Article
`A Rapid Coordinate Transformation Method Applied
`in Industrial Robot Calibration Based on
`Characteristic Line Coincidence
`
`Bailing Liu, Fumin Zhang *, Xinghua Qu and Xiaojia Shi
`
`State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Tianjin 300072,
`China; liubailing@tju.edu.cn (B.L.); quxinghua@tju.edu.cn (X.Q.); shixiaojia@tju.edu.cn (X.S.)
`* Correspondence: zhangfumin@tju.edu.cn; Tel.: +86-22-2740-8299; Fax: +86-22-2740-4778
`
`Academic Editors: Xin Zhao and Yajing Shen
`Received: 22 December 2015; Accepted: 14 February 2016; Published: 18 February 2016
`
`Abstract: Coordinate transformation plays an indispensable role in industrial measurements,
`including photogrammetry, geodesy, laser 3-D measurement and robotics. The widely applied
`methods of coordinate transformation are generally based on solving the equations of point clouds.
`Despite the high accuracy, this might result in no solution due to the use of ill conditioned matrices.
`In this paper, a novel coordinate transformation method is proposed, not based on the equation
`solution but based on the geometric transformation. We construct characteristic lines to represent the
`coordinate systems. According to the space geometry relation, the characteristic line scan is made to
`coincide by a series of rotations and translations. The transformation matrix can be obtained using
`matrix transformation theory. Experiments are designed to compare the proposed method with other
`methods. The results show that the proposed method has the same high accuracy, but the operation
`is more convenient and flexible. A multi-sensor combined measurement system is also presented
`to improve the position accuracy of a robot with the calibration of the robot kinematic parameters.
`Experimental verification shows that the position accuracy of robot manipulator is improved by
`45.8% with the proposed method and robot calibration.
`
`Keywords: coordinate transformation; robot calibration; photogrammetric system; multi-sensor
`measurement system
`
`1. Introduction
`
`Multi-sensor measurement systems usually have different coordinate systems. The original data
`must be transformed to a common coordinate system for the convenience of the subsequent data
`acquisition, comparison and fusion [1,2]. The transformation of coordinate systems is applied in many
`fields, especially vision measurement and robotics. For example, two images need to have a unified
`coordinate system for image matching [3]. In camera calibration, the coordinate systems of the image
`plane and the object plane need to be unified for the inner parameter calculation [4]. In robot systems,
`the coordinate system of the robot twist needs to be transformed to the tool center position (TCP)
`to obtain the correct pose of robot manipulators [5,6]. A minor error introduced by an imprecise
`coordinate transformation could cause problems such as the failure of image matching and track
`breaking [1]. Especially in an error accumulating system such as series industry robots, the coordinate
`transformation error would accumulate in each step and thereby decrease the position accuracy of the
`robot manipulator. Therefore, research on coordinate transformation has been of interest to researchers
`in recent years.
`
`Sensors 2016, 16, 239; doi:10.3390/s16020239
`
`www.mdpi.com/journal/sensors
`
`sensors
`
`RoboticVisionTech EX2011
`ABB v. RoboticVisionTech
`IPR2023-01426
`
`

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`Industrial robots are well-known to have weak position accuracy compared with their repeatability
`accuracy. The positioning accuracy degrades with the number of axes of the robotic arm due to error
`accumulation. Various methods have been presented to improve the position accuracy of robots,
`such as establishing a kinematic model of the robot and calibrating the kinematic parameters [7].
`Denavit and Hartenberg [8] first proposed the D-H model, which was revised to a linear model by
`Hayati [9]. It provides the basis for the kinematic calibration. Due to the geometry and non-geometry
`errors of the robot, the traditional robot self-calibration method based on the D-H model cannot
`accurately describe the robot pose. To avoid the influence of the robot body, many researchers have
`utilized external measuring instruments to calibrate the robot online [10,11]. To achieve the aim of
`calibration, the primary process is to unify the coordinate systems of a calibrated instruments and
`the robot. Only in this way is it possible to use the measurement results to correct the kinematic
`parameters of the robot. With an inaccurate coordinate transformation method, the transformation
`error might merge into the revised kinematics parameters, thereby failing to improve the positioning
`accuracy of the robot through the calibration of kinematic parameters. Therefore, an accurate method
`of coordinate transformation is indispensable in the field of robot calibration. The well-developed
`and widely-used methods of coordinate transformation at present might be classified into several
`categories: the Three-Point method, Small-Angle Approximation method, Rodrigo Matrix method,
`Singular Value Decomposition (SVD) method, Quaternion method and Least Squares method [2].
