Performance, Control, and Simulation of the Affordable ...

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2000-08

Performance, Control, and Simulation of the

Affordable Guided Airdrop System

Williams, T.

American Institute of Aeronautics & Astronautics

Dellicker, S., Benney, R., Patel, S., Williams, T., Hewgley, C., Yakimenko, O., Howard,

R., and Kaminer, I., “Performance, Control, and Simulation of the Affordable Guided

Airdrop System,” Proceedings of the AIAA Guidance, Navigation, and Control

Conference, Denver, CO, August 14-17, 2000.

http://hdl.handle.net/10945/48932

(c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

AOO-37301AIAA Modeling and Simulation

Technologies Conference and Exhibit14-17 August 2000 Denver, CO

AIAA-2000-4309

PERFORMANCE, CONTROL, AND SIMULATION OF THE AFFORDABLE GUIDEDAIRDROP SYSTEM

S.DellickerUS Army Yuma Proving Ground, Yuma, AZ

R.Benney, S.PatelUS Army SBCCOM, Natick, MA

T.Williams, C.Hewgley, O.Yakimenko*, R.Howard, I.KaminerNaval Postgraduate School, Monterey, CA

AbstractThis paper addresses the development of an autono-mous guidance, navigation and control system for a flatsolid circular parachute. This effort is a part of the Af-fordable Guided Airdrop System (AGAS) that inte-grates a low-cost guidance and control system intofielded cargo air delivery systems. The paper describesthe AGAS concept, its architecture and components. Itthen reviews the literature on circular parachute mod-eling and introduces a simplified model of a parachute.This model is used to develop and evaluate the per-formance of a modified bang-bang control system tosteer the AGAS along a pre-specified trajectory towardsa desired landing point. The synthesis of the optimalcontrol strategy based on Pontryagin's principle of op-timality is also presented. The paper is intended to be asummary of the current state of AGAS development.The paper ends with the summary of the future plans inthis area.

I IntroductionThe United States Air Force Science Advisory boardwas tasked to develop a forecast of the requirements ofthe most advanced air and space ideas to project the AirForce into the next century. The study, encompassingall aspects of Air Force operations, assessed a variety oftechnology developments critical to the Air Force mis-sion. This study culminated in a report titled "NewWorld Vistas, Air and Space Power for the 21st Cen-tury."1 The study identified a critical need to improvethe Point-of-Use Delivery; that is, getting the materielwhere it needs to be, when it needs to be there. Airdropis an important aspect of Point-of-Use Delivery. Thereport indicated that immediate improvements areneeded with emphasis provided by the statement: "In

•National Research Council Senior Research Associate at theNFS.

This paper is declared a work of the U.S. Government and isnot subject to copyright protection in the United States.

the future, the problem of airdrop should be treated asseriously as the problem of bomb drop."

The first attempts to develop such systems are asold as the introduction of gliding, maneuverable para-chutes.2 However, practical systems had to wait for thedevelopment of hi-glide parachutes, especially the ram-air inflated parafoil. In 1969 the US Army defined re-quirements for and discussed such a cargo point deliv-ery system.3 None of attempts in the 70's and 80's todevelop such a system were operationally acceptable,however nowadays such systems have been developed(see for example Ref.4).

These large-scale parafoil systems use a marker orbeacon on the ground and ensure 99% landing accuracyin a hundred-yard circle around the beacon. Therefore,they provide the accuracy required with delivery fromhigh altitude and large offset distances. The drawback isprohibitive cost for each pound of payload delivered.Alternate approaches were required to reduce systemcost. Improved Affordable Airdrop Technologies arebeing evaluated by the team of the US Army and AirForce, the Naval Postgraduate School, The BoeingCompany, and Vertigo, Incorporated. These efforts in-clude the design and development of the AGAS, whichincorporates a low-cost guidance, navigation, and con-trol system into fielded cargo air delivery systems. Thisstudy focuses on evaluating the feasibility of the AGASconcept and encompasses the design and execution of aflight test program to assess dynamic response of a flatcircular parachute, the design of initial guidance andcontrol techniques, and to evaluate the feasibility of theAGAS concept.

II AGAS concept, architecture and componentsAGAS is being evaluated as a low-cost alternative formeeting the military's requirements for precision air-drop.5'6 Designed to bridge the gap between expensivehigh glide parafoil systems and uncontrolled (ballistic)round parachutes, the AGAS concept offers the benefitsof high altitude parachute releases but cannot providethe same level of offset from the desired impact point(IP) as high-glide systems. The design goal of the

1American Institute of Aeronautics and Astronautics

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http://crossmark.crossref.org/dialog/?doi=10.2514%2F6.2000-4309&domain=pdf&date_stamp=2012-08-22


AGAS development is to provide a Guidance, Naviga-tion, and Control (GNC) system that can be placed in-line with existing fielded cargo parachute systems (G-12and G-ll) and standard delivery containers (A-22). Thesystem is required to provide an accuracy of 100m, Cir-cular Error Probable (CEP), with a desired goal of 50mCEP. No changes to the parachute or cargo system areallowed.

