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Position Feedback Control of Lagrangian Robotic Systems via an Affine PD-Based Control Law. Part 1: Design of LMI Conditions

Among the appropriate methods to describe the morphology of a robotic system, we note the development of a nonlinear dynamic model defined by the equations of the robot’s motion. Thus, the control design method can be based on the linearization of the nonlinear model. Therefore, in this work, we rely on the difference between the nonlinear model of Lagrangian robotic systems and the approximated linear dynamics to obtain a dynamical model that satisfies a certain Lipschitz condition. Thus, we introduce an affine PD-based control law in the obtained dynamic model to develop by three methods the stability conditions using the Schur complement, the Young inequality and the S-procedure lemma. These conditions are described by Linear Matrix Inequalities (LMIs). Finally, we present some results by adopting, as an illustrative example, the manipulator robot with two degrees of freedom, as a Lagrangian robotic system.