，Jian Ding，，Yingxue Yao，Zhaohong Yi，Huaijing Jingand Honggen Fang

（1．School of Mechatronics Engineering，Harbin Institute of Technology，Harbin 150001，China；2．Shanghai Aerospace Equipments Manufacturer，Shanghai 200245，China）

Accuracy Analysis for 6-DOF PKM with Sobol Sequence Based Quasi Monte Carlo Method

Jianguang Li1，Jian Ding1?，Lijie Guo2，Yingxue Yao1，Zhaohong Yi1，Huaijing Jing2and Honggen Fang2

（1．School of Mechatronics Engineering，Harbin Institute of Technology，Harbin 150001，China；2．Shanghai Aerospace Equipments Manufacturer，Shanghai 200245，China）

To improve the precisions of pose error analysis for 6-dof parallel kinematic mechanism（PKM）during assembly quality control，a Sobol sequence based on Quasi Monte Carlo（QMC）method is introduced and implemented in pose accuracy analysis for the PKM in this paper.The Sobol sequence based on Quasi Monte Carlo with the regularity and uniformity of samples in high dimensions，can prevail traditional Monte Carlo method with up to 98.59%and 98.25%enhancement for computational precision of pose error statistics. Then a PKM tolerance design system integrating this method is developed and with it pose error distributions of the PKM within a prescribed workspace are finally obtained and analyzed.

accuracy analysis；quasi monte carlo（QMC）；sobol sequence；parallel kinematic mechanism（PKM）

1 Introduction

The accuracy of positioning and orientation of a 6-dof PKM is a critically important index in the procedure of assembly quality control.How to precisely evaluate pose errors influenced by source errors and obtain distributions of pose errors are main aspects in accuracy analysis.

Statistical method with MC simulation dominates researches in this area，the basis of which is to create a model that contains all the structural errors inherent in the model and the solution to the equations is obtained with MC simulation.Wang［1］derived an affined transformation from the stationary frame to the movable frame for a Stewart Platform including all sources of errors.Zhao［2］investigated the effects of structure manufacturing tolerance and actuating error of a serialparallel machine tool with linkage kinematics analysis and differential vector methods respectively.With the simulation results，they found that the sliding errors and structural length errors are significant factors. Based on statistical method，Zhu［3-4］found a probability density function（PDF）for end pose of a robot，which was derived from joint deviation with specific distribution pattern（including normal and uniform）.They also investigated the worst positioning error due to joint clearance in a single loop linkage，and obtained the maximum error when all the clearance link were collinear by modeling the joint clearance as a small link.Huang［5］investigated the influence of joint clearance on positioning accuracy of a 2-dof parallel robot，by assuming clearance distributions a certain distribution and utilizing redundant constraints to reduce harmful clearance effects.

Monte Carlo based on statistical method has long been accepted in engineering for widely recognized due to its good reliability.However，its huge computation burden within a given precision forms a general barrier for further application.For a prescribed precision，the runs have to be increased exponentially as the problem dimension increases.Sometimes the work becomes unaffordable.So，researchers turn to QMC based on methods which is gradually developed in recent decade. Zhou and Huang［6-8］proposed a NT-net based on QMC method for tolerance analysis and believed that it possesses potential advantages in various design and manufacturing areas，Their research on problem involving no more than 10 dimensions yields a 90%-95%promotion of computational precision.However，for problems with high dimensional analysis，it cannot provide more precision results over traditional MC simulation method in practice，and the accuracy analysis for 6-dof parallel kinematic mechanism with upto 42 dimensions in this paper is very case.Thus，the QMC based on method should be investigated further before application.

In this research，the architecture and error model of a 6-dof PKM is firstly introduced，and then a Sobol based on QMC method is investigated and implemented in accuracy analysis for PKM.Finally，a tolerance design system for 6-dof PKM，integrating this method is developed，and pose error distribution of the mechanism with this system is obtained.