The current design concept includes implementa-tion of commercial Global Positioning System (GPS)receiver and a heading reference as the navigation sen-sors, a guidance computer to determine and activate thedesired control input, and the application of PneumaticMuscle Actuators (PMAs) to effect the control. Thenavigation system and guidance computer will be se-cured to existing container delivery system while thePMAs would be attached to each of four parachute ris-ers and to the container (Figure 1). Control is affectedby lengthening a single or two adjacent actuators. Theparachute deforms creating an unsymmetrical shape,essentially shifting the center of pressure, and providinga drive or slip condition. Upon deployment of the sys-tem from the aircraft, the guidance computer wouldsteer the system along a pre-planned trajectory. Thisconcept relies on the sufficient control authority to beproduced to overcome errors in wind estimation and thepoint of release of the system from the aircraft. Fol-lowing subsections discuss main AGAS components.

Figure 1. Affordable Guided Airdrop System

For an airdrop mission, the aircrew will determinethe Computed Air Release Point (CARP) based on thebest wind estimate available at that time. The aircraftwill then be navigated to that point for air delivery ofthe materiel. Should the wind estimate and calculationof the predicted release point be perfect and the aircrewgets the aircraft to the precise release point, then theparachute would fly precisely to the target without con-

trol inputs. However, wind estimation is far from a pre-cise science. The calculation of the CARP relies on lessthan perfect estimates of parachute aerodynamics andthe flight crews cannot possible precisely hit the pre-dicted release point for each airdrop mission. Therefore,the AGAS control system design must help overcomethese potential errors.

II.l ParachuteUntil now two solid flat circular parachutes C-9 and G-12 were modeled to demonstrate a feasibility of AGASconcept. (A flat circular parachute is one that when laidout on the ground forms a circle.2) Figure 2 shows adeployed configuration of C-9. Although the C-9 wasinitially designed as an ejection seat parachute, it is astandard flat circular parachute as are the larger G-lland G-12 cargo parachutes on which AGAS will ulti-mately be used. Some data on these parachutes can befound in the Table 1.

Figure 2. C-9 Parachute with 28 goresTable 1. Parachutes data2

Parameterd0 (ft)dp/d0

Number of suspension linesI0/d0

CDOParachute weight (Ibs.)Payload weight (Ibs.)Rate of descent (fps)

C-928

0.6728

0.82

0.6811.320020

G-1264

0.6764

0.80

0.73130

2,20028

Gll-A100

0.671200.90

0.68215

3,50022

Courtesy of Vertigo, Inc., Lake Elsinore, CA.

In this table dQ denotes the nominal diameter ofthe parachute, dp - inflated canopy diameter, CDO - adrag coefficient, and /0 - a suspension line length.

A cargo box is suspended from the system andhouses the remote control system, control actuators, andinstrumentation system.

II.2 ActuatorsVertigo, Incorporated developed PMAs7 to effect thecontrol inputs for this system. The PMAs are braidedfiber tubes with neoprene inner sleeves that can be pres-surized. Uninflated PMAs as installed on a scaled sys-

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tern are shown in Figure 3. Upon pressurization, thePMAs contract in length and expand in diameter.

Figure 3. Pneumatic muscle actuators

With four independently controlled actuators, twoof which can be activated simultaneously, eight differ-ent control inputs can be affected. The concept em-ployed for the AGAS is to fully pressurize all actuatorsupon successful deployment of the parachute. To affectcontrol of the system, one or two actuators are depres-surized. This action "deforms" the parachute creatingdrive in the opposite direction of the control action.

Figure 4 shows a diagram of the actuator setup inthe parachute payload provided by Vertigo, Incorpo-rated, the makers of the PMAs. The gas for filling theactuators comes from 4500psi reservoirs (the diagramshows two, but in the simulation for this study, only asingle 4500/751 reservoir is used). Each of the four ac-tuators are then connected to this same reservoir of ni-trogen gas through some piping or tubing leading to afill valve. The fill valve is opened to allow gas to fill theactuators when a command to take an actuation off isreceived. When the pressure inside the PMA reaches acertain value, a pressure switch signals the fill valve toclose.

PMAs

Vent valve

Electrical200ps/SOOpsi

—— 4500pw

Rgure 4. Vertigo, Inc. actuator system concept

Since the fill valve works with high-pressure gas ithas a small orifice and therefore opens and closes ratherquickly upon receiving the correct electrical signal. Thetime to open and close the valve is roughly 100ms.

However, the decrease in pressure of the gas tank asmore and more fills are completed slows down the ac-tual filling process. Some of this data is plotted in Fig-ure 5, showing increasing fill time as a function of de-creasing tank pressure for actuators being filled to threedifferent pressures.

