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Terminal Guidance of Autonomous

Parafoils in High Wind-To-Airspeed Ratios

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Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace

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DOI: 10.1243/09544100JAERO749

2011 225: 336Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace EngineeringN Slegers and O A Yakimenko

Terminal Guidance of Autonomous Parafoils in High Wind-To-Airspeed Ratios

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13. SUPPLEMENTARY NOTES 14. ABSTRACT Autonomous precision placement of parafoils is challenging because of their limited control authority andsensitivity to winds. In particular, when wind speed is near the airspeed guidance is further complicated bythe parafoils inability to penetrate the wind. This article specifically addresses the terminal phase anddevelops an approach for generating optimal trajectories in real-time based on the inverse dynamics in thevirtual domain. The method results in efficient solution of a two-point boundary-value problem using onlya single optimization parameter allowing the trajectory to be generated at a high rate, mitigating effects ofthe unknown winds. It is shown through simulation and experimental results that the proposed algorithmworks well even in strong winds and is robust to sensor errors and wind uncertainty.

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336

Terminal guidance of autonomous parafoilsin high wind-to-airspeed ratiosN Slegers1! and O A Yakimenko2

1Department of Mechanical and Aerospace Engineering, University of Alabama in Huntsville, Huntsville, Alabama, USA2Department of Mechanical and Astronautical Engineering, Naval Postgraduate School, Monterey, California, USA

The manuscript was received on 30 November 2009 and was accepted after revision for publication on 5 July 2010.

DOI: 10.1243/09544100JAERO749

Abstract: Autonomous precision placement of parafoils is challenging because of their limitedcontrol authority and sensitivity to winds. In particular, when wind speed is near the airspeed,guidance is further complicated by the parafoils inability to penetrate the wind. This article speci-fically addresses the terminal phase and develops an approach for generating optimal trajectoriesin real-time based on the inverse dynamics in the virtual domain. The method results in efficientsolution of a two-point boundary-value problem using only a single optimization parameterallowing the trajectory to be generated at a high rate, mitigating effects of the unknown winds.It is shown through simulation and experimental results that the proposed algorithm works welleven in strong winds and is robust to sensor errors and wind uncertainty.

Keywords: optimal control, parafoils, predictive control, trajectory optimization

1 INTRODUCTION

Precision payload delivery using guided parafoils hasextended military re-supply capability by providingaccurate and rapid response at substantial offsets witha minimal risk to the cargo delivery aircraft. Signifi-cant efforts have been underway to improve precisionairdrop with the US Department of Defense’s Joint Pre-cision Airdrop System (JPADS) programmes [1] beingan example. The JPADS self-guided programmes areseparated into five classes according to size: microlight weight (MLW) 4–70 kg, ultra light weight (ULW),100–320 kg, extra light (XL) 320–1000 kg, light (L) 2200–4500 kg, and medium (M) 6800–14 000 kg [2]. Mostexisting efforts have focused on larger systems above200 kg including; Onyx from Atair Aerospace, Pantherfrom Pioneer Aerospace, Screamer from Strong Enter-prises, and Spades from Dutch Space all demonstratedat the Precision Airdrop Technology Conference andDemonstration in 2007 [3]. Recently, there has beena new focus on smaller systems with the start of theJoint Medical Distance Support & Evacuation (JMDSE)

!Corresponding author: Department of Mechanical and Aerospace

Engineering, University of Alabama in Huntsville, Technology Hall,

Huntsville, Alabama 35899, USA.

email: [email protected]

programme in 2009.The JMDSE programme is focusedon small medical equipment delivery in the MLW andULW classes and the integration into small unmannedautonomous systems.

Autonomous parafoils for precision delivery areunique aerospace systems with challenges includ-ing limited control authority and sensitivity to winds.Parafoils typically only have lateral control usingasymmetric brake deflection, with longitudinal con-trol using symmetric deflection being very limited. Theabsence of thrust, combined with low airspeed, makeswind a significant contributor to a parafoil’s trajectory.Airspeed for parafoils can range from 6 m/s for verysmall lightly loaded systems to over 20 m/s for largersystems. This results in the common occurrence forparafoils where the wind speed may be near or exceedthe airspeed. In the case where wind speed exceedsairspeed, the parafoil cannot advance towards the tar-get if the target is upwind. This becomes increasinglylikely for parafoils in the MLW class.

Guidance strategies for parafoils vary significantlyover the spectrum of systems; however, there are somecommon aspects seen in all systems. One commonaspect is the parafoil is released upwind of the targetat a sufficient altitude to ensure that the target canbe reached. In addition, all strategies have some com-bination of homing and energy management phasesfollowed by terminal guidance. Homing refers to the

Proc. IMechE Vol. 225 Part G: J. Aerospace Engineering



Terminal guidance of autonomous parafoils in high wind-to-airspeed ratios 337

process of bringing the system from one region toanother region while energy management refers toloitering in a pattern allowing the system to reachan altitude where the parafoil can proceed to impactthe target. Assuming homing and energy managementstages bring the parafoil to a location where arriving atthe target in the presence of wind is possible, the termi-nal guidance algorithm determines the final accuracy.Homing and energy management are necessary, butnot critical, in determining final accuracy because itis relatively easy to bring the system to the target, butquite difficult to plan terminal guidance to co-ordinatearriving at the target and ground simultaneously.

