Reduction in optimal control with broken symmetry for collision and obstacle avoidance of multi-agent system on Lie groups

Efstratios Stratoglou; Alexandre Anahory Simoes; Leonardo J. Colombo; Efstratios Stratoglou; Alexandre Anahory Simoes; Leonardo J. Colombo

doi:10.3934/cam.2023001

Communications in Analysis and Mechanics

2023, Volume 15, Issue 2: 1-23. doi: 10.3934/cam.2023001

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Research article Special Issues

Reduction in optimal control with broken symmetry for collision and obstacle avoidance of multi-agent system on Lie groups

1.
Programa de Doctorado de Automática y Robótica, Universidad Politécnica de Madrid (UPM), 28006 Madrid, Spain
2.
IE School of Sciences and Technology, P. Castellana, 259E, 28046 Madrid, Spain
3.
Centre for Automation and Robotics (CSIC-UPM), Ctra. M300 Campo Real, Km 0,200, Arganda del Rey - 28500 Madrid, Spain

Received: 11 November 2022 Revised: 16 January 2023 Accepted: 28 January 2023 Published: 03 March 2023
70G45, 70H03, 70H05, 37J15, 49J15

We study the reduction by symmetry for optimality conditions in optimal control problems of left-invariant affine multi-agent control systems, with partial symmetry breaking cost functions for continuous-time and discrete-time systems. We recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Lagrangian and obtain the reduced optimality conditions from a reduced variational principle via symmetry reduction techniques in both settings continuous-time, and discrete-time. We apply the results to a collision and obstacle avoidance problem for multiple vehicles evolving on $ SE(2) $ in the presence of a static obstacle.
- Lagrangian systems,
- symmetry reduction,
- Euler-Poincaré equations,
- multi-agent control systems,
- Lie-Poisson integrators
Citation: Efstratios Stratoglou, Alexandre Anahory Simoes, Leonardo J. Colombo. Reduction in optimal control with broken symmetry for collision and obstacle avoidance of multi-agent system on Lie groups[J]. Communications in Analysis and Mechanics, 2023, 15(2): 1-23. doi: 10.3934/cam.2023001

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Abstract

We study the reduction by symmetry for optimality conditions in optimal control problems of left-invariant affine multi-agent control systems, with partial symmetry breaking cost functions for continuous-time and discrete-time systems. We recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Lagrangian and obtain the reduced optimality conditions from a reduced variational principle via symmetry reduction techniques in both settings continuous-time, and discrete-time. We apply the results to a collision and obstacle avoidance problem for multiple vehicles evolving on $ SE(2) $ in the presence of a static obstacle.

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