Towards Geometric Motion Planning for 3-link Kinematic Systems

Abstract

Geometric mechanics offers a powerful mathematical framework for studying locomotion for mobile systems. Despite the well-established literature, challenges remain when using geometric mechanics to design gaits for robots made of the multi-link chain; In this thesis, we will look at two of them. First, for a large class of nonholonomic systems, singular configurations appear when constraints are violated, resulting in infinite quantities in the equations of motion that govern the motion of the robot. Second, most geometric mechanics models rely heavily on system symmetry in SE(2) (i.e. invariance of system dynamics and constraints with respect to the system orientation) to simplify motion analysis. As a result, locomotion is rarely studied on non-flat surfaces which break the symmetry assumption.

In this work, we take initial steps in establishing a formal understanding of those two fundamentally different challenges under a common framework of the three-link kinematic system. The thesis is organized into two parts. The first part focuses on addressing the effect of singularities when designing gaits for a 3-link kinematic snake in SE(2). We show how to combine our singularity treatment with an adapted variational optimizer to find high-efficiency gait under the defined cost metrics. The second part focuses on understanding the geometric properties of a novel 3-link kinematic system on curved position space, specifically on the cylindrical surface. Interesting features arise when the surface curvature imposes additional mechanical constraints on the system, thereby making the locomotion analysis dependent on the system orientation.