`The Three-Point method uses three non-collinear points in space to construct an intermediate reference
`coordinate system [12,13]. The transformation relationship between the initial coordinate system and
`the target coordinate system is obtained by their relationship relative to the intermediate reference
`coordinate system. Depending on the choice of the public points, the accuracy of the Three-Point
`method might be unstable. The Small-Angle Approximation method means that the rotation matrix
`can be simplified by using the approximate relationship of a trigonometric function (sinθ “ θ, cosθ “ 1)
`when the angle between the two coordinate systems is small (less than 5˝). It is more suitable for the
`coordinate transformation of small angles. The Rodrigo matrix is a method of constructing a rotation
`matrix by using the anti-symmetric matrix [14,15]. Despite its high accuracy and good stability, the
`algorithm might be complex and difficult. The Singular Value Decomposition method (SVD) is a
`matrix decomposition method that can solve the minimization of the objective function based on
`the minimum square error sum [16]. The method is accurate and easy to implement, but it might be
`difficult to work out the rotation matrix under a dense public point cloud. The Quaternion method
`uses four element vectors (q0, q1, q2, q3) to describe the coordinate rotation matrix [17,18]. The aim of
`the algorithm is to solve for the maximum eigenvalue and the corresponding feature vector when
`the quadratic is minimized. It is a simple and precise method, but there might be no solution due
`to the use of ill conditioned matrices. In practice, complex calculations and unstable results would
`make the application more difficult and complicated. Therefore, researchers are searching for a simpler
`and more stable method of coordinate transformation. For example, Zhang et al. proposed a practical
`method of coordinate transformation in robot calibration [19]. This method rotates three single axes of
`the robot to calculate the normal vectors in three directions, combined with the data of the calibration
`sensor. Then, combined with the own readings of the robot, the rotation matrix and translation matrix
`are obtained. The method avoids the need to solve an equation and complex calculations, but it might
`be affected by any manufacturing errors of the robot and requires a calibration sensor with a large
`measuring range that can cover the full working range of the robot.
`
`2. Online Calibration System of Robot
`
`Industrial robots have the characteristics of high repeatability positioning accuracy and low
`absolute positioning accuracy. This is due to the structure of the robot, manufacturing errors, kinematic
`parameter error and environmental influence [10]. To improve the absolute positioning accuracy of the
`robot, the use of an external sensor to measure the position of the robot manipulator it any effective
`approach. This paper proposes an on-line calibration system for the kinematic parameters of the robot
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`using a laser tracker and a close-range photogrammetric system, as Figure 1 shows. According to the
`differential equations constructed by the kinematic parameters of each robot axis, the final mathematic
`model of kinematic parameters of the robot is established. The position errors of the robot manipulator
`are obtained by comparing the coordinates in the robot base coordinate system and the measurement
`sensor system. Then, the errors, including the coordinate transformation error, target installation
`error and position and angle errors of the robot kinematic parameters, are separately corrected. In the
`robot calibration, on the one hand, the coordinate transformation error directly affects the final error
`correction of the kinematic parameters. On the other hand, the coordinate systems of sensors are
`often required to transform in the on-line combined measurement system. Therefore, the premise of
`obtaining the position errors of a robot manipulator is to unify the coordinate systems of the various
`measurement sensors by an accurate, fast and stable coordinate transformation algorithm.
`
`Figure 1. Online calibration system of robot kinematic parameters.
`
`In combination with the characteristics of the robot, we propose a practical coordinate
`transformation method. It extracts the characteristic lines from the point clouds in different coordinate
`systems. According to the theory of space analytic geometry, the rotation and translation parameters
`needed for the coincidence of the characteristic line scan be calculated. Then, the coordinate
`transformation matrix is calculated. The coincidence of the characteristic lines represents the
`coincidence of the point clouds as well as the coincidence of the two coordinate systems.
`This method has some advantages. First, it does not require the solution of equations and
`complex calculations. Second, because the transformation matrix is obtained from the space geometry
`relationships, it would not be affected by robot errors or other environmental factors. The result is
`accurate and stable. Third, it does not require a sensor with a large field of view. Fourth, the algorithm
`is small and fast without occupying processor time and resources, and can be integrated into the host
`computer program. It could be applied easily in measurement coordinate systems that often need
`to change.
`
`3. Methods of Online Calibration System
`
`3.1. Method of Coordinate Transformation
`
`Suppose that S is a cubic point cloud in space. Point cloud M is the form of S located in the
`coordinate system of the sensor OSXSYSZS. N is the form of S located in the robot base coordinate
`system OrXrYrZr. M' represents the point cloud M transformed from the coordinate system of the
`r.
