe results for this maneuver are as follows

Final orbital elements 119886 = 719984 km 119890 = 00040Fuel consumption 0721921 kgThe propellant mass difference from the optimalmaneuver 002402 kg

Mathematical Problems in Engineering 11

Time (s)

0 05 1 15 2 25 3 35times104

002

0015

001

0005

0

minus0005

minus001

minus0015Po

siti

on d

evia

tion

(m)

Figure 21 Position deviation in maneuver 6

Vel

ocit

y d

evia

tion

(m

s)

Time (s)

0 05 1 15 2 25 3 35times104

8

6

4

2

0

minus2

minus4

times10minus3

Figure 22 Velocity deviation in maneuver 6

x axisy axisz axis

ThrustAverage

Time (s)

0 05 1 15 2 25 3 35times104

06

04

02

0

minus02

minus04

minus06

minus08

App

lied

thru

st (N

)

Figure 23 Thrust applied in maneuver 6

8 Conclusion

Primarily this paper established an algorithm to solve theproblem of orbital maneuvers using a low-thrust control fora spacecraft that is travelling around the Earth Then allthe maneuvers studied in this paper were successfully solvedwith the final conditions and constraints proposed and withreasonable fuel consumptions

It was possible to verify that maneuvers with morethrusting arcs consume less fuel as expected This occursbecause the algorithm has more variables to optimize so itis possible to reduce the fuel consumption

The second part of this work was to simulate the optimalmaneuvers found by the algorithm proposed here in a morerealistic environment which can consider errors on theactuator that was not considered in the search for the optimalmaneuver

Themaneuvers simulated in the STRSwere coherent withthe optimal maneuvers validating the integration of bothsoftwares

The error considered in the propulsion system opensthe possibility to study how the system would react tothose nonlinearities Although errors were inserted in thepropulsion system the PID control was able to correct andaccomplish the maneuver

It is possible to analyze the shifts caused by the error aswell as the deviations in semimajor axis and eccentricity It isalso possible to see that the propulsion thrust was not linear

For themaneuvers shown in Sections 735 and 736 somenonlinearities of the propulsion system such as a responsetime a rate limiter and a dead zone for the actuator signalwere considered All these errors occur in real systems Withthe help of the STRS it was possible to simulate these errorsand analyze the maneuvers studied with more accuracyThe STRS has shown to be an efficient simulator capableof simulating and studying optimal maneuvers in a morerealistic environment and to analyze many nonlinearities notconsidered in the calculation of the optimal maneuver

Evidently the phall II was more efficient than the phall Ias noticed by comparing maneuver 4 with maneuver 6 andmaneuver 5 with maneuver 7 This was expected since thephall II has a higher specific impulse and a higher averagepropulsion thrust as well Nevertheless the PID control wasable to control all the maneuvers and reaches the final orbitalconstraints

References

[1] D F Lawden ldquoMinimal rocket trajectoriesrdquo ARS Journal vol23 no 6 pp 360ndash382 1953

[2] D F Lawden ldquoFundamentals of space navigationrdquo Journal of theBritish Interplanetary Society vol 13 pp 87ndash101 1954

12 Mathematical Problems in Engineering

[3] V V Beletsky and V A Egorov ldquoInterplanetary flights withconstant output enginesrdquoCosmic Research vol 2 no 3 pp 303ndash330 1964

[4] A A Sukhanov ldquoOptimization of flights with low thrustrdquoCosmic Research vol 37 no 2 pp 182ndash191 1999

[5] A A Sukhanov ldquoOptimization of low-thrust interplanetarytransfersrdquo Cosmic Research vol 38 no 6 pp 584ndash587 2000

[6] A A Sukhanov and A F B A Prado ldquoA modification of themethod of transporting trajectoryrdquoCosmic Research vol 42 no1 pp 107ndash112 2004

[7] A A Sukhanov and A F B A Prado ldquoOptimization of lowthrust transfers in the three body problemrdquo Cosmic Researchvol 46 no 5 pp 440ndash451 2008

[8] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction Irdquo Cosmic Researchvol 45 no 5 pp 417ndash423 2007

[9] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction IIrdquo Cosmic Researchvol 46 no 1 pp 49ndash59 2008

[10] S S Fernandes and F D C Carvalho ldquoA first-order analyticaltheory for optimal low-thrust limited-power transfers betweenarbitrary elliptical coplanar orbitsrdquo Mathematical Problems inEngineering vol 2008 Article ID 525930 30 pages 2008

[11] R A Boucher ldquoElectrical propulsion for control of stationarysatellitesrdquo Journal of Spacecraft and Rockets vol 1 no 2 pp 164ndash169 1964

[12] D N Bowditch ldquoApplication of magnetic-expansion plasmathrusters to satellite station keeping and attitude control mis-sionsrdquo in Proceedings of the 4th Electric Propulsion Confer-ence AIAA-1964-677 pp 64ndash677 The American Institute ofAeronautics and Astronautics Philadelphia Pa USA Augest-September 1964

[13] S R Oleson R M Myers C A Kluever J P Riehl and F MCurran ldquoAdvanced propulsion for geostationary orbit insertionand north-south station keepingrdquo Journal of Spacecraft andRockets vol 34 no 1 pp 22ndash28 1997

[14] T A Ely and K C Howell ldquoEast-west stationkeeping of satelliteorbits with resonant tesseral harmonicsrdquoActa Astronautica vol46 no 1 pp 1ndash15 2000

[15] C Circi ldquoSimple strategy for geostationary stationkeepingmaneuvers using solar sailrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 249ndash253 2005

[16] P Romero J M Gambi and E Patino ldquoStationkeepingmanoeuvres for geostationary satellites using feedback controltechniquesrdquo Aerospace Science and Technology vol 11 no 2-3pp 229ndash237 2007

[17] V Gomes and A F B A Prado ldquoLow-thrust out-of-planeorbital station-keeping maneuvers for satellitesrdquo MathematicalProblems in Engineering vol 2012 Article ID 532708 14 pages2012

[18] B L Pierson and C A Kluever ldquoThree-stage approach to opti-mal low-thrust earth-moon trajectoriesrdquo Journal of GuidanceControl and Dynamics vol 17 no 6 pp 1275ndash1282 1994

[19] Y J Song S Y Park K H Choi and E S Sim ldquoA lunar cargomission design strategy using variable low thrustrdquo Advances inSpace Research vol 43 no 9 pp 1391ndash1406 2009

[20] S A Fazelzadeh and G A Varzandian ldquoMinimum-time earth-moon and moon-earth orbital maneuvers using time-domainfinite element methodrdquo Acta Astronautica vol 66 no 3-4 pp528ndash538 2010

[21] G Mingotti F Topputo and F Bernelli-Zazzera ldquoEfficientinvariant-manifold low-thrust planar trajectories to themoonrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 p 817 2012

[22] J R Brophy and M Noca ldquoElectric propulsion for solar systemexplorationrdquo Journal of Propulsion and Power vol 14 no 5 pp700ndash707 1998

[23] D P S Santos A F B A Prado L Casalino and G ColasurdoldquoOptimal trajectories towards near-earth-objects using solarelectric propulsion (sep) and gravity assistedmaneuverrdquo Journalof Aerospace Engineering Sciences and Applications vol 1 no 2pp 51ndash64 2008

[24] D P S Santos L Casalino G Colasurdo and A F B APrado ldquoOptimal trajectories towards near-earth-objects usingsolar electric propulsion (sep) and gravity assisted maneuverrdquoWSEAS Transactions on Applied andTheoretical Mechanics vol4 no 3 pp 125ndash135 2009

[25] D P S Santos and A F B A Prado ldquoOptimal low-thrusttrajectories to reach the asteroid apophisrdquo WSEAS MechanicalEngineering Series vol 7 pp 241ndash251 2012

[26] M C B BiggsThe Optimization of Spacecraft Orbital Manoeu-vres Part I Linearly Varying Thrust Angles The HatfieldPolytechnic Numerical Optimization Centre 1978

[27] M C B BiggsThe Optimisation of Spacecraft Orbital Manoeu-vres Part II Using Pontryaginrsquos Maximum Principle The Hat-field Polytechnic Numerical Optimization Centre 1979

[28] A F B A Prado Analise Selecao e Implementacao de Procedi-mentos que visemManobras Otimas de Satelites Artificiais [MSthesis] Instituto Nacional de Pesquisas Espaciais Sao Jose dosCampos 1988

[29] T C Oliveira Estrategias otimas para manobras orbitais uti-lizando propulsao contınua [MS thesis] Instituto Nacional dePesquisas Espaciais Sao Jose dos Campos 2012

[30] J L Ferreira G C Possa B SMoraes et al ldquoPermanentmagnethall thruster for satellite orbit maneuvering with low powerrdquoAdvances in Space Research vol 01 pp 166ndash182 2009

[31] E Stuhlinger Ion Propulsion for Space FlightMcGraw-Hill NewYork NY USA 1st edition 1964

