Globally Asymptotic Generalized Dynamic Inversion for Robot Manipulator Control Design

The generalized inverse dynamics methodology is further improved for global asymptotic tracking control of robot manipulators generalized coordinate trajectories. A scalar kinematic norm measure function of manipulators generalized coordinates deviations from their desired trajectories is defined, and a servo-constraint on robot kinematics is prescribed by zeroing the kinematic deviation function. A stable linear second order differential equation in the kinematic deviation function is evaluated along trajectory solutions of manipulators state space model equations, resulting in an algebraic relation that is linear in the control vector. The control law is designed by generalized inversion of the controls coefficient in the algebraic relation using a modified version of the Greville formula. The Moore-Penrose generalized inverse in the particular part of the modified Greville formula is scaled by a dynamic factor that uniformly decays as steady state response approaches. This yields a uniform convergence of the particular part to its projection on the range space of controls coefficients Moore-Penrose generalized inverse, resulting in asymptotically stable generalized coordinates trajectory tracking. The null-control vector in the auxiliary part of the Greville formula is taken to be linear in manipulators generalized velocities, and is constructed by means of a positive semidefinite control Lyapunov function that involves the controls coefficient nullprojector, providing a global asymptotic internal manipulator stability and a global asymptotic generalized coordinates trajectory tracking.