Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 187948, 13 pages

http://dx.doi.org/10.1155/2015/187948

## Iterative Learning Control with Desired Gravity Compensation under Saturation for a Robotic Machining Manipulator

School of Mechanical, Aeronautical and Industrial Engineering, University of the Witwatersrand, 1 Jan Smuts Avenue, Private Bag 03, Johannesburg WITS2050, South Africa

Received 10 September 2015; Revised 23 November 2015; Accepted 6 December 2015

Academic Editor: Yan-Jun Liu

Copyright © 2015 Horacio Ernesto and Jimoh O. Pedro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper proposes the design of a hybrid iterative learning controller for a four-degree-of-freedom (DOF) robotic machining manipulator (RMM). It combines a nonlinear saturated proportional + integral + derivative (PID) control with desired gravity compensation () and proportional + derivative- (PD-) based iterative learning control (ILC). The control is the primary component that maintains the local stability of the entire RMM system and the PDILC component provides robustness to parameter variations and uncertainties in the robot dynamics. Global asymptotic stability of the proposed control algorithm is conducted using Lyapunov direct method and LaSalles invariance principle. Simulation results show the effectiveness and robustness of the proposed hybrid iterative learning controller. It is also shown that the proposed controller achieved better tracking performances compared to conventional feedback controller.

#### 1. Introduction

Various iterative learning control (ILC) strategies and applications have been extensively studied in the last two decades [1–6]. Arimoto et al. [7] were the first to apply ILC for robotic manipulator tracking control. ILC is an intelligent control technique used for improving the transient performance of systems that operate repetitively over a fixed time interval [2, 5]. The main purpose of ILC is to find a control input iteratively, resulting in the plant’s ability to track the given reference signal with an output trajectory over a finite time interval. The control is the command input function defined on the unchanged time interval, which is updated by using the difference between the desired position function and the actual one. This difference is the error to be driven to zero as the number of trials increases, starting always from the same initial conditions [3, 7, 8].

The learning loop arrangement is of the pure feedback type, since the command addition is only a function of the error. This function is referred to as the learning feedback and should not be confused with* the* conventional feedback [4, 6]. A learning control layout is expected to be convergent in spite of nonpersistent and small persistent disturbances. In the learning control methods, outputs of the controlled system are recorded data at each trial of the controller with respect to the reference values [8]. The data are used to progressively reduce the errors during the subsequent trial of the controller. Most existing ILC algorithms are theoretical and difficult to implement [9].

ILC is applied for the RMM given that most industrial robots are generally used in repetitive tasks. The regulation/*tracking* problem for robotic manipulators is a basic task that can be solved either by proportional + integral + derivative (PID) control [10, 11], by proportional + derivative (PD) [12, 13], or by model-based gravity compensation [14–18] and other nonlinear adaptive intelligent control methods such as neural network-based control [19–27], fuzzy logic-based control [28–33], hybrid neurofuzzy [34], and evolutionary algorithm-based control [35, 36].

Li et al. [23] developed a stochastic adaptive reference optimal control (LQR-based) for an under-actuated inverted pendulum robot system using radial basis function neural networks. Rigorous global asymptotic stability of the system was established. Chen et al. [27] applied an adaptive radial basis function neural networks-based consensus control method for a multiple collaborative manipulators time-delay system while holding up an object or loading a workpiece. It was shown that the controlled system is asymptotically stable using Lyapunov stability theory. Shaocheng et al. [33] designed an observer-based adaptive fuzzy output feedback controller for an uncertain continuous-time multi-input-multi-output (MIMO) two-link robotic manipulator. The plant’s unknown nonlinear functions were approximated using fuzzy logic systems and state observer was used to estimate the unavailable plant’s states.

Regulation and position tracking with bounded inputs and gravity compensation for robot manipulators have been presented in the literature [17, 37, 38]. In these works, the saturation is applied by introducing nonlinear functions in the control laws. These laws require full system information for better closed-loop performance. Here we use the control laws without introducing any special function and the RMM system is driven into desired position with global and semiglobal asymptotic stability. Proportional control plus velocity feedback is the simplest closed-loop controller that may be used to control robot manipulators.

Most of the industrial robots control applications are PID [11, 39, 40], PD [12, 13, 41], and PI [42] based, due to their simplicity and clear physical meaning. Despite these advantages, the design of these controllers is still a challenge [43–45] for multi-input/multi-output (MIMO) system. A conventional PID control has no learning capability. Once the controller’s parameters are tuned, it cannot adapt if operating condition changes, as it frequently occurs in reconfigurable manufacturing approaches. The hybrid iterative control technique that searches for a desired input torque through a sequence of repetitive operations is proposed to overcome this difficulty. The controllers studied in this paper are based on control of the internal coordinates, that is, joint positions.

An increasing number of robots are employed in mechanical machining applications, especially for not too hard materials such as plastics and aluminium. Examples of processes are grinding, deburring, and polishing, while drilling and milling are less common because of the higher requirements on manipulator stiffness, bandwidth, and accuracy [46]. The reasons for using robots in machining applications are lower cost and higher flexibility in comparison with computer numerical control (CNC) machines. Moreover, the RMM has a potential to move, independent of the workpiece, giving it the ability to feed quickly on a large part as it does on a smaller, lighter part of a product being machined.

RMM was presented in [47], with conceptual design and physical and mathematical models being introduced. Note that we chose a serial-link robot, which is a more challenging case than a parallel-link robot structure, in terms of rigidity. The major contributions of the paper are as follows:(1)Design of hybrid proportional + derivative controller with desired gravity compensation and the iterative learning control approach for a 4-degree-of-freedom (DOF) nonlinear, complex, and uncertain RMM under bounded actuators’ inputs.(2)Ensuring that the system with the designed controller is globally asymptotically stable in the presence of parameter variations and external disturbances.(3)Reduction of chattering of the control inputs’ actuators so that the quality of the manufactured item will be within the acceptable limits.

The paper is organized as follows. Section 2 presents the system physical and mathematical modelling with the detailed description of some of its main dynamic properties. Designs of and hybrid ILC controllers are presented in Section 3, followed by the* stability analysis*. Section 4 is devoted to simulation and discussion of results. Section 5 gives the concluding remarks with comments on future works.

#### 2. System Overview and Modelling

##### 2.1. Physical Modelling

Figure 1 shows the physical model of the RMM. Its main purpose is the residual and new manufacturing of the small differences between variants produced using the same premachined and semifinished part. The workspace of the realized configuration is customized for small parts in the range of . This workspace is suitable for small parts in industrial production [47].