Forward Dynamics, Elimination Methods, and Formulation Stiffness in Robot Simulation

The numerical simulation problem of tree-structured multibody systems, such as robot manipulators, is usually treated as two separate problems: (i) the forward dynamics problem for computing system accelerations, and (ii) the numerical integration problem for advancing the state in time. The interaction of these two problems can be important and has led to new conclusions about the overall efficiency of multibody simulation algorithms [ClPaAs95]. In particular, the fastest forward dynamics methods are not necessarily the most numerically stable, and in ill-conditioned cases may slow down popular adaptive step-size integration methods. This phenomenon is called "formulation stiffness". In this paper, we first unify the derivation of both the composite rigid body method [WalkerOrin82] and the articulated-body method [Featherstone83,Featherstone87] as two elimination methods to solve the same linear system, with the articulated body method taking advantage of sparsity. Then the numerical instability phenomenon for the composite rigid body method is explained as a cancellation error that can be avoided, or at least minimized, when using an appropriate version of the articulated body method. Specifically, we show that the articulated-body method is better suited to deal with certain types of ill-conditioning than the composite rigid body method. The unified derivation also clarifies the underlying linear algebra of forward dynamics algorithms and is therefore of interest in its own accord.