Singularity-Free Lagrange-Poincaré Equations on Lie Groups for Vehicle-Manipulator Systems

It has been long known that the Euler–Lagrange dynamical equations of fixed-base manipulators with single-degree-of-freedom joints can be formulated on Lie groups following exponential joint parameterizations. Whereas, dynamics of symmetric vehicles can be captured using the Euler–Poincaré equations on Lie groups, with no need to choose any local parameterization. We utilize a combined form of these two geometric approaches called the Lagrange–Poincaré Equations (LPE) to develop a singularity-free Lagrangian formalism for the dynamics of vehicle-manipulator systems. We consider vehicles whose configuration manifolds are Lie subgroups of the special Euclidean group, encompassing arbitrary base vehicle motions corresponding to, e.g., ball, planar, or free joints. We revisit the Lagrange-d'Alembert principle for systems on principal bundles to derive the LPE for vehicle-manipulators with possibly symmetry-breaking externally applied wrenches. These equations effectively separate the external (locked-arm system) and internal dynamics (arm's motion) by introducing a block-diagonalized inertia matrix. We then incorporate the exponential parameterization of manipulators to explicitly formulate the reduced dynamics on Lie groups. The resulting equations are in matrix form and can be immediately implemented in simulations and model-based control strategies. The geometrical significance of the proposed formalism is further demonstrated via the step-by-step presentation of a case study.