Three necessary conditions derived from classical geometry are proposed to evaluate formulations for the simultaneous twist and wrench control of rigid bodies, and for any theory to be meaningful it must be invariant with respect to (1) Euclidean collineations, (2) change of (Euclidean) unit length, and (3) change of basis. It is demonstrated in this paper that a previously established theory of hybrid control for robot manipulators is in fact based on the metric of elliptic geometry and is thus noninvariant with respect to (1) and (2). A new alternative invariant formulation based on the metric of Euclidean geometry and an induced metric of projective geometry is presented in terms of screw theory. An example of insertion illustrates both the invariant and noninvariant methods.

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