sensors Article
Error Modeling and Experimental Study of a Flexible Joint 6-UPUR Parallel Six-Axis Force Sensor Yanzhi Zhao 1,2, *, Yachao Cao 1,2 , Caifeng Zhang 1,2 , Dan Zhang 3 and Jie Zhang 4 1 2 3 4
*
Key Laboratory of Parallel Robot and Mechatronic System of Hebei Province, Yanshan University, Qinhuangdao 066004, China;
[email protected] (Y.C.);
[email protected] (C.Z.) Key Laboratory of Advanced Forging & Stamping Technology and Science of Ministry of Education of China, Yanshan University, Qinhuangdao 066004, China Department of Mechanical Engineering, Lassonde School of Engineering, York University, 4700 Keele Street, Toronto, ON M3J1P3, Canada;
[email protected] Department of Basic Teaching, LiRen College of Yanshan University, Qinhuangdao 066004, Hebei, China;
[email protected] Correspondence:
[email protected]; Tel.: +86-335-805-7031
Received: 22 July 2017; Accepted: 19 September 2017; Published: 29 September 2017
Abstract: By combining a parallel mechanism with integrated flexible joints, a large measurement range and high accuracy sensor is realized. However, the main errors of the sensor involve not only assembly errors, but also deformation errors of its flexible leg. Based on a flexible joint 6-UPUR (a kind of mechanism configuration where U-universal joint, P-prismatic joint, R-revolute joint) parallel six-axis force sensor developed during the prephase, assembly and deformation error modeling and analysis of the resulting sensors with a large measurement range and high accuracy are made in this paper. First, an assembly error model is established based on the imaginary kinematic joint method and the Denavit-Hartenberg (D-H) method. Next, a stiffness model is built to solve the stiffness matrix. The deformation error model of the sensor is obtained. Then, the first order kinematic influence coefficient matrix when the synthetic error is taken into account is solved. Finally, measurement and calibration experiments of the sensor composed of the hardware and software system are performed. Forced deformation of the force-measuring platform is detected by using laser interferometry and analyzed to verify the correctness of the synthetic error model. In addition, the first order kinematic influence coefficient matrix in actual circumstances is calculated. By comparing the condition numbers and square norms of the coefficient matrices, the conclusion is drawn theoretically that it is very important to take into account the synthetic error for design stage of the sensor and helpful to improve performance of the sensor in order to meet needs of actual working environments. Keywords: parallel six-axis force sensor; flexible joints; error modeling; Monte Carlo method; calibration experiment
1. Introduction Compared with the traditional multi-axis force sensor, the sensor with flexible joints has advantages of fast response, small accumulated error, no mechanical friction and high measurement accuracy, so it has broad application prospects [1–5]. At present, the design of sensors with flexible joints can be divided into two categories: the majority of sensors are designed and processed based on the integral structure. The other is using the assembled structure. For the former, there have been numerous research achievements. Kerr [6] proposed that the Stewart platform with instrumented elastic legs can be used as a six-axis force sensor. Gao et al. [7] developed a six-axis controller based on the Stewart platform-based force sensor, and introduced the use of elastic joints to replace the real spherical joints which made miniaturization possible. Liang et al. [8] designed and developed a Sensors 2017, 17, 2238; doi:10.3390/s17102238
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new six-axis sensor system with a compact monolithic elastic element, which detected the tangential cutting forces along the x-, y-, and z-axes as well as the cutting torques about the x-, y-, and z-axes simultaneously. Unfortunately, restricted by their integrated structure, most of the sensors mentioned above are used in a small range of applications. In addition, the main error source of these sensors is deformation error. As for the latter assembled by flexible kinematic joints, Yang [9] developed a planar three-axis force sensor with flexible joints to diagnose and monitor bearing faults online in real time. Zhang [10] studied the model reconstruction theory of flexible assembly six-axis force sensors based on a hybrid leg spoke layout. Li [11] established an integral stiffness model of a flexible assembly six-axis force sensor based on the Stewart mechanism. These sensors are assembled traditionally. Consequently, the errors are mainly caused by the assembly process, which leads to large errors and low accuracy, so how to achieve high accuracy while taking into account a large measurement range is still a challenging problem. At present, there is limited literature available on this issue. Zhao et al. [12] proposed a large measurement range flexible joints six-axis sensor. Its mathematical modeling and calibration experiments were performed. Inevitably, the main errors of flexible assembly force sensors involve not only deformation errors, but also assembly errors. Many excellent studies [13–16] on error modeling and analysis of the parallel mechanism have been conducted so far. Arai and Ropponen [17] modeled and analyzed the error of the Stewart mechanism based on the vector algebra loop increment method. In addition, through the singular value decomposition of the force Jacobian, analytical expressions of the structural parameters of the Stewart platform, actuated error and end error were obtained. Wang and Massory [18,19] introduced the joint point error and actuated joint error, and end error of the mechanism was solved by a D-H numerical method. Wang and Ehmann [20] used a coordinate transformation method to establish input-output equations including the joint manufacturing error and positioning error, and then directly differentiated it, establishing the error model. Aimed at manufacturing error, installation error and actuator motion error of the parallel mechanism, Patel and Ehmann [21] performed an error modeling and analysis of a parallel machine in terms of route planning by means of a mechanism motion differential method and further considered the effect of joint manufacturing errors on end pose. Zou et al. [22] quantitatively analyzed the influence of characteristic parameter errors on the end pose error of the mechanism by using the error transfer matrix of the parallel mechanism. Huang [23] applied screw theory to model and analyze known size errors, control errors and kinematic joint gap errors. Ma et al. [24] established a space vector chain model and deduced the analytic mapping relationship between manufacturing errors of a parallel machine and the pose error of a moving platform. Lv et al. [25] proposed an error modeling method based on the forward kinematics problem. Unfortunately, there are few related literatures that comprehensively consider modeling the two main types of error (assembly error and deformation error), which results in some limitations to improve accuracy of large measurement range sensors. Based on the flexible joints 6-UPUR six-axis force sensor developed in the prephase, this paper focuses on establishment of the error modeling, namely, assembly error modeling and deformation error modeling. The synthetic error of the force-measuring platform is superposed by the two kinds of errors, resulting in a total pose error. Then, the corresponding first order influence coefficient matrix G0 is calculated. Meanwhile, deformation of the force-measuring platform are detected by using laser interferometry and analyzed to verify the correctness of the sensor error model, and calibration experiments are completed to obtain the first order kinematic influence coefficient matrix G0B in actual circumstances. The structure of this paper is as follows: after this Introduction, Section 2 introduces the structure of the prototype sensor and solves the theoretical first order kinematic influence coefficient. Sections 3 and 4 present the error modeling and analysis of the sensor in terms of assembly error and deformation error, respectively. Section 5 comprehensively considers the two main errors, and the first order kinematic influence coefficient when the synthetic error is taken into account is obtained. Section 6 introduces the experimental research on measurement and calibration of the sensor prototype and
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analyzes the results of the experiment. The paper is concluded in Section 7, summarizing the work that has been Sensors 2017,done. 17, 2238 3 of 20 Sensors 2017, 17, 2238
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2. Prototype of the Flexible Joints 6-UPUR Six-Axis SensorThe of the the sensor sensor prototype and analyzes the results results of the theForce experiment. The paper paper is is concluded concluded in in Section Section of prototype and analyzes the of experiment. 7, summarizing the work that has been done.
