ANovelConceptforGuidanceandControlof Spacecraft Orbital ... · Chaser-Target Line-Of-Sight (LOS),...

Research ArticleA Novel Concept for Guidance and Control ofSpacecraft Orbital Maneuvers

Matteo Dentis,1 Elisa Capello,2 and Giorgio Guglieri1

1Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy2Department of Mechanical and Aerospace Engineering, CNR-IEIIT, Politecnico di Torino, Corso Duca degli Abruzzi 24,10129 Torino, Italy

Correspondence should be addressed to Elisa Capello; [email protected]

Received 28 July 2016; Revised 21 September 2016; Accepted 16 October 2016

Academic Editor: Christian Circi

Copyright © 2016Matteo Dentis et al.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The purpose of this paper is the design of guidance and control algorithms for orbital space maneuvers. A 6-dof orbital simulator,based onClohessy-Wiltshire-Hill equations, is developed in C language, considering cold gas reaction thrusters and reactionwheelsas actuation system. The computational limitations of on-board computers are also included. A combination of guidance andcontrol algorithms for an orbital maneuver is proposed: (i) a suitably designed Zero-Effort-Miss/Zero-Effort-Velocity (ZEM/ZEV)algorithm is adopted for the guidance and (ii) a linear quadratic regulator (LQR) is used for the attitude control. The proposedapproach is verified for different cases, including external environment disturbances and errors on the actuation system.

1. Introduction

Space systems often need to be controlled by actuators withlimited output level, enforcing strict requirements in terms ofrelative position and attitude. In the past years, autonomousspacecraft rendezvous and docking (RVD) maneuvers havebeen extensively studied, in order to obtain controlled tra-jectories during which the Chaser (active) tries to docka passive Target spacecraft. The goal of this mission is tosafely and efficiently approach the Target vehicle to within afew centimeters (surface-to-surface), following a predefinedpath, which is generated by the guidance algorithm. Inthese maneuvers, the control system has to maintain thestrict requirements, despite the limitations imposed in theactuation system and the external environment disturbances.

In order to overcome inherent difficulties to achievethe desired objectives, it can be seen as a cooperation ofseveral complex subsystems, with their own dynamics. In thispaper, a combination of guidance and control algorithms isproposed, to obtain a potentially flight compliant algorithmin which some on-board limitations and errors/disturbancesare taken into account.This research is based on the previous

works of [1, 2], in which a GNC system of a ground test-bed for spacecraft rendezvous and docking experiments isdesigned.The novelty of this approach is the design and vali-dation of a suitably designed guidance system based on Zero-Effort-Miss/Zero-Effort-Velocity (ZEM/ZEV) theory for acomplete orbital maneuver, reducing the fuel consumption.

Guidance algorithms should be divided into two classes:(i) a predictive guidance scheme and (ii) a feedback-basedguidance law, which also uses on-off pulses (see Algorithm 1).In the first class, the following guidance laws are investigated:(i) the Lambert guidance [3] and (ii) time-varying statetransition matrix (STM) guidance [4]. Other guidance laws,based on the same theory, have been derived to follow definedpath to the Target, for example [5], or to intercept an asteroidwith terminal velocity direction constraints [6]. Consideringthe results obtained in [3], in which a simple 3DOF simulatoris considered for the validation of the proposed guidance, itis demonstrated that the computational effort is too high forthe implementation on-board. For this reason, in this paper,the second class of guidance laws is considered, in whichthe well-known Proportional Navigation (PN) algorithmand the ZEM/ZEV theory are included. The PN law issues

Hindawi Publishing Corporation

International Journal of Aerospace Engineering

Volume 2016, Article ID 7695257, 14 pages

http://dx.doi.org/10.1155/2016/7695257

2 International Journal of Aerospace Engineering

[Waypoint Initialization](i) set !0 position with current position(ii) set !1, !2 and !3 positions as Table 1(iii) start tracking Target position and set !4 position(iv) compute Δ#!1!2[Initialization and Coefficients evaluation](i) set $0,man = $ s, Δ$ = 1Hzif !1-!2compute Δ%",!1!2

if !2-!3compute Δ%#,!2!3

(ii) compute &1,'1,(1,)1,&2,'2,&3,'3,(3 according to (19) with initial conditions according to the current !$-!$+1 phaseEnd.[Initial Acceleration]if !0-!1not accelerate

if !1-!2while (+̇%,$ < +̇0,!1!2) accelerate to +̇0,!1!2

if !2-!3while (-̇%,$ < -̇0,!2!3) accelerate to -̇0,!2!3

if !3-!4while (+̇%,$ < 0.15) accelerate to 0.15m/s

after acceleration switch to [ZEM/ZEV Guidance][ZEM/ZEV Guidance]if !0-!1(i) null acceleration command: . = [0, 0, 0]&

