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.
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 [
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. [
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 [
After the establishment of motion reliability model, the model should be calculated to achieve evaluation. For this purpose, Kim et al. [
Since many tasks depend on the positioning operation of manipulator, path planning [
In conclusion, the paper is organized as follow. In Section
Path planning tasks are always divided into pointtopoint 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 pointtopoint 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.
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
The division of the Cartesian space.
Suppose
Define the PDF related to the position accuracy of the entire trajectory points as
In (
For
Due to
Combining
For the entire manipulator, the nominal transformation matrix from the coordinate system of the endeffector to the base coordinate system can be expressed as
Define
Combine (
Then the position deviation of the endeffector can be obtained from (
So far, relationship between deviations of kinematics parameters and position error is derived. Meanwhile, the contribution of each parameter to the position accuracy is also obtained as matrix
Influence relationships between factors and motion reliability.
Figure
In summary, combining (
The motion reliability model for continuous trajectory tracking is expressed as
So far, models of motion reliability for path planning task are obtained. According to (
In order to calculate motion reliability models of path planning task, samples of position deviations should be firstly obtained by introducing errors into factors. As a basis, PDF related to the samples can be computed and the motion reliability can be calculated by integral for PDF. For pointtopoint path planning, motion reliability can be calculated according to (
Define theorem as follows.
Suppose
Firstly, prove
Define
Because the maximum value of
Consider
The maximum value of
Combing with
According to the theorem, define
After the model is simplified, the core of motion reliability evaluation is the calculation of PDF. Generally, Monte Carlo method can be used to solve any problems about probability distribution. However, massive samples are needed and computation cost is high. In order to achieve fast calculation of PDF with high precision, the information entropy is introduced and a method based on the principle of maximum entropy is proposed. The principle of maximum entropy [
When evaluating the motion reliability of continuous trajectory tracking, the PDF can be calculated as follows. Suppose
Equation (
As the maximum entropy principle describes, the PDF is the most realistic when the information entropy is maximum. In order to find the maximum value of entropy, Lagrange multiplier method [
Take partial derivative with respect to
The expression of
Meanwhile,
In order to calculate the unknown constants
Substitute (
When
According to (
Set the initial value of
Based on the analysis above, the process for calculating the motion reliability of path planning task can be concluded, which is also shown in Figure
Motion reliability model is firstly established for a particular path planning task. For model of continuous trajectory tracking, the approach of equivalent extreme value is introduced. As a result, multidimensional integrals for PDF related to the entire trajectory points are turned into one dimensional integral for PDF related to the maximum position deviation.
Introduce errors into factors related to position accuracy of the model, and simulate the path planning task for multiple times. For pointtopoint path planning, position deviation at the target point of each simulation is obtained. For continuous trajectory tracking, position deviations at each trajectory point are obtained.
For pointtopoint path planning, the position deviation at the target point of each simulation is selected as samples. For continuous trajectory tracking, the maximum position deviation of each simulation is selected as samples.
Lagrange function about information entropy of position deviation samples is established, and the expression of PDF related to position deviation samples is derived at the state of maximum entropy.
The unknown coefficients of the PDF are calculated by using KL divergence. Then the motion reliability evaluation is achieved by one dimensional integral for PDF.
Process for calculating the motion reliability model of path planning task.
A 7DOF manipulator is used for motion reliability evaluation of path planning task. The coordinate systems are defined as shown in Figure
The nominal DH parameters of 7DOF manipulator.
Link 