;- — _ __ — set pressure100 psi -150 psi -

X/•

"

/'':

25

20

15 Sdark blue (dots) / / // . j «light blue (dashes) -r/T^ •' ,'•'•'• I ~

^.--^,'^^ff f: •'' : ,.'.'10 i

—il

•"- ^-^i:^^''-" /'" ,-:'-'.'-''

„000 4500 4000 3500 3000 2500 2000 1500 1000 500 0

Tank pressure (psi)Figure 5. Fill time versus tank pressure

The vent valve opens to empty the actuator when acommand to actuate is received. The vent valve has alarge orifice and can open quickly to vent the PMA, butrequires a certain time to vent the gas and close the ori-fice. The opening of the vent valve requires approxi-mately 100ms, but the venting process and closing ofthe valve depends on the maximum pressure of the ac-tuator fill. This process also takes a constant amount oftime (approximately) because the pressure in the actua-tors is the same upon each vent. Figure 6 includes adiagram of the computer-modeling concept for the ac-tuators.

Controller commandsips)

Valve responses(ps>>

PMA pressures(psi)

Reservoir pressure changeper PMA pressure change

Figure 6. Actuator modelController commands are input to the system for each ofthe four PMAs. The controller commands are the pres-sure signals for each PMA; with Opsi being a vent of theactuator and \15psi (or the maximum pressure of theactuator) being a fill of the PMA. These commands arethen passed through code that models the dynamics ofthe valves. This code is just a first order lag with a risetime of approximately 100ms to model the opening andclosing of the valves. The valve responses are thenpassed to a code that models the actual venting andfilling of the actuators, keeping in mind that the ventingprocess takes a constant amount of time and the fillingprocess increases with decreasing tank pressure. Onceagain this behavior is modeled as a first order lag. Thiscode outputs the derivative with respect to time of the


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PMA pressures. This is integrated to give the currentstate of each of the PMAs at a given time. PMA filltime is calculated by passing the change in PMA pres-sures through a gain that reflects data taken on the res-ervoir pressure changes per PMA pressure changes orthe amount of reservoir pressure depleted for every ac-tuator fill. This negative gain is integrated from an ini-tial reservoir pressure to give current reservoir pressure.A look-up table is used to provide a value for the filltime of the actuators based on the remaining reservoirpressure from experimental data.

Ill Parachute modelingSignificant amount of research on flat circular parachutemodeling has been done over past fifty years by re-searchers in US, Europe and Russia. It has been recog-nized that the basic parachute model is similar to that ofa conventional aircraft. However, parachutes do haveseveral unique features. First of all, they must be mod-eled as a flexible structure that includes both the para-chute and the payload. Secondly, the added masses mustalso be modeled. And lastly it appears that no one at-tempted to model the control effectiveness of the risers.This section summarizes some of the research done inthis area and introduces a simplified model used in thispaper for preliminary analysis of control strategies.

III.l Literature reviewIn 1968 White and Wolf8 developed a 5-DOF dynamicmodel (ignoring the N equation, "yaw" about thedownward z-axis), reviewing Henn (1944) and Lester(1962), noting that Lester discussed certain errors in theequations of motion by Henn. They noted that Lestercarefully derived the equations of motion but did notattempt any solutions. They used scalar values for ap-parent mass and moment of inertia and conducted adynamic stability study in which 1) steady vertical de-scent, 2) steady gliding (a * 0 , with a defined fromthe z axis), 3) large-angle pitch oscillation, and 4) con-ing are considered. A linearized analysis led to uncou-pled longitudinal and lateral motions as is typical withaircraft.

It was noted that because of the shapes of the force-coefficient curves (CN = *Jx2 +Y2(qS0)~}, the normalforce coefficient, and CT = -Z(qS0)"', the tangentialforce coefficient), most parachutes do not fall vertically,but trim at a stable glide angle a0. Even so, statically-stable parachutes may oscillate in pitch rather thanglide. The authors noted that based on data from Hein-rich and Haak9, most parachutes could not develop suf-ficient CNa to be glide stable except at high loadmasses, high descent rates, and low effective porosity.They noted the personnel guide surface canopy, which

has low CNa and high canopy-mass-to-payload-massratio, would be expected to oscillate rather than glide.Small lateral motions may result in neutrally-stable os-cillations, but coning cannot develop: the longitudinaland lateral motions are uncoupled.

A nonlinear analysis produced different results. Aparachute stable in the linearized sense will jump to alarge-angle pitch oscillation if struck with a longitudinaldisturbance greater than half of its stable angle ce0. Onthe other hand, a large lateral disturbance induces alongitudinal motion of comparable magnitude, throwingthe parachute into a uniform vertical coning motion. Tosummarize: a glide-stable parachute is neutrally stableto a small lateral disturbance, but may jump into a verti-cal coning motion if hit with a large lateral disturbance.A single coning solution was found to exist for anygiven system, and only the parameters a0 and CNa

have a great effect on coning.Doherr10 attempted to simulate the oscillatory be-

havior of a parachute and payload using two rotationalDOE for the payload and three rotational DOE for theparachute. He noted that theory predicts all parachutesto assume a stable position at or = otaable, while duringwind-tunnel tests the less stable parachutes continued tooscillate. He recommended that aerodynamic coeffi-cients be functions of both a and time, i.e.,CN=CN(a(t\t}.

