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Impact of pore solution concentration on the accelerated mortar bar alkali-silica reactivity test
Advanced Materials Research Vols 317-319 (2011) pp 1041-1044 © (2011) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMR.317-319.1041
Online: 2011-08-16
Errors Analysis and Compensation of Six-axis Force Sensor Based on Stewart Platform Zhijun Wang1,a, Jiantao Yao1,2, Yulei Hou1 and Yongsheng Zhao1,b 1
Parallel Robot & Mechatronics System Laboratory, Yanshan University, Qinhuangdao, Hebei, 066004, P. R. China
2
Jiangsu Key Lab of Digital Manufacturing, Technology Huaiyin institute of technology, Huai’an, Jiangsu, 223001, P. R.China a
b
[email protected], [email protected]
Keywords: Force sensor; Six-axis; Stewart platform; Measurement error; Compensation
Abstract. Six-axis force sensors based on Stewart platform necessitate highly accurate, sensitivity and dynamic response. In response to this need, errors analysis and compensation of the force sensor are essential. In this paper, the measurement error generated by the upper platform deformation is discussed and evaluated. Furthermore, in order to improve the precision, a real-time compensation algorithm is proposed depending on the external force applied on the force sensor. Finally, a numerical simulation example is presented, which indicates that the precision is related to the stiffness of limbs directly and improved obviously by the compensation algorithm. Introduction Scientists have been researching the design of an intelligent robot with interest, because people want to get a mechanism that can work for a human being. In order to get the intelligent robot more similar to the human behavior, the six-axis force /torque sensor which can measure three force components and three torque components simultaneously is mounted to the hand of it. The six-axis force/torque sensor is also widely used in both industries and research as well, and real-time measurement of six-axis force/torque is a foundation to realize compliance control and multi-degree-freedomcoordinated control on industrial robots. In order to achieve low uncertainties in force measurements, the performance characteristics such as elastic deformation, repeatability, linearity and creep errors influence the measurement uncertainty of force sensor directly. In the field of robot manipulator research, Zhen Huang [1] analyzed the error of position and orientation in a robot hand. In recent years, a lot of literatures on the errors analysis, and compensation of the six-axis force/torque sensor are available. Chul-Goo Kang [2] derived a closed-form solution of the forward kinematics by means of linearization of the inverse kinematics equations, and evaluated the force error of the proposed method. In Ref. [3] the position and orientation error of a six-component force sensor based on Stewart platform was analyzed. Most of researchers obtained the measurement error from calibration machine and calibration experimental [4]. In this paper, the measurement error generated by the upper platform deformation and the real-time compensation to improve the precision of the sensor are discussed. Mathematical Model of Stewart Platform-based Force Sensor Fig. 1 illustrates the structure diagram of the six-axis force sensor based on Stewart platform. It is composed of an upper platform, a base platform and six elastic limbs connecting the two platforms with spherical joints; bi (i=1,2…6) and Bi(i=1,2…6) located at the angular vertex of hexagon respectively, are the center of i-th spherical joint of the upper platform and lower platform; the Cartesian coordinate o-xyz is set up with its origin located at the geometrical center of the upper platform, the z axis is chosen to be perpendicular to the surface of the upper platform, the y axis is All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, www.ttp.net. (ID: 152.14.136.77, NCSU North Carolina State University, Raleigh, USA-13/05/15,19:53:39)
1042
Equipment Manufacturing Technology and Automation
perpendicular to the line of b1b6; the central angles between b1 and b3, b3 and b5 are 2π/3, and the central angles between B1 and B3, B3 and B5 are 2π/3 too. Each of the limb’s length is denoted by li, i=1,2,...6. When an external force and torque are applied on the upper platform of the sensor, for the equilibrium of the upper platform, the following equation can be obtained as [5] 6
6
i =1
i =1
F = ∑ fi Si , M = ∑ ri × f i Si
(1)
where F and M represent the applying force vector and the applying toque vector acted on the center of the upper platform respectively; ri is the vector from the origin o to bi; Si represents the unit line vector along the i-th limb, and fi is the reacting force of the i-th limb. Eq. (1) can be rewritten in the form of matrix equation as F =G⋅ f
(2) T
where F = Fx Fy Fz M x M y M z is the vector of six-dimension external force applied on the T upper platform; f = [ f1 f 2 f3 f 4 f5 f 6 ] is the vector composed of the reacting forces of the six limbs; G is the force Jacobin matrix.
