The Hierarchical Operational Space Formulation: Stability Analysis for the Regulation Case

The Operational Space Formulation (OSF) from the 1980s is probably the most frequently applied task-space controller in robotics. In multipriority control of redundant robots via the OSF, a feedback linearization is performed on the first hierarchy level while lower-priority tasks are executed in the dynamically consistent null space of the Jacobian matrices of all higher-priority tasks without disturbing them. However, it has been observed in the past that a formal stability analysis for the overall closed loop is rather difficult, especially for the null space dynamics. Except for exponential stability on the main task level, a complete proof is still missing when the tasks conflict with each other. Here, we provide this formal proof of asymptotic stability for the regulation case of a passivity-based OSF controller by means of conditional stability theory and semidefinite Lyapunov functions. Simulations support the intuitive, energy-based interpretation of the proof. This stability analysis lifts the widely used OSF onto a more solid foundation.