This paper describes a method for finding the location of a rigid body such that N specified points of the body lie on N given planes in space. Of special interest is the case N = 6, which is the minimum number to fully constrain the body. This geometric problem arises in two seemingly disparate contexts: metrology, as a generalization of so-called “3-2-1” locating schemes; and robotics, as the forward kinematics problem for 6ES or 6SE parallel-link platform robots. For N = 6, the geometric problem can be formulated algebraically as 3 quadratic equations having, in general, eight possible solutions. We give a method for finding all eight solutions via an 8 × 8 eigenvalue problem. We also show that for N ≥ 7, the solution can be found uniquely as a linear least squares problem.

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