Examples of simple strategies are Onyx by AtairAerospace [3, 4] and Screamer by Strong Enterprises[3] where homing takes the system to the target,energy management is a spiral around the target ata varying radius, followed by terminal guidance as astraight path near the ground. In theory, this simplestrategy limits the error to the spiral radius. How-ever, in practice the strategy is limited to the casewhere wind speed is significantly less than airspeed inorder to remain over the target and maintain guidancestability [5] making it unsuitable for high wind-to-airspeed ratios. Another limitation of the spiralingstrategy is that it cannot guarantee landing into thewind, which is preferred because it minimizes speedat impact. Several other types of guidance conceptsfor parafoils have also been developed, including opti-mal control, trajectory databases, model predictivecontrol, and direct glide slope control [6–10]. Morerecently, terminal guidance strategies using optimaltrajectories have been investigated. Carter et al. devel-oped a method for generating bandwidth limitedoptimal terminal guidance trajectories [11]. Limitingthe trajectory bandwidth results in trajectories thatare easily tracked. The trajectory was parameterizedusing five parameters with the optimization problemiteratively solved to minimize final position and head-ing error. Rademacher et al. [12] proposed a hybridapproach where optimal trajectories were found usingeither modified Dubins paths or minimum controltrajectories with multiple optimization parameters.The hybrid strategy was employed, because eachmethod had different initial conditions where nume-rical robustness was a problem or non-existence ofsolutions existed. In both references [11] and [12],kinematic parafoil models were used and the optimaltrajectories were tracked using proportional-integral-derivative (PID) controllers.

This article develops an optimal terminal guidancealgorithm that is robust in high winds even if thewind speed exceeds the parafoil’s airspeed. An opti-mal terminal trajectory is found using a kinematicmodel similar to references [11] and [12]. However,using the direct method of calculus of variations andinverse dynamics in a virtual domain, the problem isreduced to a single parameter optimization problem

[13]. The result is an extremely efficient solution that iseasily solved in real-time. In addition, since the entireterminal trajectory is solved, a model predictive con-troller (MPC) [8] is used to track the trajectory ratherthan a PID algorithm. A complete six degrees-of-freedom (DoF) model is used to evaluate the methodin simulation and demonstrate its application includ-ing parafoil turning dynamics. Finally, experimentalresults are shown where the algorithm was used on asmall 2.3 kg parafoil.

2 GUIDANCE STRATEGY

In order to ensure the parafoil system can reach adesired target it is released upwind of a target at highaltitudes. The result is typical parafoil precision place-ment algorithms having an energy management andhoming region. The proposed strategy for precisionplacement shown in Fig. 1 is divided into four phaseswith the first being energy management, the secondhoming, followed by terminal guidance comprising afinal turn and final approach (FA). Guidance geometryis defined by the target location and four parameters:an away distance, cycle distance, turn diameter, andwind heading angle.

Energy management is commenced after the sys-tem is released. The parafoil travels clockwise aroundthe rectangular region defined by the away distance,cycle distance, and turn diameter. The away distanceis selected so that in the case of wind speed exceed-ing airspeed the parafoil will still be able to reachthe target, while in energy management, the parafoilcontinually estimates descent rate and wind speed.Descent rate and wind speed estimates, combinedwith a distance from the target, are used to determinean altitude zstart where energy management is termi-nated and homing commences. In the homing phase

Fig. 1 Guidance strategy




338 N Slegers and O A Yakimenko

the system travels toward the target while continuingto estimate wind and vertical velocity. All estimateddata are used to determine a turn initiation point (TIP)at a distance Dswitch downwind of the target where ter-minal guidance will begin. Terminal guidance startswith a final turn followed by an FA into the wind.

3 TERMINAL GUIDANCE

Development of energy management and homingstrategies is flexible with the only requirement beingthat the parafoil is brought near the TIP with sufficientaltitude. In contrast, terminal guidance is the mostcritical stage of parafoil precision placement with avery strict time limitation. The parafoil can be slightlylate or earlier departing to and arriving at all otherphases, but terminal guidance ends sharply at landingrequiring special precautions to be made in buildinga control algorithm. An ideal terminal guidance tra-jectory is described in Fig. 2 where, iI, and jI , are axesof the standard North–East–Down inertial referenceframe, and iT, and jT are target axes aligned with thewind W . The parafoil is a distance L upwind of thetarget, tstart is the current time, and R is the final turnradius.

The final turn is initiated at the TIP, occurring attime t0, a distance Dswitch past the target. After theTIP is reached the final turn is defined by a com-manded yaw !(t) occurring from time t0 until the FA isreached at time t1. The resulting final turn time is thenTturn = t1 " t0. The desired FA time Tapp is t2 " t1 wheret2 is at touchdown. Transition from the final turn to theFA occurs at texit.

