Spatial operator factorization and inversion of the manipulator mass matrix

’bo 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$M$* 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 = (2+ H Q G ) D ( Z + H 9 G ) * . This leads to an immediate inversion M-’ = (2 H 4 G ) * D-l (Z H 4 G ) , where H and are given by known link geometric parameters, and G , 4 and D are obtained by a discrete-step Riccati equation driven by the link masses. The factors (Z + H 9 G ) and (Z H q 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 Choleskylike 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 $, 9 and 4, which govern the propagation of forces, velocities, and accelerations from link to link along the span of the manipulator. NOMENCLATURE Unit vector along joint axis k. Angle of link k with respect to link k + 1 about joint axis k. Fixed point on joint axis k, which can be viewed as the origin of a frame fixed in link k. Vector from O ( k ) to O(k 1). Constraint force on link k at point O ( k ) of joint k. Constraint moment on link k at joint k. Manuscript received May 11, 1988; revised June 13, 1991. This work was carried out by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. K. Kreutz-Delgado was partially supported by the National Science Foundation under Presidential Young Investigator Award IRI-9057631 and by the California Space Institute under Grant CS-22-90. G. Rodriguez is with the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109. K. Kreutz-Delgado is with the Electrical and Computer Engineering Department, University of California at San Diego, La Jolla, CA 92130-0407. IEEE Log Number 9104592. Net force on link k at link k mass center. Mass center of link k. Vector from O ( k ) to C M ( k ) . Velocity of link k at point O(k) of joint k. Angular velocity of link k. Velocity of link k at link k mass center. Mass of link k. Inertia tensor of link k at point C M ( k ) . Inertia tensor of link k at point O ( k ) . Actuated torque at joint k.