Solution of Multibody Dynamics Using Natural Factors and Iterative Refinement: Part I — Open Kinematic Loops

An O(n) methodology employing block matrix partitioning and recursive projection to solve multibody equations of motion coupled by a sparse connectivity matrix was developed in (Wehage 1988, 1989, Wehage and Shabana, 1989). These primitive equations, which include all joint generalized and absolute coordinates and constraint reaction forces, are easily obtained from free body diagrams. The corresponding recursive algorithms isolate the generalized joint accelerations for numerical integration and offer the best computational advantage when solving long kinematic chains on serial processors. Recursion, however, precludes effective exploitation of vector or parallel processors. Therefore this paper explores less recursive algorithms by applying the inverse of joint connectivity to eliminate absolute accelerations and constraint forces yielding a generalized system of equations. The resulting positive definite generalized inertia matrix is first represented symbolically as a product of sparse matrices, of which some are singular and then as the product of nonsingular factors obtained recursively. This algorithm has overhead ranging from O(n²) to O(n) depending on the degree of system parallelism. Incorporating iterative refinement and exploiting parallel and vector processing makes this approach competitive for many applications.