[American Institute of Aeronautics and Astronautics AIAA Atmospheric Flight Mechanics Conference and Exhibit - Keystone, Colorado ()] AIAA Atmospheric Flight Mechanics Conference and

Modeling and Simulation of 9-DOF Parafoil-Payload

System Flight Dynamics

Om Prakash∗and N. Ananthkrishnan†

Indian Institute of Technology - Bombay, Mumbai 400076, India

Parafoil-payload system requires a 9-DOF dynamic model representing three degreesof freedom each for rotational motion of the canopy, and payload, and three degrees offreedom for translational motion of the confluence point of the lines. The gliding flight ofparafoil-payload system is directly affected by choice of rigging angle. The glide angle iscontrolled by symmetric trailing edge (brake) deflection and turn is effected by asymmetricbrake deflection. The parafoil trim and stability characteristics are a complex function ofrigging angle and magnitude of downward deflection of left and right brakes. The presentwork uses a 9-DOF model to analyze gliding and turning flight of the parafoil-payloadsystem for different choices of design rigging angle subjected to change in left and rightbrake deflections. Also, 9-DOF model is used to show the possibility of heading controlusing Nonlinear Dynamic Inversion law by close-loop simulations

Nomenclature

A,B,C apparent mass termsb canopy spand length of brake control line pulledCD drag coefficientCL lift coefficientCY side force coefficientClp, Clr rolling moment damping coefficientsCmc/4 pitching moment coefficient at quarter chordCmq pitching moment damping coefficientCnr, Cnp yawing moment damping coefficientsc canopy chord lengthF forceg acceleration due to gravityIA, IB , IC apparent inertia termsIF parafoil apparent inertia matrixM mass matrixm massq dynamic pressure (= 1

2ρV2)

p, q, r roll, pitch, and yaw ratesSb payload (body) cross-section areaSp parafoil planform areaTb transformation matrix from inertial reference frame to payload reference frameTp transformation matrix from inertial reference frame to parafoil reference framet canopy maximum thicknessu, v, w velocity components along reference frameV total velocityX,Y, Z body-fixed reference frame

∗Ph.D. Student, Department of Aerospace Engineering; [email protected]. Student Member AIAA.†Associate Professor, Department of Aerospace Engineering; [email protected]. Senior Member AIAA.

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American Institute of Aeronautics and Astronautics

AIAA Atmospheric Flight Mechanics Conference and Exhibit21 - 24 August 2006, Keystone, Colorado

AIAA 2006-6130

Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

x, y, z position componentsα angle of attackβ sideslip angleγ glide angleδa dimensionless asymmetric deflection (= d/c)δs symmetric brake deflection angleµ rigging angleρ air densityφ, ψ, θ Euler roll, yaw, and pitch angles, respectivelySubscriptb referred to payload (body)c referred to link joint Ce referred to earth fixedp referred to parafoil (canopy)pa from parafoil CG to AC in parafoil framecb from joint C to payload CG in joint C framecp from joint C to parafoil CG in joint C framepI referred to air inside the canopyβ stability derivative with respect to βδa control derivative with respect to δaSuperscriptb referred to payload bodyl referred to linesp referred to parafoil canopyAbbreviationsAC Aerodynamic CenterCG Center of GravityDOF Degree of FreedomNDI Nonlinear Dynamic Inversion

I. Introduction

Parafoils find wide use in UAV (Unmanned Aerial Vehicle), CRV (Crew Return Vehicle), GPADS (GuidedParafoil Air Drop System) to sports activities due to their good gliding as well as control characteristics.The glide angle is controlled by symmetric trailing edge (brake) deflection and turn is effected by asymmetricbrake deflection. The canopy open leading edge, large number of lines, and payload of arbitrary shape are thedrag producing components. Also, deflection of trailing edges (brakes) for parafoil control results in increasein drag. The presence of vertical offsets in centers of various aerodynamic forces from overall system centerof gravity gives rise to nonlinear trim and stability characteristics of the parafoil-payload system. Parafoilgeometric parameters like rigging angle, have a strong effect on the trim and stability characteristics of thesystem.

Although the parafoil canopy has very small rigid mass as compared to payload mass, the includedair mass and apparent mass result in total parafoil canopy mass being comparable to payload mass. Theadditional included air mass and apparent mass have a large effect on the rotational motion of parafoilcanopy. Due to the presence of a confluence point of the line connecting the payload to the parafoil, parafoiland payload exhibit independent rotational motion. Hence, the parafoil-payload system dynamics is requiredto be modeled as a two-body problem. Thus, the parafoil-payload system requires a 9-DOF dynamic modelrepresenting three degrees of freedom each for rotational motion of the canopy and payload, and three degreesof freedom for translational motion of the confluence point.

Slegers and Costello1 used 9-DOF model to investigate control issues for a parafoil-payload system withleft and right parafoil brakes used as the control mechanism. They were able to show that parafoil-payloadsystem can exhibit two basic modes of directional control, namely, roll steering and skid steering. Mooij etal.2 presented a 9-DOF flight dynamic model of parafoil-payload system which they used to develop a flightsimulation environment for the Small Parafoil Autonomous Delivery System (SPADES). Machin el al.3 used

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a two-body 8-DOF flight dynamic model to determine the aerodynamic characteristics of parafoil recoverysystem used for safely landing a crew return vehicle. Heise and Muller4 used a nonlinear high-fidelity two-body 8-DOF model (considering 6-DOF motion of parafoil and 2-DOF relative motion of payload) to developa unified software tool for the modeling, simulation, and highly realistic visualization of parafoil system.

Iosilevskii5 used standard static stability analysis to show that most forward CG position of glidingparachute results in loss of longitudinal stability. Lingard6 has carried out longitudinal and lateral stabilityanalysis of parafoil-payload system showing effect of rigging angle and brake deflection. Brown7 illustratedthe effect of scale and wing loading on turn response of a parafoil-payload system using closed-form turnequation. Crimi8 presented a lateral stability analysis of gliding parachutes. He determined the effect ofparafoil anhedral angle, suspension line length, and glide slope in spiral divergence and oscillatory response.Machin et al.3 investigated a range of rigging angles from 4 to 16 deg through actual parafoil flight tests,with most of the testing focused on 10 and 13 deg, to get required trim glide performance. They reportedthat a parafoil rigged at higher rigging angle has higher turn performance.

Bifurcation methods are a convenient tool for numerical analysis of trim and stability of dynamicalsystems, including nonlinear effects. Bifurcation methods were introduced to flight dynamics by Carroll andMehra,9 and Zagainov and Goman.10 Bifurcation methods have been used to study complete high alphadynamics, and constrained flight maneuvers, such as level trims.11 A previous study by us12 used bifurcationmethods to show the possibility of multiple trims for a given rigging angle and symmetric brake deflectionof a parafoil-payload system using a longitudinal 4-DOF model.

