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

This paper presents a general method to compute configuration space (c-space) obstacle surfaces (c-surfaces) in dual quaternion space and for projecting points onto them. We parameterize the c-surface using the rotation angles of the object and the vector of translation parameters of the individual contacts. Once we compute the domain of the rotation parameters, 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 find the projection of a point in c-space onto the c-surface. 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 c-surface. 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