CALIBRATION OF A ROBUST 2 DOF PATH MONITORING TOOL FOR INDUSTRIAL ROBOTS AND MACHINE TOOLS BASED ON PARALLEL KINEMATICS

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1 CALIBRATION OF A ROBUST 2 DOF PATH MONITORING TOOL FOR INDUSTRIAL ROBOTS AND MACHINE TOOLS BASED ON PARALLEL KINEMATICS E. Batzies 1, M. Kreutzer 1, D. Leucht 2, V. Welker 2, O. Zirn 1 1 Mechatronics Research Group Giessen University of Applied Sciences Giessen, Germany 2 Fachbereich Mathematik und Informatik Philipps-Universität Marburg Marburg, Germany INTRODUCTION Up to the early 90s acceptance tests of machine tool and robot manipulators have been realized by static positioning accuracy measurements, e.g. using laser interferometers. With the increasing demand for higher manipulator performance dynamic trajectory monitoring became important. A first dynamic monitoring approach was the circular test using a touch trigger probe [1]. Speed and measurement accuracy are limited due to probe friction. The Double Ball Bar [2] measurement system avoids friction influences but is limited to large circle trajectories. The circular test elucidates static and dynamic manipulator performance and is meanwhile standardized [2] for machine tool inspection tests. FIGURE 1. Prototype of the parallel 2 DOF inspection system in the Giessen University machine tool laboratory - arranged beside the Lattice Grid Plate [2]. The most sophisticated inspection tool for arbitrary trajectories is the Lattice Grid Plate [1]. It yields high precision (ca. 1µm) as well as high resolution (ca. 5µm) path monitoring at velocities up to 0.5 m/s. The Lattice Grid Plate consists of a steel substrate with a grid of pattern squares. A scanning unit that has no mechanical contact to the plate measures the position with two degrees of freedom (2 DOF). The air gap between plate and scanning unit is in the range of 500 to 1000µm. Inspection devices to measure three and more DOF movements are subject of further developments [3], [1]. These high precision systems require careful use, especially in the rough manufacturing environment. Operator or path errors rectangular to the measurement plane as well as falling screws, tools, etc. can cause severe damages to the inspection devices. Especially robot inspection devices require adequate robustness against path errors and end effector orientation errors. On the other hand, the measurement precision for robot acceptance tests is much smaller than the required precision for machine tools. Experiences gained with parallel manipulators show that these kinematic principles yield very high repeatability with moderate effort [4]. Based on these experiences and the growing need to inspect robot manipulators a parallel 2 DOF inspection device was developed (see Figure 1). The end effector of the robot or the tool center point of the machine tool is attached at the top joint. The foot joints are attached to legs, mounted on two high resolution rotary encoder. REQUIREMENTS AND DESIGN The mechanical and operational requirements for the parallel 2 DOF inspection device are: rectangular measurement area (200x200 mm) to achieve adequate robot velocities 50µm absolute precision in the measurement

2 time programming. Thus it is integrated in a MAT- LAB postprocessor step as well as the graphical display of the inspection result. The application of standard components reduces the costs for the parallel inspection device to a fraction of the Lattice Grid Plate system price. Scaling up the measurement range will cause minor additional costs. FIGURE 2. Principle sketch of the parallel 2 DOF inspection system with measurement hardware and postprocessing. area 2 m/s maximum path velocity robustness against operator errors low cost design easy to use calibration method For the recommended measurement area, the ratio of device size to measurement range is high. Prolongation of the measurement area in the preferred direction parallel to the linear encoder yields better ratios. Thus the system is scalable without any changements to concept presented here. Especially for large measurement ranges, the inverse Dinosaur effect [4] of parallel manipulators leads to increasing system performance of the concept. The mechanical design of the inspection device is based on standard components. The backlash free and stiff joints are realized with magnetic ball joints. A PC interface board captures the linear encoder positions [2] (see Figure 2). Real time measurement and data recording is implemented in C. Forward transformation and error compensation requires no real CALIBRATION ISSUES Figure 3 shows a systematic picture of the measurement device: The encoders are positioned at coordinates (x 1, y 1 ) and (x 2, y 2 ) and measure the angles α 1 and α 2. For the calculation of the TCP position from angles α 1 and α 2, that is, working out the forward-transformation, the following system parameters are needed: x 1, y 1, x 2, y 2, a 1, b 1, a 2, b 2, β 1, β 2. Here, β 1 and β 2 are the zeroangle-positions, that is, the angles at which the encoders measure angles 0. The task of calibration is to calculate these system parameters from a set of measurements which give corresponding values for the TCP-Position and the rotary encoders. Y y 3 y 4 y 1 y 2 x 3 a 1 b 1 x 1 α 1 β 1 X x 2 a 2 b 2 α 2 β 2 FIGURE 3. Schematic figure of measurement device Introduction of the errors (leg lengths, tolerances of the magnetic ball joints, Encoder offsets, lattice grid plate position and orientation). NUMERICAL APPROACH TO CALIBRATION In the previous section a mathematical model was presented that, based on known system parameters, allows the calculation of Cartesian co- x 4