The present work uses a 9-DOF model to analyze effect of choice of rigging angle, and use of left and rightbrake deflections on the gliding and turning flight of parafoil-payload system. The gliding flight analysisuses bifurcation methods to determine effect of rigging angle and symmetric brake deflection on trim andstability of the system. Using best glide rigging angle obtained from gliding analysis, turning flight of theparafoil-payload system is analyzed for different left and right brake deflections.

II. Parafoil-Payload 9-DOF model

X

α

c

p

Z

Xb

V

γ

C

Zc

Y

Yb

c

d

µ

c/4Cm Xp

pZpY

b

a

b

p

Figure 1. 9-DOF parafoil-payload system

As shown in Fig. 1, the parafoil-payload system is modeled as a fixed-shape parafoil canopy of massmp, and a payload body of mass mb. The mass centers of canopy and payload are connected to joint Cthrough rigid massless links. Both the parafoil and the payload are free to rotate about joint C, but areconstrained by the internal joint force (Fxc, Fyc, Fzc) at C. The 9-DOF motion of parafoil-payload systemis described by three inertial position components of joint C (xc, yc, zc), as well as three Euler orientation

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angles of the parafoil (φp, θp, ψp) and payload (φb, θb, ψb). Formulation of equations of motion requires threereference frames, namely, parafoil reference frame (Xp, Yp, Zp) fixed to parafoil CG, payload reference frame(Xb, Yb, Zb) fixed to payload CG, and joint C fixed reference frame (Xc, Yc, Zc) parallel to inertial Earth-fixedframe (Xe, Ye, Ze). For the present study, the parafoil mass center is assumed to be at the parafoil canopymid-baseline point. Then, rigging angle µ is defined as the angle between the parafoil link and the parafoilZp axis. The rigging angle µ = 0 indicates that the forwardmost and rearmost parafoil lines, meeting atjoint C, are of equal length. The 9-DOF parafoil-payload model is similar to that in Slegers and Costello,1

except that we do not employ the spring and damper modeling of relative yawing motion between parafoiland payload due to the lines. In our model, the payload is not reoriented when the parafoil yaws during aturn as the payload aerodynamic and gravitational forces are independent of payload orientation.

A. Equations of Motion

The kinematic equations for parafoil and payload are given as :

xe

ye

ze

=

xc

yc

zc

=

uc

vc

wc

(1)

φb

θb

ψb

=

1 Sφbtθb Cφb tθb0 Cφb −Sφb0 Sφb/Cθb Cφb/Cθb

pb

qb

rb

(2)

φp

θp

ψp

=

1 Sφptθp Cφptθp0 Cφp −Sφp0 Sφp/Cθp Cφp/Cθp

pp

qp

rp

(3)

The common shorthand notation for trigonometric function is employed, where sinα ≡ Sα, cosα ≡ Cα,and tanα ≡ tα. The 9-DOF model of combined parafoil canopy and payload system in matrix form isrepresented as :

−MbRb 0 MbTb Tb

0 −(Mp +MF )Rcp (Mp +MF )Tp −TpIb 0 0 −RcbTb0 Ip + IF 0 RcpTp

ΩbΩpVc

Fc

=

B1B2B3B4

(4)

B1 = FAb + FGb − Ωb ×MbΩb ×Rcb

B2 = FAp + FGp − Ωp × (Mp +MF )Ωp ×Rcp +MFΩp × TpVc − Ωp ×MFTpVc

B3 = −Ωb × IbΩbB4 = MA

p − Ωp × (Ip + IM )Ωp

where,

Ωb× =

0 −rb qb

rb 0 −pb−qb pb 0

; Ωp× =

0 −rp qp

rp 0 −pp−qp pp 0

(5)

and

Ωb =

pb

qb

rb

; Ωp =

pp

qp

rp

; Rcb =

xcb

ycb

zcb

; Rcp =

xcp

ycp

zcp

; Vc =

uc

vc

wc

(6)

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The velocity vector and joint force vector at joint C are Vc = [uc, vc, wc]T and Fc = [Fxc, F yc, F zc]T

respectively. The aerodynamic force and weight force vector, respectively, of payload are :

FAb =12ρVbSbC

bD

ub

vb

wb

; FGb = mbg

−SθbSφbCθbCφbCθb

(7)

where only drag force CbD is modeled for payload.The aerodynamic force and moment at parafoil CG, and parafoil weight force vector are :

FAp = qpSp

CX

CY

CZ

; MAp = qpSp

bCl

cCmc/4 + xpaCZ

bCn

(8)

FGp = mpg

−SθpSφpCθpCφpCθp

(9)

The parafoil apparent mass and moment of inertia matrices are :

MF =

A 0 00 B 00 0 C

; IF =

IA 0 00 IB 00 0 IC

(10)

The matrix Tb represents the transformation matrix from an inertial reference frame to the payload bodyreference frame :

Tb =

CθbCψb CθbSψb −SθbSφbSθbCψb − CφbSψb SφbSθbSψb + CφbCψb SφbCθbCφbSθbCψb + SφbSψb CφbSθbSψb − SφbCψb CφbCθb

(11)

The matrix Tp represents the transformation matrix from an inertial reference frame to the parafoil bodyreference frame :

Tp =

CθpCψp CθpSψp −SθpSφpSθpCψp − CφpSψp SφpSθpSψp + CφpCψp SφpCθpCφpSθpCψp + SφpSψp CφpSθpSψp − SφpCψp CφpCθp

(12)

The payload and parafoil mass center velocity components in payload and parafoil frame respectively are :

ub

vb

wb

= Tb

xc

yc

zc

+ Ωb ×

xcb

ycb

zcb

(13)

up

vp

wp

= Tp

xc

yc

zc

+ Ωb ×

xcp

ycp

zcp

(14)

The apparent mass and inertia terms are based on the following formulas given by Lissaman and Brown7 :

A = ρ 0.913πt2b/4B = ρ 0.339πt2c/4C = ρ 0.771πc2b/4IA = ρ 0.630πc2b3/48IB = ρ 0.872 4c4b/48πIC = ρ 1.044πt2b3/48 (15)

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The mass mpI = ρbct/2 and moment of inertia of the included air in the parafoil canopy also need to beadded to the apparent mass and inertia terms, respectively.

B. Rigging Angle Modeling

The rigging angle µ is the angle between the line joining mid-baseline point of the canopy to joint C and theline parallel to Zp axis passing through the mid-baseline point. Therefore,

zcp = Rp cosµxcp = Rp sinµ

ycp = 0 (16)

Also, zcb = Rb, xcb = 0 and ycb = 0.