if !1-!2 or !2-!3(i) compute &1','1',(1',)1',&2','2',&3','3',(3' by (19), setting the initial conditions as the actual position and velocity(ii) compute ZEM and ZEV according to (16)-(17)-(18)-(20)-(21)(iii) compute command acceleration according to (15)(iv) if !2 or !3 are approaching goto [Stationkeeping]

if !3-!4(i) compute &1','1',(1',)1',&2','2',&3','3',(3' by (19), setting the initial conditions as the actual position and velocity(ii) compute 0̃( and Ṽ( according to (20)-(21)(iii) set 0( = [−500 + +0,!3!4($ − $0,man,!3!4), 0, 0]&(iv) set V( = [0.15, 0, 0]&(v) compute ZEM and ZEV by (16)(vi) compute command acceleration by (15)(vii) if +act < −4

stop controlling thrust on +-axis: . = [0,ZEM),ZEM#]if +act < −1stop controlling thrust: . = [0, 0, 0]

End.[Stationkeeping]if !1-!2(i) break to maintain position within a 10 × 10m box around !2 and velocity lower than 10 cm/s (three axes)

if !2-!3(i) break to maintain position within a 5 × 5m box around !3 and velocity lower than 5 cm/s (three axes)

when Stationkeeping is reached goto [ZEM/ZEV Guidance]End.

Algorithm 1: ZEM/ZEV Guidance Algorithm.

acceleration commands, perpendicular to the instantaneousChaser-Target Line-Of-Sight (LOS), which are proportionalto the LOS rate and closing velocity [7].

Bryson and Ho [6] also discussed optimal feedbackcontrol laws for a simple rendezvous problem and therelationship between optimal feedback control and PNguidance algorithm. Ebrahimi et al. [8] proposed the new

concept of the Zero-Effort-Velocity (ZEV) error, analogousto the Zero-Effort-Miss (ZEM) distance. Both these feedbackguidance laws can be implemented with on-off thrustersusing a simple Schmitt trigger or other pulse-modulationdevices as described in [9]. As clearly explained in [10],these algorithms are usually used for asteroid intercept andrendezvous missions, even if different mission requirements

International Journal of Aerospace Engineering 3

Table 1: Waypoint LVLH position.

Waypoint Description Position [m][+,3, -]LVLH!0 Initial simulation waypoint [−12000, 0, 3000]!1 Initial waypoint for Hohmann [−12000 − Δ#!1!2, 0, 3000]!2 Terminal waypoint for Hohmann/initial waypoint for Radial Boost [−3000, 0, 0]!3 Terminal waypoint for radial boost/initial waypoint for straight line [−500, 0, 0]!4 Terminal position [+target,3target, -target]and spacecraft capabilities require continued research onterminal-phase guidance laws. Motivated by this issue, theZEM/ZEV algorithm is selected and deeply analyzed. Thisalgorithm evaluates directly the control commands in termsof accelerations starting from the positions of the Chaser,measurable at each time step. In detail, as said before, theidea is to evaluate the performance of the chosen algorithm,combining it with a linear quadratic regulator (LQR) forthe attitude dynamics. Extensive theory about LQR can beeasily found in the literature and application of LQR forattitude control of spacecraft [11–14]. LQR controllers are veryeffective in controlling linear systems, but, applying a modellinearization, LQR can be also used in controlling nonlinearsystems, as in the case of spacecraft attitude dynamics. SinceLQR belongs to the class of optimal controller, attitudecontrol could be optimized to reach and maintain its goalorientation with minimum time or energy used more easilythan PID controller, even though this aspect is not covered bythe scope of this paper.

The advantage of this proposed approach lies in itssimplicity, and, in our case, the information in terms ofvelocity and position of the target are taken into account,despite the classical implementation. Using a short termprediction of terminal condition instead of the classicalterminal conditions of the complete maneuver guarantees amore precise trajectory control and earlier trajectory errorrecovery. Moreover, a reduction of the fuel consumptionis guaranteed, in particular on curvilinear maneuvers likeHohmann transfer.

The paper is organized as follows. In Section 2, a detailedspacecraft model is analyzed, including actuator models,position, and attitude dynamics.The proposed GNC strategyis developed in Section 3. The simulation results are inSection 4. Conclusions are drawn in Section 5.

2. Spacecraft Model

The spacecraftmodel, analyzed in this work, is a 6-degree-of-freedom (dof)model in which actuator errors, nonlinearities,and external disturbances are taken into account. As dis-cussed in the Introduction, a complete rendezvous maneuveris considered (see Figure 1), in which a Chaser vehicle has toreach the Target spacecraft, initially moving along a differentaltitude orbit [15]. A detailed description of maneuver phasesand waypoint positions will be provided in Section 4.

Chaser

Target S3 S2

S0S1

S4Rbar

Vbar

Vbar

Figure 1: Orbital maneuver.

