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 degrees of freedom each for rotational motion of the canopy, and payload, and three degrees of freedom for translational motion of the confluence point of the lines. The gliding flight of parafoil-payload system is directly affected by chice of rigging angle. The glide angle is controlled by symmetric trailing edge (brake) deflection and turn is effected by asymmetric brake deflection. The parafoil trim and stability charateristics are a complex function of rigging angle and magnitude of downward deflection of left and right brakes. The present work uses a 9-DOF model to analyze gliding and turning flight of parafoil-payload system for different choices of design rigging angle subjected to change in left and right brake deflections.
Nomenclature A, B, C b d CD CL CY Clp , Clr Cmc/4 Cmq Cnr , Cnp c F g IA , IB , IC IF M m q¯ p, q, r Sb Sp Tb Tp t u, v, w V X, Y, Z ∗ Ph.D.
apparent mass terms canopy span length of brake control line pulled drag coefficient lift coefficient side force coefficient rolling moment damping coefficients pitching moment coefficient at quarter chord pitching moment damping coefficient yawing moment damping coefficients canopy chord length force acceleration due to gravity apparent inertia terms parafoil apparent inertia matrix mass matrix mass dynamic pressure (= 21 ρV 2 ) roll, pitch, and yaw rates payload (body) cross-section area parafoil planform area transformation matrix from inertial reference frame to payload reference frame transformation matrix from inertial reference frame to parafoil reference frame canopy maximum thickness velocity components along reference frame total velocity body-fixed reference frame
Student, Department of Aerospace Engineering;
[email protected]. Student Member AIAA. Professor, Department of Aerospace Engineering;
[email protected]. Senior Member AIAA.
† Associate
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x, y, z α γ δa δs µ ρ φ, ψ, θ Subscript b c c p pa cp pI β δa Superscript b l Abbreviations AC CG DOF
position components angle of attack glide angle dimensionless asymmetric deflection (= d/c) symmetric brake deflection angle rigging angle air density Euler roll, yaw, and pitch angles, respectively referred to payload (body) referred to link joint C referred to earth fixed referred to parafoil (canopy) from parafoil CG to AC in parafoil frame from joint C to parafoil CG in joint C frame referred to air inside the canopy stability derivative with respect to β control derivative with respect to δa referred to payload body referred to lines Aerodynamic Center Center of Gravity Degree of Freedom
I.
Introduction
Parafoils find wide use in UAV (Unmanned Aerial Vehicle), CRV (Crew Return Vehicle), GPADS (Guided Parafoil 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 asymmetric brake deflection. The canopy open leading edge, large number of lines, and payload of arbitrary shape are the drag producing components. Also, deflection of trailing edges (brakes) for parafoil control results in increase in drag. The presence of vertical offsets in centers of various aerodynamic forces from overall system center of gravity gives rise to nonlinear trim and stability characteristics of the parafoil-payload system. Parafoil geometric parameters like rigging angle, have a strong effect on the trim and stability characteristics of the system. Although the parafoil canopy has very small rigid mass as compared to payload mass, the included air mass and apparent mass result in total parafoil canopy mass being comparable to payload mass. The additional included air mass and apparent mass have a large effect on the rotational motion of parafoil canopy. Due to the presence of a confluence point of the line connecting the payload to the parafoil, parafoil and payload exhibit independent rotational motion. Hence, the parafoil-payload system dynamics is required to be modeled as a two-body problem. Thus, the parafoil-payload system requires a 9-DOF dynamic model representing three degrees of freedom each for rotational motion of the canopy and payload, and three degrees of freedom for translational motion of the confluence point. Slegers and Costello1 used 9-DOF model to investigate control issues for a parafoil-payload system with left and right parafoil brakes used as the control mechanism. They were able to show that parafoil-payload system can exhibit two basic modes of directional control, namely, roll steering and skid steering. Mooij et al.2 presented a 9-DOF flight dynamic model of parafoil-payload system which they used to develop a flight simulation environment for the Small Parafoil Autonomous Delivery System (SPADES). Machin el al. 3 used a two-body 8-DOF flight dynamic model to determine the aerodynamic characteristics of parafoil recovery system used for safely landing a crew return vehicle. Heise and Muller4 used a nonlinear high-fidelity twobody 8-DOF model (considering 6-DOF motion of parafoil and 2-DOF relative motion of payload) to develop a unified software tool for the modeling, simulation, and highly realistic visualization of parafoil system.
