Cubic spline algorithms for orientation interpolation

This article presents a class of spline algorithms for generating orientation trajectories that approximately minimize angular acceleration. Each algorithm constructs a twice-differentiable curve on the rotation group SO(3) that interpolates a given ordered set of rotation matrices at specified knot times. Rotation matrices are parametrized, respectively, by the unit quaternion, canonical co-ordinate, and Cayley–Rodrigues representations. All the algorithms share the common feature of (i) being invariant with respect to choice of fixed and moving frames (bi-invariant), and (ii) being cubic in the parametrized co-ordinates. We assess the performance of these algorithms by comparing the resulting trajectories with the minimum angular acceleration curve.