Operational space control of constrained and underactuated systems

The operational space formulation (Khatib, 1987), applied to rigid-body manipulators, describes how to decouple task-space and null space dynamics, and write control equations that correspond only to forces at the end-effector or, alternatively, only to motion within the null space. We would like to apply this useful theory to modern humanoids and other legged systems, for manipulation or similar tasks, however these systems present additional challenges due to their underactuated floating bases and contact states that can dynamically change. In recent work, Sentis et al. derived controllers for such systems by implementing a task Jacobian projected into a space consistent with the supporting constraints and underactuation (the so called support consistent reduced Jacobian). Here, we take a new approach to derive operational space controllers for constrained underactuated systems, by first considering the operational space dynamics within projected inverse-dynamics (Aghili, 2005), and subsequently resolving underactuation through the addition of dynamically consistent control torques. Doing so results in a simplified control solution compared with previous results, and importantly yields several new insights into the underlying problem of operational space control in constrained environments: 1) Underactuated systems, such as humanoid robots, cannot in general completely decouple task and null space dynamics. However, 2) there may exist an infinite number of control solutions to realize desired task-space dynamics, and 3) these solutions involve the addition of dynamically consistent null space motion or constraint forces (or combinations of both). In light of these findings, we present several possible control solutions, with varying optimization criteria, and highlight some of their practical consequences.