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Iosilevskii5 used standard static stability analysis to show that most forward CG position of gliding parachute results in loss of longitudinal stability. Lingard6 has carried out longitudinal and lateral stability analysis of parafoil-payload system showing effect of rigging angle and brake deflection. Brown 7 illustrated the effect of scale and wing loading on turn response of a parafoil-payload system using closed-form turn equation. Crimi8 presented a lateral stability analysis of gliding parachutes. He determined the effect of parafoil 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 reported that 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 dynamical systems, including nonlinear effects. Bifurcation methods were introduced to flight dynamics by Carroll and Mehra,9 and Zagainov and Goman.10 Bifurcation methods have been used to study complete high alpha dynamics, and constrained flight maneuvers, such as level trims.11 A previous study by us12 used bifurcation methods to show the possibility of multiple trims for a given rigging angle and symmetric brake deflection of 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 right brake deflections on the gliding and turning flight of parafoil-payload system. The gliding flight analysis uses bifurcation methods to determine effect of rigging angle and symmetric brake deflection on trim and stability of the system. Using best glide rigging angle obtained from gliding analysis, turning flight of the parafoil-paylaod system is analyzed for different left and right brake deflections.
II.
Parafoil-Payload 9-DOF model Cmc/4
p
a
Xp αp γ
d
V
Yp Zp µ
C Xc Yc
Xb
b
Zc Yb
Zb
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 mass mp , and a payload body of mass mb . The mass centers of canopy and payload are connected to joint C through rigid massless links. Both the parafoil and the payload are free to rotate about joint C, but are constrained by the internal joint force (F xc , F yc , F zc ) at C. The 9-DOF motion of parafoil-payload system is described by three inertial position components of joint C (xc , yc , zc ), as well as three Euler orientation angles of the parafoil (φp , θp , ψp ) and payload (φb , θb , ψb ). Formulation of equations of motion requires three reference 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-fixed frame (Xe , Ye , Ze ). For the present study, the parafoil mass center is assumed to be at the parafoil canopy 3 of 16 American Institute of Aeronautics and Astronautics
mid-baseline point. Then, rigging angle µ is defined as the angle between the parafoil link and the parafoil Zp axis. The rigging angle µ = 0 indicates that the forwardmost and rearmost parafoil lines, meeting at joint 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 parafoil and payload due to the lines. In our model, the payload is not reoriented when the parafoil yaws during a turn as the payload aerodynamic and gravitational forces are independent of payload orientation. A.
Equations of Motion
The kinematic equations for parafoil and x˙ e y˙ e z˙e φ˙ b θ˙b ˙ ψb φ˙ p θ˙p ˙ ψp
payload are given as : x˙ c uc = = y˙ c vc z˙c wc
1 S φb t θ b = 0 C φb 0 Sφb /Cθb
1 S φp t θ p = 0 C φp 0 Sφp /Cθp
(1)
C φb t θ b pb qb −Sφb rb Cφb /Cθb
(2)
C φp t θ p pp −Sφp qp Cφp /Cθp rp
(3)
The common shorthand notation for trigonometric function is employed, where sin α ≡ S α ,cos α ≡ Cα , and tan α ≡ tα . The 9-DOF model of combined parafoil canopy and payload system in matrix form is represented as : −Mb Rb 0 M b Tb Tb Ω˙ b B1 0 −(Mp + MF )Rcp (Mp + MF )Tp −Tp Ω˙ p B2 (4) ˙ = B3 Ib 0 0 −Rcb Tb Vc 0 Ip + IF 0 Rcp Tp Fc B4 B1 = FbA + FbG − Mb Ωb Rcb Ωb B2 = FpA + FpG − Mp Ωp Rcp Ωp + MF Ωp Tp Vc −Ωp MF Tp Vc − Ωp MF Rcp Ωp B3 = −Ωb Ib Ωb B4 = MA p − Ωp (Ip + IF )Ωp