C. Aerodynamic Model

Parafoil aerodynamic force and moment coefficients are modeled as:

CL = CL(αp, δs) + CLδa δa

CpD = CpD(αp, δs) + CDδa δa

CY = CYββ + CYr (rpb/2Vp) + CYδa δa

CX = (−CpDup + CLwp)/VpCZ = (−CpDwp − CLup)/Vp

Cl = Clββ + Clpppb

2Vp+ Clr

rpb

2Vp+ Clδa δa

Cm = Cmc/4(αp, δs) + Cmqqpc

2Vp+ Cmδa δa

Cn = Cnββ + Cnpppb

2Vp+ Cnr

rpb

2Vp+ Cnδa δa (17)

Here symmetric brake deflection δs corresponds to zero, half and full brake conditions, while asymmetricbrake deflection δa is defined as δa = d/c with value 0.0-0.24, where d is length of brake control lines pulleddown.

The magnitude of total velocity vector in payload and parafoil reference frame is :

Vb = (u2b + v2

b + w2b )

12

Vp = (u2p + v2

p + w2p)

12 (18)

The payload and parafoil angles of attack, parafoil sideslip angle, and glide angle are computed, respec-tively, as

αb = tan−1(wb/ub)αp = tan−1(wp/up)βp = sin−1(vp/Vp)γ = sin−1(cosαp cosβp sin θp − sinβp sinφp cos θp − sinαp cosβp cosφp cos θp) (19)

III. Geometric and Aerodynamic Data

A. Parafoil-payload Geometry

The parafoil and payload geometric parameters, as given in Table 1, are taken from Lingard6 except whereindicated otherwise.

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Table 1. Parafoil-payload system geometry.

Parameter Value Parameter Valuec 3.75 m mb 135 kgt 0.18 c Rb 0.5 mb 7.5 m Sb 0.5 m2

mp 5 kg Sp 28 m2

mPI ρbct/2 xpa 0.25cRp -7.5 m Cmqp -1.864 ((cb/2Vp)−1)7

B. Longitudinal Aerodynamic Coefficients

The longitudinal aerodynamic characteristics consists of parafoil aerodynamic drag force coefficient CpD , liftforce coefficient CL, and pitching moment coefficient at quarter chord Cmc/4 . These are highly nonlinearfunction of angle of attack αp for zero, half and full symmetric brake deflections. These aerodynamiccoefficients as shown in Figs. 2, are taken from Lingard.6 CpD and Cmc/4 are corrected for line drag ClDacting at middle of line length using relation ClD = nRpd/Sp, where n = 40 is number of lines and diameterof a line, d = 35 mm.

C. Lateral Aerodynamic Derivatives

The parafoil lateral aerodynamic characteristics consisting of lateral stability and control derivatives aretaken from Brown7 for a similar parafoil-payload system. These are listed in Table 2.

Table 2. Lateral Derivatives.

Parameter Value Parameter ValueCYβ -0.0095 /deg CLδa 0.2350CYr -0.0060 (rb/2Vp)−1 CDδa 0.0957Clβ -0.0014 /deg CYδa 0.1368Clp -0.1330 (rb/2Vp)−1 Cmδa 0.2940Clr 0.0100 (rb/2Vp)−1 Clδa -0.0063Cnβ 0.0005 /deg Cnδa 0.0155Cnp -0.0130 (rb/2Vp)−1

Cnr -0.0350 (rb/2Vp)−1

IV. Gliding Flights

The gliding flight of parafoil-payload system is directly affected by choice of rigging angle and magnitudeof symmetric brake deflection applied. Bifurcation method is used to analyze gliding trim and stabilitycharacteristics of parafoil-payload system subject to symmetric zero, half, and full brake, for different designrigging angles. For a best glide rigging angle, effect of dynamic change of symmetric brake on glidingcharacteristics of parafoil-payload system is investigated using 9-DOF simulation.

The present bifurcation analysis makes use of 9-DOF dynamic model of parafoil-payload system unlike thelongitudinal 4-DOF model used earlier.12 The 9-DOF kinematic and dynamic equations of parafoil-payloadsystem from Eqs. (2), (3), and (4) are represented as:

χ = f (χ,U) (20)

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where χ = [φb, θb, ψb, φp, θp, ψp, pb, qb, rb, pp, qp, rp, uc, vc, wc] and U = [δs, µ], the δs is symmetric brakedeflection for zero, half and full brake configuration, and µ is rigging angle.

Using the AUTO200013 continuation algorithm, trims for gliding flight of parafoil-payload system arecomputed with zero, half, and full symmetric brakes, and different rigging angles. The trims so computedare shown in terms of bifurcation diagrams of parafoil angle of attack αp, and glide angle γ, in Figs. 3 and4, for zero, half, and full brake. The trims represented by solid lines are stable, and those represented bydashed lines are unstable. Points of onset of instabilities are bifurcation points represented by solid square.The bifurcation diagrams show possibility of multiple trims and regions of unstable gliding flight, as observedin earlier work.12

The bifurcation diagram in Fig. 3 shows two regions of glide instabilities for zero, half, and full brakeconditions. In zero brake condition, a very small region of instability occurs at α = 12− 15 deg and a largeregion of instability occurs between α = 25 deg to α = 32 deg. For the full symmetric brake, a small regionof instability between α = 8 deg to α = 15 deg and a large instability region between α = 20 deg to α = 30deg. The discontinuous bifurcation diagram (due to numerical problems) for symmetric half brake deflectionshows near about same trim and stability characteristics as for zero brake deflection indicating no effect ofbrake on system glide upto half brake deflection.

In Fig. 4 bifurcation diagram for γ shows parafoil-payload system has minimum stable glide angle atrigging angle µ = 9 deg for zero and half symmetric brake. At this rigging angle, full brake shows large glideangle γ = 55 deg. At rigging angles sufficiently larger than µ = 9 deg, full brake results in two stable trimglide angles, one showing γ ≈ 30 deg and the other at γ ≈ 50 deg. At rigging angles sufficiently smallerthan γ = 9, zero brake results in two stable trim glide angles, one showing γ ≈ 28 deg and the other atγ ≈ 50 deg. Thus, rigging angle µ = 9 deg is the optimum value of rigging angle to obtain good glide as wellas good flare characteristics (large γ under full brake conditions) for parafoil-payload system. The riggingangle µ = 9 deg is selected for further studies.

A. Simulation of Full Symmetric Brake

Bifurcation analysis only gives trims and stability of the dynamical system. To predict the transient response,one needs to carry out time simulation. This is especially important in case of large perturbations and forlarge and/or rapid control inputs. Numerical simulations of dynamics of parafoil-payload system have beencarried out using the 9-DOF dynamic model.

