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Robotics repeatability and accuracy: another approach.

Abstract. -- Repeatability is one characteristic of a robot, which is of tremendous importance. In this paper the concept of repeatability is clearly defined in terms of the standard deviation of the random component of the error of a robot in returning to a taught position and accuracy is defined in terms of the mean error as a function of three important variables. Data used to estimate repeatability and accuracy were obtained from a full-factorial experiment in which speed, payload and amount of axis movement were used as independent variables. The robot used to furnish data for this research was a PUMA 560. A regression model was developed to estimate the accuracy at various factor levels and the repeatability was determined to be 0.0036 inches. The statistical analysis clearly indicated that all three factors, as well as their interactions, affect the accuracy of the robot. The regression model indicated that approximately 35% of the radial error variability was explained by the linear model and 65% of the radial error was due to repeatability.


The performance of a robot is highly dependent upon both the repeatability and the accuracy of the robot. Repeatability is the robot's ability to return to a previously taught point (Rehg 1985). Repeatability is especially important in assembly applications of robots and has a critical effect on product quality since product tolerances are decreasing (Khouja & Kumar 1999). Specifications on robots are often obtained from robot vendors, but the problem with the use of these data is that the user does not know the conditions under which they are tenable. It is therefore necessary to investigate the interaction among various robot process variables and determine the conditions under which a given mix of values can be achieved (Offodile & Ugwu 1991).

Repeatability and accuracy are often confused and rarely defined in a clear and unambiguous way. Necessarily, both accuracy and repeatability must be estimated by using the error made by the robot when trying to return to a previously taught point. This error is defined to be the radial distance from the previously taught point to the point at which the end effector comes to rest. The method for estimating accuracy and repeatability in this research will depend upon errors obtained experimentally by varying the speed, the weight of the payload and the amount of axis movement. More specifically, for any combination of the three variables the accuracy will be defined as the mean of the distribution of errors for that combination. As a result of this definition, accuracy is constant for any fixed combination of speed, weight and amount of axis movement. Therefore, accuracy has no connection to the variability of the distribution of errors. The repeatability of a robot does depend on the variability of the distribution of errors. In fact, repeatability will be defined to be equal to three times the standard deviation of the distribution of errors.

The definitions of accuracy and repeatability in the above paragraph indicate that the mean of the distribution of errors depends on the values of the variables while the standard deviation does not. Therefore, the variation in the errors due to changes in the mean as a result of changes in the three input variables must be removed in order to estimate accuracy and repeatability. The standard mechanism for this task is a model for the means developed by using statistical techniques on data obtained from a designed experiment.


The parameters speed, payload and percentage of axis movements were varied on a PUMA 560 robot using different combinations to estimate accuracy and repeatability. A conventional X-Y-Z Cartesian coordinate measurement system was used. The points of movements to the X, Y and Z gauges were found by driving each of the six axes to different percentages of axis movements. Errors were measured using precision gauges for the X, Y and Z coordinates. A test stand was constructed for this experiment similar to the one discussed by Warnecke & Schraft (1982). The test stand was securely clamped down to a table that was leveled. The test stand allowed measurement of X, Y and Z deviations using three Mitutoyo dial indicator gauges. The three gauges used have flat faced contact plates. The resolution of the Mitutoyo gauges used is 0.0001 inches with a 0.25 inch stroke. The deviations were expected to be in the 0.001-0.004 inches range. The rule of "10" was therefore applied. This means the gages have a resolution 10 times the expected reading. The temperature in the laboratory was kept at a constant 71[degrees] F which is very close to the desired 68[degrees] F for precision measurements (DeGarmo et al. 1997).

The three parameters weight, speed and percent of movement in each axis were varied at three different levels designated low, medium and high. A total of 27 different combinations were used. The PUMA robot used had six different axes.

Weight. -- The payload of the PUMA robot used was 2.5 kg (5.5 lbs). This did not include the gripper. Four "one" lb weights and two "0.5" lb weights were used. A special designed fixture that can be attached to the wrist was used for varying the weight. It included a precision ground 0.5000 inch diameter +/- 0.0001 inch precision tooling ball. The tooling ball probe has a small "negligible" weight. The probe was locked in position so no movement was available in the X, Y and Z direction. The following loads were used in this experiment: low (1.5 lbs [approximately equal to] 30% of the payload), medium (3.0 lbs [approximately equal to] 60% of the payload) and high (4.5 lbs [approximately equal to] 90% of the payload).

