Equivalence of the Projected Forward Dynamics and the Dynamically Consistent Inverse Solution

The analysis, design, and motion planning of robotic systems, often relies on its forward and inverse dynamic models. When executing a task involving interaction with the environment, both the task and the environment impose constraints on the robot’s motion. For modeling such systems, we need to incorporate these constraints in the robot’s dynamic model. In this paper, we define the class of Task-based Constraints (TbC) to prove that the forward dynamic models of a constrained system obtained through the Projection-based Dynamics (PbD), and the Operational Space Formulation (OSF) are equivalent. In order to establish such equivalence, we first generalize the OSF to a rank deficient Jacobian. This generalization allow us to numerically handle redundant constraints and singular configurations, without having to use different controllers in the vicinity of such configurations. We then reformulate the PbD constraint inertia matrix, generalizing all its previous distinct algebraic variations. We also analyse the condition number of different constraint inertia matrices, which affects the numerical stability of its inversion. Furthermore, we show that we can recover the operational space control with constraints from a multiple Task-based Constraint abstraction.