Mathematical Model and Calibration Experiment of a Large Measurement Range Flexible Joints 6-UPUR Six-Axis Force Sensor

Nowadays improving the accuracy and enlarging the measuring range of six-axis force sensors for wider applications in aircraft landing, rocket thrust, and spacecraft docking testing experiments has become an urgent objective. However, it is still difficult to achieve high accuracy and large measuring range with traditional parallel six-axis force sensors due to the influence of the gap and friction of the joints. Therefore, to overcome the mentioned limitations, this paper proposed a 6-Universal-Prismatic-Universal-Revolute (UPUR) joints parallel mechanism with flexible joints to develop a large measurement range six-axis force sensor. The structural characteristics of the sensor are analyzed in comparison with traditional parallel sensor based on the Stewart platform. The force transfer relation of the sensor is deduced, and the force Jacobian matrix is obtained using screw theory in two cases of the ideal state and the state of flexibility of each flexible joint is considered. The prototype and loading calibration system are designed and developed. The K value method and least squares method are used to process experimental data, and in errors of kind Ι and kind II linearity are obtained. The experimental results show that the calibration error of the K value method is more than 13.4%, and the calibration error of the least squares method is 2.67%. The experimental results prove the feasibility of the sensor and the correctness of the theoretical analysis which are expected to be adopted in practical applications.


Introduction
With the ability of measuring three force components and three torque components, the six-axis force sensor is one kind of the most important and challenging sensors, used widely in many research areas such as wind tunnel balances, thrust stand testing of rocket engines, and in robotics, the automobile industry, aeronautics, etc. [1]. Compared with single-axis sensors, not only the volumes and prices of multi-axis force sensors are considered, but also their structures, in order to achieve a balance between the isotropy of force/torque and that of sensitivity [2]. Recently, researchers all over the world have done a lot of work on six-axis force sensors.
Since the Stewart platform was applied to the measurement of space six-axis forces by measuring the forces in the six legs with convection elements in 1983 [3], parallel structure have been widely used in six-axis force sensors [4], stimulated by the advantages of good stiffness, symmetric and compact the structure which included the ideal state and the state of flexibility of each flexible joint is considered in Section 3. Section 4 introduces the experimental research on static calibration of the sensor prototype. Section 5 presents the results of the experiment. The paper is concluded in Section 6, summarizing the work that has been done. Figure 1a illustrates the traditional parallel sensor diagram, which is based on the Stewart platform and composed of a measuring platform, a lower fixed platform and six elastic legs connecting the two platforms with traditional spherical joints.

Sensor Structure
6, summarizing the work that has been done. Figure 1a illustrates the traditional parallel sensor diagram, which is based on the Stewart platform and composed of a measuring platform, a lower fixed platform and six elastic legs connecting the two platforms with traditional spherical joints.

