Journal of Advanced Mechanical Design, Systems, and Manufacturing

Transcription

1 Vol., No., 7 Evaluation of Synchronous Motion in Five-axis Machining Centers With a Tilting Rotary Table * Masaomi TSUTSUMI ** Daisuke YUMIZA ** Keizo UTSUMI *** and Ryuta SATO ** **Tokyo University of Agriculture and Technology -4-6, Nakacho, Koganei Tokyo, , Japan ***Makino Milling Machine, Co. Ltd. 43 Nakatsu, Aikawa-cho, Kanagawa, 43-33, Japan tsutsumi@cc.tuat.ac.jp Abstract This paper proposes a new method for evaluating the synchronous inaccuracy of a translational axis and a rotational axis in five-axis controlled machining centers with a tilting rotary table. A circular trajectory whose shape is easy to evaluate the specific features is adopted for measuring the circular path described by the two axes. The influence of inaccurate synchronization on the circular path was simulated by changing the distance between the centers of the rotary table and the circular path, the radius of the circular path and the feed speed. Measurement conditions were determined based on the simulation results, and then ball bar measurements and machining experiments were conducted. From the simulation and experimental results, it is confirmed that the proposed method can be used for evaluating the inaccurate synchronization of a translational axis and a rotational axis. However, careful alignment of the center of the rotational axis and the machine coordinate origin is important for evaluating the synchronous accuracy. The ratio of the distance between both centers of the rotational axis and the circular path to the radius provides useful information for the evaluation. Key words: Machine Tool, Five Axis Machining Center, Synchronous Motion, Ball Bar System, Rotational Axis, Translational Axis. Introduction *Received Dec., 6 (No. T-6-34) Japanese Original : Trans. Jpn. Soc. Mech. Eng., Vol.7, No.73, C (6), pp (Received 3 Mar., 6) [DOI:.99/jamdsm..4] Five-axis machining centers have been used for manufacturing complex components, such as aero-components and turbine blades, and also have been recently employed for the machining of automobile parts and plastic molds. The machining of these components and molds requires simultaneous five-axis control motion. However, the geometric inaccuracy of such five-axis motion has not been evaluated appropriately because there are few existing methods that can perform this task. Many research works relating to performance evaluations have been conducted. Bryan () reported a telescoping magnetic ball bar instrument for measuring the positioning inaccuracy of three-dimensional measurement machines and machine tools. Kakino () enhanced the telescopic magnetic ball bar and applied it to the diagnosis of numerical controlled machine tools. Knapp (3) and Nakazawa (4) independently developed a circular test method that utilized a circular plate and a bi-directional displacement sensor. One of authors also developed an alternative method that employs a rotating one-dimensional displacement probe (5). These three methods were specified as the subclause 6.63 of ISO 3- in 996 (6). 4

2 Vol., No., 7 After that, several measuring devices and methods were reported (7,8). However, the ball bar system is still widely used for evaluating and diagnosing the geometric inaccuracy in all three methods, and this system is currently applied to the measurement of the geometric inaccuracy of five-axis controlled machining centers (9~4). As mentioned above, previous investigations have focused on geometric inaccuracy due to the inaccuracy of translational and rotational axes. Five-axis machining centers are generally used for machining complex parts. When the relative feed speed between a tool and a workpiece is held constant during the machining of complex parts, the displacement of each controlled axis is over the prescribed movement, and rapid velocity changes (5) are observed. As a result, it may be pointed out that the rapid speed changes degrade the accuracy of finished workpieces (6). Test methods for accuracy evaluation specified by Annex B of ISO 79- (7) and Clause K6 of ISO 79-6 (8) cannot be applied to five-axis machines with a tilting rotary table. These two standards were developed for five-axis machining centers with a double pivot head. The machining test that employs a cone frustum specified by NAS 979 (9) is also for a double pivot type, and has been accepted by numerous