Planning of smooth motions on SE(3)

This paper addresses the general problem of generating smooth trajectories between an initial and a final position and orientation. A functional depending an velocity and its higher derivatives involving a left invariant Riemannian metric on SE(3) is used to measure the smoothness of a trajectory. The problem of determining a smooth trajectory between two points is formulated as a variational problem on SE(3). The authors derive necessary conditions for the shortest distance and minimum jerk trajectories and solve the resulting two-point boundary value problem