Stiffness analysis and parameter optimization of six-dimensional force sensor with the novel circular flexible spherical joint

As robot haptic provider, six-dimensional force sensors are an indispensable component of industrial intelligence. To address the disadvantages of the existing six-dimensional force sensor, a novel circular flexible spherical joint is proposed, its stiffness model is derived by numerical analysis method and microelement, and the stiffness model of the circular flexible spherical joint is verified by finite element simulation. A parallel six-dimensional force sensor is designed based on a circular flexible spherical joint, and a mathematical model of the sensor was established. Based on the genetic algorithm, the structural optimization design of the circular flexible spherical joint was performed, and the structural parameters of the circular flexible spherical joints are obtained by analyzing isotropy. Finally, according to the optimization results, a sensor was developed and calibration experiments were carried out. The experimental results show that the maximum error of the sensor is 1.85%, which verifies the effectiveness of the mathematical model of the sensor, which has high measurement accuracy and can accurately measure the six-dimensional external force and moment applied to the sensor.


Introduction
With the development of science and technology, sensing technology has penetrated into all aspects of daily life and is widely used in measurement, control and obtaining relevant Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
information [1][2][3][4][5].Among the various forms of sensors, sixdimensional force sensors are a class of electrical components that convert force signals into electrical signals, and can simultaneously measure force and torque information in three directions in space.In the field of robotics, it is precisely because of the six-dimensional force sensor that the robot has a 'sense of touch' [6][7][8].At present, six-dimensional force sensors can be divided into capacitive type, piezoelectric type, photoelectric type and strain gauge type according to different measuring principles.Among the many forms of six-dimensional force transducers, strain gauge six-dimensional force sensors are widely used at this stage due to their long development history [9][10][11].Strain gauge type six-dimensional force sensor is mainly divided into integrated and parallel type, parallel six-dimensional force sensor adopts spherical joint connection mode, the force measuring branch can be regarded as only bear the axial force of the two-force rod, in theory can achieve complete decoupling, at the same time, parallel six-dimensional force sensor also has the advantages of large relative stiffness, high bearing capacity, no cumulative error, etc, has become a very successful six-dimensional force sensor structure implementation form [9,12,13].
In reality, the ideal spherical joint is not easy to achieve, parallel six-dimensional force sensor usually uses ball pair instead, due to the ball pair has processing error, theoretical gap and friction torque and other influencing factors, resulting in parallel structure six-dimensional force sensor also has the influence of interdimensional coupling and the volume of the six-dimensional force sensor is too large.In order to solve the problem caused by the ball pair, in the literature [14], the ball pair is replaced with a tapered ball pair, and a preload device is added.The conical ball head reduces the contact area with the ball socket, and the preload device reduces the gap between the ball pairs, improving the effect of friction/torque and clearance on the accuracy of the sensor, but does not change the problem of excessive sensor size.Literature [12] designed a force/torque sensor based on a Stewart plat-form in a near-singular configuration.They used flexible joints instead of rotary joints and made a finite element analysis.The result showed that a properly designed flexible joint approximated to a rotary joint and could avoid the friction and non-linearity associated with rotary joints.Literature [15] proposed a novel redundant parallel six-component force sensor with spoke structure combining parallel mechanisms with flexible mechanisms and analyzed its performances such as non-linearity, repeatability and hysteresis.Literature [16] designed a three-axial body force sensor for flexible manipulators.The upper plate and bottom plate of the sensor are connected by three flexible cantilever beams.They optimized the parameters of the flexible cantilever beams by Finite Element Method.Literature [17] established a model of the solution space, for all the sensor mechanisms, which is a novel and useful tool for investigation of the optimal sensor design.They proposed replacing the spherical joints with the elastic joints, which could realize the Stewartplatform-based sensor of being designed as small as possible.In the literature [12,[14][15][16], although the effects of machining error, theoretical clearance and friction torque of the ball pair have been improved, the problem of excessive volume of the six-dimensional force sensor has not been improved.In the literature [17], the flexible ball joint solves the problems caused by the ball pair, and can make the six-dimensional force sensor develop towards miniaturization, but the processing and installation of the flexible ball joint are difficult, thus increasing the cost of manufacture and use.In order to solve the above problem of parallel six-dimensional force sensor, this paper proposes a novel circular flexible spherical joint and applies it to the six-dimensional force sensor.The circular flexible spherical joint has the advantages of simple processing, easy installation and reduced manufacturing cost.At the same time, because the flexible spherical joint cannot be completely equivalent to the ideal spherical joint, the flexible ball hinge will cause the coupling effect between the sensor force-sensitive elements, thereby reducing the accuracy of the sensor, so it is necessary to optimize the structure of the flexible spherical joint to improve the accuracy of the sensor.
Because the stiffness of the three rotation directions of the ideal spherical joint is exactly the same, this paper takes the same stiffness of the three rotation directions of the circular flexible spherical joint as the optimization goal.Firstly, the structure of circular flexible spherical joint is proposed, and its stiffness matrix is deduced.Then, a six-dimensional force sensor is designed according to the circular flexible spherical joint, and its structural parameters are optimized by genetic algorithm.Finally, a prototype with optimized data is selected and verified by experiments.