machine tool manufacturers because The Boeing Company has specified this method as an acceptance condition of five-axis machining centers for the manufacturing of aero-components (). However, according to the latest report (), the center coordinates and the inclination angle of the cone frustum dominate the shape of the finished workpiece, so it is not easy to identify the causes of deterioration in the synchronous motion by measuring the diameter and profile of a finished cone frustum. When a circular motion test using the C and X axes was carried out, there was a condition that caused remarkable velocity changes in the two axes. As a result, there is the possibility of a response difference between the translational and rotary axes. Therefore, such a test is a first attempt and nobody has yet developed a method for measuring synchronous motion. For this reason, this paper details a newly-developed method for measuring the synchronous motion accuracy of translational and rotational axes. The synchronous motion was analyzed via simulation, measurements were carried out by a ball bar instrument, actual machining of cylindrical parts was conducted, and the effectiveness of the proposed method was discussed.. Simultaneous two-axis motion. Generation of simultaneous two-axis controlled motion The target of this research work is five-axis machining centers with a tilting rotary table. A variety of structural configurations of this type of machine have recently been manufactured. The five-axis machine used in this study has rotary and tilting axes designated as the C and A axes, respectively. +Z Figure shows a typical +C +A five-axis controlled machining Column center with a tilting rotary table +Y +X and its coordinate frame. The structural configuration is Tilting rotary described as ZX/Y A C according table to the references (~5). The bed is expressed by slash / according to the reference (3) in this paper, Bed although the bed has been Fig. Five-axis controlled machining center with expressed by a character of F or a tilting rotary table (vertical spindle) (ZX/Y A C type) 5

3 Vol., No., 7 O in several papers (4,5). Figure shows the cutter path and definition of symbols used in this paper. As shown in the figure, the X and C axes are used for describing the circular cutter path. r is the radius of the circular path described by the tool center P i (x i,y i ). X T is the distance between the center O T of the circular path and the center O of the rotary table. The coordinates of the tool center position are calculated as follows without using CAM software. (a) The circular path is equally divided into n mutual line segments. The length of segment L W is the distance between adjacent two points, P i and P i+. πr L w = () n (b) As defined in Fig., the tool center coordinates P i (x i, y i ) are expressed by ( i i T i i x, y ) = ( X + r cosφ, r sinφ ) () where, φ i = P O T P i (c) The rotational angle θ i of the C axis can be calculated from the tool center coordinates P i (x i,y i ). yi θ i = tan (3) xi Tilting rotary table (d) The command value X i of the X axis can be calculated from the tool center coordinates P i (x i,y i ) and the rotational angle θ i. Y Fig. X O θ i P i+ Pi φ i O P T Cutter path Cutter path and definition of symbols Xi cosθi = sinθi sinθi xi cosθ i yi (4). Feature of simultaneous control motion When the synchronous motion of the C axis and the X axis is conducted in accordance with the command values, which are calculated using the above procedures, the speed change of the C axis greatly depends on the values of X T and r. The maximum angular velocity of the C axis is derived from Eq. (3), which is differentiated by time t. That is, dθ = dt d dt tan X T r sinφ + r cosφ (5) The angular velocity ω is given by Eq. (6) dθ + u cosφ dφ ω = = dt u cosφ + u (6) + dt X where, u T dφ F = and = are constant. r dt 6r The angular acceleration is also derived by differentiating Eq. (6) d θ = dt usinφ ( u + ) ( u cosφ + u + ) dφ dt From Eq. (7), when sinφ =, the maximum velocity of ω is given; that is φ= π. The maximum angular velocity is expressed by Eq. (8) when X T r. (7) F ω max = (8) X 6r T r 6

