Zhi－Juan Sun，Jing Zhao，Li－Ming Li

(College of Mechanical Engineering and Applied Electronics Technology，Beijing University of Technology，Beijng 100124，China)

1 Introduction

Robotic mechanism is the main part to achieve a variety of assigned tasks，so mechanism analysis and synthesis is the key technology for robot innovation and development［1］.Robotic mechanism analysis always focuses on the relations among the structure，kinematics and dynamics， which can provide theoretical basis for the robotic comprehensive performance evaluation and mechanism synthesis.According to system engineering，robotic mechanism analysis should be based on the working tasks to ensure the system requirements［2］.Recently， mechanism analysis is mainly limited to some mechanisms for specialapplications， and the weights of single performance indexes are always determined by designer’s subjectiveperception.Becauserobotic single performance indexes are complex and diversiform， and there is no comprehensive performance evaluation theory，the relations among robotic comprehensive performance，working tasks，configuration and scales cannot be revealed effectively.Therefore，a general task-oriented method of serial robot for mechanism analysis and evaluation，which reveals the relationsamong robotic comprehensive performance，configuration，scales and working tasks， becomes an international focus of academic research.

Principal component analysis(PCA)and kernel principal component analysis(KPCA)have been respectively used to research on the linearand nonlinearrelations between multiple variables in agriculture， biology， meteorology， demography，economics and other relating fields［3］.In mechanics research，PCA and KPCA have also been successfully applied in machine’s condition monitoring and fault identification for feature extraction capabilities［4－6］.Due to the correlation and diversity of the robotic single performance indexes，PCA and KPCA can be introduced into mechanism analysis and evaluation oriented to working tasks，which may reveal the relations among robotic single performance indexes，configuration，scales and working tasks［7］.However，due to the nonlinear relations among the indexes，the contribution rates of each comprehensive performance indexes calculated by PCA are dispersive，so PCA may not evaluate mechanism comprehensive performance currently［8］.KPCA is an extension of PCA to deal with nonlinear problems.The original variables can be mapped into high-dimensional feature space，and then calculated by PCA.By selecting appropriate kernel function and kernel parameters，the contribution rate of the 1stprincipal component can always reach 85%，and the robotic comprehensive performance can be evaluated by the 1stprincipal component perfectly.Then the robotic configuration，scales and tasks of the best comprehensive performance can be obtained［9］.Compared with othermulti-objective mathematical methods，the data size of single performance indexes is not limited in PCA and KPCA，and the single indexes can be transferred into principal components.Then some redundant information is discarded by linear or nonlineartransformation，and the calculation for comprehensive performance evaluation is easier and the weights of the single indexes are more objective than other multi-objective mathematical methods.

Serial robot’s structure is relatively simple and easy to control，while its workspace is large，so it’s used widely in machine manufacture and assembly.Therefore，series robot is researched in this paper，and the single performance indexes are calculated based on PCA and KPCA respectively.Through comparing the dimension reduction effect，more suitable method for comprehensive performance evaluation is chosen.And then a task-oriented method ofserialrobotfor mechanism analysis and evaluation is proposed.So the robotic configuration，scales and task of the best comprehensive performance can be obtained，which provides scientific research basis for the mechanism synthesis and optimum task order.

2 Principles of PCA and KPCA

2.1 Principle of PCA

PCA is used to replace the primitive multidimensional variables with a small number of new independent synthetical variables (principal components) through the linear transformation (orthogonal).The different principal components have different significances to reveal basic nature of the primitive sample［10］.The calculation method is introduced in Ref.［11］.

2.2 Principle of KPCA

The key idea of KPCA is both intuitive and generic.In general，PCA can only be effectively performed on a set of observations which vary linearly.When the variations are nonlinear，the data can always be mapped into a higher-dimensional space in which they vary linearly.That is，according to support vector machine(SVM)，the nonlinear data structure in the input space is more likely to be linear after highdimensional nonlinear mapping［12］.This higherdimensional linear space is referred to the feature space F.Rdis mapped to feature space F by non-linear mapping function Φ.

