Robotica (2001) volume 19, pp. 311 321. Printed in the United Kingdom 2001 Cambridge University Press Calibration of geometric and non-geometric errors of an industrial robot Joon Hyun Jang*, Soo Hyun Kim and Yoon Keun Kwak Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373 1, Kusong-dong, Yusong-gu, Taejon, 305 701 (South Korea) (Received in Final Form: July 10, 2000) SUMMARY Inaccurate positioning of the robot end effector causes joint deformation as well as geometric errors when an industrial robot has a payload at its end effector. We propose a new approach of calibration which deals with joint angle dependent errors to compensate for these phenomena. To implement this method, we divided the robot workspace into several local regions, and built a calibration equation by generating the constraint conditions of the end effector s motion in each local region using a three-dimensional position measurement system. The parameter errors obtained this way were interpolated using the Radial Basis Function Network (RBFN) so as to estimate calibration errors in the regions that we did not measure. We used this technique to improve the performance of a six DOF industrial robot used for arc welding. KEYWORDS: Calibration; Deformation; Kinematic error; Radial Basis Function Network; Constraint equation; Compensation 1. INTRODUCTION A robot manipulator is subject to numerous sources of positioning errors. There are two types of errors: The first is geometric errors which result from imprecise manufacturing of parts and the second is non-geometric errors which result from gravity, joint compliance and gear transmission errors. 1 A robot with no load at its end effector can be easily calibrated. 2 But when it carries an object of a certain weight moving from one pose to another, a significant discrepancy develops between the desired and actual position of the robot because of deformations in links and joints. According to previous experimental results, 3,4 after eliminating geometric parameter errors, nonlinear errors in the joint and link elasticity exert the most significant influence on the static positioning error. Duelen and Schroer 3 utilized a gear model to represent joint elasticity and backlash due to link weights and an additional payload at the end effector. Judd and Knasinski 4 showed that errors due to elastic deformations can be compensated for by corresponding changes in coefficients of the gear error model. Caenen and Angue 5 took into account only angular deformations caused by gravity forces, because these forces have a dominant influence on the end effector position error. * Corresponding author; E-mail: jhjang@kaist.ac.kr Previous calibration models of non-geometric errors were complex models and were studied independently of geometric calibration models. Among the Denavit-Hartenberg parameters, 6 the joint angular error was the most significant parameter affecting the accuracy of robot end effector positioning. However, it is necessary to calibrate the joint angular error both geometrically and non-geometrically, including encoder mis-alignment with the shaft, gear drive transmission error and joint deformation due to payload. On the other hand, for other parameters it is sufficient only to know geometric errors for simplicity of modeling and identification. In this paper, we present a new method of calibration of geometric and non-geometric errors. It can simultaneously calibrate two kinds of errors for the joint parameters. However, for the rest of the parameters, we only compensated for geometric errors. The geometric model uses the modified D-H notation. 7 In addition, the joint compliance and gear transmission errors are taken into account in defining the non-geometric parameters of the model. We divided the robot workspace into several local regions and applied a new method of construction of the calibration equation by constraint conditions. We used a three dimensional position measurement system without measuring the absolute position or restricting the measuring range. The obtained parameter errors from the measurement and identification process had a discrete value in a Cartesian workspace. In order to know the correct value at any specified joint coordinate space, identified errors must be generalized or interpolated over all joint space. For this purpose, the RBFN (Radial Basis Function Network) 8 10 was used to obtain the continuous error function of the joint coordinate system. The remainder of the paper is organized as follows: In Section 2, we present a new calibration scheme with error model including interpolation techniques using the RBFN. We describe our data collection method and the experimental results for a six axis industrial robot in Section 3. Discussion and conclusions are given in Section 4. 2. CALIBRATION METHOD 2.1 Error model The model utilizes a modified Denavit-Hartenberg convention, 7 in which the transformation between consecutive coordinate frames is expressed as:
312 A i = Rot(z i, i ) Trans(a i,0,d i ) Rot(x i, i ) Rot(y i, i ) (1) where, i is the joint angle, a i is the link length, d i is the link offset, i and i are the twist angles between neighboring links around the x i and y i axis, respectively. If these parameters have errors, i, a i, d i, i and i, the transformation matrix becomes: A i = Rot(z i, i + i ) Trans(a i + a i,0,d i + d i ) Rot(x i, i + i ) Rot(y i, i + i ) (2) The gravity can produce three types of compliance on the joint of the robot manipulator. They are angular deformations around the x i, y i and z i axis due to the different gravity torques M xi, M yi and M zi, caused by the end effector payload. 5 These gravity torques produce the following angular deformation at each robot joint. A i = Rot(z, i + z i ) Trans(a i,0,d i ) Rot(x, i + x i ) Rot(y, i + y i ) (3) where x i, y i and z i are the angular deformation errors around each axis caused by gravity torques M xi, M yi and M zi, respectively. The composite error model, including geometric error, is: A i = Rot(z i, i + i ) Trans(a i + a i,0,d i + d i ) Rot(x i, i + i ) Rot(y i, i + i ) (4) where, Fig. 1. The kinematic relations. Calibration of geometric and non-geometric i = i + z i i = i + x i (5) i = i + y i The total angular error is the sum of the geometric error and the joint deformation error. The link length and link offset errors are assumed to have geometric errors only. Fig. 2. Perpendicular condition of plane and line. 2.1 Calibration method The objective of error identification is to know the effective kinematic parameters of the robot manipulator in order to control accurately the end effector position. Let be the error vector, which is assumed to be small perturbations from the nominal values, where is an n 1 vector, and n is the number of parameters. The relationship between the measured position of the robot end effector p M and the actual position of the robot end effector p is as follows (see Figure 1): p M =R B p+p B (6) where R B and p B are the rotation matrix and position vector from the measurement system coordinate to the robot base frame, respectively. The relative position between point i and j in workspace is: p M, i p M, j =R B (p i p j ) (7) Let us consider that the actual position of the robot end effector differs slightly from the nominal ones P (n) : p=p (n) +J (8) where J is the Jacobian of position changes from nominal ones for small parameter errors. Fig. 3. DR06.
Calibration of geometric and non-geometric 313 Then Eq. 7 becomes: p M, i p M, j =R B (p (n) i p (n) j )+(J i J j ) (9) The relative position of p M,1 and p M,0 is: p M, 1 p M, 0 =R B (p (n) 1 p (n) 0 )+(J 1 J 0 ) (10) Table I. Nominal Parameters of DR06. Joint i (deg) i (deg) a i (mm) d i (mm) 1 1 90 0 850 2 2 0 1050 0 3 3 90 250 0 4 4 90 0 1300 5 5 90 0 205 6 6 0 0 0 If positions p M, i, p M, j and p M, 0 lie on a plane, and the line p M, 0 to p M, 1 is perpendicular to this plane, as shown in Figure 2. (p M, 1 p M, 0 ) T (p M, i p M, j )=0 (11) then, from Eq. 9 and 10 we have: (p (n) 1 p (n) 0 ) T (p (n) i p (n) j ) + (p (n) 1 p (n) 0 ) T (J i J j ) +(p (n) i p (n) j ) T (J 1 J 0 ) =0 (12) Denoting, X i, j = (p (n) 1 p (n) 0 ) T (p (n) i p (n) j ) (13) K i, j =(p (n) 1 p (n) 0 ) T (J i J j )+(p (n) i p (n) j ) T (J 1 J 0 )} (14) then, Eqn. 12 becomes: K i, j =X i, j (15) The above calibration equation is independent of R B and p B. If we measure m points of position, we have m 1 equations. Fig. 4. Measurement system. Fig. 5. Workspace division. Fig. 6. Structure of RBFN.
314 Calibration of geometric and non-geometric Table II. Identified errors of DR06. Link i (deg) i (deg) i (deg) a i (mm) d i (mm) 1 Fig 7 0.276 2.843 1.587 2 Fig 8 9 0.384 0.654 3.476 0 3 Fig 10 11 0.203 1.647 3.428 4 0.473 0.445 2.818 0.611 5 0.307 0.143 1.879 1.329 6 0.184 0 1.834 0.584 Fig. 7. First joint angular error. 0.28 Load : 1.4 kg 0.26 0.24 δα2 (deg) 0.22 0.2 0.18 0.16 0.14-120 -110-100 -90-80 -70-60 -50 θ 2 (deg) Fig. 8. Second joint angular error when the load is 1.4 kg.
