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

Ram V. Iyer; Raymond Holsapple; David Doman

ESAIM: Control, Optimisation and Calculus of Variations (2005)

- Volume: 12, Issue: 1, page 1-11
- ISSN: 1292-8119

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topIyer, 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 (2005): 1-11. <http://eudml.org/doc/90788>.

@article{Iyer2005,

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.
},

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.; modified simple shooting method},

language = {eng},

month = {12},

number = {1},

pages = {1-11},

publisher = {EDP Sciences},

title = {Optimal control problems on parallelizable Riemannian manifolds: theory and applications},

url = {http://eudml.org/doc/90788},

volume = {12},

year = {2005},

}

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

DA - 2005/12//

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.; modified simple shooting method

UR - http://eudml.org/doc/90788

ER -

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