In terminal guidance described above, the targetlocation R, and a desired Tapp are specified, while thewind W is estimated online and the altitude z and L aremeasured. The terminal guidance problem can thenbe summarized as follows. For a parafoil in the pres-ence of wind W , at altitude z, and a distance L from

Fig. 2 Terminal guidance manoeuver

the target, find the distance Dswitch to the final TIP foran ideal impact at t2.

In general, the dynamic model of a parafoil is com-plex and non-linear, so that the problem can only besolved numerically. However, some assumptions canbe made, allowing an analytical solution to be used asa reference trajectory. These assumptions are a slowturn rate, so that the roll and sideslip angles can beignored, and nearly constant descent rate Vv and hori-zontal airspeed Vh. In this case it immediately followsthat

t2 = tstart " zstart

Vv(1)

The problem reduces to a kinematic model repre-sented by three components of the ground velocity inthe target axes

!

"xyz

#

$ =

!

""W + Vh cos !

Vh sin !

Vv

#

$ (2)

where x, y, and z are positions in the target frame. Now,the commanded turn !(t) can be chosen as any func-tion that satisfies the boundary conditions !(t0) = 0and !(t1) = "! with respect to the wind while satis-fying the constraint that the distance travelled in jT is"2R. For a constant turn rate, the solution to the finalturn time Tturn and turn rate are

! = "Vh

R(3)

Tturn = !RVh

(4)

Hence, the most straightforward algorithm to con-trol the final turn, is to track the turn rate "Vh/R.Assuming that the wind W is constant and usingthe commanded turn rate ! c = "Vh/R, integration ofinertial velocities along iT and jT from tstart to t2 yield

Dswitch +! t1

t0

x dt +! t2

t1

x dt

= Dswitch " WTturn " (Vh + W )Tapp = 0 (5)

z + Vv

%L + Dswitch

Vh " W

&+ VvTturn + VvTapp = 0 (6)

Equations (5) and (6) can then be solved for Dswitch

and Tapp resulting in

Dswitch = WTturn +%

V 2h " W 2

2Vh

&

#%"z

Vv" Tturn " L + WTturn

Vh " W

&(7)

Tapp =%

Vh " W2Vh

& %"zVv

" Tturn

&" L + WTturn

2Vh(8)

From equations (7) and (8) it can be seen that thehigher the altitude z, the larger Dswitch and Tapp become.





Fig. 3 Simulated guidance in ideal conditions (•TIP andFA W = "3.4 m/s, ! TIP and FA W = "7.7 m/s)

As the parafoil loiters upwind of the target, zstart the alti-tude at which to turn towards the target can be foundusing a desired FA time T des

app. The switching altitude toexit energy management which achieves T des

app is thengiven by solving equation (8) for z, resulting in

zstart = "Vv

%Tturn + L + WTturn

Vh " W+ 2Vh

Vh " WT des

app

&(9)

Once the system is homing towards the target thegoal is to bring the system to the point defined by x =Dswitch and y = "2R with Dswitch continually estimatedduring homing using equation (7) and current esti-mates of Vh, Vv, and W . A desired FA time T des

app is usedin equation (9) in making the decision to exit energymanagement. However, disturbances, measurementerror, and tracking error while homing will alter theideal terminal guidance resulting in an actual Tapp

being estimated during homing using equation (7).Figure 3 demonstrates trajectories starting 150 m

upwind of the target when there are no disturbancesand the commanded yaw rate is tracked preciselyfor a horizontal airspeed of 6.82 m/s, descent rate of3.05 m/s, turn radius of 37.5 m, and a desired FA timeT des

app of 7.5 s. Trajectories for two wind magnitudes areshown, "3.4 and 7.7 m/s, resulting in the wind speedbeing 50 per cent and 110 per cent of the airspeed.The manoeuvre in winds of 50 per cent of the air-speed starts at an altitude of 110 m resulting in theTIP at a distance Dswitch of "33.3 m and the FA begin-ning 25.7 m downwind of the target. In contrast, whenthe wind speed is 110 per cent of the airspeed themanoeuver starts at an altitude of 78 m with the TIPat a distance Dswitch of "137 m and the FA beginning6.0 m upwind of the target, with the parafoil travellingbackwards at impact.

Fig. 4 Final turn trajectories with initial condition errors

In practice, sensor errors, uncertain winds, andimperfect control will disturb the ideal touchdowndepicted above. Figure 4 shows Monte Carlo simula-tions with W = "3.4 m/s, where it is assumed errorin the initial horizontal position, heading, and windoccur at the beginning of terminal guidance. Thestandard deviation was assumed to be 6 m in eachco-ordinate, 10$ in heading, and 0.5 m/s in wind speed.As seen from Fig. 4, these errors result in spreadingthe arrival to the FA point. Therefore, while the sim-plified algorithm may provide a reasonable strategyfor exiting the energy management phase, an effec-tive terminal guidance strategy must compensate forsensor errors and varying winds during the final turn.