4 mm/min V X T =5 mm (a) X axis Fig. 3 Vol., No., 7 By the way, even if the feed speed F is held constant, the velocity of each axis is not constant. For this reason, the command speed of the X and C axes were calculated for different distances X T = 45,5 and 55 mm for a fixed feed speed F= mm/min along the circular path and radius r=5 mm. Figure 3 shows the relationships between the angular position φ and the velocities. As shown in Fig. 3(a), the velocity of the X axis gradually changes at X T = 5 mm. At X T =45 and 55 mm, the velocity steeply changes around φ=8 degrees from negative to positive. However, the maximum velocity is only mm/min, which is considerably lower than the maximum feed speed of 5 m/min in the machine specification. Therefore, it may be said that the X-axis will not cause the delay. On the other hand, as shown in Fig. 3(b), the maximum angular velocity of the C axis is over deg/min ( 3 min - ) at 8 degrees in the case of X T =45 mm. This speed change means that the C axis rapidly accelerates up to the maximum rotational speed (33 min - ) within a very short time. Therefore, there is the possibility of causing a delay in the C axis. In addition, two enlarged views are shown in Fig. 3(b). It can be seen from the enlarged views that the C axis rotates in one direction at X T =45 mm, while the C axis changes its direction of rotation twice at X T =55 mm. In the case of X T = 5mm, the C axis rotates at a constant speed of 573 deg/min. 3 Simulation and results 3. Simulation Simulations were conducted before experiments, and the trajectories described by the center of the tool were investigated. Figure 4 shows block diagrams for a translational axis and a rotational axis. As shown in the figure, the dynamic model for a translational axis is a conventional one, while friction and backlash were introduced into the model for the rotational axis. A worm gear model was also introduced for the rotational axis, as shown in Fig. 4(b). The reduction ratio of the worm gear R is expressed by Eq. (9). ( θ ) X T =45 mm X T =55 mm ω 3 deg/min (b) C axis R = Rall + W sin NR all m (9) Here, R all is the total reduction ratio of the worm gear, N is the number of teeth of the worm wheel, and W is the fluctuation ratio of the worm gear. The parameters, along with their units, used in the models are shown in Table. The position loop gain G p was calculated by the procedure described in the reference (6), and the velocity loop gain G v was the default. In the simulation, the distance change between both balls on the spindle side and the table side was calculated using the same method as the ball bar measurement. The deviation E from the reference circle was calculated by Eq. (), into which the output values of each axis were substituted. - ω 3 deg/min X T =45 mm X T =55 mm 573 deg/min Velocity changes of X and C axes under constant feed speed ω 3 deg/min X T =5 mm 36 7

5 out T out T out Vol., No., 7 E = ( X X cos C ) + ( X sin C ) R () In this equation, R is the reference length, which is equal to the radius of circular path, X out is the linear displacement of the X axis, and C out is the rotational angle of the C axis. Both X out and C out are used for the simulation. X Xref + K ff s f K K pp + + θ m Xout l X + Kvp T i s + Js+C s π Velocity loop Position loop (a) Translational axis C Cref + K ff s K pp + θ m θ m + + Kvp T i s Js+C Velocity loop Position loop s C R Backlash Gear Cout (b) Rotational axis Fig. 4 Block diagrams of feed drive system. Parameters are defined in Table Table Parameters for the model of feed drive systems Parameters Unit Values X-axis C -axis Equivalent inertia, J kgm.8.5 Coulomb's friction torque, f Nm Viscous damping coefficient, C Nms/rad.4.5 Velocity loop gain, G v=k vp/j rad/s 35 Position loop gain, G p=k pp (l/π) /s 4 4 Velocity integrator reset time, T i s.5. Lead of ball screw, l m Total reduction gear ratio, R all /9 Number of teeth of worm wheel, N Fluctuation of worm gear, W (clockwise) rad Simulation results Simulations were conducted, changing the distance X T while the feed speed F = mm/min along the circular path and the radius r= 5 mm are held constant. Figure 5 shows the simulated trajectories. According to the figures, periodic changes can be observed around 9 degrees and 7 degrees in all results. This is the effect of the worm gear fluctuation. A convex deviation about 5 µm in height is observed at the 8-degree position in Fig. 5(a), and a concave deviation about 5 µm in depth is observed at the 8-degree position in Fig. 5(c). However, such ruggedness is not observed in Fig. 5(b). These results indicate that quite different trajectories are obtained at the 8-degree position according to the distance X T. The ruggedness of the trajectory corresponds to a rapid change in the angular velocity of C axis. Reversal glitches of the X-axis at degrees and 8 degrees are observed in Fig. 5(a). Two steps, which are the effect of backlash in the C axis, are clearly observed at 55 degrees and 5 degrees in Fig. 5(c). If the concave and convex deviations are affected by the asynchronous motion, it is assumed that the deviation depends on the feed speed. For this reason, the simulation was 8

