Journal of Robotics

Volume 2015 (2015), Article ID 596327, 7 pages

http://dx.doi.org/10.1155/2015/596327

## Inverse Kinematic Analysis and Evaluation of a Robot for Nondestructive Testing Application

^{1}School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China^{2}School of Materials Science and Engineering, Dalian Jiaotong University, Dalian 116028, China

Received 29 December 2014; Revised 13 April 2015; Accepted 16 April 2015

Academic Editor: Nan Xiao

Copyright © 2015 Zongxing Lu 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

The robot system has been utilized in the nondestructive testing field in recent years. However, only a few studies have focused on the application of ultrasonic testing for complex work pieces with the robot system. The inverse kinematics problem of the 6-DOF robot should be resolved before the ultrasonic testing task. A new effective solution for curved-surface scanning with a 6-DOF robot system is proposed in this study. A new arm-wrist separateness method is adopted to solve the inverse problem of the robot system. Eight solutions of the joint angles can be acquired with the proposed inverse kinematics method. The shortest distance rule is adopted to optimize the inverse kinematics solutions. The best joint-angle solution is identified. Furthermore, a 3D-application software is developed to simulate ultrasonic trajectory planning for complex-shape work pieces with a 6-DOF robot. Finally, the validity of the scanning method is verified based on the C-scan results of a work piece with a curved surface. The developed robot ultrasonic testing system is validated. The proposed method provides an effective solution to this problem and would greatly benefit the development of industrial nondestructive testing.

#### 1. Introduction

Ultrasonic testing as an important nondestructive inspection method is adopted in numerous applications to test the internal defects of composite materials with simple structures. However, the manufacturing technology of composite materials with a complex shape, such as a curved shape, variable thickness, and complex rotary structure, has developed rapidly. Thus, the automatic inspection of complex structures has become a challenge.

The joint robot with six degrees of freedom (DOF) can reach any position and orientation in its operating range; the inspection problem can be solved by combining this joint robot with ultrasonic testing technology [1]. In such an automatic ultrasonic testing system, optimization and simulation of the inspection path are important so that the robot that scans these complex structures can be operated according to reasonable tracks. Many researchers [2–5] have worked on the inverse kinematics of the general 6R serial robot. Robot simulation was based on robot kinematics. Yu et al. [6] built a 6R serial robot using OpenGL and conducted motion simulation. Lee and Liang [7] provided a resultant elimination procedure by using complex number method and vector theory. However, the geometric interpretation of their elimination procedure was not completely revealed because of its complexity. Establishing a common inverse algorithm so that the analytical method [8, 9], geometric method [10, 11], and numerical method [12] are all developed to address this problem is difficult. Wang et al. [13] provided a preliminary inverse kinematics method for the Staubli robot. However, this method cannot explain the optimization of these solutions.

A new evolutionary arm-wrist separateness method is used in the present study to formulate the kinematics equations of the general 6-DOF robot. Inverse kinematics only has eight solutions; hence, the difficulty in solving is reduced. With the optimized method, the optimal solution for a trajectory can be identified. A 3D-application software is created to simulate ultrasonic trajectory planning. Based on the feature that the subsequent three joint axes intersect at one point, the arm-wrist separateness method is adopted in Section 2 to solve the inverse kinematics of the robot. Additionally, the shortest distance rule is adopted in Section 3 to optimize the results [14]. A 3D-application software is also developed to simulate ultrasonic trajectory planning for complex-shape work pieces. Finally, the work piece with a curved surface is detected with this robot. The experiment results verify the effectiveness and feasibility of the proposed method. The simulation and experiment results are shown in Section 4, and the paper ends with concluding remarks in Section 5.

#### 2. Inverse Kinematics Solution

Denavit and Hartenberg established the D-H method [15, 16], which is commonly utilized in robot kinematics models. The inverse kinematics solution uses the position and orientation of robot end-effector, which has been known to solve the joint angles . An arm-wrist separateness method was used in this study.

In the TX90 XL robot, the axes of the last three joints intersect at one point, which is referred to as point . The position of point is independent of the last three joints , , and . Therefore, only the three previous joints should be considered when solving the position of point . These joints are the main content of the arm-wrist separateness method. The position and orientation are , and matrix can be achieved from the D-H model. The position of is denoted as .

The position of point can be described as

##### 2.1. Solutions of Arm Joint Angles , ,

The position of point can be determined from the homogeneous transformation matrix, which is derived from , , , . Consider

In the expressions above,

The elements of can be drawn as follows:

The following can be obtained by calculating (4) × + (5) × . Consider

The solution of can be obtained through solving (7). Therefore,

The following can be obtained by calculating (4) × (5) × . Moreover,

In the TX90 XL robot, can be obtained from (9) as follows:

We can obtain by solving (6) as follows:

Substituting the equation with (10) and (11) yields

The solution of is achieved by solving (12). Considerwhere

The result of performing the calculation of (10)/(11) is

The solution of (15) is , and we have

##### 2.2. Solutions of Wrist Joint Angles

The orientation of the robot is controlled by the rotation matrix, and the orientation of is described by . The orientation of the tool end-effector is described by . The relationship between and is . Matrix can be described as

can then be calculated as follows:

When ,

When , the arms are at the singular position where link axes 4, 5, and 6 are collinear. This condition has only one motion form of the robot tool side, the orientation of which is calculated by the sum or difference of and . Oftentimes, the current value of is used.

According to (17), when ,

When , the current value of is also utilized because the case is similar to the value determination of .

#### 3. Optimization of the Inverse Kinematics Solution

The obtained equations, , show two roots; thus, the robot has eight groups of inverse kinematics solutions that correspond to the same position and orientation. For the kinematics of the robot, which has numbers of solutions, a suitable algorithm is required to select a set of values as the inverse solution of a robot.

For the eight group solutions, several solutions are real solutions, but some may be imaginary solutions. In the traditional inverse kinematics calculation method, the result of double variables arctangent function is in the range of . If the path trajectory has a singular point, then the joint will have a jump of 180°. Achieving a smooth motion is difficult because a long time is spent to rotate the joint of the robot by 180°, and the entire movement process is slow. We presented a type of inverse optimization method according to the incomplete solution of the inverse function. Comparison of the results of the joint with the actual range of the corresponding joint variables showed that the joint results may be . All of the trigonometric functions were not changed. This condition prevents the singular point jump and maintains the continuity of the trajectory. The actual range of the joint is shown in Table 1. The optimized process is shown in Figure 1.