We investigate the effect of applying full brake on the dynamics of parafoil-payload system with riggingangle µ = 9 deg. A typical maneuver is simulated which is shown in Figs. 5 and 6 with the following sequenceof control inputs. The system is trimmed at the zero brake condition, then a ramp input over a time of 1 secis applied from zero brake to full brake starting at t = 5 sec. The full brake input is held fixed until t = 30sec, when a ramp input over a time of 1 sec is applied to recover from full brake to zero brake condition.

When parafoil brake is deflected from zero to full starting at t = 5 sec, the system takes 1 − 2 cycles ofoscillations over a period of about 8 sec before it reaches a post-stall trim state. This trim state has a steepglide angle γ = 52 deg, and high αp = 38 deg. This is the post-stall trim state computed in bifurcationdiagram of Figs. 3 and 4 for µ = 9 deg. Thus, for full brake deflection, the parafoil system glides withcanopy-stalled configuration. Effectively, with full brake deflection, the parafoil-payload system descendsmuch like a conventional circular parachute with L/D ratio less than 1.

When the parafoil canopy reverts back to zero brake from full brake deflection starting t = 30 sec, thesystem takes at least 3 cycles of oscillation over more than 20 sec to successfully recover to the original zerobrake trim.

B. Simulation of Different Choice of Rigging Angles

We investigate the effect of three different choices of rigging angles on dynamic maneuver as well as stallrecovery. The simulation is started with initial condition as parafoil-payload system in trim glide with zerobrake deflection and drop altitude 1000 m.

The simulation results are presented in Fig. 7 for three choices of rigging angles, i.e., µ = 6, 9 and 12deg, and discussed below. As shown in bifurcation diagrams, Figs. 3 to 4, µ = 6 deg corresponds to parafoil-payload system in post-stall zero brake and post-stall full brake trim, µ = 9 deg corresponds to pre-stall(minimum glide slope) zero brake and post-stall full brake trim, and µ = 12 deg corresponds to pre-stall zerobrake and pre-stall full brake trim.

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Flare maneuver : On applying symmetric brake at t = 5 sec, the dynamics is characterized by parafoilpitch oscillation coupled with pendulum effect of payload.

The rigging angle µ = 6 deg achieves highest αp but least damped oscillation. For µ = 6 and 9 deg, thesystem shows large reduction in forward (horizontal) velocity and large negative glide slope, because systemfinds post-stall trims at these rigging angles. On the other hand, for µ = 12 deg, the system shows smallerdecrease in forward velocity and minor increase in negative glide slope as the system finds only pre-stall trimin full brake condition. After 2.5 sec of the ramp input initiation at t = 5 sec, the system reaches maximumpitch up angle. The full brake is usually used for dynamic flare maneuver, therefore, it is the peak transientthat is important, not, the trim value. On application of the ramp input, at the point of maximum positivepitch up, rigging angle µ = 6 deg shows maximum reduction in forward velocity to uc ≈ 4 m/s, whereasµ = 9 and 12 deg show reduced forward velocity of uc ≈ 6.5 m/s and uc ≈ 8 m/s, respectively. Hence, thesmaller rigging angle µ = 6 deg is most promising to obtain good flare maneuver, followed by µ = 9, degwhich is only marginally worse and also corresponds to minimum glide slope for zero brake flight.

Stall recovery : As the rigging angles µ = 6 and 9 deg for full brake condition put the parafoil-payloadsystem in post-stall trim, a ramp input over 1 sec at t = 30 sec is applied to symmetric brake from full to zeroto check for stall recovery. This results in stall recovery for µ = 9 deg with lightly damped phugoid oscillation,but, the system with µ = 6 deg is not able to recover from stall due to the post-stall trim characteristics atthis rigging angle. Whereas parafoil-payload system with µ = 12 deg does not show post-stall trim in fullbrake, so there is no need for stall recovery.

Thus, although the rigging angle µ = 6 deg looks most promising for flare maneuver, but due to inabilityof stall recovery on redeploying zero brake it is not advisable for practical purpose. Therefore, the riggingangle µ = 9 deg which shows adequate flare maneuver and good stall recovery, as well best glide slope, isadvisable for practical purpose.

V. Turning Flight

Parafoils are capable of performing circular turns using either asymmetric brake alone (i.e., either left orright brake deflection), or asymmetric in combination with symmetric brake deflection (i.e., differential brakedeflection). Unlike aircraft which obtain turn rate due to the lift component created by banking, parafoilsachieve required turn rate due to differential drag force generated by asymmetric brake deflection on leftand right canopy. The asymmetric brake not only causes differential drag but affects lift force too, therebyaltering L/D ratio and glide slope of the system. Thus, asymmetric deflection of left and right parafoilbraking not only affects lateral-directional (turn and bank) response but also longitudinal response of theparafoil-payload system.

The parafoil circular turn maneuver is accompanied by loss of height while turning for a sufficiently largetime interval. Therefore, parafoil turn maneuver is performed in order to lose excess height over the landingzone so as to reach the appropriate height to execute the flare maneuver. A larger turn rate gives a smallcircular turn radius which helps the system to stay close to the landing zone.

A. Full Asymmetric Brake

The 9-DOF parafoil-payload model derived in chapter 3 is used to investigate the turn performance usingdifferent deflections of right and left brake. The system is rigged at µ = 9 deg, which gave optimum glideperformance in the previous chapter. Full right and left brake deflections correspond to δa = +0.24 andδa = −0.24, respectively.

The simulation for turning flight of parafoil-payload is carried out separately for full right and full leftbrake deflections and results are shown in Figs. 8, 9 and 10. The system is allowed to glide with zero leftand zero right brake deflection for 25 sec. Then, full right or left brake, as appropriate, is deflected overan interval of 1 sec and remains deflected thereafter. The system loses about 100 m height and covers adistance of about 275 m range before asymmetric brake deflection is initiated, it then takes approximatelytwo cycles of lateral oscillations over 10 sec to achieve a trimmed turning flight.

1. Figure 8 shows trajectories of the system for left and right full brakes. The system descends executingperfect symmetric left and right circular turns for the full left and right brake deflections, respectively.Right brake deflection causes parafoil to turn towards right showing positive cross range and performcircular turns. This is because right brake down gives increased drag on right side of the canopy

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generating positive yawing moment leading to right turn. The left brake shows similar turning responsetowards left side showing negative cross range. The system loses around 450 m height in approximatelythree and a half circular turns of 75 m radius for both left as well as right brake deflection.

2. For the two cases of deflection, Fig. 9 shows symmetric parafoil yaw and bank response. The sideslipangle is typically small. The symmetric left and right turn response is expected due to symmetriclateral aero data used in the 9-DOF dynamic model. The large angle of attack trim on application offull asymmetric brake shows that the canopy is stalled during turn leading to low L/D, hence steeperglide slope. Thus, the system descends faster during turning with full asymmetric brake. This could behelpful when parafoil-payload systems are required to execute circular turns in order to reduce excessheight over landing zone.