6 9º 9º Vol., No., 7 9º 8º º 8º º 8º º 7º 7º 7º ω max = 459 deg/min ω max = 573 deg/min ω max = 459 deg/min (a) X T =45 mm (b) X T =5 mm (c) XT =55 mm Fig. 5 Simulated trajectories (ω max: maximum angular velocity) Deviation µm 5 5 F=5 mm/min F= mm/min F=5 mm/min (a) r=5mm, X T =45mm (b) r= mm, X T =9 mm Fig. 6 Synchronous accuracy of X and C axes conducted for feed speeds F = 5, 5 and mm/min without considering the backlash. Moreover, in order to investigate the cause of the generation of the servo delay due to the rapid speed change of the axes, two conditions were compared; the speed of the X axis is constant and only the angular velocity of the C axis is changed. That is, at X T /r<, the angular velocity of the C axis in the condition of (r,x T )=(,9) is a half of the condition of (r,x T )=(5, 8º Deviation µm 9º F=5 mm/min division=5 µm º F= mm/min F=5 mm/min º 9º º 7º 7º (a) Only feed forward (b) Feed forward control and control was introduced velocity loop gain adjusted Fig. 7 Simulated trajectories of X and C axes under feed forward control (at X T =45 mm) 45), and the translational velocity in the two conditions does not change. As shown in Fig. 6, which is the simulation results, the maximum value at 8 degrees increases with increasing feed speeds. Furthermore, the maximum value of the conditions shown in Fig. 6(a) is larger than that shown in Fig. 6(b). This means that the deviation around 8 degrees increases for faster feed speeds and also increases for the conditions in Fig. 6(a), where the angular velocity of the C axis is double. Therefore, it can be said that convex and concave deviations shown in Fig. 5 are greatly associated with the angular velocity of the C axis. The delay can be canceled by introducing feed forward control if the convex and concave deviations are generated by a servo delay of the C axis. And, as a result, the synchronous motion accuracy is thought to be repairable. The feed forward control was introduced into the block diagrams shown in Fig. 4. As shown in Fig, 7(a), which is the result of the simulation, the deviation around 8 degrees is considerably improved, but the distortion remains in the trajectory even after the feed forward control is introduced. Then, the velocity loop gain of the C axis was matched to the gain of the X-axis. As shown in Fig

7 Vol., No., 7 7(b), it can be seen that the distortion around 8 degrees disappeared completely. It may be said that the distortion of the trajectory around 8 degrees is generated by the asynchronous motion of two axes. 4. Ball bar measurements and machining tests 4. Experimental methods Machining tests were then carried out. In the machining tests, the center of a workpiece was set on O T shown in Fig.. The radius r =5 mm of the tool path and two distances X T =45 and 55 mm between the centers were chosen. The other cutting conditions are shown in Table. The shape of the cylindrical part of the test piece was measured by a roundnessmeasuring machine. Ball bar measurements were also carried out under the same conditions as the machining tests except for the spindle rotation. It is important to center the ball on the spindle side with the center of the C axis of rotation to minimize the eccentricity at the ball bar measurement. 4. Experimental results Figure 8 shows the roundness curve of the finished workpiece and the measured trajectory with the ball bar at X T /r=.9. In this condition, the influence of the backlash of the C axis on both roundness curves does not appear because the C axis continuously rotates in one direction. According to the figures, the curve largely projects outside between 6 degrees and degrees and sharply falls into the center at the 8-degree position. At the degree position, a quadrant glitch due to the reversal motion of the X-axis is observed in both trajectories. Moreover, periodic changes are observed between 45 degrees and 35 degrees and between 5 degrees