Tory and Ayres11 used a complete 6-DOF model.Differences they pointed out with White and Wolfsmodel are the additional DOE, forces considered on thepayload, and the consideration of the apparent mass as atensor. They showed plots of lift, drag and moment on aflat circular canopy determined from wind-tunnel tests,which will be useful for aerodynamic modeling. Theymodeled the apparent masses and moments of inertiabased on Lamb's expressions and the assumption of thecanopy being shaped as a 2:1 ellipsoid. They comparedtheir results to some flight data, and noted that apparentinertia may not be as critical a parameter as had beensuggested. A simulated drop exhibited oscillations of±30° with a period of 5 seconds, coincident with theplane of the trajectory (therefore all pitch - no coning).Actual tests showed oscillations of 20-30° magnitudewith a period of about 4-5 seconds.

Eaton and Cockrell12 described a series of planneddrop tests, from which the center of rotation was to bedetermined and the apparent mass estimated, but notests were apparently conducted.

As noted by Lester and presented in the open lit-erature by Doherr and Saliaris13, in a 6-DOF system inwhich apparent mass and inertia terms are important, anapparent mass tensor can be represented by the coeffi-cients dy to form a symmetric matrix, which for the


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rigid parachute-payload system satisfies the followingequalities: atj = ajt, a22 = oru ^ 0 , a33 ^ 0,#44 = #55 * 0, als = -QT24 * 0. The rest of the ele-ments in this case are equal to zero.

Doherr and Saliaris13 noted that predicted dynamicstability depends strongly on the math model of appar-ent mass. They detailed errors of Henn as pointed outby Lester. It is interesting that though White and Wolfdiscussed Lester's corrections of Henn's equations, theychose to use a scalar model of apparent mass and mo-ment of inertia, rather than include the apparent-mass-tensor terms as documented by Doherr and Saliaris.Doherr and Saliaris use the set of apparent-mass termsin a small-disturbance 3-DOF model of dynamic stabil-ity. For apparent mass ma they use half the mass of theair included within the canopy hemisphere, "arbitrarily"as noted later by Cockrell and Haidar14. The apparent-mass terms are given by the following equations:

a22=an, #33=20?,,,=f"n = — p—m ——" 4 3 2

=0.04Saudp and =±0.2anl0. (1)Doherr and Saliaris considered five cases: i)

au = a33 = als = ass = 0 (all apparent mass terms be-ing neglected); ii) a33 =aH and or15 =0 (apparentmass constant, without the coupling term); iii)Gf33 = 2an (tangential apparent mass is twice normalapparent mass); iv-v) aJ5 is negative and positive.

The authors noted the importance of nonlinearforce terms and apparent-mass effects. They referred tothe Yavuz and Cockrell15 study which showed that theapparent-mass terms are not constant, but are dependentupon the acceleration modulus (for example, wdPV~2

for «33).Cockrell and Doherr16 referred to White and Wolf

as a widely accepted and employed parachute modelingmethod, yet lamented the lack of validation from flighttest data. They aggressively pointed out that the equa-tions of Tory and Ayres were developed incorrectly,though the particular errors were not noted. Cockrelland Doherr showed the same 3-DOF equations as Do-herr and Saliaris denoting them as Lester's equations.They also listed the full 6-DOF form of the equationsand referred to Eaton17 for their derivation.

Yavuz and Cockrell15 repeated the same 6-DOFequations as Cockrell and Doherr with the same re-maining independent apparent-mass terms: au, a33,a55 , and a,5 . Experiments are described that allowedthe authors to obtain values of the apparent mass coeffi-cient ktj , where ktj = atj (pVml }~l , and Vml is the vol-ume of the fluid displaced by the body - this case, the

volume of a hemisphere. Values of ktj are presented asfunctions of angle of attack and acceleration modulus(VdpV2). Values of kv may vary by a factor of fourwith acceleration modulus and the same with angle ofattack. No estimations were made of practical values ofacceleration moduli for dropped parachutes.

Eaton17 noted in his paper that the complete formof the added mass tensor for a rigid axisymmetric para-chute is obtained and implemented correctly for the firsttime. The error in Tory and Ayres is discussed as beinga reversion to Henn's original faulty assumption of areal physical distinction between "included" and "ap-parent" mass. In Baton's analysis, the actual masses ofthe canopy, rigging lines, and payload appear in thegoverning equations along with the four added masscomponents. Experiments led Eaton to run simulationswith values of an to be 0.0 to 0.5 of its analogoussolid-body inertial tensor component, with a baselinevalue of 0.2, and of a33 to be 0.0 to 1.0 of its analogoussolid-body inertial tensor component, with a baselinevalue of 0.4. It is noted that the "current order-of-magnitude estimates (of atj) are very inadequate forstability studies on personnel-type, low-porosity sys-tems."

Cockrell and Haidar14 began the popular later ap-proach of developing higher-order models, regardingthe canopy and payload as coupled sub-systems. As foradded mass effects, the authors followed Doherr andSaliaris, taking a representative value for all a^ I ma of1.0.

Later papers continue to develop higher-ordermodels (separate DOF for payload and canopy, rangingfrom 9 to 15 DOF).

