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 Ropf
ⁿ
. 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.