[32] R G Jahn Physics of Electric Propulsion McGraw-Hill NewYork NY USA 1st edition 1968

[33] D G Luemberger Linear and Non-Linear ProgrammingSpringer Science amp Business Media New York NY USA 3rdedition 2008

[34] M S Bazaraa and C M ShettyNonlinear Programming-Theoryand Algorithms JohnWilley amp Sons New York NY USA 1979

[35] V V Zhurin H R Kaufman and R S Robinson ldquoPhysics ofclosed drift thrustersrdquo Plasma Sources Science and Technologyvol 8 no 1 pp R1ndashR20 1999

[36] E M Rocco ldquoPerturbed orbital motion with a PID controlsystem for the trajectoryrdquo in Proceedings of the 14th ColoquioBrasileiro De Dinamica Orbital Aguas de Lindoia AnaisUNESP Guaratingueta Brazil November 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Range angle (degrees)

30002500200015001000500

Pitc

h an

gle

(deg

rees

)

0

0

5

10

15

minus10

minus15

minus5

Figure 8 Optimal pitch angles for maneuver 4

Range angle (degrees)

Pitc

h an

gle

(deg

rees

) 03

010

02

0405

1400120010008006004002000

minus01minus02

minus03minus04minus05

Figure 9 Optimal pitch angles for the first arc of the maneuver 5

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

1

0

05

15

2

40003800360034003200300028002600

minus05

minus1

minus15

minus2

Figure 10 Optimal pitch angles for the second arc of maneuver 5

Total thrust = 0252 Newton specific impulse 1083 s

Solution

Final orbital elements after the first arc 119886 =716530 km 119890 = 00039Final orbital elements after the second arc 119886 =720000 km 119890 = 00040Fuel consumption 1013402 kgDuration of the maneuver 4267954 s

Note that the use of two arcs generates savings in the fuelconsumption as expected

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

0

5

10

15

22001800 2000160014001000 1200800600400200

minus5

minus10

minus15

Figure 11 Optimal pitch angle for maneuver 6

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

02

0

0604

081

1000900600 700 8005004003002001000

minus02

minus06

minus1

minus04

minus08

Figure 12 Optimal pitch angles for the first arc of maneuver 7

Range angle (degrees)

Pitc

h an

gle

(deg

rees

)

0

05

1

15

240023002100 2200200019001800170016001500

minus05

minus1

minus15

Figure 13 Optimal pitch angles for the second arc of maneuver 7

723 Maneuver 6 For this maneuver it was considered theexistence of three Phall II thrusters The propulsion force ofeach one is assumed to be the average measured value Theparameters considered in this maneuver are as follows

Total thrust = 0360 Newton specific impulse 1600 s

Solution

Final orbital elements 119886 = 720000 km 119890 = 0004Fuel consumption 0697930 kgDuration of the maneuver 3039873 s

Mathematical Problems in Engineering 9

Time (s)

minus5

minus10

0

0

5

05 1 15 2 25 3 35 4 45times104

Sem

imaj

or a

xis

dev

iati

on(m

)

Figure 14 Semimajor axis deviation in maneuver 4

02468

1012

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus2minus4minus6minus8

Ecc

entr

icit

y d

evia

tion

times10minus7

Figure 15 Eccentricity deviation in maneuver 4

0

0005

001

0015

Posi

tion

dev

iati

on (m

)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus0005

minus001

Figure 16 Position deviation in maneuver 4

Note that the use of the Phall II generates savings inthe fuel consumption since it has better parameters whencompared to the Phall I device

724 Maneuver 7 Thismaneuver has the same conditions ofmaneuver 6 but now with two thrust arcs instead of oneTheparameters considered in this maneuver are as follows

Total thrust = 0360 Newton specific impulse 1600 s

Solution

Final orbital elements after the first arc 119886 =716777 km 119890 = 00039

0

1

2

3

4

5

Vel

ocit

y d

evia

tion

(m

s)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus1

minus2

minus3

times10minus3

Figure 17 Velocity deviation in maneuver 4

Final orbital elements after the second arc 119886 =720000 km 119890 = 00040Fuel consumption 0686314 kgDuration of the maneuver 2989285 s

Note that the use of two arcs generates savings in the fuelconsumption one more time as expected

73 Optimal Maneuvers Simulations Using the STRS Nowwith the help of the optimal maneuvers found previously weshall simulate thosemaneuvers in the STRS using the optimalangles Some different aspects of the propulsion system wereconsidered to simulate amore realistic environment and alsothe PID control was considered

In this section for the propulsion system it was consid-ered that at each inertial direction 119909 119910 119911 has an error on theactuator signal (ms) with mean equal to zero and varianceequal to 10minus10 that is it was inserted an error in the velocity ofthe satelliteThis error has amplitude approximately close to 5times 10minus3N for each thrust which corresponds to the error of thePMHT (Ferreira et al [30]) There is a PID control to correctthe errors due the actuator but it is still possible to analyzethe deviations and the increase of the fuel consumption dueto the errors

731 Maneuver 4 in STRS In this case it is consideredmaneuver 4 including the solution found there

Final orbital elements 119886 = 719985 km 119890 = 0004Fuel consumption 10470 kgPropellant mass difference from the optimal maneu-ver 666 times 10minus4 kg

The error inserted in the propulsion system resulted in adeviation of the orbital elements as we can see in Figures 14and 15 for the semimajor axis and eccentricity If there wasnot any error at the propulsion system the deviation wouldbe zero all the time for the orbital elements In Figures 16 and17 we can see also the effect of the error in the velocity andposition as a function of these parameters Finally the thrusterror is shown in Figure 18

10 Mathematical Problems in EngineeringA

pplie

d th

rust

(N)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

0504

03

02

01

0

minus01

minus02

minus03

minus04

x axisy axisz axis

ThrustAverage

Figure 18 Thrust applied in maneuver 4

0

5

10

Time (s)

0 05 1 15 2 25 3 35times104

minus5

minus10

minus15

Sem

imaj

or a

xis

dev

iati

on(m

)

Figure 19 Semimajor axis deviation in maneuver 6

732 Maneuver 5 in STRS In this case maneuver 5 isconsidered including the solution found

Final orbital elements after the first arc 119886 =716523 km 119890 = 00039

Final orbital elements after the second arc 119886 =719980 km 119890 = 00040

Fuel consumption 1013362 kg

Propellant mass difference from the optimal maneu-ver 6384 times 10minus4 kg

733 Maneuver 6 in STRS Now maneuver 6 is considered

Final orbital elements 119886 = 719982 km 119890 = 0004

Fuel consumption 06979 kg

Propellant mass difference from the optimal maneu-ver 219 times 10minus4 kg

The same analysis for the figures made in Section 731 fitshere Figures 19ndash23 show the deviations and the error thatoccurred on the propulsion system

Time (s)

0 05 1 15 2 25 3 35times104

Ecc

entr

icit

y d

evia

tion

times10minus6

2

15

1

05

0

minus05

minus1

minus15

Figure 20 Eccentricity deviation in maneuver 6

734 Maneuver 7 in STRS Maneuver 7 is now studiedFinal orbital elements after the first arc 119886 =716770 km 119890 = 00039Final orbital elements after the second arc 119886 =719983 km 119890 = 00040Fuel consumption 0686279 kgPropellant mass difference from the optimal maneu-ver 21469 times 10minus4 kg

735 Maneuver 8 in STRS This maneuver uses the optimalangles given in maneuver 4 but includes the errors proposedfor the propulsion system In addition it was consideredother nonlinearities for the actuator two seconds delay torespond the control signal a rate limiter for the variationof the satelite velocity which the absolute limit of the rateis 000025ms dead zone where the propulsion system isnot turned on if the required velocity increment is lowerthan 00002ms Considering these nonlinearities for thepropulsion system the PID control was able to achievethe maneuver although the fuel consumption was slightlyincreased The results for this simulation are as follows

Final orbital elements 119886 = 719986 km 119890 = 0004Fuel consumption 11276322 kgPropellant mass difference from the optimal maneu-ver 0081429 kg

736 Maneuver 9 in STRS The maneuver simulated hereuses the optimal angles given in maneuver 6 with the errorsproposed for the propulsion system In addition it wasproposed the samenonlinearities (rate limiter dead zone andtime delay) of maneuver 8 Considering these nonlinearitiesfor the propulsion system the PID control was able to achievethe maneuver although the fuel consumption was increasedThe results for this maneuver are as follows

Final orbital elements 119886 = 719984 km 119890 = 00040Fuel consumption 0721921 kgThe propellant mass difference from the optimalmaneuver 002402 kg

Mathematical Problems in Engineering 11

Time (s)

0 05 1 15 2 25 3 35times104

002

0015

001

0005

0

minus0005

minus001

minus0015Po

siti

on d

evia

tion

(m)

Figure 21 Position deviation in maneuver 6

Vel

ocit

y d

evia

tion

(m

s)

Time (s)

