# A Robust \(H_{\infty}\) Application for Motor-Link Control Systems of Industrial Manipulators
## 1 Introduction
For example, in [15], preliminary results are provided for a robust control approach for multi-DOF manipulators, performing tasks requiring frequent interactions with the environment; in [16], a robust discontinuous controller algorithm is proposed for a 5 DOF laboratory robot, whereas, in [17], an interesting adaptive proportional integral robust control scheme is developed for \(n\)-link robotic manipulators with model uncertainties and time-varying disturbances, but only simulation results for a two-link robot are available. The \(H_{\infty}\) approach is rarely applied in the industrial context because there are not predefined procedures to build weighting functions able to automatically guarantee the fulfillment of the control specifications. The design of the weighting functions is a slow iterative process that can take a long time, and this is not acceptable in an industrial reality that asks for valid results in a short time. Even if the research world counts many studies referring to the \(H_{\infty}\) method, as previously discussed, most research is focused on the advantages of its robustness properties, without guiding the reader in the development of the implementation steps. In the meantime, the most frequently implemented solutions still rely on control schemes of PID type, in which techniques such as Linear Quadratic Theory (LQT) are applied to achieve optimal tracking results (see [18] and the references therein for insight into LQT and the generalized PID controllers thus derived, denoted as \(\mathrm{PI}^{m}\mathrm{D}^{n-1}\)). This paper reports the results of an academic-industrial research activity, carried out in collaboration between Politecnico di Torino and COMAU S.p.A. The goal is to investigate the adoption of \(H_{\infty}\) approaches in the control software architecture of industrial manipulators, providing a proof-of-concept demonstration potentially useful as a basis for a concrete and complete robust control scheme alternatives to PID or generalized \(\mathrm{PI}^{m}\mathrm{D}^{n-1}\) ones. The design of the control system for a single axis of a 6 DOF manipulator is presented, describing all the steps that have to be fulfilled to reach satisfying time results, with a particular attention to the construction of two weighting functions summarizing the frequency requirements of the system. No specific assumption is made on the trajectories that will be applied, since in practice they are provided by the internal jerk bounded reference trajectory generator of the robot constructor, i.e., COMAU in our case. Our target to is to prove the potentiality of the application of the complex \(H_{\infty}\) theory to a much more complex world, which is the robotic one, where systems are not linear and where the models are continually affected by external disturbances. It is interesting to highlight that the application of robust control techniques have been recently investigated in other areas of the robotic world, such as in [19], where resilient interval consensus problems for continuous-time, time-varying, multi-agent systems are addressed. In this paper, starting from a realistic case, it will be shown how weighting functions defined according to standard procedures are not suitable for controlling the considered plant. General guidelines for the design of new weighting functions incorporating plant dynamic characteristics are exploited in order to obtain a control system responsive to both motor vibration, captured by sensors (as in the standard industrial configuration) and link vibration, on which no information is available. Matlab has been adopted as computing environment, using the Simulink toolbox to carry out simulation tests before moving to experimental tests on a real robot, whose preliminary but significant results are reported and discussed. The paper is organized as follows: Section 2 illustrates the basics of the \(H_{\infty}\) approach, with a particular focus on the definition of the involved weighting functions.