Force/Moment Isotropy of 8/4-4 Parallel Six-Axis Force Sensor Based on Performance Atlases

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College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P.R.China

Abstract: A six-axis force sensor with parallel 8/4-4 structure is introduced and its measurement principle is analyzed. Based on condition numbers of Jacobian matrix spectral norm of the sensor, the relationship between the force and moment isotropy and some structural parameters is deduced. Orthogonal test methods are used to determine the degree of primary and secondary factors that have significant effect on sensor characteristics. Furthermore, the relationship between each performance index and the structural parameters of the sensor is analyzed by the method of the atlas, which lays a foundation for structural optimization design of the force sensor.

Key words: six-axis force sensor; Jacobian matrix; condition number; isotropy; orthogonal test; indices atlases

0 Introduction

Six-axis force sensors can measure three-axis force and three-axis moment information in space at the same time and have widespread applications in various fields such as national defense science and technology[1], automotive electronics, robotics[2,3], automation[4]and machining[5].

From the elastomer material, six-axis force sensors can be divided into piezoelectric quartz crystal type[6], strain gauge type[7,8]and so on. From the overall structure of view, six-axis force sensors can be divided into Stewart-type parallel structure[9-11], cross beam structure[12-15], spoke type structure[8, 16]and so on. The isotropy of force and moment is the vital performance evaluation of six-axis force sensors[17]. The six-axis force transducer with a high isotropy of force and moment has the advantages of the same performance index and the same measurement accuracy in each direction. Masaru et al.[18]defined the isotropic evaluation index of a six-axis force sensor as the condition number of the Jacobian matrix spectral norm. Jin et al.[19]defined the translational stiffness and torsional stiffness of the sensor and analyzed the isotropy of the sensor in its solution space through the indices atlases. Yao et al.[17]put forward a kind of overall preloading six-axis force sensor, which can improve the dynamic stiffness of the sensor. Yao et al.[20]analyzed the forward and reverse isotropy of the six-branch Stewart six-axis force sensor by mathematical analysis method, and concluded that this sensor can’t satisfy the requirement that both the reversed force and the reversed moment are isotropic at the same time. Tong et al.[21-23]described the Gough-Stewart parallel force sensor with single-leaf hyperboloid and composite single-leaf hyperboloid, and gave closed and analytic quasi-isotropic mathematical expressions on the premise of orthogonal characteristics. Jia, Li et al.[6,24]analyzed the isotropy of six-axis heavy force sensor, and combined the atlases analysis and genetic algorithm to optimize the structural parameters of the sensor. However the structural parameters of the spectrum analysis are less than those of all structural parameters, which makes structure optimization insufficient.

In this paper, an 8/4-4 structure parallel six-axis force sensor is introduced, and it has the advantage of achieving force and moment isotropy simultaneously by parameter optimization. Based on the principle of minimum number of condition numbers of the Jacobian matrix spectral norm, the isotropy of force and moment of the sensor is expressed. Through the numerical optimization method, the optimization of structural parameters of the force sensor under optimal isotropic performance is solved. Orthogonal experiments are used to determine the degree of primary and secondary factors that achieve a significant influence on the performance index of the sensor, and then the influence of structural parameters changes on the performance index of sensor is analyzed clearly by the method of spectrum optimization.

1 Measurement Principle of 8/4-4 Parallel Six-Axis Force Sensor

1.1 Mathematical model

Fig.1 Schematic diagram of 8/4-4 six-axis force sensor

Specifying the force and moment applied to the upper platform asFandM, the static balance equations for the upper platform can be expressed as

(1)

(2)

Eqs. (1) and (2) can be rewritten in the form of a matrix, which is

FW=J·f

(3)

where

Jacobian matrixJis shown as

(4)

1.2 Isotropic indicators of force and moment

Isotropy of force and moment is an important performance evaluation index of six-axis force sensors. The Jacobian matrixJcan be expressed in the form of [JFJM]Tbecause of different dimensions of the first three lines and the last three lines. From Eq. (3), we know that the smaller the condition number of Jacobian matrixJis, the better the numerical stability could be. The smaller the influence of the small change offon theFWis, the less influence of various errors in all directions on the measurement result can get. The better the force transmission performance of the sensor is, that is, physical properties of the sensor tend to be consistent in all directions, the better the isotropic degree of force and moment will be.