`sensor OSXSYSZS to the robot base coordinate system OrXrYrZr with the transformation matrix TS
`r. Then,
`The difference between N and M' is the transformation error caused by the transfer matrix TS
`
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`the coincidence of the coordinate systems OSXSYSZS and OrXrYrZr can be expressed as the coincidence
`of the two point clouds N and M'. For simplifying this mathematical model of the transformation
`process, we establish several characteristic lines instead of each point cloud. As verified by experiment,
`at least two characteristic lines are required to ensure the transformation accuracy.
`In Figure 2, two points A1 and A2 are chosen to be linked to the characteristic line A. Points B1
`and B2 form characteristic line B. Similarly, in point cloud N, the corresponding points A1'and A2'
`form line A', and points B1' and B2' form line B'. To achieve the coincidence of lines A and A', line A
`must be rotated around an axis in space. The rotated axis is the vector C which is perpendicular to the
`plane constructed by lines A and A'. As Figure 3 shows, the process of a vector rotating around an
`arbitrary axis can be divided into a series of rotations around the axis X, Y, Z. The following are the
`decomposition steps.
`
`Figure 2. The schematic diagram of coordinate transformation method.
`
`Figure 3. Schematic diagram of a vector rotated around an arbitrary axis.
`
`Take the first coincidence of Lines A and A' as an example:
`
`(a)
`
`Tpx1, y1, z1q “
`
`Translate the rotation axis to the coordinate origin. The corresponding transformation matrix can
`be calculated as:
`0 ´a0
`0
`0 ´b0
`0
`1
`1 ´c0
`0
`0
`0
`0
`0
`1
`where, (a0, b0, c0) is the coordinates of the center point of line A.
`
`»———– 1
`
`fiffiffiffifl
`
`(1)
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`Rotate the axis α1 degrees to Plane XOZ.
`
`»———– 1
`
`Rxpα1q “
`
`fiffiffiffifl
`
`5 of 16
`
`(2)
`
`c1b
`2 ` c1
`b1
`, where, (a1, b1, c1) are the coordinates of vector C, as Figure 3b shows.
`
`,
`
`2
`
`sinα1 “
`
`0
`0
`0
`cosα1 ´sinα1
`0
`0
`0
`sinα1
`cosα1
`0
`0
`0
`0
`1
`α1 is the angle between the axis and plane XOZ. It can be obtained by cosα1 “
`b1b
`2 ` c1
`2
`b1
`Rotate the axis β1 degrees to coincide with Axis Z.
`
`$’’’’’&’’’’’%
`
`»———–
`
`fiffiffiffifl
`
`0
`cosβ1
`1
`0
`´sinp´β1q 0
`0
`0
`
`sinp´β1q 0
`0
`0
`cosβ1
`0
`0
`1
`
`a1
`
`2
`
`.
`
`»———– cosθ1
`
`´sinθ1
`0
`0
`
`Rzpθ1q “
`
`fiffiffiffifl
`
`sinθ1
`cosθ1
`0
`0
`
`0
`0
`1
`0
`
`0
`0
`0
`1
`
`(b)
`
`(c)
`
`(d)
`
`(e)
`
`(f)
`
`Ryp´β1q “
`b
`b
`where, β1 is the angle between the rotation axis and axis Z. It can be obtained by
`2 ` c1
`2
`b1
`cosp´β1q “ cosβ1 “
`b
`2 ` b1
`2 ` c1
`a1
`sinp´β1q “ ´sinβ1 “ ´
`2 ` b1
`2 ` c1
`2
`a1
`Rotate the axis θ1 degrees around Axis Z, as shown in Figure 3d.
`
`where θ1 is the angle between lines A and A', which can be obtained by θ1 “ă Ñ
`¨ Ñ
`A1
`arccosp
`A1
`Rotate the axis by reversing the process of Step (c)
`0 ´sinβ1
`1
`0
`0
`cosβ1
`0
`0
`
`0
`sinβ1
`0
`
`ˇˇˇˇˇˇˇˇÑ
`
`ˇˇˇˇq.
`
`Rypβ1q “
`
`ˇˇˇˇÑ
`
`A
`
`ÑA
`
`»———– cosβ1
`»———– 1
`
`where, β1 is as the same as in step (c).
`Rotate the axis by reversing the process of Step (b).
