Development of Control Algorithm for the Autonomous Gliding Delivery System

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17th AIAA Aerodynamic Decelerator Systems Technology Conference and Seminar AIAA 2003-2116 19-22 May 2003, Monterey, California

AIAA-2003-2116

DEVELOPMENT OF CONTROL ALGORITHM FOR THE AUTONOMOUS GLIDING DELIVERY SYSTEM

Isaac I. Kaminer† and Oleg A. Yakimenko‡ Department of Aeronautics and Astronautics, Naval Postgraduate School, Monterey, CA

Abstract. foil from the initial position to the desired landing The paper considers the development and simulation point. Finally, it is the responsibility of the control sys- testing of the control algorithms for an autonomous tem to track this trajectory using the information pro- high-glide aerial delivery system, which consists of vided by the navigation subsystem and onboard actua- 650sq.ft rectangular double-skin parafoil and 500lb tors. payload. The paper applies optimal control analysis to In the past decade, several GNC concepts for glid- the real-time trajectory generation. Resulting guidance ing parachute applications have been developed and 13-22 and control system includes tracking this reference tra- published. Most of them were tested in a simulation jectory using a nonlinear tracking controller. The paper environment, some in flight test. presents the description of the algorithm along with Some of the published papers include analytical simulation results and ends with guidelines for the fur- development of feasible trajectories. In Ref.23, for ex- ther implementation of the developed software aboard ample, minimum glide angle and descent rate, mini- the real system. mum turn radius and maximum gliding turn rate, reach- able ground area, minimum altitude loss problems were I. Introduction considered and solved analytically for a simplified 1-3 three-degree-of-freedom (3DoF) model with a parabolic Maneuverable ram-air parafoils are widely used to- lift-drag polar. The problems of the maximum time to day. The list of users includes skydivers, smoke jump- stall the glider at the predefined point in a horizontal ers and special forces. Furthermore, their extended plane and a maximum range for the appropriate simpli- range (compared to that of round canopy parachutes) fied model with constraints were also considered. Op- makes them very practical for payload delivery. Recent timization included no wind scenario. introduction of the Global Positioning System (GPS) Assuming the wind velocity to be constant Ref.24 made the development of fully autonomous ram-air synthesized zero-lag control of the gliding parachute parafoils possible. Moreover, ram-air parafoils are system with the constant rate of descent. This interest- nowadays considered a vital element of the unmanned ing work uses a compound performance index blending air vehicles (UAVs) capability and of International 4-12 time, kinetic energy, and controls expenditure. The op- Space Station (ISS) crew rescue vehicle. timal problem was solved by applying Pontrjagin's Autonomous parafoil capability implies delivering maximum principle.25 The resulting two-point bound- the system to a desired landing point from an arbitrary ary-value problem was solved using Krylov- release point using onboard computer, sensors and ac- Chernous'ko method.26 tuators. This requires development of a guidance, navi- However, majority of the published results on gation and control (GNC) system. The navigation sub- GNC system development for parafoils sue heuristic system manages data acquisition, processes sensor data rules and classical control algorithms. It is appropriate and provides guidance and control subsystems with to mention some of them here emphasizing a scale of information about parafoil states. Using this informa- the system, sensor suite specs and control approach. tion along with other available system data (such as Probably one of the first GNC algorithms for small local wind profiles), the guidance subsystem plans the ram-air parachutes is presented in Ref.13. Computer mission and generates feasible (physically realizable programs were written using 6DoF models of small and mission compatible) trajectory that takes the para- parafoils, such as Ram-air, Para-commander, and Stan- dard Round. The aerodynamic models were simplified, but they included the non-linearities and severe con- .This paper is declared a work of the U.S. Government and is straints common to all parachutes as compared to air- not subject to copyright protection in the United States. craft. The algorithms used GPS position and velocity

†Associate Professor; Senior Member AIAA. data provided at 1Hz to control both toggle lines. Hom- ing and circling algorithms were simulated using direc- ‡Research Associate Professor, Associate Fellow AIAA.