Configuration of the novel circular flexible spherical joint
The flexible spherical joints are components that weaken their local stiffness through notches and undergo bending deformation under the action of torque.Their bending deformation mainly occurs at the weakest notch [18].The structure of a circular flexible spherical joint proposed based on the definition of a flexible spherical joint is shown in figure 1.
In figure 1, the circular through hole in the middle is the connecting part, and the connected part can be connected with the circular flexible spherical joint with bolts.O is the center of the circular through hole on the upper surface.O 1 is the center of a circular notch.The distance from the center of the circle O to the outer circle of the circular notch is t.The radius of the circular notch is R and the distance between its two endpoints is l.The upper and lower circular notch are symmetrically distributed at the center of the flexible spherical joint.The distance between the weakest point of the upper and lower circular notch is t.The distance from the center O of the upper surface to the outermost circle of the incision is r.The flexible spherical joint has the advantages of convenient processing, easy installation and rotation, and wide application occasions.

Stiffness modeling of the novel circular flexible spherical joints
The circular flexible spherical joint is shown in figure 1.In order to model its stiffness, the flexible spherical joint is first divided into several parts by using the segmentation method.The segmented figure can be regarded as a compliant mechanism, and the specific form is shown in figure 2 below.
Firstly, calculate the stiffness matrix of the compliant mechanism using numerical analysis method [19].Secondly, coordinate transformation is performed through the pose transformation matrix.Finally, since several compliant  mechanisms have the same reference coordinate system, the stiffness matrix of the overall annular flexible spherical joint can be obtained by adding the stiffness matrices of the same reference coordinate system.The stress of compliant mechanism is mainly concentrated in the groove, so the deformation caused by the stress is mainly concentrated in the groove, and the rest can be regarded as a rigid body.From this, an approximate equation for the deflection curve of the beam bending deformation can be obtained Here, E donates elastic modulus. 12.
The calculation and solution of this differential equation is very complex, and approximate solutions can be obtained using numerical analysis method.
3l 8Eb ) l 3l(Fzl+2Mx) 2Eb ( Corresponding relationship between x i and ∂ i . ) The corresponding relationship between x i and θ i is shown in table 1. Here,

