A quasi-linear method for computing and projecting onto c-surfaces: planar case

This report presents a general method to compute configuration space (c-space) obstacle surfaces (c-surfaces) in planar quaternion space and for projecting points onto them. We parameterize the general c-surface using the rotation angle and the vector of translation parameters of the individual contacts. Once we compute the domain of the rotation parameter, we can setup the translation parameters in a linear equation. The singular value decomposition of this equation gives us with the exact parameters of translation. We extend the theory to project a point in c-space onto the csurface. We implement our theory on the Assembly Plan from Observation (APO) system. The APO observes discrete instants of an assembly task and reconstructs the compliant motion plan employed in the task. We compute the contacts at each observed instant and the corresponding csurface. We then interpolate the path on each c-surface to obtain segments of the path. The complete motion plan will be the concatenation of the connected path segments.