and 35 degrees. From these points of view, it can be said that the trajectory measured by the ball bar coincides with the roundness curves of the machined workpiece. The roundness curve sharply falls at the 35-degree position. This is due to the effect of the machining start point. Figure 9 also shows the experimental results at X T /r=.. It can be seen that two big steps are formed at 55 degrees and 5 degrees and a sharp cusp is also formed at 8 degrees. The big step that appears when the direction of rotation reverses is due to the backlash of the C axis. The height of the step is about 5 µm in the cutting result, but its value is lower than that of the ball bar measurement. Other features of the trajectories also coincide for the two results. Figure shows the enlarged photographs of the machined workpiece surface at the 8-degree position, where the rapid velocity change of the C axis is present. A ruggedness pattern can be confirmed in both photographs. 8º Table Cutting conditions Work piece Aluminum alloy flute end mill (HSS) Right hand herical tooth Tool Diameter: mm Length of cutting teeth:38 mm Cutting method Dry cutting, Down cut milling Feed speed mm/min Spindle speed 5 min - Radial depth of cut 5 µm Axial depth of cut 5 mm 9º 7º (a) Cutting division=5 µm º 8º 9º 7º (b) Ball-bar Fig. 8 Experimental results of synchronous motion of X and C axes (at X T/r=.9) º 3

8 The short periodic deviations observed around 9 and 7 degrees in Figs. 8 and 9 are due to the influence of the worm gear fluctuation, which drives the C axis. The periodic deviations are same as the simulated trajectories. The difference between the simulation and the experiment is only quadrant glitch at the 8-degree position, where the thin ruggedness changes internally and externally in Figs. 8 and 9, respectively. These are caused by the reversal motion of the X-axis. The quadrant glitch in the simulation is smaller than that in the experiment. In addition, the influence of the backlash appeared on the trajectories at both the 55 degree and 5 degree positions, as shown in Fig. 9, when the C axis was reversed. If only the evaluation of the synchronous accuracy of a translational axis and a rotational axis is the purpose of an acceptance test, it is convenient that the influence of backlash is not included in the measurement results. Consequently, the measurement condition of X T /r=.9 is more suitable for the 8º Roundness µm º (a) Cutting Fig. 9 Simulation with backlash Simulation without backlash Ball-bar measurement Vol., No., 7 7º (b) Ball-bar X T /r Fig. Effect of X T/r on roundness of motion (r=5 mm,f= mm/min) acceptance test than that of X T /r=.. Next, the influence of X T on the roundness of the circular path was investigated to determine the suitable distance for evaluating the synchronous accuracy. 4.3 Influence of X T on the roundness It has been pointed out in the preceding chapter that the angular velocity rapidly increases when X T /r approaches one in X T /r<., as shown in Eq. (8). In order to investigate the influence of the distance X T on the roundness of the circular path, measurements with a ball bar as well as simulations were conducted. The radius r=5 mm and the feed rate F= mm/min were held constant. The backlash of the worm gear was also introduced into the simulation. Figure shows the relationships between X T /r and roundness. There is no effect of the backlash in the range of X T /r<., because the C axis rotates in one direction. When X T /r approaches one in X T /r<., the roundness measured with the ball bar and the simulated trajectory with backlash rapidly increases. In the range of X T /r >., the value of the roundness first decreases and then increases when X T /r is increased in both the measured result and the simulated results considering the backlash, as shown in Fig.. The roundness takes a minimum value. The influence of the 9º º (a) X T =45 mm Fig. mm division=5 µm 9º 8º Experimental results of synchronous motion of X and C axes (at X T/r=.) mm (b) X T =55 mm Photograph of machined surface near the position of 8º º 3

