Total Orientation and Total Pose

Abstract

Trajectories in SO(n) and SE(n) arise naturally in applications, such as robot motion. The terminal states of trajectories in these groups do not record information about the trajectories that led to them. But can finite-dimensional algebraic and geometric structures capture information about trajectories in a way that behaves well under trajectory concatenation ? This leads to the concepts of total orientation and total pose described in this talk. In the planar case, these quantities are simply the universal covering groups of the Lie groups SO(2) and SE(2). It is shown how these universal covers can not only add value, but are actually simpler objects to deal with than their more familiar counterparts. In contrast, for rotation and motion in three dimensions, the universal covers do not provide all of the desired properties. Therefore, new algebraic and geometric structures that generalize better to the three-dimensional case are discussed. These include monoids built on sheet counting and extensions of the iterated integrals of angular velocity introduced by Magnus and Chen in their hierarchies. It is shown that truncated versions of those hierarchies' form finite-dimensional Lie groups.

Biography

Gregory S. Chirikjian is Professor of Robotics at MBZUAI. He received undergraduate degrees from Johns Hopkins University in 1988 and his Ph.D. from the California Institute of Technology in 1992. He served on the Mechanical Engineering faculty at Johns Hopkins from 1992 to 2021, becoming full professor in 2001 and serving as department chair from 2004 to 2007. He became Head of Mechanical Engineering at the National University of Singapore in January 2019 and Mechanical Engineering department chair at the University of Delaware in January 2024. He joined MBZUAI in January 2026.