Chapter 4: From Trajectories to Velocity: Forward Kinematics of Rigid Surfaces with Rolling Contact

In this chapter, we develop the forward kinematics of two rigid surfaces with rolling contact when the contact trajectory on each surface is present. We use the moving-frame method and the adjoint approach to obtain the instantaneous forward kinematics in terms of geometric invariants: rolling rate, normal curvature, geodesic curvature, and geodesic torsion. This approach is coordinate-invariant in the sense that the equations are invariant w.r.t. the Special Euclidean group in ℝ³.