Singularity-Free Dynamic Equations of Open-Chain Mechanisms With General Holonomic and Nonholonomic Joints

Standard methods to model multibody systems are aimed at systems with configuration spaces isomorphic to Ropfⁿ. This limitation leads to singularities and other artifacts in case the configuration space has a different topology, for example, in the case of ball joints or a free-floating mechanism. This paper discusses an extension of classical methods that allows for a more general class of joints, including all joints with a Lie group structure as well as nonholonomic joints. The model equations are derived using the Boltzmann-Hamel equations and have very similar structure and complexity as obtained using classical methods. However, singularities are avoided through the use of global non-Euclidean configuration coordinates, together with mappings describing a local Euclidean structure around each configuration. The resulting equations are explicit (unconstrained) differential equations, both for holonomic and nonholonomic joints, which do not require a coordinate atlas and can be directly implemented in simulation software.