TY - JOUR

T1 - Optimal control problems on parallelizable riemannian manifolds

T2 - Theory and applications

AU - Iyer, Ram V.

AU - Holsapple, Raymond

AU - Doman, David

PY - 2006/1

Y1 - 2006/1

N2 - 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.

AB - 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.

KW - Calculus of variations

KW - Modified simple shooting method

KW - Numerical solution

KW - Regular optimal control

KW - Simple mechanical systems

UR - http://www.scopus.com/inward/record.url?scp=33645768520&partnerID=8YFLogxK

U2 - 10.1051/cocv:2005026

DO - 10.1051/cocv:2005026

M3 - Article

AN - SCOPUS:33645768520

VL - 12

SP - 1

EP - 11

JO - ESAIM - Control, Optimisation and Calculus of Variations

JF - ESAIM - Control, Optimisation and Calculus of Variations

SN - 1292-8119

IS - 1

ER -