Direct Trajectory Optimization Using a Variable Low-Order Adaptive Pseudospectral Method

Darby, Christopher L.; Hager, William W.; Rao, Anil V.

doi:10.2514/1.52136

Cited by 210 publications

(177 citation statements)

References 28 publications

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“…; K the nonzero elements are defined by the matrix given in Eq. (18). Next, the discretized path constraints of Eq.…”

Section: Radau Pseudospectral Methodsmentioning

confidence: 99%

“…In a p method, the state is approximated using few mesh intervals (often a single mesh interval is used), and convergence is achieved by increasing the degree of the polynomial [9][10][11][12][13][14][15][16]. In an hp method, both the number of mesh intervals and the degree of the polynomial within each mesh interval is varied, and convergence is achieved through the appropriate combination of the number of mesh intervals and the polynomial degrees within each interval [17,18].…”

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confidence: 99%

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Exploiting Sparsity in Direct Collocation Pseudospectral Methods for Solving Optimal Control Problems

Patterson¹,

Rao²

2012

Journal of Spacecraft and Rockets

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In a direct collocation pseudospectral method, a continuous-time optimal control problem is transcribed to a finitedimensional nonlinear programming problem. Solving this nonlinear programming problem as efficiently as possible requires that sparsity at both the first-and second-derivative levels be exploited. In this paper, a computationally efficient method is developed for computing the first and second derivatives of the nonlinear programming problem functions arising from a pseudospectral discretization of a continuous-time optimal control problem. Specifically, in this paper, expressions are derived for the objective function gradient, constraint Jacobian, and Lagrangian Hessian arising from the previously developed Radau pseudospectral method. It is shown that the computation of these derivative functions can be reduced to computing the first and second derivatives of the functions in the continuous-time optimal control problem. As a result, the method derived in this paper reduces significantly the amount of computation required to obtain the first and second derivatives required by a nonlinear programming problem solver. The approach derived in this paper is demonstrated on an example where it is found that significant computational benefits are obtained when compared against direct differentiation of the nonlinear programming problem functions. The approach developed in this paper improves the computational efficiency of solving nonlinear programming problems arising from pseudospectral discretizations of continuous-time optimal control problems.

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“…; K the nonzero elements are defined by the matrix given in Eq. (18). Next, the discretized path constraints of Eq.…”

Section: Radau Pseudospectral Methodsmentioning

confidence: 99%

mentioning

confidence: 99%

Exploiting Sparsity in Direct Collocation Pseudospectral Methods for Solving Optimal Control Problems

Patterson¹,

Rao²

2012

Journal of Spacecraft and Rockets

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“…In a direct collocation method, the state and/or the control is approximated using trial (basis) functions and the continuous problem is transcribed to a finite-dimensional nonlinear programming problem (NLP). A particular class of direct collocation methods that has become popular in the last decade is that of pseudospectral methods [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. In a pseudospectral method, the state is approximated using a basis of Lagrange polynomials and the differential-algebraic constraint equations are enforced at a finite set of collocation points.…”

mentioning

confidence: 99%

“…In a pseudospectral method, the state is approximated using a basis of Lagrange polynomials and the differential-algebraic constraint equations are enforced at a finite set of collocation points. The three most commonly used sets of collocation points for pseudospectral methods are the Legendre-Gauss (LG) [10][11][12]14,18], Legendre-Gauss-Radau (LGR) [13][14][15]17,18], and Legendre-Gauss-Lobatto (LGL) [1,3,4,9,19] points. These points are the roots of the linear combinations of a Legendre polynomial and/or its derivatives and correspond to the three different types of Gaussian quadrature.…”

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confidence: 99%

Costate Estimation using Multiple-Interval Pseudospectral Methods

Darby

Garg

Rao

2011

Journal of Spacecraft and Rockets

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A method is presented for costate estimation in nonlinear optimal control problems using multiple-interval collocation at Legendre-Gauss or Legendre-Gauss-Radau points. Transformations from the Lagrange multipliers of the nonlinear programming problem to the costate of the continuous-time optimal control problem are given. When the optimal costate is continuous, the transformed adjoint systems of the nonlinear programming problems are discrete representations of the continuous-time first-order optimality conditions. If, however, the optimal costate is discontinuous, then the transformed adjoint systems are not discrete representations of the continuous-time firstorder optimality conditions. In the case where the costate is discontinuous, the accuracy of the costate approximation depends on the locations of the mesh points. In particular, the accuracy of the costate approximation is found to be significantly higher when mesh points are located at discontinuities in the costate. Two numerical examples are studied and demonstrate the effectiveness of using the multiple-interval collocation approach for estimating costate in continuous-time nonlinear optimal control problems. = initial time ut = control on time domain t 2 t 0 ; t f u = control on time domain 2 1; 1 U j = control approximation at time point j w j = jth Legendre-Gauss or Legendre-Gauss-Radau quadrature weight X = state approximation on time domain 2 1; 1 xt = state on time domain t 2 t 0 ; t f x = state on time domain 2 1; 1 X j = state approximation at time point j = Lagrange multiplier associated with discretized inequality path constraint = Lagrange multiplier associated with inequality path constraint ij = Kronecker delta function = accuracy tolerance j = Lagrange multiplier of discretized dynamic constraint at j = continuous costate on the time domain 2 1; 1 j = costate approximation at time point j = Lagrange multiplier associated with quadrature constraint = jump in the costate at mesh point = transformed time domain 1; 1 = Mayer cost = boundary condition function = Lagrange multiplier associated with the discretized boundary condition = Lagrange multiplier associated with the boundary condition

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“…The optimal aeroassisted orbital transfer problem is posed as a nonlinear multiple-phase optimal control problem, and the optimal control problem is solved via direct collocation using the Radau collocation hp-adaptive 24 version of the open-source optimal control software GPOPS. [25][26][27] The overall performance of the vehicle is analyzed as a function of the number of atmospheric passes, required inclination change, and maximum allowable heating rate.…”

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confidence: 99%

A Preliminary Analysis of Small Spacecraft Finite-Thrust Aeroassisted Orbital Transfer

Rao

2012

AIAA/AAS Astrodynamics Specialist Conference

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The problem of finite-thrust small spacecraft minimum-fuel heat rateconstrained aeroassisted orbital transfer between two low-Earth orbits with inclination change is considered. The trajectory design for a vehicle of small mass and moderate thrust is described in detail and the aeroassisted orbital transfer is posed as a nonlinear optimal control problem. The optimal control problem is solved using a previously developed open-source pseudospectral optimal control software. It is found for heating rate-unconstrained cases that the fuel consumption is insensitive to the number of atmospheric passes, whereas for heating rate-constrained cases the fuel consumption decreases as a function of the number of atmospheric passes. The key features of the optimal trajectories are identified. The result of this research demonstrates the efficiency of an aeroassisted orbital transfer as compared with an all-propulsive transfer and show the more realistic results obtained when compared with an aeroassisted orbital transfer where it is assumed that the thrust is impulsive.

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Direct Trajectory Optimization Using a Variable Low-Order Adaptive Pseudospectral Method

Cited by 210 publications

References 28 publications

Exploiting Sparsity in Direct Collocation Pseudospectral Methods for Solving Optimal Control Problems

Exploiting Sparsity in Direct Collocation Pseudospectral Methods for Solving Optimal Control Problems

Costate Estimation using Multiple-Interval Pseudospectral Methods

A Preliminary Analysis of Small Spacecraft Finite-Thrust Aeroassisted Orbital Transfer

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