Calibration of geometric and non-geometric 315 K =X (16) where K=[K 1,2 K 2,3 K 3,4...K i, j...k m 1, m ] T (17) X=[X 1,2 X 2,3 X 3,4...X i, j...x m 1, m ] T (18) Perpendicular constraints between the plane and line can be obtained by 3D position measurement systems as explained in Section 3. 3. EXPERIMENTS 3.1 Measurement system Experiments were also conducted using the DR06, a six axes industrial robot manipulator with all revolute joints as shown in Figure 3. The kinematic parameters are listed in Table I. A camera type measurement system, OPTOTRAK, manufactured by Northern Digital Inc. was used to construct the constraint conditions in local regions of the workspace. These systems, as shown in Figure 4, consist of a camera, infrared LED(IrLED) and a system controller. The camera has photosensitive detector that is sensitive to light in the infrared range. The IrLED is capable of accurately determining the location of the center of the spot of light that has been projected onto its surface. This system has a repeatability of 0.01 mm and an accuracy of 0.05 mm in the 1 m 1 m 1 m measuring range, while the maximum range is 3 m 3 m 3 m. Making constraints in the workspace using the 3D coordinate measuring system is as follows: (1) Pose the robot end effector in the position of any range to be measured and read the positioning coordinate from measurement system, via the computer. Fig. 9. Second joint angular error when loads are 6.0 kg and 15.2 kg.
316 Calibration of geometric and non-geometric 0.34 Load : 1.4 kg 0.32 0.3 δθ3 (deg) 0.28 0.26 0.24 0.22 0.2 20 25 30 35 40 45 50 55 60 65 θ 3 (deg) Fig. 10. Third joint angular error when the load is 1.4 kg. (2) Move the robot end effector to the next position. (3) Move the robot end effector and let the z axis value from the measurement system agree with the pervious value. (4) Repeat (2) (4) process. (iv) The twist angle errors are functions of the neighboring joint angle and mass. Based on the above assumptions, the functional representations are: Then the points lie in an arbitrary x y plane because of the same z coordinate values. The method of obtaining one vertical line with respect to this plane is the reverse of the above method, i.e., move the robot end effector only along with the z-axis coordinate while fixing x and y axis values of the measurement system. The measurement system is used as a tool to construct a plane and perpendicular line in the workspace, not used to measure the value of the position coordinate. 1 =f( 1,m) 2 =f( 2, 3,m) 3 =f( 2, 3,m) 1 =f( 1, 2,m) 2 =f( 2, 3,m) 2 =f( 2, 3,m) (19) 3.2 Experimental results As a rule of thumb, in order to suppress the influence of measurement noise, the number of measurement equations should be two or three times larger than the number of parameters to be estimated. 11,12 If the workspace is divided into too many regions, there are too many points to measure, and that is time consuming and difficult to do. We divided the workspace in 22 regions as shown in Figure 5 under the following assumptions: (i) (ii) The major axis, i.e., first, second and third link parameters, have the influence on the end effector positioning accuracy among all parameters. Therefore, we assumed that the minor links, the fourth, fifth and sixth links, have fixed parameter errors. The first joint angular error is only function of first joint angle and mass, not a function of other joint angles. (iii) The second and third joint angular errors are dependent on the second, third joint angle and mass. where m is the mass at the robot end effector. The volume of serviceable workspace of the DR06 is 700 1500 mm of x-direction, 700 700 mm of y-direction and 500 1500 mm of z-direction. The volume is divided into twenty regions in the x and z directions, and seven regions in the ydirection. The results of identifying parameter errors after measuring seven data points per region are shown in Table II. Joint angular error i and twist angular errors i, i have some variations according to the robot configuration as mentioned in the previous section. We divided the whole workspace into several local regions in the Cartesian space and obtained calibrated error parameters in each local region. Continuous parameter error functions are obtained by using functions capable of interpolation, the RBFN (Radial Basis Function Network). The concept of the interpolation problem can be stated as follows (see Figure 6). Given a set of N different points x i and N real numbers, y i, construct a smooth function F satisfying the following two equations:
Calibration of geometric and non-geometric 317 F(x i )=y i, i=1,2,...n (20) N F(x) = w i ( x x i ) (21) i=1 where the function is a basis function that is continuous, w i denotes weights associated with each of the N basis functions which overlap at position x, and the vector x i are the measured data points themselves, called the center of the basis function. Thus, this equation represents a basic function approach. A very popular choice for the radial basis function is the Gaussian function. 13 ( x x i )=exp x x i 2 (22) 2 If the basis functions have global support, adjusting F(x) requires all weight values w i to be changed. However, if the basis function decays to zero with an increasing distance from its center, and if the value of the basis function is sufficiently small at a certain distance from its center, so that the RBFN can be cut off, local support can be achieved. Since we assumed that the first joint angular error 1 was a function of the first joint angle, this error can be fitted shown in Figure 7. Asterisks represent calibrated errors from the measurement. The solid line is fitted by using the RBFN and the dotted line by a third order polynomial function. We think that the source of this variation of error was in the reduction gear in the joint. The variations of 2, 3 over the second and third joint space are shown in Figure 8 and 10, respectively. 2 is only dependent on the second joint angle and 3 is on the third joint angle when there is a negligible payload. These errors seem to be non- Fig. 11. Third joint angular error when loads are 6.0 kg and 15.2 kg.
318 Calibration of geometric and non-geometric δαt (deg) 0.28 0.279 0.278 0.277 0.276 0.275 0.274 0.273 0.272 0.271 0.27-40 -30-20 -10 0 10 20 30 40 θ 1 (deg) 0.28 0.279 0.278 0.277 δα1 (deg) 0.276 0.275 0.274 0.273 0.272 0.271 0.27-120 -110-100 -90-80 -70-60 -50 θ 2 (deg) Fig. 12. First twist angular error. Fig. 13. Test cube and test plane.
Calibration of geometric and non-geometric 319 Fig. 14. Correction functions of joint angles.
320 Calibration of geometric and non-geometric Table III. Comparison of distance accuracy. Path Programmed Pre-calibration Post-calibration Measured Accuracy Measured Accuracy (mm) (mm) (%) (mm) (%) 1 2 500 504.156 0.831 500.361 0.072 2 3 500 495.814 1.163 499.628 0.074 3 4 500 496.182 0.764 501.645 0.329 4 1 500 503.620 0.724 500.914 0.183 5 6 500 504.580 0.916 499.541 0.034 6 7 500 505.113 1.023 499.319 0.136 7 8 500 505.515 1.103 501.171 0.034 8 5 500 495.920 0.816 501.876 0.375 Mean 0.920 0.154 geometric errors caused by irregularities of the gear transmission in the joints. The changes of in the second and third joint angular errors are shown in Figures 9 and 11 when the load at the robot end effector weighed 6.0 kg and 15.2 kg, respectively. The first twist angular error 1 along with 1 and 2 has a randomly distributed with 0.276 deg average value as shown in Figure 12. This random phenomenon is supposed measurement noises; therefore we conclude that 1 has a fixed value of 0.276 deg. 2 and 2 have also fixed values as the same reason as a result of analyzing of measurement data. This is contrary to our first assumption that these parameters are functions of the neighboring joint angles and masses. 3.3 Compensation We conducted some experiments to evaluate the calibration performance. The distance accuracy expresses the deviation between the programmed distance and the average of the measured distance. Path accuracy is the maximum deviation from the commanded path obtained by following the same path several times in the same direction. We compared the calibration performance before and after the calibration. The measurement points and paths used for testing were located inside a cube within the workspace of the robot. Figure 13 illustrates a test cube and test plane. The robot was programmed to move to the eight vertices of the test cube of 500 mm 500 mm 500 mm. The DR06 was set to move to each vertex in turn with a 10.6 kg load and then pause until the point was read from the robot controller and coordinate measurement system. This process was repeated three times both before and after calibration. For the post-calibration measurements, the twist angles were not modified because of simplicity of inverse kinematics. The comparison of distance accuracy was evaluated for eight paths. The distance accuracy showed a significant improvement from a mean value of 0.920% to 0.154%, as shown in Table III. The test plane for path accuracy is one of the four possible side planes of the test cube, as shown in Fig. 13(b). For the post-calibration measurements, the joint angle Pre-Calibration Post-Calibration x (mm) y (mm) Fig. 15. Path accuracy.