`
`Rxp´α1q “
`
`0
`cosα1
`0
`0 ´sinα1
`0
`0
`
`0
`sinα1
`cosα1
`0
`
`0
`0
`0
`1
`
`where, α1 is as the same as in step (b).
`
`0
`0
`0
`1
`
`fiffiffiffifl
`fiffiffiffifl
`
`(3)
`
`(4)
`
`ÑA
`
`1 ą“
`
`(5)
`
`(6)
`
`A,
`
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`Sensors 2016, 16, 239
`
`(g)
`
`Rotate the axis by reversing the process of Step (a)
`
`fiffiffiffifl
`
`0
`1
`0
`0
`
`0
`0
`1
`0
`
`a0
`b0
`c0
`1
`
`6 of 16
`
`(7)
`
`»———– 1
`
`0
`0
`0
`
`Tp´x1,´y1,´z1q “
`
`where, (a0, b0, c0) is as the same as in step (a).
`Combining all of the previous steps, the final transformation matrix Trt1 of the first parallel (lines
`A and A') is expressed as:
`Trt1 “ Tp´x1,´y1,´z1q ¨ Rxp´α1q ¨ Rypβ1q ¨ Rzpθ1q ¨ Ryp´β1q ¨ Rxpα1q ¨ Tpx1, y1, z1q
`
`(8)
`
`Through the rotation matrix Trt1 calculated by Equation (8), the points Pi(x, y, z) in point cloud M
`can generate a new point cloud M1 by Equation (9).
`P1
`i px, y, zq “ Trt ¨ Pi px, y, zq
`Then, the characteristic line A of the new point cloud M1 is parallel with the characteristic line A'
`of point cloud N, as Figure 4a shows.
`
`(9)
`
`Figure 4. The processes of the coordinate transformation method.
`
`Based on the new point cloud M1 and point cloud N, the rotation matrix Trt2, which make the Line
`B of Point cloud M1 parallel with Line B' of Point cloud N, can be calculated through Equations (1)–(8):
`Trt2 “ Tp´x2,´y2,´z2q ¨ Rxp´α2q ¨ Rypβ2q ¨ Rzpθ2q ¨ Ryp´β2q ¨ Rxpα2q ¨ Tpx2, y2, z2q
`
`(10)
`
`Through the rotation matrix Trt2, the points Pi(x, y, z) in point cloud M1 can generate a new point
`cloud M2 again by Equation (9). Then, the characteristic line B of the new point cloud M2 is parallel
`with the characteristic line B' of point cloud N, as Figure 4b shows.
`Since the point clouds are cubic, the characteristic lines are the diagonal lines. So, BKA, B'KA'.
`Since, B//B'. Then, BKA', B'KA. Therefore, the parallel Line B and B' are perpendicular to Line A and
`
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`A'. There is an angle θ between Line A of point cloud M2 and Line A' of point cloud N, so the Line
`B of point cloud M2 is chosen as the rotation axis. The angle between Line A of point cloud M2 and
`Line A' of point cloud N is chosen as the rotation angle. The point cloud M2 is rotated by the above
`parameters. Then, the Line A of point cloud M2 and Line A' of point cloud N are parallel, like Line B
`of point cloud M2 and Line B' of point cloud N. Similarly, the rotation matrix Trt3 can be calculated by
`Equations (1)–(8):
`Trt3 “ Tp´x3,´y3,´z3q ¨ Rxp´α3q ¨ Rypβ3q ¨ Rzpθ3q ¨ Ryp´β3q ¨ Rxpα3q ¨ Tpx3, y3, z3q
`
`(11)
`
`The points Pi(x,y,z) in point cloud M2 can generate a new point cloud M3 by Equation (9), which
`are parallel with the point cloud N, as Figure 4c shows. In order to make coincident the point cloud
`M3 and point cloud N, the translation matrix Tr needs to be calculated by the two center points of Line
`A and A'. The new point cloud M' can be generated after translated by Tr. Therefore, through a series
`of simple rotations and translation, the two point clouds N and M' are coincident, as Figure 4d shows.
`The final transformation matrix is shown as Equation (12). The result, as a necessary preparation step,
`can then be used in robot calibration:
`
`Trt “ Trt3 ¨ Trt2 ¨ Trt1 ` Tr
`
`(12)
`
`3.2. Method of Robot Calibration
`
`»———– r1p
`
`Assume that Bp “
`
`The actual kinematic parameters of the robot deviate from their nominal values, which is referred
`to as kinematic errors [10]. The kinematic parameter calibration of a robot is an effective way to improve
`the absolute position accuracy of the robot manipulator. A simple robot self-calibration method based
`on the D-H model is described as follows. Reference [20] gives a more detailed description.
