The Role of Symmetry in Constructing Geometric Flat Outputs for Free-Flying Robotic Systems

TL;DR

This paper explores how symmetry properties of mechanical systems can be used to construct geometric flat outputs, simplifying motion planning for free-flying robots by leveraging the inherent structure of their configuration spaces.

Contribution

It introduces the concept of geometric flat outputs derived from symmetry, providing conditions for their existence and demonstrating their application to various robotic systems.

Findings

Symmetry can be directly employed to construct flat outputs for robotic systems.

Geometric flat outputs are often global or almost-global, unlike traditional flat outputs.

Application examples include planar rocket, aerial manipulator, and quadrotor.

Abstract

Mechanical systems naturally evolve on principal bundles describing their inherent symmetries. The ensuing factorization of the configuration manifold into a symmetry group and an internal shape space has provided deep insights into the locomotion of many robotic and biological systems. On the other hand, the property of differential flatness has enabled efficient, effective planning and control algorithms for various robotic systems. Yet, a practical means of finding a flat output for an arbitrary robotic system remains an open question. In this work, we demonstrate surprising new connections between these two domains, for the first time employing symmetry directly to construct a flat output. We provide sufficient conditions for the existence of a trivialization of the bundle in which the group variables themselves are a flat output. We call this a geometric flat output, since it is equivariant (i.e. maintains the symmetry) and is often global or almost-global, properties not typically enjoyed by other flat outputs. In such a trivialization, the motion planning problem is easily solved, since a given trajectory for the group variables will fully determine the trajectory for the shape variables that exactly achieves this motion. We provide a partial catalog of robotic systems with geometric flat outputs and worked examples for the planar rocket, planar aerial manipulator, and quadrotor.