Motion Planning and Differential Flatness of Mechanical Systems on Principal Bundles

Mechanical systems often exhibit physical symmetries in their configuration variables, allowing for significant reduction of their mathematical complexity arising from characteristics such as underactuation and nonlinearity. In this paper, we exploit the geometric structure of such systems to explore the following motion planning problem: given a desired trajectory in the workspace, can we explicitly solve for the appropriate inputs to follow it? We appeal to results on differential flatness from the nonlinear control literature to develop a general motion planning formulation for systems with symmetries and constraints, which also applies to both fully constrained and unconstrained kinematic systems. We conclude by demonstrating the utility of our results on several canonical mechanical systems found in the locomotion literature. INTRODUCTION In the analysis of mechanical systems, one often desires to express the equations of motion in a manner conducive to analysis and control. For a certain class of systems, this motivation often amounts to reducing the equations by exploiting a system’s natural symmetries and the corresponding invariants due to Noether’s theorem. The configuration variables of such systems often constitute a principal fiber bundle structure defined by a Lie group (the non-actuated symmetry directions) and a shape manifold (the system’s actuated internal configuration). For ex∗Address all correspondence to this author. FIGURE 1: TWO-WHEEL DIFFERENTIAL-DRIVE CAR. ample, the two-wheeled mobile robot shown in Fig. 1 locomotes on the plane, parameterized by the Lie group SE(2) (position and orientation of the robot), and has a shape manifold parameterized by its two wheel angles ψ1 and ψ2. In recent years, many results have analyzed the properties and structure of the reduced equations that emerge from such a splitting. These formulations have been shown to accommodate nonholonomic constraints as well as locomotion in high and low Reynolds number fluids. The equations’ structure, and in particular the kinematic form, has been shown to be conducive to the analysis of forward motion planning, which has found much success in periodic inputs known as gaits. In this paper, we attack the motion planning problem headon; instead of deriving gaits to achieve a desired motion, we are interested in situations for which the inputs can be directly solved to follow a complete trajectory in the position variables. This problem bears similar overtones to that of exploiting differential flatness in systems theory, a property useful for trajec1 Copyright c © 2015 by ASME tory planning for nonlinear dynamical systems. To the authors’ knowledge, commonalities between these systems and those with reducible geometric structure have been little explored in the locomotion literature, and here we try to clarify some of these links in the motion planning context. We structure our paper as follows. We first present an overview of relevant work in the geometric mechanics and differential flatness communities. Following this, we present the necessary mathematical tools and summarize relevant results from prior work. In the next section, we explicitly state the motion planning problem and present the general solution for mixed nonholonomic systems followed by the special cases of principally kinematic and purely mechanical systems. We finish by considering the specific case of systems on SE(2) and applying our technique to several examples from the literature.