`pxp
`r2p
`r3p
`r5p
`r6p
`pyp
`r4p
`r8p
`r9p
`pzp
`r7p
`0
`0
`0
`1
`of the photogrammetric system, where r1p „ r9 p are the attitude parameters and pxp „ pzp are the
`position parameters. Through transformation from the coordinate system of the measurement sensor
`pxo
`r2o
`r3o
`r5o
`r6o
`pyo
`r8o
`r9o
`pzo
`0
`0
`1
`
`fiffiffiffifl is the pose of a certain point in the coordinate system
`»———– r1o
`fiffiffiffifl
`
`r4o
`r7o
`0
`
`OpXpYpZp to the robot base coordinate system OoXoYoZo, the point pose Bo “
`
`Bo “ Trt ˆ Bp
`where, Trt is the transformation matrix, which can be obtained by the method described in Section 3.1.
`Given the six DOF robot in the lab, the transformation matrix from the robot tool coordinate
`system to the robot base coordinate system is expressed as:
`0 “ T1
`1 L Tnn´1 L TNN´1 pN “ 6q
`
`TN
`0 T2
`In this system, the cooperation target of the measurement sensor, which is set up at the end axis
`of the robot, should be considered as an additional axis, Axis 7. Then, the transformation matrix from
`Axis 6 to Axis 7 is:
`
`can be obtained by Equation (12):
`
`
`
`»———– 1
`
`0
`0
`0
`
`0
`1
`0
`0
`
`0
`0
`1
`0
`
`fiffiffiffifl
`
`tx
`ty
`tz
`1
`
`6 “
`T7
`
`(13)
`
`(14)
`
`(15)
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`where tx, ty, tz are the translation vectors, which can be measured previously. Therefore, according to
`the kinematic model of the robot, the typical coordinates of the robot manipulator in the robot base
`coordinate system OOXOYOZO is expressed as:
`
`¸
`
`Ti
`i´1
`
`¨ Bt
`
`(16)
`
`˜
`
`7ÿ
`
`i“1
`
`Bo “
`
`where Bt is the point pose in the robot tool coordinate system, and Bo is the point pose from the robot
`tool coordinate system to the robot base coordinate system.
`In the robot calibration, the kinematic parameters are the most significant impact factors, which
`usually means the link parameters of the robot. In the D-H model, the link parameters include the
`length of the link a, the link angle α, the joint displacement d and the rotation angle of the joint θ. With
`the disturbances of the four link parameters, the position error matrix for adjacent robot axes dTi
`i´1
`can be expressed as:
`
`∆ai ` BTi
`∆αi ` BTi
`∆θi ` BTi
`i´1 “ BTi
`i´1Bdi
`i´1Bai
`i´1Bαi
`i´1Bθi
`i´1q´1 ¨ BTi
`i´1Bqi
`where ∆θi,∆αi,∆ai and ∆di are the small errors of link parameters. Suppose that Aqi “ pTi
`where, q represents the link parameters (a, d, α, θ).
`If every two adjacent axes are influenced by the link parameters, the transformation matrix from
`the robot base coordinate system to the coordinate system of the robot manipulator can be expressed as:
`
`dTi
`
`∆di
`
`(17)
`
`,
`
`0 “ Nź
`
`0 ` dTN
`TN
`
`i“1
`
`´
`
`
`Ti
`
`i´1 ` dTii´1
`
`¯
`
`´
`
`“ Nź
`
`i“1
`
`¯
`
`
`Ti
`
`i´1 ` Tii´1
`
`∆i
`
`pN “ 6q
`
`(18)
`
`(19)
`
`»—– kx
`
`»—– kx
`
`kx
`tx
`ky
`tx
`kz
`tx
`
`kx
`ty
`ky
`ty
`kz
`ty
`
`kx
`tz
`ky
`tz
`kz
`tz
`
`where, TN
`0 is the typical transformation matrix from the robot base coordinate system to the coordinate
`system of the robot manipulator and dTN
`0 is the error matrix caused by the link parameters. Through
`expanding dTN
`0 and performing a large number of simplifications and combinations, Equation (18)
`can be simplified as:
`0 “ T1
`∆θ1 ` T1
`∆α1 ` T1
`∆a1 ` T1
`∆d1
`dTN
`0 Aθ1TN
`0 Aα1TN
`0 Aa1TN
`0 Ad1TN
`1
`1
`1
`1
``T2
`∆θ2 ` T2
`∆α2 ` T2
`∆a2 ` T2
`∆d2
`0 Aθ2TN
`0 Aα2TN
`0 Aa2TN
`0 Ad2TN
`2
`2
`2
`2
`
`
`
`
`0 AαN∆αN ` TN`L ` TN0 AθN∆θN ` TN 0 AaN∆aN ` TN0 AdN∆dN
`Suppose that kiq “ Ti
`0AqiTN
`1 , where, q represents the four link parameters. The position error of
`the robot manipulator can be simplified as given in Equation (20):
`∆p “ rdtxdtydtzsT
`L kx
`kx
`kx
`kx
`kx
`1a
`1d
`1α
`6θ
`2θ
`1θ
`“
`ky
`ky
`ky
`L ky
`ky
`ky
`1a
`1d
`1α
`1θ
`6θ
`2θ