1 This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. tion sines and cosines. Wind was included in all algo- Another GNC system incorporating appropriate rithms. Homing algorithms were included that in- data acquisition system was developed to support high- creased the allowable offset, minimized the miss dis- altitude-high-opening insertion into hostile environ- tance and reduced the impact velocity. These were ments and was supposed to be used with a 500lb para- based on two-axis proportional navigation (PN), and foil. The GPS guidance mode was designed to steer a also upon upward-biased rate-integrating navigation in glider to the LZ using only GPS data or GPS plus up to glide angle. The latest produced final flare and nearly eight additional sensors. The GPS data was provided at vertical landing even from a long initial offset. For the 1Hz and inertial sensors at 200Hz. Control steering was case of high initial altitude and small offset circling accomplished by actuating one of the two steering con- algorithms were also introduced. Finally, combined trols, which allowed four flight modes: straight flight, downwind, upwind, and guidance algorithms were steer left or right, hard turn left or right, and flare. No simulated to emulate manned parachute procedures. test results however are available so far. Another approach was employed in Ref.16 where Another project was reported by the German Aero- Pseudo-Jumper (PJ) guidance and control algorithms space Center. It aimed to identify behavior of a small based on the professional jump procedures were intro- (200lbs) and extensively instrumented parafoil-based duced. The main phases of the mission after release system and to investigate GNC-concepts for the were defined as follows: autonomous landing.29,30 Three and four degree of free- "Cruise" phase consisting of aiming for the holding dom models were developed and parameters of these area way-points (WPs) by one or more types of hom- models were identified during this study. ing algorithms; The world's largest parafoil was designed for the "Hold" phase: As the parachute gets close enough to U.S. Army by Pioneer Aerospace Corp. This was a part the homing WPs, it is caused to perform S-turns just of a contract undertaken by Pioneer and the U.S. Army downwind of the approach zone. The pattern is ad- Soldier Systems Command and was tested in 2001- justed to suit the wind speed so that it stays near the 2002 at the U.S. Yuma Proving Grounds, AZ (YPG) as landing zone (LZ) and has just enough upwind dis- an aid for ISS "lifeboat". The resulting parafoil was tance left for the approach and flare; ultimately intended to evacuate a seven-person crew "Center" phase: at some point, governed mainly by from the station in case of an emergency (Fig.1).11-12 altitude (time-to-go), the PJ must aim for the center The 1,600lb (50 times the weight of a typical per- of the drop zone so that it is heading towards the sonal parachute) 7,500sq.ft parafoil wing has the wing- touchdown target. Enough time must be allowed to span wider than a football field and a canopy area of complete any ongoing S-turn; more than one and a half times that of the wings of a "Approach" phase: at some point, the PJ has to stop 747 jumbo jet and is designed to carry weights of up to trying to center and get lined up with the surface 42,000lbs, the heaviest cargo ever transported by a wind direction and to minimize toggle actions past parafoil. The system is able to move anything from that point; supplies to armored vehicles into strategic areas. Other "Flare" phase is triggered with rapid full braking to applications could include mentioned recovery of high- arrest the vertical and/or horizontal inertial velocities value assets for the space program, and humanitarian in an optimal control fashion. aid, such as the delivery of food, supplies and medica- Perfect knowledge of parafoil's relative location, tions to war-torn areas like Bosnia, Sarajevo, Afghani- i.e. availability of a GPS sensor and of sonic altimeter stan, and Iraq. as well as of winds below 300ft was assumed. The Pioneer Aerospace Corporation is an industry Recent work by the Iowa State University leader in parachute design. They have provided recov- (http://cosmos.ssol.iastate.edu/RGS)27,28 and Atair ery parachutes for NASA's space shuttle, the Air Aerospace (http://www.extremefly.com/aerospace) Force's B-1A Bomber and parachutes for other NASA includes development of a parafoil-based Recovery and Department of Defense programs. Since testing on Guidance System to return payloads from a high- the Guided Parafoil Aerial Delivery System (GPADS) altitude balloon back to the ground safely and predicta- program began in January 1994, Pioneer holds records bly. The parafoil is supposed to accompany the balloon for the largest parafoil, highest altitude (22,000ft), and and to be cut free of the balloon at a predetermined heaviest weight (15,500lbs) airdropped. time. It then guides the payload to a designated LZ. As The GNC system for GPADS7,8 was developed by reported, fully automated guidance hardware and soft- the European Space Agency.22,29-33 The Parafoil Tech- ware used DGPS (differential GPS) data, a digital com- nology Demonstrator program was set up and con- pass, a linear quadratic regulator and PN guidance to ducted under ESA to demonstrate feasibility of guiding control and navigate the parafoil. Only simulation re- a large-scale parafoil autonomously to a predefined sults have been reported so far. point and to perform a flared landing within a specific range. Another objective was to enable further investi-

