MAPAN - Journal of The Metrology Society of of Parallel India, Vol. Model 26, Coordinate No. 1, 2011; Measuring pp. 47-53 Machine ORIGINAL ARTICLE The of Parallel Model Coordinate Measuring Machine KOSUKE IIMURA *, EIICHIRO KATAOKA, MIYU OZAKI and RYOSHU FURUTANI Tokyo Denki University Kanda Nishiki-cho 2-2, Chiyoda, 101-8457 Tokyo, Japan *e-mail:10kmm03@ms.dendai.ac.jp [Received: 09.02.2011 ; Revised: 03.03.2011 ; Accepted: 04.03.2011] Abstract The Stewart platform is one of 6-DOF (6 degrees of freedom) parallel mechanism. A parallel mechanism is a mechanism to connect six actuators in parallel between the base and the stage. It is necessary to calibrate the kinematic parameters that compose the mechanism in order to use the parallel mechanism as the coordinate measuring machine. Additionally, it is necessary to estimate the uncertainty about the measurement results. In this study, the kinematics parameters are estimated by using the calibration point in one point, the uncertainty is estimated at several measurement coordinates from kinematic parameters and postures angle. The verification method is shown following; i) The amount of the actuator expansion and contraction in the parallel mechanism is assumed to be a value obtained with the sensor and, the kinematic parameters are estimated from the values by using the least squares method. ii)the coordinates of the stylus are calculated from the estimated kinematic parameters by forward kinematics. The uncertainty is estimated from the coordinates of the stylus. As a result, the influence of the uncertainty is reported when the target coordinates are measured. 1. Indroduction Recently, the size and the form of machine parts are required to be measured accurately for its high accuracy. The serial mechanism has been conventionally used for coordinate measuring machines (CMMs), machine tools and industrial robots because of easiness in its structure and control. However, this structure has the problem that the deflection is caused by the weight of itself at each joint. Therefore, a parallel mechanism is expected to be applied to CMM as it has positioning, and more rigid structure. The CMM with parallel mechanism is called parallel model CMM. As for a CMM, more accurate positioning is required. However, it is Metrology Society of India, All rights reserved 2011. difficult to manufacture a CMM as exactly as a design value, and the various errors arise. Therefore, the errors should be compensated and the calibration work of a mechanism is important. In this study, the kinematic parameter was estimated by the selfcalibration method using sensor information of actuator on the measuring instrument. As a result, the target coordinates disposed in measuring volume are measured using the kinematic parameters estimated at the only one calibration point. The uncertainty of measurement is reported when the target coordinates are measured. 2. Model of 6 DOF Parallel CMM In this study, the Stewart platform, the representative of six DOF parallel mechanism, is dealt. The model is 47
Kosuke Iimura, Eiichiro Kataoka, Miyu Ozaki and Ryoshu Furutani shown in Fig. 1 and its top view is shown in Fig. 2. This mechanism consists of the base plate and stage. Both of the base plate and stage have six spherical joints. The base plate and stage are connected with six links which ended at each spherical joint. The links have the actuators and the stage could change the location and orientation in world coordinate system by expanding and contracting the actuators. The definition of kinematic parameters is shown in Table 1. In that case, six parameters of the base parameters are used to express the local coordinate system of itself and six parameters of the stage parameters are also used to express the local coordinate system. Some parameters are freely chosen to define coordinate system. Therefore, these parameters(b x1, b y1, b z1, b z4, b y6, b z6, s x1, s y1, s z1, s z4, s y6, and s z6 ) are redundant and are possible to be Link Actuator Base Stage Shere Joint zero. The Eq. (1) is used in the simulation is expressed for one of six equations of closed links. One closed link is shown in Fig. 3 where P is a vector of the stylus, R is a coordinate transformation matrix expressing the orientation of the stage, S is a vector of the joint of the stage, B is a vector of the joint of the base plate and A 'n is offset of an actuator.. A = P+ RS - B A ' (1) n n n n Fig. 3. One link Fig. 1. Parallel mechanism S n =[X n, Y n, Z z ] T : Coordinates of stage joints B n =[X n, Y n, Z z ] : Coordinates of base joints CC CSS SC CSC + SS R = S C S S S + C C S S C C S Sθ CθSϕ CθCϕ Rotating matrix φ θ φ θ ϕ φ ϕ φ θ ϕ φ ϕ φ θ φ θ ϕ φ ϕ φ θ ϕ φ ϕ : φ : Turning angle around z axis [deg], θ: Turning angle around y axis [deg], ϕ: Turning angle around x axis [deg] Fig. 2. Top view A 'n : Offset of an actuator, A n : Amount of actuator expansion and contraction n=1 6 48