7, physical summarizing the work been done. A prototype ofthat the has large measurement range 6-UPUR six-axis force sensor with flexible joints2.was manufactured, as shown in Figure 1. Considering the manufacturing process and economic Prototype of of the the Flexible Flexible Joints Joints 6-UPUR 6-UPUR Six-Axis Six-Axis Force Force Sensor Sensor 2. Prototype cost, the material properties of the sensor are listed in Table 1. The main parameters of the sensor are A physical physical prototype prototype of of the the large large measurement measurement range range 6-UPUR 6-UPUR six-axis six-axis force force sensor sensor with with flexible flexible as follows:A radius of the force-measuring platform is 550 mm; radius of the fixed platform is 550 mm; joints was manufactured, as shown in Figure 1. Considering the manufacturing process and economic joints was manufactured, as shown in Figure 1. Considering the manufacturing process and economic the vertical distance between the two platforms is 300 mm; measuring range are: Fx : ±10,000 N, Fy : cost, the the material material properties properties of of the the sensor sensor are are listed listed in in Table Table 1. 1. The The main main parameters parameters of of the the sensor sensor are are as as cost, ±10,000 N, Fzradius : ±10,000 N,force-measuring Mx : ±5,000 Nplatform m, My : ± 5,000 N m, Mz : ± 5,000 N mplatform and overload capacity follows: of the is 550 mm ; radius of the fixed is 550 mm; follows: radius of the force-measuring platform is 550 mm ; radius of the fixed platform is 550 mm; is 120%. the vertical distance between the two platforms is 300 mm; measuring range are: Fx: ±10,000 N, the vertical distance between the two platforms is 300 mm; measuring range are: Fx: ±10,000 N, Fyy:: ±10,000 ±10,000 N, N, F Fzz:: ±10,000 ±10,000 N, N, M Mxx:: ±5,000 ±5,000 N N m, m, M Myy:: ±5,000 ±5,000 N m, M Mzz:: ±5,000 ±5,000 N Nm m and and overload overload capacity capacity F Table 1. Material propertiesNofm, the sensor. is 120%. is 120%. Components Materials Elastic Modulus Poisson Ratio Table 1. 1. Material Material properties properties of of the the sensor. sensor. Table Force-measuring platform Hard aluminum alloy 70 Gpa 0.30 Components Materials Elastic Modulus Poisson Poisson Ratio Components Materials Elastic Modulus Flexible joints 40CrNiMoA 206 Gpa 0.30Ratio Force-measuring platform Hard aluminum alloy 70 Gpa 0.30 Fixed platform Q235 210 Gpa 0.25 Force-measuring platform Hard aluminum alloy 70 Gpa 0.30 Flexible joints joints 40CrNiMoA 206 Gpa Gpa 0.30 Flexible 40CrNiMoA 206 0.30 Fixed platform platform Q235 210 Gpa Gpa 0.25 Fixed Q235 210 0.25
Density 2700 kg/m3 Density Density 7830 kg/m3 3 3 2700 kg/m 3 7850 kg/m 2700 kg/m 7830 kg/m kg/m33 7830 7850 kg/m kg/m33 7850
A 3D model of the six-axis force sensor with flexible joints is shown in Figure 2. The structure Ajoints 3D model model of the the six-axis six-axis force sensor withdegree flexibleofjoints joints is shown shown in Figure Figure 2. 2.Each The structure structure where allA are flexible joints force withsensor a single freedom is adopted. leg is a split 3D of with flexible is in The where all joints are flexible joints with a single degree of freedom is adopted. Each leg is split structure. upper block of of two flexible and of the where The all joints arepositioning flexible joints withisa composed single degree freedom is rotation adopted. joints, Each leg is aaone split structure. The upper positioning block is composed of two flexible rotation joints, and one of the joints upperspherical positioning block is composed of two flexible rotation and one of the joints jointsstructure. forms a The flexible joint with the flexible universal joint byjoints, an assembling relationship. forms aa flexible flexible spherical spherical joint with the the flexible universal joint joint by by an an assembling assembling relationship. relationship. The The forms with universal The middle part of the leg is joint mounted by flexible a single-axis force sensor. The lower part is composed of middle part of the leg is mounted by a single-axis force sensor. The lower part is composed of a flexible middle part of the leg is mounted by a single-axis force sensor. Thepositioning lower part is composed of aelastic flexibleleg is a flexible universal joint with an integral structure and a lower block. Each universal joint joint with with an an integral integral structure structure and and aa lower lower positioning positioning block. block. Each Each elastic elastic leg leg is is connected connected to to universal connected to the measuring-force and fixed through platforms through thelower upperpositioning and lower positioning blocks the measuring-force and fixed platforms the upper and blocks by bolts, the measuring-force and fixed platforms through the upper and lower positioning blocks by bolts, by bolts, respectively. decomposition of theexternal six-axis external force six legs is realized. respectively. Thus, Thus, decomposition of the six-axis force to the six legstoisthe realized. respectively. Thus, decomposition of the six-axis external force to the six legs is realized.
Figure 1. Physical prototype of the 6-UPUR six-axis force sensor with flexible joints.
Figure 1. Physical prototypeofofthe the6-UPUR 6-UPUR six-axis with flexible joints. Figure 1. Physical prototype six-axisforce forcesensor sensor with flexible joints.
Figure 2. 3D 3D Model of the6-UPUR 6-UPUR six-axis six-axis force force sensor with flexible joints. Figure 2. 3D Model ofof the forcesensor sensorwith with flexible joints. Figure 2. Model the 6-UPUR six-axis flexible joints.
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2， ， 6) Figure 3 illustrates the sensor structure based on 6-UPUR parallel mechanism. Bi ( i = 1，
stands for center point of the first revolute joint axis on the lower positioning block, which is Figure 3 illustrates the sensor structure based on 6-UPUR parallel mechanism. Bi (i = 1, 2, · · · , 6) 2， ， 6 ) denotes center point of the revolute joint axis on the adjacent to the fixed platform. bi ( i = 1， stands for center point of the first revolute joint axis on the lower positioning block, which is adjacent upper positioning block. Their matrices expressed as rB joint and axis rb , on respectively. to the fixed platform. bi ( i = 1, 2, ·coordinate · · , 6) denotes center are point of the revolute the upper According spaceTheir staticcoordinate equilibrium conditions, the following can be obtained by screw positioningtoblock. matrices are expressed as rBequation and rb , respectively. According to theory [26]: equilibrium conditions, the following equation can be obtained by screw theory [26]: space static 6
F = 6 faii$i Fww = ∑ i =1 f a $i
(1) (1)
i =1
where f ai represents magnitude of axial tension/compression force on the i-th measuring leg; $i where f ai represents magnitude of axial tension/compression force on the i-th measuring T leg; $i represents the unit line vector along the i-th measuring leg, expressed as $i = (Si S0 i ) ; F Tw is represents the unit line vector along the i-th measuring leg, expressed as $i = ; Fw S S 0i referred to generalized external force vector on center of the measuring platform, iexpressed as is referred to generalized external force vector on center of the measuring platform, expressed as T Fw = (fw m w ) , then, T it can be obtained as: Fw = fw mw , then, it can be obtained as: 6 fai Si fw = 6 i =1 (2) fw = ∑6 f ai Si m =i=1 f i S (2) 6 a 0i w mw = i =1 ∑ f ai S0i i =1
where Si = rb ( : i ) - rB ( : i ) rb ( : i ) - rB ( : i ) ; S0 i = rb ( : i ) × rB ( : i ) rb ( : i ) - rB ( : i ) . where Si = [rb (: i ) − rB (: i )]/|rb (: i ) − rB (: i )|; S0i = [rb (: i ) × rB (: i )]/|rb (: i ) − rB (: i )|. Then, Equation (1) can be rewritten in form of matrix expression as: Then, Equation (1) can be rewritten in form of matrix expression as: Fw = GFa (3) Fw = GFa (3) where Fa represents axial tension/compression force of all legs, expressed as 1 F 2 represents 3 4 5axial 6 tension/compression force of all legs, where expressed as Fa = a Fa = ( fa fa fa fa fa fa)T ; G denotes the first order kinematic influence coefficient matrix ; G denotes the first order kinematic influence coefficient matrix which f a1 isf a2alsof a3called f a4 Jacobian f a5 f a6 matrix: which is also called Jacobian matrix: " S S S # G = S11 S22 · · · 6S6 (4) G = S01 S02 S06 (4) S01 S02 · · · S06 T
The Jacobian matrix The Jacobian matrix directly directly determines determines many many characteristics characteristics of of the thesensor, sensor, such such as as tis tis isotropy, isotropy, stiffness, sensitivity, etc. It is the foundation to study the performance and structure design of the stiffness, sensitivity, etc. It is the foundation to study the performance and structure design of the sensor. sensor.