4 OPTIMAL TERMINAL GUIDANCE

To overcome tracking errors and unaccounted winds,the following two-point boundary-value problem(TPBVP) is formulated. Staring at t = 0 with the statevector defined as x0 = [x0, y0, !0]T, the system influ-enced by the last known wind needs to be broughtto another point, xf = [(Vh + W )T des

app, 0, "!]T at t = tf .Hence, they need to find the trajectory that satisfiesthese boundary conditions along with a constraintimposed on yaw rate, |! | ! !max, while finishing themanoeuver in exactly Tturn seconds.The optimal !(t) isthen tracked by the guidance unit using any appropri-ate control algorithm. Errors and unaccounted windswill not allow exact tracking of the calculated optimaltrajectory; therefore, computation of an optimal tra-jectory will be updated during the final turn, each timestarting from the current state, requiring the systemto arrive at xf = [(Vh + W )T des

app, 0, "!]T in Tturn secondsfrom the beginning of the turn. The remaining time





until arrival at the FA is computed as

tf = " zVv

" T finalapp (10)

where T finalapp is the final calculated approach time from

equation (8) before entering the final turn.Development of the optimal trajectory algorithm

begins by recalling from equation (2) the target framehorizontal trajectory kinematics

'xy

(=

'"W + Vh cos !

Vh sin !

((11)

Recognizing that if the final turn horizontal trajec-tory is given by equation (11) the yaw angle alongthis trajectory is related to the change of inertialco-ordinates as

! = tan"1 yx + W

(12)

Differentiating equation (12) provides the yaw ratecontrol

! = y(x + W ) " xy(x + W )2 + y2

(13)

required to follow the reference final turn trajectoryin presence of constant wind W . The inertial speedalong the trajectory will also depend on the currentyaw angle

VG =)

x2 + y2 =*

V 2h + W 2 " 2VhW cos ! (14)

Now, following the general idea of direct methodsof calculus of variations [13] the solution of theTPBVP will be represented as functions of some scaledargument " = "/"f % [0; 1]

x(" ) = P1(" ) = a10 + a1

1 " + a12 "

2 + a13 "

3

+ b11 sin(!" ) + b1

2 sin(2!" )

y(" ) = P2(" ) = a20 + a2

1 " + a22 "

2 + a23 "

3

+ b21 sin(!" ) + b2

2 sin(2!" )(15)

Coefficients a#i and b#

i (# = 1, 2) are defined by theboundary conditions up to the second-order deriva-tive at " = 0 and " = "f (" = 1). According to the prob-lem formulation and equation (11), these boundaryconditions are

'xy

(

"=0

='

x0

y0

(,

'xy

(

"=0

='"W + Vh cos !0

Vh sin !0

(,

'xy

(

"=0

='"!0Vh sin !0

!0Vh cos !0

((16)

'xy

(

"="f

='(Vh + W )T des

app

0

(,

'xy

(

"="f

='"W " Vh

0

(,

'xy

(

"="f

='

00

((17)

While final condition equation (17) will be constant,initial conditions will reflect the current state of thesystem at each cycle of optimization. Differentiationof equation (15) two times with respect to " results in

"f P &#(" ) = a#

1 + 2a#2 " + 3a#

3 "2 + !b#

1 cos(!" )

+ 2!b#2 cos(2!" )

" 2f P &&

# (" ) = 2a#2 + 6a#

3 " " !2b#1 sin(!" )

" (2!)2b#2 sin(2!" ) (18)

Equating equation (18) at the terminal point tothe known boundary conditions equations (16) and(17) yields a system of linear algebraic equations forcoefficients a#

i and (b#i # = 1, 2). For instance, in the

x-co-ordinate

!

++++++"

1 0 0 0 0 01 1 1 1 0 00 1 0 0 ! 2!

0 1 2 3 "! 2!

0 0 2 0 0 00 0 2 6 0 0

#

,,,,,,$

!

++++++++"

a10

a11

a12

a13

b11

b12

#

,,,,,,,,$

=

!

++++++++"

x0

xf

x&0"f

x&f "f

x&&0"

2f

x&&f "

2f

#

,,,,,,,,$

(19)

Resolving equation (19) yields

a10 = x0, a1

1 = "(x0 " xf ) +(2x&&

0 + x&&f )"

2f

6,

a12 = x&&

0"2f

2, a1

3 = "(x&&

0 " x&&f )"

2f

6

b11 =

2(x&0 " x&

f )"f + (x&&0 + x&&

f )"2f

4!

b12 =

12(x0 " xf ) + 6(x&0 + x&

f )"f + (x&&0 " x&&

f )"2f

24!