Sensor Structure
Considering the traditional spherical joints are difficult to processed into flexible joints in a large measurement range situation, and the measurement accuracy will be affected, which restricts its application in six-axis force sensors based on flexible parallel mechanism, this paper proposes a kind of structure model of six-axis force sensor based on a parallel 6-UPUR mechanism. As shown in Figure 1b, it consists of a measuring platform, a fixed platform and six measuring legs divided into three groups of legs and two legs in each group are located in a vertical plane. Each measuring leg contains a single-axis force sensor and connects the fixed platform with flexible universal joint, and connects the measuring platform with combined spherical joint. Through the improvement of the traditional Stewart platform mechanism, and the introduction of flexible joints which have the peculiarities of non-clearance, friction-less and high sensitivity to replace the traditional spherical joints, it is possible to develop a sensor with a large measurement range. The characteristic parameters of the parallel 6-UPUR six-axis force sensor include radius R of the measuring platform, radius r of the lower platform, the positioning angle A, B, C, the center distance l1 of the U joint and location center distance l2 of the R joint, as shown in Figure 2.  Considering the traditional spherical joints are difficult to processed into flexible joints in a large measurement range situation, and the measurement accuracy will be affected, which restricts its application in six-axis force sensors based on flexible parallel mechanism, this paper proposes a kind of structure model of six-axis force sensor based on a parallel 6-UPUR mechanism. As shown in Figure 1b, it consists of a measuring platform, a fixed platform and six measuring legs divided into three groups of legs and two legs in each group are located in a vertical plane. Each measuring leg contains a single-axis force sensor and connects the fixed platform with flexible universal joint, and connects the measuring platform with combined spherical joint. Through the improvement of the traditional Stewart platform mechanism, and the introduction of flexible joints which have the peculiarities of non-clearance, friction-less and high sensitivity to replace the traditional spherical joints, it is possible to develop a sensor with a large measurement range.
The characteristic parameters of the parallel 6-UPUR six-axis force sensor include radius R of the measuring platform, radius r of the lower platform, the positioning angle A, B, C, the center distance l 1 of the U joint and location center distance l 2 of the R joint, as shown in Figure 2. is considered in Section 3. Section 4 introduces the experimental research on static calibration of the sensor prototype. Section 5 presents the results of the experiment. The paper is concluded in Section 6, summarizing the work that has been done. Figure 1a illustrates the traditional parallel sensor diagram, which is based on the Stewart platform and composed of a measuring platform, a lower fixed platform and six elastic legs connecting the two platforms with traditional spherical joints.

Sensor Structure
Considering the traditional spherical joints are difficult to processed into flexible joints in a large measurement range situation, and the measurement accuracy will be affected, which restricts its application in six-axis force sensors based on flexible parallel mechanism, this paper proposes a kind of structure model of six-axis force sensor based on a parallel 6-UPUR mechanism. As shown in Figure 1b, it consists of a measuring platform, a fixed platform and six measuring legs divided into three groups of legs and two legs in each group are located in a vertical plane. Each measuring leg contains a single-axis force sensor and connects the fixed platform with flexible universal joint, and connects the measuring platform with combined spherical joint. Through the improvement of the traditional Stewart platform mechanism, and the introduction of flexible joints which have the peculiarities of non-clearance, friction-less and high sensitivity to replace the traditional spherical joints, it is possible to develop a sensor with a large measurement range. The characteristic parameters of the parallel 6-UPUR six-axis force sensor include radius R of the measuring platform, radius r of the lower platform, the positioning angle A, B, C, the center distance l1 of the U joint and location center distance l2 of the R joint, as shown in Figure 2.

Force Analysis
There are reacting forces on measuring legs when an external force loaded on the measuring platform of the parallel sensor whose surface is assumed to be friction-less and continuous, and the reacting force is considered along the axis of each measuring leg. According to the space static balance conditions of the measuring platform, the following equation can be obtained on screw theory [24]: where f i represents the reacting force on the measuring legs; S i represents the unit line vector along the ith measuring leg; F and M, respectively, represent the ith loaded force vector and toque vector on the center of the measuring platform; P is the dual sign.
Equation (1) can be rewritten in the form of matrix expression as: where F ω " " F x F y F z M x M y M z ‰ T is the vector of six-axis external force applied on the measuring platform; f " r f 1 f 2 f 3 f 4 f 5 f 6 s T is the vector composed of the reacting forces of the six measuring legs; G F f is the first-order static influence coefficient matrix. If G F f is non-singular matrix, the inverse mapping between F ω and f is: where, G f F "´G F f¯´1 is the force Jacobi matrix, and it is the mapping matrix from the external force loaded on the measuring platform of the sensor to the force produced on the six measuring leg. G F f is closely related with the geometric structure of the sensor, and the characteristics of G F f determine stiffness, isotropy, sensitivity and many other features of the sensor.

