Objective Functions of Principal Contact Estimation from Motion Based on the Geometrical Singular Condition

In this paper, we propose objective functions to estimate the principal contact between a unknown manipulated target object and its unknown surroundings from the motion of the object. We derived the objective functions based on the fact that contact involves a pair of geometrical primitives (a point of vertex, a line of edge, and a plane of face) for the singular condition of the calculation for their intersection or their spanned space from the point of view of geometrical algebra. The minimization of the proposed objective functions, which are differential quadratic forms of the Kronecker product of geometrical parameters, efficiently provided us the contact geometries that constrained the object movements. Additionally, the proposed objective functions are fundamentals for identifying contact during compliant manipulation, and we showed that the objective functions provide a clue for contact identification via experiments.