The of Parallel Model Coordinate Measuring Machine Table 1 Parameter definition Name Letter Minion parameter Minion coordinates P Attitude angle Rotating matrix R parameter Sensor parameter Amount of actuator A n expansion and contraction Mechanism Base joints B n parameter coordinates S n Stage joints coordinates Offset of an actuator 3. Calibration Method A 'n In general calibration, the calibrated artifact is used for calibration of CMM. In this study, the selfcalibration method is applied, which uses a non calibrated artefact. The position and orientation of the parallel mechanism are determined by the amount of the expansion and contraction of six actuators. Therefore, the kinematic parameters are estimated by least squares method using six actuators lengths to be possible to be so. In least squares method, it is necessary to measure more than to determine the unknown kinematic parameters. The measurement frequency should be determined as the actuator sensor. The measurement frequency should be determined as the actuator sensor information are more than the number of the unknown kinematics parameter as shown in Eq. (2). The left side in Eq.(2) shows the number of conditions depending on the measurement frequency and the number of links of parallel mechanisms. The first term in the right side is a number of estimated kinematics parameters. The second term is a number of fixed point coordinates. The third term shows a number of orientations. 6 nm > 30 + 3n + 3nm (2) where n is number of minion coordinates and m is measurement frequency. When a fixed point of the stylus is one point, m becomes more than nine, and the measurement frequency becomes ten times or more. 4. Estimation of The target coordinates of 125 points are disposed in the measuring volume. The uncertainty was estimated by measuring the target coordinates 100 times respectively. Each target coordinates are measured 100 times and acquired data is x i. The average value is x. n is the number of measurement data, the experimental standard deviation is the mean value of the data from the Eq. (3). Next, the value acquired by Eq. (3) is assigned to Eq. (4) formulas, and standard uncertainty is estimated. Finally, expanded uncertainty was estimated from the Eq. (5) using a coverage factor k = 2. 1 = 2 S ( xi x) n 1 u c = S n n i= 1 (3) (4) U = k u c (5) 5. Simulation The kinematic parameters estimated at one calibration point are used, and all target coordinates disposed in the measuring volume are measured. The uncertainty of measurement is then estimated. First of all, it is necessary to calculate the amount of the actuator expansion and the contraction by using the reverse kinematics. Here, the symbols in Fig. 2 for the simulation are set to values in Table 2. The value when the attitude angle in the stage is changed at random by ± 30 and a set values of the mechanical parameters and one true coordinates are prepared. The amount of the actuator expansion and contraction is calculated. To estimate the kinematic parameter from more amounts of the actuator expansion and contraction, the measurement frequency was set to 100 times. The random errors of mm of standard deviation and an average of 0 were added to the calculated amount of actuator expansion and contraction as a reading error of an actuator sensor in simulation. From the acquired value, kinematics parameters are estimated using a least-squares method. Next, 125 target points are measured using order kinematics from the estimated mechanical 49
Kosuke Iimura, Eiichiro Kataoka, Miyu Ozaki and Ryoshu Furutani parameters 100 times. Fig. 4 shows the target coordinates where the uncertainty is estimated. Fig. 5 shows the front target coordinates as shown in Fig. 4. Fig. 6 shows the left target coordinates as shown in Fig. 4. The target coordinates are measured sequentially. The pitch of target coordinate in X-Y direction is 200mm and the pitch of target coordinate in Z direction is 100mm. Moreover, the random errors of mm of standard deviation and an average of 0mm were added to the amount of the actuator expansion and the contraction as a reading error of an actuator. Furthermore, target coordinates were measured using the kinematic parameters estimated using one calibration point. It is done purposely to examine how the uncertainty of the measurement is influenced by one calibration point. Therefore, the attitude angle in the stage of the measurement was kept 0 degree. The position of an artifact point is changed and measured. The uncertainty of measurement is estimated from the deviation between the target coordinates and the measured coordinates. The used calibration points are shown in Table 3, and the calibration point number from No.1 to No. 4 is No. 1 or 21,101,121 in the target coordinates of Figs. 5 and 6. Table 2 Parameter setting value Base diameter:d 1 1000mm Stage diameter:d 2 500mm Default of minion coordinates:p 400mm Offset of an actuator:a 'n 400mm Angle of base joints:α 1 90 Angle of base joints:α 2 30 Angle of stage joints :β 1 90 Angle of stage joints :β 2 30 (-400,-400,400) Z Y 800mm X 400mm 800mm Fig. 4. Target coordinates (400,400,400) (400,-400,400) 5 30 55 8 0 1 0 5 Z 1 26 51 76 101 X Fig. 5. Target coordinates (X-Z Plane) Calibration point number Table 3 Calibration coordinates 1 (-400,-400,400) 2 (-400,400,400) 3 (400,400,400) 4 (400,-400,400) Fig. 6. Target coordinates (Y-Z Plane) 50