(5)
where, 0 −rb qb Ωb = rb 0 −pb ; −qb pb 0
0 −rp qp Ωp = rp 0 −pp −qp pp 0
(6)
and Rcb
0 −zcb ycb = zcb 0 −xcb ; −ycb xcb 0
Rcp
0 −zcp ycp = zcp 0 −xcp −ycp xcp 0
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(7)
The velocity vector and joint force vector at joint C are Vc = [uc , vc , wc ]T and Fc = [F xc , F yc , F zc ]T respectively. The aerodynamic force and weight force vector, respectively, of payload are : b −Sθb Cαb CD G FbA = q¯b Sb ; F = m g (8) S φb C θ b 0 b b b Sα b C D C φb C θ b b where only drag force CD is modeled for payload. The aerodynamic force and moment at parafoil CG, and parafoil weight force vector are : bCl CX A A ; Mp = q¯p Sp Fp = q¯p Sp cCm + xpa CX CY bCn CZ
FpG
−Sθp = mp g S φp C θ p C φp C θ p
The parafoil apparent mass and moment of A 0 MF = 0 B 0 0
(9)
(10)
inertia matrices are : 0 IA 0 0 IF = 0 I B 0 ; 0 C 0 0 IC
(11)
The matrix Tb represents the transformation matrix from an inertial reference frame to the payload body reference frame : C θ b C ψb C θ b S ψb −Sθb (12) T b = S φ b S θ b C ψb − C φ b S ψb S φ b S θ b S ψb + C φ b C ψb S φ b C θ b C φ b S θ b C ψb + S φ b S ψb C φ b S θ b S ψb − S φ b C ψb C φ b C θ b The matrix Tp represents the transformation matrix from an inertial reference frame to the parafoil body reference frame :
Tp
=
C θ p C ψp S φ p S θ p C ψp − C φ p S ψp C φ p S θ p C ψp + S φ p S ψp
The payload and parafoil mass x˙ c ub = Tb y˙ c vb z˙c wb
center velocity xcb + Ωb ycb zcb
C θ p S ψp S φ p S θ p S ψp + C φ p C ψp C φ p S θ p S ψp − S φ p C ψp
components in payload and x˙ c up = Tp ; y˙ c vp z˙c wp
−Sψp S ψp C θ p C ψp C θ p
parafoil frame xcp + Ωb ycp zcp
(13) respectively are : (14)
The apparent mass and inertia terms are based on the following formulas given by Lissaman and Brown 7
: A = ρ 0.913πt2b/4 B = ρ 0.339πt2c/4 C IA
= ρ 0.771πc2b/4 = ρ 0.630πc2b3 /48
IB IC
= ρ 0.872 4c4 b/48π = ρ 1.044πt2b3 /48
(15)
The mass mpI = ρbct/2 and moment of inertia of the included air in the parafoil canopy also need to be added to the apparent mass and inertia terms, respectively. 5 of 16 American Institute of Aeronautics and Astronautics
B.
Aerodynamic Model
Parafoil aerodynamic force and moment coefficients are modeled as: CL p CD
= CL (αp , δs ) + CLδ δa p = CD (αp , δs ) + CDδ δa
CY
= CY β β + CY r (rp b/2Vp ) + CY δ δa
CX
p = (−CD up + CL wp )/Vp
CZ
p = (−CD wp − CL up )/Vp
Cl Cm Cn
rp b pp b + Clr + Clδ δa 2Vp 2Vp qp c = Cmc/4 (αp , δs ) + Cmq + Cmδ δa 2Vp rp b pp b + Cnr + Cnδ δa = Cnβ β + Cnp 2Vp 2Vp = Clβ β + Clp
(16)
Here symmetric brake deflection δs corresponds to zero, half and full brake conditions, while asymmmetric brake deflection δa is defined as δa = d/c with value 0.0-0.24, where d is length of brake control lines pulled down. The magnitude of total velocity vector in payload and parafoil reference frame is : 1
Vb
= (u2b + vb2 + wb2 ) 2
Vp
= (u2p + vp2 + wp2 ) 2
1
(17)
The payload and parafoil angles of attack, parafoil sideslip angle, and glide angle are computed, respectively, as αb
= tan−1 (wb /ub )
αp βp
= tan−1 (wp /up ) = sin−1 (vp /Vp )
γ
= sin−1 (cos αp cos βp sin θp − sin βp sin φp cos θp − sin αp cos βp cos φp cos θp )
III. A.
(18)
Geometric and Aerodynamic Data
Parafoil-payload Geometry
The parafoil and payload geometric parameters, as given in Table 1, are taken from Lingard 6 except where indicated otherwise. Table 1. Parafoil-payload system geometry.
