A graph-theoretic approach to large scale multibody systems

Mathematical models of complicated mechanical systems such as multibody systems with kinematic constraints, whether holonomic or nonholonomic, are generally represented by implicit nonlinear Differential-Algebraic Equations (DAEs). For the numerical integration of the constrained multibody dynamics represented by the DAEs, we eventually need to calculate inversion of a large scale sparse Jacobian matrix during Newton iteration at each time step. This directly causes in taking much CPU time, since the Jacobian matrix of such mathematical models has generally the characteristic of random sparseness and hence it is very difficult to solve the matrix inversion fast unless employing some efficient sparse matrix technique based on topological structure of the DAEs. In this paper, we propose a graph theoretic approach to symbolic generation of sparse matrix inversion for large scale multibody systems, by which one can " symbolically " calculate matrix inversions of random sparse Jacobian matrices associated to nonlinear implicit differential equations for the sake of fast numerical integrations. It is noteworthy that one can easily calculate the sparse matrix inversion without numerical Gaussian elimination at each time step. To do this, we first show how kinematical and dynamical relations can be effectively set up by introducing connection matrices by mechanical analogy with KCL and KVL constraints in circuit theory. Second, we show how to assign input-output relations among variables in all the kinematical and dynamical relations that appear in the mathematical models of DAEs by introducing bipartite graphs. Then, we demonstrate how one can choose pivots of the Jacobian matrix associated to the kinematical and dynamical relations by using input-output relations in the context of bipartite graphs. Furthermore, we explain solvability for the sparse Jacobian matrix inversion associated to the DAEs by using the bipartite graph. Finally, we propose symbolic generation of the sparse matrix inversion for the Jacobian matrix, which is to be explicitly done by symbolic manipulation and we demonstrate the validity of the proposed approach in numerical efficiency by an example of the Stanford manipulator.