Efficient solving methods exploiting sparsity of matrix in real-time multibody dynamic simulation with relative coordinate formulation

In this paper, new methods for efficiently solving linear acceleration equa tions of multibody dynamic simulation exploiting sparsity for real-time sim ulation are presented. The coefficient matrix of the equations tends to hav e a large number of zero entries according to the relative joint coordinate numbering. By adequate joint coordinate numbering, the matrix has minimum off-diagonal terms and a block pattern of non-zero entries and can be solve d efficiently. The proposed methods, using sparse Cholesky method and recur sive block mass matrix method, take advantages of both the special structur e and the sparsity of the coefficient matrix to reduce computation time. Th e first method solves the n x n sparse coefficient matrix for the accelerat ions, where n denotes the number of relative coordinates. In the second met hod, for vehicle dynamic simulation, simple manipulations bring the origina l problem of dimension n x n to an equivalent problem of dimension 6 x 6 to be solved for the accelerations of a vehicle chassis. For vehicle dynamic simulation, the proposed solution methods are proved to be more efficient t han the classical approaches using reduced Lagrangian multiplier method. Wi th the methods computation time for real-time vehicle dynamic simulation ca n be reduced up to 14 per cent compared to the classical approach.