The of Parallel Model Coordinate Measuring Machine 0 0 0 Fig. 7. Calibration coordinates (-400,-400,400) 0 Fig. 10. Calibration coordinates (-400,400,400) 0.5 2.0 0.4 0.3 Fig. 8. Calibration coordinates (-400,400,400) 1.5 1.0 0.5 Fig. 11. Calibration coordinates (400,400,400) 0 0 0 0 Fig. 9. Calibration coordinates (-400,-400,400) Fig. 12. Calibration coordinates (400,-400,400) 51
Kosuke Iimura, Eiichiro Kataoka, Miyu Ozaki and Ryoshu Furutani Joint Table 4 Tedious parameter Coordinate 1 X,Y,Z 4 Z 6 Y,Z Fig. 13. Calibration coordinates (400,400,400) Fig. 14. Calibration coordinates (400,-400,400) 6. Consideration Figures 9-10 and Figs.13-14 show the uncertainty of measurement when the target coordinates are measured by using the mechanical parameters estimated by each calibration point. When target coordinates are measured without any errors to the estimated kinematic parameters, the standard deviation of the difference between the target coordinates and the actual measured coordinates is shown in Figs. 7-8 and Figs.11-12, respectively (Scale:mm). The line in Figs. 9, 10, and13, 14 is proved to be lower right. It is proved that the uncertainty becomes smaller as the target coordinates apart from the origin in the base plate coordinate system. The joint with redundant parameters is shown in Fig. 15 and the removed parameter is shown in Table 4. Fig. 15. Position of base and stage joints was influenced by having used many redundant parameters to 1, 4, and 6 joint. The calibration points are marked as red in Fig.7, Fig8, Fig.11 and Fig.12.When target coordinates are different from the calibration point on all the conditions, the difference between the target coordinates and the actual measured coordinates became large. When the target points are measured after the kinematics parameter are calibrated around the target point from the abovementioned result, it is proved that the difference between the target coordinates and the actual measured coordinates becomes small. 7. Conclusion The following conclusions are drawn from the studies; The uncertainty of measurement becomes small when the target coordinates is apart from the origin of the base plate coordinate system. 52
The of Parallel Model Coordinate Measuring Machine The uncertainty of the measurement is influenced by the method vide which the redundant parameters are specified. As a result of measurement using one calibration point, when target coordinates are apart from the calibration point, it turned out that the difference between target coordinates and an actual measured coordinates becomes large. When the target points are measured after the kinematics parameter are calibrated around the target point from the above-mentioned result, it is proved that the difference between the target coordinates and the actual measured coordinates becomes smaller. References [1] K. Takamasu, R. Furutani, K. Shimojima and O. Sato, Artifact Calibration of Coordinate Measuring Machine - Kinematic Calibration, Journal of JSPE, 69 (2003) 851-855 (In Japanese). [2] K. Takamasu, O. Sato, C. Sinlapeecheewa, K. Shimojima and R. Furutani, Artifact Calibration of Coordinate Measuring Machine(2 nd Report)- Self Calibration of Redundant Coordinate Measuring Machine, Journal of JSPE, 70 (2004) 711-715 (In Japanese). [3] K. Takamasu, O. Sato, K. Shimojima and R. FurutaniI, Artifact Calibration of Coordinate Measuring Machine(3nd Report)-Estimation of of Measurements After Calibration, Journal of JSPE, 71 (2005) 890-894 (In Japanese). [4] K. Takamasu, G. Bi-Wei, R. Furutani and S. Ozono, Basic Concept of Feature Based Metrology, Journal of JSPE, 64 (1998) 94-98 (In Japanese). [5] O. Sato, K. Shimojima, R. Furutani and K. Takamasu, Artifact Calibration of Parallel Mechanism (1 st Report)-Kinematic Calibration with a Priori Knowledge, Journal of JSPE, 70 (2004) 96-100 (In Japanese). [6] O. Sato, G. Olea, M. HirakI and K. Takamasu, Calibration of Parallel Coordinate Measuring Machine-Kinematic Calibration with Multi- Links, Journal of JSPE, 70 (2004) 214-218 (In Japanese). [7] K. Umetsu and R. Furutani, Calibration and of CMM based on Estimation of Geometric Errors, Journal of JSPE, 69 (2003) 64-68 (In Japanese). 53