Fast functional decomposition of sine-cosine-polynomials

To solve the inverse kinematics problem in symbolic form it is necessary to determine the roots of so-called sine-cosine-polynomials (SC-polynomials). A functional decomposition of an SC-polynomial reduces this task to the solution of two equations of degree less than or equal to half of the original degree. Thus, decompositions can significantly improve symbolic solutions of the inverse kinematics problem. An earlier algorithm only detects a special type of decomposition, is of exponential complexity, and fails for complicated kinematical problems. The algorithm presented here finds all decompositions and reduces the complexity by magnitudes, thus satisfying all needs in kinematics. The algorithm is applicable as well to symbolic solutions of the direct position problem of parallel manipulators. By combination with the so-called specialized analysis technique, it is possible to scan all different symbolic solutions of any manipulator and to determine the particular solution that allows a maximum decomposition. This is an important step toward finding optimal symbolic solutions for kinematic equation systems