(20)

The only problem is that derivatives in equation (20)are taken in the virtual domain, while the actualboundary conditions are given in the physical domain.Mapping between the virtual domain [0; "f ] and phys-ical domain [0; tf ] is addressed by introducing a speedfactor $

$ = d"

dt(21)

Using this speed factor the authors may nowcompute corresponding derivatives in the virtual





domain using the differentiation rules valid for anytime-variant parameter %

% = $% &, % = $($&% & + $% &&) (22)

Inverting equation (22) yields

% & = $"1% , % && = $"2% " $&$"1% (23)

needed to transfer the boundary conditions.The speedfactor $ simply scales the entire problem – the higherspeed factor $ is, the larger "f it results in reference[14] – one may let $ = 1 and $& = 0 at the boundaryconditions which means one can safely assume atthe boundary conditions % & = % and % && = % .

Finding the optimal solution among all candidatetrajectories described by equation (15) begins by firstguessing a value of the only varied parameter "f . Forthe initial value of "f one can take the length of thecircumference connecting terminal points as

"f = !

2

*(xf " x0)2 + (yf " y0)2 (24)

Coefficients of the candidate trajectory are com-puted using equation (20) with the boundary condi-tions equations (16) and (17) converted to the virtualdomain. Having an analytical representation of thecandidate trajectory equations (15) and (18) definesthe values of xj , yj , x&

j , and y &j , j = 1, . . . , N over a fixed

set of N points spaced evenly along the virtual arc [0; "f ]with the interval

&" = "f (N " 1)"1 (25)

so that

"j = "j"1 + &" , j = 2, . . . , N , ("1 = 0) (26)

Next, for each node j = 2, . . . , N compute

&tj"1 =-

(xj " xj"1)2 + (yj " yj"1)2

V 2h + W 2 " 2V W

h cos !j"1(27)

(!1 ' !0), and

$j = &"&t"1j"1 (28)

The yaw angle ! can now be computed using thevirtual domain version of equation (12)

!j = tan"1$jy &

j

$jx&j + W

(29)

Finally, the yaw rate ! is evaluated simply as

!j = (!j " !j"1)&t"1j"1 (30)

Fig. 5 Comparison of optimal and constant turn rate

When all parameters are computed at each of theN points, one can compute the performance index

J =

.

/N"10

j=1

&tj " Tturn

1

22

+ k!& (31)

where

& = maxj

30;

44!j

44 " !j max52

(32)

with k! being a weighting coefficient.The optimal trajectory problem based on the direct

method has now been turned into a single-parameteroptimization problem where "f is found such thatequation (31) is minimized subject to the system inequations (2) and (11). The problem is solved usinga non-gradient optimization numerical algorithmbased on the golden section search and parabolicinterpolation method [15]. Reduction of finding anoptimal terminal trajectory to a single-parameter opti-mization results in short computation times. To bemore specific, a 16-bit 80 MHz microprocessor allowedcomputation of a 17.5 s turn manoeuver with N = 25in ten iterations, in only 0.07 s.

As a demonstration of the proposed approach, Fig. 5shows an example of the optimal final turn for the casein Fig. 3 when W = "3.4 m/s. The optimal final turnand constant turn rate commands are similar; how-ever, as opposed to the constant turn rate, the yawangle changes smoothly at departure from the TIP andarrival at the FA. As a result, the optimal turn trajectoryis slightly different but, more importantly, the systemstill captures the FA in exactly Tturn seconds. Therefore,the authors have a tool allowing construction of theoptimal trajectory from any initial point to the pre-determined final. To this end, Fig. 6 shows that usingthe same initial errors from Fig. 4 the optimal turn isfound so that the systems still arrive at the FA in thepredetermined time, Tturn.

Implementation of the algorithm on actual parafoilsystems requires three additional components tomake it more robust in practice: compensation forfull dynamic motion, wind disturbance mitigation,and terminal trajectory tracking errors. The kinematic





Fig. 6 Optimal guidance with an error in the TIP

model in equation (2) does not account for parafoilturning dynamics and assumes that the sideslip androll angles are small. When the turn rate is sufficientlysmall, or the radius R is large, the model in equation (2)provides sufficient accuracy. However, in order to trackthe desired trajectory for a wide range of R, the errorfrom sideslip and turning dynamics can be compen-sated by adding an additional commanded !c forthe first tpre seconds after the TIP has been reached.Specifically, for the first tpre seconds of the maneuverthe command yaw !c produced by equation (29) isaugmented as

!c = !c " KturnVh

R(33)

where Kturn is the correction gain and Vh/R is therequired constant turn rate in equation (4). Distur-bances from wind and sensor errors during trackingof the optimal trajectory will result in errors in theFA arrival. These errors can be accommodated byupdating the optimal trajectory from the current con-ditions to the desired FA point during the final turn.Re-computation of the optimal trajectory is scheduledso that a specified number of updates occur at equalintervals over the period from t0 to texit. Finally, theFA location, xf in equation (17), assumes that duringthe FA the parafoil travels directly to the target by theshortest path. In practice, disturbances and the guid-ance algorithm results in a trajectory with less thanperfect efficiency at reaching the target. In order tocorrect for realistic errors during the FA the location xf

is corrected to be

xf = VhT desapp 'final (34)

where 'final is the FA efficiency.