Parameter c t b mp mP I Rp
Value 3.75 m 0.18 c 7.5 m 5 kg ρbct/2 -7.5 m
Parameter mb Rb Sb Sp xpa Cmqp
Value 135 kg 0.5 m 0.5 m2 28 m2 0.25c -1.864 ((cb/2Vp )−1 )7
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B.
Longitudinal Aerodynamic Coefficients
p The longitudinal aerodynamic characteristics consists of parafoil aerodynamic drag force coefficient C D , lift force coefficient CL , and pitching moment coefficient at quarter chord Cmc/4 . These are highly nonlinear function of angle of attack αp for zero, half and full symmetric brake deflections. These aerodynamic p l coefficients as shown in Figs. 2, are taken from Lingard.6 CD and Cmc/4 are corrected for line drag CD l acting at middle of line length using relation CD = nRp d/Sp , where n = 40 is number of lines and diameter of a line, d = 35 mm.
C.
Lateral Aerodynamic Derivatives
The parafoil lateral aerodynamic characteristics consisting of lateral stability and control derivatives are taken from Brown7 for a similar parafoil-payload system. These are listed in Table 2. Table 2. Lateral Derivatives.
Parameter CY β CY r Clβ Clp Clr Cnβ Cnp Cnr
Value -0.0095 /deg -0.0060 (rb/2Vp )−1 -0.0014 /deg -0.1330 (rb/2Vp )−1 0.0100 (rb/2Vp )−1 0.0005 /deg -0.0130 (rb/2Vp )−1 -0.0350 (rb/2Vp )−1
IV.
Parameter CLδa CDδa C Y δa Cmδa Clδa Cnδa
Value 0.2350 0.0957 0.1368 0.2940 -0.0063 0.0155
Gliding Flights
The gliding flight of parafoil-payload system is directly affected by choice of rigging angle and magnitude of symmetric brake deflection applied. Bifurcation method is used to analyze gliding trim and stability characteristics of parafoil-payload system subject to symmetric zero, half, and full brake, for different design rigging angles. For a best glide rigging angle, effect of dynamic change of symmetric brake on gliding characteristics 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 the longitudinal 4-DOF model used earlier.12 The 9-DOF kinematic and dynamic equations of parafoil-payload system from Eqs. (2), (3), and (4) are represented as: χ˙ = f (χ, U )
(19)
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 brake deflection for zero, half and full brake configuration, and µ is rigging angle. Using the AUTO2000 continuation algorithm, trims for gliding flight of parafoil-payload system are computed with zero, half, and full symmetric brakes, and different rigging angles. The trims so computed are shown in terms of bifurcation diagrams of parafoil angle of attack αp , and glide angle γ, in Figs. 3 and 4, for zero, half, and full brake. The trims represented by solid lines are stable, and those represented by dashed 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 observed in earlier work.12 The bifurcation diagram in Fig. 3 shows two regions of glide instabilities for zero, half, and full brake conditions. In zero brake condition, a very small region of instability occurs at α = 12 − 15 deg and a large region of instability occurs between α = 25 deg to α = 32 deg. For the full symmetric brake, a small region of instability between α = 8 deg to α = 15 deg and a large instability region between α = 20 deg to α = 30
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deg. The discontinous bifurcation diagram (due to numerical problems) for symmetric half brake deflection shows near about same trim and stability characteristics as for zero brake deflection. In Fig. 4 bifurcation diagram for γ shows parafoil-payload system has minimum stable glide angle at rigging angle µ = 9 deg for zero and half symmetric brake. At this rigging angle, full brake shows large glide angle γ = 55 deg. At rigging angles sufficiently larger than µ = 9 deg, full brake results in two stable trim glide angles, one showing γ ≈ 30 deg and the other at γ ≈ 50 deg. At rigging angles sufficiently smaller than γ = 9, zero brake results in two stable trim glide angles, one showing γ ≈ 28 deg and the other at γ ≈ 50 deg. Thus, rigging angle µ = 9 deg is the optimum value of rigging angle to obtain good glide as well as good flare characteristics (large γ under full brake conditions) for parafoil-payload system. The rigging angle µ = 9 deg is selected for further studies. A.
