Using Gröbner bases to generate efficient kinematic solutions for the dynamic simulation of multi-loop mechanisms

Many mechanical systems of practical interest contain closed kinematic chains, and are most conveniently modeled using a set of redundant generalized coordinates. The governing dynamic equations for systems with more coordinates than degrees-of-freedom are differential-algebraic, and can be difficult to solve efficiently yet accurately. In this work, the embedding technique is used to eliminate the Lagrange multipliers from the dynamic equations and obtain one ordinary differential equation for each independent acceleration. Gröbner bases are then generated to triangularize the kinematic constraint equations, thereby producing recursively solvable systems for calculating the dependent generalized coordinates given values of the independent coordinates. For systems that can be fully triangularized, the kinematic constraints are always satisfied exactly and in a fixed amount of time. Where full triangularization is not possible, a block-triangular solution can be obtained that is still more efficient than using existing techniques. The proposed approach is first applied to the Gough–Stewart platform, whose fully triangular solution motivates the block-triangular solution strategy for a five-link suspension system. Finally, a fully triangular solution is obtained for an aircraft landing gear mechanism.