On the Hybrid Minimum Principle On Lie Groups and the Exponential Gradient HMP Algorithm

TL;DR

This paper derives a geometric version of the Hybrid Minimum Principle for systems on Lie groups, extending the HMP algorithm to an Exponential Gradient method with convergence analysis and simulations.

Contribution

It introduces a geometric derivation of the HMP on Lie groups and extends the HMP algorithm to an Exponential Gradient approach with convergence guarantees.

Findings

The extended algorithm converges based on LaSalle Invariance Principle.

Simulation results demonstrate the effectiveness of the proposed method.

The geometric derivation provides new insights into hybrid systems on Lie groups.

Abstract

This paper provides a geometrical derivation of the Hybrid Minimum Principle (HMP) for autonomous hybrid systems whose state manifolds constitute Lie groups $(G,\star)$ which are left invariant under the controlled dynamics of the system, and whose switching manifolds are defined as smooth embedded time invariant submanifolds of $G$. The analysis is expressed in terms of extremal (i.e. optimal) trajectories on the cotangent bundle of the state manifold $G$. The Hybrid Maximum Principle (HMP) algorithm introduced in \cite{Shaikh} is extended to the so-called Exponential Gradient algorithm. The convergence analysis for the algorithm is based upon the LaSalle Invariance Principle and simulation results illustrate their efficacy.