Performance Analysis and Optimization of Sizable 6-axis Force Sensor Based on Stewart Platform

To improve the performances of the parallel 6-axis force sensor with Stewart platform, the paper plots the indices atlases based on the screw theory and the theory of physical model of the solution space, and summaries the laws how the structure parameters affect the indices. Then, the sizable parallel 6-axis force sensor's structure parameters are optimized in nonlinear single objective and multi-objective respectively, in order to obtain better performances than that of the initial ones. So, the powerful basis and method are raised for design and optimization of the sizable parallel Stewart platform 6-axis force sensor with general spherical joint.


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presents the static mathematics model of the 6-axis force sensor with screw theory. The static force influence coefficient matrix and the generalized force Jacobian matrix of the 6axis force sensor are derived. Based on the screw theory and the theory of physical model of the solution space, some performances indices are defined. The force isotropy, torque isotropy, force sensitivity isotropy and torque sensitivity isotropy indices atlases of the 6axis force sensor are plotted, and the rules how structure parameters affect the performances indices are summarized in Section 3. The optimization method of sizable parallel 6-axis force sensor's structure parameters is proposed, and an optimization numerical example is demonstrated in nonlinear single objective and multi-objective in section 4, respectively. Based on the result of the performances analysis and optimization, the section 5 presents a novel sizable 6-axis force sensor with flexible joints, which can avoid effectively the friction and the clearance in general spherical joint and has a wider application foreground. The research result reported of the chapter is concluded in section 6, future research in section 7, acknowledgement in section 8, and references in section 9.

Static mathematics model of 6-axis force sensor
The Stewart platform 6-axis force sensor is a kind of special parallel mechanism that is symmetrical design. Fig.1 is the sketch of the mechanism and forces acted on the platform. The platforms of the upper and lower platform are shown in Fig.2 The former three rows of the matrix G is the force transmitting factor of the parallel sensor, while the latter three rows is the torque-transmitting factor. The factors having different unit, which the former is dimensionless, while the latter has length unit, the matrix G is disintegrated into the static force influence coefficient matrix 1 G and the static torque where, 1 − = J G is the generalized force Jacobian matrix of the parallel 6-axis sensor. Similarly, the generalized force Jacobian matrix J is disintegrated into the force Jacobian matrix 1 J and the torque Jacobian matrix 2 J , that is

Physical model of the solution space theory
The physical model of the solution space theory has the ability to show all possible size combination of the mechanism. It is convenient to obtain the law of the sensor's indices following the changing of the element structure parameters. From the static mathematics model of the force sensor above, the 6-axis force sensor based on Stewart platform contains four structure parameters. That is the radius a R of the upper platform, the radius b R of the lower platform, the height H between platforms, and the angle difference ab between the corresponding twin link of the upper and the lower platform. With the precondition of ab θ is changeless, let ab RRHT + +=, then (5) gives Therefore, all possible parameters combination of the 6-axis force sensor based on Stewart platform are included in the triangle ocd. In other words, each point in the triangle ocd corresponds with a set of structure parameters. With the physical model of the solution space theory, selecting parameters and optimization structure design are convenient greatly.

Performances atlases analysis
The indices evaluating the performances of the 6-axis force sensor are the foundation of the performance evaluating and the optimization design. As for the parallel 6-axis force sensor, it should have high force isotropy, torque isotropy, and force/torque (F/T) sensitivity isotropy, in addition to the high sensitivity, precision, signal noise ratio (SNR) and speedy response. The performances atlases are plotted in the area of the physical model triangle ocd, based on the static mathematics model above and the defining of the performances indices given by Uchiyama and Hakomori (1985), Xiong (1996) and Jin (2003)  Considering the latter optimization design of the structure parameters, the performances spacial and planar atlases are plotted as shown in Fig. 4-11, respectively, when the coordinate system fixed on the center point of the lower platform and 60 ab θ = . It can be easily gotten the indices distributing laws with the performances atlases of force isotropy, torque isotropy, force sensitivity isotropy and torque sensitivity isotropy, especially in the planar atlases of as shown in Fig.5, Fig.7, Fig.9 and Fig.11. From the influence that the structure parameters act on the sensor's performances indices shown in Fig. 4-11, the laws guiding the optimization design can be concluded as following. The plot of the force isotropy distributes parabola approximately in the area of the physical model as shown in Fig. 4 and Fig. 5. The force isotropy will becomes higher in the middle and lower area of the physical model. The corresponding structure parameters can be selected, when the index of the force isotropy should be attached importance to design.