Position and Orientation Error of the upper Platform If the force F is applied on the upper platform of the sensor, the i-th limb will generate the straight displacement ∆li (i = 1 ~ 6) along its axial direction which will produce the infinitesimal displacement of the upper platform ∆D . Fig. 2 shows the deformation diagram when an external force applied on the force sensor.
Fig 1 Schematic diagram of the 6-SPS Stewart platform-based force sensor
Fig 2. Deformation diagram of an external force applied on the force sensor
The strain equation on the i-th limb can be derived using the tension and compression strain equations as follows f i = ki ⋅ ∆li = Ei Ai / li ⋅ ∆li
(3)
where ∆li is the axial elastic deformation of the i-th limb. ki is the stiffness coefficient of the i-th limb, Ei is the modulus of longitudinal elasticity, Ai is the cross-sectional area of the i-th limb. And we write the infinitesimal displacement of the upper platform ∆D in the form as ∆D = [∆d ∆θ ]T
where ∆d is the linear displacement, ∆θ is the rotational displacement. Based on kinematics of the Stewart platform, we have:
(4)
Advanced Materials Research Vols. 317-319
1 1 ∆li = Si ⋅ (∆d + ∆θ × ri ) , ∆Si = Si × (∆d + ∆θ × ri ) × Si = ( ∆d + ∆θ × ri − ∆li Si ) li li
1043
(5)
Eq.(5) denotes the relations between the displacements of the upper platform and the axial elastic deformation of the i-th limbs. We can obtain the displacements of the upper platform and express in the form of matrix as ∆D = (G T )−1 ∆l
(6)
where ∆l=[∆l1 ∆l2 ∆l1 ∆l4 ∆l5 ∆l6]T. According to Eq.(6), when every measured force of the limbs is given, the axial elastic deformation ∆li can be determined by Eq.(3). Then, for known changes in the length of the limbs, the position and orientation deviations of the upper platform can be determined. So after the external force applied on the upper platform, the final state force Jacobin matrix G ' can be expressed as follow
S' S2' ... S6' G' = ' 1 ' ' ' ' ' r1 × S1 r2 × S 2 ... r6 × S6
(7)
where Si' = Si + ∆Si , ri' = ri + ∆li Si , and then ri' × Si' = (ri + ∆li Si ) × ( Si + ∆Si ) = ri × ( Si + ∆Si ) . In conclusion, the final state force Jacobin matrix is determined by incorporating the displacements of all line vectors along limbs into Eq.(7). Then we defined the error by the position and orientation of the upper platform as follow
E=
F − Fc F
× 100%
(8)
where F is the factual force applied on the upper platform; Fc = G ⋅ f is the calculated force obtained by multiplying the force Jacobin matrix and the stress on limbs. It can be known from above that the displacement of Si can be obtained if the axial deformation of limbs ∆l is known, when an external force applied on the upper platform. And then the final state force Jacobin matrix G ' is derived by Eq.(7), which is changed by the external force. So, the position and orientation error can be eliminated by the compensation matrix G ' .