b5
b4
b6
b3
b1
b2 B5
B6
B4 B3
B1
B2
Figure 3. The The force force sensor sensor structure based on 6-UPUR parallel mechanism. Figure
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3. Assembly of the the 6-UPUR 6-UPUR Force Force Sensor Sensor Based Based on on Imaginary Imaginary Kinematic Kinematic Joint 3. Assembly Error Error Modeling Modeling of Method Joint Method
In the the last lastsection, section,the theJacobian Jacobian matrix the six-axis external force exerted the G between matrix G between the six-axis external force exerted on the on sensor sensor andtension/compression axial tension/compression force the measuring is a definite But in practice and axial force on theon measuring legs islegs a definite value.value. But in practice due duethe to deformation the deformation caused manufacturing, assembly andcalibration, calibration,the themechanical mechanical part part will to caused by by manufacturing, assembly and suffer a certain certain deviation. deviation. Thus, Thus, the the transformation transformation relation relation in in different coordinate frames of the sensor is changed, which leads to a change of the originally set sensor working position and forms a measurement error. Consequently, in this this section section the the assembly error of the 6-UPUR parallel six-axis force sensor is modeled. This part mainly aims at radius errors of the force-measuring platform and platform,errors errorsof of axial clearances forlower the lower positioning block and theuniversal middle fixed platform, twotwo axial clearances for the positioning block and the middle universal joint and installation error of single-axis force sensor. The deformation error model of the joint and installation error of single-axis force sensor. The deformation error model of the sensor is sensor is established insection. the next section. established in the next The working byby thethe five errors of The working position positionerror errorof ofthe theforce-measuring force-measuringplatform platformisisaccumulated accumulated five errors one corresponding leg. To establish the sensor error model easily, the fixed coordinate frame and of one corresponding leg. To establish the sensor error model easily, the fixed coordinate frame and moving coordinate coordinate frame frame are are defined defined as as shown shown in in Figure Figure 4. 4. moving
Figure 4. Diagram structure. Diagram of fixed coordinate frame and moving coordinate frame of the sensor structure.
= 11, ， 2， forfor center point positioning 2,· ， ·6·), 6stands BBii((ii = center pointofofthe thefirst firstrevolute revolutejoint jointaxis axis on on the the lower lower positioning ) stands fixed platform. platform. These six points can theoretically compose a planar block, which is adjacent to the fixed frame named named OOBB−- XXBBYYB BZZ hexagon. A fixed coordinate frame is attached attached to to the the geometric center point B B is O The ZZBB-axis -axis isisarranged arrangedon onthe thenormal normal direction of the fixed plane; OBB of of the the hexagon. The direction of the fixed basebase plane; the the XB X -axis is perpendicular to connection between two points B and B ; the Y -axis is determined by the B -axis is perpendicular to connection between two points B11 and 2B2 ; theB YB -axis is determined by right-hand rule. Similarly, bi (i = 1, 2, · · · , 6) stands for center point of the revolute joint axis on the 2， ， 6 ) stands for center point of the revolute joint axis on the the right-hand rule. Similarly, bi ( i = 1， upper positioning block, and a moving coordinate frame Ob − Xb Yb Zb is established. upper positioning block, and a moving framecoordinate Ob - XbYb Z b is established. Applying the D-H method [27], we coordinate establish a local frame on the i-th measuring leg as i i Applying the D-H method [27], we establish a local coordinate onlink the on i-ththe measuring leg shown in Figure 5. S j , a j( j+1) respectively refer to the axial vector offrame the j-th i-th leg and i i S j , between a j ( j+1) respectively refer to the axial vector of the j-th as shownnormal in Figure common line5.vector two adjacent axes, which can be expressed as:link on the i-th leg and common normal line vector between two adjacent axes, which can be expressed as: 0 i i S j = T j−1 − sin0α( j−1) j i cos iiαα( j−1)j S j = Tj-1 -sin ( j-1) j i cos α( j-1) j
(5)
(5)
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6 of 20 cos iθj i a j ( j+1) = Tj-1 cos i α( j-1) j sin i θj (6) i cos i θ i sin α( j-1) j sinj θj i a j( j+1) = T j−1 cos i α( j−1) j sin i θ j (6) i α coordinate a local where i Tj denotes rotation transform matrix of sin sin i θ frame of the j-th link on the i-th
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( j −1) j
j
leg relative to the fixed coordinate frame O - X YB ZB , which can be obtained as: where i T j denotes rotation transform matrixBof aBlocal coordinate frame of the j-th link on the i-th leg relative to the fixed coordinate framei TOB=− X Y Z , which a j ( Bj+1)B SBj × a j ( j+1) can obtained as: S j be (7) j h i i i a j( j+1) line Sj i a = a j( j+ (7) 1) S j ×normal common The setover along i S j of twoT jadjacent ( j-1) j and a j ( j+1 ) is denoted by i
S j . The length of the common normal line and rotation angle are denoted by i a j ( j+1) and The setover along i S j of two adjacent common normal line i a( j−1) j and i a j( j+1) is denoted by i S j . i θ j , length respectively. The of the common normal line and rotation angle are denoted by i a and i θ , respectively. j ( j +1)
j
Figure 5. D-H coordinate frame of the i-th leg. i As is well known, representsthe theaxis axisof ofthe therevolute revolutejoint. joint. If If there there exists exists rotation rotation around around As is well known, i SS11 represents i the XXBB-axis, -axis, itit can can directly directly map map to to i SS11. . However, However, if if there there exists exists translation XBB -axis, -axis, that that the translation along along the the X is is taken into account, it will lacklack certain definition. For is to say, say, the theradius radiuserror errorofofthe thefixed fixedplatform platform taken into account, it will certain definition. this this purpose, a new error modeling mechanism method is proposed. That is, is, thethe radius error ofofa For purpose, a new error modeling mechanism method is proposed. That radius error afixed fixedplatform platformisisrepresented representedby byan animaginary imaginaryprismatic prismaticjoint jointwhich whichisismounted mounted on on the connection between the leg and the fixed fixed platform. platform. We We define define its its motion motionalong alongpositive positivehalf halfof ofthe the XBB -axis as the positive direction, namely, namely, there there exists exists a positive radius error, error, and and the corresponding coordinate i i a i i Yi i S is established. By the same reason, the radius error of the force-measuring frame O − force-measuring OBB - a0101Y0 0S0 0 is established. By the same reason, the radius error platform byby an an imaginary prismatic jointjoint and the coordinate frame platform isisalso alsorepresented represented imaginary prismatic andcorresponding the corresponding coordinate i a i Y i i S iis established. i O − These imaginary prismatic joints and coordinate frames are illustrated b 67b - 7 a677 Y7 S7 is established. These imaginary prismatic joints and coordinate frames are frame O in Figure 6. illustrated in Figure 6.