Russian scientific school (Refs. 18-20) proceed withthe flexible structure parachute-payload and uses appar-ent-mass terms, obtained from the numerical simulation.

Summarizing, over a 30-year period, investigatorshave expressed concern over the lack of accurate dy-namic modeling of apparent-mass effects, yet it appearsthat no studies have estimated practical values of accel-eration moduli for coning, oscillating parachutes, orsuccessfully implemented values for atj that treat themas functions of a or of acceleration module.

III.2 Simplified modelFor preliminary study discussed in this paper the fol-lowing simplified three-degree-of-freedom (3-DOF)model was used:

uV

wM -l •

~(}CDQS0

VA

uV

w

' 0 "0

mg

recant (2)


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where u, v, and w are the airspeed components on theaxes of the body coordinate frame, au , i = 1,3 are theapparent masses, m is the mass of payload and para-chute assembly, q is the dynamic pressure,S0 = 0.25;zrf o is a canopy surface area, g is accelerationdue to gravity, and F™"' are the force effect of thecontrol actuators (in just the x and y directions). Theapparent mass terms are based on the included canopyair mass ma and are computed from the Eqns.l:

The wind speed is added to the airspeed to givegroundspeed:

VG=VA+VW (3)Control forces are calculated based on the pressure

of the four actuators and the assumption (based on flighttest data) that one control input at a time causes a 0.4glide ratio and two control inputs at a time causes a 0.2glide ratio. This control force is then used in the calcu-lation of the linear accelerations of the parachute byEqn.2, along with other parachute properties such as itsmass, size, and weight, and the dynamic pressure of theatmosphere which is dependent on altitude. Linear ac-celeration is integrated to give airspeeds. Groundspeedis integrated to give true positions in x, y, and z coordi-nates of the parachute. The parachute also has a con-stant yaw rate (y/ = 0.03s"1) with small perturbationsfrom this constant, and zero pitch and roll rates. Theseangular rates are integrated to give the Euler angles ofthe parachute, which are used to transform the coordi-nate axes of the parachute from the body to inertial co-ordinates or vice versa.

IV Derivation of the optimal control strategy

IV.l General statement of optimization problemBased on the AGAS concept introduced above, the op-timal control problem for determination of parachutetrajectories from a release point to the target point canbe formulated as follows: among all admissible trajec-tories that satisfy the system of differential equations,given initial and final conditions and constraints oncontrol inputs determine the optimal trajectory thatminimizes a cost function of state variables z andcontrol inputs ii

J =) fo(t,z,u)dt (4)

and compute the corresponding optimal control.For the AGAS, the most suitable cost function J is

the number of actuator activations. Unfortunately thiscost function cannot be formulated analytically in theform given by expression (4). Therefore, we investi-gated other well-known integrable cost functions and

used the results obtained to determine the most suitablecost function for the problem at hand.IV.2 Application of Pontryagin's maximum princi-ple of optimalityTo determine the optimal control strategy we appliedPontryagin's principle21 to a simplified model of para-chute dynamics. This model essentially represents para-chute kinematics in the horizontal plane (Figure 7):

i = u cos iff - v sin y/y = usiny/ + vcosy/ (5)iff = C = const

Each of four actuators in two control channels canbe activated in the manner allowing the following dis-crete speed components in the axis of the parachuteframe: «,ve [-V;0;Vj. We considered these speedcomponents as controls for the task at hand.

pjvaeinrfeTarget

Two pairs ofrotating controls

XFigure 7. Projection of the optimization task onto the horizontal plane

The Hamiltonian21 for the system (5) can be writtenin the following form:

i / px cosy/ + p siny/ \H=(u,v\ +1 -pxsmy/ + py cosy/j

where equations for ajoint variables px, py, and p¥

are given by

p C-f0(6)v

Px=0 Py=Q

i>r=(p*>i(7)

(8)

We consider two cost functions/0 = 1 - minimum time/o = |M|+ H • minimum 'fuel'

According to Ref.21, the optimal control is deter-mined as uopt = argmaxrl(p,z,u). Therefore, for thetime-minimum problem the optimal control is given by

-^..(9)

Figure 8 shows the graphical interpretation of theseexpressions. In general, the vector \px,py] defines a


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direction towards the target and establishes a semi-planeperpendicular to itself that defines the nature of controlactions. Specifically, if an actuator happens to lie withina certain operating angle A with respect to the vector(px,py ) it should be activated. For a time-optimumproblem since A = n two actuators will always be ac-tive. Parachute rotation determines which two. (We donot address the case of singular control, which in gen-eral is possible if the parachute is required to satisfy afinal condition for heading). Figure 9 shows an exampleof time-optimal trajectory. It consists of several arcs anda sequence of actuations (for this example \jf = 0.175s'1

and V =5m/ s ).For the 'fuel'-minimum problem we obtain analo-

gous expression for optimal control inputs:=> u = V

pxcosi//+ py

pycosy/>Vpy

u = usc

v-Vv = -V (10)

In this case actuators will be employed when appropri-ate dot products will be greater than some positivevalue. Obviously, this narrows the value of the angle A.In fact, for this particular cost function A —» 0. In gen-eral any cost function other than minimum-time willrequire an operating angle A < n (Figure 10).

t.^(cos^,sin^)

Figure 8. Time-optimal controlTarget point

s«sy,<.