`L kz
`kz
`kz
`kz
`kz
`kz
`1a
`1d
`1α
`6θ
`2θ
`1θ
`r∆θ1∆α1∆a1∆d1∆θ2 ¨¨¨ ∆d6∆tx∆ty∆tzsT
`“ Bi∆qi
`where, ∆ p is the position error of the robot manipulator. dtx, dty, dtz are the Cartesian coordinate
`L kx
`tz
`1θ
`components of the position error and Bi “
`M O M
`kz
`L kz
`tz
`1θ
`
`fiffifl¨
`fiffifl is the parameter matrix related to the
`
`(20)
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`typical position value of the robot manipulator. In this paper, because the DOF of the series robot
`is 6, ∆qi “ r∆θ1 „ ∆tzs includes 24 kinematics parameters of the robot a1 „ a6, d1 „ d6, α1 „ α6,
`θ1 „ θ6 and three translation error variables of T6
`7. Therefore, there are 27 parameters of the robot
`that need to be calibrated. In Equation (20), the left side of equation is the position error at each
`point, as measured by the measurement sensor, and the right side is the kinematics errors that need
`to be corrected. These errors can be revised by the least squares method in the generalized inverse
`matrix sense.
`
`4. Experiments and Analysis
`
`Through the designed experiments, we show how to use the proposed coordinate transformation
`method to achieve the coordinate transformation of the on-line robot calibration system. Using
`verification experiments, we determine the result of the robot calibration using the proposed method.
`For evaluating the performance of the proposed method, it is compared with four other common
`methods of coordinate transformation under the same experimental conditions.
`
`4.1. Coordinate Transformationin an On-line Robot Calibration System
`
`The on-line robot calibration system we constructed includes an industrial robot, a photographic
`system and a laser tracker as shown in Figure 4. The model of the robot in lab is the KR 5 arc from
`KUKA Co. Ltd. (Augsburg, Germany), one of the world's top robotic companies. Its arm length is
`1.4 m and the working envelope is 8.4 m3. For covering most of the robot working range, the close
`range photogrammetric system in the lab, TENYOUN 3DMoCap-GC130 (Beijing, China), requires a
`field of view of more than 1 mˆ 1 mˆ 1 m without any dead angle. To achieve the goal of on-line
`measurement, a multi-camera system is needed. We used a multi-camera system symmetrically formed
`by four CMOS cameras with fixed focal lengths of 6 mm. The laser tracker in the lab, FARO Xi from
`FARO Co, Ltd. (Lake Mary, FL, USA) is a well-known high accuracy instrument whose absolute
`distance measurement (ADM) is 10 µm ˘ 1.1 µm/mL. The laser beam can easily be lost in tracking
`because of barriers or the acceleration of the target, which would cause minor errors. Therefore, we
`combine the laser tracker with the photographic system to improve the measurement accuracy and
`stability and thereby make full use of the advantages of the high accuracy of the laser tracker and the
`free light-of-sight of the photographic system. After proper data fusion, the two types of data from
`the photographic system and the laser tracker can be gathered together. The method of data fusion
`and the verified experimental result are detailed in reference [21]. In the experiment, 80 points in the
`public field of the robot and the photogrammetric system are picked to build a cube of 200 mm ˆ
`200 mm ˆ 200 mm. The reason for building a cube is to facilitate the selection of characteristic lines
`and the calculation of coincidence parameters. The two targets of the photogrammetric system and
`laser tracker are installed together with the end axis of the robot by a multi-faced fixture. To obtain
`accurate and stable data, the robot stops for 7 s at each location, and the sensors measure each point
`20 times, providing an adequate measurement time for the photographic system and laser tracker.
