Modelling multibody systems with indirect coordinates

We present an automated formulation that generates multibody kinematic and dynamic equations in terms of indirect coordinates, a general set of motion variables that can be used to represent the relative motion of any two bodies in a system. This new formulation is achieved by combining classical mechanics with linear graph theory and the concept of a “virtual joint” between any two bodies. By selecting this joint into the spanning tree of a graph-theoretic representation of the system, the governing equations are systematically generated in terms of the indirect coordinates corresponding to the virtual joint. Different trees may be selected for translational and rotational motions, which can result in fewer equations to be solved than those obtained using joint coordinates. As shown in the examples, the use of indirect coordinates can also lead to a reduction in equation complexity and an increase in the computational efficiency of numerical simulations.