Simulation Result
Figure 5 shows simulation results showing effect of varying symmetric brake deflections on parafoil-payload system parameters for rigging angle µ = 9 deg. All the simulations begin with parafoil-payload system gliding from 1000 m height and 12 m/s velocity in zero brake condition. In zero brake condition, the parafoil-payload system glides at parafoil angle of attack αp = 8 deg with glide angle γ = 18 deg. The parafoil has pitch angle θp = −10 deg and payload pitch angle θb = −2 deg. At this angle of attack, parafoil shows maximum lift to drag ratio (L/D) of 3.25 and it is flying about 2 deg below stall angle of attack. After 5 sec of zero brake glide, symmetric half or full brakes are applied over 1 sec and kept fixed upto 30 sec. Thereafter, parafoil is brought back to zero brake condition by releasing brakes over 1 sec. The system lateral variables are seen to remain zero for all simulations. On application of symmetric half brake, the parafoil-payload system trims after 2-3 cycles of oscillation of period nearly 7-8 sec. The system in half brake shows negligible changes in trim glide angle from zero brake condition, showing only 2-3 deg reduction of parafoil angle of attack α p . Same result can be seen from bifurcation diagrams in Figs. 3 and 4 for half symmetric brakes corresponding to µ = 9 deg. The reason for no change in trim can be seen from L/D plot in Fig. 2 which shows same L/D of the parafoil-payload system for zero and half symmetric brake. On application of symmetric full brake, the parafoil-payload system trims after 1-2 cycles of oscillation of period nearly 7 sec. The parafoil-payload system in full brake shows large changes in trim states from zero brake condition, showing change of glide angle from 18 deg to 54 deg, angle of attack from 8 deg to 38 deg, and pitch angle from -10 deg to -17 deg. The system lift to drag ratio reduces from 3.25 in zero brake to 0.75 in symmetric full brake. This large change in parafoil trim γ and αp due to full brake could also be seen in bifurcation diagrams in Figs. 3 and 4 for full symmetric brakes corresponding to µ = 9 deg. The reason for large change in glide trim is due to significant change in L/D of the parafoil-payload system as seen in Fig. 2 for zero and full symmetric brake. This results in the steeper descent trajectory in Fig. 5. When half or full symmetric brake are released back to zero brake condition, the parafoil-payload system takes two and half cycles of longitudinal oscillations to trim to zero brake condition. Thus, the parafoilpayload system is able to recover from stall.
V. A.
Turning Flights
Simulation
Figure 6 shows the response of parafoil-payload system for three different combinations of left and right brake deflections, namely, 0L50R, 0L100R, and 50L100R. The 0L50R is zero symmetric brake (i.e., δ s = zero brake), and half asymmetric brake (i.e., δa = 0.12), indicating only right brake half down. 0L100R is zero symmetric brake (i.e., δs = zero brake) and full asymmetric brake (i.e., δa = 0.24) indicating right brake full down. 50L100R is half symmetric deflection (i.e., δs = half brake) and half asymmetric deflection (i.e,. δa = 0.12) indicating left brake half down and right brake full down. For 0L50R and 0L100R brake combinations, parafoil-payload system is allowed to glide from initial height of 1000 m, 12 m/s velocity with zero brake deflection for 25 sec, then, half or full right brake is deflected over interval of 1 sec. For 50L100R brake combination, initially, parafoil-payload system is allowed to trim for half symmetric brake over 25 sec, Then, right brake is further deflected half down over interval of 1 sec. In all the three cases, the parafoil takes 1-3 cycles of lateral oscillation over period of ≈ 8 sec to achieve lateral trim.
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The 0L50R, 0L100R and 50L100R brake deflections show parafoil-payload in perfect turns taking 1, 3 loops in 75 sec. 0L50R and 0L100R show parafoil turn rate rp of 7 deg/sec and 17 deg/sec respectively with parafoil bank angle φp of 7 deg and 18 deg respectively. Also, 0L100R and 0L50R result in increase in parafoil glide angle γ from 18 deg to 30 deg and 25 deg respectively. The reason for large effect on glide angle is that the 0L100R and 0L50R deflections result in parafoil glide angle of attack α p = 16 − 18 deg in post-stall region thereby reducing system L/D ratio hence, increasing glide angle. The 0L50R, 50L100R show approximately same turn rate rp = 7 − 8 deg/sec and bank angle φ = 7 − 9 deg, but 50L100R does not affect parafoil glide angle. This is because the parafoil glide angle of attack does not cross stall and remains around 9 deg thereby not changing L/D and hence glide angle. Parafoil-payload system in turning flight under 0L50R, 0L100R and 50L100R brake deflections shows small sideslip angle βp = 0.5 − 1.75 deg suggesting little skidding. As compared to 0L100R, the 0L50R gives approximately same glide angle but with much smaller turn rate, bank angle and sideslip. Therefore, 0L50R is ideal for parafoil-payload system to descend fast over a target. As 50L100R gives small glide angle with approximately same turn rate as 0L50R, therefore it is ideal for parafoil-payload system to hover over a larger area for longer time. B.
