Advanced $$\mathcal{H}_{\infty }$$ Synthesis of Fully Actuated Robot Manipulators with Frictional Joints

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Motivated by the presence of nonsmooth phenomena caused by dry friction forces, in this chapter we undertake the study of local nonsmooth \(\mathcal{H}_{\infty }\) synthesis of a multilink manipulator with frictional joints. The manipulator is required to follow a desired trajectory and particularly to move from an initial position to a desired one. Since in robotic applications velocity sensors are often omitted to save considerably in cost, volume, and weight, the position is assumed to be the only available measurement on the system. Well-known static and dynamic friction models are reviewed here. For certainty, the frictional forces that occur in the manipulator joints are represented by the Dahl model augmented with viscous friction. This simplest dynamic model captures all the essential features of the general treatment and allows one to straightforwardly apply the proposed nonsmooth \(\mathcal{H}_{\infty }\) synthesis that proves capable of counting for the nonsmooth terms of the Dahl friction model. Along with the velocity compensator, the resulting nonsmooth \(\mathcal{H}_{\infty }\) synthesis necessarily includes friction compensator design, thereby yielding a \(\mathcal{H}_{\infty }\) controller of a higher order compared to that of the plant. To avoid having to implement the friction compensator, thus saving in the computational cost and physical volume of the controller, we develop ad hoc an alternative discontinuous \(\mathcal{H}_{\infty }\) design based on the discontinuous static Coulomb model that can be viewed as a singularly perturbed version of the Dahl model with an infinitely large stiffness coefficient. Performance issues of the controllers thus developed are illustrated in an experimental study made for a 3-degree-of-freedom (DOF) robot manipulator with frictional joints. We should point out that implementation of the discontinuous local \(\mathcal{H}_{\infty }\) controller is of the same level of simplicity as that of the aforementioned nonlinear \(\mathcal{H}_{\infty }\) controller, whereas implementation of the nonsmooth local \(\mathcal{H}_{\infty }\) controller is of a higher complexity level due to the need to implement the dynamic friction compensator.