. The values in table 1 can be approximated by Lagrange
Polynomial Interpolation to obtain θ (x).θ (x) is the angle of rotation around x, expressed as equation ( 3) ( The equation at the end is 3) is the expression of the rotation angle of the compliant mechanism around the x-axis, that is, the expression of the end.So the end expression is equation ( 4) Using the formula of Runge-Kutta again for ∂ ′ z = θ (x), the corresponding relationship between x i and ∂ i is obtained, as shown in table 2.
Perform Lagrange Polynomial Interpolation on table 2 to obtain the displacement expression (5) at the end along the zaxis.Equation ( 5) is as follows ( By combining equations ( 4) and ( 5), the relationship between F z , M z , and θ x , ∂ z can be obtained as follows Similarly, the relationship between F x , M z , and θ z , ∂ x can be obtained as follows The axial tensile deformation of the compliant mechanism under the action of F y is: It is difficult to directly obtain the result of this integral, which can be achieved by compound parabolic integration formula Here After the above analysis, it can be concluded that ∂ y is: It can be concluded that F y is: The axial torsional deformation of the compliant mechanism under the action of M y is: Calculate the torsional deformation using the compound parabolic integration formula as equation ( 12) It can be concluded that M y is: Here, G = E/ 2 (1 + µ).By using the corresponding relationship obtained above, the overall corresponding relationship can be obtained Here, K 1 is the stiffness matrix of compliant mechanism , , .
The circular flexible spherical joint can be seen as composed of multiple flexible structures as shown in figure 2. The reference coordinate system is located at the geometric center of the upper surface of the flexible spherical joint, and the angle of rotation of the reference coordinate system of the compliant mechanism around the Z-axis relative to the fixed coordinate system at any position is set as γ。 According to the coordinate system pose transformation formula, the stiffness matrix mapping formula from coordinate system i to coordinate system j is as follows.The reference coordinate system is located at the geometric center of the upper surface of the flexible spherical joint Here, K i represents the stiffness matrix in the reference coordinate system i, T j i represents the transformation matrix from reference coordinate system i to coordinate system j.R j i represents the rotation matrix from reference coordinate system i to coordinate system j.r j i represents the position coordinate of reference coordinate system i to coordin-ate system.S r j i represents a diagonal symmetric matrix of positional coordinates.
Bringing the values of the coordinate system into the formula yields the following expression Here, r is the radius of the outer circle of the groove.m is the overall thickness According to the pose transformation formula, the stiffness matrix K 2i of any compliant mechanism can be obtained Now, the flexible spherical joint is divided into 720 parts, each of which can be approximately regarded as a compliant mechanism as shown in figure 2. The side length of compliant mechanism isb = 2π d/ 720.Since each compliant mechanism has the same reference coordinate system, the overall stiffness matrix of the circular flexible spherical joint is equal to the sum of the stiffness matrices of each compliant mechanism.So: Due to order of magnitude differences, discarding a portion of the values will not have a significant impact on the final result.Take a larger coefficient for the stiffness coefficient of movement in the direction of x, y, z, and a smaller coefficient for the stiffness coefficient of rotation in the direction of x, ym, z.When a corresponding force or torque is applied, the main change occurs as rotation or movement in the corresponding direction.So only the values of the main diagonal elements of the stiffness matrix are retained, and the final determination of the stiffness matrix K 2 is as follows

Verification through the finite element method
The stiffness of the flexible spherical joint is analyzed by finite element simulation, and the above stiffness modeling results are checked at the same time.Set the structural parameters of the flexible spherical joint, each structural parameter is shown in the following table 3. Alloy 7075Al is selected as the material of the circular flexible spherical joint.
Because the 3D modeling process of ANASYS is relatively complicated, this paper uses SolidWorks to establish a 3D model of the flexible spherical joint, save the 3D model after modeling in the form of '.x t', and import the saved model into ANASYS.In order to make the simulation results more accurate, 0.5 mm is used for mesh division, and the finite element model of the annular flexible spherical hinge is shown in figure 3.
The analysis of the model first determines the constraints, the constraints are fixed at one end, the other end is loaded with torque load, by detecting the corresponding rotation angle, the stiffness of the direction can be determined, this paper first fixes the bottom end of the circular flexible spherical joint, and applies torque inside the circular groove at the other end.From this, the degree of consistency between the simulation results and the theoretical analysis results is judged.
A torque of 30 N•m around the z-axis is applied to the upper surface of the circular flexible spherical joint, and the deformation result is shown in figure 4. Through the simulation results, when the same size of torque is applied in different directions, the rotation angle around the z axis is smaller than the rotation angle around x and y, and the relationship between the rotational stiffness coefficient corresponding to the z direction is greater than the rotational direction stiffness coefficient corresponding to x and y.In summary, the derived annular flexible ball hinge stiffness matrix has certain theoretical and practical reference value.