Speed. -- Maximum speed of this robot was 0.5m/sec, which is equivalent to an external program speed of 100. The following speeds were used: low (30% of the maximum speed), medium (60% of the maximum speed) and high (90% of the maximum speed).

Percent of range in each axis. -- The maximum range of motion for each of the axes was as follows: Joint 1: 320 degrees (waist), Joint 2: 250 degrees (shoulder), Joint 3: 270 degrees (elbow), Joint 4: 280 degrees (wrist 1), Joint 5: 200 degrees (wrist 2) and Joint 6: 520 degrees (wrist 3). The following ranges of motion were used: low (10% of the total range), medium (30% of the total range) and high (50% of the total range). The three ranges of the total motion used in this study are given in Table 1. The 50% axis movement was not exceeded because the return approach of the robot would have been unpredictable.

The robot was operated for a 15 minute warm up period before the data gathering began. The point called GAUGE to which the end effector was programmed to return was located near one of the extreme points of the axis system. This extreme point was determined by rotating joints 1, 3 and 5 of the robot the maximum amount in the negative direction and joints 2, 4 and 6 the maximum in the positive direction. The fixture with the three gauges was located at the point called GAUGE and contact was made with the tooling ball to accurately zero the three gauges. The PUMA 560 Victor Assembly Language was used to create a program that drove the end effector to one of the three locations determined by the chosen values for the variable, amount of axis movement, and returned it to the point GAUGE. This movement was repeated ten times for each of the twenty-seven combinations for levels of speed, weight and amount of axis movement. The radial error was measured each time the tooling ball returned to the point GAUGE. The total of 270 data measurements met the minimum for the twenty-seven factor level combinations according to the ANSI/RIA R15.02-2 standard (ANSI 1992).


There were 10 measurements taken at each of the 27 factor level combinations. Therefore, the experiment is considered a full-factorial experiment with 10 replications. The response variable was the radial distance R from the point gauge to the location of the center of the tooling ball. This distance was computed from the errors in the X, Y and Z directions by R = [([X.sup.2] + [Y.sup.2] + [Z.sup.2]).sup.1/2]. After the experiment was designed and the 270 data points were obtained, data analysis was performed to determine if the three variables used in the experiment were all significant in determining the mean of error R. The analysis of the data using the Minitab software package yielded the main effects plots shown in Figure 1. These main effects plots indicate that each of the three variables was significant in determining the mean error. In general, the mean error was increased when any of the variables were changed from their medium or zero setting which indicated a quadratic relationship. Further evidence of the influence of the variables can be seen in Figure 2 which shows a graph of the error data in groups of ten replicates. This graph clearly indicates that the replicates produced tightly grouped errors but changes in levels of the three factors produced large changes in the magnitudes of the errors. Much of the variability in the values of the errors reflects changes in the factor levels. In order to get to the component of the data that reflects the repeatability of the robot, regression techniques with a linear model were used to remove the variation due to changes in factor levels. Figure 3 reveals the distribution of the random components of the data that determines the repeatability characteristic of the robot. This graph indicates an approximately normally distributed random pattern of error variation about the mean for the particular factor level combination at which the readings were taken. The computations in the analysis of variance (ANOVA) and the linear regression yielded the following equation to predict the accuracy:




RMEAN(W,S,A) = 11.8 - 3.25 S + 1.44 A + 1.85 W - 3.32 W*S - 3.02 W*A + 4.85 S*A + 2.23 [W.sup.2] + 8.59 [S.sup.2] + 4.21 [A.sup.2]

Where: W = weight, S = speed and A = percent of axis movement.