`The experimental parameters of the photogrammetric system are an exposure time of 15 us, a frequency
`of 10 fps and a gain of 40, based on experience.
`According to Equations (1)–(7) and the experimental data, we can obtain the parameters of
`the transformation matrix shown in Table 1, where ai–θi. are the parameters for the coincidence of
`characteristic lines in Equations (1)–(7).
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`Table 1. Calculated Results of Coincidence Parameters (Units: mm, ˝).
`
`Robot to Laser Tracker
`Robot to Photogrammetric System
`a1
`c1
`b1
`a1
`b1
`c1
`α1
`β1
`θ1
`α1
`β1
`θ1
`´2588.9 81326 ´62472 127.53˝ 1.4461˝ 256.282˝ ´3.0135 11.132 ´5.8921 117.89˝ 13.456˝ 0.007˝
`a2
`b2
`c2
`b2
`c2
`a2
`α2
`β2
`θ2
`α2
`β2
`θ2
`87.979˝ 37.588˝ 208.161˝ ´6.143
`0.26899 ´5.9267 177.4˝
`45.997˝ 0.004˝
`´30491 ´39586 1397.2
`a3
`b3
`c3
`b3
`c3
`a3
`α3
`β3
`θ3
`α3
`β3
`θ3
`38.555˝ 40.198˝ 176.953˝ 200.03
`159.99 ´200.07 141.35˝ 37.984˝ 0.005˝
`210.37 ´155.16 194.69
`
`According to Equations (8)–(10), the transformation matrices from the robot base coordinate
`system to the coordinate system of the sensors are calculated as:
`
`»– 0.99917951
`
`0.03352107
`´0.02272970
`0
`
`Trtrp “
`
`´0.02245170
`0.03370790
`´0.99940061 ´0.00864662
`´0.99971054
`0.00788692
`0
`0
`
`1003.54380
`167.88234
`984.54935
`1
`
`fifl Trtrl “
`
`»– 0.999178
`
`0.033635 ´0.02262 ´832.501
`´0.9994
`0.007803
`131.4773
`0.033819
`´0.02235 ´0.00856 ´0.99971
`1004.76
`0
`0
`0
`1
`
`fifl
`
`where, Trtrp is the transformation matrix from the robot base coordinate system to the coordinate
`system of the photogrammetric system. Trtrl is the transformation matrix from the robot base
`coordinate system tothe coordinate system of the laser tracker.
`By means of the above transformation matrix, we can obtain the point cloud coordinates
`transformed from the coordinate system of the robot to that of the sensors by Equation (12). Both
`the origin coordinates before and after transformation as well as the transformation error are shown
`in Table 2, where, Px,Py,Pz and Rx,Ry,Rz are three components of the original coordinates in two
`different coordinate systems. Tx,Ty,Tz are the coordinates of points transformed from the robot base
`coordinate system to the sensor coordinate system, and ∆x, ∆y, ∆ z are the three components of the
`transformation error.
`
`Table 2. Coordinates of Point clouds after Transformation (Units: mm).
`
`Rz
`875
`875
`875
`875
`
`Robot to Photogrammetric System
`Photogrammetric system
`Robot
`Rx
`Px
`Py
`Pz
`Ry
`895
`42.728
`138.567 109.566
`30
`961
`108.751 140.724 108.418
`30
`1028
`175.846 143.007 107.153
`30
`1095
`242.882 145.388 105.845
`30
`76 points are ignored
`error
`Transformation result
`∆y
`∆x
`∆z
`Tx
`Ty
`Tz
`138.590 109.751 ´0.247 ´0.023 ´0.185
`42.975
`108.921 140.822 108.277 ´0.170 ´0.098
`0.141
`175.866 143.088 106.780 ´0.020 ´0.081
`0.373
`242.812 145.355 105.282
`0.070
`0.033
`0.563
`76 points are ignored
`
`Rz
`875
`875
`875
`875
`
`Robot to Laser Tracker
`Laser tracker
`Robot
`Ly
`Ry
`Lx
`1048.620 29.944
`30
`1114.679 29.985
`30
`1181.646 29.955
`30
`1248.689 29.935
`30
`
`Rx
`Lz
`895
`875.077
`961
`874.995
`1028
`874.989
`1095
`874.791
`76 points are ignored
`error
`Transformation result
`∆x
`∆y
`Ty
`Tz
`∆z
`Tx
`875.024 ´0.004 ´0.023
`1048.624 29.967
`0.053
`0.029 ´0.017
`1114.625 29.956
`875.012
`0.054
`0.011 ´0.012
`1181.624 29.944
`875.001
`0.022
`0.003 ´0.099
`1248.624 29.932
`874.890
`0.065
`76 points are ignored
`
`It is can be calculated from Table 2 that the average values of the transformation error between
`the coordinate systems of the robot and photogrammetric system are ∆ x = 0.106 mm, ∆y = ´0.062 mm
`and ∆z = 0.013 mm. The average values of the transformation error between the coordinate systems of
`the robot and laser tracker are ∆x = ´0.015 mm, ∆y = 0.041 mm and ∆z =0.023 mm. Figures 5 and 6
`show that the transformation error of the photogrammetric system is approximately 10 times greater
`than that of the laser tracker. As in the earlier presentation, the nominal measurement accuracy of the
`photogrammetric system is 10´2 mm and that of the laser tracker is 10´3 mm. The results illustrate
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`that the transformation accuracy has the same order of magnitude as that of the measurement sensor.