5 PRECISION PLACEMENT SIMULATION

In the previous development of an optimal terminaltrajectory a kinematic model was used and it wasassumed the trajectory was tracked perfectly. How-ever, actual parafoil systems have turning dynamicsresulting in side slip and varying descent rates. Evalu-ation of the proposed algorithm will be done using ahigher fidelity model. The combined parafoil canopyand payload can be modelled using six DoF, whichinclude three inertial position components of the sys-tem mass centre as well as the three Euler orientationangles of the parafoil–payload system, yaw ! , pitch ( ,and roll ). The 6 DoF model described in reference [9]is used in all simulations with a constant canopy inci-dence angle * , payload drag included in the systemaerodynamic coefficients, and the canopy apparentmass approximated by assuming the canopy has twoaxes of symmetry.

The guidance strategy outlined in sections 2 to 4requires a controller that tracks a commanded yaw ! c ,using the brake deflection +a. A simple two DoF modelof the roll and yaw dynamics can be used for develop-ing a yaw controller since for parafoils, pitch and speedare not typically controllable. The state vector for thetwo DoF rotational model is x = [,, ! , p, r]T, wherep and r are the parafoil roll and pitch rates. A widerange of trajectory tracking controllers used on UAVscould be used [8, 16], however, MPC is selected sincethe optimal terminal guidance prescribes a desired !c

over the terminal trajectory horizon. The rotationalkinematics and dynamics described in reference [9]are non-linear and thus linearized to take advantage ofwell-developed control techniques for linear systems.Assuming that the aerodynamic velocity Va is con-stant, the non-linear model with parameters listed inAppendix 2 can be numerically linearized resulting ina linear discrete model for the parafoil with samplingperiod of 0.5 s

!

++"

,

!

pr

#

,,$

k+1

=

!

++"

0.962 0 0.153 0.0120.0078 1 "0.011 0.043"0.103 0 0.033 0.0040.0191 0 "0.0023 "0.003

#

,,$

!

++"

,

!

pr

#

,,$

k

+

!

++"

"0.0060.0501

"0.01310.1098

#

,,$ +a;k (35)

For a given desired discrete output vector !c =[!c

k+1, !ck+2, . . . , !c

k+Hp]T that spans the prediction hori-

zon Hp, MPC solves for the discrete optimal controlvector U = [uk , uk+1, . . . , uk+Hp"1]T governed by thelinear plant xk+1 = Axk + Buk with output yk = Cxk

[8, 17]. The single-input single-output parafoil modelabove has the output matrix C = [0 1 0 0]. The discretelinear model can be used to estimate the future state





of the system as

!c = KCAxk + KCABU (36)

from the current state vector xk , and the control vectorU, where

KCA =

!

+++"

CACA2

...CAHp

#

,,,$(37)

KCAB =

!

++++++"

CB 0 0 0 0CAB CB 0 0 0CA2B CAB CB 0 0

......

.... . . 0

CAHp"1B · · · CA2B CAB CB

#

,,,,,,$(38)

In MPC, the optimization problem is cast as a finite-time discrete optimal control problem. To computethe control input at a given time instant, a quadraticcost function is minimized through the selection ofthe control history over the control horizon. The costfunction can be written as

J =6!c " !

c7T

Q6!c " !

c7

+ U TRU (39)

where Q and R are symmetric positive semi-definitematrices of size Hp # Hp that penalize tracking errorand control, respectively. The control U , which mini-mizes equation (39), can be found analytically [17] as

U =3KT

CABQKCAB + R5"1

KTCABQ

3!c " KCAxk

5(40)

Equation (40) contains the optimal control inputsover the entire control horizon. In practice, only thefirst control is applied and the problem is updated atthe next time update.

Implementation of the terminal guidance algorithmrequires both determining the optimal terminal tra-jectory using the method outlined in section 4 andthe trajectory tracking controller in equation (40). Thefirst step in the complete terminal guidance algorithmis to use the current measured parafoil position andvelocity along with the known FA point to define theboundary conditions to the TPBVP (equations (16),(17) and (34)). Once the TPBVP has been defined, theguidance algorithm must determine the optimal solu-tion among all candidate trajectories equation (15) byinitializing the only varied parameter "f according toequation (24). The optimal trajectory is found usingthe single-parameter optimization problem outlinedin section 4 (equations (24) to (32)) with a non-gradient optimization numerical algorithm based onthe golden section search and parabolic interpolationmethod [15]. Upon completion of the optimizationproblem, the desired yaw angle trajectory is known

from equation (29). Finally, the discrete optimal yawangle can be combined with the correction in equation(33) and used in the proposed MPC algorithm inequation (40) to determine the final control.