Figure 9. Example of the time-optimal trajectory and time-optimalcontrols

Figure 11 shows the effect of operating angle onthe flight time, 'fuel' and number of actuator activations.It is clearly seen that the nature of the dependence of thenumber of actuations on the operating angle is the sameas that of the time of flight. This implies that by solvingthe time minimum problem we automatically ensure aminimum number of actuations. Moreover, it is alsoseen that the slope of these two curves in the intervalA 6 [0.57r; n\ is flat. This implies that small changes ofan operation angle from its optimal value will result innegligible impact on the number of actuations. There-fore, changing the operating angle to account for therealistic actuator model will not change the number ofactuations significantly.

Figure 10. Generalized case of optimal control400

<8 300

200

100

*—a— 'fuel'— £— numb, of switches

s\-l— — -

i '' >i•j i^. ,...„..: f """*••-

^—^*

^or"

-.... _ _

-••'*

j^^^ 1

.................. ...,,'J

25 .x:

- 20 ..H

13 o

o>- 1 0 f

z- 5

) 50 100 150 200Operating angle A <deg)

Figure 11. Influence of operating angle

IV.3 Control strategyPreceding analysis suggested that the shape of optimalcontrol is bang-bang. Therefore, for preliminary nu-merical simulation in presence of wind the control strat-egy was established as follows.

Considering the relatively low glide ratio demon-strated in flight test (approximately 0.4-0.5) with a de-scent rate of approximately 25ft/s#, the AGAS couldonly overcome a twelve foot per second (approximatelyIkns) wind. It is therefore imperative that the controlsystem steers the parachute along a pre-specified tra-jectory obtained from most recent wind predictions.This can be done by comparing the current GPS posi-

Equilibrium velocity is given by the formula2mg

sired altitude.

, where p is a mass density of air at de-


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tion of the parachute with the desired one at a givenaltitude to obtain the position error( Pe = (z f - zn 1 ). Furthermore, to eliminate actuator

' ^ i " i\h=fix

'oscillations', a tolerance cone is established around theplanned trajectory (Figure 12) starting at 600ft. at thebeginning of the trajectory and gradually decreasing to60ft. at ground level. Should the position error be out-side this tolerance, a control is activated to steer thesystem back to the planned trajectory. When the systemis within 30ft. of the planned trajectory the control isdisabled and the parachute drifts with the wind. Thirtyfeet was selected to encompass approximately one-sigma of the GPS errors (Selective Availability off).

TargetPoiat

Compiled AirRelease Point

• Generate predicted trajectory based onwiud estimate•Establish control tolerance• Compare actual position to predicted trajectory• 'Drive' to predicated trajectory

Figure 12. Control concept

As outlined above, the control system relies on thecurrent horizontal position error to determine whetherthe control input is required. This position error is com-puted in inertial coordinate system and is then con-verted to the body axis using an Euler angle rotationwith heading only. The resulting body-axis error ( P b ) isthen used to identify which control input must be acti-vated

input=*R (ID

Trying to account for maximum refill time and sen-sors errors we chose A = 2.5 instead of A = n (Figure13). This allows the activation of a single control inputor two simultaneous control inputs.

Figure 13. Control activationBoth the tolerance cone and the operating angle con-straints must be active for a given PMA to be activated.

Figure 14 shows results of a simulation run that pro-vides an insight into this control logic. The simulationuses a wind prediction profile that matches the windprofile used in the actual parachute simulation. Theparachute is released at an offset from the ideal droppoint of 2500ft. The plots show that the proper PMAsare activated (vented) when the tolerance cone and theoperating angle constraints are active. One can see thatat the end of the simulation that the parachute has justmade it within 100m of the target. This brings up theconcept of the "feasibility funnel." The feasibility fun-nel is defined as the set of points maximum distanceaway from the predicted trajectory for which the vehiclestill has sufficient control authority to land within acertain distance from the target. The third plot in Figure9 shows a line in the "feasibility funnel."s 1» 0.5x•5 0E -0.5Z -1

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Time (s)Figure 14. Example of control histories

V Simulation resultsInvestigation of the anticipated performance of the en-tire system was conducted using computer simulationincorporating the actuator model and control strategydescribed above. The first goal of this investigation wasto determine the effectiveness of the described "trajec-tory-seeking" control strategy versus a control strategythat simply seeks the target landing position withoutusing any knowledge of the winds. The second goal ofthis effort was to estimate the impact of changing thecharacteristics of the actuator system on the overallsystem performance.

One prerequisite to both avenues of investigationwas to obtain a complete set of wind information for thedrop zone. Wind information was gathered from theYuma Proving Ground (YPG) "Tower M" drop zoneusing eleven Rawinsonde balloons released at one-hourintervals throughout the day on March 7th, 2000. Themagnitude and direction of the wind measured by theseballoons is shown in Figure 15.