3. Figure 10 shows reduced horizontal velocity, increased vertical velocity, and cross velocity of oppositesigns at joint C for right and left full brakes. Note that the joint C reference frame and payload referenceframe do not reorient during turning flight, therefore, components of joint forces Fcx, Fcy exchangehorizontal joint force among themselves showing false cyclic oscillations. For the same reason, payloadoutward swing motion during turning flight is exchanged between pitch and bank angles showing falseoscillation. The small payload yawing angle build up is due to false pitch and bank angles and mustbe ignored. As the aerodynamics of payload is independent of bank and yaw angles, it does not affectthe dynamics of the overall system.

B. Various Right Brakes

Figure 11 shows turning flight simulation for various right brake deflections (positive asymmetric brakedeflections), i.e., δa = 0.05, 0.12 and 0.24 keeping δs = 0.

1. As compared to δa = +0.24 (full) asymmetric brake, parafoil-payload system with δa = +0.12 (half)asymmetric brake shows approximately half the yaw rate, bank angle and sideslip angle. This isexpected as the lateral aerodynamics is modeled as a linear function of δa (i.e., no angle of attackdependency). Thus, system executes a turn double the radius in turning flight with half brake. Thelongitudinal trims show post-stall angle of attack for both full and half asymmetric brake. Thus, theglide slope with half brake is only a little less than for full brake, and the system loses just a little lessheight as with full brake during 360 deg turn.

2. As compared to δa = +0.24 (full), +0.12 (half) asymmetric brakes, the small asymmetric deflectioncase, i.e., δa = 0.05, shows parafoil angle of attack in pre-stall region during turn. Thus, system turnswith glide angle and descent rate nearly same as zero brake deflection.

Thus, parafoil-payload system with smaller asymmetric brake deflection remains in pre-stall angle ofattack region giving a shallow glide slope, and hence descends slower taking larger turn radius. On theother hand, system with large symmetric deflections trim post-stall angles of attack giving steeper glideslopes, and hence descend faster with smaller turn radius. Therefore, parafoil-payload system can make useof large asymmetric brake deflection to descend faster while staying nearby in turn of small radius over thelanding zone. This is useful during excess height loss phase before touchdown. But, at small symmetricbrake deflection, turn rate is very small, and it takes a large turn radius, consequently a large time intervalis needed to complete one turn. This is useful for sport hovering over a region for longer time.

VI. Closed-Loop Simulations

Autonomous parafoil-payload system is required to deliver payload to a desired landing point from anarbitrary release point using GPS-based technologies. For the large part of the autonomous flight phase,parafoil-payload system is expected to mainly maintain desired heading angle leading to landing zone (LZ).After reaching the LZ, the parafoil-payload system is required to lose the excess height by flying in S-turnsor circular turns.

The basic requirement of a control law for a flight vehicle like parafoil-payload system is that it shouldbe able to attain and hold the vehicle in a desired attitude and heading using available controls, in thiscase, left and right brakes. Also, if the parafoil-payload system encounters undesirable flight conditions, the

10 of 26


controller should be able to bring it back to some desirable safe situation. Bifurcation and simulation resultsshow that parafoil-payload system may be expected to exhibit high-angle-of-attack, nonlinear dynamics, fora typical choices of given combination of control input and rigging angle.

Nonlinear Dynamic Inversion (NDI) is a method to access closed-loop characteristics of a system priorto design and testing of a control law. NDI method does not require to linearize the vehicle dynamics, andcan be used to develop closed-loop system dynamics even at nonlinear aerodynamic regimes. Thus, NDI isused to design simple generic nonlinear control law for the whole region without detailed analysis. In recenttimes, NDI-based control law has been suggested as effective means for flight control problems, e.g. Steer14

has applied NDI flight control method to a second generation supersonic aircraft, and Mulder et al.15 havedesigned the flight controller for a re-entry vehicle using NDI.

A. Nonlinear Dynamic Inversion (NDI) Theory

NDI is based on the concept of canceling the original dynamics of the system and replacing it with thedesired dynamics designed by the designer. This is achieved by inverting the governing equations of thesystem dynamics based on the measured system states and input commands. The forces and the momentsare modeled as functions of measured state variables. Opposing forces and moments are then commandedto negate the undesired contributions (forces and moments acting on the body) followed by the applicationof the desired forces and moments needed to accomplish the maneuver. The net result is that the outputshould perfectly match the input, if the inversion is exact and actuator saturation and rate limits are ignored.Usually dynamic inversion requires equal number of inputs and states. If the number of inputs are less thanthe available states, then all the states cannot be controlled. The states which remain uncontrolled, thenbehave according to the nature of the open-loop dynamics. Multiple time-scale using inner loop inversionfor fast system dynamics and outer loop inversion for slow system dynamics allows to control more numberof states for a given control input. The dynamic inversion has been used for aircraft longitudinal as well aslateral control at high angle of attack.

B. NDI Applied to Parafoil Turning Flight

The 9-DOF, nonlinear dynamic model with complete nonlinear longitudinal aerodynamics from -10 deg to+80 deg angle of attack is used to implement the NDI for heading control of parafoil-payload system. Theparafoil-payload system is assumed to be in trimmed gliding flight with symmetric brake deflection, δs, keptzero and asymmetric brake command, δa, used as the variable control for the forward simulation. Theasymmetric brake deflection, δa, is used to command parafoil turn angle, ψp. The external command is therequired heading angle ψcp, which is required to be converted into an asymmetric brake deflection demand δdain the inversion procedure. This is achieved through a two-step process consisting of an outer loop inversionand an inner loop inversion as shown in Fig. 12. The outer loop calculates the desired parafoil yaw rate rcpbased on the parafoil turn command ψcp. The inner loop then calculates the control demand, δda based onthe yaw rate rcp calculated in the outer loop. Note that pp, qp loops are not closed, hence the qp, rp obtaineddue to rcp are accepted.

