IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 13, NO. 4, AUGUST 1997 567 Learning Approximation of Feedforward Control Dependence on the Task Parameters with Application to Direct-Drive Manipulator Tracking Dimitry Gorinevsky, Member, IEEE, Dirk E. Torfs, Associate Member, IEEE, and A. A. Goldenberg, Fellow, IEEE Abstract—This paper presents a new paradigm for model-free design of a trajectory tracking controller and its experimental implementation in control of a direct-drive manipulator. In ac- cordance with the paradigm, a nonlinear approximation for the feedforward control is used. The input to the approximation scheme are task parameters that define the trajectory to be tracked. The initial data for the approximation is obtained by performing learning control iterations for a number of selected tasks. The paper develops and implements practical approaches to both the approximation and learning control. As the initial feedforward data needs to be obtained for many different tasks, it is important to have fast and robust convergence of the learning control iterations. To satisfy this requirement, we propose a new learning control algorithm based on the on-line Leven- berg–Marquardt minimization of a regularized tracking error index. The paper demonstrates an experimental application of the paradigm to trajectory tracking control of fast (1.25 s) motions of a direct-drive industrial robot AdeptOne. In our experiments, the learning control converges in five to six iterations for a given set of the task parameters. Radial Basis Function approximation based on the learning results for 45 task parameter vectors brings an average improvement of four times in the tracking accuracy for all motions in the robot workspace. The high performance of the designed approximation-based controller is achieved despite nonlinearity of the system dynamics and large Coulomb friction. The results obtained open an avenue for industrial applications of the proposed approach in robotics and elsewhere. I. INTRODUCTION T HIS PAPER considers a learning control approach to output tracking in a nonlinear system. The term learning control appears in the title of many papers and denotes one of a few different approaches applicable in the absence of a system Manuscript received October 31, 1994; revised September 29, 1996. The work of D. E. Torfs was supported by the Prof. R. Snoeys Foundation, University of Toronto. The material in this paper was presented in part by the 1995 American Control Conference, San Francisco, CA, June 1995. This paper was recommended for publication by Associate Editors A. De Luca and Y. Nakamura and Editor A. J. Koivo upon evaluation of the reviewers’ comments. D. Gorinevsky was with the Robotics and Automation Laboratory, Univer- sity of Toronto, Toronto, Ont., Canada M5S 1A4. He is now with Honeywell Measurex, North Vancouver, B.C., Canada V7J 3S4. D. E. Torfs is with Trasys Space, Horizon Center, B-1930 Zaventem, Belgium. A. A. Goldenberg is with the Robotics and Automation Laboratory, Department of Mechanical Engineering, University of Toronto, Toronto, Ont., Canada M5S 1A4. Publisher Item Identifier S 1042-296X(97)05908-9. dynamics model, where the control or the system is ‘learned’ on the basis of the past operational data for the system. Early work in the learning control systems developed into the modern adaptive control theory, e.g., see [43]. Recently, many adaptive control approaches employing iterative estimation of the system dynamics in the neural network of fuzzy system context have been called learning control. In this paper, we particularly refer to the learning control approach introduced in the works by Arimoto and others (e.g., see [3], [4]), mostly for robotics applications. The referenced and many other related papers consider one motion of a non- linear system (manipulator) that is repeatedly executed with updated feedforward input until a desired tracking performance is achieved. The main advantage of such approach is that it does not require an accurate model of the system dynamics. The major practical drawback is that the feedforward control is obtained only for a single given task. Should the trajectory change, even slightly, the learning process has to be re- peated anew. We remove this barrier by designing an efficient learning-based feedforward controller that works for a range of the task parameters. Such task parameters comprise the initial and the final setpoints of the system and define the trajectory to be tracked. Our approach is based on a paradigm of a nonlinear approximation of the feedforward control dependence on these task parameters. The initial data for the approximation is obtained by performing learning control iterations for a set of selected task parameters within a given range. The paradigm and techniques for obtaining the approx- imation of the feedforward control are the first and main contribution of this paper. Motivation and application ex- amples for the concept of approximating the dependency of the feedforward control on the task parameters can be found in [16], [18], [20], and [23]. In this paper, we use a radial basis function (RBF) network approximation [35], [36]. RBF approximation has a number of very attractive properties such as excellent accuracy, algorithmic simplicity, and efficient handling of vector-valued functions. It has recently become a much used tool in control engineering applications, where it is often used in the neural network or fuzzy system context. We would like to note that Arimoto’s work, as well as many subsequent papers present human motor skill learning as a 1042–296X/97$10.00  1997 IEEE