Control of locomotion systems and dynamics in relative periodic orbits

Fassò Francesco; Passarella Simone; Zoppello Marta

doi:10.3934/jgm.2020022

Article Contents

2020, Volume 12, Issue 3: 395-420. Doi: 10.3934/jgm.2020022

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Control of locomotion systems and dynamics in relative periodic orbits

1.
Università di Padova Dipartimento di Matematica "Tullio Levi-Civita", Via Trieste 63, 35121 Padova, Italy
2.
Politecnico di Torino, Dipartimento di Scienze Matematiche, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Dedicated to James Montaldi

Received: December 2019

Revised: June 2020

Early access: July 2020

Published: September 2020

FF has been partially supported by the MIUR-PRIN project 20178CJA2B New Frontiers of Celestial Mechanics: theory and applications. MZ gratefully acknowledges support from the MIUR grant Dipartimenti di Eccellenza 2018-2022 (CUP: E11G18000350001)

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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\to\pm\infty $ ('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).

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Mathematics Subject Classification: 37N35, 37C80, 34H05, 70Q05, 70E60.

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Figure 1. The phase

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Figure 2. Images of gaits

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Figure 3. The car robot

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Figure 4. Four trajectories of a point of the car's frame in the $ (x,y) $-plane. The gaits have $ \dot\psi_2^\ell = 1 $ and $ \phi^\ell $ as shown in the insets. The coordinates in the insets' plots are time (horizontal) and $ \phi^\ell $ (vertical). In all cases $ \lambda = 2.5 $, $ a = 0.4 $ and the initial configuration of the car is $ (\theta_0,x_0,y_0) = (\pi/4,0,0) $. The value of $ \theta^\ell(T) $ is $ 0 $ in (a), $ 2\pi $ in (b), approximately $ 0.262\,\pi $ in (c) and approximately $ 0.727\pi $ in (d)

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