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(b)
(a)
Figure 6. Diagram of imaginary prismatic joints. (a) Coordinate frame establishment of the fixed Figure 6. Diagram of imaginary prismatic joints. (a) Coordinate frame establishment of the fixed platform radius error; (b) Coordinate frame establishment of the measuring platform radius error. platform radius error; (b) Coordinate frame establishment of the measuring platform radius error. i
Rj denotes the position vector of the origin i Oj of the j-th link on the i-th leg expressed in the denotes the position vector of the origin i O j of the j-th link on the i-th leg expressed in the fixedj coordinate frame. It can be calculated by the following equation:s, fixed coordinate frame. It can be calculated by the following equations, i i i i i i i i i R j = S1 S1 + a12 a12 + S2 S 2 + a23 a23 + + i Sj i S j (8) i (8) R j = i S1 i S1 + i a12 i a12 + i S2 i S2 + i a23 i a23 + · · · + i S j i S j i P denotes position vector of the origin Ob of the force-measuring platform expressed in the iR
i P denotes fixed coordinate frame.vector It can be obtained equation:platform expressed in the fixed position of the originusing Ob ofthe thefollowing force-measuring i iobtained i i using the i i i i equation: i coordinate frame. It ican be following P = S S + a a + + S S + a a + S i S + i a i a + iS i S (9) 1
1
12
12
5
5
56
56
6
6
67
67
7
7
i i i i i i i i i According P =toi SEquations a12 + and · · · +combining S5 S5 + i athe + i a67 i a67coefficient + i S7 i S7 theory, the(9) 56 akinematic 56 + S6 S6influence 1 S1 + a12(5)–(9) i R rotation influence coefficient sub-matrix G3×7 and translation influence coefficient sub-matrix i According to Equations (5)–(9) combining kinematicthe influence coefficient theory, can be solved. Forand general parallelthe mechanisms, following relationship existsthe G3P×7 of each legs i GR i GP i i i i rotation influence coefficient sub-matrix and translation influence coefficient sub-matrix i R i P 3×7 3×7 a , S , θ , α [28]: between the matrices G3×7 , G3×7 and parameters of each legs can be solved. For general parallel mechanisms, the following relationship exists between i iPθ, i α [28]: ∂ iG P i a, =i S, the matrices i G3R×7 , i G3P×7 and parameters Ga j ∂ i aj i P i P ii PP G∂ G ∂ ∂∂i aijS == GSa jj j i P∂i GP (10) i P i R ∂G ∂i Sij == iG GθsP jj , ∂ G i = iGθR j (10) ∂i∂GPθj i P ∂i GR ∂ θij R = Gθ j , ∂ii θ R = Gθ j i P∂i θ G i iPj = iGαP j , ∂ i Gj i = iGαR j ∂ ∂α ∂ αj R ∂ iG j= i GαP j , ∂ iGR = i G αj ∂α ∂α j
j
Then, all the corresponding influence coefficient matrices iGaP , iGSP , iGθR , iGθP , iGαR and iGαP i P i P i R i P i R i P all the corresponding influence coefficient of Then, each error source can be solved by Equation (10). matrices Ga , GS , Gθ , Gθ , Gα and Gα of each error source be solved by actual Equation (10). errors, vectors i S1 , i S2 , i S3 , i S4 , i S5 , and i S6 are not Due tocan existence of the assembly i , i S , i S , i S , i S i , and i S are not coplanar. Due to existence of the actual assembly errors, vectors 2 3 5S1 ( i = 1， 4 26， ， 6 ) can be coplanar. By the space geometry and sensor accuracyS1requirements, By the space geometry and sensor accuracy requirements, i S1 (i = 1, i2, · · · , 6) can be assumed in the 2， ， 6 ) of the revolute assumed in the plane XBYB , as shown Figure 7, as is the axial vector S ( i = 1， plane XB YB , as shown Figure 7, as is the axial vector i S6 (i = 1, 2, · · · , 66) of the revolute joint on the joint on the upper positioning block. upper positioning block. 1 Taking S6 for example, according to the design and processing requirements of the sensors, 1 Taking S6 for example, according to the design and processing requirements of the sensors, the 1 1 1 the directions S3 are identical. Meanwhile, S3 is taken as the direction that 0joint b1' 1 directions of 1 S6 of andS16 Sand 3 are identical. Meanwhile, S3 is taken as the direction that joint b1 points at points at joint B1 , i.e.: joint B1 , i.e.: 0 ' b1 − B1 1 b B ( ) S = (11) 1 S3 3= 1 b10 1 − B (11) b1' 1- B1
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Figure 7. 7. Diagram equivalent angles. angles. Figure Diagram of of equivalent 1
and 1S represent both axes of the universal joint on the lower positioning block, so they and 1 S2 2represent both axes of the universal joint on the lower positioning block, so they meet 1 1 1 i meet the relationship: the structure ofsensor, the sensor, bethat seeni Sthat S3 . From S4 , ii S1 1 S = 1S i 1 ×. From the relationship: S21 =× S1 S the structure of the it canitbecan seen 2 3 4 , S1 and S2 , i i iand in samedue direction due todirection identicalof direction the two joints. universal joints. S5 are respectively S5 areSrespectively in same direction to identical the two of universal So far, all 2 , So far, the axis the first measuring legfound have been found and the other vectors can the axisall vectors on vectors the firston measuring leg have been out, and theout, other vectors can be obtained be obtained the same way. by the same by way. 3π i Then, twist angles of all axes can be obtained as: i α01i = 0◦ , i αi12 = 3π , α = ππ2 , i αi 34 = ππ2 , Then, twist angles of all ◦axes can be obtained as: α01 = 0 , α12 = 2 , i23 α23 = , α34 = , π i 3π i iα 2 2 2 45 = 2 , α56 = 2 , α67 = 0 . Meanwhile, other D-H parameters are further obtained by the π 3π i i following , i α67 = 0 . Meanwhile, other D-H parameters are further obtained by the α45 = , equation: α56 = i i i 2 2 a j( j+1) = S j × S( j+1) following equation: ia × i a( j −1) j (12) i θ = j ( j +1) cos j ia i i i a i × a j ( j+1) = S j ×j( j+S1()j+1) ( j−1) j Consequently, the error influence coefficientsi aof each including rotation influence coefficient × i aleg, (12) j ( j+1) ( j-1) j P i GR i Gj3=×7 iwill be icalculated according to kinematic influence cos θ 3×7 and translation influence coefficient a j ( j+1of a( j-1i-th ) × the ) j leg is established. coefficient theory [26] after the D-H coordinate frame Error integrations of each leg can be expressed as in vector form: ∆i a, ∆i S, ∆i θ and ∆i α. Consequently, the error influence coefficients i of each leg, including rotation influence Consideringi the working principle of the sensor, ∆ θ which is indirectly determined by other coefficient G3R×7 and translation influence coefficient iG3P×7 will be calculated according to parameters has no realistic meaning in the course of error analysis. kinematic influenceerror coefficient theory [26] after the D-H coordinate of the i-th leg is If the position and attitude error of the force-measuring platformframe are expressed as vectors T T established. ∆P = [ ∆Px ∆Py ∆Pz ] and ∆δ = [ ∆δx ∆δy ∆δz ] . Then, for the i-th leg, they can obtained as: Error integrations of each leg can be expressed as in vector form: Δ i a , Δ i S , Δ iθ and Δ i α . ( i Considering the working principle ofP the sensor, determined by other i i P Δ θ i which i Pis indirectly i ∆Pi = i G a × ∆ a + GS × ∆ S + Gα × ∆ α (13) parameters has no realistic meaning course of error analysis. iα ∆δi = iin GαRthe ×∆ If the position error and attitude error of the force-measuring platform are expressed as vectors T ΔP =If[ΔP ΔPy ΔPof δ = [ Δδ Δδyis taken Δδz ]T into . Then, for thethe i-th leg, they obtained thex influence legs’Δerror sources account, vectors are can rewritten as: as: z ] alland x S1 1S 1
(
)
i P i i6 P i i P6 i 6 αP i Δ P Si+ Gα × Δ 1 Pi = iGaP × Δi a + GS i × Δ i ∆P = 6 ∑ i GRa ·∆ ai + ∑ GS ·∆ S + ∑ Gα ·∆ α (13) δii==1 Gα × Δ α i=1 i =1 Δ (14) 6 ∆δ = 16 ∑ i GαR ·∆i α If the influence of all legs’ errori=sources is taken into account, the vectors are rewritten as: 1 6 6 1 6position force-measuring platform i P i error and i P attitude i i P of ithe Furthermore, the comprehensive error ΔP = Ga ⋅ Δ a + GS ⋅ Δ S + Gα ⋅ Δ α 6 i =1 i =1 i =1 q are defined as: 2 (14) 2 2 6 1 ∆P = ∆P + ∆P + ∆P ( ) ( ) | | y z i R i x Δδ = Gαq⋅ Δ α (15) 2 6 i =1| = (∆δ )2 + ∆δ + (∆δ )2 |∆δ
x
y
z
Furthermore, the comprehensive position error and attitude error of the force-measuring platform are defined as:
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ΔP = Δδ =
( ΔPx ) ( Δδx )
(
)
(
)
2
+ ΔPy
2
+ Δδy
2
+ ( ΔPz )
2
+ ( Δδz )
2
2
(15) 9 of 20
