The direct kinematics of parallel manipulators under joint-sensor redundancy

We study, for each of the possible joint-sensor layouts, the subspaces into which the motion of the hip-attachment points of parallel manipulators are completely measured. The projection of the motion of these points onto their subspaces allows us to write the underlying direct kinematics as a linear algebraic system constrained by the proper orthogonality of the rotation matrix. Although the solution of this problem requires a nonlinear technique, we propose a linear procedure that provides what we term a polar least square estimate. The resulting procedure is fast, robust to measurement noise, and produces estimates with about the same accuracy as a nonlinear procedure