The sample points in feature space F are expressed as φ(xi)，which fulfillsKPCA finds a computationally tractable solution through a simple kernel function which intrinsically constructs a nonlinear mapping from the input space to the feature space.As a result，KPCA performs a nonlinear PCA in the input space.The commonly used kernel functions are as follows:

1)Polynomial kernel function:

2)Gauss kernel function:

3)Multilayer perception kernel function:

In Eqs.(1－3)，a，b，c and d are selected parameters.

3 PCA and KPCA of Serial Robot for Mechanism Analysis and Evaluation

In general，the task-oriented method of serial robot for mechanism analysis and evaluation based on PCA is uesd to deal with single performance indexes of different tasks，configuration and scales by PCA.The contribution rate of each principal component reflects the amount of information extracted from the original single indexes.The larger the contribution rate is，the more original information is contained in the principal component.The cumulative contribution rate is generally demanded to reach 85%to ensure enough information ofprincipalcomponents.So the 1stprincipal component，whose contribution rate is greater than 85%，can be considered as a comprehensive performance index with definite mechanism meaning.Further，the relation among robotic single performance indexes，different tasks，configurations and scales can be revealed，and the working task，configuration and scales with bestcomprehensive performance are obtained simultaneously.

When the contribution rate of the 1stprincipal component is smaller than 85%， the single performance indexes’data compression is insufficient.Nonlinear characteristics of the single performance indexes are also difficult to extract，so KPCA is applied to the serial robot’s mechanism analysis and synthesis.Through nonlinear transformation， the original single indexes’data can be mapped into highdimensional feature space，and then calculated by PCA to realize dimensionality reduction effectively.The calculation flowchart is shown in Fig.1.

There are some advantages of this method.

1)The single performance indexes can be selected as many as possible.All sample data can be calculated by PCA or KPCA，and the impacts between single performance indexes can be eliminated.

Fig.1 Flowchart of the task-oriented method of serial robot for mechanism analysis and evaluation

2)After the original sample data of single performance indexes are changed to principal components，weights of the indexes are got to score the comprehensive performance，which are more objective than human decision.

3)The nonlinear relations among single performance indexescan berevealed byKPCA，providing a valid comprehensive performance analysis and evaluation method.

4)Compared with PCA，by selecting appropriate kernel function and its parameters，the 1stprincipal component’s contribution rate of KPCA can always reach 85%，avoiding to evaluating comprehensive performance partly.

3.1 Single Robotic Performance Indexes

The robotic mechanism comprehensive performance is analyzed by PCA or KPCA based on single performance indexes，and the commonly used robotic single performance indexes are condition number(x1)， directional manipulability(x2)，kinematics manipulability (x3)， isotropic index(x4)［14］and other index(x5)［15］.The indexes are shown in Table 1.

Table 1 Commonly used single performance indexes

In addition to the robotic single performance indexes above，according to different tasks，more single indexes can be selected.

3.2 Kernel Function

When KPCA is applied for serial robotic mechanism comprehensive performance evaluation，kernel function and its parameters impact on the results directly.As commonly used kernel functions，polynomial kernel function has good overall properties，which is able to promote the extrapolation.Gauss kernel function is localized，and its inner learning ability is weakened with the increase of the parameter c.Multilayer perception kernel function can realize multilayer perception including a hidden layer［16］.

The contribution rates of principal components have nonlinear relation with the kernel function’s type and parameters.Theprocessto determine kernel function’s type and parameters is also a nonlinear optimization process.So the contribution rate of the 1stprincipal component is the optimization objective，and kernel function’s type and parameters are variables.Then the optimization function is established to determine appropriate kernel function’s type and parameters［17］.

4 PCA for Serial Robot

Robotic single performance indexes are analyzed based on PCA.The relations among the single indexes，configurations，scales and tasks are revealed，and the comprehensive performance is evaluated.