0 05 1 15 2 25 3 35times104

8

6

4

2

0

minus2

minus4

times10minus3

Figure 22 Velocity deviation in maneuver 6

x axisy axisz axis

ThrustAverage

Time (s)

0 05 1 15 2 25 3 35times104

06

04

02

0

minus02

minus04

minus06

minus08

App

lied

thru

st (N

)

Figure 23 Thrust applied in maneuver 6

8 Conclusion

Primarily this paper established an algorithm to solve theproblem of orbital maneuvers using a low-thrust control fora spacecraft that is travelling around the Earth Then allthe maneuvers studied in this paper were successfully solvedwith the final conditions and constraints proposed and withreasonable fuel consumptions

It was possible to verify that maneuvers with morethrusting arcs consume less fuel as expected This occursbecause the algorithm has more variables to optimize so itis possible to reduce the fuel consumption

The second part of this work was to simulate the optimalmaneuvers found by the algorithm proposed here in a morerealistic environment which can consider errors on theactuator that was not considered in the search for the optimalmaneuver

Themaneuvers simulated in the STRSwere coherent withthe optimal maneuvers validating the integration of bothsoftwares

The error considered in the propulsion system opensthe possibility to study how the system would react tothose nonlinearities Although errors were inserted in thepropulsion system the PID control was able to correct andaccomplish the maneuver

It is possible to analyze the shifts caused by the error aswell as the deviations in semimajor axis and eccentricity It isalso possible to see that the propulsion thrust was not linear

For themaneuvers shown in Sections 735 and 736 somenonlinearities of the propulsion system such as a responsetime a rate limiter and a dead zone for the actuator signalwere considered All these errors occur in real systems Withthe help of the STRS it was possible to simulate these errorsand analyze the maneuvers studied with more accuracyThe STRS has shown to be an efficient simulator capableof simulating and studying optimal maneuvers in a morerealistic environment and to analyze many nonlinearities notconsidered in the calculation of the optimal maneuver

Evidently the phall II was more efficient than the phall Ias noticed by comparing maneuver 4 with maneuver 6 andmaneuver 5 with maneuver 7 This was expected since thephall II has a higher specific impulse and a higher averagepropulsion thrust as well Nevertheless the PID control wasable to control all the maneuvers and reaches the final orbitalconstraints

References

[1] D F Lawden ldquoMinimal rocket trajectoriesrdquo ARS Journal vol23 no 6 pp 360ndash382 1953

[2] D F Lawden ldquoFundamentals of space navigationrdquo Journal of theBritish Interplanetary Society vol 13 pp 87ndash101 1954

12 Mathematical Problems in Engineering

[3] V V Beletsky and V A Egorov ldquoInterplanetary flights withconstant output enginesrdquoCosmic Research vol 2 no 3 pp 303ndash330 1964

[4] A A Sukhanov ldquoOptimization of flights with low thrustrdquoCosmic Research vol 37 no 2 pp 182ndash191 1999

[5] A A Sukhanov ldquoOptimization of low-thrust interplanetarytransfersrdquo Cosmic Research vol 38 no 6 pp 584ndash587 2000

[6] A A Sukhanov and A F B A Prado ldquoA modification of themethod of transporting trajectoryrdquoCosmic Research vol 42 no1 pp 107ndash112 2004

[7] A A Sukhanov and A F B A Prado ldquoOptimization of lowthrust transfers in the three body problemrdquo Cosmic Researchvol 46 no 5 pp 440ndash451 2008

[8] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction Irdquo Cosmic Researchvol 45 no 5 pp 417ndash423 2007

[9] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction IIrdquo Cosmic Researchvol 46 no 1 pp 49ndash59 2008

[10] S S Fernandes and F D C Carvalho ldquoA first-order analyticaltheory for optimal low-thrust limited-power transfers betweenarbitrary elliptical coplanar orbitsrdquo Mathematical Problems inEngineering vol 2008 Article ID 525930 30 pages 2008

[11] R A Boucher ldquoElectrical propulsion for control of stationarysatellitesrdquo Journal of Spacecraft and Rockets vol 1 no 2 pp 164ndash169 1964

[12] D N Bowditch ldquoApplication of magnetic-expansion plasmathrusters to satellite station keeping and attitude control mis-sionsrdquo in Proceedings of the 4th Electric Propulsion Confer-ence AIAA-1964-677 pp 64ndash677 The American Institute ofAeronautics and Astronautics Philadelphia Pa USA Augest-September 1964

[13] S R Oleson R M Myers C A Kluever J P Riehl and F MCurran ldquoAdvanced propulsion for geostationary orbit insertionand north-south station keepingrdquo Journal of Spacecraft andRockets vol 34 no 1 pp 22ndash28 1997

[14] T A Ely and K C Howell ldquoEast-west stationkeeping of satelliteorbits with resonant tesseral harmonicsrdquoActa Astronautica vol46 no 1 pp 1ndash15 2000

[15] C Circi ldquoSimple strategy for geostationary stationkeepingmaneuvers using solar sailrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 249ndash253 2005

[16] P Romero J M Gambi and E Patino ldquoStationkeepingmanoeuvres for geostationary satellites using feedback controltechniquesrdquo Aerospace Science and Technology vol 11 no 2-3pp 229ndash237 2007

[17] V Gomes and A F B A Prado ldquoLow-thrust out-of-planeorbital station-keeping maneuvers for satellitesrdquo MathematicalProblems in Engineering vol 2012 Article ID 532708 14 pages2012

[18] B L Pierson and C A Kluever ldquoThree-stage approach to opti-mal low-thrust earth-moon trajectoriesrdquo Journal of GuidanceControl and Dynamics vol 17 no 6 pp 1275ndash1282 1994

[19] Y J Song S Y Park K H Choi and E S Sim ldquoA lunar cargomission design strategy using variable low thrustrdquo Advances inSpace Research vol 43 no 9 pp 1391ndash1406 2009

[20] S A Fazelzadeh and G A Varzandian ldquoMinimum-time earth-moon and moon-earth orbital maneuvers using time-domainfinite element methodrdquo Acta Astronautica vol 66 no 3-4 pp528ndash538 2010

[21] G Mingotti F Topputo and F Bernelli-Zazzera ldquoEfficientinvariant-manifold low-thrust planar trajectories to themoonrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 p 817 2012

[22] J R Brophy and M Noca ldquoElectric propulsion for solar systemexplorationrdquo Journal of Propulsion and Power vol 14 no 5 pp700ndash707 1998

[23] D P S Santos A F B A Prado L Casalino and G ColasurdoldquoOptimal trajectories towards near-earth-objects using solarelectric propulsion (sep) and gravity assistedmaneuverrdquo Journalof Aerospace Engineering Sciences and Applications vol 1 no 2pp 51ndash64 2008

[24] D P S Santos L Casalino G Colasurdo and A F B APrado ldquoOptimal trajectories towards near-earth-objects usingsolar electric propulsion (sep) and gravity assisted maneuverrdquoWSEAS Transactions on Applied andTheoretical Mechanics vol4 no 3 pp 125ndash135 2009

[25] D P S Santos and A F B A Prado ldquoOptimal low-thrusttrajectories to reach the asteroid apophisrdquo WSEAS MechanicalEngineering Series vol 7 pp 241ndash251 2012

[26] M C B BiggsThe Optimization of Spacecraft Orbital Manoeu-vres Part I Linearly Varying Thrust Angles The HatfieldPolytechnic Numerical Optimization Centre 1978

[27] M C B BiggsThe Optimisation of Spacecraft Orbital Manoeu-vres Part II Using Pontryaginrsquos Maximum Principle The Hat-field Polytechnic Numerical Optimization Centre 1979

[28] A F B A Prado Analise Selecao e Implementacao de Procedi-mentos que visemManobras Otimas de Satelites Artificiais [MSthesis] Instituto Nacional de Pesquisas Espaciais Sao Jose dosCampos 1988

[29] T C Oliveira Estrategias otimas para manobras orbitais uti-lizando propulsao contınua [MS thesis] Instituto Nacional dePesquisas Espaciais Sao Jose dos Campos 2012

[30] J L Ferreira G C Possa B SMoraes et al ldquoPermanentmagnethall thruster for satellite orbit maneuvering with low powerrdquoAdvances in Space Research vol 01 pp 166ndash182 2009

[31] E Stuhlinger Ion Propulsion for Space FlightMcGraw-Hill NewYork NY USA 1st edition 1964

[32] R G Jahn Physics of Electric Propulsion McGraw-Hill NewYork NY USA 1st edition 1968

[33] D G Luemberger Linear and Non-Linear ProgrammingSpringer Science amp Business Media New York NY USA 3rdedition 2008

[34] M S Bazaraa and C M ShettyNonlinear Programming-Theoryand Algorithms JohnWilley amp Sons New York NY USA 1979

[35] V V Zhurin H R Kaufman and R S Robinson ldquoPhysics ofclosed drift thrustersrdquo Plasma Sources Science and Technologyvol 8 no 1 pp R1ndashR20 1999

