Euclidean metrics for motion generation on

Previous approaches to trajectory generation for rigid bodies have been either based on the so-called invariant screw motions or on ad hoc decompositions into rotations and translations. This paper formulates the trajectory generation problem in the framework of Lie groups and Riemannian geometry. The goal is to determine optimal curves joining given points with appropriate boundary conditions on the Euclidean group. Since this results in a two-point boundary value problem that has to be solved iteratively, a computationally efficient, analytical method that generates near-optimal trajectories is derived. The method consists of two steps. The first step involves generating the optimal trajectory in an ambient space, while the second step is used to project this trajectory onto the Euclidean group. The paper describes the method, its applications, and its performance in terms of optimality and efficiency. 3401 Walnut St., Philadelphia, PA 19104-6228, Tel:(215) 898-5814 Notation IRn Euclidean space of dimension n SO(n) special orthogonal group SE(n) special Euclidean group GL(n) general linear group of dimension n GA(n) the affine group so(n) the Lie algebra of SO(n) se(n) the Lie algebra of SE(n) fFg reference (fixed) frame fMg mobile frame O origin of fFg O0 origin of fMg R rotation matrix (2 SO(n)) A homogeneous transformation matrix (2 SE(n)) B affine transformation (2 GA(3)) M nonsingular matrix (2 GL(3))) d position vector in fFg exponential coordinates on SO(3) TPM tangent space at P 2M to manifold M ! angular velocity in fMg v linear velocity in fMg S twist 2 se(3) X; Y tangent vectors V velocity vector A acceleration vector < ; > Riemmanian metric G matrix of metric in SO(3) ~ G matrix of metric in SE(3) Tr matrix trace L0i basis vector in so(3) Li basis vector in se(3) x; y; z Cartesian axes D dt covariant derivative