Journal of Robotics

Volume 2018 (2018), Article ID 8219123, 10 pages

https://doi.org/10.1155/2018/8219123

## Nonlinear Friction and Dynamical Identification for a Robot Manipulator with Improved Cuckoo Search Algorithm

^{1}College of Mechanical Engineering, Jiangsu University of Technology, Changzhou 213001, China^{2}Department of Industrial and System Engineering, The Hong Kong Polytechnic University, Kowloon 999077, Hong Kong

Correspondence should be addressed to Li Ding; moc.361@ildaaun

Received 17 September 2017; Accepted 28 November 2017; Published 8 January 2018

Academic Editor: Yangmin Li

Copyright © 2018 Li Ding et al. 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 concerns the problem of dynamical identification for an industrial robot manipulator and presents an identification procedure based on an improved cuckoo search algorithm. Firstly, a dynamical model of a 6-DOF industrial serial robot has been derived. And a nonlinear friction model is added to describe the friction characteristic at motion reversal. Secondly, we use a cuckoo search algorithm to identify the unknown parameters. To enhance the performance of the original algorithm, both chaotic operator and emotion operator are employed to help the algorithm jump out of local optimum. Then, the proposed algorithm has been implemented on the first three joints of the ER-16 robot manipulator through an identification experiment. The results show that (1) the proposed algorithm has higher identification accuracy over the cuckoo search algorithm or particle swarm optimization algorithm and (2) compared to linear friction model the nonlinear model can describe the friction characteristic of joints better.

#### 1. Introduction

Modern industry is increasingly oriented towards the production of small batches of a large variety of products, asking for flexibility and automation in the manufacturing systems [1, 2]. Additionally, the increasing quality standards, international competition, and economic reasons put higher requirements on reliability and accuracy and especially on the speed of production processes. In this context, industrial serial robot manipulators have become an indispensable means of automation to increase productivity and flexibility of production units. Robots are programmed by teaching the sequence of the attitude and position which are necessary to execute the desired task. To reach a sufficient accuracy, this teaching is mostly done on-site and relies on a good repeatability, rather than on a good absolute accuracy. To improve the operation accuracy, a precise dynamical model is essential for accurate offline programming.

In the academic aspect, a typical manipulator identification procedure consists of dynamic modelling, excitation trajectory design, data collection, signal preprocess, parameter identification, and model validation [3]. When a priori knowledge is available about the robot system, parametric models can be derived based on the laws of physics and mechanics resulting in a set of differential equations. The unknown dynamical parameters have a physical meaning and can be identified by several approaches. Atkeson et al. [4] used the least square method (LS) to implement the load estimation of dynamical parameters on a PUMA600 robot. According to weighted least squares method (WLS), Gautier and Poignet [5] proposed a dynamical identification approach only from the torque data, without other sensors. Grotjahn et al. [6] used the two-step method to execute the friction and rigid body identification of robot dynamics. Considering the effect of measurement noise, Olsen and Petersen [7] used the maximum likelihood estimating (MLE) method for parameters identification of an industrial robot with a statistical framework. Recently, some novel dynamical identification methods for robot manipulators have been reported using intelligence algorithms. For instance, Bingül and Karahan [8] integrated the particle swarm optimization (PSO) algorithm with LS algorithm to estimate the dynamical parameters of Staubli RX-60 robots. In the identification experiment, the velocity and acceleration are measured by three high-speed cameras, and the joint torques are measured by six load cell sensors. Without velocity and acceleration sensors, Ding et al. [9] used the motor current and joint positions to calculate the torques, velocity, and acceleration of joints and proposed an artificial bee colony algorithm (ABC) to obtain the unknown dynamical parameters. Nevertheless, when handling complicated and high-dimensional parameters identification problems, the flaw that the premature convergence can make those intelligence computation algorithms stuck in a local optimum.

