SPATIAL OPERATOR FACTORIZATION AND INVERSION OF THE MANIPULATOR MASS MATRIX

Two new recursive factorizations are developed of the mass matrix for fixed-base and mobile-base manipulators. First, the mass matrix M is shown to have the factorization M = H-phi-M-phi*H*. This is referred to here as the Newton-Euler factorization because it is closely related to the recursive Newton-Euler equations of motion. This factorization may be the simplest way to show the equivalence of recursive Newton-Euler and Lagrangian manipulator dynamics. Second, the mass matrix is shown to have a related innovations factorization M = (I + H-PHI-G)D(I + H-PHI-G)*. This leads to an immediate inversion M-1 = (I - H-PSI-G)* D-1 (I - H-PSI-G), where H and PHI are given by known link geometric parameters, and G, PSI and D are obtained by a discrete-step Riccati equation driven by the link masses. The factors (I + H-PHI-G) and (I - H-PSI-G) are lower triangular matrices that are inverses of each other, and D is a diagonal matrix. Efficient order N inverse and forward dynamics algorithms are embedded in the two factorizations. Moreover, the factorizations provide a high-level architectural understanding of the mass matrix and its inverse, which is not available readily from the detailed algorithms. The two factorizations are model-based in the sense that the manipulator model itself determines the sequence of computations. This makes the two factorizations quite distinct from more traditional Cholesky-like numerical factorizations of positive definite matrices. Because the manipulator model is used, every computational step has a corresponding physical interpretation. This adds a substantial amount of robustness, and numerical errors can be detected by physical intuition. Development of the factorizations is made simple by the use of spatial operators, such as phi, PHI and PSI, which govern the propagation of forces, velocities, and accelerations from link to link along the span of the manipulator.