Performance Analysis and Optimization of Sizable 6-axis Force Sensor Based on Stewart Platform 211
The plot of the torque isotropy distributes beeline approximately in the area of the physical model as shown in Fig. 6 and Fig. 7. The torque isotropy will becomes lower in the right side and upper area of the physical model. The corresponding structure parameters should be eliminated, when the index of the torque isotropy should be attached importance to design. The plot of the force sensitivity isotropy distributes beeline approximately in the area of the physical model as shown in Fig. 8 and Fig.9. The force sensitivity isotropy will change rapidly by the x axis in the physical model. The corresponding structure parameters should be eliminated in design. In the upper most area, the index of the force sensitivity isotropy is smaller. The force sensitivity isotropy distributing resembles the torque isotropy distributing of the force sensor. The plot of the torque sensitivity isotropy distributes parabola approximately in the area of the physical model as shown in Fig. 10 and Fig. 11. The torque sensitivity isotropy will becomes higher in the middle part of the physical model. The corresponding structure parameters can be selected, when the index should be attached importance to design.

Optimization design of sizable parallel 6-axis force sensor 4.1 Optimization objective function
In the sensor's practical application, the request for the performances indices varies with the practical application cases. Some performance index should be considered principally in some cases, while the comprehensive performance index is pivotal in some cases. The paper optimizes the existing sensor's structure parameters in nonlinear single objective and multiobjective respectively, in order to obtain better performances than that of the initial ones. As the restriction of mechanical special model, the constraint equation 0≤ ab θ ≤120° should be applied. In the single objective optimum, the objective functions are chosen as following The objective functions in the above equation (8)-(11) are the reciprocals of the force isotropy u 1 , the torque isotropy u 2 , the force sensitivity isotropy u 3 and the torque sensitivity isotropy u 4 , respectively. When the objective function reaches the minimum, the corresponding performance index attains the maximum. When the comprehensive performance is pivotal, the multi-objective optimum would be executed to obtain the sensor with the high performances. The corresponding objective function can be expressed as

Optimization numerical examples
Considering the practical structure parameters of the sizable parallel 6-axis force sensor, the initial parameters are set as In the multi-objective optimization, the comprehensive performances indices should be taken into account synthetically. With the weight matrix k= [1111], the optimal performance indices with respect to the comprehensive parameters and the corresponding initial indices are shown in Table 1. It is obvious that the performances of the sizable parallel 6-axis force sensor are improved. The corresponding performance indices of the initial structure parameters in Table 1 are shown in the planar atlas Fig. 5, Fig. 7, Fig. 9, and Fig. 11 with the " * " symbol.

A novel sizable 6-axis force sensor with flexible joints
Based on the above analysis, optimization design and considering the machining technics synchronously, we design the a novel sizable 6-axis force sensor structure with flexible joints as shown in Fig. 12. Each branch is composed of UUR flexible joints and a standard pull and press force sensor. The axe of the flexible R joints go through the near U flexible joint, which can be considered as a sphere joint. The flexible joints here are the novel flexible joints which can carry the biggish loading. The six branches are divided the same 3 groups. The first U joints of the branches in some group are made in a whole material, similary as the last R joints of the branches in some group. Another design project with the same 6 unitary branch is shown as in Fig. 13.

Future research
The performance indices of the 6-axis force sensor based on Stewart platform shouled be further analyzed, especially dynamic performance index. The novel sizable 6-axis force sensor with flexible joints should be futher optimized, especially the stucture parameters of the flexible joints. The manufacture and calibration of the sizable 6-axis force sensor with flexible joints are also the future research.

Conclusion
The paper plots the indices atlases based on the screw theory and definition of the performances indices, and summaries the law how structure parameters affect the indices. With the constructed optimization objective functions, the sizable parallel 6-axis force sensor's structure parameters are optimized in nonlinear single objective and multi-objective respectively. The corresponding optimal structure parameters are obtained. A novel sizable 6-axis force sensor with flexible joints is developped. So, the powerful basis and method are raised for design and optimization of sizable parallel 6-axis force sensor based on Stewart platform.

Acknowledgment
The research work reported here is supported by National Natural Science Foundation of China (NSFC) under Grant No.50375134 and No.50675191.