In this paper, we introduce the generalized Stewart-Gough platform (GSP) consisting of two rigid bodies connected with six distance and/or angular constraints between six pairs of points, lines, and/or planes in the base and the moving platform, respectively. We prove that there exist 3850 possible forms of GSPs. We give the upper bounds for the number of solutions of the direct kinematics for all the GSPs. We also obtain closed-form solutions and the best upper bounds of real solutions of the direct kinematics for a class of 1120 GSPs.

A basic idea of geometric constraint solving is to divide a large geometric constraint problem into several subproblems according to certain patterns and then to merge these subproblems to obtain a solution to the original problem. In this paper, we give a classification of the basic merge patterns in 2D when solving a constraint problem with generalized construction sequences. We also obtain the analytical solutions to all eleven cases of merge patterns, which is to assemble two rigid bodies connected with three constraints. We also give conditions to solve these patterns with ruler and compass constructions.