According to the minimum principle of conditional number of Jacobian matrix spectral norm, the isotropic degrees of force and momentμFandμMare expressed respectively as the reciprocal of the conditional number of Jacobian matrix of force and moment, i.e.

(5)

In order to make the sensor isotropic, the sensor should be structural symmetric, the isotropic degree of force and moment is 1, and the condition number of Jacobian matrix is also 1 at this moment. That is, cond(JF)=cond(JM)=1.JJTmust be a generally diagonal matrix, which can be expressed as

(6)

(7)

(8)

WhenγF1=γF2=γF3,γM1=γM2=γM3, the sensor satisfies the force and moment isotropy. At this moment, both condition numbers ofJFandJMare equal to 1.

SimplifyingJJTas

(9)

where

FromX1=X2=0，RB1andRB2can be respectively expressed as

(10)

SubstitutingRB1andRB2intoγM1-γM2=0, we can obtain

(11)

Thus,hcan be expressed as

(12)

SubstitutingRB1,RB2andhintoγF1-γF2=0,we can obtain

(13)

In Eq.(13)，cos(2θ1-2θ2)-1≠0，that is，θ1≠θ2.

It can be seen that, given the range ofRaandθ2, Eq.(13) can be solved by the numerical method, andθ1,Raandθ2can be brought into the expression ofRB1,RB2andh. Thus all structural parameters of the sensor meeting the force and moment isotropic can be gained.

2 Determination of Main Relation of Each Parameter Based on Orthogonal Experiment

From Eqs.(5-9), it can be seen that the force and moment isotropy of the sensor are related to multiple factors of the sensor, includingRa,RB1,RB2,h,θ1andθ2. In order to study the degree of primary and secondary factors that have a significant influence on the sensor characteristics, the method of orthogonal experiment[25,26]is used in combination with the experiment.

2.1 Orthogonal design

The orthogonal experiment includes six factors, which areRa,RB1,RB2,h,θ1andθ2, respectively. The last blank column is left as the error column. Four levels are selected in each factor[27]. According to the limited physical size of the sensor, the level parameter settings of the orthogonal test are shown in Table 1.

Table ofL32(47) is selected for orthogonal experiment design. Tables 2 and 3 give only the first six experimental data. The combination of structural parameters for each test case is expressed in terms of levels. There are four levels for each factor, and each level appears eight times in 32 trials.

Table 1 Orthogonal factor level table

2.2 Analysis of orthogonal experiment results

In this example, within the given range of each factor level,Ra(factor A),RB1(factor B),RB2(factor C),h(factor D),θ1(factor E) andθ2(factor F), the order of the six factors that influence the force isotropicμFcan be arranged(primary→secondary): D，A，E，B，F(xiàn)，C. Using the same method to make orthogonal experiment onμM, the main and secondary order of factors that influence the moment isotropicμMcan be arranged (primary→secondary): D,E,B,C,F,A.

Since the factor A is limited strictly in practice and cannot be changed within a wide range, it has a weak influence even though it ranked in front on μF. From the above analysis, it is found that the influence of factors B, D and E (the corresponding structural parameters ofRB1,handθ1) onμF,μMis the most significant.

Table 2 Orthogonal experiment designs table of μF

Table 3 Orthogonal experiment design table of μM

Performance atlases from the overall point of view more vividly show the distribution of the various structural parameters of the sensor on the performance index. Therefore, with the help of space model theory[26], some atlases of the variation of the sensor’s performance index with structural parameters (RB1,handθ1) are plotted on the plane in Section 3.

3 Performance Atlases

LetRB2=Pd·RB1(Pdis a constant, and specified to be 0.38 as an example), the parameter ranges are limited to 0.1m

Based on the analysis of the performance atlases, we can get the following results:

(1) When the sensor height needs to be as small as possible, a smallerθ2-θ1can be taken.

(2) Sensor structure parameters should be designed by selecting areas where the contour changes are relatively flat.

(3) For force isotropic design,θ<20°andRB1<0.16 m should be selected andθ2-θ1should be increased moderately. The bright color region broadens, making the optional parameter range of the sensor larger. For moment isotropic design,θ1can be chosen between -9° and -45°, andRB1can be chosen between 0.12 m and 0.14 m. IfRB1needs to be smaller,θ2-θ1can be increased appropriately.