9 Vol., No., 7 synchronous motion on the roundness appears to be more significant than the influence of the backlash up to the minimum value. This means that the influence of the backlash of the C axis becomes larger than the influence of the synchronous motion over the minimum value. It is clear that the roundness curve without considering the backlash decreases when X T /r is increased. In X T /r <., both the measurement and simulations curves rapidly increase as X T /r approaches.. The value of the roundness decreases with decreasing X T /r. However, the difference between the measurement and the simulation is about 3 µm. It is thought that this difference is an uncertain value that can change easily when the ball bar is set. Moreover, the measured value is about 5 µm at X T /r=.5. In the accuracy measurement of the machine tools used for a general purpose, a tolerance less than 5 µm cannot be defined. It is, therefore, preferable to set X T /r as close to one as possible if the synchronous accuracy of a translational axis and a rotation axis is evaluated. 4.4 Influence of eccentricity of the center of C axis In this paper, the influence of measurement conditions have been investigated based on the assumption that the machine coordinate origin O was exactly aligned with the C axis centerline. Therefore, the C axis center was carefully aligned with the machine coordinate origin by using a displacement sensor before every measurement. For instance, before a cutting experiment, a test mandrel with the same diameter as the end mill used is fixed to the tool holder, and the stylus of a displacement sensor mounted on the rotary table is touched to the cylindrical part of the test mandrel. The C axis is then rotated and the center of the C axis of rotation is adjusted until the reading of the displacement sensor becomes zero. Before the ball bar measurement, alignment work was carefully performed. That is, the ball on the table side was fixed to a position 5 mm away from the C axis center, and the C axis was rotated 36 degrees. The rotation of the table was measured by the ball bar in the radial direction. Eccentric components of the circular trajectory were calculated and then corrected. However, it is not easy to match the machine coordinate origin O and the C axis center exactly. The influence of the eccentricity on the measurement results could be very large if this alignment is inadequate. As a result, it may be impossible to evaluate the synchronous accuracy of a translational axis and a rotary axis. Then, the machine coordinate origin O and the C axis center were intentionally moved and the simulation and measurement were conducted in order to investigate the influence of eccentricity on the roundness of the circular trajectories. The eccentricity in the XY plane is expressed as (dx, dy). The deviation E was calculated by Eq. (). E = a + b R () where, {( X dx cosc + dy C dx) } a = X out T ) out sin out () b = ( X T dy)sin Cout dy coscout dy (3) Figure shows the trajectory measured with the ball bar and the trajectory simulated with an eccentricity in the X direction. The trajectory without eccentricity is also shown in the figure. As shown in Fig. (a), both the simulated and measured trajectories take a minimum value when there is an eccentricity of dx=3 µm. However, when there is no eccentricity, the trajectory takes a maximum value at 8 degrees. This means that the existence of a positive eccentricity of dx may deny the maximum value, though its value is actually taken at 8 degrees due to the servo delay of the C axis. The existence of a negative eccentricity of dx makes the maximum value larger, as shown in Fig. (b). 3

10 Vol., No., 7 Deviation µm 4 Simulation (dx= µm) Deviation µm 4 Simulation (dx=-3 µm) - Ball-bar - Simulation (dx= µm) Ball-bar Simulation (dx=3 µm) (a) dx=3 µm (b) dx= 3 µm Fig. Effect of center offset in the X direction Deviation µm 4 - Simulation (dy= 3µm) Simulation (dy= µm) Ball-bar (a) dy=3 µm Fig. 3 - Simulation (dy= µm) (b) dy= 3 µm An eccentricity in the Y direction greatly influences the trajectory, as shown in Fig : dx The influence of the eccentricity in the Y 4 : dy direction is larger than that of the eccentricity in the X direction and that of the C axis servo delay. This means that it is more difficult to evaluate the synchronous accuracy because the influence of the eccentricity in the Y direction Eccentricity µm is larger than that of the servo delay of the C Fig. 4 Effect of eccentricity on roundness axis. Next, the eccentricities of dx and dy were changed every 5 µm from -3 µm to +3 µm in order to make clear the range of allowable eccentricity. Measurements with the ball bar were conducted. Figure 4 shows the relationship between the eccentricity and the