Control of locomotion systems and dynamics in relative periodic orbits

Francesco Fassò (1) , Simone Passarella (1) , Marta Zoppello (2)

1 Unipd - Università degli Studi di Padova = University of Padua
2 Dipartimento di Matematica Pura e Applicata [Padova]

The connection between the dynamics in relative periodic orbits of vector fields with noncom-pact symmetry groups and periodic control for the class of control systems on Lie groups known as '(robotic) locomotion systems' is well known, and has led to the identification of (geometric) phases. We take an approach which is complementary to the existing ones, advocating the relevance-for trajectory generation in these control systems-of the qualitative properties of the dynamics in relative periodic orbits. There are two particularly important features. One is that motions in relative periodic orbits of noncompact groups can only be of two types: either they are quasi-periodic, or they leave any compact set as t ± ∞ ('drifting motions'). Moreover, in a given group, one of the two behaviours may be predominant. The second is that motions in a relative periodic orbits exhibit 'spiralling', 'meandering' behaviours, which are routinely detected in numerical integrations. Since a quantitative description of meandering behaviours for drifting motions appears to be missing, we provide it here for a class of Lie groups that includes those of interest in locomotion (semidirect products of a compact group and a normal vector space). We illustrate these ideas on some examples (a kinematic car robot, a planar swimmer).

Mots clés: 34H05; 37C80; Robotic locomotion systems Relative periodic orbits Geometric phases Reconstruction Equivariant dynamics Scallop theorem Trajectory generation MSC: 37N35; 70E60; 70Q05

Domaines: Physique mathématique [math-ph]; Systèmes dynamiques [math.DS]

Francesco Fassò, Simone Passarella, Marta Zoppello. Control of locomotion systems and dynamics in relative periodic orbits. 2020. ⟨hal-02546098v2⟩