3 ordinates of the tool center point from the angles measured by the position monitoring system. The numerical approach to calibration is based on this forward transformation. Starting from an estimate of the system parameters the forwardtransformation is calculated for each pair of α 1 and α 2 in the calibration dataset. A merit function is defined as the sum of squares of differences between the Cartesian coordinates calculated by the forward-transformation and the coordinates given by the comparison measurement system. The optimized parameter set is determined by localizing the minimum of the merit function for all given calibration positions. From the set of mathematical algorithms applicable to this situation we have chosen the Levenberg-Marquardt-Algorithm [5], known as a robust solver for least square optimization problems. The results obtained by numerical calibration are disappointing. The measurement error of the position monitoring tool after calibration averages around 200µm. But an analysis of the mechanical construction leads to an expected measurement error 50µm. In addition, it turns out that the determined system parameters differ significantly from their measured physical dimensions. Particularly, the calculated lengths of the legs b 1 and b 2 differ more than 10% from their actual lengths. FIGURE 4. Measurement error depending on the parameters b 1 and b 2. gorithms can only be performed by extraordinary charges of computing time or fails at all. In the sequel we show that an analysis of this failure leads to a hybrid method combining the numerical with the symbolic approach from the next section. A close examination of the model leads to the conclusion that the location of minima obstructs the success of the numerical algorithm. First, we examine the situation when only the system parameters b 1 and b 2 are considered. Figure 4 shows a 3D-plot of the merit-function. The axes labelled b 1 and b 2 are scaled by the difference of the parameter value and its physical value. Figure 4 shows that there is a unique minimum at the position where the system-parameters b 1 and b 2 equal the physical length. Thus the minimum can be easily localized by any numerical least square optimization method. Next, we study the situation when only the system parameters b 1 and β 1 are considered. Using conventions analogous to Figure 4 in Figure 5 the reason why numerical calibration fails becomes transparent. The merit-function has numerous local minima which are distributed along a a flat valley. Therefore, optimization with numerical al- FIGURE 5. Measurement error depending on the parameters β 1 and b 1. The situation is significantly different in case the estimates of the system parameters are very accurate. The next section describes how this can be achieved by symbolic methods. Experiments show that a numerical optimization based on very good estimates of the system parameters can further reduce the measurement error by involving additional calibration measurements. The remaining calculation complexity is small since nearly accurate system parameters guarantee fast convergence. The measurement errors of the Levenberg-Marquardt algorithm with different estimates of the system parameters are as follows:

4 Error rough symbolic method estimates estimates Average 200µm 20µm Maximum 1000µm 200µm SYMBOLIC APPROACH TO CALIBRATION The symbolic approach to calibration is based on the solution of polynomial dependencies between the measured data and the calibration values. In order to obtain polynomial dependencies, we introduce the following symbols: p 1 := a 1 cos(β 1 ), p 2 := a 2 cos(β 2 ), q 1 := a 1 sin(β 1 ), q 2 := a 2 sin(β 2 ), u 1 := cos(α 1 ), u 2 := cos(α 2 ), v 1 := sin(α 1 ), v 2 := sin(α 2 ). Exploiting the measurement geometry, we obtain the following equation: x y p q 2 1 b u 1 ((x 1 p 1 + y 1 q 1 ) + 2v 1 ( x 1 q 1 + y 1 p 1 ) 2(u 1 X + v 1 Y ) + 2( v 1 X + u 1 Y )q 1 2Xx 1 2Y y 1 + X 2 + Y 2 = 0 (1) This equation involves only variables which belong to the left part of the device, that is, the legs with lengths a 1 and b 1. There is an analogous equation for the right part. Therefore, the calibration problem separates into two identical problems for the two parts. These consist of a list of equations of the above type (1), one equation for each single measurement of the calibration process. Here, for all single measurements of the calibration process, the system parameters x 1, y 1, p 1, q 1 and b 1 are the same and unknown, while for each single measurement of the calibration process the values of the variables u 1, v 1, X and Y are measured. In the symbolic approach for each measurement we introduce a new set of these variables. We successfully applied two different methods for solving the system in case of 7 and 5 measurements. 7 measurements: Using appropriate transformations which successively eliminate all nonlinear terms from the equations, it is possible to deduce a system of 4 linear equations for the unknowns x 1, y 1, p 1 and q 1. Solving the system by standard methods after replacing the measurement variables by the actual values and then successively solving equation (1) for b 1 leads to a solution. 5 measurements: By calculating Gröbner bases using SINGULAR [6] for the system of 5 measurements it can be shown that the system indeed can be solved using 5 measurements only, which clearly is the best one can hope for. After some simple transformations of the system by hand it was possible to calculate another Gröbner basis using an elimination order. The resulting Gröbner bases then reduces the calibration to finding zeros of univariate polynomials of degree 4 and a simple elimination of the geometrically inconsistent roots. Both solutions lead to an algorithm which has been implemented in MATLAB. We apply the algorithm on randomly chosen sets of 7 (resp. 5) measurement points and chose the best results with regard two maximum errors between Lattice Grid Plate TCP position measurements and TCP - positions as calculated from forward transformation based on the calibration result. Figure 6 shows a plot of these two grids of TCP positions in the case a 7 measurements. The resulting errors were as follows: Average error: 50µm Maximum error: 200µm FIGURE 6. Comparison of TCP-position grids: Crosses represent measured positions, circles represent positions calculated from encoder values using forward transformation based on calibrated values. MEASUREMENT RESULTS The exemplary measurements at the robot shown in Figure 7 demonstrate, that dynamic path errors

5 FIGURE 8. ABB-robot. Fast double corner test path for the REFERENCES [1] Weikert, S.: Dynamic Accuracy Monitoring. Proceedings ASPE 14th Annual Meeting, FIGURE 7. Prototype applied on the ABB-robot in the Giessen University robot laboratory. of industrial robots are in the range of several millimeters (see Figure 8). Thus the achieved calibration accuracy of 50µm satisfies the needs of robot trajectory monitoring quite well. CONCLUSION The parallel 2 DOF inspection device is a easy to use trajectory monitoring tool robot manipulators. The high repeatability of parallel mechanisms and suitable error compensation yield sufficient accuracy in the range of 50µm. Here, the presented symbolic methods yield more accurate and more reliable calibration results than numerical approaches. The system has a simple and low cost mechanical design. It is robust against operator and path errors as well as manufacturing environment influences. The new inspection tool will be used for dynamic robot manipulator measurements to validate simulation models. [2] N.N.: ISO Test code for machine tools. International Organization of Standardization, [3] Zirn, O.; Weikert, S.: Dynamic Accuracy Monitoring for the Comparison and Optimization of Fast Axis Feed Drives. Proceedings ASPE 12th Annual Meeting,, Norfolk, [4] Zirn, O.; Treib, T.: Similarity Laws of Parallel and Serial Manipulators for Machine Tools. Proceedings, MOVIC 98, IfR ETH Zürich, [5] Mor, J.J: The Levenberg-Marquardt algorithm: implementation and theory. in Lecture Notes in Mathematics 630: Numerical Analysis, G.A. Watson (Ed.), Springer- Verlag: Berlin, 1978, pp [6] Greuel, G.-M.; Pfister, G.; Schönemann H.: SINGULAR 3.0. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern (2005). ACKNOWLEDGMENTS The presented results have been worked out in the applied research project HWP/Innovationsbudget founded by the Hessian Ministry of Science and Art.

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