Lyapunov–Based Output Feedback Learning Control of Robot Manipulators K. Merve Dogan, Enver Tatlicioglu ⋆ , Erkan Zergeroglu, and Kamil Cetin Abstract— This paper address the output feedback learning tracking control problem for robot manipulators with repetitive desired joint level trajectories. Specifically, an observer–based output feedback learning controller for periodic trajectories with known period have been proposed. The proposed learning controller guarantees semi–global asymptotic tracking despite the existence of parametric uncertainties associated with the robot dynamics and lack of velocity measurements. A learning– based feedforward term in conjunction with a novel observer formulation is designed to obtain the aforementioned result. The stability of the controller–observer couple is guaranteed via Lyapunov based arguments. Numerical studies performed on a two link robot manipulator are also presented to demonstrate the viability of the proposed method. I. I NTRODUCTION The main purpose of using robotic automation in nearly all different fields industry, is to perform repetitious tasks. Therefore robots used in an industrial application mostly perform a predefined task over and over again. Given the nonlinear nature of the robot dynamics, the need to achieve better tracking performance despite system uncertainties and periodic disturbances related to the periodic task, learning controllers among other nonlinear model based controllers are the preferred controller choice. Moreover, compared to other controller formulations, repetitive learning controllers are computationally efficient, can compensate disturbance terms without the need of high frequency or high gain feedback terms and can deal with time–varying disturbances. Some of the initial work on repetitive learning control research for robotic systems was made by [1], [2], and [3]; where asymptotic convergence of the aforementioned control schemes can only be guaranteed under restrictive conditions on the plant dynamics. Later to enhance the robustness of [1], [2] modified the repetitive update rule to include the so–called Q–filter. In an attempt to increase the robustness of the previously proposed repetitive learning algorithm [4] and [5] proposed a scheme that exploited the use of kernel functions in the update rule. Sadegh et al. in [6] also proposed to enhance the robustness of the repetitive ⋆ To whom all the correspondence should be addressed. This work is funded by The Scientific and Technological Research Council of Turkey via grant number 113E147. K. M. Dogan, and E. Tatlicioglu are with the Department of Electrical & Electronics Engineering, Izmir Institute of Technology, 35430, Urla, Izmir, Turkey (Phone: +90 (232) 7506536; Fax: +90 (232) 7506599; E- mail: [mervedogan,enver]@iyte.edu.tr). E. Zergeroglu is with the Department of Computer Engineering, Gebze Institute of Technology, 41400, Gebze, Kocaeli, Turkey (Email: ez- erger@bilmuh.gyte.edu.tr). K. Cetin is with the Department of Electrical & Electronics Engineer- ing, Gediz University, 35665, Menemen, Izmir, Turkey (Email: kamil- cetin@gediz.edu.tr). learning controllers by using a saturated update rule. In [7], authors presented a full state feedback learning controller that achieves asymptotic tracking backed up by a Lyapunov based stability analysis. All of the controllers mentioned above are full state feedback controllers, that is the controller formulation re- quires both the position and velocity measurements. However nearly all industrial robots only have position sensors. And it is a known fact that using numerical differentiation to numerically form the velocity signal from position informa- tion introduces extra noise to the system. Therefore many researchers were also motivated to design output feedback learning controllers that does not require link velocity mea- surements. To name a few, in [8] and [9] neural network based reinforcement–learning controllers were presented for different classes of nonlinear discrete–time systems. In [10], a learning controller for a class of single–input, single– output, minimum phase, nonlinear, time–invariant systems with unknown output–dependent nonlinearities, unknown parameters and known relative degree ρ is considered. In this study, by making use of a model free observer together with a novel feedforward learning term, we were able to design an output feedback repetitive learning type controller for robotic manipulators with periodic joint level trajectories. The proposed method ensures asymptotic track- ing despite the uncertainties associated with the robot dy- namics and lack of velocity measurements. Overall stability of the observer–controller couple was ensured via the use of Lyapunov based arguments. The rest of the text is or- ganized in the following manner; The robot dynamics and its properties are given in Section 2; The observer–controller design and closed loop definitions are presented in Section 3. Stability analysis of the overall closed loop system is detailed in Section 4 while the numerical studies performed on a two link planar robot manipulators are presented in Section 5. Concluding remarks are given in Section 6. II. SYSTEM MODEL AND PROPERTIES The dynamic model of an n degree of freedom, direct drive robot manipulator is given in the following form [11], [12] M (q)¨ q + V m (q, ˙ q)˙ q + G (q)+ F d ˙ q = τ (1) where q (t), ˙ q (t), ¨ q (t) ∈ R n denote the joint positions, ve- locities, and accelerations, respectively, M (q) ∈ R n×n is the positive–definite and symmetric inertia matrix, V m (q, ˙ q) ∈ R n×n is the centripetal–Coriolis terms, G(q) ∈ R n is the gravitational effects, F d ∈ R n is the constant, diagonal, positive–definite, viscous frictional effects, and τ (t) ∈ R n is 2015 American Control Conference Palmer House Hilton July 1-3, 2015. Chicago, IL, USA 978-1-4799-8684-2/$31.00 ©2015 AACC 5337