[36] E M Rocco ldquoPerturbed orbital motion with a PID controlsystem for the trajectoryrdquo in Proceedings of the 14th ColoquioBrasileiro De Dinamica Orbital Aguas de Lindoia AnaisUNESP Guaratingueta Brazil November 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Time (s)

minus5

minus10

0

0

5

05 1 15 2 25 3 35 4 45times104

Sem

imaj

or a

xis

dev

iati

on(m

)

Figure 14 Semimajor axis deviation in maneuver 4

02468

1012

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus2minus4minus6minus8

Ecc

entr

icit

y d

evia

tion

times10minus7

Figure 15 Eccentricity deviation in maneuver 4

0

0005

001

0015

Posi

tion

dev

iati

on (m

)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus0005

minus001

Figure 16 Position deviation in maneuver 4

Note that the use of the Phall II generates savings inthe fuel consumption since it has better parameters whencompared to the Phall I device

724 Maneuver 7 Thismaneuver has the same conditions ofmaneuver 6 but now with two thrust arcs instead of oneTheparameters considered in this maneuver are as follows

Total thrust = 0360 Newton specific impulse 1600 s

Solution

Final orbital elements after the first arc 119886 =716777 km 119890 = 00039

0

1

2

3

4

5

Vel

ocit

y d

evia

tion

(m

s)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

minus1

minus2

minus3

times10minus3

Figure 17 Velocity deviation in maneuver 4

Final orbital elements after the second arc 119886 =720000 km 119890 = 00040Fuel consumption 0686314 kgDuration of the maneuver 2989285 s

Note that the use of two arcs generates savings in the fuelconsumption one more time as expected

73 Optimal Maneuvers Simulations Using the STRS Nowwith the help of the optimal maneuvers found previously weshall simulate thosemaneuvers in the STRS using the optimalangles Some different aspects of the propulsion system wereconsidered to simulate amore realistic environment and alsothe PID control was considered

In this section for the propulsion system it was consid-ered that at each inertial direction 119909 119910 119911 has an error on theactuator signal (ms) with mean equal to zero and varianceequal to 10minus10 that is it was inserted an error in the velocity ofthe satelliteThis error has amplitude approximately close to 5times 10minus3N for each thrust which corresponds to the error of thePMHT (Ferreira et al [30]) There is a PID control to correctthe errors due the actuator but it is still possible to analyzethe deviations and the increase of the fuel consumption dueto the errors

731 Maneuver 4 in STRS In this case it is consideredmaneuver 4 including the solution found there

Final orbital elements 119886 = 719985 km 119890 = 0004Fuel consumption 10470 kgPropellant mass difference from the optimal maneu-ver 666 times 10minus4 kg

The error inserted in the propulsion system resulted in adeviation of the orbital elements as we can see in Figures 14and 15 for the semimajor axis and eccentricity If there wasnot any error at the propulsion system the deviation wouldbe zero all the time for the orbital elements In Figures 16 and17 we can see also the effect of the error in the velocity andposition as a function of these parameters Finally the thrusterror is shown in Figure 18

10 Mathematical Problems in EngineeringA

pplie

d th

rust

(N)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

0504

03

02

01

0

minus01

minus02

minus03

minus04

x axisy axisz axis

ThrustAverage

Figure 18 Thrust applied in maneuver 4

0

5

10

Time (s)

0 05 1 15 2 25 3 35times104

minus5

minus10

minus15

Sem

imaj

or a

xis

dev

iati

on(m

)

Figure 19 Semimajor axis deviation in maneuver 6

732 Maneuver 5 in STRS In this case maneuver 5 isconsidered including the solution found

Final orbital elements after the first arc 119886 =716523 km 119890 = 00039

Final orbital elements after the second arc 119886 =719980 km 119890 = 00040

Fuel consumption 1013362 kg

Propellant mass difference from the optimal maneu-ver 6384 times 10minus4 kg

733 Maneuver 6 in STRS Now maneuver 6 is considered

Final orbital elements 119886 = 719982 km 119890 = 0004

Fuel consumption 06979 kg

Propellant mass difference from the optimal maneu-ver 219 times 10minus4 kg

The same analysis for the figures made in Section 731 fitshere Figures 19ndash23 show the deviations and the error thatoccurred on the propulsion system

Time (s)

0 05 1 15 2 25 3 35times104

Ecc

entr

icit

y d

evia

tion

times10minus6

2

15

1

05

0

minus05

minus1

minus15

Figure 20 Eccentricity deviation in maneuver 6

734 Maneuver 7 in STRS Maneuver 7 is now studiedFinal orbital elements after the first arc 119886 =716770 km 119890 = 00039Final orbital elements after the second arc 119886 =719983 km 119890 = 00040Fuel consumption 0686279 kgPropellant mass difference from the optimal maneu-ver 21469 times 10minus4 kg

735 Maneuver 8 in STRS This maneuver uses the optimalangles given in maneuver 4 but includes the errors proposedfor the propulsion system In addition it was consideredother nonlinearities for the actuator two seconds delay torespond the control signal a rate limiter for the variationof the satelite velocity which the absolute limit of the rateis 000025ms dead zone where the propulsion system isnot turned on if the required velocity increment is lowerthan 00002ms Considering these nonlinearities for thepropulsion system the PID control was able to achievethe maneuver although the fuel consumption was slightlyincreased The results for this simulation are as follows

Final orbital elements 119886 = 719986 km 119890 = 0004Fuel consumption 11276322 kgPropellant mass difference from the optimal maneu-ver 0081429 kg

736 Maneuver 9 in STRS The maneuver simulated hereuses the optimal angles given in maneuver 6 with the errorsproposed for the propulsion system In addition it wasproposed the samenonlinearities (rate limiter dead zone andtime delay) of maneuver 8 Considering these nonlinearitiesfor the propulsion system the PID control was able to achievethe maneuver although the fuel consumption was increasedThe results for this maneuver are as follows

Final orbital elements 119886 = 719984 km 119890 = 00040Fuel consumption 0721921 kgThe propellant mass difference from the optimalmaneuver 002402 kg

Mathematical Problems in Engineering 11

Time (s)

0 05 1 15 2 25 3 35times104

002

0015

001

0005

0

minus0005

minus001

minus0015Po

siti

on d

evia

tion

(m)

Figure 21 Position deviation in maneuver 6

Vel

ocit

y d

evia

tion

(m

s)

Time (s)

0 05 1 15 2 25 3 35times104

8

6

4

2

0

minus2

minus4

times10minus3

Figure 22 Velocity deviation in maneuver 6

x axisy axisz axis

ThrustAverage

Time (s)

0 05 1 15 2 25 3 35times104

06

04

02

0

minus02

minus04

minus06

minus08

App

lied

thru

st (N

)

Figure 23 Thrust applied in maneuver 6

8 Conclusion

Primarily this paper established an algorithm to solve theproblem of orbital maneuvers using a low-thrust control fora spacecraft that is travelling around the Earth Then allthe maneuvers studied in this paper were successfully solvedwith the final conditions and constraints proposed and withreasonable fuel consumptions

It was possible to verify that maneuvers with morethrusting arcs consume less fuel as expected This occursbecause the algorithm has more variables to optimize so itis possible to reduce the fuel consumption

The second part of this work was to simulate the optimalmaneuvers found by the algorithm proposed here in a morerealistic environment which can consider errors on theactuator that was not considered in the search for the optimalmaneuver

Themaneuvers simulated in the STRSwere coherent withthe optimal maneuvers validating the integration of bothsoftwares

The error considered in the propulsion system opensthe possibility to study how the system would react tothose nonlinearities Although errors were inserted in thepropulsion system the PID control was able to correct andaccomplish the maneuver

It is possible to analyze the shifts caused by the error aswell as the deviations in semimajor axis and eccentricity It isalso possible to see that the propulsion thrust was not linear

For themaneuvers shown in Sections 735 and 736 somenonlinearities of the propulsion system such as a responsetime a rate limiter and a dead zone for the actuator signalwere considered All these errors occur in real systems Withthe help of the STRS it was possible to simulate these errorsand analyze the maneuvers studied with more accuracyThe STRS has shown to be an efficient simulator capableof simulating and studying optimal maneuvers in a morerealistic environment and to analyze many nonlinearities notconsidered in the calculation of the optimal maneuver

Evidently the phall II was more efficient than the phall Ias noticed by comparing maneuver 4 with maneuver 6 andmaneuver 5 with maneuver 7 This was expected since thephall II has a higher specific impulse and a higher averagepropulsion thrust as well Nevertheless the PID control wasable to control all the maneuvers and reaches the final orbitalconstraints

References

[1] D F Lawden ldquoMinimal rocket trajectoriesrdquo ARS Journal vol23 no 6 pp 360ndash382 1953

[2] D F Lawden ldquoFundamentals of space navigationrdquo Journal of theBritish Interplanetary Society vol 13 pp 87ndash101 1954