4.1 Typical Robot and Tasks

According to intended working tasks，2 typical robots are selected;PUMA space 3R robot and another space 3R robot are shown in Figs.2 and 3 respectively.

Fig.2 A PUMA space 3R robot

Fig.3 Another space 3R robot

The robots complete circular tasks in 18 different locations，and the coordinates of the circular task’s center is［1.35，1.35，1］，and the radius is 0.35 unit length.The tasks can be described by Eq.(4).

where γiis the angle of each 18 discrete points on the circle in their workspace.

The operation direction of ending actuator is along［1，1，1］.When the ending actuator reaches the locations described by Eq.(4)，there are many different scales of the PUMA space 3R robot and the other space 3R robot which can complete tasks.In Figs.2 and 3，L1is 1 unit length;L2is discrete of 0.1 unit length in the range of 0.6 to 2 units length;while L3is 1 unit length of the other space 3R robot.To ensure the length level of the robots with the two configurations and a variety of scales are all the same in XY plane，the PUMA space 3R robot satisfies L2+ L3=2.6 units length，and the other space 3R robot satisfies L2+L4=2.6 units length.Furthermore，to ensure the 2 robots completing the working tasks described by Eq.(4)，the discrete working points and L2influence each other.When the points are discrete largely in workspace，the changing interval of L2becomes smaller;while the points are discrete densely in workspace，the changing interval of L2becomes larger.

Compared withPUMA space3R robot，the analytical solution of the other space 3R robot’workspace isdifficultto obtain，so the robots’workspaces are drawn with a random scale by Monte Carlo methods［18］.All the tasks and the workspace projected in XY plane are shown in Figs.4 and 5.

Fig.4 Workspace and tasks of PUMA space 3R robot

The sample directly affects the computation time and evaluation result of PCA.There’re no special requirements for the samples in mechanism analysis and evaluation based on PCA.But the samples’sequence should be bounded，while the quantity should be more than 30［19］.So the robotic arm length is discrete of 0.1 unit length，then the values of condition number(x1)， directional manipulability(x2)，kinematics manipulability(x3)，isotropic index(x4) and other index(x5)for the PUMA space 3R robot and the other space 3R robot of 15 different groups of scales according to 18 different tasks described by Eq.(4) can be calculated.All the values of the 540 samples are shown in Table 2.

Fig.5 Workspace and tasks of another space 3R robot

Table 2 Values of robotic single kinematic dexterity indexes

4.2 PCA Result

As condition number(x1)，isotropic index(x4) and other index(x5)are moderate index，the data need to be processed positively before PCA.Due to the different measurement units and changing ranges，the data should be standardized by zero-mean normalization.Then the correlation coefficients between each index should be calculated forPCA.The correlation coefficients between each index are shown in Table 3.

Table 3 Correlation coefficient matrix

As shown in Table 3，each index processed positively has positive correlations.Then the PCA results calculated from correlation coefficient matrix are shown in Table 4.

As shown in Table 4，instead of the 5 single indexes，the 1stprincipal component can be used as a comprehensive performance index reflecting the balance of original single indexes，and also to evaluate the comprehensive performance， which can be calculated as follows:

In Eq.(5)，zxiis the processed data of each single index.The comprehensive performance scores of the 540 samples of different configurations，scales and tasks are calculated，which are shown in Fig.6.The higher the score is， the better comprehensive performance is.Thus the configuration，scales and task of the serialrobotwith the bestcomprehensive performance can be got.

However，as seen in Table 4， the PCA contribution rate of the 1stprincipal component is 74.344%.Therefore Eq.(5)doesn’t contain enough information of the original single indexes，and cannot reflect the comprehensive performance.Thus，KPCA is applied in serial robot’s mechanism evaluation and optimal working task selection.

5 KPCA for Serial Robot

5.1 KPCA Result

Optimization model should be constructed to select properkernelfunction and its parameters.The contribution rate of the 1stprincipal component is the optimization objective，and the kernel function’s type and parameters are variables.In this case，polynomial kernel function with a=3 and b=20 in Eq.(1)is selected.Then the results of KPCA are compared with the PCA results in Table 4.