The structure of six-dimensional force sensor with the circular flexible spherical joint
According to the structure of the novel circular flexible spherical joint, the six-dimensional force sensor designed is shown in figure 5.The sensor consists of an upper platform, a lower platform, six circular flexible ball joints, six columns, and six measurement branches.The upper and lower platforms have the same structure and have three identical columns, the columns on the platform are distributed in a circumferential manner, and circular flexible spherical joints are present on each column.At the same time, the flexible spherical joints of the upper and lower platforms are distributed in a circle and correspond to each other.Among the six force measurement branches, it is placed horizontally and vertically, the horizontally placed force measurement branch is connected to the spherical joint on the column, the vertical force measurement branch is connected to the spherical joint on the platform.The horizontal and vertical force measurement branches are distributed symmetrically in a circle.When the acting force is x and y as the acting force and the moment in the z direction, it is measured by the force measuring branch placed horizontally; when the acting force is the acting force in the z direction and the x and y directions are moments, the force measuring branch placed vertically branch measurement.This structure can not only measure accurately in principle, but also realize partial decoupling to a certain extent.

Mathematical model of six-dimensional force sensor
The schematic diagram of the structure of the new sixdimensional force sensor with circular flexible spherical joint is established, as shown in figure 6. a 1 a 2 a 3 means the upper platform, A 1 A 2 A 3 means the lower platform.a 1 A 1 , a 2 A 2 , a 3 A 3 represent vertically placed force measuring branches, a 4 A 4 , a 5 A 5 , a 6 A 6 represent horizontally placed force measuring branches, the angle between any two points in a 1 , a 2 , a 3 and the origin o is 120 • .The axes of the three horizontal force-measuring branches are tangent to the same circle center o ′ , the tangent point is the key point of the force-measuring When external forces and moments act on the upper platform, according to the static equilibrium equation, it can be inferred that: Here, F = F x F y F z T a is the external force acting on the upper platform.M = M x M y M z T is the external moment acting on the upper platform.
is the axial force acting on each force-measuring branch.G is the force mapping matrix.Its specific form can be expressed according to the spiral theory, and the specific form is as follows: It is calculated that: It can be seen from equation ( 22) that when the force is F x , it is measured by the 4 and 5 force measurement branches; When the force is F y or M z , it is measured by the 4, 5, and 6 force measurement branches; When the force is F z or M y , it can be measured by the force measuring branch 1, 2, 3, and when the force is M x , it can be measured by the force measuring branch 1, 2.

Selection of elastomers
Traditional force measuring components have problems such as difficult selection of patch positions, high manufacturing difficulty, cumbersome process, and low precision.Cylindrical one-dimensional force sensors have the advantages of high measurement accuracy and easy installation, so cylindrical one-dimensional force sensors are used instead.Traditional force measuring components.The selected model is SBT641A, the size is φ 10 × 22 mm, and the screw port is M3.

Dimensional design of circular flexible ball joint
According to the selected cylindrical single-dimensional force sensor, the screw of the cylindrical single-dimensional force sensor is M3, the measuring range is F max = 500 N, from the consideration of reducing the processing difficulty and reducing the processing difficulty error, so the radius of outermost circle of the circular flexible spherical joint is selected r = 18 mm.
When calculating the stiffness matrix of the circular flexible spherical joint, the microelement method was used for calculation.Based on this, the analogy method is used to calculate the size of the circular flexible spherical joint.Divide the circular flexible spherical joint into 720 parts, and each small part is approximately equivalent to a compliant mechanism.From the perspective of force, since the stiffness of the circular flexible spherical joint is greater than the overall stiffness of the 720 compliant mechanism, when a single flexible mechanism meets the force requirements, the circular flexible spherical joint also meets the force requirements.
The 7075 Aluminum Alloy is selected as the material of the annular flexible ball joint, and its physical and mechanical properties are shown in the following tables 4 and 5.
The structural diagram of the compliant mechanism is shown in figure 2, When the cylindrical one-dimensional force sensor bears the axial force, it is equivalent to the force in the z direction of the compliant mechanism.According to the measurement range of the sensor and the strength of the material, it can be obtained as follows, Here, the t is the distance between the weakest point of the upper and lower circular notch.F s is the shear force borne by the compliant mechanism.n is the safety factor of the sensor, where n = 2 is taken.The r is the outermost circle of the circular flexible spherical joint.τ is the allowable shear stress of aluminum alloy 7075.
After calculation, the result can be obtained as follows, The t is the length of the shortest part in the compliant mechanism.When the length of the shortest part meets the strength requirement, according to the analogy method, the rest of the part also meets the strength requirement.

