# Optimal control problems on parallelizable riemannian manifolds : theory and applications

• [1] U.S. Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio 45433-7531, USA.
• Volume: 12, Issue: 1, page 1-11
• ISSN: 1292-8119

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## Abstract

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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 $SE\left(3\right),$ 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.

## How to cite

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Iyer, Ram V., Holsapple, Raymond, and Doman, David. "Optimal control problems on parallelizable riemannian manifolds : theory and applications." ESAIM: Control, Optimisation and Calculus of Variations 12.1 (2006): 1-11. <http://eudml.org/doc/244946>.

@article{Iyer2006,
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.},
affiliation = {U.S. Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio 45433-7531, USA.},
author = {Iyer, Ram V., Holsapple, Raymond, Doman, David},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {regular optimal control; simple mechanical systems; calculus of variations; numerical solution; modified simple shooting method; Regular optimal control},
language = {eng},
number = {1},
pages = {1-11},
publisher = {EDP-Sciences},
title = {Optimal control problems on parallelizable riemannian manifolds : theory and applications},
url = {http://eudml.org/doc/244946},
volume = {12},
year = {2006},
}

TY - JOUR
AU - Iyer, Ram V.
AU - Holsapple, Raymond
AU - Doman, David
TI - Optimal control problems on parallelizable riemannian manifolds : theory and applications
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2006
PB - EDP-Sciences
VL - 12
IS - 1
SP - 1
EP - 11
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.
LA - eng
KW - regular optimal control; simple mechanical systems; calculus of variations; numerical solution; modified simple shooting method; Regular optimal control
UR - http://eudml.org/doc/244946
ER -

## References

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