2 Architecture and Error Modeling of PKM

This mechanical architecture of the considered PKM in Fig.1 is used to dock spacecrafts on obit，as a particular 6-dof PKM，it can be divided into two related parts：a typical 6-dof architecture and a set of translational chain system.The orientation and position of the upper moving platform is controlled by six screw links，and each link is also driven by the lower translational chain involving a series of gear pairs and motors.However，due to source error accumulation from translational chain system，as the end part of chain，the pose errors of upper moving platform may be out of the prescribed accuracy.An effective method to solve this problem is to figure out all the main source structural errors and influences on the pose error.Once the source errors are under control in the manufacturing and assembly process，the pose errors can be guaranteed.

Fig.1 PKM used as docking mechanism

Fig.1（b）shows the simplified architecture of 6-dof PKM.A coordinates｛A｝is attached at the center of the upper moving platform，and with itsZaxis that perpendicular to the platform.The coordinates｛B｝is located at the center of the base platform.Both platforms are connected with 6 links and 12 ball joints. A closed loop vector chain can be obtained as follows：

where，Ai，Bidenote centers of thei-th joint connecting both platforms，andLiis a vector fromBitoAi，RandPare definitions as orientation and translation of｛B｝with respect to｛A｝．Eq.（1）can be expressed as functions as follows：

where，x，y，z，α，β，γdenote center position and orientation of the moving platform.The error model is yield by deviation Eq.（2），and can be got as Eq.（3），and represents in matrix form as Eq.（4）

There are forty two kinematic parameters as source errors including 6 link errors dLi，and 36 joint location errors dAix，dAiy，dAiz，dBix，dBiy，dBiz，i＝1，2，…，6.When the pose of the PKM is given，Eq.（4）becomes a determined tolerance system，which reflects the relation between the output and input errors.

3 Sobol Sequence Based QMC Method

A Quasi-Monte Carlo method is introduced to deal with multi-dimensional simulation，since it leads significant improvement in precision than that with Monte Carlo method.The core is that random sequence applied in standard Monte Carlo is replaced with deterministic sequence，named low discrepancysequence（LD sequence），the deterministic sequence have the property that the sampled points are evenly dispersed throughout the domain of desired cubic space.Theoretical research［9-10］proved that the QMC can attain a convergence rate ofO（N-1logsN）in dimensions，which is better than MC only ifNgrows exponentially with dimensions．

3.1 Sobol Sequence Construction

There are several kinds of LD sequence such as Halton［11］，Hua［12］，Hammersley，Niederreiter［13］，Sobol［14］sequence.Among these sequences，Sobol sequence may provide better performance in high dimensional simulation than the other ones［15］，which is constructed with a set of direction numbervi，

where，miis an odd integer，direction numberviderives from a primitive polynomialP（x），

whereai（1＜i＜d-1）as coefficients ofP（x）with degreedare 0 or 1，for a problem withsdimensions，Sobol suggests multiple sequences can be constructed withSdifferentaito achieveO（logsN）discrepancy.

Eq.（7）is then used to yield direction numberviafter determiningP（x）：

where⊕denotes a bit-by-bit exclusive or operation. For instance，1⊕0＝1，0⊕1＝1，0⊕0＝0，1⊕1＝1. Thenmican be determined by Eq.（8），

Using a primitive polynomial of degreed，the value ofm1，m2，…，mdcan be freely given on condition thatmiis odd andmi≤2i．

For instance，the initial values are set asm1＝1，m2＝3 andm3＝7，choose ap（x）of degree 3，

The coefficients ofp（x）area1＝0，a2＝1 anda3＝1，and thenmican be obtained from Eq.（10）

Thus，m4＝12⊕8⊕1＝［1100］2⊕［1000］2⊕［0001］2＝［0101］2＝5，sincevi＝mi／2i，and thenv4＝0.010 1，correspondingly，

Finally，then-th value of Sobol sequence can be denoted as

where，…b3b2b1is the binary form ofn．For more uniformity of the sequence，it is suggested that Sobol sequence should begin withb4-1，whereinbis then-th prime，andnis the dimension of the problem.In this problem，the mechanism involves 42 dimensions，thus，nis 42，and thenbis 181，and the sequence started with 1814-1.