12 Mathematical Problems in Engineering

[3] V V Beletsky and V A Egorov ldquoInterplanetary flights withconstant output enginesrdquoCosmic Research vol 2 no 3 pp 303ndash330 1964

[4] A A Sukhanov ldquoOptimization of flights with low thrustrdquoCosmic Research vol 37 no 2 pp 182ndash191 1999

[5] A A Sukhanov ldquoOptimization of low-thrust interplanetarytransfersrdquo Cosmic Research vol 38 no 6 pp 584ndash587 2000

[6] A A Sukhanov and A F B A Prado ldquoA modification of themethod of transporting trajectoryrdquoCosmic Research vol 42 no1 pp 107ndash112 2004

[7] A A Sukhanov and A F B A Prado ldquoOptimization of lowthrust transfers in the three body problemrdquo Cosmic Researchvol 46 no 5 pp 440ndash451 2008

[8] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction Irdquo Cosmic Researchvol 45 no 5 pp 417ndash423 2007

[9] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction IIrdquo Cosmic Researchvol 46 no 1 pp 49ndash59 2008

[10] S S Fernandes and F D C Carvalho ldquoA first-order analyticaltheory for optimal low-thrust limited-power transfers betweenarbitrary elliptical coplanar orbitsrdquo Mathematical Problems inEngineering vol 2008 Article ID 525930 30 pages 2008

[11] R A Boucher ldquoElectrical propulsion for control of stationarysatellitesrdquo Journal of Spacecraft and Rockets vol 1 no 2 pp 164ndash169 1964

[12] D N Bowditch ldquoApplication of magnetic-expansion plasmathrusters to satellite station keeping and attitude control mis-sionsrdquo in Proceedings of the 4th Electric Propulsion Confer-ence AIAA-1964-677 pp 64ndash677 The American Institute ofAeronautics and Astronautics Philadelphia Pa USA Augest-September 1964

[13] S R Oleson R M Myers C A Kluever J P Riehl and F MCurran ldquoAdvanced propulsion for geostationary orbit insertionand north-south station keepingrdquo Journal of Spacecraft andRockets vol 34 no 1 pp 22ndash28 1997

[14] T A Ely and K C Howell ldquoEast-west stationkeeping of satelliteorbits with resonant tesseral harmonicsrdquoActa Astronautica vol46 no 1 pp 1ndash15 2000

[15] C Circi ldquoSimple strategy for geostationary stationkeepingmaneuvers using solar sailrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 249ndash253 2005

[16] P Romero J M Gambi and E Patino ldquoStationkeepingmanoeuvres for geostationary satellites using feedback controltechniquesrdquo Aerospace Science and Technology vol 11 no 2-3pp 229ndash237 2007

[17] V Gomes and A F B A Prado ldquoLow-thrust out-of-planeorbital station-keeping maneuvers for satellitesrdquo MathematicalProblems in Engineering vol 2012 Article ID 532708 14 pages2012

[18] B L Pierson and C A Kluever ldquoThree-stage approach to opti-mal low-thrust earth-moon trajectoriesrdquo Journal of GuidanceControl and Dynamics vol 17 no 6 pp 1275ndash1282 1994

[19] Y J Song S Y Park K H Choi and E S Sim ldquoA lunar cargomission design strategy using variable low thrustrdquo Advances inSpace Research vol 43 no 9 pp 1391ndash1406 2009

[20] S A Fazelzadeh and G A Varzandian ldquoMinimum-time earth-moon and moon-earth orbital maneuvers using time-domainfinite element methodrdquo Acta Astronautica vol 66 no 3-4 pp528ndash538 2010

[21] G Mingotti F Topputo and F Bernelli-Zazzera ldquoEfficientinvariant-manifold low-thrust planar trajectories to themoonrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 p 817 2012

[22] J R Brophy and M Noca ldquoElectric propulsion for solar systemexplorationrdquo Journal of Propulsion and Power vol 14 no 5 pp700ndash707 1998

[23] D P S Santos A F B A Prado L Casalino and G ColasurdoldquoOptimal trajectories towards near-earth-objects using solarelectric propulsion (sep) and gravity assistedmaneuverrdquo Journalof Aerospace Engineering Sciences and Applications vol 1 no 2pp 51ndash64 2008

[24] D P S Santos L Casalino G Colasurdo and A F B APrado ldquoOptimal trajectories towards near-earth-objects usingsolar electric propulsion (sep) and gravity assisted maneuverrdquoWSEAS Transactions on Applied andTheoretical Mechanics vol4 no 3 pp 125ndash135 2009

[25] D P S Santos and A F B A Prado ldquoOptimal low-thrusttrajectories to reach the asteroid apophisrdquo WSEAS MechanicalEngineering Series vol 7 pp 241ndash251 2012

[26] M C B BiggsThe Optimization of Spacecraft Orbital Manoeu-vres Part I Linearly Varying Thrust Angles The HatfieldPolytechnic Numerical Optimization Centre 1978

[27] M C B BiggsThe Optimisation of Spacecraft Orbital Manoeu-vres Part II Using Pontryaginrsquos Maximum Principle The Hat-field Polytechnic Numerical Optimization Centre 1979

[28] A F B A Prado Analise Selecao e Implementacao de Procedi-mentos que visemManobras Otimas de Satelites Artificiais [MSthesis] Instituto Nacional de Pesquisas Espaciais Sao Jose dosCampos 1988

[29] T C Oliveira Estrategias otimas para manobras orbitais uti-lizando propulsao contınua [MS thesis] Instituto Nacional dePesquisas Espaciais Sao Jose dos Campos 2012

[30] J L Ferreira G C Possa B SMoraes et al ldquoPermanentmagnethall thruster for satellite orbit maneuvering with low powerrdquoAdvances in Space Research vol 01 pp 166ndash182 2009

[31] E Stuhlinger Ion Propulsion for Space FlightMcGraw-Hill NewYork NY USA 1st edition 1964

[32] R G Jahn Physics of Electric Propulsion McGraw-Hill NewYork NY USA 1st edition 1968

[33] D G Luemberger Linear and Non-Linear ProgrammingSpringer Science amp Business Media New York NY USA 3rdedition 2008

[34] M S Bazaraa and C M ShettyNonlinear Programming-Theoryand Algorithms JohnWilley amp Sons New York NY USA 1979

[35] V V Zhurin H R Kaufman and R S Robinson ldquoPhysics ofclosed drift thrustersrdquo Plasma Sources Science and Technologyvol 8 no 1 pp R1ndashR20 1999

[36] E M Rocco ldquoPerturbed orbital motion with a PID controlsystem for the trajectoryrdquo in Proceedings of the 14th ColoquioBrasileiro De Dinamica Orbital Aguas de Lindoia AnaisUNESP Guaratingueta Brazil November 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in EngineeringA

pplie

d th

rust

(N)

Time (s)0 05 1 15 2 25 3 35 4 45

times104

0504

03

02

01

0

minus01

minus02

minus03

minus04

x axisy axisz axis

ThrustAverage

Figure 18 Thrust applied in maneuver 4

0

5

10

Time (s)

0 05 1 15 2 25 3 35times104

minus5

minus10

minus15

Sem

imaj

or a

xis

dev

iati

on(m

)

Figure 19 Semimajor axis deviation in maneuver 6

732 Maneuver 5 in STRS In this case maneuver 5 isconsidered including the solution found

Final orbital elements after the first arc 119886 =716523 km 119890 = 00039

Final orbital elements after the second arc 119886 =719980 km 119890 = 00040

Fuel consumption 1013362 kg

Propellant mass difference from the optimal maneu-ver 6384 times 10minus4 kg

733 Maneuver 6 in STRS Now maneuver 6 is considered

Final orbital elements 119886 = 719982 km 119890 = 0004

Fuel consumption 06979 kg

Propellant mass difference from the optimal maneu-ver 219 times 10minus4 kg

The same analysis for the figures made in Section 731 fitshere Figures 19ndash23 show the deviations and the error thatoccurred on the propulsion system

Time (s)

0 05 1 15 2 25 3 35times104

Ecc

entr

icit

y d

evia

tion

times10minus6

2

15

1

05

0

minus05

minus1

minus15

Figure 20 Eccentricity deviation in maneuver 6

734 Maneuver 7 in STRS Maneuver 7 is now studiedFinal orbital elements after the first arc 119886 =716770 km 119890 = 00039Final orbital elements after the second arc 119886 =719983 km 119890 = 00040Fuel consumption 0686279 kgPropellant mass difference from the optimal maneu-ver 21469 times 10minus4 kg

735 Maneuver 8 in STRS This maneuver uses the optimalangles given in maneuver 4 but includes the errors proposedfor the propulsion system In addition it was consideredother nonlinearities for the actuator two seconds delay torespond the control signal a rate limiter for the variationof the satelite velocity which the absolute limit of the rateis 000025ms dead zone where the propulsion system isnot turned on if the required velocity increment is lowerthan 00002ms Considering these nonlinearities for thepropulsion system the PID control was able to achievethe maneuver although the fuel consumption was slightlyincreased The results for this simulation are as follows