Ideal Mathematical Model
Each leg of the parallel 6-UPUR mechanism can be seen as a series open-chain mechanism which is composed of six links and six joints, and the U joint is equivalent to two joints whose axes are vertical and intersect. The base is called link 0, which is not included in the six links. The first link is connected to the base by the first rotational joint; the second link is connected to the first link by the second rotational joint, and so on.
The coordinate system attached to the base fixedly is referred to as {0}; the coordinate system on the reference point of the end of the leg is referred to as {P}; coordinate system attached to the ith link fixedly is referred to as{i}, as shown in Figure 3. There are reacting forces on measuring legs when an external force loaded on the measuring platform of the parallel sensor whose surface is assumed to be friction-less and continuous, and the reacting force is considered along the axis of each measuring leg. According to the space static balance conditions of the measuring platform, the following equation can be obtained on screw theory [24]: where fi represents the reacting force on the measuring legs; Si represents the unit line vector along the ith measuring leg; F and M, respectively, represent the ith loaded force vector and toque vector on the center of the measuring platform;  is the dual sign.
Equation (1) can be rewritten in the form of matrix expression as: ] is the vector of six-axis external force applied on the measuring platform; = [ ] is the vector composed of the reacting forces of the six measuring legs; is the first-order static influence coefficient matrix. If is non-singular matrix, the inverse mapping between Fω and f is: where, = ( ) is the force Jacobi matrix, and it is the mapping matrix from the external force loaded on the measuring platform of the sensor to the force produced on the six measuring leg. is closely related with the geometric structure of the sensor, and the characteristics of determine stiffness, isotropy, sensitivity and many other features of the sensor.

Ideal Mathematical Model
Each leg of the parallel 6-UPUR mechanism can be seen as a series open-chain mechanism which is composed of six links and six joints, and the U joint is equivalent to two joints whose axes are vertical and intersect. The base is called link 0, which is not included in the six links. The first link is connected to the base by the first rotational joint; the second link is connected to the first link by the second rotational joint, and so on.
The coordinate system attached to the base fixedly is referred to as {0}; the coordinate system on the reference point of the end of the leg is referred to as {P}; coordinate system attached to the ith link fixedly is referred to as{i}, as shown in Figure 3.   The differential motion vector of central reference point P which is on the measuring platform is: where: δ h " " δ hx δ hy δ hz ı is the angle variable, d P "´d P x d P y d P z¯i s the displacement variable, G P φ is the first-order static influence coefficient matrix, ı T is the differential motion of the six legs: . . S 6 are the directions of joints' axes of each leg, and they are: T j is rotational transformation matrix: a jpj`1q is the direction of the common normal line between adjacent axes: pr´1qr a pr´1qr`d r S r¯`Tj P p6q (9) where a pr´1qr is length of the common normal line between adjacent axes, d r is offset along the axis of rotation. P p6q is the representation of the point P in coordinate at the end of the leg, and the left leg and the right leg are separately expressed as: where: According to the above equations, can be obtained, separately: Because the z axis of the fixed coordinate system O-xyz which is on the center of the lower platform is the direction of S 1 , and selection of coordinate system has a direct impact on the first-order static influence coefficient matrix, so it is necessary to take the symmetry of the load platform structure into account, thus: When movement of the upper platform is known, movement of the joints in each leg can be written: When six active members of the mechanism are determined, equations of the six active movements from the above equations are taken out and expressed as: where . φ paq α is the differential motion of the six legs, its subscripts are leg's number and joint's number respectively, represents the α-th line of the inverse matrix Combining the above six equations to constitute a matrix expression, like this: The inverse solution is D P " . Furthermore, the first-order static influence coefficient matrix can be obtained: Thus, the force Jacobi matrix is G , and that is the mapping matrix from the external force applied on the measuring platform of the sensor to the force produced on the six elastic legs in the ideal condition.