2 gation into the flight dynamics of the parafoil systems. supposed to be completely automated, although the The first automatically-guided, parafoil based descent crew will have the capability to switch to backup sys- and landing of a test vehicle was carried out in support tems and steer the parafoil manually, if necessary. of ESA's development in 1997. The spacecraft was

Figure 1 The X-38 vehicle under the world’s largest parafoil (courtesy of NASA)

The prototype reference trajectory (RT) consisted will lead to a touchdown at LZ. Seven guidance modes of the following nominal phases are available: Acquisition - trajectory initialization and initial turn; Automatic trim; Homing - travel to intercept Energy Management Maneuver to WP; Circle (EMC); Transition to hold; EMC Entry, EMC and EMC Exit - the energy man- Holding pattern (to maintain turn radius); agement circling maneuver; Maneuver to target; Final - final approach to target on desired heading; Landing approach; The GNC system employs laser or radar altimeter Landing flare. to trigger flare and is required to steer the parafoil to The control system may switch between any of these within 100m of a preselected desired impact point (DIP) modes sequentially or bypass some of them. (so far the accuracy of 200m was achieved on a smaller The guidance concept presented in Ref.22 follows scaled prototypes). generally the same scheme. The energy management To support the development of GPADS software concept is adopted motivated by the way novice sky- and to evaluate its expected performance on ram-air divers learn the landing approach.16 The trajectory is parafoils with payload capacities ranging from less than planned using WPs. The guidance works as follows: 200lbs to 40,000lbs, Draper Laboratory also developed Homing: from the current position the energy man- a GNC package.17-19 This software was demonstrated agement center (EMC) is approached if possible. If using NASA Dryden Flight Research Center airdrop the remaining distance is not long enough to reach test article known as the "Wedge 3" which utilizes an the desired impact point, the WP is moved in the di- 88sq.ft ram-air parafoil and carries a payload of 175lbs. rection to the final turn point (FTP); Same GNC algorithms were also tested in simulation Energy Management: after reaching EMC, surplus using the model of a larger 3,600sq.ft guided parafoil altitude is reduced by flying S-patters to the energy air delivery system with a 13,088lb payload. Upon full management turn points (EMTP). If the remaining deployment of the parafoil canopy the GNC algorithm distance becomes too small to reach DIP, the EMTP is activated. It determines the desired heading angle that is shifted along the energy management axis towards the EMC;

