Curvature in force/position control

If a rigid body is subject to a set of holonomic constraints then the set of allowed positions and orientations comprise a subspace in the group of rigid body motions. Consideration of the geometry of these constraint subspaces leads us to consider their second fundamental form. This allows us to predict the constraint forces and torques generated by a constrained motion. The result has implications for the control of robots that have constrained end-effectors. A slight modification is proposed to the standard methods of shared control and hybrid control. Effectively this modification accounts for the curvature of the constraint space. Finally, the situation where the constraints are in the form of a physical surface in space is considered. In particular some comments are made regarding the problem of measuring the curvature of the surface using a robot