Controlled Lagrangians and stabilization of Euler-Poincaré mechanical systems with broken symmetry. II: Potential shaping. (English) Zbl 1494.93080

Contreras, César; Ohsawa, Tomoki

Math. Control Signals Syst. 34, No. 2, 329-359 (2022).

Summary: We apply the method of controlled Lagrangians by potential shaping to Euler-Poincaré mechanical systems with broken symmetry. We assume that the configuration space is a general semidirect product Lie group \(\mathsf{G}\ltimes V\) with a particular interest in those systems whose configuration space is the special Euclidean group \(\mathsf{SE}(3) = \mathsf{SO}(3)\ltimes\mathbb{R}^3\). The key idea behind the work is the use of representations of \(\mathsf{G}\ltimes V\) and their associated advected parameters. Specifically, we derive matching conditions for the modified potential exploiting the representations and advected parameters. Our motivating examples are a heavy top spinning on a movable base and an underwater vehicle with non-coincident centers of gravity and buoyancy. We consider a few different control problems for these systems, and show that our results give a general framework that reproduces our previous work on the former example and also those of Leonard on the latter. Also, in one of the latter cases, we demonstrate the advantage of our representation-based approach by giving a simpler and more succinct formulation of the problem.
For Part I, see [the authors, “Controlled Lagrangians and stabilization of Euler-Poincaré mechanical systems with broken symmetry”, Preprint, arXiv:2003.10584].

MSC:

93D05	Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
70Q05	Control of mechanical systems
34H15	Stabilization of solutions to ordinary differential equations
70H33	Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics

Keywords:

stabilization; controlled Lagrangians; potential shaping; Euler-Poincaré mechanical systems; broken symmetry; semidirect product

Software:

GEOMetrics

References:

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