3 Landing: the landing phase is divided into 4 sub- sible trajectory is generated in real-time that takes the phases: (1) transition into landing corridor, (2) ap- parafoil from initial release point to touch down at a proach of final turn point (FTP), (3) turn into wind desired impact point. It is assumed that only the direc- and (4) flare. tion of the wind at LZ is known. The trajectory consists The Backup mode was also developed as follows: of an initial glide, spiral descent and of final glide and if altitude and subsequently remaining distance are too flare. The final glide and flare are directed into the wind small to reach the landing point during homing phase, at LZ. This structure is motivated by optimal control the vehicle keeps heading the FTP until a specified de- analysis carried out on a simplified model. The result- cision altitude is reached. Then the vehicle turns against ing trajectory is tracked using a nonlinear algorithm wind independently of its position and prepares for with guaranteed local stability and performance proper- landing. ties (see Ref.34). The control algorithm converts trajec- The guidance system output consists of a heading tory-tracking errors directly into control actuator com- command computed using current position and next mands. Therefore, only GPS position and velocity are WP. For wind accommodation, the trajectory is planned needed to implement it. in the wind fixed coordinate system.21 Assuming con- As expected the basic trajectory structure is similar stant horizontal wind, the actual position is shifted to a to the ones reported in the literature. However, as virtual position by the predicted wind drift, the vehicle shown in Section III, a single smooth inertial trajectory will be affected during the remaining flight until touch is generated using simple optimization and is tracked down. Within this wind fixed system the guidance can throughout the drop by the same control algorithm. This be done as if there is no wind. Depending on the sign of eliminates the need for multiple modes, extensive the heading error, either the left or the right actuator is switching logic and wind information throughout the activated, leading to a left or a right turn. A saturation drop. The latter is made possible by the nonlinear con- limited PN controller computes the desired actuator trol algorithm that tracks the inertial trajectory directly position. For instance, with an amplification of 4, a and treats wind as a disturbance. commanded turn of 90° first activates one actuator to This specific delivery system considered in this pa- full stroke until the heading difference goes below 15° per is called Pegasus. It consists of 650sq.ft span rec- and the proportional part of the controller takes over. tangular double-skin parafoil and 500lb payload (Fig.2) According to Ref.22 after selective availability has and was originally manufactured by the FXC Corp. It been switched off (in May 2000), plain GPS data with- can be controlled by symmetric and differential flap out differential corrections is accurate enough for tra- deflections that occupy outer four (out of eight on each jectory planning. The GPS altitude, with errors of about side) cells of the parafoil. ±20m, is combined with the pressure altitude. Heading This paper is organized as follows. Section II con- information is supposed to be computed by applying a tains a brief description of the system and of the com- complementary filter to blend azimuth and yaw rate as plete 6DoF model used to support the development of described in Refs.14-15. Simulation results also show the GNC algorithm. Section III introduces optimal con- that possible wind mispredictions have more influence trol strategy for a simplified parafoil model using on the landing accuracy, than incorrect estimates of Pontrjagin's principle of optimality. Section IV dis- model parameters and measurement errors. However, as cusses real-time trajectory generation for the control long as the errors stay in the normal range (position algorithm. Section V discusses the development of the ±10m, altitude ±5m, heading ±10°) the GNC algorithm tracking control algorithm. Section VI introduces simu- is robust enough to land the vehicle within a range of lation results of the complete guidance and control sys- ±50m around the predefined landing point, provided tem. Finally, Section VII contains the main conclusions. that the wind is perfectly known. As seen above in most GNC concepts, the guidance II. System Architecture and Modeling itself is divided into three phases: homing, energy man- To support the development of the GNC algorithm, a agement and landing, where each of them can consist of complete 6DoF model of Pegasus was developed and further subphases. Homing, i.e. flying towards the LZ, tuned using flight test data provided by the YPG. Fig.3 and landing are similar in almost all guidance concepts. outlines Pegasus dimensions. Energy management is done by flying a certain pattern The description of the 6DoF model used is given in close to the landing point, until some criteria is satis- Ref.35. This model matches flight test data and is char- fied, switching to the landing phase. Most differences acterized by the following integral parameters: the av- among the realized guidance concepts are concerned erage descent rate is 3.7-3.9m/s, glide ratio is about 3.0, with the shape of the pattern and the criteria for the the turn rate of ~6°/s corresponds to the full deflection landing decision. of one flap. An interesting feature from the control Present paper addresses the problem of GNC de- standpoint is that the system exhibits almost no flare velopment for a parafoil system as follows. First, a fea- capability (flaps deflection results in almost no change

4 in the descent rate). Therefore, it is very important to land it into the wind.