Numerical Example and Experimental Results For the six-axis force sensor based on Stewart platform, force Jacobin matrix is closely related with the five structural parameters, which include the radii of the base platform R, the radii of the upper platform r, the distance of the two platforms h, and the directional angles α, β. The preliminary values of the structural parameters are given as below
R = 100 mm, r = 60 mm, h = 70 mm, α = π 2, β = π 6 According to the structural parameters, the initial state force Jacobin matrix G can be calculated. Provided that the stiffness coefficients of all limbs are identical, k = 2.8×105N/m, and the external forces applied on the upper platform is Fx=100N, Fy=100N, Fz=100N, Mx=5Nm, My=5Nm, Mz=5Nm. The Experimental result is also obtained as follow. When external force Fx=100N applied on the upper platform of the given force sensor, we obtained the final state force Jacobin matrix G ' by Eq.(7). Fig.3 shows the measurement errors difference between external force and limb’s stiffness coefficient. It is evident that the differences of the error are quite small when the stiffness coefficient of limb is more than 106N/m. And the measurement errors also relate to the external force linearly, the
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Equipment Manufacturing Technology and Automation
bigger force applied the bigger errors produced. The measurement errors obtained by the six axes forces applied on the sensor are shown in Fig.4 respectively. Comparing the error without compensated with the compensated one, it can be seen the measurement error is reduced distinctly. 0.08 uncompensated 0.04 0.02 0 3 Stif 2 fnes 1 s (10 6 coeffici N/m ent )
compensated
0 20
40
80 60 Fx (N)
Error (%)
0.06
100
Fig 3 Measurement error difference between external force and limb’s stiffness coefficient
Fig 4 Measurement errors on six axes
Furthermore, it has been shown that the proposed real-time compensation algorithm is effective for reducing the effect of cross-axis error on measurements, which is significant to improve the measurement accuracy of the six-axis force sensor. Conclusions In this paper, the error analysis and compensation research of a six-axis force sensor based on Stewart platform has been conducted. The initial and final state force Jacobin matrixes are established depending on the external force applied on the force sensor. Numerical example and experimental results have shown that errors in force measurements generated by the upper platform deformation are quite small when the limb’s stiffness coefficient is more than 106N/m, but the errors become much bigger with increasing external force, and the compensation algorithm is effective for improving the measurement precision. This fact shows the validity of the proposed compensation method using state force Jacobin matrix instead of using initial state matrix. Acknowledgements This work was financially supported by the National Natural Science Foundation of China under Grant No. 50975245, Natural science foundation of Hebei province No. E2011203184, The research and development project of science and technology of Qinhuangdao city under Grant No. 201001A11, and Jiangsu province key lab of digital manufacturing open research No. HGDML-0911. References [1] Zhen Huang: Mech. Mach. Theory Vol. 22(6) (1987), p. 577 [2] Chul-Goo Kang: Sens. Actuator A Vol. 90 (2001), p. 31 [3] Yulei Hou, Jiantao Yao, Ling Lu, et al: Mech. Mach. Theory Vol. 44 (2009), p. 359 [4] Jiantao Yao, Yulei Hou, Jie Chen, et al: Sens. Actuator A Vol. 150 (2009), p. 1 [5] Zhen Huang, Yongsheng Zhao, Tieshi Zhao: Advanced Spatial Mechanism, Higher Education Press, Beijing, (2006). (In Chinese)
Equipment Manufacturing Technology and Automation 10.4028/www.scientific.net/AMR.317-319
Errors Analysis and Compensation of Six-Axis Force Sensor Based on Stewart Platform 10.4028/www.scientific.net/AMR.317-319.1041 DOI References [1] Zhen Huang: Mech. Mach. Theory Vol. 22(6) (1987), p.577. http://dx.doi.org/10.1016/0094-114X(87)90053-X [2] Chul-Goo Kang: Sens. Actuator A Vol. 90 (2001), p.31. http://dx.doi.org/10.1016/S0924-4247(00)00564-1 [3] Yulei Hou, Jiantao Yao, Ling Lu, et al: Mech. Mach. Theory Vol. 44 (2009), p.359. http://dx.doi.org/10.1016/j.mechmachtheory.2008.03.008 [4] Jiantao Yao, Yulei Hou, Jie Chen, et al: Sens. Actuator A Vol. 150 (2009), p.1. http://dx.doi.org/10.1016/j.sna.2008.11.030
Online: 2011-08-16
Errors Analysis and Compensation of Six-axis Force Sensor Based on Stewart Platform Zhijun Wang1,a, Jiantao Yao1,2, Yulei Hou1 and Yongsheng Zhao1,b 1
