Generalized coordinate partitioning for complex mechanisms based on kinematic substructuring

Abstract

Generalized Coordinate Partitioning (GCP), a partitioning of the Jacobian matrix, allows for the automatic identification of dependent and independent variables and primary and secondary systems of equations for constrained multibody mechanisms. The GCP method is achieved through the row and column permutations associated with applying Gaussian Elimination with Complete Pivoting (GECP) to the constraint Jacobian matrices. GCP forms the basis of robust and efficient kinematic path planning and solution of dynamic equations of motion, with the potential to achieve orders of magnitude speed up over least squares and iterative methods. Despite these benefits, a significant disadvantage is that the GCP method is numerically expensive. This paper presents a technique to automatically rearrange groups of intersecting kinematic loops into non-overlapping kinematic substructures, which effectively block-diagonalizes the Jacobian Matrix. The method is applied to a hydraulic excavator, an illustrative example of a complex mechanism, and an approximate order of magnitude reduction in floating point operations required to perform the GCP method is demonstrated. Numerical results comparing the run time of the kinematic substructuring technique to sparse matrix methods and full dense GECP are provided. Furthermore, a compact representation of the equations of motion is formulated, accounting for the kinematic substructures identified in the mechanism.