JOURNAL OF AIRCRAFT Vol. 42, No. 5, September–October 2005 Simultaneous-Perturbation-Stochastic-Approximation Algorithm for Parachute Parameter Estimation Govindarajan Kothandaraman ∗ and Mario A. Rotea † Purdue University, West Lafayette, Indiana 47907-1282 This paper presents an algorithm to estimate unknown parameters of parachute models from flight-test data. The algorithm is based on the simultaneous-perturbation-stochastic-approximation method to minimize the prediction error (difference between model output and test data). The algorithm is simple to code and requires only the model output. Analytical gradients are not necessary. The algorithm is used to estimate aerodynamic and apparent mass coefficients for an existing parachute model. Nomenclature A = apparent mass tensor B, C , D = points on parachute C d = drag coefficient C m = moment coefficient c ref = reference length e 0 , e 1 , e 2 , e 3 = quaternions F x , F y , F z = force along x , y , and z directions H = cost function I xx , I yy , I zz = moment of inertia about x , y , and z axes M x , M y , M z = moment about x , y , and z axes m = mass p = roll rate q = pitch rate r = yaw rate S ref = reference area U = velocity along X direction u, v, w = velocity along the x , y , and z axes V = velocity, velocity along Y direction X , Y , Z = inertial coordinates x , y , z = body coordinates α = angle of attack α T = total angle of attack β = angle of sideslip θ = pitch angle ϑ = unknown parameters ρ = density of air φ = roll angle ψ = yaw angle Introduction T HIS paper describes a simple algorithm for parameter esti- mation that can be used with nonlinear dynamic parachute models. Model parameters are determined by minimizing the pre- diction error obtained by comparing model output with test data. Minimization is accomplished using the simultaneous-perturbation- stochastic-approximation (SPSA) algorithm developed by Spall. 1,2 The SPSA algorithm is an iterative method for optimization, with randomized search direction, that requires at most three function Presented as Paper 2003-2118 at the AIAA 17th Aerodynamic Decelerator Systems Technology Conference and Seminar, Monterey, CA, 19–22 May 2003; received 22 June 2004; revision received 15 September 2004; accepted for publication 24 September 2004. Copyright c  2004 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0021-8669/05 $10.00 in correspondence with the CCC. ∗ Graduate Student, School of Aeronautics and Astronautics. † Professor, School of Aeronautics and Astronautics; rotea@purdue.edu. Member AIAA. (model) evaluations at each iteration. Hence, execution time per it- eration does not increase with the number of parameters. The method can handle nonlinear dynamic models, nonequilibrium transient test conditions, and data obtained in closed loop. For this reason, this method is suitable for the estimation of parameters in guided parachute models. The present paper has three main sections. The first section de- scribes the model whose parameters are to be determined. The model is for a G-12 parachute, and it was developed at the Naval Postgrad- uate School (NPS). 3,4 The second section explains the basic param- eter estimation approach. This section includes a simple description of the SPSA algorithm. In the third section we give the numerical results corresponding to the determination of three aerodynamic co- efficients, four apparent mass coefficients, and the initial states for the G-12 parachute model. Conclusions and recommendations for further work are the end of the paper. Parachute Model A six-degrees-of-freedom model of a fully deployed G-12 parachute was developed at NPS. 3,4 Figure 1 gives a schematic of the G-12 parachute. This model assumes the following: 1) The parachute canopy and payload form one rigid system. 2) The aerodynamic forces and moments of the payload are negligible. 3) The aerodynamic forces act at the center of pressure of the canopy, which is nothing but the centroid of the air in the canopy. 4) The G-12 system is symmetrical about the axis joining the canopy centroid to the payload centroid. 5) The parachute is fully deployed. Equations of Motion Let m be the total mass of the parachute system. Let u, v, and w be the components of the ground velocity of the parachute in the body coordinate system (see Fig. 1). Let p, q , and r be the components of the angular velocity of the parachute expressed in body coordinates. Then, the equations of motion of the parachute are as follows. 3 F x = (m + A 11 )( ˙ u − vr ) + (m + A 33 )q w + ( J 1 + A 15 )( ˙ q + rp) (1a) F y = (m + A 11 )( ˙ v + ur ) − (m + A 33 ) pw − ( J 1 + A 15 )( ˙ p − qr ) (1b) F z = (m + A 33 ) ˙ w − (m + A 11 )(uq − v p) − ( J 1 + A 15 )( p 2 + q 2 ) (1c) M x = ( I xx + A 55 ) ˙ p − ( J 1 + A 15 )( ˙ v − pw + ur ) − ( I yy + A 55 − I zz )qr + ( A 33 − A 11 )vw (2a) 1229