Symbolic Processing of Multiloop Mechanism Dynamics Using Closed-Form Kinematics Solutions

This paper describes a method for the automated symbolic generation of the equations of motion of multibody systems using closed-form solutions of the kinematics at the position, velocity and acceleration levels where possible. The basic idea of the method is to employ the set of smallest independent loops of the system as building blocks for the overall kinematics of the system. Using a geometric-algebraic approach, closed-form solutions are detected and generated for each loop where this is possible. These local solutions are then assembled at the global level, yielding a block diagram from which closed-form solutions for the overall system are produced where possible. The equations of motion of minimal order are then generated by sums and products involving matrices from the kinematic processing. The resulting expressions are fully symbolic and do not contain redundant computations. The method was implemented in Mathematica and was applied to several mechanisms of practical relevance. A comparison of closed-form solutions with iterative solutions shows that the closed-form solutions are more efficient by a factor of up to 8.