Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 937565, 13 pages

http://dx.doi.org/10.1155/2015/937565

## Motion Reliability Modeling and Evaluation for Manipulator Path Planning Task

School of Automation, Beijing University of Posts and Telecommunications, Beijing 100876, China

Received 11 October 2014; Accepted 2 March 2015

Academic Editor: Zhan Shu

Copyright © 2015 Tong Li 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

Motion reliability as a criterion can reflect the accuracy of manipulator in completing operations. Since path planning task takes a significant role in operations of manipulator, the motion reliability evaluation of path planning task is discussed in the paper. First, a modeling method for motion reliability is proposed by taking factors related to position accuracy of manipulator into account. In the model, multidimensional integral for PDF is carried out to calculate motion reliability. Considering the complex of multidimensional integral, the approach of equivalent extreme value is introduced, with which multidimensional integral is converted into one dimensional integral for convenient calculation. Then a method based on the maximum entropy principle is proposed for model calculation. With the method, the PDF can be obtained efficiently at the state of maximum entropy. As a result, the evaluation of motion reliability can be achieved by one dimensional integral for PDF. Simulations on a particular path planning task are carried out, with which the feasibility and effectiveness of the proposed methods are verified. In addition, the modeling method which takes the factors related to position accuracy into account can represent the contributions of these factors to motion reliability. And the model calculation method can achieve motion reliability evaluation with high precision and efficiency.

#### 1. Introduction

As a kind of complicated multichain structure, manipulator can achieve various operations and has good environmental adaptability, which makes it widely used in industrial manufacturing, medical and aerospace fields, and so forth. In order to guarantee the positioning accuracy requirement of various tasks and complicated environment, the motion safety and reliability of manipulator attract extensive attention [1–4]. As a result, motion reliability [5–9] which concentrates on the motion accuracy is used to give a quantitative description for the motion performance of manipulator. Generally, the motion reliability is defined as the probability that the position guarantees the accuracy requirement under the affection of various factors [10]. Since manipulator achieves various tasks via positioning operation, the operation status of manipulator can be reflected in numerical by evaluation of motion reliability.

In order to evaluate motion reliability, the model of motion reliability should be established firstly, during which factors related to position accuracy should be considered. Firstly, the relationship between factors and position accuracy should be derived. Zhuang et al. [11] and Chen et al. [12] established the relationship between position error and parameters deviation of manipulator, respectively. Wu [13, 14] analyzed uncertain factors influencing the mechanical positioning accuracy with interval method. The methods mentioned above can be referenced in establishing motion reliability model. Pandey and Zhang [15] discussed the motion reliability of manipulator considering the clearance in joint independently based on probability theory. The clearance is transformed into errors of joint angles to establish the relationship with motion reliability. However, single factor considered is not enough for the modeling of motion reliability. Rao and Bhatti [16] evaluated the kinematics reliability and dynamics reliability of a two-link structure. Unfortunately, factors related to motion reliability were not taken into account. Moreover, the reliability evaluation for manipulator is more complicated than two-link structure, which needs further discussion.

Based on the analysis above, factors related to motion reliability are not given enough attention. And a universal model which reflects the relationship between factors and motion reliability is not achieved. The model of motion reliability should take the contribution of factors to position accuracy into account. Actually, factors related to position accuracy are various, such as clearance [17], friction [18], and wear [19]. Many of them are unobservable and uncontrollable and even have coupling relationship between each other, which makes it difficult to establish a model including all factors. In order to reflect the influence on position accuracy caused by clearance, friction, wear, and so forth into model, kinematics parameters and joint angles can be taken as intermediate variables to deliver the influence, which has been implemented in [15]. In this way, the motion reliability model can be established based on the deviations of kinematics parameters and joint angles.

