Computational Intelligence and Neuroscience

Volume 2019, Article ID 5426986, 10 pages

https://doi.org/10.1155/2019/5426986

## A Repeatable Motion Scheme for Kinematic Control of Redundant Manipulators

Department of Information and Electronic Engineering, Zhejiang University of Science and Technology, Hangzhou, China

Correspondence should be addressed to Kong Ying; moc.361@888-gniygnok

Received 29 May 2019; Accepted 28 July 2019; Published 18 September 2019

Academic Editor: Daniele Bibbo

Copyright © 2019 Kong Ying 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

To achieve closed trajectory motion planning of redundant manipulators, each joint angle has to be returned to its initial position. Most of the repeatable motion schemes have been proposed to solve kinematic problems considering only the initial desired position of each joint at first. Actually, it is very difficult for various joint angles of the robot arms to be positioned in the expected trajectory before moving. To construct an effective kinematic model, a novel optimal programming index based on a recurrent neural network is designed and analyzed in this paper. The repetitiveness and timeliness are presented and analyzed. Combining the kinematic equation constraint of manipulators, a repeatable motion scheme is formulated. In addition, the Lagrange multiplier theorem is introduced to prove that such a repeatable motion scheme can be converted into a time-varying linear equation. A finite-time neural network solver is constructed for the solution of the motion scheme. Simulation results for two different trajectories illustrate the accuracy and timeliness of the proposed motion scheme. Finally, two different repetitive schemes are compared and verified the optimal time for the novelty of the proposed kinematic scheme.

#### 1. Introduction

Robot manipulators have been playing an important role in various kinds of engineering fields. They have been widely used to perform effective and high-intensive repetitive work, such as car assembling, logistics handling, and sculpturing [1–3]. Robot arms usually have two types in needs of special operators. One is the redundant manipulators which have more degrees of freedom (DOF) than the given tasks required while the other one is the ordinary nonredundant manipulators which can complete the objective directly. Correspondingly, nonredundant manipulators refer to the one that has no additional DOF when performing a given task. Redundant manipulators are more flexible and advantageous for its redundant DOF. That is the reason why it is increasingly important in practical engineering applications [4, 5].

One of the fundamental problems in redundant manipulators motion controlling is redundancy solutions, known as inverse kinematics, which attracts many researchers’ interests [6, 7]. Given the position and pose of the end-effector, calculating the homologous trajectories of joint angles with the redundant manipulators is named inverse kinematics. Liegeios-Chauvel et al. put forward a gradient projection method based on inverse kinematics solution to divide the particular motion controlling into zero space by solving the optimization goal to regulate the solutions for redundancy [8]. From then on, many researchers study pseudo inverse approaches for kinematic equation of redundant manipulators [9, 10]. However, these approaches do not only take a heavy burden of calculations, but also require the Jacobian of manipulators to be full rank, which is hard to realize.

With the development of neural networks, recurrent neural networks based on negative gradient directions are emerged. Due to the high efficient ability for computing, gradient neural networks (GNNs) are widely used in identifications and matrix equations [10]. GNN is set up by establishing non-negative direction function and to obtain a scheme for kinematic controlling. When applied to motion planning of redundant manipulators, the convergent errors generated by GNN are always lagged behind the ideal one. That is, every joint angle of manipulators cannot return to their initial position in the end, which may cause nonrepetition phenomenon in trajectory planning of the robot manipulators. With the deepening of research studies in repeatable controlling of redundant manipulators, various velocity schemes based on online quadratic optimization have been developed. Such optimal schemes incorporate the equality and inequality constraints and avoid the limitation of joint angles and joint velocities. For efficient repetitive tasks, Zhang et al. firstly introduced a repetitive motion index as the optimization criterion, using Zhang neural network (ZNN) to solve the redundancy problems [11–13]. Then, various motion schemes combining the physical joint constraints are formulated as an optimal programming index subjected to the kinematic equation of manipulators. In addition, theoretical analyses prove that the motion schemes can be converted into time-varying equations by using the Lagrange multiplier. When considering necessary conditions of Karush-Kuhn-Tucker (KKT) for nonlinear optimization problems, such an optimal programming index also can be converted into linear variational inequality (LVI) and a piecewise linear projection equation (PLPE) [14, 15]. Various neural networks have spung up to solve the LVI and PLPE. Simulations on different types of redundant manipulators are studied and different shapes of the trajectory tasks are given, which verified the effectiveness and superiority of the proposed optimal programming index for repeatable motion planning as well as the corresponding neural solvers [16, 17].

