This paper presents a general method to compute configuration space (c-space) obstacle surfaces (c-surfaces) in planar quaternion space. We extend the method to find the projection of a given point in c-space onto the c-surface We parameterize the general c-surface using a rotation angle and the vector of translation parameters of the individual contacts. We first compute the domain of the rotation parameter. Then, we can setup the translation parameters in a linear equation. The solution of this equation using singular value decomposition gives us the exact parameters of translation. We can extend this quasi-linear method to project 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.
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