Co-chairs: Maani Ghaffari, Ram Vasudevan

Abstract:
Lie groups serve as a powerful mathematical framework for modeling the kinematics and dynamics of robotic systems composed of 3D rigid bodies. However, the nonlinear nature of rigid body motion introduces significant challenges, making problems such as motion planning, feedback control, and state estimation inherently nonconvex and difficult to solve.

In this thesis, I first present a unified modeling framework based on matrix Lie groups, which captures the rich algebraic and geometric structures of rigid body systems to enable systematic analysis and computation. Building on this foundation, I exploit the underlying polynomial structures of matrix Lie groups to develop globally optimal and certifiable solutions to these robotics problems via convex optimization. Notably, I address the kinodynamic motion planning problem — ensuring both kinematic and dynamic feasibility — for 3D rigid body systems with global optimality guarantees. I then introduce the Generalized Moment Kalman Filter, a novel extension of Kalman filtering theory from linear Gaussian systems to nonlinear polynomial systems with arbitrary noise distributions. This generalization enables more robust and accurate state estimation in complex robotic systems. Finally, I leverage the geometric properties of matrix Lie groups to design efficient local gradient-based solvers tailored for rigid body systems. In particular, I present the Riemannian Direct Trajectory Optimization framework, which ensures geometric compatibility while enabling fast and efficient motion planning.

Together, these contributions provide a comprehensive set of tools for certifiable, efficient, and geometrically consistent motion planning and state estimation in rigid body robotic systems.

1000 Robotics Atrium,
https://umich.zoom.us/j/98764011774
Meeting ID: 987 6401 1774
Passcode: 83112