Earth

Docking axis

Orbit

T

C

+Vbar

Figure 2: LVLH frame definition.

2.1. Position Mathematical Model. For circular orbits, thetranslational dynamics is written in local-vertical local-horizontal (LVLH) frame (see Figure 2).The +-axis %bar is inthe direction of the orbital velocity vector and the 3-axis4baris in the opposite direction of the angular momentum vectorof the orbit, while the --axis 5bar is radial from the spacecraftcenter of mass to the Earth center of mass.


Clohessy-Wiltshire-Hill equations [15] are implementedto describe the relative motion of the two bodies in neighbor-ing orbits: +̈ = 7"8* + 290-̇,3̈ = 7)8* − 9203,-̈ = 7#8* − 290+̇ + 3920-,

(1)

where + = [+,3, -]& ∈ R3 is the position vector,8* ∈ R is theChaser mass (known and varying with time), 90 = √


!on

!off

!

t0 t

Tmax

Figure 4:Thrust provided by the Qth thruster switched on at time $0.magnitude. As will be detailed in the following, in eachrequired control direction a pair of actuators is used andthese thrusters coupled by direction are always switchedon together. This precise choice of design for the actuationsystem guarantees a nominal zero moment due to thethrusters in the ideal case, when no thrusters errors occur.Each thruster is characterized by a fixed output and theindividual thruster can provide either the maximum amountof thrust when switched on or no force when switched off.

Further characteristics of each thruster are given by thetime duration of the pulse width Ron$ and the thruster zerotime before turning on again Roff $. In fact, if the controllerswitches on the Qth thruster at time $0, this actuator providesthe maximum thrust Smax$ for a time Ron$ and then the samethruster cannot be turned on again for a time Roff $, as shownin Figure 4 and according to the following:

S+$ = {{{Smax$, if $ ∈ ($0, $0 + Ron$) ,0, if $ ∈ ($0 + Ron$, $0 + Ron$ + Roff $) , (9)where Ron$ and Roff $ are known and constant for each thruster.The constant R$ = Ron$ + Roff $ can be computed and its inverseconstitutes the maximum allowed frequency at which the Qththruster can be switched on.

To model the different sources of uncertainty, the totalthruster force of the Qth thruster is described as7thr! = W$Smag!Xthr! , Q = 1, . . . ,Ythr, (10)where Ythr is the total number of thrusters. The vector W ∈R2thr is a Boolean vector related to the thruster switchingon/off. The magnitude Smag! ∈ R of the force applied bythe Qth thruster is modeled considering the maximum thrustand a magnitude error applied by each thruster. The vectorXthr! ∈ R3 is the vector representing the shoot direction ofthe Qth thruster: that is,Xthr! = X0thr! + ΔXthr! , (11)where X0thr! ∈ R3 is the vector of the nominal directionof thrust for each individual thruster and ΔXthr! ∈ R3 is

the shooting direction error due to nonperfect assembly.Thedesign of both shooting and magnitude thruster errors isbased on the theory of [17] and it includes bias and randomcomponents.

The resulting total force applied by thrusters is given by

7thr = 2thr∑$=17thr! . (12)The total moment due to the thrusters is evaluated asDthr = 2thr∑$=17thr! × ℓ$, (13)where ℓ$ ∈ R3 is the Qth thruster position vector for Q =1, . . . ,Ythr.The vector ℓ$ changes during themaneuver, due tothe variation of the spacecraftCoM position, and it is definedin terms of +,3, - position with respect to the CoM in bodyreference frame. To model this variation, we writeℓ$ = ℓ0$ + Δℓ$, (14)where ℓ0$ ∈ R3 represents the vector of initial position of thethrusters and Δℓ$ ∈ R3 is due to CoM position variation.

See Figure 5 for the thruster distribution on the spacecraftand for the scheme of thruster positions.

The attitude control is based on three axial reactionwheels with inertia CRW, driven by electric motors poweredby the spacecraft electrical power supply. They are managedand controlled by the on-board attitude control computer. Areaction wheel actuator produces a moment DRW, causingits angular momentum to increase. A saturation on themaximum-minimum torque assigned by the RWs is alsoincluded. The angular velocities of the reaction wheels 9RWand the momentDRW are calculated in body reference frame(described in Section 2.2).