Mathematical Model of the Sensor with Flexible Joints
The six-axis force sensor adopts the structure that all joints are flexible joints with single degree of freedom, and its 3D model is shown in Figure 4. Each leg is a split structure. The lower part of the leg is composed of a flexible universal joint with an integral structure and a lower positioning block. Each elastic leg is connected to the measuring and fixed platforms through the upper and lower positioning blocks by bolts. Thus, it's realized that the decomposition of the six-axis external force to the six legs. Each elastic leg is connected to the measuring and fixed platforms through the upper and lower positioning blocks by bolts. Thus, it's realized that the decomposition of the six-axis external force to the six legs. The lower positioning block is an integral structure, which is composed of two symmetrical flexible universal joints, with the same structure as the flexible universal joint in the upper part of the leg. Its front view and graphic model of a single joint are shown in Figure 6. By using the knowledge of material mechanics, the rotational stiffness of a single joint is obtained: k2 = 4.0852 × 10 4 (N·m/rad).  The upper positioning block is composed of two flexible rotation joint, which form a flexible spherical joint with composite flexible universal joint by assembling relationship. Its front view and graphic model are shown in Figure 5. By using the knowledge of material mechanics, the rotational stiffness of the flexible rotary joint is obtained: k 1 = 8.592ˆ10 4 (N¨m/rad).
Each elastic leg is connected to the measuring and fixed platforms through the upper and lower positioning blocks by bolts. Thus, it's realized that the decomposition of the six-axis external force to the six legs. The lower positioning block is an integral structure, which is composed of two symmetrical flexible universal joints, with the same structure as the flexible universal joint in the upper part of the leg. Its front view and graphic model of a single joint are shown in Figure 6. By using the knowledge of material mechanics, the rotational stiffness of a single joint is obtained: k2 = 4.0852 × 10 4 (N·m/rad).  The lower positioning block is an integral structure, which is composed of two symmetrical flexible universal joints, with the same structure as the flexible universal joint in the upper part of the leg. Its front view and graphic model of a single joint are shown in Figure 6. By using the knowledge of material mechanics, the rotational stiffness of a single joint is obtained: k 2 = 4.0852ˆ10 4 (N¨m/rad). Each elastic leg is connected to the measuring and fixed platforms through the upper and lower positioning blocks by bolts. Thus, it's realized that the decomposition of the six-axis external force to the six legs. The lower positioning block is an integral structure, which is composed of two symmetrical flexible universal joints, with the same structure as the flexible universal joint in the upper part of the leg. Its front view and graphic model of a single joint are shown in Figure 6. By using the knowledge of material mechanics, the rotational stiffness of a single joint is obtained: k2 = 4.0852 × 10 4 (N·m/rad).  Considering the effect of elastic deformation of joints on G f F , so the following will re-establish the complete force Jacobian matrix. where, O-xyz is the fixed coordinate system, O i1 -x i1 y i1 z i1 , i = 1,2,3, . . . ,6 is the local coordinate system which is established on the reference point of the lower positioning block, O i2 -x i2 y i2 z i2 is the local coordinate system which is established on the reference point of the standard one-axis force sensor, O i3 -x i3 y i3 z i3 is the local coordinate system which is established on the reference point of the flexible U joint, O i4 -x i4 y i4 z i4 is the local coordinate system which is established on the reference point of the integrated positioning block, O ip -x ip y ip z ip is the reference coordinate system which is established on the reference point of the end of a flexible series leg.
