Bézier motions with end-constraints on speed

A free-form motion can be considered as a smoothly varying rigid-body transformation. Motions can be created by establishing functions in an appropriate space of matrices. While a smooth motion is created, the geometry of the motion itself is not always immediately clear. In a geometric algebra environment, motions can be created using extensions of the ideas of Bézier and B-spline curves and the geometric significance of the construction is clearer. A motion passing through given precision poses can be obtained by direct analogy with the curve approach. This paper considers the more difficult problem of dealing additionally with velocity constraints at the ends of the motion: here the analogy is less obvious. A geometric construction for the end pairs of control poses is established and is demonstrated by creating motions satisfying given pose and velocity constraints.