The sensor error sources analyzed in the above includes the radius errors of the force-measuring platform and fixed platform, errors of the axial clearances for the lower The sensor error sources analyzed in the above includes thetwo radius errors of the force-measuring positioning block and the middle universal joint and installation error of single-axis force and sensor, platform and fixed platform, errors of the two axial clearances for the lower positioning block the i i i i i which correspond to the five D-H parameters , , , and , respectively. Δ a Δ a Δ a Δ a Δ S 67 01 12 45 3 middle universal joint and installation error of single-axis force sensor, which correspond to the five i a , ∆processing i a , ∆i a and According to the accuracy of the sensor, the tolerance ranges of processing each error D-H parameters ∆inine a67 , ∆stage ∆i S3 , respectively. According to the nine stage 01 12 45 TΔ i a = 155ranges μm , ofTeach = 130 μm , T = 36 μm , T μm and source are respectively: i i i i = 36 accuracy of the sensor, the tolerance error source are respectively: T = 155 µm, Δ a01 Δ a12 Δ ∆ a45 a 67 67 TT∆iia ==87130 μmµm, . T∆i a12 = 36 µm, T∆i a45 = 36 µm and T∆i S3 = 87 µm. Δ S01 3 Now, the Monte Carlo simulation analysis method [29] is adopted to simulate and analyze the Now, the Monte Carlo simulation analysis method [29] is adopted to simulate and analyze the pose error of the force-measuring platform caused by assembly of 6-UPUR six-axis force sensor with pose error of the force-measuring platform caused by assembly of 6-UPUR six-axis force sensor with flexible joints. Firstly, the error sources with different distribution characteristics are sampled. From the flexible joints. Firstly, the error sources with different distribution characteristics are sampled. From theory of mechanical technology, when the workpiece is produced in single batch and small-scale the theory of mechanical technology, when the workpiece is produced in single batch and production, the dimension error is a normal distribution in its tolerance range T. According to ±3σ small-scale production, the dimension error is a normal distribution in its tolerance range T . principle [30], standard deviation of each error source can be obtained as: According to ±3σ principle [30], standard deviation of each error source can be obtained as: T σσ = (16) =T (16) 66
sampling value value of of these these error error sources sources is is calculated calculated by by the thefollowing followingequation: equation: Then the sampling p −2lnμ cos ( 2πμ ) ΔW 2lnµ11 cos(2πµ22 ) ∆W ==σσ −
(17) (17)
where both μ and μ2 are the random numbers between 0–1. where both µ1 1and µ2 are the random numbers between 0–1. By MATLAB, the sample sizes of these error sources are all 100. Substituting in Equation (15), By MATLAB, the sample sizes of these error sources are all 100. Substituting in Equation (15), then the position error and attitude error are statistically simulated. Figures 8 and 9 show the then the position error and attitude error are statistically simulated. Figures 8 and 9 show the influence influence of all the five error sources on the comprehensive position error and the comprehensive of all the five error sources on the comprehensive position error and the comprehensive attitude error attitude error of the force-measuring platform, respectively. It should be noted that in the legend, of the force-measuring platform, respectively. It should be noted that in the legend, REM, REF, ECU, REM, REF, ECU, ECP and IES indicate the radius errors of the force-measuring platform and fixed ECP and IES indicate the radius errors of the force-measuring platform and fixed platform, errors of platform, errors of two axial clearances for the middle universal joint and the lower positioning two axial clearances for the middle universal joint and the lower positioning block and installation block and installation error of single-axis force sensor, respectively. error of single-axis force sensor, respectively.
Figure 8. 8. The The influence influence of of the the five five error Figure error sources sources on on the the comprehensive comprehensive position position error error of of the the platform. platform.
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Figure 9. The influence of the five error sources on on the the comprehensive comprehensive attitude attitude error error of of the the platform. platform.
It can can be be seen seen that It that the the installation installation error error of of single-axis single-axis force force sensor, sensor, among among the the five five error error sources, sources, has the the greatest greatest influence influence on on the the comprehensive comprehensive position position and and attitude attitude error. error. Due Due to to the the cumulative cumulative has amplification of of errors, errors, the the radius radius error error of of the the fixed fixed platform platform and and error error of of the the two two axial amplification axial clearances clearances on on the lower positioning block also have great impact. Comparatively, the other two error the lower positioning block also have great impact. Comparatively, the other two error sources sources have have less impact. impact. Meanwhile, less Meanwhile, the the radius radius error error of of the the force-measuring force-measuring platform platform has has aa huge huge influence influence on on the comprehensive attitude error. Therefore, conclusions can be drawn that the radius accuracy the comprehensive attitude error. Therefore, conclusions can be drawn that the radius accuracy of of force-measuring platform platformand andfixed fixed platform mounting accuracy of single-axis force force-measuring platform andand axialaxial mounting accuracy of single-axis force sensor sensor particularly are ensured in themanufacturing sensor manufacturing particularly are ensured in the sensor process.process. Force Sensor Sensor 4. Deformation Error Modeling of the 6-UPUR Force
In the theworking workingprocess processofof sensor, elastic deformation of flexible is objective. The In thethe sensor, the the elastic deformation of flexible legs islegs objective. The actual actual working position of a reference point on the force-measuring platform will also change working position of a reference point on the force-measuring platform will also change accordingly, accordingly, which seriously affects the static of performance which seriously affects the static performance the sensor. of the sensor. When a six-dimensional external force vector is exerted exerted at at the the end end of of the i-th flexible series leg, can be be obtained obtained as as follows follows by by the the principle principle of of virtual virtual work: work: it can iTi T ! i i ih i i i i 1,1,2,2, · ,6 Δx ij ,Δy , 6 i , ∆z i , ∆α i , ∆α i , ∆α i j S j =∆x j ,Δz j , Δα x j ,Δα y j ,Δα z j ii = = · · ∆Sij Δ=S jJij=SijJ = ij , ∆y xj yj zj j j Fi =FjJi i= FJiFji F i jj ==1,1,2,2,3,3,4 4
j
(18) (18)
Fj
where ∆S ΔSi ijdenotes denotes the deformation vector end reference point caused elastic deformation where the deformation vector at at thethe end reference point caused byby elastic deformation of j i i leg. S stands for the elastic deformation vector of the j-th basic flexible element for the i-th the j-th basic flexible element for the i-th leg. S stands for the elastic deformation vector produced by j j
i the end force Fi atend theforce end ofFthe for theelement i-th leg.for Fij refers counterforce produced by the atj-th the basic end offlexible the j-thelement basic flexible the i-thtoleg. Fji refers
i i stands for the iposei vector at the endvector of theat j-th element by the produced end force F to counterforce thebasic end flexible of the j-th basic produced flexible element by. Jthe j end force F . J j i transformation matrix. JFj denotes the force transformation matrix. stands for the pose transformation matrix. JFji denotes the force transformation matrix. According to the superposition principle of deformation, the total deformation vector ∆Si ofi the According to the superposition principle of deformation, the total deformation vector ΔS of flexible leg end is obtained as follows: the flexible leg end is obtained as follows: 4
4
4
4
j =1
j=1
j =1
j= 1
i i i i i ii i i i i i i i ∆Si =Δ∑ Jij SijJ = J1 SJ11S+1 +J2JS22S2++· · ·++J 6iJS6i 6S 2,， S i ∆S = ΔS∑ 1， 21, ， 6· )· · , 6) (6i(=i = j = j = j Sj =
(19) (19)
Under the definition of the stiffness matrix, the relationship between the leg end force F i and the total deformation vector ΔSi is:
F i = K i ΔSi ( i = 1， 2， ， 6)
(20)
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Under the definition of the stiffness matrix, the relationship between the leg end force Fi and the total deformation vector ∆Si is: Fi = Ki ∆Si (i = 1, 2, · · · , 6) (20) where Ki denotes stiffness matrix at the end of the flexible leg. Similarly, the counterforce vector Fij at the end of the j-th basic flexible element can be expressed as: Fij = Kij ∆Sij
(21)
Combining the above equations, the total deformation vector can be rewritten as: −1 ∆Si = Ki Fi =
4
4
j =1
j =1
∑ Jij ∆Sij = ∑ Jij
Kij
−1
Fij =
4
∑ Jij
Kij
−1
JiFj Fi (i = 1, 2, · · · , 6)
(22)
j =1
where Kij refers to the stiffness matrix of the j-th basic flexible element: 4
∑ Jij
Ki =
Kij
−1
! −1 JiFj
(i = 1, 2, · · · , 6)
(23)
j =1
Then the stiffness matrix Ki can be expressed easily. Based on the stiffness model of each leg, the overall stiffness matrix of flexible joints 6-UPUR six-axis force sensor can be obtained. At the same time, we assume that the force-measuring platform stiffness reaches infinity and the small deformation produced by the external force is ignored. When a six-dimensional external force vector Fw is exerted, the geometric compatibility condition between the end of the i-th leg and reference point of the force-measuring platform is as follows: ∆S =
∆x ∆y ∆z ∆αx ∆αy ∆αz
Op T Oip R = 03×3
O
−Opip RT S(ri ) Op T Oip R
·
∆xi ∆yi ∆zi ∆αix ∆αiy ∆αiz
= Ji ∆Si
(24)
where ∆S stands for the deformation vector at the center reference point of the force-measuring platform. ∆x and ∆xi refer to the linear displacement vector of the force-measuring platform and the i-th leg along x-axis, respectively. Similarly, ∆y, ∆yi , ∆z and ∆zi denote those along the y-, and z-axis, respectively. ∆α x and ∆αix refer to the angular displacement vector of the force-measuring platform and the i-th leg along x-axis, respectively. Similarly, ∆αy , ∆αiy ∆αz and ∆αzi denote those along the y-, and z-axis, respectively.