6 SIMULATION RESULTS

In all simulations the full non-linear parafoil modelis numerically integrated using a fourth-order Runge–Kutta algorithm with time step of 0.05 s. The MPCcontrol algorithm described previously is used to track!c. Figures 7 and 8 show examples of the complete ter-minal guidance strategy for an R of 37.5 m in winds of"3.4 and "7.7 m/s. In both cases Tapp is 7.5 s, 'final is0.95, t1 " texit is 3.0 s, tpre is 6.0 s, Kturn is 1.0, !j max is20 deg/s, and k! is 400, with two trajectory updatesmade during the final turn. Results are shown fromthe TIP (x0, y0, z0), which are ("33, 75, and "93 m) in

Fig. 7 Optimal real-dynamics-augmented guidance forW = "3.7 m/s (" trajectory update)

Fig. 8 Optimal real-dynamics-augmented guidance forW = "7.7 m/s (" trajectory update)





Fig. 9 Monte Carlo simulation dispersion

Fig. 8 and ("137, 75, and "93 m) in Fig. 9. In both casesthe parafoil initially tracks the optimal trajectory; how-ever, errors slowly build due to mismatch in the actualdynamics and the model. At each update a new tra-jectory is calculated that results in arrival at the FA.Impact errors are 0.4 m in Fig. 7 and 0.5 m in Fig. 8.

The complete precision placement algorithm com-bines energy management, an exit from energy man-agement based on equation (9), estimation of a TIPduring homing using equation (7), and the optimalturn followed by the FA. Monte Carlo simulations of100 drops were done using the complete precisionplacement algorithm with an away distance of 450 m,a cycle distance of 125 m, an R of 37.5 m, and the tar-get at the origin. The initial position and altitude werenominally 760 m upwind of the target, at an altitude of700 m, both normally distributed with a 50 m standarddeviation. Gaussian noise was injected into measuredposition, altitude, and inertial sensors as summarizedin Table 1.

In addition to sensor errors, two sources of windvariation were added to the simulation, varying direc-tion and varying ground wind magnitude. Wind isassumed to have a constant magnitude prior to ter-minal guidance but varies linearly from W at the TIP

Table 1 Error statistics

Parameter 1-GPS bias (m) 2.0GPS noise (m) 0.5Altitude bias (m) 2.0Altitude noise (m) 0.5Roll, pitch, and yaw bias (degree) 2.0Roll, pitch, and yaw noise (degree) 1.0u, v, and w bias (m/s) 0.1u, v, and w noise (m/s) 0.2p, q, and r bias (degree/s) 1.0p, q, and r noise (degree/s) 1.0

Fig. 10 Test results of precision placement algorithm

to W + WF at the ground. The wind W is normallydistributed with 4.75 m/s mean and 2.0 m/s standarddeviation while WF is 1.5 m/s (1- ). For the exampleparafoil, this results in a nominal wind-to-airspeedratio of 0.70 with 25 per cent exceeding 1.0. Prevailingwind is assumed by the system to travel down rangeat a heading of zero degrees while the true wind head-ing varies 15$(1- ) in direction. Dispersion results areshown in Fig. 9. The resulting circular error probable(CEP) shown by the circle is 16.8 m and is defined bythe radius which includes 50 per cent of the impacts.

7 EXPERIMENTAL RESULTS

Test results of the small 2.3 kg parafoil outlined inAppendix 2 are shown in Fig. 10 with an away dis-tance of 200 m, cycle distance of 120 m, R of 30 m,and the target at the origin. The parafoil is released330 m down range and "240 m cross range at an alti-tude of 610 m above ground. Wind was from 310$

with magnitude varying from 4.9 to 6.7 m/s resultingin a wind-to-airspeed ratio of 70 per cent to 98 percent. Energy management is exited at 211 m, 97 s afterrelease. Homing proceeds until the parafoil is 60 mupwind of the target, 121 s after release, at an alti-tude of 103 m. The final turn lasts 9.0 s until the FAbegins 58 m above ground. The impact occurs 7.4 maway from the target 146 s after release.

8 CONCLUSIONS

A terminal guidance algorithm was developed forparafoil precision placement in high wind-to-airspeedratios. Using the direct method of calculus of variationsand inverse dynamics in a virtual domain, the TPBVP





is reduced to single-parameter optimization problem.The result is an extremely efficient solution that iseasily solved in real-time. It was shown, that assum-ing a constant final turn rate, resulted in an analyticsolution that could be used for calculating guidancedecisions prior to the final turn. Uncertainty duringterminal guidance from wind, sensors, and guidanceresulted in the system arriving only approximately atthe predicted location. Efficient computation allowedthe trajectory to be recomputed during the final turnadding robustness to changing winds and trackingerrors. The algorithm was verified using a full 6 DoFparafoil model and Monte Carlo simulations result-ing in a circular error probable of 16.8 m. The finaltest results were presented for a small 2.3 kg systemin strong winds where the final error was 7.4 m.

© Authors 2011

REFERENCES

1 Benney, R., McGrath, J., McHugh, J., Meloni, A.,Noetscher, G., Tavan, S., and Patel, S. DOD JPADS pro-grams overview and NATO activities. In Proceedings ofthe 19th AIAA Aerodynamic Decelerator Systems Tech-nology Conference and Seminar, Williamsburg, Virginia,21–24 May 2007.