Parallel Robot & Mechatronics System Laboratory, Yanshan University, Qinhuangdao, Hebei, 066004, P. R. China
2
Jiangsu Key Lab of Digital Manufacturing, Technology Huaiyin institute of technology, Huai’an, Jiangsu, 223001, P. R.China a
b
[email protected], [email protected]
Keywords: Force sensor; Six-axis; Stewart platform; Measurement error; Compensation
Abstract. Six-axis force sensors based on Stewart platform necessitate highly accurate, sensitivity and dynamic response. In response to this need, errors analysis and compensation of the force sensor are essential. In this paper, the measurement error generated by the upper platform deformation is discussed and evaluated. Furthermore, in order to improve the precision, a real-time compensation algorithm is proposed depending on the external force applied on the force sensor. Finally, a numerical simulation example is presented, which indicates that the precision is related to the stiffness of limbs directly and improved obviously by the compensation algorithm. Introduction Scientists have been researching the design of an intelligent robot with interest, because people want to get a mechanism that can work for a human being. In order to get the intelligent robot more similar to the human behavior, the six-axis force /torque sensor which can measure three force components and three torque components simultaneously is mounted to the hand of it. The six-axis force/torque sensor is also widely used in both industries and research as well, and real-time measurement of six-axis force/torque is a foundation to realize compliance control and multi-degree-freedomcoordinated control on industrial robots. In order to achieve low uncertainties in force measurements, the performance characteristics such as elastic deformation, repeatability, linearity and creep errors influence the measurement uncertainty of force sensor directly. In the field of robot manipulator research, Zhen Huang [1] analyzed the error of position and orientation in a robot hand. In recent years, a lot of literatures on the errors analysis, and compensation of the six-axis force/torque sensor are available. Chul-Goo Kang [2] derived a closed-form solution of the forward kinematics by means of linearization of the inverse kinematics equations, and evaluated the force error of the proposed method. In Ref. [3] the position and orientation error of a six-component force sensor based on Stewart platform was analyzed. Most of researchers obtained the measurement error from calibration machine and calibration experimental [4]. In this paper, the measurement error generated by the upper platform deformation and the real-time compensation to improve the precision of the sensor are discussed. Mathematical Model of Stewart Platform-based Force Sensor Fig. 1 illustrates the structure diagram of the six-axis force sensor based on Stewart platform. It is composed of an upper platform, a base platform and six elastic limbs connecting the two platforms with spherical joints; bi (i=1,2…6) and Bi(i=1,2…6) located at the angular vertex of hexagon respectively, are the center of i-th spherical joint of the upper platform and lower platform; the Cartesian coordinate o-xyz is set up with its origin located at the geometrical center of the upper platform, the z axis is chosen to be perpendicular to the surface of the upper platform, the y axis is All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, www.ttp.net. (ID: 152.14.136.77, NCSU North Carolina State University, Raleigh, USA-13/05/15,19:53:39)
1042
Equipment Manufacturing Technology and Automation
perpendicular to the line of b1b6; the central angles between b1 and b3, b3 and b5 are 2π/3, and the central angles between B1 and B3, B3 and B5 are 2π/3 too. Each of the limb’s length is denoted by li, i=1,2,...6. When an external force and torque are applied on the upper platform of the sensor, for the equilibrium of the upper platform, the following equation can be obtained as [5] 6
6
i =1
i =1
F = ∑ fi Si , M = ∑ ri × f i Si
(1)
where F and M represent the applying force vector and the applying toque vector acted on the center of the upper platform respectively; ri is the vector from the origin o to bi; Si represents the unit line vector along the i-th limb, and fi is the reacting force of the i-th limb. Eq. (1) can be rewritten in the form of matrix equation as F =G⋅ f
(2) T
where F = Fx Fy Fz M x M y M z is the vector of six-dimension external force applied on the T upper platform; f = [ f1 f 2 f3 f 4 f5 f 6 ] is the vector composed of the reacting forces of the six limbs; G is the force Jacobin matrix.
