On a Rigid Body Subject to Point-Plane Constraints

This paper investigates the location of a rigid body such that N specified points of the body lie on N given planes in space. Variants of this problem arise in kinematics, metrology, and computer vision, including some, such as the motion of a spherical four-bar, that are not at first glance point-plane contact problems. The case N = 6, the minimum number to fully constrain the body, is of special interest: we give an eigenvalue method for finding all solutions, which may number up to eight. For N ≥ 7 there are, in general, no solutions, but if the constraints are compatible and not degenerate, we show how to find the unique solution by a linear least-squares method. For N ≤ 5, the body is underconstrained, having in general 6 − N degrees of freedom; we determine the degree of the general motion for each case. We also examine the workspace of a particular three-degree-of-freedom parallel-link tripod mechanism.