Improved “Order-N” Performance Algorithm for the Simulation of Constrained Multi-Rigid-Body Dynamic Systems

This paper presents an algorithm for the efficient numerical analysis and simulation of modest to heavily constrained multi-rigid-body dynamic systems. The algorithm can accommodate the spatial motion of general multi-rigid-body systems containing arbitrarily many closed loops in O(n+m) operations overall for systems containing n generalized coordinates, and m independent algebraic constraints. The presented approach does not suffer from the performance (speed) penalty encountered by most other of the so-called “O(n)” state-space formulations, when dealing with constraints which tend to actually show O(n + m + nm + nm + m) performance. Additionally, these latter formulations may require additional constraint violation stabilization procedures (e.g. Baumgarte’s method, coordinate partitioning, etc.) which can contribute significant additional computation. The presented method suffers less from this difficulty because the loop closure constraints at both the velocity and acceleration level are directly embedded within the formulation. Due to these characteristics, the presented algorithm offers superior computing performance relative to other methods in situations involving both large n and m. Nomenclature a Matrix representation of acceleration of center of mass k∗ in the Newtonian reference frame N . at Acceleration remainder term associated with body k in N ; This is all terms of a k which are not explicit in u̇’s. A The generalized acceleration matrix of body k in N . Ā That portion of the generalized acceleration matrix of body k in N which is explicit in the unknown state derivatives u̇. At That portion of the generalized acceleration matrix of body k in N which is not explicit in the unknown state derivatives u̇. 0i A The generalized acceleration matrix of ki in reference frame 0i, which are associated with closed loop i. 0ī A That portion of the generalized acceleration matrix of ki in 0i which is explicit in the unknown state derivatives u̇. 0i At That portion of the generalized acceleration matrix of ki in 0i which is not explicit in the unknown state derivatives u̇. C Invertible transformation matrix relating q̇ to u. C k Direction cosine matrix relating the basis vectors fixed in body k to those in proximal body Pr[k]. Dist[k] Distal body set associated with body k. D Matrix used in relating q̇ to u, and commonly associated with prescribed motions. Ei Expansion matrix which converts mi constraint load measure numbers to constraint load matrix Fci . F Recursive generalized force matrix for body k. ∗Associate Professor. [email protected] †Doctoral Student. [email protected]