Symbolic Geometric Modelling of Tree-Structure Robotic Mechanisms Using Lie Groups and Graph Theory

The present work describes a symbolic formulation based on Lie groups and graph theory to obtain the dynamic equations of tree-structure robotic mechanisms (TRM)s. The resulting equation is the equivalent geometric form of the classical Newton-Euler's equation of motion. Such formulation is valid for any TRM without closed kinematic chains and whose joints have one degree of freedom (revolute and/or prismatic). For instance, open chain manipulator, robotic hands or humanoids are included within TRM. The structure of the proposed formulation allows analytic differentiation, which is required for techniques such as optimal control or robust control. Moreover, the assembly parameters are explicitly held in matrices, so that they can be mathematically manipulated in order to be adapted for simulation, analysis or identification algorithms. Finally, the formulation is applied to a 7DOF manipulator and to a two arm torso of 16 degrees of freedom.