Control of locomotion systems and dynamics in relative periodic orbits

Abstract

The connection between the dynamics in relative periodic orbits of vector fields with noncompact 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 orbit 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).

First Integrals and symmetries of nonholonomic systems

“…the dimension of the maximal abelian subgroup of G. If the symmetry group G is not compact, the complete dynamics can be either quasi-periodic on tori or an unbounded copy of R, depending on the symmetry group. Some details on these aspects are reviewed in Appendix B, but see also [2,33]. We thus show how the broad integrability of the complete dynamics of these type of systems is deeply related to their symmetries, that are able to produce, not only the right amount of dynamical symmetries, but also the complementary number of first integrals.…”

“…In particular, if the symmetry group G is compact, the reconstructed dynamics is quasi-periodic on tori of dimension at most r + 1, where r is the rank of the symmetry group G, and the phase space inherits the structure of a torus bundle. If the symmetry group is not compact, the situation is less simple, but still understood: the complete dynamics is either quasi-periodic or diffeomorphic to R, and whether one or the other case is more frequent (or generic) depends on the symmetry group (see Section 4.2, Appendix B and [2,33]).…”

Mean-field and kinetic limit of affine control systems: optimal control through leaders

“…A less explored strategy is instead to think that the controller acts on the system by directly assigning the values of the coordinates of some of the agents, regarded as control parameters. This can happen for example in the collective motion of the so called robotic locomotion systems, introduced in [15], where even the single agent dynamics is controlled assigning the evolution of some coordinates.…”

Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction

“…Functions with the structure of x(ξ, •) in (1.1) are sometimes called derivo-periodic [1,2] (referring to the fact that they are primitives of periodic functions) or running-periodic [17,19]. Moreover, they can be identified as a class within the more general family of relative-periodic functions [6], noticing that they can be decomposed as a periodic change in shape plus a translation in the position of the locomotor. This decomposition is classical in the modelling of locomotion [15] and will be crucial also in our paper, as we will soon show discussing equation (1.2).…”