1  0  0.6  0  90 
2  90  0.5  0  −90 
3  0  0.5  5  0 
4  0  0.5  5  0 
5  0  0.5  0  90 
6  −90  0.5  0  −90 
7  0  0.6  0  0 
The coordinate systems of 7DOF manipulator.
The initial configuration of manipulator is set as
During the motion reliability evaluation of task, errors are introduced into kinematics parameters. Errors following normal distribution
With the samples of maximum position deviation, PDF of the continuous trajectory tracking task is calculated. The initial value of
The PDF related to position accuracy threshold.
The motion reliability varies with accuracy requirement.
Meanwhile, the PDF and probability curves calculated by Monte Carlo method are also provided in Figures
Besides, huge computation cost and samples requirement limit Monte Carlo method in computing the PDF of trajectory tracking task. Nearly 3000 seconds are used to handle the 10000 samples to achieve the PDF. However, the method based on the maximum entropy principle needs fewer samples (just 500 samples), which greatly reduces the computation cost. In conclusion, the motion reliability evaluation method based on the maximum entropy principle is feasible with high efficiency and computation precision.
When evaluating motion reliability of continuous tracking task, the number of trajectory points will have influence on the motion reliability. If the number is small, the characteristics of the trajectory cannot be reflected completely, which leads to a credible evaluation result. In addition, with the number increase, the computation cost will become higher. Therefore, proper number of trajectory points should be selected by taking both computation efficiency and precision into account.
The trajectory
The motion reliability varies with trajectory points.
Besides, trajectory points are determined by task cycle and control cycle. The motion reliability evaluation will benefit from the proper number of trajectory points, which also indicates that the arrangement for task cycle and control cycle of the task is reasonable. In this way, reasonable arrangement for trajectory points is the basis to achieve fast and precise motion reliability evaluation.
Factors influencing the position accuracy of manipulator can be included into the model of motion reliability. Via introducing errors into them, motion reliability of a particular path planning task can be evaluated, which is achieved in Section
Decrease in motion reliability caused by factors with the same error.
Errors introduced into
Errors introduced into
It is obvious that decreases caused in motion reliability are different within different position accuracy threshold for one factor. Compared with the matrix
Decrease in motion reliability caused by single factor and combined effect.
Angular factors  Stand. deviation of error (°)  Decrease of reliability  Coupling effect 


0.01  0.08733  0.3809 

0.01  0.1197  


Length factors  Stand. deviation of error (m)  Decrease of reliability  Coupling effect 



0.001  0.1174  0.3608 

0.001  0.1121 
In Table
PDF and decease in motion reliability caused by different standard deviations and accuracy thresholds.
Decrease in motion reliability
Thus the contributions of kinematics parameters to motion reliability are analyzed, and the coupling effect of parameters is discussed, too. Meanwhile, the variation according to the position accuracy threshold and standard deviation of errors is derived. As a result, the importance of each kinematics parameter in improving motion reliability can be obtained. When improving the motion reliability of manipulator, factors with high contribution should be given high priority to handle. According to the different contributions of parameters, the strategy to achieve motion reliability improvement is discussed in the next section.
Since
Errors following normal distribution with different standard deviation are introduced into
The motion reliability varies with errors of different standard deviation.
Because of the timevarying characteristics of gear clearance, transmission error, and friction, the motion errors at different joint angles are different, which can be established as
Task constraints should be fulfilled firstly; namely, the position accuracy should be guaranteed in the entire trajectory, which can be expressed as
In trajectory
Motion reliability as a criterion can reflect the motion performance of manipulator synthetically. With the modeling and evaluation of motion reliability, the status of systems can be obtained and monitored in numerical during task operation. This paper establishes a universal model for motion reliability of a particular path planning task. The model which takes the factors related to position accuracy (such as clearance, wear, and friction) into account can reflect the contribution of these factors to motion reliability. Meanwhile, motion reliability can be evaluated by calculating the model. For model calculation, a model simplification method based on the approach of the equivalent extreme value is introduced. With this method, the multidimensional integral for PDF can be converted into one dimensional integral for PDF. Further, a method based on the maximum entropy principle is proposed to obtain the PDF. This method needs less samples and computation cost than the traditional Monte Carlo method, which makes it successful to achieve motion reliability evaluation with high precision and efficiency.
In the paper, the authors mean to establish a motion reliability model which can reflect the influence of factors related to position accuracy, such as clearance, wear, and friction,. Since it is difficult to derive the relationship between these factors and position accuracy, kinematics parameters are considered to be the intermediate variables to deliver the influence of the factors. Thus, the relationship between deviations of kinematics reliability and position accuracy is derived, based on which the model is established. However, the relationships between factors and kinematics parameters are not discussed in the paper, which should be devoted to further research.
From the model and motion reliability evaluation results, the contribution of each kinematics parameter to motion reliability can be concluded. According to the contribution, the importance of each kinematics parameter in improving motion reliability can be decided. When improving the motion reliability of manipulator, factors with high contribution should be given priority to handle. For example, in Section
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the National Key Basic Research Program of China (2013CB733000), the National Natural Science Foundation of China (61175080), and Specialized Research Fund for the Doctoral Program of Higher Education (20120005120004).