Event

This dissertation develops a geometric framework for modeling and control of human agents through the use of Lie groups. Recognizing that traditional models of robotic systems with rigid body components based on the special euclidean group (SE(3) for manipulator joint kinematics and dynamics, SE(2) for planar wheeled mobility) are ill-suited to capture the inherent complexities in human motion (e.g. planar locomotion, handwriting, 3D arm movements), we investigate an alternative approach focused on treating human agents as moving points on the special affine (or equi-affine) group SA(2). In particular, we exploit the idea that trajectories of agents in the affine plane with constant equi-affine speed naturally satisfy the two-thirds power law, a well-documented regularity observed in human movement (e.g. handwriting, planar locomotion).

 
We begin by discussing existing models of human motion, and then using Cartan’s method of moving frames, we set up control systems on the Lie groups SE(2) and SA(2), with geometric invariants such as euclidean curvature and affine curvature as controls.

 
Next we use the constructed motion models to solve optimal control problems on SA(2) and its subgroup SL(2) for a single agent by appeal to the maximum principle of Pontryagin and coworkers, and Lie-Poisson symmetry reduction. The resulting hamiltonian systems on duals of Lie algebras are analyzed (using the structure of co-adjoint orbits and associated Casimir invariants) to compute extremals which respect the two-thirds power law. We compare these extremals with ones for robotic agents modeled on SE(2).

Extending these results, we explore multi-agent systems, specifically the landscape of relative equilibria for pairs of agents in SA(2) (and make comparisons with earlier results for SE(2)). We then construct feedback control laws based on the indirect method of Lyapunov to stabilize a multi-agent system to a specific relative equilibrium. Again we make comparisons between SA(2) and SE(2) settings.

This work lays a mathematical framework for designing motion algorithms for robotic systems that emulate the underlying geometry of human movement, thereby contributing to the broader goal of achieving effective human-robot collaboration.