<tex>$L_{1}$</tex> Robustness of Computed Torque Method for Robot Manipulators

Abstract

This paper revisits computed torque method for robot manipulators and aims at developing its new framework based on the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L_{1}$</tex> robustness, in which the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L_{\infty}$</tex> norm together with its induced norm is employed to characterize model uncertainties and a performance measure. More precisely, we consider the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L_{1}$</tex> robust stability and performance for a given robot manipulator with a computed torque controller. We first show that the modelling errors in the computed torque method can be divided into an exogenous disturbance and a multiplicative model uncertainty, which are bounded in terms of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L_{\infty}$</tex> norm and its induced norm, respectively. It is next shown that the robot manipulator with the computed torque controller can be equivalently represented by an interconnection of a continuous-time linear time-invariant (LTI) nominal plant and a stabilizing controller together with the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L_{\infty}$</tex> -induced norm bounded model uncertainty. Based on the interconnected representation, the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L_{1}$</tex> robust stability condition and an upper bound of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L_{1}$</tex> performance against the exogenous disturbance with respect to all model uncertainties in a class of a bounded <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L_{\infty}$</tex> -induced norm are dealt with by using the small-gain theorem. Finally, the effectiveness of the theoretical results is demonstrated through some experiment results.