New, Fast Numerical Method for Solving Two-Point Boundary-Value Problems

Holsapple, Raymond; Venkataraman, R.; Doman, David B.

doi:10.2514/1.1329

Cited by 29 publications

(26 citation statements)

References 6 publications

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“…We set up a two-point boundary value problem using Theorem 2.1, and then obtain a numerical solution using the Modified Simple Shooting method [8].…”

Section: Rigid Body Translation and Rotationmentioning

confidence: 99%

“…We used a modified shooting method technique [8] to numerically solve for the unknown "Lagrange multipliers" (ξ,v 1 , η,v 2 ) at initial time. The first equation (10) is a matrix equation that we integrated using the well-known Rodriguez's formula [14]:…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…In this paper, we use the indirect approach to the solution of an optimal control problem. Among the indirect methods, the Modified Simple Shooting Method (MSSM) has been shown to be more accurate and resulting in faster computation times than other methods such as the Multiple Shooting (MSM), Finite Difference and Collocation methods [8]. In related research, it was found that the earlier MSM failed to converge numerically for an optimal control problem on T SO(3) while using frame co-ordinates, while the MSSM converged successfully [7].…”

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confidence: 99%

“…The main motivation for the choice of frame co-ordinates, is that the resulting first-order necessary conditions can be solved using a numerical method called the Modified Simple Shooting Method [8]. Though numerical methods for optimal control problems on local co-ordinates have a long history, there is a dearth of literature on numerical methods using frame co-ordinates.…”

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confidence: 99%

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Optimal control problems on parallelizable Riemannian manifolds: theory and applications

2005

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Abstract. The motivation for this work is the real-time solution of a standard optimal control problem arising in robotics and aerospace applications. For example, the trajectory planning problem for air vehicles is naturally cast as an optimal control problem on the tangent bundle of the Lie Group SE(3), which is also a parallelizable Riemannian manifold. For an optimal control problem on the tangent bundle of such a manifold, we use frame co-ordinates and obtain first-order necessary conditions employing calculus of variations. The use of frame co-ordinates means that intrinsic quantities like the Levi-Civita connection and Riemannian curvature tensor appear in the equations for the co-states. The resulting equations are singularity-free and considerably simpler (from a numerical perspective) than those obtained using a local co-ordinates representation, and are thus better from a computational point of view. The first order necessary conditions result in a two point boundary value problem which we successfully solve by means of a Modified Simple Shooting Method.Mathematics Subject Classification. 49K99, 49M05, 49Q99, 90C99, 93C10.

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“…We set up a two-point boundary value problem using Theorem 2.1, and then obtain a numerical solution using the Modified Simple Shooting method [8].…”

Section: Rigid Body Translation and Rotationmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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Optimal control problems on parallelizable Riemannian manifolds: theory and applications

2005

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“…Originally created to solve two-point boundary value problems (TPBVPs), the Modified Simple Shooting Method (MSSM) has been shown to be superior, both in speed and accuracy, to known methods for solving TPBVPs [2]. Since optimal control problems can be formulated with differential equations and boundary conditions, it seems feasible to propose that the MSSM could be used to solve problems in optimal control.…”

Section: Introductionmentioning

confidence: 99%

Trajectory generation using a modified simple shooting method

Trent

Venkataraman

Doman

2004 IEEE Aerospace Conference Proceedings (IEEE Cat. No.04TH8720)

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Abstract-In this paper, we combine the theoretical result of Pontryagin's Minimum Principle and a new numerical method to obtain a fast algorithm for the trajectory design problem for a reusable launch vehicle. We work with the outer-loop equations for an aircraft and consider the angle-of-attack to be the input. We cast the trajectory design problem as an optimal control problem and use Pontryagin's Minimum Principle to obtain first order necessary conditions. These are in the form of a two-point boundary-value problem (TPBVP) and we solve them by means of the Modified Simple Shooting Method (MSSM). In recent work, the MSSM has been shown to be superior, both in speed and accuracy, for TPBVPs.

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