The computations confirmed that the three factors, as well as their interactions, are statistically significant (P < 0.05) in the mean of the radial error values. Analysis of variance (ANOVA) computations produced a computed value of 98.4 for the variance of the random component of the radial error values. The square root of the variance yields the standard deviation of the random component to be 9.9. The repeatability of a robot was defined to be three times the standard deviation of the random component of the radial error. Therefore, the estimate for the repeatability of the Puma 560 turns out to be 29.7. Since measurements were in 0.0001 inch units, the repeatability estimate would be stated as 0.00297 inch. The estimate is somewhat smaller than the [+ or -]0.004 inch specified by the manufacturer. If regression techniques had not been used to remove the variability due to the changes in the factor levels, the standard deviation of the raw data would be 12.12. This standard deviation yields 0.001212 when the units are changed to inches and a corresponding repeatability estimate of 0.0036 inch. When rounded to the nearest thousandth of an inch, this estimate agrees with the manufacturer's estimate.

The adequacy of such a model is usually judged by [R.sup.2], the coefficient of determination, because it gives the fraction of the total variation in the radial error data explained by the model. This model developed for predicting the accuracy of the robot had an [R.sup.2] value of 35.2%. The statistical analysis clearly indicates that all three factors, as well as their interactions, affect the accuracy of the robot. However, the relationship between these factors and the accuracy is such that the standard linear regression techniques will not produce models which account for more than approximately 35% of the radial error variability, leaving approximately 65% of the radial error variability due to repeatability. When using a robot, the accuracy of the robot at a particular setting of the parameters can be determined by the regression model and adjustments can be made to compensate for the predicted mean radial error. However, the portion of the radial error which is due to repeatability must be tolerated without recourse. Manufacturers should therefore concentrate on giving more information about the accuracy of a robot. Since they have extensive test data for each model of robot, the manufacturer could provide a linear model for the purposes of predicting accuracy of the robot as well as an estimate of the constant repeatability.
Table 1. Ranges of motion used for each of the six axes for each of the
three levels of axis movement.

 Low (10%) Medium (30%) High (50%)
Joint 1: 32 degrees Joint 1: 96 degrees Joint 1: 160 degrees
Joint 2: 25 degrees Joint 2: 75 degrees Joint 2: 125 degrees
Joint 3: 27 degrees Joint 3: 81 degrees Joint 3: 135 degrees
Joint 4: 28 degrees Joint 4: 84 degrees Joint 4: 140 degrees
Joint 5: 20 degrees Joint 5: 60 degrees Joint 5: 100 degrees
Joint 6: 52 degrees Joint 6: 156 degrees Joint 6: 260 degrees


We thank the administration of Midwestern State University and especially Dr. Norman Horner in providing the funds to perform this research. We further like to thank Mr. Andy Webb for the construction of the table and the testing fixture. We also thank Mrs. Lois Moore and Dr. Michael Shipley for their advice. This paper would not have been possible without the help of all these people and MSU institutional support.


ANSI. 1992. American National Standard for Industrial Robots and Robot Systems-Path-Related and Dynamic Performance Characteristics-Evaluation-ANSI/RIA 15.05-2-1992. American National Standards Institute, New York, 45 pp.

DeGarmo, P., J. T. Black & R. Kohser. 1997. Materials and Processes in Manufacturing, 8th ed. Prentice Hall, Upper Saddle River, New Jersey, 1259 pp.

Khouja M. J. & R. L. Kumar. 1999. An options view of robot performance in a dynamic environment. Int. J. Prod Res., 37 (6):1244-1250.

Offodile, F. & K. Ugwu. 1991. Evaluating the effect of Speed and Payload on Robot Repeatability. Robot. Comput.-Integr. Manuf., 8(1):27-28.

Rehg, J. 1985. Introduction to Robotics: A Systems Approach. Prentice Hall, Inc., Englewood Cliffs, New Jersey, 230 pp.

Warnecke, H. J. & R. D. Schraft. 1982. Industrial Robots, Application Experience. I.F.S. Publications Ltd., Kempston, Bedford, England, 298 pp.

Jan Brink, Bill Hinds* and Alan Haney

Department of Manufacturing Engineering Technology and

*Department of Mathematics, Midwestern State University

Wichita Falls, Texas 76308

JB at:
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Author:Brink, Jan; Hinds, Bill; Haney, Alan
Publication:The Texas Journal of Science
Date:May 1, 2004
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