In order to develop the NDI law, the 9-DOF equations of motion of parafoil-payload system from Eq. (4)are represented as:

AX = B0 + Bδaδa (21)

where

B0 =

FAb + FGb − Ωb ×MbΩb ×Rcb

FAp0 + FGp − Ωp × (Mp +MF )Ωp ×Rcp +MFΩp × TpVc − Ωp ×MFTpVc

−Ωb × IbΩbMA

p0 − Ωp × (Ip + IM )Ωp

FAp0 = qpSp

−CpD(αp, δs)up + CpL(αp, δs)wp/VpCY ββ + CY rrp

b2Vp

−CpD(αp, δs)wp − CpL(αp, δs)up/Vp

;

11 of 26


MAp0 = qpSp

bClββ + Clpppb

2Vp+ Clrrp

b2Vp

cCmc/4(αp, δs) + xpaCZ + Cmqqp

c2Vp

bCnββ + Cnppp

b2Vp

+ Cnrrpb

2Vp

and

Bδa =

03×1

FApδa03×1

MApδa

FApδa = qpSp

(−CDδaup + CLδawp)/VpCY δa(−CDδawp − CLδaup)/Vp

; MA

pδa = qpSp

bClδacCmδa

bCnδa

C. Inner Loop Inversion

The equation of motion for dynamics of yaw rate is

rp = a(12)B0 + Bδaδa (22)

where a(12) is a row vector corresponding to 12th row of matrix A−1 in Eq. (21).The yaw rate command, rcp, and the feedback, rp are used to calculate the error er = (rcp − rp), and the

desired yaw acceleration rdp is modeled as,

rdp = ωr(rcp − rp) (23)

where ωr is the bandwidth.The inversion function, by manipulation of Eq. (22), leads to the following expression:

δca =rdp − a(12)B0

a(12)Bδa

(24)

This is the asymmetric brake deflection required to achieve the demanded yaw acceleration, rdp . SubstitutingEq. (24) in Eq. (22) for δa, we get

rp = ωr(rcp − rp) (25)

Thus, the yaw rate dynamics has been converted into a first-order system with time constant T = 1/ωr. If(rcp − rp) is denoted as ∆rp and if rcp is a constant command (i.e., rcp = 0), we can get

∆rp = −ωrp∆rp (26)

D. Outer Loop Inversion

The outer loop produces yaw rate command rcp by inversion of the ψp equation. The design of the inversionlaw for the slow state, ψp, is based on the assumption that the fast state, rp responds much faster than theslow state ψp. It is assumed that the transient dynamics of the fast state occurs so quickly that it has anegligible effect on ψp.

The Equation for dynamics of yaw angle is given by third row of Eq. (3) as:

ψp = qpsinφpcos θp

+ rpcosφpcos θp

(27)

12 of 26


When the commanded parafoil yaw angle, ψcp, is compared with the feedback, ψp, we get ψdp as

ψdp = ωψ(ψcp − ψp) (28)

where ωψp is the bandwidth. The inversion function derived after manipulation of Eq. (27) is as follows:

rcp =(ψdp − qp

sinφpcos θp

)/

(cosφpcos θp

)(29)

The calculated yaw rate is fed as the command to the inner loop. In order to obtain the yaw rateinstantly, the inner loop is attributed a higher bandwidth, i.e. ωr = 5 whereas ωψ = 2, hence faster responseas compared to outer loop.

If the inversion is exact and ωψωr

∼ 0, it is equivalent to substituting rcp in Eq. (27), giving the equation:

ψp = ψdp = ωψp(ψcp − ψp) (30)

If ψcp is a constant command (i.e., ψp = 0), then Eq. (30) becomes

∆ψp = −ωψp∆ψp (31)

where ∆ψp = (ψcp − ψp).Equation (31) suggests that the outer-loop inversion is that of a first-order system with time constant

T = 1/ωψ. However in practice, the inner and outer loops couple, ans the obtained dynamics is not preciselyas given in Eqs. (25) and (30).

E. Turn Response using Inner Loop Alone

Effectiveness of NDI inner loop controller to generate required turn rate rp is analyzed using 9-DOF simula-tion. First, the simulation is carried out without NDI for full asymmetric deflection δa = 0.24, which showssteady yaw rate of 15.6 deg/sec. Then corresponding yaw rate is given as input yaw rate command, i.e.,rcp = 15.6 deg/sec in cosed loop using NDI inner loop only. In both cases the input command in the form oframp input for 1 second is initiated at 25 sec of simulation and kept fixed for all time. The results obtainedfrom both the simulations are compared in Fig. 13.

The NDI inner loop is able to generate commanded yaw rate efficiently with only a minor error in steadystate in all the variables. As the NDI inner loop works by inverting the rp equation alone, the other statesof the system respond according to their open loop dynamics. Thus, although the NDI inversion ought toresult in linear response of the inverted dynamics, due to coupling with other states, the yaw rate dynamicsalso responds to the dynamics of the other states. Nevertheless, using NDI, the system not only closelyachieves the commanded turn rate, but also, the corresponding longitudinal post-stall characteristics seen inthe open-loop simulations. The radius of turn for closed-loop is little bigger than open-loop with little lagas seen from height vs range plot.

F. Heading Tracking using both Loops

Using NDI with inner as well as outer inversion loops, the parafoil-payload system should be able to trackthe commanded heading. There is no control on the pitch and roll attitudes or rate variables, thereforethey will be free to respond according to their open-loop dynamics. Figure 14 shows how perfectly theNDI-controller is able to track a series of heading command, i.e., ψp by generating required rp and henceasymmetric brake deflection δa. The other state variables respond according to their own natural dynamics.The system is required to undergo four turning commands followed by straight gliding flight. During eachturn the system undergoes post-stall angle of attack flight and subsequently recovers from stall when inputcommand is withdrawn. The large variations in lateral-directional states is due to momentarily large brakedeflection input generated by NDI law. Whereas, longitudinal states show large variation due to post-stallangle of attack flight corresponding to large brake deflections.

13 of 26


VII. Conclusions

We have been successful in employing a bifurcation and continuation algorithm to compute all trim statesand their stability with varying rigging angle for three different symmetric brake settings. The results areall plotted on a single diagram making it easy to draw conclusions about the parafoil dynamic behavior. Forinstance, it is clearly seen that the best glide angle is obtained for an angle of attack within 1-2 deg of thestall angle of attack, when rigged correctly. Also, the glide ratio is not appreciably affected for symmetricbrake deflection up to 50 per cent. For larger than optimum rigging angle, the post-stall trim is at a relativelylower angle of attack and glide ratio, thereby giving poor flare maneuver. All these conclusions match withobservations made in the literature from previous experiences with designing parafoil systems.

Through simulations, we have demonstrated that the choice of rigging angle from our bifurcation studiesis indeed the correct one. All maneuvers, trims, glides, flare, and turns, are seen to be cleanly carried out forthe chosen best rigging angle. For a lower value of rigging angle, our simulations clearly show poorer glideand inability to recover from post-stall trims on removal of full brake deflection.

An NDI control strategy has been proposed and successfully implemented in closed-loop simulations.Our simulations with the NDI control law are able to cleanly track yaw rate and heading angle commands.In fact, it is clear from our simulations that a properly designed and rigged parafoil system needs only theyaw loops to be closed, the pitch and roll response show good behavior in open loop including post-stallflight phases and recovery.

References

1Slegers, N., and Costello, M., “Aspects of Control for a Parafoil and Payload System,” Journal of Guidance, Control andDynamics, Vol. 26, No. 6, 2003, pp. 898-905.

