Task Space Control of the Non-Linearly Parametrisable Euler-Lagrange Dynamic Systems

A robust-adaptive second order sliding mode control (SMC) law is proposed for multi-input multi-output Euler-Lagrange (MIMO E-L) non-linear systems with uncertain both (non-linear) output (kinematic) and dynamic equations. Moreover, external unknown disturbances are allowed to act on the E-L system when tracking a desired trajectory prescribed in Cartesian (task) space. To cope with both parametric non-linearity of the control matrix of the input-to-output form of the E-L system and unknown (linearly and/or non-linearly parametrisable) as well as unstructured disturbances, we introduce a novel auxiliary control matrix and a second order task space integral sliding manifold (ISM). Based on a suitably defined Lyapunov function candidate, we derive a new control law whose convergence to zero is asymptotically stable. The proposed controller involves both an adaptive part to compensate for parametrically linear and/or non-linear uncertainties of the E-L dynamic system and a robust one compensating the (unstructured) disturbances. An illustrative example demonstrates the advantages of the approach. Numerical comparison of our strategy with that known from the literature is also given.