Sensors 2016, 16,1271 8 of 20 Considering the effect of elastic deformation of joints on , so the following will re-establish the complete force Jacobian matrix. Figure 7 is the schematic diagram of flexible element coordinate systems' establishment on a leg. where, O-xyz is the fixed coordinate system, Oi1-xi1yi1zi1, i = 1,2,3,…,6 is the local coordinate system which is established on the reference point of the lower positioning block, Oi2-xi2yi2zi2 is the local coordinate system which is established on the reference point of the standard one-axis force sensor, Oi3-xi3yi3zi3 is the local coordinate system which is established on the reference point of the flexible U joint, Oi4-xi4yi4zi4 is the local coordinate system which is established on the reference point of the integrated positioning block, Oip-xipyipzip is the reference coordinate system which is established on the reference point of the end of a flexible series leg. When there is the six-axis external force loaded on the end of the flexible series leg, by the principle of virtual work, the following equation can be obtained: where According to the deformation superposition principle of series leg, the total deformation vector of the end of flexible series leg's reference point caused by movement and rotation deformation vector of every basic flexible unit can be obtained: It can be known by the definition of the stiffness matrix, the relationship between the six-axis external force at the end of the leg reference point and the end deformation of the leg is: (19) where K is the stiffness matrix of the end of flexible series leg.
Substituting Equations (17) and (20) into Equation (21), the following equation can be obtained: (20) where Kj ( When there is the six-axis external force loaded on the end of the flexible series leg, by the principle of virtual work, the following equation can be obtained: where ∆Xj is deformation vector of the end of leg's reference point, Xj is the elastic deformation vector of the end of the j-th basic flexible unit, J j is the pose transformation matrix, is the six-axis non-coplanar force, F j " " f x j , f y j , f z j , m x j , m y j , m z j ı T is the reaction force on the end of the j-th basic flexible unit, J Fj is the force transformation matrix.
According to the deformation superposition principle of series leg, the total deformation vector of the end of flexible series leg's reference point caused by movement and rotation deformation vector of every basic flexible unit can be obtained: It can be known by the definition of the stiffness matrix, the relationship between the six-axis external force at the end of the leg reference point and the end deformation of the leg is: where K is the stiffness matrix of the end of flexible series leg. Substituting Equations (17) and (20) into Equation (21), the following equation can be obtained: On the assumption that the stiffness of the upper platform and all of branches is infinite, neglecting the minor deformation caused by force and the thickness of upper platform, when there is a six-axis external force, the geometric compatibility conditions between the end reference point and the reference point of the upper platform of the ith flexible series leg: where ΔX is displacement vector at the reference point of the upper platform. According to the synthesis principle of the space force system, the relationship between the external force vector at the reference point of the upper platform and the reaction force vector on the six flexible series branches is established by the following expression: According to the definition of stiffness matrix of flexible parallel mechanism, there is: The stiffness matrix of the upper platform reference point can be obtained by the above equation: The relationship between the end force of each flexible element and the end force of the leg is: On the assumption that the stiffness of the upper platform and all of branches is infinite, neglecting the minor deformation caused by force and the thickness of upper platform, when there is a six-axis external force, the geometric compatibility conditions between the end reference point and the reference point of the upper platform of the ith flexible series leg: where ∆X is displacement vector at the reference point of the upper platform. According to the synthesis principle of the space force system, the relationship between the external force vector at the reference point of the upper platform and the reaction force vector on the six flexible series branches is established by the following expression: According to the definition of stiffness matrix of flexible parallel mechanism, there is: The stiffness matrix of the upper platform reference point can be obtained by the above equation: The relationship between the end force of each flexible element and the end force of the leg is: The relationship between the end force of each flexible element and the six-axis external force acting on the sensor's upper platform is: The relationship between the six forces expressed in the local coordinate system and the six external forces acting on the sensor platform is: For Equations (21) . . .
F s 1 is the mapping matrix from the external force/toque applied on the measuring platform of the sensor to the force produced on the six elastic legs considering elastic deformation of the flexible joints.