Figure 2 Double-skin 650sq.ft parafoil with 500lb payload

Figure 3 The 500lb parafoil geometry

5 III. Optimal Control Synthesis This differential equation gives two sinusoids Consider the following kinematic model of a parafoil in (shifted with respect to abscise axis by ±Ξ−1 ) as solu- the horizontal plane. Suppose we have a constant glide tions for the general (non-singular) case −1 ratio and by pulling risers we can control its yaw rate. pCψ = 1sin(Ξ+tC2) ±Ξ , (6) Mathematically, this is expressed by the following sim- where C and C are constants defined by the concrete plified equations: 1 2 −1 xV = cosψ , yV = sinψ , ψ = v , (1) boundary conditions. If C1 ≠Ξ the parafoil model −1 where v ∈−[ Ξ;Ξ] is the only control. moves along a descending spiral. It takes 2πΞ sec- −1 The Hamiltonian for the system (1) for a time- onds to make a full turn with a radius of V Ξ . If −1 optimal control can now be written as: C1 = Ξ there exists a possibility of singular control. cosψ  This is caused by the fact that there exists a point in Hp=+vVp,p −1, (2) ψ ()xy time where both p and p are zero as can be seen sinψ ψ ψ where equations for adjoint variables p , p and p from (6). x y ψ Consider singular control for this model. By defini- are given by tion it means that p ≡ 0 . For the time-optimal prob- pp==0, 0, ψ xy lem from the Hamiltonian (2) and third equation in (3) sinψ  (3) (of course keeping in mind the first two) it follows that pVψ = ()pxy,.p −cosψ for a singular control case −1 −1 The optimal control for the time-minimum problem pVx = cosψ , pVy = sinψ , ψ = const . (7) now is given by Expressions (7) imply that singular control corresponds to motion with a constant heading ( v ≡ 0 ). It vs=Ξ ign(pψ ). (4) may not however be realized. Instead, the parafoil By differentiating the expression for pψ (3) and model may switch from right-handed spiral to a left- combining it with Hamiltonian (2) for both cases when handed one or vice versa. Planar projections of possible pψ > 0 and pψ < 0 we can get a set of equations for trajectories are shown on Fig.4. Concrete boundary conditions define which case will be realized. pψ : 2 ppψψ+Ξ ∓ Ξ=0 . (5)

Figure 4 Planar projections of possible trajectories

To summarize, Figs.5,6 show examples of three- its potential energy at the beginning of descent or at the dimensional (3D) time-optimal trajectories computed in end. It may depend on the tactical conditions in the LZ, the inertial frame for the gliding system (1). As sug- terrain, or some other factors. In a general case parafoil gested by the Pontrjagin's principle of optimality they could spend its potential energy in the vicinity of some would consist of helices (spiral descent) and straight other WP differing from the start and final portions of descent segments. The basic difference between two the trajectory as it shown in Fig.6. In any case, since the trajectories shown on Fig.5 is whether parafoil expends control system cannot meet any time constraints due to

6 unavailability of thrust on a parafoil system, the com- inertial frame and are time independent. mon feature of these RTs is that they are defined in the