3. Guidance, Navigation, andControl Algorithms

Proximity operations and docking require extremely deli-cate and precise translational and rotational maneuvering.During the orbital maneuver and, in particular, in the finalapproach of the proximity operations phase, the relativeposition, velocity, attitude, and angular rates between thetwo spacecrafts must be precisely controlled in order toobtain the required docking interface conditions. Thus, wepropose an on-board Guidance, Navigation, and Control(GNC) concept to perform various rendezvous functionsincluding translational and rotational control, targeting, andrelative navigation, automatically andwith extreme precision.As first approximation, we assume that an extended Kalmanfilter is implemented that filters all the sensor noise and allthe measurement disturbances. The role of each element ofthe GNC system is

(i) to receive and elaborate signals from sensors (Naviga-tion),


Forces Thrusters

X

Y

Z

2X2Y

2Z

1X

1Y

1Z

3X3Y

3Z4X

4Y4Z

2X2Y

2Z

1Y

1Z

1X 3X

3Z

4X

4Z

+X−X+Y−Y+Z−Z

4Y3Y(CoG)c

Figure 5:Thrust provided by the Qth thruster switched on at time $0.(ii) to compute the required acceleration or force needed

to change, if required, the actual orbit and to follow apredefined path (Guidance),

(iii) to control the actuators enforcing the correct trajec-tory of the Chaser spacecraft pointing to the Targetvehicle (Control).

In the next section each function of the GNC software,developed in the present work, is deeply described.

3.1. Guidance Algorithm. The guidance function can be con-ceptualized as the trajectory generator. If a simple example isconsidered, the guidance law defines the trajectory that thespacecraft has to follow to reach the desired final position.The selected algorithm is theZEM/ZEV, extensively discussedin [10, 18], and it corresponds to an optimal control only inthe presence of uniform gravitational field, as in the case ofLowEarthOrbit environment.The general formulation of theZEM/ZEV control law is. = 6$2goZEM − 2$goZEV, (15)where . ∈ R3 is the required control acceleration, $go = $( − $is the time-to-go, and $( is the final maneuver time.The time$go represents a parameter related to the expected time, themaneuver is concluded, and it is chosen a priori.The time $ isthe elapsing simulation time.

Expanding the ZEM and ZEV terms we obtain

ZEM = 0( − 0̃(,ZEV = V( − Ṽ(, (16)

where 0( ∈ R3 and V( ∈ R3 are the desired final position andvelocity and 0̃( ∈ R3 and Ṽ( ∈ R3 are the predicted positionand velocity at $ = $go.

Choosing properly the final maneuver time $( is not atrivial task; indeed, a bad choice of $( could cause extremelyhigh values of the command acceleration, since 1/$go tendsto infinity as the time $ increases. To fix this behaviour it ispossible to set a limiting value to $go, once it becomes smallerthan a specific limit value.

The approach proposed in the present paper is as follows.Instead of computing the final position and velocity vectorsrequired as terminal conditions of the entire maneuver,the ZEM/ZEV terms in (16) are implemented as positionsand velocities that the spacecraft should have while it isfollowing an ideal maneuver. As a consequence, 0( and V(are the position and velocity vectors propagated forewordby a defined and constant Δ$ using the Clohessy-Wiltshireequations [15]. Considering 0( = [0(", 0(), 0(#]& and V( =[V(", V(), V(#]&, the new required final position is0(" = &1 sin (90Rpred) − '1 cos (90Rpred) + (1Rpred+)1,0() = &2 cos (90Rpred) + '2 sin (90Rpred) ,0(# = &3 cos (90Rpred) + '3 sin (90Rpred) + (3,

(17)

and, computing the time derivative, the new required finalvelocity is obtained

V(" = &190 cos (90Rpred) + '190 sin (90Rpred) + (1,V() = −&290 sin (90Rpred) + '290 cos (90Rpred) ,V(# = −&390 sin (90Rpred) + '390 cos (90Rpred) . (18)


The coefficients of (17)-(18) can be easily computed as&1 = 4+̇090 − 6-0,'1 = 2-̇090 ,(1 = 690-0 − 3+̇0,)1 = +0 + 2-̇090 ,&2 = 30,'2 = 3̇090 ,&3 = 2+̇090 − 3-0,'3 = -̇090 ,(3 = 4-0 − 2+̇090 ,

(19)

where +0 = [+0,30, -0]& ∈ R3 is the initial position vectorof the spacecraft and +̇0 = [+̇0, 3̇0, -̇0]& ∈ R3 is the initiallinear velocity vector. Note that Rpred = ($ − $0,man) + Δ$ is theelapsed time from the beginning of themaneuver ($0,man) andΔ$ is the prediction horizon.

The ideal trajectory and speed profiles are computedfrom (17) to (18). To compute the predicted position andvelocity, respectively, 0̃( and Ṽ(, Clohessy-Wiltshire equationsare propagated over Δ$, so we have0̃(" = &1' sin (90Δ$) − '1' cos (90Δ$) + (1'Δ$+)1',0̃() = &2' cos (90Δ$) + '2' sin (90Δ$) ,0̃(# = &3' cos (90Δ$) + '3' sin (90Δ$) + (3',

(20)

Ṽ(" = &1' cos (90Δ$) + '1' sin (90Δ$) + (1',Ṽ() = −&2' sin (90Δ$) + '2' cos (90Δ$) ,Ṽ(# = −&3' sin (90Δ$) + '3' cos (90Δ$) , (21)

where all the coefficients are computed by (19) using theactual position and velocity as initial conditions.