Lateral Trims (δs = zero brake)
Figure 7 shows trim values obtained from 9-DOF simulation for varying asymmetric brake deflection δ a keeping symmetric brake δs = 0 with rigging angle µ = 9 deg. The asymmetric brake deflection δa is varied from -0.24 to +0.24 indicating negative values of δa as left brake down and positive values of δa as right brake down. At first, lateral trims were obtained for rigging angle µ = 9 deg as it gave minimum glide angle for zero brake condition in bifurcation analysis. Then, two more rigging angles µ = 6, 12 deg were selected to see effect of rigging angle on lateral trims. Parafoil-payload system shows symmetric lateral and longitudinal trims for same amount of left or right brake down. The negative values of turn rate, bank angle indicate parafoil-payload system turning and banking towards left. Parafoil-payload system with increasing asymmetric brake deflection upto δ a = ±0.24 shows linear increase in parafoil turn rate upto ±16 deg/sec and parafoil bank angle φp = ±16 deg. The gliding trims are same for negative or positive vales of δa . The gliding trims for µ = 9 deg show large variation of glide angle γ from -18 deg to -29 deg because of αp variation in parafoil stall region. For rigging angle µ = 6, 12 deg, parafoil lateral trims in yaw rate rp , bank angle φp , and sideslip angle βp are same as for µ = 9 deg, whereas rigging angle µ = 6 deg shows trim values of α p = 18 − 23 deg in post-stall region thereby resulting in high negative values of trim γ = 27 − 31 deg. Rigging angle µ = 12 deg shows trim αp = 6 − 9 deg within the stall limits thereby resulting in low values of negative trim γ = 1820 deg. Therefore, rigging angle higher than minimum glide µ = 9 is required to achieve better glide during parafoil turn. C.
Lateral Trims (δs = half brake)
Figure 8 shows comparison of trim values for varying asymmetric brake deflection δ a keeping symmetric brake δs at zero and half brake, with rigging angle µ = 9 deg. The asymmetric brake deflection δ a is varied from -0.12 to +0.12. Varying asymmetric brake deflection with half symmetric brake results in approximately same lateral trims as with zero symmetric brake, but trims for glide angle, parafoil angle of attack and pitch angle show very small variation from minimum glide angle. This is because, half symmetric brake results in lower parafoil angle of attack than zero symmetric brake thereby shifting the angle of attack range to well within stall. Thus, asymmetric brake deflection while holding half symmetric brake does not affect parafoil glide and gives better glide performance while parafoil-payload is in turning flight.
VI.
Conclusions
A nonlinear 9-DOF flight dynamic model is used to determine trim and stability characteristics of a parafoil-payload system. The aerodynamic model uses longitudinal aerodynamic coefficients in tabular lookup form, including all nonlinearities, with angle of attack range of -10 to +80 deg. The lateral aerodynamics uses standard stability and control derivatives as available in literature for similar parafoil-payload system.
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The model also uses apparent mass terms from standard literature. A bifurcation analysis using a continuation algorithm was used to study longitudinal trim and stability characteristics for zero brake, half brake, and full brake deflections, for different choices of rigging angle. The gliding flight analysis reveals regions of unstable glide angles of attack as well as the best glide rigging angle. The turning flight analysis shows the effect of asymmetric brake on parafoil turn. The asymmetric brake deflection shows posibility of parafoil turning flight in post-stall region. The post-stall turn can be prevented by using either rigging angle higher than the best glide rigging angle, or using half symmetric brake with different asymmetric brake deflections.