Final orbital elements 119886 = 719986 km 119890 = 0004Fuel consumption 11276322 kgPropellant mass difference from the optimal maneu-ver 0081429 kg

736 Maneuver 9 in STRS The maneuver simulated hereuses the optimal angles given in maneuver 6 with the errorsproposed for the propulsion system In addition it wasproposed the samenonlinearities (rate limiter dead zone andtime delay) of maneuver 8 Considering these nonlinearitiesfor the propulsion system the PID control was able to achievethe maneuver although the fuel consumption was increasedThe results for this maneuver are as follows

Final orbital elements 119886 = 719984 km 119890 = 00040Fuel consumption 0721921 kgThe propellant mass difference from the optimalmaneuver 002402 kg

Mathematical Problems in Engineering 11

Time (s)

0 05 1 15 2 25 3 35times104

002

0015

001

0005

0

minus0005

minus001

minus0015Po

siti

on d

evia

tion

(m)

Figure 21 Position deviation in maneuver 6

Vel

ocit

y d

evia

tion

(m

s)

Time (s)

0 05 1 15 2 25 3 35times104

8

6

4

2

0

minus2

minus4

times10minus3

Figure 22 Velocity deviation in maneuver 6

x axisy axisz axis

ThrustAverage

Time (s)

0 05 1 15 2 25 3 35times104

06

04

02

0

minus02

minus04

minus06

minus08

App

lied

thru

st (N

)

Figure 23 Thrust applied in maneuver 6

8 Conclusion

Primarily this paper established an algorithm to solve theproblem of orbital maneuvers using a low-thrust control fora spacecraft that is travelling around the Earth Then allthe maneuvers studied in this paper were successfully solvedwith the final conditions and constraints proposed and withreasonable fuel consumptions

It was possible to verify that maneuvers with morethrusting arcs consume less fuel as expected This occursbecause the algorithm has more variables to optimize so itis possible to reduce the fuel consumption

The second part of this work was to simulate the optimalmaneuvers found by the algorithm proposed here in a morerealistic environment which can consider errors on theactuator that was not considered in the search for the optimalmaneuver

Themaneuvers simulated in the STRSwere coherent withthe optimal maneuvers validating the integration of bothsoftwares

The error considered in the propulsion system opensthe possibility to study how the system would react tothose nonlinearities Although errors were inserted in thepropulsion system the PID control was able to correct andaccomplish the maneuver

It is possible to analyze the shifts caused by the error aswell as the deviations in semimajor axis and eccentricity It isalso possible to see that the propulsion thrust was not linear

For themaneuvers shown in Sections 735 and 736 somenonlinearities of the propulsion system such as a responsetime a rate limiter and a dead zone for the actuator signalwere considered All these errors occur in real systems Withthe help of the STRS it was possible to simulate these errorsand analyze the maneuvers studied with more accuracyThe STRS has shown to be an efficient simulator capableof simulating and studying optimal maneuvers in a morerealistic environment and to analyze many nonlinearities notconsidered in the calculation of the optimal maneuver

Evidently the phall II was more efficient than the phall Ias noticed by comparing maneuver 4 with maneuver 6 andmaneuver 5 with maneuver 7 This was expected since thephall II has a higher specific impulse and a higher averagepropulsion thrust as well Nevertheless the PID control wasable to control all the maneuvers and reaches the final orbitalconstraints

References

[1] D F Lawden ldquoMinimal rocket trajectoriesrdquo ARS Journal vol23 no 6 pp 360ndash382 1953

[2] D F Lawden ldquoFundamentals of space navigationrdquo Journal of theBritish Interplanetary Society vol 13 pp 87ndash101 1954

12 Mathematical Problems in Engineering

[3] V V Beletsky and V A Egorov ldquoInterplanetary flights withconstant output enginesrdquoCosmic Research vol 2 no 3 pp 303ndash330 1964

[4] A A Sukhanov ldquoOptimization of flights with low thrustrdquoCosmic Research vol 37 no 2 pp 182ndash191 1999

[5] A A Sukhanov ldquoOptimization of low-thrust interplanetarytransfersrdquo Cosmic Research vol 38 no 6 pp 584ndash587 2000

[6] A A Sukhanov and A F B A Prado ldquoA modification of themethod of transporting trajectoryrdquoCosmic Research vol 42 no1 pp 107ndash112 2004

[7] A A Sukhanov and A F B A Prado ldquoOptimization of lowthrust transfers in the three body problemrdquo Cosmic Researchvol 46 no 5 pp 440ndash451 2008

[8] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction Irdquo Cosmic Researchvol 45 no 5 pp 417ndash423 2007

[9] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction IIrdquo Cosmic Researchvol 46 no 1 pp 49ndash59 2008

[10] S S Fernandes and F D C Carvalho ldquoA first-order analyticaltheory for optimal low-thrust limited-power transfers betweenarbitrary elliptical coplanar orbitsrdquo Mathematical Problems inEngineering vol 2008 Article ID 525930 30 pages 2008

[11] R A Boucher ldquoElectrical propulsion for control of stationarysatellitesrdquo Journal of Spacecraft and Rockets vol 1 no 2 pp 164ndash169 1964

[12] D N Bowditch ldquoApplication of magnetic-expansion plasmathrusters to satellite station keeping and attitude control mis-sionsrdquo in Proceedings of the 4th Electric Propulsion Confer-ence AIAA-1964-677 pp 64ndash677 The American Institute ofAeronautics and Astronautics Philadelphia Pa USA Augest-September 1964

[13] S R Oleson R M Myers C A Kluever J P Riehl and F MCurran ldquoAdvanced propulsion for geostationary orbit insertionand north-south station keepingrdquo Journal of Spacecraft andRockets vol 34 no 1 pp 22ndash28 1997

[14] T A Ely and K C Howell ldquoEast-west stationkeeping of satelliteorbits with resonant tesseral harmonicsrdquoActa Astronautica vol46 no 1 pp 1ndash15 2000

[15] C Circi ldquoSimple strategy for geostationary stationkeepingmaneuvers using solar sailrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 249ndash253 2005

[16] P Romero J M Gambi and E Patino ldquoStationkeepingmanoeuvres for geostationary satellites using feedback controltechniquesrdquo Aerospace Science and Technology vol 11 no 2-3pp 229ndash237 2007

[17] V Gomes and A F B A Prado ldquoLow-thrust out-of-planeorbital station-keeping maneuvers for satellitesrdquo MathematicalProblems in Engineering vol 2012 Article ID 532708 14 pages2012

[18] B L Pierson and C A Kluever ldquoThree-stage approach to opti-mal low-thrust earth-moon trajectoriesrdquo Journal of GuidanceControl and Dynamics vol 17 no 6 pp 1275ndash1282 1994

[19] Y J Song S Y Park K H Choi and E S Sim ldquoA lunar cargomission design strategy using variable low thrustrdquo Advances inSpace Research vol 43 no 9 pp 1391ndash1406 2009

[20] S A Fazelzadeh and G A Varzandian ldquoMinimum-time earth-moon and moon-earth orbital maneuvers using time-domainfinite element methodrdquo Acta Astronautica vol 66 no 3-4 pp528ndash538 2010

[21] G Mingotti F Topputo and F Bernelli-Zazzera ldquoEfficientinvariant-manifold low-thrust planar trajectories to themoonrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 p 817 2012

[22] J R Brophy and M Noca ldquoElectric propulsion for solar systemexplorationrdquo Journal of Propulsion and Power vol 14 no 5 pp700ndash707 1998

[23] D P S Santos A F B A Prado L Casalino and G ColasurdoldquoOptimal trajectories towards near-earth-objects using solarelectric propulsion (sep) and gravity assistedmaneuverrdquo Journalof Aerospace Engineering Sciences and Applications vol 1 no 2pp 51ndash64 2008

[24] D P S Santos L Casalino G Colasurdo and A F B APrado ldquoOptimal trajectories towards near-earth-objects usingsolar electric propulsion (sep) and gravity assisted maneuverrdquoWSEAS Transactions on Applied andTheoretical Mechanics vol4 no 3 pp 125ndash135 2009

[25] D P S Santos and A F B A Prado ldquoOptimal low-thrusttrajectories to reach the asteroid apophisrdquo WSEAS MechanicalEngineering Series vol 7 pp 241ndash251 2012

[26] M C B BiggsThe Optimization of Spacecraft Orbital Manoeu-vres Part I Linearly Varying Thrust Angles The HatfieldPolytechnic Numerical Optimization Centre 1978

[27] M C B BiggsThe Optimisation of Spacecraft Orbital Manoeu-vres Part II Using Pontryaginrsquos Maximum Principle The Hat-field Polytechnic Numerical Optimization Centre 1979

[28] A F B A Prado Analise Selecao e Implementacao de Procedi-mentos que visemManobras Otimas de Satelites Artificiais [MSthesis] Instituto Nacional de Pesquisas Espaciais Sao Jose dosCampos 1988