In order to generate the ideal trajectory that the Chasershall follow for the phase of rendezvous and docking maneu-ver, Clohessy-Wiltshire equations have to be propagatedforward with time from a starting time $0 to a finish time$( which varies according to the specific maneuver. In theideal case, an impulsive maneuver of orbit altitude raisingends after half an orbital period, while a continuous thrustmaneuver (with the same initial and terminal orbit altitude)ends after one orbital period. As depicted in Figure 1 the

whole maneuver is composed of different phases (Hohmann,R-bar pulse (radial boost), etc.). This means that the initialconditions of the generated ideal maneuver have to bechanged when a different phase is considered [15]. As a con-sequence, the GNC software has to compute the coefficientsof (19) using different initial conditions for each phase beforestarting running the ZEM/ZEV algorithm and to generatethe guidance law. So, $0,man is initialized at each waypoint !$after the station keeping conditions are reached. Finally, it ispossible to generate the trajectory and the velocity profile thatthe Chaser should have Δ$ steps forward the current elapsedtime $ − $0,man from the beginning of the maneuver.

To compute ZEM and ZEV errors, it is required toestimate the predicted terminal conditions, which, in thisproposed approach, are the position 0̃( and the velocity Ṽ(computed forward in time by Δ$. In this case, differentlyfrom the generation of the ideal trajectory, initial conditionsused to compute coefficients of (19) are position and velocityvectors of the Chaser at the current simulation time $.Thus, it is possible to predict positions and velocities ofthe spacecraft after Δ$ in free dynamics (no control actionand disturbances are applied). The coefficients of (20) arecomputed continuously during each phase of the simulation.

3.2. Attitude Control. Attitude control is performed by bothactivating reaction wheels and reaction control thrusters orReaction Control System (RCS). Usually, fine and continuousattitude control is executed by RW, while the use of RCS,due to its impulsive behaviour, is required when the controltorque is high (compared to the maximum torque generatedRW) or to desaturate RW.

Our idea, to satisfy pointing requirements and con-straints, is to implement a LQR control system. For thedefinition of the LQR gains, a standard continuous time-invariant state-space formulation is evaluated:+̇ = &+ + '^, (22)where & ∈ R6,6 is the state matrix, ' ∈ R6,3 isthe control matrix, and the state is defined by + =[F1, F2, F3,91,92,93]& ∈ R6, with F1, F2, and F3 being thevectors of the first three components of the vector F (definedin Section 2.3). 91,92,93 are, respectively, the +, 3, and -components of the angular speed vector 90 of the spacecraft.The control vector ^ ∈ R3 is the control torque. Note thatFrel and 9rel are relative quaternion and angular rates writtenin LVLH frame. Since the attitude dynamics model of thespacecraft is computed relative to the ECI frame, Frel and 9relhave to be derived in the inertial frame.The angular speed ofthe spacecraft relative to the ECI frame is9-30 = 9-34 + 9-40, (23)where 9-34 is the angular speed of the LVLH frame relativeto ECI and 9-40 is the angular speed of the spacecraftwith respect to LVLH frame. Rewriting the equation andexpressing the rotation matrix from LVLH to body frame 5-.we obtain 9-40 = 9-30 − 5-. (F) 9.34, (24)


where, under the assumption of circular orbit, angular veloc-ity of the LVLH frame is9.34 = [0,90, 0]& , (25)where90 is the angular velocity of the LVLH frame, as definedin Section 3.1.The rotation matrix 5-.(F) can be defined usingthe quaternion Frel5-. (F)= [[[[F20 + F21 − F22 − F33 2 (F1F2 + F0F3) 2 (F1F3 − F0F2)2 (F1F2 − F0F3) F20 − F21 + F22 − F33 2 (F2F3 + F0F1)2 (F1F3 + F0F2) 2 (F2F3 − F0F1) F20 − F21 − F22 + F33]]]] .

(26)

In order to design the LQR controller, a linearization of thedynamic equations has to be applied. Bryson’s rule [19] isconsidered for the evaluation of the controller gains. Thedesired quaternion Frelid is thenFrelid = [1, 0, 0, 0]& (27)and the resulting rotation matrix is

5-. (Frelid) = [[[1 2F3 −2F2−2F3 1 2F12F2 −2F1 1 ]]] . (28)

The linearized angular velocities 9.34 and 9-40 are9.34 = [−2F390,−90, 2F190]& , (29)9-40 = 9-30 − [[[

−2F390−902F190 ]]] , (30)with time derivative

9̇-40 = 9̇-30 − [[[−2Ḟ39002Ḟ190 ]]] . (31)