After the establishment of motion reliability model, the model should be calculated to achieve evaluation. For this purpose, Kim et al. [20] used the first order reliability method (FORM) to calculate reliability. Generally, FORM can only solve reliability problem related to a few performance functions. The performance functions which are established with position accuracy related to various factors are too many for FORM to be applied. Kumar et al. [21] and Sharma et al. [22] used genetic algorithms and fuzzy methodology to analyze the reliability, respectively. Dashuang et al. [23] used the traditional Monte Carlo method to calculate the dynamic precision reliability of six degrees of freedom (DOF) mechanism. And the authors [24] have analyzed the motion reliability of a 8-DOF modular robot with Monte Carlo method based on a simple model. However, the recursive relationship between factors and motion reliability is not systematically derived in the model. And too many samples are needed in motion reliability calculation with Monte Carlo method. Actually, the fuzzy methodology and Monte Carlo method are quite complex, which leads to high computation cost during reliability computation. Novi Inverardi and Tagliani [25] calculated the probability density function with the principle of maximum entropy, which greatly decreased the samples in computation compared with Monte Carlo method. The method with maximum entropy shows a new way to achieve motion reliability calculation with high efficiency. Considering the practical application, a simple and feasible method for calculating motion reliability model with high precision should be proposed.

Since many tasks depend on the positioning operation of manipulator, path planning [26] takes an important role in achieving various operations. The paper aims at establishing the motion reliability models of path planning task and achieving evaluation. The motion reliability model of path planning task represents the probability that the position of manipulator guarantees the accuracy threshold during the task. The modeling method proposed in the paper takes the contributions of various factors to position accuracy into account. And, in order to reflect the influence of various factors, kinematics parameters and joint angles are taken as intermediate variables to deliver the influence. As a basis, the motion reliability model can be established based on the devotions of kinematics parameters and joint angles. Considering the complex of multidimensional integrals in the model of continuous trajectory tracking, the approach of equivalent extreme value is introduced to achieve model simplification. As a result, multidimensional integrals for PDF are converted into one dimensional integral. Since huge samples and computation time are needed in traditional method for PDF calculation, a method based on the maximum entropy principle is proposed to achieve calculation with high precision and efficiency. So far, the evaluation of motion reliability can be achieved, and an intuitionistic expression for the performance of manipulator can be obtained with the evaluation result. Meanwhile, calibration and control can be devoted to improving the motion reliability of manipulator for further discussion.

In conclusion, the paper is organized as follow. In Section 2, the mathematical relationship between motion reliability and position accuracy is firstly derived. The influence of the factors related to the position accuracy is delivered to the deviations of kinematics parameters and joint angles, based on which the motion reliability models of path planning task are established. In Section 3, the approach of equivalent extreme value is introduced to simplify the motion reliability model, and method based on the maximum entropy principle is proposed to achieve model calculation. In Section 4, simulations are carried out to verify the effectiveness and correctness of the proposed modeling and evaluation method. And strategies for improving the motion reliability are discussed further based on the evaluation result. The last part is summary.

#### 2. Motion Reliability Models of Path Planning Task

Path planning tasks are always divided into point-to-point path planning and continuous trajectory tracking, and their motion reliability models have different focuses. For the model of continuous trajectory tracking, position accuracy of the entire trajectory should be considered, while, for the model of point-to-point path planning, position accuracy of target point is paid more attention. Aiming at the two kinds of path planning, motion reliability models are established based on the analysis of factors related to position accuracy.

##### 2.1. The Mathematical Expression for Motion Reliability of Path Planning

The motion reliability for a single trajectory point is discussed. Set position accuracy threshold (namely, the maximum acceptable value of the deviation between the actual and desired position) as at each direction of the point. Then an enveloping space is formed as a sphere. The desired position is set as center and is set as radius. As a result, the Cartesian space at the end-effector can be divided into three subspaces shown in Figure 1. The actual position locating in the sphere is considered to be motion reliable, while it is considered to be motion unreliable when actual position locates outside the sphere. The actual positions locating on the shell are critical positions, which can either be reliable or unreliable at the next movement. Based on the analysis above, the mathematical expression when path planning task is motion reliable can be achieved.