The layout of Sobol sequence are more regularly and uniformly distributed over the united square than random points，while other LD points in high dimension projection exhibits either clustering or relative sparsely.The enhancement uniformity of the points can lead to a higher rate of convergence as the dimension increases，as shown in Fig.2.

Fig.2 Two-dimensional projection of random and Sobol sequence

3.2 Analysis Procedure

1）Sobol sequence with 42 dimensions（Si（1），Si（2），…，Si（42））are produced within a super cubic［0，1］42，n＝1，2，…，N，whereNdenotes sampling scale；

2）Transform the sequence into sampling points in terms of intervals of input errors；

3）Obtain the output results as pose errors at a given pose configuration of the mechanism；

從道路運輸行業(yè)安全監(jiān)管來說，存在著以下問題：①安全監(jiān)管邊界劃分不明晰。大多數(shù)交通運輸主管部門以及道路運輸管理機構(gòu)，沒有做好安全監(jiān)管法律法規(guī)等的全面梳理，沒有明晰安全監(jiān)管工作邊界，比如行業(yè)監(jiān)管和專項監(jiān)管邊界等，使得監(jiān)管責任不明晰、邊界模糊問題，常見越位情況或者缺位情況，無法落實行業(yè)監(jiān)管責任。②事前預防、事中監(jiān)管以及事后應(yīng)急轉(zhuǎn)變的理念缺乏，難以擴大道路運輸安全監(jiān)管范圍。③道路運輸安全監(jiān)管手段不夠全面。從實際情況來說，監(jiān)管手段比較少，加之重點不突出，缺乏強有力的方法，難以保證監(jiān)管工作的高質(zhì)量落實。

4）Compute statistics（mean valueμand standarddeviation（STD）σ）of pose error components.

3.3 Verifications

Eq.（4）gives the relation between pose errors and source errors.The QMC method is implemented in accuracy analysis in contrast with MC method in this section.The structure parameters of the PKM are listed in Tables 1 and 2.All the source errors are assumed with toleranceT（±0.5 mm）conform to uniform distribution，so the variance of each error isT2／12．Since tolerances of errors are symmetrically distributed，the mean valuesμof all the pose errors are zeros according to Eq.（4）.The source errors are independent with each other，thus，the covariance values are zeros，and the variance of pose errors can be formulated as follows：

Table 1 Joint locations of the moving platform（mm）

Table 2 Joint locations of the base platform（mm）

3.3.1 Pose 1

The selected pose is listed in Table 3，and the theoretical values can be got from Eq.（12）and listed in Tables 4 and 5.Pose error estimations of mean values and STDs with different sample size（1 000，2 000，3 000，4 000 and 5 000）and with Sobol sequence based on QMC method and MC based method are shown in Fig.3 and partly listed in Tables 6 and 7.

Table 3 Pose of the PKM

Table 4 Mean valuesμof pose errors

Table 5 STDsσof pose errors

Table 6 Mean valuesμof pose errors with MC and QMC in comparison

Table 7 STDsσof pose errors with MC and QMC in comparison

Fig.3 Convergence contrast with MC and QMC method on pose errors

Convergence in Fig.3 shows that the convergence rates of mean valueμand STDσwith Sobol sequence based on QMC method are steadier than that with MC based method.Taken the results with a sample size of 5 000 as examples，the enhancements in computational precision comparatively with MC and QMC are tabulated in Tables 8 and 9.