As known, the friction is a major source of disturbances affecting motion quality. Therefore, it must be included as an additional component in robot modelling. In robot identification applications, a model including Coulomb and viscous friction is frequently applied [10, 11]. With such linear model, the parameters estimation is significantly simplified. However, this friction model is not capable of describing the experimentally measured friction characteristic, especially the static model at joint reversal. Aiming at these problems, we add a nonlinear friction model into the dynamical model of a 6-DOF industrial serial manipulator and use an improved cuckoo search (ICS) algorithm for dynamical identification. The idea of the method is to measure the positions and gravitational torques of different joints through designing Fourier series as excitation trajectories. The collected values are used to calculate the dynamical parameters based on ICS. In ICS, considering the outstanding performance of chaotic operator and emotion operator, these improved operators are used to enhance the performance of the classical CS. And the comparison of three different identification methods, CS, PSO, and ICS, illustrated the superiority of our proposed algorithm in the application of dynamical identification.

This article is organized as follows. The dynamical model with a nonlinear friction model of a robot manipulator is given in Section 2. For the unknown dynamical parameters, the improved cuckoo search algorithm is introduced to realize the parameters estimation in Section 3. Then, the design of excitation trajectory, data collection, and preprocessing are presented in Section 4. Later, an ER-16 robot is used as a test platform for identification experiment, and the results are analyzed in Section 5. In addition, the superiority of linear and nonlinear friction model has been compared through the model validation in Section 6. Finally, Section 7 discussed the key findings and prospective research target.

#### 2. Dynamic Modelling

According to the literature [12], a -DOF serial manipulator is described as a kinematic chain of several rigid bodies. Hence, we can utilize the Newton-Euler method to deduce the dynamical model of the manipulator:where joint torque , joint position , joint velocity , and joint acceleration are -dimensional vector. denotes the -dimensional joint friction vector. represents inertial matrix, is a -dimensional vector including Coriolis and centrifugal forces, and is -dimensional gravity vector.

Equation (1) except for friction torques can be rewritten as a linear form if using the modified Newton-Euler parameters [13] or the barycentric parameters [11]:where only contains the motion data, which can be treated as identification matrix or observation matrix. is the barycentric parameter vector. This conversion vastly reduces the complexity of parameters identification. In addition, the dynamical parameters of link are governed by the formwhere is the inertial tensor of the link . Similarly, is the mass of the link and is the inertia moment.

Generally, the identification matrix in (2) is not full of rank, that is, not all dynamical parameters give contribution to the joint torques. In the literature [14], some methods like case-by-case analysis or singular value decomposition are adopted to eliminate the redundant parameters. And the barycentric parameter vector can be replaced by a vector of minimal barycentric parameters with . Hence, (2) can be transformed into another formwhere denotes the observation or identification matrix and is the number of minimal barycentric parameters.

Except for the dynamical parameters in (2), there also exists friction torques and extra torques caused by inertias of motor rotors. In general, the inertias of motor rotors are provided by manufacturers, and corresponding torques can easily be compensated to the dynamical model. As for joint friction model, it is regarded as a complex nonlinear model. To simplify the model, the Coulomb and viscous friction were used to describe the friction model. But the researchers found [15] that the friction torques of some joints exceeded their full speed range, and the simple friction model could not cover the characteristics, especially at motion reversal. A better description of the joint friction characteristics may be based on the following nonlinear equations:where is the zero drift error of friction torque, is the Coulomb friction coefficient, is the viscous friction coefficient, and and are the experiential friction coefficients. It should be noted that this model has a discontinuity at zero velocity.

In summary, using the nonlinear friction model yields the whole dynamical model of the 6-DOF robot aswhere denotes the friction torques vector and is the number of friction coefficients. Obviously, the classical least square method could not solve the above nonlinear equation. Hence, applying an intelligence algorithm for solving this problem may be a feasible method.

#### 3. Identification Algorithm

##### 3.1. Introduction to Chaos Theory

Chaos theory is epitomised by the so-called “buttery” detailed by Lorenz [16]. He discovered that tiny changes in an initial state would make a radically different final result and typically rendering long-term prediction impossible. Chaotic map has ergodic and stochastic properties, which is regarded as a bounded nonlinear system with deterministic dynamical behavior. Moreover, it has a very sensitive dependence on initial conditions. There are many forms about chaotic map, such as tent map, Gauss map, logistic map, and tent map [17]. Considering the high robustness and stability of tent map, we choose it to generate the chaotic sequence. The description of the tent map is written aswhere is iterations and denotes a positive number. The value of is updated with the initial condition . A tent map for after 500 iterations is shown in Figure 1. As the value of increases, gets a new value. In this paper, the parameter is chosen as 0.6 after many trials.