Op Oip R
stands for the rotation matrix of the measuring platform expressed in a local coordinate frame where the moving coordinate frame Op is relative to the local coordinate frame Oip . S(ri ) refers to the vector of the platform expressed in the fixed coordinate frame. According to the principle of spatial force system synthesis, the relationship between the six-dimensional external force vector Fw and the counterforce vector Fi at the end of the i-th leg can be established as: Fw =
fx fy fz mx my mz
Op 6 Oip R = ∑ i = 1 S ( r i )O p R Oip
Op R Oip 03×3
f xi f yi f zi mix miy miz
6 = ∑ JiF Fi i =1
(25)
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In addition, according to the definition of stiffness matrix of the flexible parallel mechanism, the six- dimensional external force vector Fw is: 6
Fw = K∆S =
−1 6 6 i i i i i i i i J F = J K ∆S = J K ∑ F ∑ F ∑ F J ∆S
i =1
i =1
(26)
i =1
Then the stiffness matrix K of the reference point is expressed as: 6
K=
∑ JiF Ki
−1 Ji
(27)
i =1
When an external force Fw exerted on the platform changes by δFw , the micro displacement vector of the reference point is: δD = K−1 δFw (28) Then, the deformation error of the platform caused by elastic deformation of the flexible legs can be solved by Equation (28) when the external force Fw exerted on the platform changes. When the external force fw or the torque mw exerted on the platform change by 1000 N or 1000 Nm, the corresponding deformation vectors calculated by Equation (28) are shown as Table 2. Table 2. Theoretical calculation value of reference point deformation of the force-measuring platform. Force/Torque Variation (N/N·m)
Force along X Axis
Force along Y Axis
Force along Z Axis
Torque around X Axis
Torque around Y Axis
Torque around Z Axis
1000
230 µm
190 µm
27 µm
82 arc s
79 arc s
67 arc s
5. Synthetic Error of the 6-UPUR Parallel Six-Axis Force Sensor Assume the assembly error and deformation error are expressed as ∆D1 and ∆D2 , respectively. ∆D1 is obviously a function with respect to the sensor structure parameters, which is certain for the processed sensor. When the external force exerted on the platform is certain, that is to say, ∆D2 is assured, then, the synthetic error ∆D of the platform is the deformation coupling resulting from the assembly and exerted force, which can expressed as: ∆D = ∆D1 + ∆D2 =
h
∆Px
∆Py
∆Pz
∆δx
∆δy
∆δz
i
(29)
As is well known, when the synthetic error of the platform is taken into account, the homogeneous transformation matrix with respect to ideal position of the platform is: " ∆T =
∆R 01×3
∆P 1
# (30)
where ∆P is translational component of the force-measuring platform, ∆P = ∆R can be expressed by the RPY description method:
cos ∆δz cos ∆δy ∆R = cos ∆δz sin ∆δy sin ∆δx − sin ∆δz cos ∆δx cos ∆δz sin ∆δy cos ∆δx + sin ∆δz sin ∆δx
h
∆Px
sin ∆δz cos ∆δy sin ∆δz sin ∆δy sin ∆δx + cos ∆δz cos ∆δx sin ∆δz sin ∆δy cos ∆δx − cos ∆δz sin ∆δx
∆Py
∆Pz
iT
;
− sin ∆δy cos ∆δy sin ∆δx cos ∆δy cos ∆δx
Then, the transform matrix of the force-measuring platform after deformation is: " T = ∆TT0 =
R0 01×3
P0 1
# (31)
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where T0 represents the pose transformation matrix of the ideal position of the force-measuring platform expressed in the fixed coordinate frame. whereHere, T0 represents the pose transformation matrix the position of the force-measuring G' of taking into account the synthetic error, can beideal calculated by Equation (32): platform expressed in the fixed coordinate frame. ' S' G S'20 can beScalculated Here, taking into account the synthetic error, by Equation (32): 6 G' = 1' (32) ' ' S S S " 01 02 06 # S10 S20 · · · S60 G0 = (32) 0 0 0 S01 S02 · · · S06 6. Deformation Measurement and Calibration Experiments 6. Deformation Measurement and Calibration This experimental equipment consists of aExperiments hardware and software system. The former mainly includes a hydraulic loading system, loading calibration signal processing device, data This experimental equipment consists of a hardware andbench, software system. The former mainly acquisition device, data processor, etc. The hydraulic loading system provides the loading force. By includes a hydraulic loading system, loading calibration bench, signal processing device, data calibrating device, the twodata hydraulic cylinders thehydraulic loading calibration bench,provides which transmit force force. to the acquisition processor, etc. in The loading system the loading measuring platform, and adjusting the installation positions of the two loading units every time, six By calibrating the two hydraulic cylinders in the loading calibration bench, which transmit force dimensional forces and torques can be exerted on the platform. There are eight output signal to the measuring platform, and adjusting the installation positions of the two loading units every channels from the single-axis sensor the calibration experiments are time, six dimensional forces andtension-compression torques can be exerted on when the platform. There are eight output performed. The signals are transmitted to the computer by a signal processing device and data signal channels from the single-axis tension-compression sensor when the calibration experiments acquisition card, and then processed by theto calibration software are performed. The signals are transmitted the computer by a system. signal processing device and data In the loading process of the deformation measurements, one or two loading units should be acquisition card, and then processed by the calibration software system. chosen according the loading The measurements, specific implementation is as follows: a loading In the loadingto process of thedirection. deformation one or two loading units shouldunit be X -axis. By adjusting the tension/compression is installed on one upright column side along the B chosen according to the loading direction. The specific implementation is as follows: a loading unit is
mode ofon theone hydraulic thealong loading along XB -axis be achieved. The same installed upright cylinder, column side theforce XB -axis. Bythe adjusting thecan tension/compression modeis true of the loading along the Y -axis. Both loading units are installed on two upright column of the hydraulic cylinder, the loading force along the XB -axis can be achieved. The same is trueends of B the alongof thethe YB -axis. Both loading units are installed column ends in Both the in loading the direction XB -axis, then the loading force alongon thetwo ZBupright -axis can be achieved. direction the X then the force along the sides ZB -axisalong can be achieved. units B -axis, loading of units are installed on loading two upright column the YB -axis.Both By loading adjusting the are installed on two upright column sides along the YB -axis. By adjusting the tension/compression tension/compression mode of the hydraulic cylinder, the loading torque along the XB -axis can be mode of the hydraulic cylinder, the loading torque along the XB -axis can be achieved. Similarly, both achieved. Similarly, both loading units are installed on two upright column sides in the direction of loading units are installed on two upright column sides in the direction of the XB -axis, and then the the X -axis, and then the loading torque along the Y -axis can be obtained. Two loading units are loadingB torque along the YB -axis can be obtained. TwoB loading units are installed on two upright installed on twosides upright column different in or thethedirection the XB -axis or the the loading YB -axis, column different in the direction of the sides XB -axis YB -axis, of respectively. Then respectively. Then the loading along the ZB -axis can be obtained. torque along the ZB -axis can be torque obtained. Based ofof force and torque mentioned above, an optical lenslens is mounted on Basedon onthe theloading loadinglocation location force and torque mentioned above, an optical is mounted the platform. The position of a laserofinterferometer is adjusted is and then theand deformation on measuring the measuring platform. The position a laser interferometer adjusted then the ofdeformation the platform measured. The laser interferometer and optical lens are ofcan the be platform can be measured. The laser interferometer andinstallation optical lenslocation installation shown in are Figures 10 in and 11, respectively. location shown Figures 10 and 11, respectively.