Although, most of the aforementioned approaches for repeatable motion planning of redundant manipulators are effective, the convergent time of the dynamical equations has not been ensured. That is the optimal programming index for motion controlling using neural solvers can make the joint angles of the manipulators back to initial desired position as long as time goes infinity. For the perspective of finite-time convergence, Li et al. first proposed a finite-time neural network (FTNN) model to solve the repetitive motion scheme based on ZNN in order to ensure that the convergent time is finite [18]. Then, around this FTNN, various revised FTNN models are constructed to accelerate the convergent rate of ZNN [19–21]. In the literature [22], a motion scheme of mobile robot arms based on finite-time convergence property has been set up to apply in the grasping work of the manipulators.

A redundant manipulator is a part of a robot. It is hard to locate every joint angle in the desired state at first. It is not efficient for adjusting the positions of joint angles through self-movement. However, most of the above repetitive motion plans and different neural solvers, which are mainly for ideal initial state, do not consider the deviations of manipulators. These models of repetitive motion planning based on pseudo inverse and asymptotic convergent dynamic recurrent neural networks have been studied by many researchers. Few studies are reported for finite-time repeatable motion controlling when the joint angles are deviated from the initial desired position at first [23]. In the literature [24], only a new type of terminal neural network is researched for solving motion scheme of ZNN, which has been designed from the perspective of controlling. In [25], initial deviations of joint angles are considered. The optimization performance indices is still based on infinite-time convergence, and only the neural solver for motion scheme is designed for finite-time convergence.

The remainder of this paper is organized as follows. The kinematic equations of redundant manipulators are established in Section 2. Section 3 gives out the optimal programming index, and a repeatable motion scheme of manipulators is formulated and analyzed. In Section 4, a terminal recurrent neural network algorithm is proposed to solve the mentioned motion scheme. In Section 5, simulation results on two different path trajectories with Katana6M180 manipulators verified the effectiveness and superiority of the repeatable motion scheme and the terminal neural solver. Finally, remarks and conclusions are presented in Section 6. The main contributions of this paper are summarized in the following aspects:(1)A new optimal programming index for repeatable motion planning of redundant manipulators is exploited. It is the first time to use this performance index, which can ensure the joint angles back to their initial desired positions in finite time no matter considering the initial state of each joint of the manipulators.(2)A special kind of neural model based on recurrent neural networks is presented to solve the repeatable motion scheme. The activation of the neural solvers has adopted limited-value function, which is applicable in practical application problems.(3)Two different tracking tasks with redundant manipulator Katana6M180 are introduced to illustrate the superiority and effectiveness of the proposed repeatable motion scheme. Comparison results of various repetitive motion schemes are visualized in the end.

#### 2. Kinematic Structure of Katana6M180 Robot Arm

In this section, a redundant manipulator Katana6M180 model has been set up for illustrating the effectiveness of the proposed repeatable motion scheme. The mechanical structure of the Katana6M180 robot arm is shown in Figure 1. The Katana6M180 robot arm is composed of five degrees of freedom (DOF) with three-DOF elbow for revolute joints, two-DOF wrist for revolute joints, and a gripper connected to the end-effector. The range of angular motion of Katana6M180 is displayed in Table 1. From the Table 1, joint is the angle between horizontal line and link 1 , joint is the angle between link 2 and link 3, joint is the angle between link 3 and link 4, and joint is the angle between link 4 and link 5. Joint is between motor 5 and motor 6. In Table 1, it is shown that the rotation and extension of the joint angles are limited by the mechanical arm itself.