1. To clarify the statements, we present the following simple, closed-loop system where x(t) is a tracking error signal, is an unknown nonlinear function, and is a learning-based estimate It is assumed that is periodic with a known period For the above system, the standard repetitive update rule is given by With regard to the above error system, Messner et al. noted that the techniques they presented, could not be used to show that is bounded if it is generated using the standard repetitive update rule Introduction - Previous Research

2. To address the boundedness problem associated with the standard repetitive update rule, Sadegh et al. proposed the following update rule and hence, guarantee that is bounded for all time Introduction - Previous Research It is well known how one can apply a projection algorithm to the adaptive estimates of a gradient adaptive update law and still accommodate the Lyapunov-based stability analysis Unfortunately, it is not clear from the analysis by Sadegh et al. how the Lyapunov-based stability analysis accommodates the saturation of the standard repetitive update rule To address the boundedness problem, we propose the following update rule

3. Consider the following error dynamics General Problem and where are bounded provided the error system is bounded, is a learning-based estimate of Assumption 1: The origin of the error system is uniformly asymptotically stable for and there exists a positive- definite function, a symmetric matrix, and a known matrix such that Assumption 2: The unknown periodic function has a period of and we assume that where is a vector of known, positive bounding constants

4. The learning-based estimate is designed as follows Based on the definition of, we can prove the following inequality To facilitate subsequent analysis, we develop the following relationship where we utilized the fact that Learning-Based Estimate Formulation

5. Stability Analysis Theorem 1: The learning based estimate designed previously, ensures that Proof: To prove Theorem 1, we define the following non-negative function where was defined in Assumption 1 Signal Chasing Arguments Barbalat’s Lemma

6. Dynamic Model Dynamic equation for an n-DOF revolute robot : link position, velocity, and acceleration : inertia matrix : centripetal-Coriolis matrix : gravity vector : viscous friction coefficient matrix : constant, diagonal, static friction matrix : torque control input Non-periodic Effects

7. Dynamic Model Properties Inertia matrix is positive-definite and symmetric where are known positive constants Skew Symmetry Property Linearity in the parameters The centripetal-Coriolis matrix, gravity vector, and dynamic friction matrices can be upper bounded as follows

8. parameter estimation vector To quantify our objective to design a global link position tracking controller, we define the link position tracking error as follows where the desired trajectory and its first two time derivatives are assumed to be bounded, periodic functions of time with a known period such that Since this objective is to be met despite parametric uncertainty in the dynamic model, we define the following parameter estimate error Control Objective unknown constant vector desired trajectory

9. To facilitate the subsequent control development and stability analysis, we reduce the order of the dynamic model by defining a filtered tracking error-like variable as follows the following open-loop dynamics for the filtered tracking error can be obtained where Control Formulation known, positive bounding function known, positive bounding constant

10. Control Formulation Given the previous open-loop error system, we design the following torque control input where are positive constant control gains, is generated on-line according to the following learning-based algorithm is a positive, constant control gain, is defined previously, and the parameter estimate vector is generated on-line according to the following gradient-based adaptation algorithm where is a constant, diagonal, positive-definite, adaptation gain matrix.

11. Control Formulation The following closed-loop error system can now be formulated where is defined as follows Substituting the learning-based estimate in the above expression yields where we used the fact that

12. Stability Analysis Theorem 2: The proposed hybrid adaptive/learning controller ensures global asymptotic link position tracking in the sense that provided the control gains are selected as follows Proof: To prove Theorem 2, we define the following non-negative function

13. Signal Chasing Arguments Barbalat’s Lemma Stability Analysis After taking the time derivative of the following expression is obtained

14. Experimental Results The following controller was implemented on a two-link direct-drive, planar robot manipulator manufactured by Integrated Motion Inc. The two-link robot is directly actuated by switched-reluctance motors. A Pentium 266 MHz PC running RT-Linux (real-time extension of Linux OS) hosted the control algorithm. The Matlab/Simulink environment with Real-Time Linux Target for RT-Linux was used to implement the controller. The Servo-To-Go I/O board provided for data transfer between the computer subsystem and the robot. learning-based estimate feedback term

15. The two-link IMI robot has the following dynamic model The reference trajectory was selected as follows Experimental Results