Figure10. 10.Laser Laserinterferometer interferometerand andoptical opticallens lensinstallation installationlocation. location. Figure
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(a)
(b)
(c)
(d)
Figure 11. Optical lens installation measuring in six directions. (a) Linear displacement measurement
Figure 11. Optical lens installation measuring in six directions. (a) Linear displacement measurement along the X / YB -axis. (b) Linear displacement measurement along the ZB -axis. (c) Pitching angle along the XB /YBB-axis; (b) Linear displacement measurement along the ZB -axis; (c) Pitching angle measurement around the X / YB -axis. (d) Swing angle measurement around the ZB -axis. measurement around the XB /YBB -axis; (d) Swing angle measurement around the ZB -axis.
Each axial force/torque within the sensor range is divided into 10 load points in two positive
Each axial force/torque within theassensor is 3. divided intoor10torque load points in two positive and negative directions, respectively, shownrange in Table Load force in a corresponding and negative directions, respectively, showndirection in Tableof3.loading Load force orConversely torque in the a corresponding direction are applied according to theaspositive points. reversely load is applied in descending we save the data of the laser interferometer loaded direction are applied according order. to the Then, positive direction of loading points. Conversely theevery reversely time. We follow the experimental steps described in [12], and then check and process the data andevery load is applied in descending order. Then, we save the data of the laser interferometer loaded decoupled calculation and result analysis are carried out. time. We follow the experimental steps described in [12], and then check and process the data and decoupled calculation and result analysis are carried out. Table 3. Loading points of calibration force/torque.
1 2 3of calibration 4 5 force/torque. 6 7 8 9 10 Table 3. Loading points Loading Points 11 12 13 14 15 16 17 18 19 20 positive 1000 30003 5000 7000 90005 7000 5000 3000 1 2 4 6 7 1000 80 Force (N) Loading Points negative −1000 −3000 −5000 −7000 −9000 −7000 −5000 −3000 −1000 0 11 12 13 14 15 16 17 18 positive 1000 3000 5000 7000 9000 7000 5000 3000 1000 0 Torque (N·m) positive 1000 3000 −3000 5000 9000−70007000 5000−10003000 negative −1000 −5000 7000 −7000 −9000 −5000 −3000 0 Force (N) negative −1000 −3000 −5000 −7000 −9000 −7000 −5000 −3000 positive 1000 3000 6.1. Measurement Results and Analysis Torque (N·m) negative −1000 −3000
5000 −5000
7000 −7000
9000 −9000
7000 −7000
5000 −5000
3000 −3000
9
10
19
20
1000 −1000
0 0
1000 −1000
0 0
The linear displacement or pitching angle comparisons of the platform between the calibration deformation measurement results and the theoretical calculation results of the synthetic error are 6.1. Measurement Results and Analysis made as shown in Figures 12–17. XB -axis,between so in the the theoretical Since the sensor structure is theoretically theplatform The linear displacement or pitching angle symmetrical comparisonsabout of the calibration calculation, when the force is exerted along the XB -axis, the linear displacement along the positive deformation measurement results and the theoretical calculation results of the synthetic error are made and negative half12–17. of the XB -axis is symmetrical about the XB -axis, and with any increase of the as shown in Figures are loading force, the structure linear displacements alongsymmetrical the positive and negative of so theinXthe Since the sensor is theoretically about the XBhalf -axis, theoretical B -axis 2327.3 μm and linearlywhen increased. The theoretical values ofthe the linear maximum displacement are the calculation, the force is exerted calculation along the X displacement along positive B -axis, 2327.3 2079.88 μm and −half respectively. The maximum positive and negative measurements are negative ofμm the, X -axis is symmetrical about the X -axis, and with any increase of the loading B B −2129.72 μm as shownalong in Figure 12. force,and the linear displacements the positive and negative half of the XB -axis are linearly increased. The theoretical calculation values of the maximum displacement are 2327.3 µm and −2327.3 µm, respectively. The maximum positive and negative measurements are 2079.88 µm and −2129.72 µm as shown in Figure 12.
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15 15 of of 20 20 15 15 of of 20 20
Figure -axis. Figure 12. 12. Linear Linear displacement displacement comparison comparison along along the the XXBB-axis. Figure Figure 12. 12. Linear Linear displacement displacement comparison comparison along along the the X BB -axis. -axis.
Figure 13. Linear displacement comparison along the YB -axis. Figure -axis. Figure 13. Linear displacement comparison along the YYBBB-axis. -axis. Figure 13. 13. Linear Linear displacement displacement comparison comparison along along the the
Similarly, the theoretical calculation value of the maximum displacement along positive and Similarly, the theoretical calculation value of the maximum displacement along positive and Similarly, theoretical value the maximum displacement along positive and μmofand −1714.85 μm negative half the of the Y -axiscalculation are 1714.85 , respectively. The maximum negative half of the YBBB -axis are 1714.85 μm and −1714.85 μm , respectively. The maximum negative half of the Y -axis are 1714.85 µm and − 1714.85 µm, respectively. The maximum B measurements are 1799.87 μm and −1838.26 μm (Figure 13). The theoretical value of the 1799.87 μm −1838.26 μm (Figure measurements are are1799.87 and 13). The theoretical value of the measurements µm and 1838.26 µm 13). Z The theoretical value of the maximum maximum displacement along the−positive and(Figure negative B -axis are 336.5μm and −336.5 μm , −336.5 μm , maximum displacement along the negative ZB -axis andrespectively. displacement along the positive andpositive negativeand ZB -axis are 336.5 µm are and 336.5μm −336.5 µm, respectively. The maximum measurements are 428.21μm andB −206.99 μm (Figure 14). The maximumThe measurements are 428.21 µmare and 428.21μm −206.99 µm (Figure 14).μm (Figure 14). −206.99 respectively. maximum measurements and
Figure 14. Linear displacement comparison along the ZB -axis. Figure 14. Linear displacement comparison along the Z -axis. Figure -axis. Figure 14. 14. Linear Linear displacement displacement comparison comparison along along the the ZBBB-axis.
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Figure 15. Pitching angle comparison around the X B -axis. Figure 15. Pitching angle comparison around the X B -axis. Figure 15. Pitching angle comparison around the XB -axis. Figure 15. Pitching angle comparison around the X B -axis.
Figure 16. Pitching angle comparison around the YB-axis. -axis. Figure 16. 16. Pitching Pitchingangle anglecomparison comparison around around the the YY B -axis. Figure B Figure 16. Pitching angle comparison around the YB -axis.
Figure 17. Pitching angle comparison around the Z -axis. Figure 17. Pitching angle comparison around the ZBB -axis. Figure 17. Pitching angle comparison around the ZB -axis.