The time derivative of the relative quaternion is

Ḟrelid = 12 [[[[[[0 −91 −92 9391 0 93 −9292 −93 0 9193 92 −91 0

]]]]]][[[[[[F0F1F2F3]]]]]] (32)

and, after linearization, the result isḞrel = 12 [91,92,93, 0]& . (33)

The linearized dynamic model is obtained substituting(30) and (31) in (4).Then, it is possible to use the state-spacerepresentation (22) defining & ∈ R6,6 and ' ∈ R6,3 matrix by&14 = &25 = &36 = 12 ,&41 = −2920 (C)) − C##C"" ) ,&46 = −(C)) − C##C"" + 1)90,&63 = −2920 (C"" − C))C## ) ,&64 = −(1 + C"" − C))C## )90,'41 = 1C"" ,'52 = 1C)) ,'63 = 1C##

(34)

in which gyroscopic torque due to RW is neglected due tolimitation in software implementation.The other elements ofthe state and control matrices are zero.The design of the LQRcontroller consists in generating a control torque equal to^ = −c+, (35)where + is the state and c = 5−1'&d is the static gainobtained by the solution d of the associated Algebraic RiccatiEquation (ARE):d& + &&d − d'5−1'&d + e = 0, (36)where e and 5 are weighting matrix of suitable dimensions.4. Results

The simulation model is tested for a generic Chaser-Targetcombination involved in sequential flight phases, as in Fig-ure 1. The Chaser is considered in stable initial conditionsalong an orbit of height ℎ = 500 km and the Target center ofmass is located at the origin of the LVLH reference frame, andit is 12 km far from theChaser. A cubic-shapeChaser (1.2m) isconsidered with an initial mass of 600 kg.The layout adoptedfor thrusters is as in Figure 5, as discussed in Section 2.4.Theparameters and the initial conditions are selected accordingto [1] and to the approaching maneuver examples in [15],including safety requirements.

Position of waypoints relative to LVLH frame is summa-rized in Table 1, while the complete rendezvous and dockingmaneuver is divided into four leading phases as summarizedin Table 2.


qBHq 0

2000 4000 6000 8000 10000 120000Time (s)

012

×10−7

2000 4000 6000 8000 10000 120000Time (s)

−101

q 1

×10−12

−101

q 2

2000 4000 6000 8000 10000 120000Time (s)

×10−5

−202

q 3

2000 4000 6000 8000 10000 120000Time (s)

(a) Relative quaternion 5rel

×10−8

2000 4000 6000 8000 10000 120000Time (s)

−5

0

5

×10−13

−1

0

1

2000 4000 6000 8000 10000 120000Time (s)

×10−4

2000 4000 6000 8000 10000 120000Time (s)

−2

0

2

$ X$ Y

$ Z

$rel

(b) Relative angular velocity 6relMGNC req

×10−4

2000 4000 6000 8000 10000 120000Time (s)

−202

×10−11

−202

2000 4000 6000 8000 10000 120000Time (s)

−0.010

0.01

2000 4000 6000 8000 10000 120000Time (s)

MZ

MY

MX

(c) RW torque command

−10000 −8000 −6000 −4000 −2000 0 2000−12000−500

0

500

1000

1500

2000

2500

3000Vbar/Rbar trajectory

R bar

(m)

Vbar (m)

(d) 7bar-8bar trajectoryHILL positions (m)

×104

2000 4000 6000 8000 10000 120000Time (s)

×10−3

2000 4000 6000 8000 10000 120000Time (s)

2000 4000 6000 8000 10000 120000Time (s)

ZH

ILL

−5000

0

5000

−5

0

5

Y HIL

L

−2

0

2

XH

ILL

(e) (",), #)LVLH versus time

×10−5

−2

0

2

2000 4000 6000 8000 10000 120000Time (s)

−2

0

2

2000 4000 6000 8000 10000 120000Time (s)

HILL velocities (m/s)

Vfree drift

0 2000 4000 6000 8000 10000 12000Time (s)

−10

0

10

V HIL

LZ

V HIL

LY

V HIL

LX

VX HILL

(f) ("̇, )̇, #̇)LVLH versus timeFigure 6: Continued.


−10

0

10

2000 4000 6000 8000 10000 120000Time (s)

×10−4

−2

0

2

2000 4000 6000 8000 10000 120000Time (s)

2000 4000 6000 8000 10000 120000Time (s)

−10

0

10

F ZF Y

F X

(g) Thrust force in LVLH frame

−400 −300 −200 −100−500 0−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8


R bar

(m)

Vbar (m)

(h) Final approach 7bar-8bar highlightFigure 6: Ideal case results: controller frequency 100Hz.

Table 2: Rendezvous and docking maneuver phase description.