2Mooij, E., Wijnands, Q.G.J. and Schat, B., “9-dof Parafoil/Payload Simulator Development and Validation,” AIAAModeling and Simulation Technologies Conference and Exhibit, Austin, TX, August 11-14, 2003.

3Machin, R. A., Iacomini, C. S, Cerimele, C. J., and Stein, J. M. “Flight Testing the Parachute System for the SpaceStation Crew Return Vehicle,” Journal of Aircraft, Vol. 38, No. 5, 2001, pp. 786-799.

4Heise, M., and Muller, S., “Dynamic Modeling and Visualization of Multi-Body Flexible Systems,” AIAA Modeling andSimulation Technologies Conference and Exhibit, Providence, Rhode Island, August, 2004.

5Iosilevskii, G., “Center of Gravity and Minimal Lift Coefficient Limits of a Gliding Parachute,” Journal of Aircraft, Vol.32, No. 6, 1995, pp. 1297-1302.

6Lingard, J. S., “The Performance and Design of Ram-Air Parachutes,” Precision Aerial Delivery Seminar, TechnicalReport, Royal Aircraft Establishment, Aug 1981.

7Brown, G. J., “Parafoil Steady Turn Response to Control Input,” 12th AIAA Aerodynamic Decelerator Systems Tech-nology Conference and Seminar, London, UK, May 10-13 1993, pp. 248-254.

8Crimi, P., “Lateral Stability of Gliding Parachute,” Journal of Guidance, Control and Dynamics, Vol. 5, No. 5, 1982,pp. 529-536.

9Carroll, J. V., and Mehra, R. K., “Bifurcation Analysis of Nonlinear Aircraft Dynamics,” Journal of Guidance, Controland Dynamics, Vol. 5, No. 5, 1982, pp. 529-536.

10Zagainov, G. I., and Goman, M. G., “Bifurcation Analysis of Critical Aircraft Flight Regimes,” International Council ofthe Aeronautical Sciences, September 1984, pp. 217-223.

11Ananthkrishnan, N., and Sinha, N. K., “Level Flight Trim and Stability Analysis using Extended Bifurcation and Con-tinuation Procedure,” Journal of Guidance, Control and Dynamics, Vol. 24, No. 6, 2001, pp. 1225-1228

12Prakash, O., Daftary A. and Ananthkrishnan, N, “Trim and Stability Analysis of Parafoil/Payload System using Bifur-cation Methods,” 18th AIAA Aerodynamic Decelerator Systems Technology Conference and Seminar, Munich, Germany, May2005.

13Doedel, E. J., Paffenroth, R. C. Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B., and Xang, X.,“AUTO2000: Continuation and Bifurcation Software for Ordinary Differential Equations (with Hom Cont),” Technical Re-port,California Inst. of Technology, Pasadena, CA, USA, 2001.

14Steer, A. J., “Application of NDI Flight Control to a Second Generation Supersonic Transport Aircraft,” Computing andControl Engineering Journal, June 2001, pp. 108-114.

15daCosta, R. R., Chu, Q. P., and Mulder, J. A., “Reentry Flight Controller Design Using Nonlinear Dynamic Inversion,”Journal of Spacecraft and Rockets, Vol. 40, No. 1, Jan-Feb 2003, pp. 64-71.

14 of 26


−20 0 20 40 60 800

0.2

0.4

0.6

0.8

1

1.2

1.4

Angle of Attack, αp

−20 0 20 40 60 800

0.2

0.4

0.6

0.8

1

−20 0 20 40 60 80−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

−20 0 20 40 60 800

0.5

1

1.5

2

2.5

3

3.5

*o

. Zero Brake Half Brake Full Brake

C

Cm

CD

c/4

L

L/D

p




Figure 2. Longitudinal aerodynamics data

15 of 26


-10. -5. 0. 5. 10. 15.

0.

10.

20.

30.

40.

50.

60.

70.

80.

Rigging Angle, µ

α p

Full Brake

Zero Brake

Half Brake

Half Brake

Figure 3. Bifurcation diagram of the parafoil angle of attack (Full line: stable trim, dashed line: unstabletrim, solid square: Hopf bifurcation point)

-10. -5. 0. 5. 10. 15.

10.

20.

30.

40.

50.

60.

70.

80.

Rigging Angle, µ

γ

Full Brake

Half Brake

Half Brake

Zero Brake

Figure 4. Bifurcation diagram of glide angle of parafoil-payload system (Full line: stable trim, dashed line:unstable trim, solid square: Hopf bifurcation point)

16 of 26


0 20 40 60−1

−0.5

0

0.5

1

Time, sec

φp,d

eg

0 20 40 60−40

−20

0

20

Time, sec

θ p,deg

0 20 40 60−1

−0.5

0

0.5

1

Time, sec

r p

0 20 40 600

10

20

30

40

Time, sec

α p,deg

0 20 40 60−1

−0.5

0

0.5

1

Time, sec

β p,deg

0 20 40 60−60

−40

−20

0

Time, sec

γ,de

g

0 200 400 600700

800

900

1000

Hei

ght,

m

Range, m0 200 400 600

−1

−0.5

0

0.5

1

Range, m

Cro

ssR

ange

m

0 20 40 600

50

100

Time, sec

δ s,deg

0 20 40 60−1

−0.5

0

0.5

1

Time, sec

δ a,deg

Figure 5. Dynamic effect of full symmetric brake δs deflections with µ = 9 deg parafoil canopy variables

17 of 26


0 20 40 600

5

10

15

u c,m/s

Time in sec

0 20 40 60−1

−0.5

0

0.5

1

v c,m/s

Time in sec

0 20 40 600

2

4

6

8

wc,m

/s

Time in sec

0 20 40 60−1000

−500

0

500

Fcx

,N

Time in sec

0 20 40 60−1

−0.5

0

0.5

1

Fcy

,N

Time in sec

0 20 40 601000

1200

1400

1600

1800F

cz,N

Time in sec

0 20 40 60−1

−0.5

0

0.5

1

φ b,deg

Time in sec0 20 40 60

−40

−20

0

20

θ b,deg

Time in sec

0 20 40 60−1

−0.5

0

0.5

1

ψb,d

eg

Time in sec0 20 40 60

−1

−0.5

0

0.5

1

ψp,d

eg

Time in sec

Figure 6. Dynamic effect of full symmetric brake δs deflections with µ = 9 deg connection point and payloadvariables including parafoil yaw angle

