Combinatorial methods

Since the constraints are expressed with polynomials, it may not be surprising that the computational algebraic geometry methods of Section 6.4 can solve the general motion planning problem with closed kinematic chains. Either cylindrical algebraic decomposition or Canny's roadmap algorithm may be applied. As mentioned in Section 6.5.3, an adaptation of the roadmap algorithm that is optimized for problems in which $m < n$ is given in [76].