`This indicates that the transformation error is so small that it would not influence the accuracy of the
`sensors. The transformation method can also make the error distribution of the low precision sensor
`more uniform to improve the transformation accuracy and the accuracy of the robot calibration.
`
`Figure 5. Transformation error from robot to photogrammetric system.
`
`Figure 6. Transformation error from robot to laser tracker.
`
`4.2. Position Error of Robot after Coordinate Transformation and Calibration
`
`Experiments are designed to calibrate the kinematic parameters of the robot. The measurement
`system is shown in Figure 7. Sixty points in space are used for the calibration. After the coordinate
`transformation, the position errors between the sensor and the robot manipulator are obtained.
`A constraint method based on the minimum distance error is adopted to calibrate the robot kinematic
`parameters [20,21]. Twenty seven kinematic parameters, including 24 link parameters and three
`parameters of the fixture, are corrected. Then, 60 correct positions are calculated using the calibrated
`robot kinematic parameters. To evaluate the performance of the proposed method, we use a
`group of calibration results, which adopts a different coordinate transformation method and the
`same robot calibration algorithm, as a comparison. The position errors of the robot after the
`coordinate transformation and calibration are shown in Figure 8. δ1 is the position error after the
`coordinate transformation with the other method [19], and δ2 is the position error after the coordinate
`transformation with the proposed method (Characteristic Line method).
`
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`Figure 7. Measurement system.
`
`Figure 8. Position errors after robot calibration.
`
`In position measurement, the distance between two points is often used to evaluate the position
`b
`accuracy, which is called the root mean square (RMS) error expressed by:
`pxi
`1 ´ xiq2 ` pyi
`1 ´ yiq2 ` pzi
`1 ´ ziq2
`RMSdi “
`It can be indicated from Figure 5 that the average RMS using the other coordinate transformation
`method is δ1 “ 0.436 mm. while the average RMS using the proposed method is δ2 “ 0.200 mm.
`The position accuracy is improved by 45.8% using the Characteristic Line method.
`To evaluate the accuracy distribution of the robot for different areas of the working range, a new
`set of testing data are utilized in a demonstration experiment. The coordinates of the center of robot
`calibration region O is (750, 0, 1000) in the robot base coordinate system. Taking O as the center of the
`circle, 200 mm as the radius, in this region the positioning accuracy of the robot would be the highest.
`To verify the distribution of the robot accuracy in the non-calibration regions, five positions O1 (1000,
`150, 550), O2 (780, 710, 900), O3 (780, 870, 410), O4 (600, ´800, 860), O5 (940, ´960, 450) are chosen.
`
`(21)
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`Taking the five positions as the centers of the circle, 200 mm as the radius, in each region 60 points
`are chosen to calibrate robot as same as the previous calibration experiment. The position errors in
`different regions are shown in Table 3.
`
`Table 3. The RMS of position error calibrated in the different regions.
`
`Region
`Position error/mm
`
`O
`0.200
`
`O1
`0.330
`
`O2
`0.360
`
`O3
`0.271
`
`O4
`0.335
`
`O5
`0.319
`
`It is indicated from Table 3 that the average RMS of robot position error within the calibration
`region is 0.200 mm. The position error outside the calibration region is about 0.323 mm. It is proved
`that the calibration accuracy isn't consistent in the whole working range of robot. Therefore, this
`calibration method is more applicable to a smaller working range of the robot.
`
`4.3. Accuracy of Coordinate Transformation Method
`
`To obtain the accuracy of the proposed coordinate transformation method, an experiment is
`designed using the laser tracker. The laser tracker is placed at two different stations to m