Optimization of circular flexible spherical joint
In the design and application research of six-dimensional force sensors, performance indicators are an important criterion for optimizing sensor design.At present, the performance indicators of six-dimensional force sensors are diverse, and there is no universal definition of performance indicators.As an important indicator of six-dimensional force sensors, isotropy has been recognized by researchers from all walks of life [20,21].In the six-dimensional force sensor with the novel circular flexible spherical joint, the ideal spherical joint is replaced by a circular flexible spherical joint.It is precisely because of the presence of spherical joints that the force measuring element can be regarded as a two-force rod that only bears axial forces.In order to make the circular flexible spherical joint more have the performance target of the ideal spherical joint, the isotropy of the three rotation directions of the circular flexible spherical joint is taken as the optimization goal.

Selection of optimization algorithm.
Genetic algorithms are commonly used as optimization algorithms in engineering.Because it does not rely on other auxiliary knowledge in the optimization process, only needs to optimize the objective function and fitness function, so that the genetic algorithm does not depend on the specific field of the problem, and has strong adaptability to the optimization of the problem, so the genetic algorithm is used as the optimization algorithm of the annular flexible ball hinge, and this paper completes the optimization of the genetic algorithm by writing out the optimization objective function, programming calculation, and obtaining the optimized target parameters.

Optimizing the objective function.
In order to make the force-measuring element of the six-dimensional force sensor a better two-force rod, the stiffness of the three rotation directions of the circular flexible spherical joint is the same as the optimization goal.Due to the symmetry and particularity of its own structure, the circular flexible spherical joint has the same stiffness in the x and y rotation directions, and the rotation stiffness in the z direction is always greater than the stiffness in the x and y directions.For the optimization objective function, as long as the rotation stiffness coefficient in the z direction is subtracted from the rotation stiffness coefficient in the x or y direction, the construction of the optimization objective function can be completed.Taking the minimum value of the optimization objective function can be regarded as the result of the best performance, and the parameter corresponding to the minimum value is the parameter value to be selected.Let θ be 1/2 of the angle between the two ends of the circular groove and the center of the circle, so l = 2R sin θ.For the convenience of calculation, the optimization objective function F t R l is converted into F t R θ .The specific optimization objective function can be obtained follows, Here, 720H 1 is the stiffness factor of the z-direction of rotation.360G 1 is the stiffness factor in the x or y direction.

Parameter value.
The design principle of the sensor is that the space size and spatial distribution should be as compact as possible, and the distance between the upper and lower platforms should be the length of the single-dimensional force tension and compression sensor, so as to ensure the strength of each measurement branch and the absence of interference phenomenon while reducing mass and easy processing.Under the restrictions of the above conditions, the design parameters should meet the following conditions, 2 mm ⩽ t ⩽ 10 mm 1 mm ⩽ R ⩽ 4 mm 0.1π ⩽ θ ⩽ 0.5π .

Optimization results.
By writing the genetic algorithm optimization program, multiple sets of optimized values of each parameter were obtained, as shown in table 6 below.The optimization process of the genetic algorithm is to write the fitness function and the cross-variation probability, and finally obtain the optimal fitness value, that is, the best optimization result, but due to the influence of the cross-mutation probability, there are certain differences in each optimization result, but they are reasonable optimization data, and the final parameters of the spherical joint are t = 2 mm, R = 4 mm, θ = 0.2660π .From l = 2R sin θ, it can be seen that when θ = 0.2660π ; to facilitate processing and minimize processing errors, l is rounded and l = 6 mm is taken.

