IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 1 Passive Finite Dimensional Repetitive Control of Robot Manipulators Josip Kasac, Branko Novakovic, Dubravko Majetic and Danko Brezak Abstract—In this paper a new class of finite dimensional repet- itive controllers for robot manipulators is proposed. The global asymptotic stability is proved for the unperturbed system. The passivity-based design of the proposed repetitive controller avoids the problem of tight stability conditions and slow convergence of the conventional, internal model-based, repetitive controllers. The passive interconnection of the controller and the nonlinear mechanical systems provides the same stability conditions as the controller with the exact feed-forward compensation of robot dynamics. The simulation results on a three degrees of freedom spatial manipulator illustrate the performances of the proposed controller. Index Terms—Repetitive control, passive system, manipulators, robot dynamics, stability I. I NTRODUCTION An important subject in the control of mechanical systems is tracking periodic reference signals and attenuating periodic disturbances. Many tracking systems, such as computer disk drives [1], rotation machine tools [2], or robots [3], have to deal with periodic reference and/or disturbance signals. A promising control approach to achieving the tracking of periodic reference signals is learning control or repetitive control. In most of the conventional approaches to robot trajectory control, including parametric adaptive control, it is necessary to compute in real time the so-called inverse dynamics equa- tions of the robot or regression matrix. However, due to the model uncertainties, it is difficult to derive the exact descrip- tion of the system. Also, using neural networks for learning feed-forward control has some drawbacks: slow convergence and relatively large tracking errors. There have been many studies on the topic of repetitive con- trol for controlling mechanical systems in an iterative manner. In contrast with the conventional approaches to robot trajectory control, repetitive control schemes are easy to implement and do not require the exact knowledge of the dynamic model. Repetitive controllers can be classified as being either internal model-based or external model-based [4]. Controllers using the internal model are linear and have periodic signal generators [5], [6]. In the external model controllers the disturbance model is placed outside the basic feedback loop [3], [7]. The internal model controllers are based on a delayed integral action of the form (1 − exp(−sT )) -1 which produces an infinite number of poles on imaginary axes. However, the This work was supported by the National Scientific Foundation of the Republic of Croatia under Grant No. 120-1201842-3048. The authors are with the Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, HR-10000 Zagreb, Croatia (e- mail: josip.kasac@fsb.hr, branko.novakovic@fsb.hr, dubravko.majetic@fsb.hr, danko.brezak@fsb.hr). asymptotic convergence can only be guaranteed under restric- tive conditions in the plant dynamics - zero relative degree or direct transmission term. These conditions are generally not satisfied in robot control applications because they imply acceleration measurement. Further, the positive feedback loop used to generate the periodic signal decreases the stability margin. So, the repetitive controller is likely to make the system unstable. To enhance the robustness of these repetitive control schemes, the repetitive update rule is modified to include the so-called Q-filter [5], [6]. Unfortunately, the use of the Q-filter eliminates the ability of tracking errors to converge to zero. Therefore, the trade-off between stability and tracking performance has been considered to be an important factor in the repetitive control system. Another problem is that, due to infinite dimensional dynam- ics of delayed line, a large memory space is required for digital implementation of the control law. To overcome this problem, in [8] a finite dimensional approximation of delayed line is proposed in the form of a cascade connection of N harmonic oscillators and one integrator. The advantages of the internal model controllers are that they are linear, making analysis and implementation easier. The disadvantages are that the stability is almost entirely governed by the feedback loop of the repetitive compensator. The frequency response of the system is altered and robustness to noise and unmodelled dynamics is reduced. The external model controllers are based on the feedforward compensation of inverse dynamics. The disturbance model is adjusted adaptively to match the actual disturbance. The central idea in [3] is that the disturbance can be represented as a linear combination of basis functions like Fourier series expansion. In this way, an adaptive control law with regressor matrix containing basis functions is obtained. In [7] unknown disturbance functions are represented by integral equations of the first kind involving a known kernel and unknown influence functions. The learning rule indirectly estimates the unknown disturbance function by updating the influence function. The main advantage of the external model approach is that there is no significant influence on the stability conditions of the control system. The map between the feedforward function error and the tracking errors is strictly passive. Thus, the control system is robust to the imprecise estimation of the robot inverse dynamics. The disadvantage is that the analysis and implementation are more complex than for the internal model-based algorithms. In this paper a new class of internal model-based repetitive controllers for robot manipulators is proposed. The proposed finite dimensional repetitive controller is founded on the passivity-based design and has a structure in the form of a parallel connection of N linear oscillators and one integrator. The passive interconnection of the proposed controller with