Phase Name Initial condition Units!0-!1 Free drift +0 = −12, 30 = 0, -0 = 3 [km]+̇0 = 3290-0, 3̇0 = -̇0 = 0 [m/s]!1-!2 Hohmann +0 = −3 − Δ#!1!2, 30 = 0, -0 = 3 [km]+̇0 = 3290-0 + Δ%",!1!2, 3̇0 = -̇0 = 0 [m/s]!2-!3 Radial boost +0 = −3, 30 = 0, -0 = 0 [km]+̇0 = 3̇0 = 0, -̇0 = Δ%#,!2!3 [m/s]!3-!4 Straight line +0 = −500, 30 = 0, -0 = 0 [m]+̇0 = 0.15, 3̇0 = 0, -̇0 = 0 [m/s]Note that, with the exception of the starting point of the

Hohmannmaneuver (!1), the initial points of free drift, radialboost, and straight linemaneuvers correspond to the positionof the waypoints, respectively, !0, !2, and !3.The position of!1 results from computing Δ%",!1!2 required to raise the orbitof the Chaser to the Target orbit (3 km in this case). WhenΔ%",!1!2 is known, we evaluate the distance along%bar that theChaser will cover during the maneuver, and %bar position of!1 is computed as Δ#!1!2 behind !2. Δ%",!1!2 and Δ#!1!2 arecalculated in the following form:Δ%",!1!2 = 904 -0,Δ#!1!2 = 3g90 Δ%",!1!2, (37)using the value of -0 corresponding to !1-!2. Once thedistance between two consecutive waypoints !2 and !3 isfixed, it is possible to compute the required Δ%",!2!3 tocomplete the radial boost maneuver, soΔ%#,!3!4 = 904 (!2" − !3") . (38)

Table 3: Ideal case terminal conditions.

Position errorΔ+( = 3.0 ⋅ 10−4

[m]Δ3( = 2.5 ⋅ 10−3Δ-( = −7.4 ⋅ 10−3Velocity error

Δ+̇( = 0.15[m/s]Δ3̇( = 2.5 ⋅ 10−7Δ-̇( = 2.2 ⋅ 10−3

Propellant 8' = 15.52 [kg]The following cases are analyzed:

(1) Ideal case: in the ideal case, no external disturbancesdue to the LEO environment and errors inducedby RCS are considered (see Table 3). The maneuverdescribed in Table 2 is executed completely, includingreaching station keeping conditions before switchingto the next phase, as described in the guidancealgorithm. The frequency of the guidance controlloop is 1Hz and both the attitude control and thesimulation frequency are set to 100Hz. The resultsobtained are presented in Figure 6. As shown inFigure 6(d) ideal trajectories (green and magentadashed lines) are well followed. The performanceof the control law can be evaluated by analyzingFigure 6(g): except for initial acceleration boosts, thetrajectory is controlled with few pulses along %bar(7") and 5bar (7#), with the presence of disturbancethrusts along4bar (7)) due to attitude misalignment.The two-sided chattering is due to the magnitudeand sign of the control acceleration command of (15).The deadband is managed by the PWM function:the thrusters are switched on only if the accelerationrequired by the guidance algorithm is higher than theminimum feasible acceleration pulse.


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(h) Final approach 7bar-8bar highlightFigure 7: Real case results: controller frequency 100Hz.

(2) Real case: in the second case both atmospheric dragdisturbances and RCS errors are activated. The samemaneuver as in the ideal case is analyzed. In thenext considered cases, the frequency of the guidancecontrol loop is 1Hz. We analyze two subcases:

(2.A) We set the frequency of the control loop at100Hz (as in case (1)), to evaluate the per-formance of GNC algorithms as in the idealcase but introducing leading disturbances. SeeTable 4 for the results. In Figures 7(a) and7(c) the effort required by the attitude controlto counteract disturbance torque due to thrusterrors is shown. Such misalignment causes alsoa greater effort for the guidance loop withrespect to the ideal case, mainly highlightedby the presence of pulsed thrust along 4bar inFigure 7(g).The request of a larger control acti-vation is translated into a larger consumption offuel (see Table 4). Better results are obtained interms of Δ3̇( as the control along4bar is active,instead of the ideal case, in which ZEM/ZEVerrors along 4bar are not large enough (freedynamics motion) to require the activation ofthe control law.

(2.B) The control loop frequency is 50Hz. The sim-ulation frequency is 100Hz and the guidanceloop frequency is 1Hz. As it is shown in Figures8(a), 8(c), and 8(g) the required control effortby both guidance and control is greater than thetwo previous cases, even though the Chaser isable to complete the entire maneuver even witha degradation of performance. This is due tothe “low” update frequency of the controller incomparison with the high dynamics variationoccurring during the maneuver. This is also

Table 4: Real case terminal conditions: attitude control at 100Hz.