18 of 26


0 10 20 30 40 50−30

−20

−10

0

10

20

30

Time in sec

θb,d

eg

0 10 20 30 40 50−40

−30

−20

−10

0

10

Time in sec

θ p,deg

0 10 20 30 40 50−1

−0.5

0

0.5

1

Time in sec

0 10 20 30 40 500

10

20

30

40

50

60

Time in sec

αp,d

eg

0 10 20 30 40 500

20

40

60

80

100

Time in sec

δs,d

eg

0 10 20 30 40 50−80

−60

−40

−20

0

Time in sec

γ,d

eg

0 10 20 30 40 502

4

6

8

10

12

14

Time in sec

uc,

m/s

0 10 20 30 40 500

2

4

6

8

10

Time in sec

wc,

m/s

δ a

Time in sec

Time in sec

Figure 7. Dynamic effect of full brake for different rigging angles Solid line: 12 deg, Thick solid line: 9 deg, andDashed line: 6 deg. (All angles in deg)

19 of 26


0

50

100

150

200

250

300−60

−40

−20

0

20

40

60

600

700

800

900

1000

Cross Range, Yc

Forward Range, Xc

Hei

ght,

H

Figure 8. Turning flight for full left, and full right brake deflections with µ = 9 deg. Solid line: δa = +0.24;Dotted line: δa = −0.24

20 of 26


0 20 40 60 80 100−20

−10

0

10

20

Time, sec

φp,d

eg

0 20 40 60 80 100−15

−10

−5

0

Time, sec

θp,d

eg

0 20 40 60 80 100−20

−10

0

10

20

Time, sec

rp,d

eg

0 20 40 60 80 1005

10

15

20

Time, sec

α p,deg

0 20 40 60 80 100−4

−2

0

2

4

Time, sec

βp,d

eg

0 20 40 60 80 100−35

−30

−25

−20

−15

Time, sec

γ,de

g

0 100 200 300 400400

600

800

1000

Hei

ght,

m

Range, m0 100 200 300 400

−100

−50

0

50

100

Range, m

Cro

ssR

ange

m

0 20 40 60 80 100−1

−0.5

0

0.5

1

Time, sec

δs

0 20 40 60 80 100−0.4

−0.2

0

0.2

0.4

Time, sec

δa

Figure 9. Turning flight for full left, and full right brake deflections with µ = 9 deg. Solid line: δa = +0.24;Dashed line: δa = −0.24, parafoil canopy variables

21 of 26


0 20 40 60 80 1008

9

10

11

u c,m/s

Time in sec

0 20 40 60 80 100−5

0

5

v c,m/s

Time in sec

0 20 40 60 80 1003

4

5

6

wc,m

/s

Time in sec

0 20 40 60 80 100−500

0

500

Fx

Time in sec

0 20 40 60 80 100−500

0

500

Fy

Time in sec

0 20 40 60 80 1001150

1200

1250

1300

1350F

z

Time in sec

0 20 40 60 80 100−20

−10

0

10

20

φ b,deg

Time in sec0 20 40 60 80 100

−20

−10

0

10

20

θ b,deg

Time in sec

0 20 40 60 80 100−100

−50

0

50

100

ψb,d

eg

Time in sec

0 20 40 60 80 100−2000

−1000

0

1000

2000

ψp,d

eg

Time in sec

Figure 10. Turning flight for full left, and full right brake deflections with µ = 9 deg. Solid line: δa = +0.24;Dashed line: δa = −0.24, connection point and payload variables including parafoil yaw angle

22 of 26


0 50 100 1500

5

10

15

20

Time, sec

φp,d

eg

0 50 100 150−15

−10

−5

0

Time, sec

θ p,deg

0 50 100 1500

5

10

15

20

Time, sec

r p

0 50 100 1505

10

15

20

Time, sec

α p,deg

0 50 100 150−2

0

2

4

Time, sec

β p,deg

0 50 100 150−35

−30

−25

−20

−15

Time, sec

γ,de

g

0 100 200 300 400 500200

400

600

800

1000

Hei

ght,

m

Range, m0 100 200 300 400 500

0

100

200

300

400

Range, m

Cro

ssR

ange

m

0 50 100 150−1

−0.5

0

0.5

1

Time, sec

δ s,deg

0 50 100 1500

0.1

0.2

0.3

0.4

Time, sec

δ a,deg

Figure 11. Turning flight for various asymmetric brake deflections with µ = 9 deg. (Solid line: δa = +0.05;Dashed line: δa = +0.12, Thick solid line: δa = 0.24.)

23 of 26


r

r

9−DOFModelc

p

p

InversionInner−Loop

DynamicsInversion

ψp

ω

Inner−LoopBandwidth

ωrψBandwidthOuter−Looppψ d Outer−Loop

Dynamics

ψp

c

Inner−Loop

−

++

−

Outer−Loop NDI

r dp

..

δ = 0s

Xadδ

Fig

ure

12.

Blo

ck

dia

gram

ofN

DI

law

24

of26

Am

erican

Institu

teofA

eronautics

and

Astro

nautics

0 20 40 60 80 100−5

0

5

10

15

20

Time in sec

φ p,deg

0 20 40 60 80 100−14

−12

−10

−8

−6

−4

−2

Time in sec

θ p,deg

0 20 40 60 80 100−10

−5

0

5

10

15

20

Time in sec

r p,deg

0 20 40 60 80 1008

10

12

14

16

18

20

Time in sec

α p,deg

0 20 40 60 80 100−1

0

1

2

3

4

Time in sec

β p,deg

0 20 40 60 80 100−35

−30

−25

−20

−15

Time in sec

γ,de

g

0 100 200 300 400500

600

700

800

900

1000

Hei

ghtm

Range m

0 100 200 300 4000

20

40

60

80

Range m

Cro

ssR

ange

m

input: δa=0.24

input: rp

=15.6 deg/s

Figure 13. Turn rate response of parafoil-payload system (Solid line: δca = 0.24; Dashed line: rc

p = 15.66 deg/s).

25 of 26


0 50 100 150−20

−10

0

10

20

Time in sec

φ p,deg

0 50 100 150−15

−10

−5

0

Time in sec

θ p,deg

0 50 100 150−20

−10

0

10

20

Time in sec

rp.d

eg

0 50 100 1505

10

15

20

Time in sec

α p,deg

0 50 100 150−4

−2

0

2

4

Time in sec

β p,deg

0 50 100 150−40

−30

−20

−10

Time in sec

γ,de

g

0 50 100 150−0.4

−0.2

0

0.2

0.4

Time in sec

δ a,deg

0 50 100 150−100

−50

0

50

100

Time in sec

0 200 400 600 800400

600

800

1000

Hei

ght,

m

Range, m

0 200 400 600 8000

100

200

300

400

Range, m

Cro

ssR

ange

,mψ

p,deg

Figure 14. Tracking capability of closed-loop parafoil-payload system (Dashed line: commanded yaw angle ψcp;

Solid line: obtained yaw angle ψp).

26 of 26


Documents

[American Institute of Aeronautics and Astronautics AIAA Atmospheric Flight Mechanics Conference and Exhibit - Keystone, Colorado ()] AIAA Atmospheric Flight Mechanics Conference and