Position and Orientation Error of the upper Platform If the force F is applied on the upper platform of the sensor, the i-th limb will generate the straight displacement ∆li (i = 1 ~ 6) along its axial direction which will produce the infinitesimal displacement of the upper platform ∆D . Fig. 2 shows the deformation diagram when an external force applied on the force sensor.
Fig 1 Schematic diagram of the 6-SPS Stewart platform-based force sensor
Fig 2. Deformation diagram of an external force applied on the force sensor
The strain equation on the i-th limb can be derived using the tension and compression strain equations as follows f i = ki ⋅ ∆li = Ei Ai / li ⋅ ∆li
(3)
where ∆li is the axial elastic deformation of the i-th limb. ki is the stiffness coefficient of the i-th limb, Ei is the modulus of longitudinal elasticity, Ai is the cross-sectional area of the i-th limb. And we write the infinitesimal displacement of the upper platform ∆D in the form as ∆D = [∆d ∆θ ]T
where ∆d is the linear displacement, ∆θ is the rotational displacement. Based on kinematics of the Stewart platform, we have:
(4)
Advanced Materials Research Vols. 317-319
1 1 ∆li = Si ⋅ (∆d + ∆θ × ri ) , ∆Si = Si × (∆d + ∆θ × ri ) × Si = ( ∆d + ∆θ × ri − ∆li Si ) li li
1043
(5)
Eq.(5) denotes the relations between the displacements of the upper platform and the axial elastic deformation of the i-th limbs. We can obtain the displacements of the upper platform and express in the form of matrix as ∆D = (G T )−1 ∆l
(6)
where ∆l=[∆l1 ∆l2 ∆l1 ∆l4 ∆l5 ∆l6]T. According to Eq.(6), when every measured force of the limbs is given, the axial elastic deformation ∆li can be determined by Eq.(3). Then, for known changes in the length of the limbs, the position and orientation deviations of the upper platform can be determined. So after the external force applied on the upper platform, the final state force Jacobin matrix G ' can be expressed as follow
S' S2' ... S6' G' = ' 1 ' ' ' ' ' r1 × S1 r2 × S 2 ... r6 × S6
(7)
where Si' = Si + ∆Si , ri' = ri + ∆li Si , and then ri' × Si' = (ri + ∆li Si ) × ( Si + ∆Si ) = ri × ( Si + ∆Si ) . In conclusion, the final state force Jacobin matrix is determined by incorporating the displacements of all line vectors along limbs into Eq.(7). Then we defined the error by the position and orientation of the upper platform as follow
E=
F − Fc F
× 100%
(8)
where F is the factual force applied on the upper platform; Fc = G ⋅ f is the calculated force obtained by multiplying the force Jacobin matrix and the stress on limbs. It can be known from above that the displacement of Si can be obtained if the axial deformation of limbs ∆l is known, when an external force applied on the upper platform. And then the final state force Jacobin matrix G ' is derived by Eq.(7), which is changed by the external force. So, the position and orientation error can be eliminated by the compensation matrix G ' .