References 1 Slegers, N., and Costello, M., “Aspects of Control for a Parafoil and Payload System,” Journal of Guidance, Control and Dynamics, Vol. 26, No. 6, 2003, pp. 898-905. 2 Mooij, E., Wijnands, Q.G.J. and Schat, B., “9-dof Parafoil/Payload Simulator Development and Validation,” AIAA Modeling and Simulation Technologies Conference and Exhibit, Austin, TX, August 11-14, 2003. 3 Machin, R. A., Iacomini, C. S, Cerimele, C. J., and Stein, J. M. “Flight Testing the Parachute System for the Space Station Crew Return Vehicle,” Journal of Aircraft, Vol. 38, No. 5, 2001, pp. 786-799. 4 Heise, M., and Muller, S., “Dynamic Modeling and Visualization of Multi-Body Flexible Systems,” AIAA Modeling and Simulation Technologies Conference and Exhibit, Providence, Rhode Island, August, 2004. 5 Iosilevskii, G., “Center of Gravity and Minimal Lift Coefficient Limits of a Gliding Parachute,” Journal of Aircraft, Vol. 32, No. 6, 1995, pp. 1297-1302. 6 Lingard, J. S., “The Performance and Design of Ram-Air Parachutes,” Precision Aerial Delivery Seminar, Technical Report, Royal Aircraft Establishment, Aug 1981. 7 Brown, G. J., “Parafoil Steady Turn Response to Control Input,” 12th AIAA Aerodynamic Decelerator Systems Technology Conference and Seminar, London, UK, May 10-13 1993, pp. 248-254. 8 Crimi, P., “Lateral Stability of Gliding Parachute,” Journal of Guidance, Control and Dynamics, Vol. 5, No. 5, 1982, pp. 529-536. 9 Carroll, J. V., and Mehra, R. K., “Bifurcation Analysis of Nonlinear Aircraft Dynamics,” Journal of Guidance, Control and Dynamics, Vol. 5, No. 5, 1982, pp. 529-536. 10 Zagainov, G. I., and Goman, M. G., “Bifurcation Analysis of Critical Aircraft Flight Regimes,” International Council of the Aeronautical Sciences, September 1984, pp. 217-223. 11 Ananthkrishnan, N., and Sinha, N. K., “Level Flight Trim and Stability Analysis using Extended Bifurcation and Continuation Procedure,” Journal of Guidance, Control and Dynamics, Vol. 24, No. 6, 2001, pp. 1225-1228 12 Prakash, O., and Ananthkrishnan, N, “Trim and Stability Analysis of Parafoil/Payload System using Bifurcation Methods,” 18th AIAA Aerodynamic Decelerator Systems Technology Conference and Seminar, Munich, Germany, May 2005. 13 Doedel, 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 Report,California Inst. of Technology, Pasadena, CA, USA, 2001.
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1.4
. Zero Brake
1.2
* Half Brake o Full Brake
1
1 0.8
p
0.6
CD 0.8
CL
0.6
0.4
0.4 0.2 0.2 0 −20
0
20
40
60
0 −20
80
0
Angle of Attack, α
0
3.5
−0.1
3
−0.2
2.5
−0.3
L/D
Cm
1.5
−0.5
1
−0.6
0.5 20
40
60
60
80
2
c/4−0.4
0
40
Angle of Attack, αp
p
−0.7 −20
20
80
0 −20
0
20
40
60
Angle of Attack, αp
Angle of Attack, αp
Figure 2. Longitudinal aerodynamics data
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80
80. 70. Half Brake
60. 50. αp
Full Brake
40.
(deg) 30.
20.
Zero Brake
10. Half Brake
0. -10.
-5.
0. Rigging Angle, µ (deg)
5.
10.
15.
Figure 3. Bifurcation diagram of the parafoil angle of attack (Full line: stable trim, dashed line: unstable trim, solid square: Hopf bifurcation point)
80. 70.
Half Brake
Full Brake
60. 50. γ (deg) 40. Zero Brake
30. 20. 10. -10.
Half Brake
-5.
0.
5.
10.
15.