Comparative Analysis of Numerical Simulation and Mathematical Models
In the above two sections, the mathematical models of the two cases are obtained. By using the ANSYS finite element simulation software platform, the external force  Figure 9 and Table 1. In the ideal case, the theoretical value of the mathematical model is recorded as the first theoretical value, and the theoretical value considering elastic deformation of flexible joints is recorded as the second theoretical values. 5000] are respectively loaded in the geometry center of the measuring platform in the three-dimensional model of the sensor, and the size of the force produced on the six elastic legs are obtained by the simulation calculation. The accuracy comparison of the theoretical values and simulation values is shown in Figure 9 and Table 1. In the ideal case, the theoretical value of the mathematical model is recorded as the first theoretical value, and the theoretical value considering elastic deformation of flexible joints is recorded as the second theoretical values.  As shown in Table 1, the measurement errors of the six elastic legs are reduced to 10% after considering the deformation stiffness error of flexure joints, which proves that the mathematical model is effective.   As shown in Table 1, the measurement errors of the six elastic legs are reduced to 10% after considering the deformation stiffness error of flexure joints, which proves that the mathematical model is effective.

Prototype
In order to prove the superiority of the proposed sensor structure and prove the correctness of the theoretical analysis, a prototype of the large measurement range six-axis force sensor of 6-UPUR parallel mechanism with flexible joints was manufactured, as shown in Figure 10. Considering the manufacturing process and economic cost, the material property of each component was selected as shown in Table 2. The main structure parameters and the measuring range of the sensor are shown in Tables 3 and 4.

Prototype
In order to prove the superiority of the proposed sensor structure and prove the correctness of the theoretical analysis, a prototype of the large measurement range six-axis force sensor of 6-UPUR parallel mechanism with flexible joints was manufactured, as shown in Figure 10. Considering the manufacturing process and economic cost, the material property of each component was selected as shown in Table 2. The main structure parameters and the measuring range of the sensor are shown in Tables 3 and 4. Figure 10. Prototype of the large measurement range six-axis force sensor of 6-UPUR parallel mechanism with flexible joints.

Calibration Experiment
To measure the loaded external force accurately, the sensor should be calibrated by using a special calibration system, which can generate forces and torques in directions of x, y, z separately. In this paper the calibration system, which mainly included a hydraulic loading system, calibration

Calibration Experiment
To measure the loaded external force accurately, the sensor should be calibrated by using a special calibration system, which can generate forces and torques in directions of x, y, z separately. In this paper the calibration system, which mainly included a hydraulic loading system, calibration platform, parallel 6-UPUR six-axis force sensor with flexible joints, signal processing device, data acquisition device, data processor, calibration software system and so on, as shown in Figures 11 and 12, was designed and manufactured.
platform, parallel 6-UPUR six-axis force sensor with flexible joints, signal processing device, data acquisition device, data processor, calibration software system and so on, as shown in Figures 11 and  12, was designed and manufactured.
Considering that the structure of the six-axis force sensor of 6-UPUR parallel mechanism with flexible joints is very complex and the presence of measuring error can't be ignored because of the influence of these factors such as the design principles, manufacturing and processing errors, so multiple point loading in the sensor range and the method of least squares linear fit are needed in calibration experiments. Thus, the linear relationship between inputs and outputs of the large measurement range 6-axis force sensor of 6-UPUR parallel mechanism with flexible joints can be calculated. Figure 11. Structure of the calibration experiment system. (1) Each axis force/torque within the sensor range is divided into 10 load points in two positive and negative directions, as shown in Table 5; Figure 11. Structure of the calibration experiment system. platform, parallel 6-UPUR six-axis force sensor with flexible joints, signal processing device, data acquisition device, data processor, calibration software system and so on, as shown in Figures 11 and  12, was designed and manufactured.
Considering that the structure of the six-axis force sensor of 6-UPUR parallel mechanism with flexible joints is very complex and the presence of measuring error can't be ignored because of the influence of these factors such as the design principles, manufacturing and processing errors, so multiple point loading in the sensor range and the method of least squares linear fit are needed in calibration experiments. Thus, the linear relationship between inputs and outputs of the large measurement range 6-axis force sensor of 6-UPUR parallel mechanism with flexible joints can be calculated. Figure 11. Structure of the calibration experiment system. (1) Each axis force/torque within the sensor range is divided into 10 load points in two positive and negative directions, as shown in Table 5; Considering that the structure of the six-axis force sensor of 6-UPUR parallel mechanism with flexible joints is very complex and the presence of measuring error can't be ignored because of the influence of these factors such as the design principles, manufacturing and processing errors, so multiple point loading in the sensor range and the method of least squares linear fit are needed in calibration experiments. Thus, the linear relationship between inputs and outputs of the large measurement range 6-axis force sensor of 6-UPUR parallel mechanism with flexible joints can be calculated.
Detailed experimental steps are described are as follows:

Experimental Method
In the analysis of experimental data, the forces that are loaded on the prototype during the experiment can be described as six linearly independent vectors, denoted as:

Experimental Method
In the analysis of experimental data, the forces that are loaded on the prototype during the experiment can be described as six linearly independent vectors, denoted as: When the external forces are loaded on the measuring platform of the sensor, the six single-axis force sensors will sense the force produced on the measuring legs, and then the change of output voltage of the Wheatstone bridges can be measured out. We can get a set of relationships: where G is the calibration matrix between the loaded forces and the output voltages; V is the output voltage matrix. Thus the calibration matrix between the loaded forces and the output voltages of the six measuring legs can be calculated: Besides, the error matrix evaluating the accuracy of the six-axis force sensor is defined as follows: where F FS is the full range of the sensor, F 6ˆ6 can be calculated based on the mapping matrix between the loaded force and the output voltage by Equation (32); the error matrix E rr is a comprehensive evaluation. The diagonal components of the error matrix separately represent the measurement errors of the six different directions of the loaded external force, and other components represent the interference errors between different directions.

Calibration Results and Analysis
Through the calibration system described above, the calibration experiments based on the sensor prototype are carried out, and the data of the six measuring legs are obtained. Taking a complete static calibration experimental data and using the K value method and least square method to substitute the loading forces and the voltages data which are collected from six measuring legs into the corresponding equation, the static calibration matrix G K , G 2C and the error matrix E rrK , E rr2C can be obtained by the analysis and process of the experimental data, therefore, the performance of force-measuring of the sensor is obtained: From Equation (34), it can be noted that when the K value method is used to decouple, the calibration error of each axis is: F x (4.31%), F y (2.64%), F z (3.33%), M x (2.08%), M y (1.63%), M z (0.76%). The maximum error of the I kind is 4.31%, which appears in F x ; the maximum error of the II kind is 13.40%, which is appears in F y , when loaded in M y . So the calibration error obtained by this method is 13.40%. From Equation (36), it can be noted that when the least squares method is used to decouple, the calibration error of each axis is: F x (1.22%), F y (0.66%), F z (0.89%), M x (0.39%), M y (0.59%), M z (0.42%).The maximum error of the I kind is 1.22%, which appears in F x ; the maximum error of the II kind is 2.68%, which appears in F y , when loaded in M y , so the calibration error obtained by this method is 2.68%.
From the results of the calibration experiment, the calibration error comparison of the two kinds of decoupling methods is as shown in Table 6. The results show that the accuracy of least squares method is much better than that of the K value calibration method.

Linearity Analysis
The input signal and the output signal of the sensor are not completely linear, and there is always an error. The ratio of the error to the measurement range is called the linearity of the sensor. When a certain force/torque is loaded to one direction of the sensor, the output voltage of the six legs will change with the change of the loading force/torque. Figure 11 shows the changing curves between the sensor's measuring force and the standard loading force when the force or torque is loaded in one direction. Figures 14 and 15 are the variations of the voltage of each leg with the loading force/torque changing, when the force/torque is loaded in the three directions of X, Y and Z.   Table 7 is the linearity of each leg obtained by least square method. As shown in Table 6, the linearity of each leg of the sensor is less than 1%, which shows that the sensor has a good linearity.  Table 7 is the linearity of each leg obtained by least square method. As shown in Table 6, the linearity of each leg of the sensor is less than 1%, which shows that the sensor has a good linearity.

Conclusions
In this paper, to overcome the influence of the gap and friction of the traditional joints on the parallel six-axis sensors' precision and stability, we have successfully demonstrated a kind of six-axis force sensor based on 6-UPUR parallel mechanism with flexible joints, which has large measurement range and high accuracy. The force mathematical model of the sensor is established on the screw theory; according to the relations of the stiffness and deformation compatibility condition, the stiffness matrix considering flexibility of each flexible joint is built up; then the complete mathematical model is established. The sensor prototype and the calibration system are manufactured and static calibration experiments were carried out on the sensor. The results show that the measurement error is less than 2.68%, which shows that the sensor has high measuring accuracy and good linearity. The experimental results prove the feasibility of the large measurement range six-axis force sensor of 6-UPUR parallel mechanism with flexible joints. The design and loaded research of the parallel six-axis force sensor has reference significance.