Fig. 2 Force isotropic atlas μF(θ2-θ1=20°) Fig. 3 Moment isotropic atlas μM(θ2-θ1=20°) Fig. 4 Comprehensive atlas of μF and μM(θ2-θ1=20°)

Fig.5 Force isotropic atlas μF(θ2-θ1=40°) Fig. 6 Moment isotropic atlas μM(θ2-θ1=40°) Fig.7 Comprehensive atlas of μF and μM(θ2-θ1=40°)

Fig.8 Force isotropic atlas μF(θ2-θ1=50°) Fig. 9 Moment isotropic atlas μM(θ2-θ1=50°) Fig.10 Comprehensive atlas of μF and μM(θ2-θ1=50°)

(4) Aiming at the higher design requirements of force and moment isotropy, the structure parameter selection should be in the region “a” shown in Figs. 4, 7 and 10. The isotropy of the force and moment in region “a” is greater than 0.8, and that in region “b” is not greater than 0.8.

4 Simulation Verification

Fig. 11 Comprehensive atlas of μF and μM(θ2-θ1=31°)

According to the optimization method mentioned in Section 3, by combining the orthogonal experiment analysis and the performance atlases as shown in Fig.11, a set of sensor structural parameters with excellent force and moment isotropy is selected as follows.

Ra=0.1m;RB1=0.15m;RB2=0.057m;

h=0.04m;θ1=-11°;θ2=20°

(14)

Bringing these structural parameters into the theoretical model of Section 1, the force and moment isotropy of the sensor are calculated as

μF=0.998 7,μM=0.990 2

(15)

The load range of the sensor isFx=Fy=Fz=1 kN andMx=My=Mz=100 Nm. Replacing each branch of the sensor with a spring, a virtual prototype of the 8/4-4 parallel six-axis force sensor is built in automatic dynamic analysis of mechanical systems(ADAMS). Now the lower platform of the sensor is fixed, and the generalized six-axis force in each direction is applied to the center point of the upper platform. Each force component is loaded successively from small to large. After reaching the maximum value, it is unloaded from large to small, and the eight branching forces of the sensor in the ADAMS simulation are recorded. According to the method in Ref.[28], the calibration matrixGacan be calculated as

(16)

According to Eq.(5), the force and moment isotropy of the six-axis force sensor obtained by ADAMS simulation can be calculated as

μFa=0.996 2,μMa=0.999 7

(17)

It can be seen from the comparison between Eq. (15) and Eq. (17) that the isotropic degrees of force and moment calculated according to the simulation and the theoretical model are very approximate. The mathematical model and the force and torque isotropic optimization algorithm of the 8/4-4 parallel six-axis force sensor are proved to be correct and accurate.

5 Conclusions

In this paper, the isotropic properties of force and moment are analyzed based on 8/4-4 parallel six-axis force sensor. A mathematical model of 8/4-4 parallel six-axis force sensor is established. Based on the definition of each performance index of sensor, the numerical optimization method is used to select the combination of six-axis force sensor structure parameters that meet the requirements of performance index. Through orthogonal experiment, the primary and secondary influences of each structural parameter on the six performance indexes are analyzed, and then the performance model is drawn and analyzed by using the spatial model theory. From the performance atlases of sensor, the regularity between the isotropy of the force and moment and the structural parameters of the sensor is analyzed, which lays a good foundation for the reasonable selection of the sensor parameters with the expected performance. The accuracy of mathematical model and optimization algorithm of the 8/4-4 parallel six-axis force sensor are confirmed by ADAMS simulation. The conclusions based on the performance atlas can be summarized as follows:

(1) The general trend of the isotropic curve is that the smallerθ2-θ1is, the smaller the distance between two platforms can get when the force and the moment reach isotropy.

(2) Whenθ2-θ1is large,his between 0.03 m and 0.05 m, and the contour is relatively flat, which shows that the slight change of structural parameters in this area has little effect on the performance index. Designing in this area allows a slightly larger tolerance to structural parameters and helps to reduce processing and assembly costs.

(3) When the force reaches isotropy, the distance between two platforms increases generally, which is in favor of sensor processing. When the moment reaches isotropy,RB1generally shows a decreasing trend, which is beneficial to make the sensor compact.

Acknowledgements

This work was supported by the Open Foundation of Graduate Innovation Base (Laboratory) of Nanjing University of Aeronautics and Astronautics (No.kfjj20170512) and the National Natural Science Foundation of China (No.51175263).