Calibration of geometric and non-geometric 321 correction functions of 1, 2 and 3 along with the path were introduced as shown in Figure 14. The correction values in the case of a 10.6 kg payload were interpolated using 6.0 kg and 15.2 kg curves. The maximum deviation was 25.87 mm, and path error profiles are shown in Figure 15. 4. CONCLUSIONS The purpose of this study was to improve the positioning accuracy of the industrial robot when considering joint deformation due to a payload. We built a composite error model, which takes into account geometric errors and nongeometric errors. The angular errors of three main joints of a serial robot were carefully investigated. Since the joint angular error is dependent on the robot configuration, we measured and calibrated geometric and non-geometric errors in each local region after dividing the workspace into several local regions. IrLED sensor and camera systems were originally used for position measurement for the robot manipulator. We applied these systems for making constraint conditions to obtain a calibration equation in several local regions, without measuring absolute position and without restricting the measurement range. Because the obtained parameter errors are discrete in the workspace, we applied the RBFN algorithm to interpolate the workspace to get continuous functions of joint angles. The width and learning rate were adequately selected to get the optimal solution with fast convergence and satisfactory given error bound. The experimental results showed that robot positioning accuracy can be improved for joint deformations around an axis which are due to the payload at the robot end effector. A three dimensional position measurement system was used to compare the calibration effect for improving the accuracy of the industrial robot. Distance accuracy and path accuracy on the workspace was evaluated. The result proved that the proposed calibration scheme was useful to improve the accuracy of the robot. Acknowledgment This work was supported in part by the Brain Korea 21 project. References 1. D.E. Whitney, C.A. Lozinski and J.M. Rourke, Industrial Robot Forward Calibration Method and Results, ASME J. Dynamic Systems, Measurement and Control 108(1), 1 8 (1986). 2. W.K. Veitschegger and C.H. Wu, Robot Accuracy Analysis Based on Kinematics, IEEE J. Robotics and Automation RA 2(2), 171 179 (1986). 3. G. Duelen and K. Schroer, Robot Calibration Method and Results Robotics and Computer Integrated Manufacturing 8, 223 231 (1991). 4. R.P. Judd and A.B. Knasinski, A Technique to Calibrate Industrial Robots with Experimental Verification, IEEE Trans. Robotics and Automation 6(1), 20 30 (1990). 5. J.L. Caenen and J.C. Angue, Identification of Geometric and Non-geometric Parameters of Robots, Proc. 1990 IEEE Conf. Robotics and Automation (1990) pp. 1032 1073. 6. J. Denavit and R.S. Hartenberg, A Kinematic Notation for Lower-pair Mechanisms Based on Metrics, ASME J. of Applied Mechanics 22, 215 221 (1955). 7. S.A. Hayati and M. Mirmirani, Improving the Absolute Positioning Accuracy of Robot Manipulator, J. Robotic Systems 2, 397 413 (1985). 8. P. Kluk, G. Misiurski and R.Z. Morawski, Total Least Square versus RBF Neural Networks in Static Calibration of Transducers IEEE Instrumentation and Measurement Tech. Conf. (1997) pp. 19 21. 9. W.H. Liao and J.K. Aggarwal, Curve and Surface Interpolation Using Rational Radial Basis Functions, Proc. ICPR 96 (1996) pp. 8 13. 10. J. Park and I.W. Sandberg, Universal Approximation Using Radial Basis Function Networks, Neural Computation 3, 246 257 (1992). 11. H. Zhuang and Z.S. Roth, Modeling Gimbal Axis Misalignments and Mirror Center Offset in a Single Beam Laser Tracking Measurement System, Int. J. Robotics Research 14(3), 211 224 (1995). 12. H. Zhuang, K. Wang and Z.S. Roth, Simultaneous Calibration of a Robot and a Hand-Mounted Camera, IEEE Trans. on Robotics and Automation 11(5), 649 660 (1995). 13. S. Haykin, Neural Network A Comprehensive Foundation (Macmillan, London, 1994).