Determination of other values of sensors.
For a sixdimensional force sensor, its structural schematic diagram is an important manifestation of its structural characteristics; Considering that there may be some errors in the machining process of six-dimensional force sensors.Therefore, only certain values in the schematic diagram of the six-dimensional force sensor structure are provided here.The specific values are shown in table 7.

Prototype experiments
From the above optimization results and sensor parameters, this paper develops a six-dimensional force sensor prototype, which is shown in figure 7 below.In the process of calibration experiment, in order to make the sensor more in line with the actual working environment, a 6-degree-of-freedom ABB manipulator was selected for the calibration experiment, and the X-direction calibration experiment is shown in figure 8.
During the calibration process, 20 loading points are evenly divided within the range of the sensor to load and unload experiments sequentially.Repeat the calibration steps to get the output voltage corresponding to the loading value.In order to judge the accuracy of the six-dimensional force sensor, it is necessary to obtain its calibration matrix.In this paper, the   least squares fitting principle used to obtain its calibration matrix.The principle is as follows, Here, F s denotes a matrix of load values; G denotes calibration matrix of the six-axis force sensor; U denotes a matrix of the voltage signal values corresponding to the load values.
For equation ( 26), the calibration matrix G can be obtained by the least squares fitting principle, and the result can be obtained as follows, where U − is the pseudoinverse matrix of U, U = U T UU T −1 , The matrix of G can be obtained as follows, G 6×n is calibration matrix, which is also a decoupling matrix.In order to obtain a more accurate calibration matrix, the number of calibrations must be much greater than the dimension of the calibration matrix.
The calibration matrices of six dimensional force sensor calculated by the above formula areG can be obtained as follows, The measured value F c can be obtained according to the calibration matrix G and the measurement voltage matrix U, and its mathematical expression is as follows, The measured value compared to the loaded value is shown in figures 9(a) and (b), from which the accuracy of the sensor is obtained.
The remaining orientation test results are similar to the x direction, and the test results are shown in table 8.It can be seen from table 8 that after parameter optimization, the  six-dimensional force sensor has high accuracy in all directions, and its maximum error is 1.85%, which shows that the mathematical model of six-dimensional force sensor with the novel circular flexible spherical joint established is correct.

Conclusion
By analyzing the characteristics of flexible spherical joint and compliant mechanism, this paper designed a novel circular flexible spherical joint with simple processing technology and convenient installation.Its stiffness matrix model is derived by numerical analysis method and micro element method, and the correctness of the stiffness matrix is verified by finite element simulation.A parallel six-dimensional force sensor was designed based on a circular flexible spherical joint, and a mathematical model of the sensor was established.Genetic algorithm was used to optimize the flexible spherical joint of the sensor, and prototype development and calibration experiments were conducted based on the optimization results.The experimental results show that the six-dimensional force sensor using a circular flexible spherical joint has high measurement accuracy, with a maximum measurement error of 1.85%.This indicates that circular flexible spherical joint can replace ideal ball joints in certain situations for use on six-dimensional force sensors, while reducing the processing difficulty and actual volume of the sensor.It has certain guiding significance for the development of six-dimensional force sensors.

Figure 1 .
Figure 1.Configuration of the novel circular flexible spherical joint.

Figure 3 .
Figure 3. Finite element model of circular flexible spherical joint.

Figure 4 .
Figure 4. Deformation distribution diagram when torque is applied in the z direction.

Figure 6 .
Figure 6.Schematic diagram of the structure of the six-dimensional force sensor.

5. 1 .
Determination of sensor parameters 5.1.1.Selection of structural parameters for circular flexible spherical joints.Based on the above optimization results, the parameters t and R are taken as 2 mm and 4 mm respectively.

Figure 8 .
Figure 8.The calibration device of the six-dimensional force sensor installed on the manipulator.

Figure 9 .
Figure 9.Comparison of x-direction force loading results.

Table 1 .
Corresponding relationship between x i and θ i .

Table 3 .
Structural parameters of the circular flexible spherical joint.

Table 5 .
Aluminum alloy mechanical performance parameters.

Table 7 .
Structural parameters of six dimensional force sensor.