[29] T C Oliveira Estrategias otimas para manobras orbitais uti-lizando propulsao contınua [MS thesis] Instituto Nacional dePesquisas Espaciais Sao Jose dos Campos 2012

[30] J L Ferreira G C Possa B SMoraes et al ldquoPermanentmagnethall thruster for satellite orbit maneuvering with low powerrdquoAdvances in Space Research vol 01 pp 166ndash182 2009

[31] E Stuhlinger Ion Propulsion for Space FlightMcGraw-Hill NewYork NY USA 1st edition 1964

[32] R G Jahn Physics of Electric Propulsion McGraw-Hill NewYork NY USA 1st edition 1968

[33] D G Luemberger Linear and Non-Linear ProgrammingSpringer Science amp Business Media New York NY USA 3rdedition 2008

[34] M S Bazaraa and C M ShettyNonlinear Programming-Theoryand Algorithms JohnWilley amp Sons New York NY USA 1979

[35] V V Zhurin H R Kaufman and R S Robinson ldquoPhysics ofclosed drift thrustersrdquo Plasma Sources Science and Technologyvol 8 no 1 pp R1ndashR20 1999

[36] E M Rocco ldquoPerturbed orbital motion with a PID controlsystem for the trajectoryrdquo in Proceedings of the 14th ColoquioBrasileiro De Dinamica Orbital Aguas de Lindoia AnaisUNESP Guaratingueta Brazil November 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Time (s)

0 05 1 15 2 25 3 35times104

002

0015

001

0005

0

minus0005

minus001

minus0015Po

siti

on d

evia

tion

(m)

Figure 21 Position deviation in maneuver 6

Vel

ocit

y d

evia

tion

(m

s)

Time (s)

0 05 1 15 2 25 3 35times104

8

6

4

2

0

minus2

minus4

times10minus3

Figure 22 Velocity deviation in maneuver 6

x axisy axisz axis

ThrustAverage

Time (s)

0 05 1 15 2 25 3 35times104

06

04

02

0

minus02

minus04

minus06

minus08

App

lied

thru

st (N

)

Figure 23 Thrust applied in maneuver 6

8 Conclusion

Primarily this paper established an algorithm to solve theproblem of orbital maneuvers using a low-thrust control fora spacecraft that is travelling around the Earth Then allthe maneuvers studied in this paper were successfully solvedwith the final conditions and constraints proposed and withreasonable fuel consumptions

It was possible to verify that maneuvers with morethrusting arcs consume less fuel as expected This occursbecause the algorithm has more variables to optimize so itis possible to reduce the fuel consumption

The second part of this work was to simulate the optimalmaneuvers found by the algorithm proposed here in a morerealistic environment which can consider errors on theactuator that was not considered in the search for the optimalmaneuver

Themaneuvers simulated in the STRSwere coherent withthe optimal maneuvers validating the integration of bothsoftwares

The error considered in the propulsion system opensthe possibility to study how the system would react tothose nonlinearities Although errors were inserted in thepropulsion system the PID control was able to correct andaccomplish the maneuver

It is possible to analyze the shifts caused by the error aswell as the deviations in semimajor axis and eccentricity It isalso possible to see that the propulsion thrust was not linear

For themaneuvers shown in Sections 735 and 736 somenonlinearities of the propulsion system such as a responsetime a rate limiter and a dead zone for the actuator signalwere considered All these errors occur in real systems Withthe help of the STRS it was possible to simulate these errorsand analyze the maneuvers studied with more accuracyThe STRS has shown to be an efficient simulator capableof simulating and studying optimal maneuvers in a morerealistic environment and to analyze many nonlinearities notconsidered in the calculation of the optimal maneuver

Evidently the phall II was more efficient than the phall Ias noticed by comparing maneuver 4 with maneuver 6 andmaneuver 5 with maneuver 7 This was expected since thephall II has a higher specific impulse and a higher averagepropulsion thrust as well Nevertheless the PID control wasable to control all the maneuvers and reaches the final orbitalconstraints

References

[1] D F Lawden ldquoMinimal rocket trajectoriesrdquo ARS Journal vol23 no 6 pp 360ndash382 1953

[2] D F Lawden ldquoFundamentals of space navigationrdquo Journal of theBritish Interplanetary Society vol 13 pp 87ndash101 1954

12 Mathematical Problems in Engineering

[3] V V Beletsky and V A Egorov ldquoInterplanetary flights withconstant output enginesrdquoCosmic Research vol 2 no 3 pp 303ndash330 1964

[4] A A Sukhanov ldquoOptimization of flights with low thrustrdquoCosmic Research vol 37 no 2 pp 182ndash191 1999

[5] A A Sukhanov ldquoOptimization of low-thrust interplanetarytransfersrdquo Cosmic Research vol 38 no 6 pp 584ndash587 2000

[6] A A Sukhanov and A F B A Prado ldquoA modification of themethod of transporting trajectoryrdquoCosmic Research vol 42 no1 pp 107ndash112 2004

[7] A A Sukhanov and A F B A Prado ldquoOptimization of lowthrust transfers in the three body problemrdquo Cosmic Researchvol 46 no 5 pp 440ndash451 2008

[8] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction Irdquo Cosmic Researchvol 45 no 5 pp 417ndash423 2007

[9] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction IIrdquo Cosmic Researchvol 46 no 1 pp 49ndash59 2008

[10] S S Fernandes and F D C Carvalho ldquoA first-order analyticaltheory for optimal low-thrust limited-power transfers betweenarbitrary elliptical coplanar orbitsrdquo Mathematical Problems inEngineering vol 2008 Article ID 525930 30 pages 2008

[11] R A Boucher ldquoElectrical propulsion for control of stationarysatellitesrdquo Journal of Spacecraft and Rockets vol 1 no 2 pp 164ndash169 1964

[12] D N Bowditch ldquoApplication of magnetic-expansion plasmathrusters to satellite station keeping and attitude control mis-sionsrdquo in Proceedings of the 4th Electric Propulsion Confer-ence AIAA-1964-677 pp 64ndash677 The American Institute ofAeronautics and Astronautics Philadelphia Pa USA Augest-September 1964

[13] S R Oleson R M Myers C A Kluever J P Riehl and F MCurran ldquoAdvanced propulsion for geostationary orbit insertionand north-south station keepingrdquo Journal of Spacecraft andRockets vol 34 no 1 pp 22ndash28 1997

[14] T A Ely and K C Howell ldquoEast-west stationkeeping of satelliteorbits with resonant tesseral harmonicsrdquoActa Astronautica vol46 no 1 pp 1ndash15 2000

[15] C Circi ldquoSimple strategy for geostationary stationkeepingmaneuvers using solar sailrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 249ndash253 2005

[16] P Romero J M Gambi and E Patino ldquoStationkeepingmanoeuvres for geostationary satellites using feedback controltechniquesrdquo Aerospace Science and Technology vol 11 no 2-3pp 229ndash237 2007

[17] V Gomes and A F B A Prado ldquoLow-thrust out-of-planeorbital station-keeping maneuvers for satellitesrdquo MathematicalProblems in Engineering vol 2012 Article ID 532708 14 pages2012

[18] B L Pierson and C A Kluever ldquoThree-stage approach to opti-mal low-thrust earth-moon trajectoriesrdquo Journal of GuidanceControl and Dynamics vol 17 no 6 pp 1275ndash1282 1994

[19] Y J Song S Y Park K H Choi and E S Sim ldquoA lunar cargomission design strategy using variable low thrustrdquo Advances inSpace Research vol 43 no 9 pp 1391ndash1406 2009

[20] S A Fazelzadeh and G A Varzandian ldquoMinimum-time earth-moon and moon-earth orbital maneuvers using time-domainfinite element methodrdquo Acta Astronautica vol 66 no 3-4 pp528ndash538 2010

[21] G Mingotti F Topputo and F Bernelli-Zazzera ldquoEfficientinvariant-manifold low-thrust planar trajectories to themoonrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 p 817 2012

[22] J R Brophy and M Noca ldquoElectric propulsion for solar systemexplorationrdquo Journal of Propulsion and Power vol 14 no 5 pp700ndash707 1998

[23] D P S Santos A F B A Prado L Casalino and G ColasurdoldquoOptimal trajectories towards near-earth-objects using solarelectric propulsion (sep) and gravity assistedmaneuverrdquo Journalof Aerospace Engineering Sciences and Applications vol 1 no 2pp 51ndash64 2008

[24] D P S Santos L Casalino G Colasurdo and A F B APrado ldquoOptimal trajectories towards near-earth-objects usingsolar electric propulsion (sep) and gravity assisted maneuverrdquoWSEAS Transactions on Applied andTheoretical Mechanics vol4 no 3 pp 125ndash135 2009

[25] D P S Santos and A F B A Prado ldquoOptimal low-thrusttrajectories to reach the asteroid apophisrdquo WSEAS MechanicalEngineering Series vol 7 pp 241ndash251 2012

