Stability and Control of Chaplygin Beanies Coupled to a Platform Through Nonholonomic Constraints

Abstract

Many multi-agent systems in nature are comprised of agents that interact with, and respond to, the dynamics of their environment. In this paper, we approach the study of such agent environment interactions through the study of passively compliant vehicles coupled to their environment via simple nonholonomic constraints. We first consider a single passively compliant Chaplygin beanie atop a platform having translational compliance, introduce the reduced equations for the system using the notion of nonholonomic momentum, and provide proof for its stability under arbitrary deformations of the elastic element modeling its compliance. We then direct our focus to results concerning the frequency response and control of passive Chaplygin beanies under actuation of the platform, discuss rich dynamical features arising from periodic actuation, and develop rules by which control can be exerted to collect and disperse multiple passive vehicles. We then discuss how the latter of these results clarifies the extent to which stable behavior can be excited in the system through exogenous control.

The Geometric Structure of Externally Actuated Planar Locomoting Systems in Ambient Media

“…In prior work we posed the problem of actuating a three-link wheeled snake robot via an external platform as a specific deviation from the usual geometric assumptions [31]. We also investigated an externally actuated version of the Chaplygin beanie in [2]. These two example systems belong to different classes in that the external actuation of one can be studied using a principal connection formulation, while the other demands reduction by means of nonholonomic momenta.…”

“…Its configuration space can naturally be written as Q w = G w × B w , where g w = (x, y, θ) T ∈ G w are the position and orientation fiber variables, and b w = (α 1 , α 2 ) T are the joint, or shape, variables. 1 The fiber G w is the Lie group of planar translations and rotations SE (2). The body velocities ξ are related to the world velocities ġw through the mapping T e L g .…”

“…(3) over a given joint trajectory, one can obtain a measure of displacement corresponding to the body frame directions. For SE (2), this measure provides the exact rotational displacement, i.e., θ = ξ θ for the third row, and an approximation of the translational component for the first two rows.…”

“…This relationship signifies a dependence on θ, the orientation of the robot relative to the platform (and inertial frame). However, if we know or can prescribe the platform's position relative to the robot's body frame 2 , we can directly use the external, mechanical connection −A ext p (b w ) without accounting for the effect of θ on the robot-platform interaction. In other words, we can choose to specify the platform's velocity using the variables…”

“…We make an explicit choice to consider only translational motion of the platform, engendering symmetries that we leverage in developing the tools we use to accomplish motion planning and control. This paper juxtaposes results from a recent conference paper [2] with complementary new material to construct a more unified narrative concerning externally actuated nonholonomic planar locomoting systems in ambient media. We first present results that draw on familiar tools for the three-link kinematic snake robot on a platform for which locomotion is governed purely by a principal connection.…”