As sensor structure is symmetrical theoretically about the X B -axis As shown shown in inFigure Figure1515since sincethe the sensor structure is symmetrical theoretically about the X Figure 17. Pitching angle comparison the Zof As increase shown in 15 since thethe sensor structure isaround symmetrical theoretically theangle XBB B -axis. with the of Figure the loading torque, theoretical calculation values the maximum about pitching -axis with the increase of the loading torque, the theoretical calculation values of the maximum -axis with the increase of the torque, the theoretical of the maximum around the X are 766.2 arc sloading and −766.2 arc s. The maximumcalculation positive andvalues negative measurements B -axis aresensor 766.2 structure arc s andis−766.2 arc s. The maximumabout positive and pitching angle around the15Xsince B -axis As shown in Figure the symmetrical theoretically the and XofB XBs,-axis are 766.2For arcthe s same and −766.2 The maximum positive pitching angle around thearc are 729.4 arc s and −700.57 respectively. reason,arc thes.theoretical calculation value -axis with thepitching increaseangle of the loading torque, the values the maximum the maximum around the positive and theoretical negative YBcalculation -axis are 731.4 arc sof and −731.4 arc s, pitching angle around the XB -axis are 766.2 arc s and −766.2 arc s. The maximum positive and
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respectively. The maximum measurements are 612.75 arc s and −697.37 arc s (Figure 16). The theoretical value of the maximum pitching angle around the positive and negative ZB -axis are 685.75 arc s and −685.75 arc s, respectively. The maximum measurements are 635.94 arc s and −674.47 arc s (Figure 17). From Figure 17, it can be seen that when the force is exerted along the ZB -axis, deformation of the force-measuring platform has an obvious nonlinear relationship with the magnitude of the force, and the deviation is larger, compared with the theoretical result. The main reason is that the loading force along the ZB -axis is achieved by two loading units, which are installed at both ends of the loading benches along the XB -axis, rather than loading the platform directly along the ZB -axis as in the theoretical analysis. Because of manufacturing errors, it is difficult to achieve complete symmetry of the sensor structure, so the measurement value will produce a deviation with the theoretical value. When the force/torque is exerted along the other directions, the deformation of the force-measuring platform basically has a linear relationship with the magnitude of the force/torque, and the measured results are basically consistent with the theoretical results. Then, the correctness of synthetic error model is verified. At the same time, the deformation error of the flexible leg is the main error factor that affects sensor accuracy and with increase of the loading force/torque, so the proportionality is more obvious. 6.2. Calibration Results and Analysis In this section, the actual first order kinematic influence coefficient matrix GB0 is obtained by the calibration results. Then we compare it with theoretical first order kinematic influence coefficient matrix G0 and the first order kinematic influence coefficient matrix G when the synthetic error is taken into account. The relationship between external force and output voltage matrix is Fw = GB V. Then the −1 calibration matrix can be expressed as GB = Fw VT VVT by the least squares method [14]. Next, GB will be transformed into the transfer relation matrix between the external force and the measuring force, that is, the actual first order kinematic influence coefficient matrix G0B . From the technical parameters of the force sensitive element, the spokewise single-axis force sensor, it can be known that its range is 2 t; the sensitivity is 2.0 ± 0.01mV/V and supply voltage is DC10 V, so when the sensor is loaded by 2 t, the output signal of the sensor is 20mV. Assume that f0a represents the actual axial force of the single-axis force sensor, whose units are N or N m. The actual output signal of the single-axis sensor is expressed as V0 , whose units are mV. Here, the relationship between them is V0 = f0a /980. On the other hand, due to circuit amplification and denoising, the relationship between V and V0 is V = kV0 (k stands for voltage amplification factor). Therefore, the transfer relationship between the external force and the actual axial force is: Fw = GB V = kGB V0 =
k GB f0a 980
(33)
Afterwards, the actual first order kinematic influence coefficient matrix G0B is: G0B =
k GB 980
(34)
So far, the theoretical first order kinematic influence coefficient matrix G, the first order kinematic influence coefficient matrix G0 when the synthetic error is taken into account and the actual first order kinematic influence coefficient matrix G0B can be calculated easily by Equations (4), (32) and (34), respectively. As is well known the condition number [31] of the first order kinematic
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influence coefficient matrix is one of the indices to measure isotropy for a force sensor. Consequently, their condition numbers are calculated as: cond(G)= 6.3336 (35) cond G0 = 11.7549 cond G0 = 10.7808 B From Equation (35), it can be seen that condition number of G0 is more close to that of G0B than G’s. By calculation, the relative errors are 9.03% and 41.25%, respectively. Obviously, G0 is similar to G0B . On the other hand, the square norm [27] of a channel output signal vector is used to measure sensitivity of a generalized force component. The sensitivity SFx , SFy and SFz of the three force components can be expressed as: SFx = kJ1 k2 (36) SFy = kJ2 k2 S = kJ k Fz 3 2 The sensitivity S Mx , S My and S Mz of the three torque components are expressed as: S Mx = kJ4 k2 S My = kJ5 k2 S = kJ k Mz 6 2
(37)
where Ji (i = 1, 2, · · · , 6) stands for column vector of force Jacobian matrix. All the component sensitivities of the three force Jacobian matrices are calculated as shown in Table 4. It can be seen that sensitivity of J0 is more close to that of J0B than J’s. Their square norm relative errors can be seen in Table 5. Here their relative errors are defined as Type 1 error and Type 2 error. Table 4. All the component sensitivities of the three force Jacobian matrices. Sensitivity
SFx
SFy
SFz
SMx
SMy
SMz
J J0 J0B
0.2964 1.0459 1.4205
0.5472 0.8465 0.9930
0.5473 0.9009 1.8518
1.5025 6.8417 8.2074
1.3973 3.7478 3.2378
1.3974 6.0042 6.9202
Table 5. The two type relative errors of all the component sensitivity. Sensitivity
SFx (%)
SFy (%)
SFz (%)
SMx (%)
SMy (%)
SMz (%)
Type 1 error Type 2 error
26.37 79.13
14.75 44.89
51.35 70.44
16.64 81.69
15.75 56.84
13.24 79.81
Table 5 shows that the Type 1 error is less than the Type 2 error. That is to say, G0 is more close to than G. It is worth noting that Type 1 SFz has a larger relative error. The main reason is that the two loading units are installed at both ends of loading benches along the XB -axis to provide the ZB -axis loading force, which is explained in the previous section. Consequently, we will not bore readers with a very detailed analysis to explain the reason any more. Obviously, the effectiveness of the error model is clarified and it is very important to take into account synthetic errors for the design stage of the sensor and this is helpful to improve the performance of the sensor in order to meet the needs of actual working environments. G0B
7. Conclusions In this paper, assembly error and deformation error are comprehensively taken into account based on the flexible joints 6-UPUR parallel six-axis force sensor developed with a large measurement range and high accuracy in the prophase. Their error models are respectively established. The synthetic
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error of the platform is deformation coupling resulting from assembly and exerted force. Then the first order kinematic influence coefficient matrix when the synthetic error is taken into account is solved. Measurements and calibration experiments are carried out. Forced deformation of the force-measuring platform is detected by using a laser interferometer and analyzed to verify the correctness of the synthetic error model. In addition, the first order kinematic influence coefficient matrix in actual circumstances is calculated. Condition numbers and square norms of the coefficient matrices are compared, which shows theoretically that it is very important to take into account the synthetic error for the design stage of the sensor and this is helpful to improve the performance of the sensor in order to meet needs of actual working environments. Acknowledgments: The authors would like to acknowledge the project supported by the National Natural Science Foundation of PR China (NSFC) (Grant No. 51105322), the Natural Science Foundation of Hebei Province (Grant No. E2014203176), the Natural Science Research Foundation of Higher Education of Hebei Province (Grant No. QN2015040) and China’s Post-doctoral Science Fund (No. 2016M590212). Author Contributions: All authors contributed extensively to the study presented in this manuscript. Yanzhi Zhao conceived and designed the flexible joints 6-UPUR parallel six-axis force sensor; Yachao Cao, Jie Zhang and Caifeng Zhang performed the experiments; Yanzhi Zhao, Yachao Cao, Jie Zhang and Caifeng Zhang established the error model, analyzed the data and wrote the paper; Jie Zhang and Dan Zhang revised the paper and polished the language. All authors contributed with valuable discussions and scientific advices in order to improve the quality of the work, and also contributed to write the final manuscript. Conflicts of Interest: The authors declare no conflict of interest.
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