Table 8 Computational precision enhancement for mean valueμ%

Table 9 Computational precision enhancement for STDσ%

3.3.2 Pose 2

This selected pose is not symmetric and parameters are listed in Table 10，and the theoretical values are computed and listed in Tables 11 and 12. Pose error estimations of mean valueμand STDσwith different sample size（1 000，2 000，3 000，4 000 and 5 000）and with Sobol sequence based on QMC method and MC based method are shown in Fig.4 and partly listed in Tables 13 and 14.

Fig.4 Convergence contrast with MC and QMC method on pose errors

Table 10 Pose of the PKM

Table 11 Mean values μ of the pose errors

Table 12 STDs σ of the pose errors

Table 13 Mean values μ of pose errors with MC and QMC in comparison

Table 14 STDsσof pose errors with MC and QMC in comparison

Convergence curves in Fig.4 shows that the convergence rates of mean valueμand STDσwith Sobol sequence based on QMC method are steadier than that with MC based method.These results are in accordance with those in Fig.3.The superiority of QMC based accuracy analysis to the MC method in convergence precision has been verified till now.Taken the results with a sample size of 5 000 as examples，the enhancements in computational precision comparatively with MC and QMC are tabulated in Tables 15 and 16.

Table 15 Computational precision enhancement for mean valueμ%

Table 16 Computational precision enhancement for STDσ%

4 System Development

The PKM tolerance design system consists of four main modules：parameters setting module，kinematics simulation module，accuracy analysis module and tolerance synthesis module.The main interface is shown in Fig.5.The parameters setting module is used to input the structure and pose parameters of the robot investigated，and also for accuracy analysis and tolerance synthesis.The kinematics simulation module provides a vivid kinematics animation of the mechanism for user with a better understanding.

Fig.6 shows the interface of accuracy analysis；and the left side items are used to input the scale of 7 types of source errors involving dL，dAx，dAy，dAz，dBx，dBy，dBz，and the button‘Accuracy Analysis’as a main function carries out QMC simulation，and then output the errors distribution patterns of the moving platform.The end of the interface shows the statistic characters of all pose errors（mean values and STDs）which reflect the width［μΔ-6σΔ，μΔ+6σΔ］．

Fig.5 Interface of the PKM tolerance design system

Fig.6 Interface of accuracy analysis module

5 Application

Pose error distribution of the moving platform within prescribed working space is important in accuracy analysis，predictions and dimensional synthesis of a 6-dof docking mechanism.In general，statistical approach is more reliable than other ones. The moving platform is parallel to the base platform，the working space is a square plane region ofxandyis［-100，100］mm×［-100，100］mm and the heightzis 1 500 mm.The distribution of pose errors is shown in Fig.7.

In Fig.7，the position errorsxandyincrease as the moving platform distances away from original symmetrical pose，and are larger than position errorz，and the orientation errorγis larger than of errorαandβ．

Fig.7 STD distributions of pose errors at the level of z＝1 500 mm

6 Conclusions

A Sobol sequence based on QMC method is proposed focusing on high dimensional simulation problem（d＞10）and implemented for accuracy analysis of a 6-dof PKM.Comparison shows that the convergence rates with Sobol sequence based on QMC method are steadier than that with MC based method. The precision enhancements in mean value and STD of the pose errors range from 92.34%to 99.54%and from 42.2%to 98.25%respectively，and it indicates that the proposed method allows for achieving the same precision with a relatively smaller sample size in comparison with MC based method.

A tolerance design system integrating this method is then developed，and with it，pose error distribution of the PKM is analyzed within a section of work space. Results show that the position errorxandyare symmetrically distributed and larger than errorz，and the orientation errorγis larger than errorαandβ．Now the system has been put into use in Shanghai Aerospace Equipments Manufacturer.

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TH164

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：1005-9113（2015）05-0001-08

10.11916／j.issn.1005-9113.2015.05.001

2014-09-05.

Sponsored by the National Defense Basic Scientific Research Program（Grant No.A0320110019），and the Shanghai Science and Technology Innovation Action Plan（Grant No.11DZ1120800）.

?Corresponding author.E-mail：dingjianhit＠126.com.