Table 4 PCA and KPCA results

As shown in Table 4，the KPCA cumulative contribution of the 1stprincipal component rate is 86.477%.The dimension reduction effect of KPCA method is relatively more significant than PCA，and more information of the 1stprincipal component can be retained by KPCA.So the 1stprincipal component calculated by KPCA can be used to evaluate the serial robot’s comprehensive performancecredibly.Then the configuration，scales and task of the serial robot with the best comprehensive performance can be got.

5.2 Compare of KPCA Result and PCA Result

The originaldata ofthe single indexesare projected to the transform space， and then comprehensive performance scores ofrobots with different configurations，scales and tasks can be got by KPCA.The 1stprincipal component scores of KPCA and PCA are compared in Fig.6.

Fig.6 Comprehensive performance scores of PCA and KPCA

As shown in Fig.6，the distribution trends of the comprehensive performance scores of the 540 samples calculated by PCA and KPCA are similar.According to the results，No.192 sample has the best comprehensive performance whileNo.282 samplehas theworst comprehensive performance. It’ s proved the effectiveness and practicality of PCA and KPCA used for serial robot’s comprehensive performance analysis and evaluation.But KPCA can increase the gradient of comprehensive performance scores，especially the best and the worst sample，while the other samples’scores are changing continuously and are relative similar，but these result data are still different to reflect the samples’comprehensive performance.The pros and cons of KPCA line is more apparent in Fig.6，so it’s easy to select the configuration，scales and task of the serial robot with the best comprehensive performance.And the information retained by dimension reduction of KPCA is much more than PCA’s，which mean the KPCA results should be more effective.Furthermore，a special sample interval can be calculated by KPCA for study comprehensive performance.

The No.192 sample with the best comprehensive performance is PUMA space 3R robot for the No.12 task，while L2is 1.6 units length，and L3is 1 unit length，as shown in Fig.7.While No.282 sample with the worst comprehensive performance is the other space 3R robot for the No.12 task，while L2is 0.6 units length，and L3is 2 units length，as shown in Fig.8.The values of single performance indexes of the robots with the best and the worst comprehensive performance are compared in Table 5.

Fig.7 Robot with the best comprehensive performance

Fig.8 Robot with the worst comprehensive performance

Table 5 Single performance indexes values of robots with the best and worst comprehensive performance

As shown in Table 5，all the single performance indexes of No.192 sample are better than No.282 sample’s.While the comprehensive performance score of No.192 sample is significantly higher than No.282 sample’s.This result is not only consistent with the better kinematic dexterity of PUMA space 3R robot than the other space 3R robot for plane tasks，but also consistentwith the betterkinematic dexterity of completing the task in the middle of the workspace.Thus，KPCA is a credible and efficient method to evaluate the serial robot’s comprehensive performance.Then configuration， scales and task with best comprehensive performance can be selected directly.

6 Conclusions

Comprehensive performance of space 3R serial robots with different configurations，scales and required tasks are analyzed and evaluated based on single performance indexes by PCA and KPCA.The correlations among single indexes can be calculated，and the relations among robotic comprehensive performance，configuration，scales and working tasks can also be revealed.Further，the comprehensive performance scoresofserialrobotswith different configurations，scales and working tasks are calculated byPCA and KPCA respectively，and then the comprehensive performance can be measured.Therefore，the effectiveness of PCA and KPCA for comprehensive performance evaluation is proved.

Compared the dimension reductions of PCA and KPCA，it indicates that KPCA can effectively deal with nonlinear relations between the single performance indexes，and provide more information through the 1stprincipal component， so the results are more reasonable.Thus，a task-oriented method of serial robotfor mechanism analysis and evaluation is proposed based on various single performance indexes，which provides scientific research basis for the mechanism synthesis and optimum task order.In future work，PCA and KPCA would be used to evaluate comprehensive performance ofmultiple robots in engineering projects，and the results also would be compared with the global performance to verify the effectiveness.

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