On the generation of smooth three-dimensional rigid body motions

TLDR

The paper seeks to generate smooth trajectories between specified initial and final poses in space. The authors formulate a variational problem on the Lie group of rigid body displacements, defining a smoothness functional based on velocity derivatives and a Riemannian metric that incorporates system inertia. They demonstrate that suitable smoothness measures enable trajectories satisfying velocity or higher‑order boundary conditions, and derive necessary conditions for shortest‑distance, minimum‑acceleration, and minimum‑jerk paths.

Abstract

This paper addresses the problem of generating smooth trajectories between an initial and a final position and orientation in space. The main idea is to define a functional depending on velocity or its derivatives that measures smoothness of trajectories and find a trajectory that minimizes this functional. In order to ensure that the computed trajectories are independent of the parametrization of positions and orientations, we use the notions of Riemannian metric and covariant derivative from differential geometry and formulate the problem as a variational problem on the Lie group of spatial rigid body displacements. We show that by choosing an appropriate measure of smoothness, the trajectories can be made to satisfy boundary conditions on the velocities or higher order derivatives. Dynamically smooth trajectories can be obtained by incorporating the inertia of the system into the definition of the Riemannian metric. We state the necessary conditions for the shortest distance, minimum acceleration and minimum jerk trajectories.