Numerical Example and Experimental Results For the six-axis force sensor based on Stewart platform, force Jacobin matrix is closely related with the five structural parameters, which include the radii of the base platform R, the radii of the upper platform r, the distance of the two platforms h, and the directional angles α, β. The preliminary values of the structural parameters are given as below
R = 100 mm, r = 60 mm, h = 70 mm, α = π 2, β = π 6 According to the structural parameters, the initial state force Jacobin matrix G can be calculated. Provided that the stiffness coefficients of all limbs are identical, k = 2.8×105N/m, and the external forces applied on the upper platform is Fx=100N, Fy=100N, Fz=100N, Mx=5Nm, My=5Nm, Mz=5Nm. The Experimental result is also obtained as follow. When external force Fx=100N applied on the upper platform of the given force sensor, we obtained the final state force Jacobin matrix G ' by Eq.(7). Fig.3 shows the measurement errors difference between external force and limb’s stiffness coefficient. It is evident that the differences of the error are quite small when the stiffness coefficient of limb is more than 106N/m. And the measurement errors also relate to the external force linearly, the
1044
Equipment Manufacturing Technology and Automation
bigger force applied the bigger errors produced. The measurement errors obtained by the six axes forces applied on the sensor are shown in Fig.4 respectively. Comparing the error without compensated with the compensated one, it can be seen the measurement error is reduced distinctly. 0.08 uncompensated 0.04 0.02 0 3 Stif 2 fnes 1 s (10 6 coeffici N/m ent )
compensated
0 20
40
80 60 Fx (N)
Error (%)
0.06
100
Fig 3 Measurement error difference between external force and limb’s stiffness coefficient
Fig 4 Measurement errors on six axes
Furthermore, it has been shown that the proposed real-time compensation algorithm is effective for reducing the effect of cross-axis error on measurements, which is significant to improve the measurement accuracy of the six-axis force sensor. Conclusions In this paper, the error analysis and compensation research of a six-axis force sensor based on Stewart platform has been conducted. The initial and final state force Jacobin matrixes are established depending on the external force applied on the force sensor. Numerical example and experimental results have shown that errors in force measurements generated by the upper platform deformation are quite small when the limb’s stiffness coefficient is more than 106N/m, but the errors become much bigger with increasing external force, and the compensation algorithm is effective for improving the measurement precision. This fact shows the validity of the proposed compensation method using state force Jacobin matrix instead of using initial state matrix. Acknowledgements This work was financially supported by the National Natural Science Foundation of China under Grant No. 50975245, Natural science foundation of Hebei province No. E2011203184, The research and development project of science and technology of Qinhuangdao city under Grant No. 201001A11, and Jiangsu province key lab of digital manufacturing open research No. HGDML-0911. References [1] Zhen Huang: Mech. Mach. Theory Vol. 22(6) (1987), p. 577 [2] Chul-Goo Kang: Sens. Actuator A Vol. 90 (2001), p. 31 [3] Yulei Hou, Jiantao Yao, Ling Lu, et al: Mech. Mach. Theory Vol. 44 (2009), p. 359 [4] Jiantao Yao, Yulei Hou, Jie Chen, et al: Sens. Actuator A Vol. 150 (2009), p. 1 [5] Zhen Huang, Yongsheng Zhao, Tieshi Zhao: Advanced Spatial Mechanism, Higher Education Press, Beijing, (2006). (In Chinese)
Equipment Manufacturing Technology and Automation 10.4028/www.scientific.net/AMR.317-319
Errors Analysis and Compensation of Six-Axis Force Sensor Based on Stewart Platform 10.4028/www.scientific.net/AMR.317-319.1041 DOI References [1] Zhen Huang: Mech. Mach. Theory Vol. 22(6) (1987), p.577. http://dx.doi.org/10.1016/0094-114X(87)90053-X [2] Chul-Goo Kang: Sens. Actuator A Vol. 90 (2001), p.31. http://dx.doi.org/10.1016/S0924-4247(00)00564-1 [3] Yulei Hou, Jiantao Yao, Ling Lu, et al: Mech. Mach. Theory Vol. 44 (2009), p.359. http://dx.doi.org/10.1016/j.mechmachtheory.2008.03.008 [4] Jiantao Yao, Yulei Hou, Jie Chen, et al: Sens. Actuator A Vol. 150 (2009), p.1. http://dx.doi.org/10.1016/j.sna.2008.11.030