Finding the Common Tangents to Four Spheres via Dimensionality Reduction

One of the advantages of formulating position analysis problems in terms of distances is that the dimension of the problem can be reduced by projecting the problem onto a subspace. Although, in general, this operation does not provide a significant advantage, when parallelism or alignment constraints must be enforced, a proper projection results in an important simplification. This is the case when computing the common tangents to four spheres in \({\mathbb R}^3\) . In this paper, we first show how this problem can be formulated in terms of just five points in \({\mathbb R}^2\) thanks to projection, and then this is applied to solve the forward kinematics of the 4-SPC parallel robot.