Rigging Angle, µ (deg)
Figure 4. Bifurcation diagram of glide angle of parafoil-payload system (Full line: stable trim, dashed line: unstable trim, solid square: Hopf bifurcation point)
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20
5 Zero Brake Half Brake Full Brake
15
Zero Brake Half Brake Full Brake
0
10
−5 5
−10
θ , deg
θ , deg
0
−15
p
b
−5 −10
−20
−15
−25
−20
−30
−25 −30 0
10
20
30
40
−35 0
50
10
20
(a) Payload pitch angle
Zero Brake Half Brake Full Brake
50
Glide angle γ, deg
30
αp, deg
25 20 15
40
30
20
10
10
10
20
30
40
0 0
50
10
20
Time in sec
30
40
50
Time in sec
(c) Parafoil angle of attack
(d) System glide angle
3.5
1000
950
2.5
900
Altitude , m
3
L/D
50
60
Zero Brake Half Brake Full Brake
35
2
1.5
Zero Brake Half Brake Full Brake
850
800 Zero Brake Half Brake Full Brake
1
0.5 0
40
(b) Parafoil pitch angle
40
5 0
30
Time in sec
Time in sec
10
20
30
40
750
700 0
50
Time in sec
100
200
300
400
Horizontal distance , m
(e) System lift to drag ratio
(f) Altitude vs Range
Figure 5. Dynamics for varying symmetric brake deflections
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500
600
5 0
0
20
40 60 Time in sec
80
100
20 15 10 5 0
0
20
40 60 Time in sec
80
100
Parafoil pitch angle, θp (deg)
p
10
Parafoil turn rate, rp (deg/s)
2 1 0 0
20
40 60 Time in sec
80
−15
0
20
40
60
80
100
80
100
80
100
Time in sec
20
15
10
5
0
20
40
60
−15 −20 −25 −30 0
20
40
60
Time in sec 150 Cross Range m
900 Height m
−10
−35
100
1000
800 700 600 500
−5
−10
3
−1
0L50R 0L100R 50L100R
Time in sec
4 Glide angle, γ (deg)
Parafoil side slip angle βp (deg)
15
0
Parafoil angle of attack, αp (deg)
Parafoil bank angle, φ (deg)
20
0
100
200
300
400
100 50 0 −50
0
100
Range m
200 Range m
Figure 6. Dynamics for different right and left brake deflections
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300
400
15 10
−7 µ=6 deg µ=9 deg µ=12 deg
Parafoil pitch angle, θp (deg)
Parafoil bank angle, φp (deg)
20
5 0 −5 −10 −15 −20 −0.4
−0.2
0
0.2
−8 −9 −10 −11 −12 −13 −0.4
0.4
0.4
Parafoil angle of attack, α (deg)
25
p
15 10 5
p
Parafoil turn rate, r (deg/sec)
20
0 −5 −10 −15 −0.2
0
0.2
20
15
10
5 −0.4
0.4
−0.2
0
0.2
0.4
Asymmetric deflection (δ )
Asymmetric deflection (δa)
a
2
−18
1.5
−20
1
Glide angle, γ (deg)
Parafoil side slip angle βp (deg)
0.2 a
a
0.5 0
−0.5 −1
−22 −24 −26 −28 −30
−1.5 −2 −0.4
0
Asymmetric deflection (δ )
Asymmetric deflection (δ )
−20 −0.4
−0.2
−0.2
0
0.2
0.4
−32 −0.4
−0.2
0
0.2
Asymmetric deflection (δa)
Asymmetric deflection (δ ) a
Figure 7. Trims for varying asymmetric brake and zero symmetric brake
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0.4
20
10
δs: Zero Brake δs: half Brake
−9 Parafoil pitch angle, θp (deg)
Parafoil bank angle, φp (deg)
15
−8
−10
5
−11
0
−12
−5
−13
−10
−14
−15 −20 −0.4
−0.2
0
0.2
−15 −0.4
0.4
Parafoil angle of attack, αp (deg)
Parafoil turn rate, rp (deg/sec)
10 5 0 −5 −10 −15 −0.2
0
0.2
15
10
5 −0.4
0.4
Asymmetric deflection (δa)
−0.2
0
0.2
0.4
Asymmetric deflection (δa)
2
−18
1.5
−20
1
Glide angle, γ (deg)
p
0.4
20
15
Parafoil side slip angle β (deg)
0.2 a
20
0.5 0
−0.5 −1
−22 −24 −26 −28
−1.5 −2 −0.4
0
Asymmetric deflection (δ )
Asymmetric deflection (δa)
−20 −0.4
−0.2
−0.2
0
0.2
0.4
Asymmetric deflection (δa)
−30 −0.4
−0.2
0
0.2
Asymmetric deflection (δa)
Figure 8. Trims for varying asymmetric brake and half symmetric brake
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0.4