2 Benney, R., Henry, M., Lafond, K., Meloni, A., and Patel,S. DOD new JPADS programs and NATO activities. InProceedings of the 20th AIAA Aerodynamic Decelera-tor SystemsTechnology Conference and Seminar, Seattle,Washington, 4–7 May 2009.

3 Benney, R., Meloni, A., Cronk, A., and Tiaden, R. Preci-sion airdrop technology conference and demonstration.In Proceedings of the 20th AIAA Aerodynamic Decelera-tor SystemsTechnology Conference and Seminar, Seattle,Washington, 4–7 May 2009.

4 Calise, A. and Preston, D. Design of a stability augmenta-tion system for airdrop of autonomous guided parafoils.In Proceedings of the AIAA Guidance, Navigation, andControl Conference and Exhibit, Keystone, Colorado,21–24 August 2006.

5 Calise, A. and Preston, D. Approximate correction ofguidance commands for winds. In Proceedings of the20th AIAA Aerodynamic Decelerator Systems Technol-ogy Conference and Seminar, Seattle, Washington, 4–7May 2009.

6 Gimadieva, T. Z. Optimal control of a guiding glidingparachute system. J. Math. Sci., 2001, 1(103), 54–60.

7 Carter, D., George, S., Hattis, P., McConley, M., Ras-mussen, S., Singh, L., and Tavan, S. Autonomous largeparafoil guidance, navigation, and control system designstatus. In Proceedings of the 19th AIAA AerodynamicDecelerator Systems Technology Conference and Sem-inar, Williamsburg, Virginia, 21–24 May 2007.

8 Slegers, N. and Costello, M. Model predictive control of aparafoil and payload system. J. Guid. Control Dyn., 2005,4(28), 816–821.

9 Slegers, N., Beyer, E., and Costello, M. Use of dynamicincidence angle for glide slope control of autonomousparafoils. J. Guid. Control Dyn., 2008, 3(31), 585–596.

10 Slegers, N. and Costello, M. Aspects of control for aparafoil and payload system. J. Guid. Control Dyn., 2003,6(26), 898–905.

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12 Rademacher, B., Lu, P., Strahan, A., and Cerimele, C. In-flight trajectory planning and guidance for autonomousparafoils. J. Guid. Control Dyn., 2009, 6(32), 1697–1712.

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APPENDIX 1

Notation

A, B, C discrete system state space matricesA, B, C element of the apparent mass matrixb canopy spanc canopy mean chordd maximum brake deflectionDswitch distance of the turn initial point with

respect to the targetHP discrete prediction horizoniI , jI , kI inertial frame unit vectorsiT , jT , kT target frame unit vectorsk! turn rate penalty coefficientKturn final turn correction gainL distance to target along target linep, q, r parafoil roll, pitch, and yaw rates in the

body frameP, Q, R elements of the apparent inertia matrixQ, R positive semi-definite error and control

penalty matricesR final turn radiusSp parafoil canopy areatexit exit time for final turntpre length of time to apply final turn

correctionTapp, T des

app final approach time and desired finalapproach time

T finalapp last calculated final approach time





Tturn final turn timeu, v, w parafoil velocities expressed in the

body frameU discrete optimal control vectorVa canopy airspeedVG parafoil ground speedVh, Vv parafoil horizontal and vertical speedW , WF target frame wind speed and ground

wind speedx, y, z parafoil inertial positions in the target

framezstart altitude to exit energy management

. canopy incidence angle+a asymmetric brake deflection'final final approach efficiency$ speed factor" virtual time"f virtual time at completion of final turn" virtual time scaled by the final virtual

time,, ( , ! parafoil Euler roll, pitch, and yaw

angles!c commanded final turn angle!max maximum desired final turn rate!c, !

cdesired and estimated output vectors

APPENDIX 2

Model parameters

System mass m (kg) 2.3Steady-state aerodynamic

velocity Vh (m/s)6.82

Canopy reference areaSp(m2)

1.1

Canopy span b (m) 1.35Canopy chord c (m) 0.75Incidence angle .

(degree)"12

Inertia matrix elements(kg m2)

Ixx = 0.423, Iyy = 0.401,Izz = 0.052, Ixz =0.027

Elements of theapparent mass matrix(kg)

A = 0.012, B = 0.032,C = 0.423

Elements of theapparent inertiamatrix (kg m2)

P = 0.054, Q = 0.13,R = 0.0024

x-distance from masscentre to apparentmass centre xBM (m)

0.05

z-distance from masscentre to apparentmass centre zBM (m)

"1.1

Maximum brakedeflection d (m)

0.13

Aerodynamiccoefficients

CD0 = 0.25, CD/2 = 0.12,CY 0 = 0.23, CL0 =0.091, CL/ = 0.90

Cm0 = 0.35, Cm/ ="0.72, Cmq = "1.49

Cl0 = "0.036, Clp ="0.84, Clr = "0.082,Cl+a = "0.0035,

Cn0 = "0.0015, Cnp ="0.082, Cnr = "0.27,Cn+a = 0.0115.




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