Figure 5 3D representation of possible inertial RTs

Figure 6 General-case trajectory Another critical issue is that in order to meet a soft- ment. The radius of the spiral is adjusted to provide landing requirement the parafoil at the LZ must align smooth transition between each segment. The rest of itself into the wind. the section derives the mathematical representation of the complete RT. IV. Real-Time Trajectory Generation Let γ denote the desired flight path angle for the RT1 The optimal control analysis of the previous section first straight-line segment of the trajectory, similarly let motivated the basic RT structure as is shown in Fig. 7. γ denote the flight path angle for the spiral-descent This RT consists of three segments: initial straight-line RT2 segment and, finally, let γ represent the flight path glide (segment 1), spiral descent (segment 2) and final RT3 glide and flare (segment 3). Since it is required that the angle for the final straight-line segment. Furthermore, parafoil is aligned into the wind at LZ, the final glide let ψ f represent the wind direction at the LZ, segment ends at the desired impact point (DIP) and is T T directed into the wind. Furthermore, the final glide p00= (x ,,yz00)is the RP, and p ff= ()x ,,yzff de- starts an offset distance doffset away and a certain height notes the DIP. Define vector T above DIP. The height is determined by the flight path i =+cos(ψπ),sin(ψπ+),0 to be the unit vector angle of the segment 3. Similarly, the first segment ( ff) starts at the parafoil release point (RP) and ends at a that points into the wind at the LZ and vector point defined by the flight path angle of the first seg- ∗ T i =−( sin(ψπf+ ),cos(ψf+π),0) to be the unit vec- ment at the spiral descent segment. The first and last segments are fused together by the spiral descent seg-

7 tor orthogonal to i . Then the initial point T pi=−dz+ 0,0, +dtanγ . (8) T 3 offset ( f offset RT3 ) p33= (x ,,yz33) of the third segment is given by

Figure 7 Graphical representation of the parafoil trajectory

Now, let pRT denote the position on the RT, s de-

SSeegment 3 p3 note the path length traveled along the RT and VRT - parafoil's velocity along this path. Then the expression p f for the third segment of the desired trajectory is Segment 2 North p1 pRT ()s =+p33Ts , s = VRT , (9) where r ψ 2 T d T =−cosψγcos ,−sinψγcos ,−sinγ(10) 3 ()fRT3fRT3RT3 ψ 1 ∆ψ p is the unit vector that points along the line connecting 2 SSeegmgmenent 1 p3 to p f . Note, that the length of the third segment p s =−pp. Similarly, let s denote the length of the 0 33f 1 Figure 8 Top view of the parafoil RT first segment and s2 - of the second. Then, along the third segment ( s12+

8 This fact implies that at p3 the horizontal projec- shown in Ref.34 trimming trajectories consist of tions of the commanded velocity vectors for segments 2 straight lines and helices. Therefore, this methodology and 3 are equal and are independent of the choice of r. is particularly suitable for the problem at hand. Finally, the radius of the spiral descent r is selected The key ideas of the design methodology devel- to guarantee that at p segments 1 and 2 merge oped in Ref.34 include the following five steps: 2 1) reparameterize trimming trajectory using the ar- smoothly. This is done numerically by solving a single clength s, thus eliminating time as an independent variable constrained optimization problem. Note that at variable; p2 2) resolve the position and velocity errors in so-called T Frenet frame (Ref.34); ppRT ==21p+()rrsin ∆,−cos ∆,0 . (20) ss= 1 3) form error dynamics for the system consisting of Therefore, let the trajectory and parafoil model, where the posi- T tion and velocity error states are resolved in Frenet er() pp21−+()rsin∆,−rcos∆,0 , (21) frame; then the desired value of r is redes = arg min (r) . Fig.9 4) design a linear tracking controller for the lineariza- rrmin ≤≤rmax tion of the system along the trimming trajectory; includes a plot of e(r) versus r, which indicates that 5) implement this controller with the true nonlinear there are multiple values of rdes . Therefore the values of plant using a nonlinear transformation provided in rmin and rmax are selected to provide a unique solution Ref.34. This implementation guarantees that the that is consistent with physical limitations of the para- linearization along the trimming trajectory of the foil. Note, in steady state turn the bank angle of the feedback system consisting of the nonlinear plant parafoil is defined by the following expression: and nonlinear controller preserves the eigenvalues and transfer functions of the feedback interconnec- V ψ V 2 cosγ φ ==tan−−11RT RT tan RT RT2 , (22) tion of the linearized error dynamics and linear RT  grg tracking controller (see Ref.36). where g is acceleration due to gravity. By construction V.1 Derivation of the errors in Frenet frame the horizontal projections of T and T at p are 1 2 2 As shown in Section IV the desired RT is parameter- aligned, i.e. TThor = hor . ized using the path/arclength parameter s. 12s==sss 11 T Let p = (x,,yz) denote current position of the parafoil. The methodology developed in Ref.34 requires