[26] M C B BiggsThe Optimization of Spacecraft Orbital Manoeu-vres Part I Linearly Varying Thrust Angles The HatfieldPolytechnic Numerical Optimization Centre 1978

[27] M C B BiggsThe Optimisation of Spacecraft Orbital Manoeu-vres Part II Using Pontryaginrsquos Maximum Principle The Hat-field Polytechnic Numerical Optimization Centre 1979

[28] A F B A Prado Analise Selecao e Implementacao de Procedi-mentos que visemManobras Otimas de Satelites Artificiais [MSthesis] Instituto Nacional de Pesquisas Espaciais Sao Jose dosCampos 1988

[29] T C Oliveira Estrategias otimas para manobras orbitais uti-lizando propulsao contınua [MS thesis] Instituto Nacional dePesquisas Espaciais Sao Jose dos Campos 2012

[30] J L Ferreira G C Possa B SMoraes et al ldquoPermanentmagnethall thruster for satellite orbit maneuvering with low powerrdquoAdvances in Space Research vol 01 pp 166ndash182 2009

[31] E Stuhlinger Ion Propulsion for Space FlightMcGraw-Hill NewYork NY USA 1st edition 1964

[32] R G Jahn Physics of Electric Propulsion McGraw-Hill NewYork NY USA 1st edition 1968

[33] D G Luemberger Linear and Non-Linear ProgrammingSpringer Science amp Business Media New York NY USA 3rdedition 2008

[34] M S Bazaraa and C M ShettyNonlinear Programming-Theoryand Algorithms JohnWilley amp Sons New York NY USA 1979

[35] V V Zhurin H R Kaufman and R S Robinson ldquoPhysics ofclosed drift thrustersrdquo Plasma Sources Science and Technologyvol 8 no 1 pp R1ndashR20 1999

[36] E M Rocco ldquoPerturbed orbital motion with a PID controlsystem for the trajectoryrdquo in Proceedings of the 14th ColoquioBrasileiro De Dinamica Orbital Aguas de Lindoia AnaisUNESP Guaratingueta Brazil November 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[3] V V Beletsky and V A Egorov ldquoInterplanetary flights withconstant output enginesrdquoCosmic Research vol 2 no 3 pp 303ndash330 1964

[4] A A Sukhanov ldquoOptimization of flights with low thrustrdquoCosmic Research vol 37 no 2 pp 182ndash191 1999

[5] A A Sukhanov ldquoOptimization of low-thrust interplanetarytransfersrdquo Cosmic Research vol 38 no 6 pp 584ndash587 2000

[6] A A Sukhanov and A F B A Prado ldquoA modification of themethod of transporting trajectoryrdquoCosmic Research vol 42 no1 pp 107ndash112 2004

[7] A A Sukhanov and A F B A Prado ldquoOptimization of lowthrust transfers in the three body problemrdquo Cosmic Researchvol 46 no 5 pp 440ndash451 2008

[8] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction Irdquo Cosmic Researchvol 45 no 5 pp 417ndash423 2007

[9] A A Sukhanov andA F B A Prado ldquoOptimization of transfersunder constraints on the thrust direction IIrdquo Cosmic Researchvol 46 no 1 pp 49ndash59 2008

[10] S S Fernandes and F D C Carvalho ldquoA first-order analyticaltheory for optimal low-thrust limited-power transfers betweenarbitrary elliptical coplanar orbitsrdquo Mathematical Problems inEngineering vol 2008 Article ID 525930 30 pages 2008

[11] R A Boucher ldquoElectrical propulsion for control of stationarysatellitesrdquo Journal of Spacecraft and Rockets vol 1 no 2 pp 164ndash169 1964

[12] D N Bowditch ldquoApplication of magnetic-expansion plasmathrusters to satellite station keeping and attitude control mis-sionsrdquo in Proceedings of the 4th Electric Propulsion Confer-ence AIAA-1964-677 pp 64ndash677 The American Institute ofAeronautics and Astronautics Philadelphia Pa USA Augest-September 1964

[13] S R Oleson R M Myers C A Kluever J P Riehl and F MCurran ldquoAdvanced propulsion for geostationary orbit insertionand north-south station keepingrdquo Journal of Spacecraft andRockets vol 34 no 1 pp 22ndash28 1997

[14] T A Ely and K C Howell ldquoEast-west stationkeeping of satelliteorbits with resonant tesseral harmonicsrdquoActa Astronautica vol46 no 1 pp 1ndash15 2000

[15] C Circi ldquoSimple strategy for geostationary stationkeepingmaneuvers using solar sailrdquo Journal of Guidance Control andDynamics vol 28 no 2 pp 249ndash253 2005

[16] P Romero J M Gambi and E Patino ldquoStationkeepingmanoeuvres for geostationary satellites using feedback controltechniquesrdquo Aerospace Science and Technology vol 11 no 2-3pp 229ndash237 2007

[17] V Gomes and A F B A Prado ldquoLow-thrust out-of-planeorbital station-keeping maneuvers for satellitesrdquo MathematicalProblems in Engineering vol 2012 Article ID 532708 14 pages2012

[18] B L Pierson and C A Kluever ldquoThree-stage approach to opti-mal low-thrust earth-moon trajectoriesrdquo Journal of GuidanceControl and Dynamics vol 17 no 6 pp 1275ndash1282 1994

[19] Y J Song S Y Park K H Choi and E S Sim ldquoA lunar cargomission design strategy using variable low thrustrdquo Advances inSpace Research vol 43 no 9 pp 1391ndash1406 2009

[20] S A Fazelzadeh and G A Varzandian ldquoMinimum-time earth-moon and moon-earth orbital maneuvers using time-domainfinite element methodrdquo Acta Astronautica vol 66 no 3-4 pp528ndash538 2010

[21] G Mingotti F Topputo and F Bernelli-Zazzera ldquoEfficientinvariant-manifold low-thrust planar trajectories to themoonrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 p 817 2012

[22] J R Brophy and M Noca ldquoElectric propulsion for solar systemexplorationrdquo Journal of Propulsion and Power vol 14 no 5 pp700ndash707 1998

[23] D P S Santos A F B A Prado L Casalino and G ColasurdoldquoOptimal trajectories towards near-earth-objects using solarelectric propulsion (sep) and gravity assistedmaneuverrdquo Journalof Aerospace Engineering Sciences and Applications vol 1 no 2pp 51ndash64 2008

[24] D P S Santos L Casalino G Colasurdo and A F B APrado ldquoOptimal trajectories towards near-earth-objects usingsolar electric propulsion (sep) and gravity assisted maneuverrdquoWSEAS Transactions on Applied andTheoretical Mechanics vol4 no 3 pp 125ndash135 2009

[25] D P S Santos and A F B A Prado ldquoOptimal low-thrusttrajectories to reach the asteroid apophisrdquo WSEAS MechanicalEngineering Series vol 7 pp 241ndash251 2012

[26] M C B BiggsThe Optimization of Spacecraft Orbital Manoeu-vres Part I Linearly Varying Thrust Angles The HatfieldPolytechnic Numerical Optimization Centre 1978

[27] M C B BiggsThe Optimisation of Spacecraft Orbital Manoeu-vres Part II Using Pontryaginrsquos Maximum Principle The Hat-field Polytechnic Numerical Optimization Centre 1979

[28] A F B A Prado Analise Selecao e Implementacao de Procedi-mentos que visemManobras Otimas de Satelites Artificiais [MSthesis] Instituto Nacional de Pesquisas Espaciais Sao Jose dosCampos 1988

[29] T C Oliveira Estrategias otimas para manobras orbitais uti-lizando propulsao contınua [MS thesis] Instituto Nacional dePesquisas Espaciais Sao Jose dos Campos 2012

[30] J L Ferreira G C Possa B SMoraes et al ldquoPermanentmagnethall thruster for satellite orbit maneuvering with low powerrdquoAdvances in Space Research vol 01 pp 166ndash182 2009

[31] E Stuhlinger Ion Propulsion for Space FlightMcGraw-Hill NewYork NY USA 1st edition 1964

[32] R G Jahn Physics of Electric Propulsion McGraw-Hill NewYork NY USA 1st edition 1968

[33] D G Luemberger Linear and Non-Linear ProgrammingSpringer Science amp Business Media New York NY USA 3rdedition 2008

[34] M S Bazaraa and C M ShettyNonlinear Programming-Theoryand Algorithms JohnWilley amp Sons New York NY USA 1979

[35] V V Zhurin H R Kaufman and R S Robinson ldquoPhysics ofclosed drift thrustersrdquo Plasma Sources Science and Technologyvol 8 no 1 pp R1ndashR20 1999

[36] E M Rocco ldquoPerturbed orbital motion with a PID controlsystem for the trajectoryrdquo in Proceedings of the 14th ColoquioBrasileiro De Dinamica Orbital Aguas de Lindoia AnaisUNESP Guaratingueta Brazil November 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of