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different control strategies were named "trajectory-seek" and "target-seek," respectively.

40 60 80 100Velocity (ftfs)

-180 -150 -120 -90 -60 -30 0Direct] on (degrees)

Figure 15. Measured wind velocity and direction versus altitude

In these plots, the measured wind magnitude anddirection from the first balloon of the day is shown us-ing a heavier line, and the data from the subsequentballoon releases are shown as lighter lines. Qualitativeobservations of wind magnitude and velocity can begained from these plots. For the altitude range of inter-est, which was zero to 10,000/t, the horizontal windvelocity changed by up to 20Jps during the time span ofthis experiment, and wind direction in this altitude bandchanged by as much as 100°.

Using this wind information, a series of computersimulations of the parachute drop were conducted. Thefirst set of simulations was run in order to visualize theeffects of having a wind prediction that differs from theactual wind. The first balloon data from the day wasused as a wind prediction to calculate the nominal tra-jectory the parachute is supposed to follow. All the sub-sequent balloon data were used as the actual winds, soten actual trajectories were computed. The predictedtrajectory and the ten actual trajectories are shown in athree-dimensional plot in Figure 16.

The release point of these simulated drops is di-rectly above the origin of the horizontal plane, at analtitude of 9,500ft. All of the actual parachute trajecto-ries were computed using no control of the parachutefall. The impact point (zero altitude) of the predictedtrajectory has the highest positive value on the north-south axis. From the plot, it can be seen that the pre-dicted trajectory lands north and west of the drop point,whereas the simulated drop using the last wind obser-vation of the day lands south and west of the drop point.This observation is consistent with the wind directiondata presented in Figure 16; the wind shifted from thesoutheast to the northeast during the course of the windmeasurement experiment.

Next, a set of computer drop simulations was per-formed in order to assess the advantage of a controlstrategy that relies on a predicted trajectory based onwind information that is not current versus a controlstrategy that ignores the wind prediction and simplydrives toward the target impact location. These two

Figure 16. Sample parachute trajectoriesA total of 437 simulation runs were made. On each

run, the drop trajectories were computed for parachutesusing both the trajectory-seek and target-seek strategies,as well as for a parachute that falls with no control in-puts. Each simulation was conducted with one of theeleven Rawinsonde data files used as the actual wind forthat simulation. A wind prediction was chosen at ran-dom from among the Rawinsonde data files that weregathered earlier in the day than the one selected for theactual winds. Therefore, if the data from the secondballoon release of the day were chosen as the actualwind, then the data from the first balloon release of theday had to be chosen as the prediction, guaranteeing aprediction only one hour old. On the other hand, if thedata from the last balloon release of the day were cho-sen as the actual wind, then a wind prediction from oneto ten hours old could be used to compute the predictedtrajectory. The predicted wind information was used inorder to determine the CARP and also to determine thepredicted fall trajectory for the trajectory-seek controlstrategy.

Another variable in these simulations was that theparachutes were dropped from a point somewhat offsetfrom the CARP. The offset from the CARP was meantto simulate that the releasing aircraft did not hit theCARP exactly. The offsets were modeled as independ-ent normally distributed random variables in the north-south and east-west directions. In order to determinehow much of an offset to use from the CARP, an ex-periment was done to determine the size of the "area ofattraction." This area is defined as the area around theCARP in the horizontal plane within which the para-chute can be dropped and still land to within 100m ofthe target position using the onboard control system. Inorder to simplify this first set of experiments, the area ofattraction was calculated without factoring in the effectof wind, so that the area is symmetric: a circle aroundthe CARP in the horizontal plane. The standard devia-tions of the distributions of the two offsets were set atone-fourth of the radius of the area of attraction. An


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overhead polar plot of the actual release and impactpositions for the trajectory-seek control strategy isshown in Figure 17.

330A knpMt fdM :® RdeamPatft \•* CARP / *- »

276

Figure 17. Trajectory-seek simulations

This plot shows that many of the release positionswere southeast of the target point, relying on the pre-dicted southeasterly winds to carry the parachute to thetarget point. Since the winds shifted from southeast tonortheast during the day, many of the parachutes landedsouth of the target zone. Each of the range rings on thispolar plot represents 2,000ft. A histogram of the simu-lation results is shown as Figure 18.

No-control startegyTrajectory-seek strategyTarget-seek strategy

Landing miss distance (m)Figure 18. Histogram of miss distances

From this histogram, it can be seen that the trajec-tory-seek strategy showed a noticeable improvement inlanding accuracy over the target-seek strategy. For thetrajectory-seek strategy, over half of the simulation runslanded within 100m of the target, meeting the systemperformance goal of 100m of CEP. One factor affectingthe data for these simulation runs was that the age of thewind predictions was not uniform. Figure 19 shows theage distribution of the wind predictions.

One can infer from this histogram that the trajec-tory-seek algorithm did have an advantage over the tar-

get-seek algorithm due to the fact that a substantial pro-portion of the simulation runs used wind data for thepredicted trajectory that was fairly current.