that position and velocity commands pRT (s∗ ), VRT ()s∗ used to compute position and velocity errors,

ppRT ()s∗ − and VsRT ()∗ −V respectively, correspond to the point on the RT that is nearest to current position of the vehicle. This is done by first determining the value of the path/arclength parameter 2 s∗ =arg min pRT (s) −p and then using it to compute s position and velocity commands. In Ref.34 the problem

of determining s∗ is reduced to a constrained optimization problem. However, computing limitations of the onboard processor have imposed a need to develop analytical techniques for the computation of s , discussed ∗ Figure 9 A representative plot of e(r) versus r next.

An exact analytical expression for s∗ can be de- rived for straight-line segments 1 and 3. Recall Eqns. V. Integrated Guidance and Control Algorithm (15) and (11). Then The development of the integrated guidance and control algorithm is based on the work reported in Ref.34, where authors have proposed a new technique for tracking so-called trimming trajectories for UAV's. As

9 pT01+−ssp,0≤≤s1 s∗ = arg min  (23) s pT+−s ss−−p, s+s

Π zΠ

z2 Π((pRT s)) p ()s p RT proj p

Π∩()BC d∗

Π()p proj γ RT2

Cylinder C 2π r x Π Figure 10 Graphical representation of Π(C)

Consider Fig.11. Note for the case illustrated in zz=−+mod(zz,2πγrtan ), 10nproj nproj RT2 this figure, the zΠ coordinate of Π((p s∗ )) is bounded (29) 2 zz11pp= roj+−znzprojcos γ RT, above by z1n and below by z1p . Using the basic ge- 2 where ometry shown in Fig.11, expressions for z1n and z1p are:

10 zz=−ssinγ , Similarly, 0n 1 ∗n RT2 zz= − 2tπ ranγ , x (30) 21nn RT2 ∗ proj  (31) s =−ρ mod(∆,2π ) . 2 ∗nRT zz= −−zzcos γ . r 22pproj n proj RT2

zΠ

z2 z0n Π((pRT s))

γ RT2 Π((p s∗ )) z1p

Π()p proj

z1n

z2 p z1 z2n

2π r x Π Figure 11 Graphical representation of the range of the function Π

∗ The variables z2 p and z1p represent the z-  cos( ργRT (ss∗ −+1 ) ∆)cos RT 2 (p ) coordinates of the projections of Π proj onto the two ∗ T21()ss∗∗== sin()ργRT ( − s)+ ∆ cosRT , nearest legs of Π((p s)). These variables can now be 2 −sinγ RT2 used to find the approximation of zΠ , zapp :  ∗  −−sin ()ρRT (ss∗ lim ) +∆ zz11p ,ifp −

zz1 − app cosψγcos s = , (33) 2 RT1 −sinψ 2 ∗ sinγ  RT2 T = sinψγcos , N = cosψ . (36) 12RT1 12 which follows from the definition of p(s) for the spiral  −sinγ 0 RT1 descent segment. This value of s can now be used to ∗ And for segment 3, compute the unit basis vectors T2, N2, and B2 of the −cosψγfRcos T sinψ Frenet frame {F} for segment 2 at s = s 3 f ∗  T =−sinψcosγ, N =−cosψ . (37) dp dT 3 fRT3 3 f  c 2  ds ds TN22× −sinγ 0 T = , N = , B = (34) RT3 2 dp 2 dT 2 TN× c 2 22 The basis vectors B1 and B3 are computed exactly ds ds F as B2. Now the rotation matrix LTP Ri from LTP to Fre- or in the final form net frame for the i-th segment is given by F T LTP Ri = (TNi ,,i Bi ) and the corresponding position and velocity errors are

