Forward displacement analysis of general six-in-parallel SPS (Stewart) platform manipulators using soma coordinates

General six-in-parallel SPS platform manipulators are constructed of six telescoping legs, each connecting a stationary base platform to a moving platform via spherical joints. These are often termed ''generalized Stewart platforms.'' Given the lengths of the six legs, the forward displacement problem is to find the location of the end platform relative to the base platform. It was first demonstrated numerically that the problem may in general have at most 40 nonsingular solutions and this bound has been verified using several different mathematical arguments. The problem is reformulated in this report using a classical representation of rigid-body displacements: Study's soma coordinates, or equivalently, dual quaternions. This provides a much simpler analytical proof of the upper bound of 40. Moreover, the simple form of the equations may be useful in further studies of the problem.