[m]Δ3( = 7.2 ⋅ 10−4Δ-( = 7.4 ⋅ 10−3Velocity error

Δ+̇( = 0.15[m/s]Δ3̇( = 4.3 ⋅ 10−5Δ-̇( = 2.1 ⋅ 10−3

Propellant 8' = 15.73 [kg]Table 5: Real case terminal conditions: attitude control at 50Hz.


[m]Δ3( = 1.8 ⋅ 10−3Δ-( = 7.2 ⋅ 10−3Velocity error

Δ+̇( = 0.15[m/s]Δ3̇( = 1.6 ⋅ 10−4Δ-̇( = 2.2 ⋅ 10−3

Propellant 8' = 16.17 [kg]translated in a high consumption of fuel (seeTable 5) that summarizes the numerical results.

5. Conclusions

In the present work, a new approach for the development ofa GNC algorithm has been proposed.The implementation ofa modified version of ZEM/ZEV algorithm is novel as thisinnovative guidance method still finds limited applicationamong the available references. The guidance design iscombined with a classical LQR attitude control.This solutionis novel and also very effective in driving a Chaser spacecraftalong a complete rendezvous and docking maneuver, evenin presence of disturbances due to external perturbations


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and to the imperfections of the actuation system.The impactof these disturbances and modeling errors was investigatedsimulating a reference maneuver with different controllerfrequency. The results confirm that the proposed solution ispromising in terms of disturbance rejection, robustness, andperformance, providing strict compliance of terminal condi-tions with nominal reference states, as a major achievementof ZEM/ZEV algorithm implementation.

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

References

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[2] E. Capello, F. Dabbene, G. Guglieri, E. Punta, and R. Tempo,“Rendez-vous and docking position tracking via sliding modecontrol,” in Proceedings of the American Control Conference(ACC ’15), pp. 1893–1898, IEEE, Chicago, Ill, USA, July 2015.

[3] S. J. Maclellan, Orbital rendezvous using an augmented lambertguidance scheme [M.S. thesis], MIT, 2005.

[4] D. Vallado, Fundamentals of Astrodynamics and Applications,Microcosm Press, Hawthorne, Calif, USA, 2007.

[5] M.-G. Yoon, “Relative circular navigation guidance forthree-dimensional impact angle control problem,” Journal ofAerospace Engineering, vol. 23, no. 4, pp. 300–308, 2010.

[6] A. E. Bryson and Y. C. Ho, Applied Optimal Control: Optimiza-tion, Estimation, and Control, John Wiley & Sons, New York,NY, USA, 1975.

[7] P. Zarchan, Tactical and Strategic Missile Guidance, Progress inAstronautics and Aeronautics, AIAA, Washington, DC, USA,2012.

[8] B. Ebrahimi, M. Bahrami, and J. Roshanian, “Optimal sliding-mode guidance with terminal velocity constraint for fixed-interval propulsive maneuvers,” Acta Astronautica, vol. 62, no.10-11, pp. 556–562, 2008.

[9] B. Wie, Space Vehicle Dynamics and Control, AIAA EducationSeries, Reston, Va, USA, 2008.

[10] M. Hawkins, Y. Guo, and B. Wie, “Spacecraft guidance algo-rithms for asteroid intercept and rendezvous missions,” Inter-national Journal of Aeronautical and Space Sciences, vol. 13, no.2, pp. 154–169, 2012.

[11] Y. Yang, “Analytic LQR design for spacecraft control systembased on quaternion model,” Journal of Aerospace Engineering,vol. 25, no. 3, pp. 449–453, 2012.

[12] Y. Yang, “Spacecraft attitude and reaction wheel desaturationcontrol,” IEEETransactions onAerospace and Electronic Systems,vol. 53, no. 1, 2017.

[13] Y. Yang, “Quaternion based model for momentum biased nadirpointing spacecraft,” Aerospace Science and Technology, vol. 14,no. 3, pp. 199–202, 2010.

[14] N. Jovanović, Aalto-2 satellite attitude control system [M.S.thesis], Aalto University, Espoo, Finland, 2014.

[15] W. Fehse, Automated Rendezvous and Docking of Spacecraft,Cambridge University Press, New York, NY, USA, 2003.

[16] J. W. Cornelisse, H. F. R. Schoyer, and K. F. Wakker, RocketPropulsion and Spacecraft Dynamics, Pitman Publishing, Lon-don, UK, 1979.

[17] E. Wilson, D. W. Sutter, and R. W. Mah, “Motion-based mass-and thruster-property identification for thruster-controlledspacecraft,” in Proceedings of the AIAA Infotech@AerospaceConference, Arlington, Va, USA, 2005.

[18] M. Hawkins, New near-optimal feedback guidance algorithms[Ph.D. dissertation], Iowa State University, Ames, Iowa, USA,2013.

[19] G. Franklin, J. Powell, and A. Emami-Naeini, Feedback Controlof Dynamic Systems, Prentice Hall, Upper Saddle River, NJ,USA, 2002.

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