Hybrid Dynamics for Closed-Loop Multibody Systems

Abstract

A popular method for solving the dynamics of closed-loop systems is the {\it tree-augmented (TA)} dynamics algorithm. In this method, a minimal set of the inter-body constraints are ``cut'' to convert the system into a tree-topology system. The inter-body constraints within the tree are treated as hinges parameterized with a minimal set of coordinates. The overall dynamics model formulation consists of the minimal-coordinate dynamics model for the tree system together with the minimal set of bilateral closure constraints for the tree system. A DAE solver is needed for solving the equations of motion in this formulation. An alternative method that avoids DAEs altogether is the {\it constraint embedding (CE)} dynamics algorithm \cite{Jain2011b}. This technique uses the TA model as a starting point. However, TA model's bilateral constraints are eliminated by aggregating bodies affected by the bilateral constraint into compound bodies. The result system topology is once again a tree with only inter-body hinges and no bilateral constraints. The benefit of this approach is that the structure-based tree algorithms can be directly used to solve the dynamics, and this formulation requires the simpler ODE solver. While the CE algorithm is undoubtedly superior for systems with small loops - and especially ones where the loop kinematics has analytical solution. However when the loops become larger, or more complex (eg. meshes), the benefits of the CE approach over the TA approach decrease. For systems involving both small and large loops, the analyst is left with the unsatisfactory situation of choosing between either the TA or CE approaches. The main contribution of this paper is the development of a {\it hybrid dynamics method} which allows the use of TA and CE approaches to different loops in the multibody system. This allows the user to judiciously choose the better of the TA or CE approaches for each of the loops in the system.