11 pp=−F Rs(()p)=:()0,y,zT , only use GPS position and velocity for feedback. This eLTPiRT∗ ee (38) constraint is motivated by the requirement to keep the FF ppe =−LTP Rsi ((RT ∗)V)+LTP Rsi (pRT (∗)−p). cost of the onboard avionics low, therefore, only GPS is Notice that by construction the x-component of the po- available for control. sition error vector p is zero, see Ref.34. These error During segments 1 and 2 the control system uses e differential flaps to drive the lateral components of the vectors were used to design a tracking control system position and velocity error vectors to zero. On the other discussed in the next section. hand, the last segment is tracked using both symmetric

V.2 Control System Design and differential flaps, denoted here by δ S and δ D , re- The trajectory tracking controller design methodology spectively. The structure of tracking controller for the proposed in Ref.34 is now applied to the design of a lateral channel shown is Fig.12. tracking controller for the trajectory generated using the algorithm developed in Section IV. The controller can

p ye Computation K y of the position and velocity error y δ V vectors e D K y

−1

trajectory - T Reference parameters trajectory

generator −1 z Figure 12 Lateral channel tracking controller structure

The gains K y and K y shown in Fig.12 were se- used to track the final segment of the trajectory. This lected to provide stability and performance for the lat- decision was motivated by the fact that symmetric flaps have negligible control authority in spiral descent. Fur- eral channel of the feedback system. The variables z−1 thermore, deflecting symmetric flaps tends to increase and T denote backward shift and sampling period. The the descent rate of the parafoil while reducing the flap sampling period for this problem was dictated by the deflection budget available for differential flaps com- onboard GPS receiver, whose update rate is 2Hz. mand and, therefore, symmetric flaps were not used The structure of the vertical channel controller is during segment 1 as well. shown in Fig.13. This controller uses symmetric flaps

δ S to drive the vertical channel error to 0 and was only

p ze Computation of the position δ PID controller S and velocity error z V vectors e

trajectory Reference parameters trajectory generator

Figure 13 Vertical channel tracking controller structure

VI. Simulation Results son between flight test data collected during Pegasus The trajectory tracking control system discussed in Sec- drop at the YPG and model's response to the wind col- tion V has been tested in simulation using a nonlinear lected during the same drop. Clearly, the 6DOF model 6DOF model of the Pegasus parafoil that also includes provides a reasonable approximation of the parafoil the available actuator model.35 Fig.14 shows a compari- dynamics.

Fig.15 shows results of an extensive simulation analysis of the trajectory tracking control system using the 6DOF model discussed above. Each of 40 simulation runs included the wind profile obtained at YPG. The GPS position and velocity errors were modeled as white noise processes with zero mean and standard de- viations shown in Tab.1. Finally, landing performance statistics shown in Tab.2 indicate that system's performance exceeded the required circular error probable (CEP) of 100m.

Table 1 GPS errors (STD) x-direction y-direction z-direction Position (m) 5 5 10 Velocity (m/s) 0.3 0.3 0.5

Table 2 Touchdown errors

Figure 14 Comparison of the flight test data and simula- CEP (m) Mean value (m) STD (m) tion results 70.7 67.5 20.5

Altitude (m) 3000

2000

1000 -2000 -1500 0 -1000 2000 DIP 1000 -500 0 0 -1000 South-North (m) East-West (m) -2000 500

Figure 15 Simulation results of the trajectory tracking control system

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