Planar, spherical, and spatial linkage design requires the formulation of synthesis equations. These equations are often polynomials or can be converted to polynomials. The degree of these polynomial systems increases rapidly with mechanism complexity. For example, the synthesis equations of a planar RR constraint used to construct a four-bar linkage that reaches its maximum of five positions is of degree 4. Meanwhile, the lowest computed degree of the synthesis equations for the Watt I six-bar that reaches its maximum of eight positions is 34 billion. See Plecnik et. al.
Resultant methods are unable to keep up with this increase in complexity and optimization methods do not provide a complete solution set. Homotopy continuation and interval analysis are methods capable of providing complete or practical sets of solutions. My research focuses on the former.
In particular, the polynomial continuation software Bertini provides an efficient module for solving high degree systems. Using the High Performance Computing cluster at UC Irvine, we have computed systems of degree 4 million. The degree of a system is equal to the number of homotopy paths a computer is required to track. Tracking millions of paths is computationally expensive. As well, kinematics equations tend to have a sparse monomial structure, indicating that the number of roots is actually much smaller than the Bezout and multihomogeneous bounds compute. If all the roots for a system with a given monomial structure can be found once, those results can be used to construct parameter homotopies which can be used to efficiently solve systems of the same structure in the future. In short, a polynomial system would require one big initial solve (on the order of hours/days), and then all future solve are computed quickly (on the order of minutes).
M. Plecnik, J. M. McCarthy, and C. W. Wampler, 2014. “Kinematic synthesis of a Watt I six-bar linkage for body guidance,” Advances in Robot Kinematics. Springer International Publishing, 317-325.
D. J. Bates, J. D. Hauenstein, A. J. Sommese, and C. W. Wampler, 2013. Numerically Solving Polynomial Systems with Bertini. SIAM Books, Philadelphia, PA.