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Age of wind prediction (hours)Figure 19: Distribution of wind predictions

VI Conclusions and recommendations for furtherresearch

The results presented in this paper indicate that a simplebang-bang control strategy may be sufficient to achievethe target lOOm CEP for AGAS landing dispersion.However, further research is warranted. In particular,we propose to choose a 5-DOF model similar to theapproach of White and Wolf, but with the tensor modelof apparent masses and proper form of the equationsfrom Eaton. There appears to be no current ability toresolve an accurate apparent-mass tensor as a functionof anything more complex than parachute geometry. Aconstant yaw rate determined from flight test will beused in lieu of the 6* DOF. As a first step, the line-arized 5-DOF equations will be programmed and com-pared to the current 2-DOF model in controlled per-formance. Then the nonlinear equations will be used tocomplete the modeling.

Aerodynamic terms will be extracted from theavailable terms in the literature reviewed above. Controlterms will be determined from flight test data. It is ofinterest to estimate the relative accuracy of the current3-DOF and the proposed 5-DOF models with regards tocontrol strategy and required inputs. Should the addedcomplexity of the 5-DOF model prove to contribute noapparent benefit in fidelity, the 3-DOF model will beretained.

Further analysis of the control-strategy will seek todetermine the most effective operating angle A (be-cause of changing aerodynamics maybe it could be bet-ter to use A < 0.5;r). Furthermore, the optimal operat-ing angle may be asymmetric because of the differentfill and vent times (Figure 21).

Figure 21. Questions to be answered in the further analysis


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We also intend to perform additional work on de-termining the proper region of attraction and tolerancecone.

References1. "Summary Report: New World Vistas, Air and Space

Power for the 21st Century," United States Air ForceScience Advisory Board, 1997.

2. Knacke, T.W., "Parachute Recovery Systems DesignManual," Para Publishing, Santa Barbara, CA, 1992.

3. Forehand, I.E., "The Precision Drop Dlider (PDG),"Proceedings of the Symposium on Parachute Technologyand Evaluation, El Centra, CA, April 1969 (USAF reportFTC-TR-64-12, pp.24-44)

4. Final Report: Development and Demonstration of aRam-Air Parafoil Precision Guided Airdrop System,Volume 3, Draper Laboratory under Army ContractDAAK60-94-C-0041, October 1996

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6. Dellicker, S., "Low Cost Parachute Navigation Guidanceand Control," MS thesis, Naval Postgraduate School,September 1999.

7. Benney, R., Brown, G. and Stein, K., "A New PneumaticActuator: Its Use in Airdrop Applications," AIAA 99-1719, 15th CAES/AIAA Aerodynamic Decelerator Sys-tems Technology Conference, June 1999.

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9. Heinrich, H.G. and Haak, E.L., "Stability and Drag ofParachutes with Varying Effective Porosity," Wright-Patterson AFB, ASD-TDR-62-100, Sept. 1962.

10. Doherr, K.-F., "Theoretical and Experimental Investiga-tion of Parachute-Load-System Dynamic Stability,"AIAA-75-1397, 1975.

11. Tory, C. and Ayres, R., "Computer Model of a FullyDeployed Parachute," Journal of Aircraft, Vol.14, No.7,1977, pp.675-679.

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13. Doherr, K.-F. and Saliaris, C., "On the Influence of Sto-chastic and Acceleration Dependent Aerodynamic Forceson the Dynamic Stability of Parachutes," AIAA-81-1941, Oct. 1981.

14. Cockrell, D.J. and Haidar, N.I.A., "Influence of the Can-opy-Payload Coupling on the Dynamic Stability in Pitchof a Parachute System," AIAA-93-1248, May 1993.

15. Yavus, T. and Cockrell, D.J., "Experimental Determina-tion of Parachute Apparent Mass and Its Significance inPredicting Dynamic Stability," AIAA-81-1920, Oct.1981.

16. Cockrell, D.J. and Doherr, K.-F., "Preliminary Consid-eration of Parameter Identification Analysis from Para-chute Aerodynamic Flight Test Data," AIAA-81-1940,Oct. 1981.

17. Eaton, J.A., "Added Mass and the Dynamic Stability ofParachutes," Journal of Aircraft, Vol.19, No.5, 1982,pp.414-416.

18. Antonenko, A., Rysev, O., Fatyhov, F., Churkin, V. andYurcev, Y., "Flight Dynamics of Parachute Systems,"Mashinostroenie, Moscow, 1982.

19. Shevljakov, Y., Temnenko, V. and Tischenko, V., "Dy-namics of Parachute Systems," Vischa Shcola, Odessa1985.

20. Rysev, O., Belozerkovsky, S., Nisht, M. and Ponomarev,A., "Parachutes and Hang-Gliders Computer Investiga-tions," Mashinostroenie, Moscow, 1987.

21. Pontrjagin, L., Boltjanskiy, V., Gamkrelidze, R. andMishenko, E., "Mathematical Theory of Optimal Proc-esses," Nayka, Moscow, 1969.


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