roundness. It is found that a larger negative eccentricity in the X direction gives a larger roundness value, and a larger positive eccentricity in the Y direction also gives a larger value. As compared with the results in the two directions, the influence of the eccentricity in the Y direction is larger than that in the X direction. It is, however, found that there is a condition in which a certain eccentricity can improve the roundness. This is because the eccentricities in X and Y directions deny the projection caused by a synchronous error. It can be said that the machine coordinate origin and the C axis center must be centered within ±5 µm when the synchronous motion is measured. In addition, it is necessary to choose the conditions needed to detect the synchronous motion error easily. As the measurements with the ball bar have an advantage in that the eccentricities can be immediately adjusted from the measurement data, it is easy to adjust both centers of the machine coordinate origin and the C axis. On the other hand, it is necessary to confirm whether the alignment work has been appropriately done by measuring the roundness of the machined test piece in the machining test. Therefore, it is difficult to correct the eccentricity immediately. Deviation µm 4 Roundness µ m Ball-bar Effect of center offset in the Y direction Simulation (dy=-3 µm) 5 Conclusions This paper detailed a newly developed method for evaluating the synchronous motion of a translational axis and a rotational axis in five-axis machining centers with a tilting 33

11 Vol., No., 7 rotary table using a circular motion. Simulation, measurement with a ball bar instrument and machining tests were conducted using the proposed method. It was confirmed that the tool path could be measured by the ball bar and the shape of the finished workpiece could be measured by a roundness measuring machine. It is pointed out that there are several conditions that should be considered when the measurements are performed. As an example for evaluating the synchronous accuracy of a translational axis and a rotational axis, the X and C axes were simultaneously controlled and the obtained trajectories were precisely investigated by varying several conditions. The following conclusions were obtained. () Simultaneous two axis controlled motion is greatly affected by the ratio of the distance X T between the centers of the table rotation and the circular path and the radius r of the circular path. It is necessary to cause a rapid speed change in the rotary axis in order to evaluate the synchronous accuracy of two axes. () The synchronizing error of the translational axis and the rotary axis is observed on the side closer to the center of the rotary table, and the profile of the trajectory is convex at X T /r< and is concave at X T /r>. In addition, the influence of the synchronous motion at X T /r<.5 does not appear on the trajectory. (3) If the eccentricity of the center of rotation from the machine coordinate origin is large, it is difficult to evaluate the synchronous motion accuracy correctly because the roundness of the measured circular trajectory is greatly affected by the eccentricity. It can be concluded that the proposed method is applicable for evaluating the synchronous motion accuracy of a translational axis and a rotation axis. Acknowledgements This research has been supported by the Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), #73659, 5, and also supported by the MAZAK Foundation. References () Bryan, B. J., A simple method for testing measuring machines and machine tools, part, Principles and applications, Precision Engineering, vol. 4, No., (98), pp () Kakino, Y., Ihara, Y., Kamei, A. and Ise, T., Study on the motion accuracy of NC machine tools (st report), - The measurement and evaluation of motion errors by double ball bar test, Journal of the Japan Society of Precision Engineering, vol. 5, No. 7, (986), pp (3) Knapp, W., Circular test for three coordinate measuring machines and machine tools, Precision Engineering, vol. 5, No. 3, (983), pp (4) Nakazawa, H. and Ito, K., Measurement system of contouring accuracy on NC machine tools, Bulletin of Japan Society of Precision Engineering, vol., No. 4, (978), pp (5) Tsutsumi, M. and Sakai, K., New measuring method of circular movement of NC machine tools: Development of alternative method for standardization, JSME International Journal, Series C, vol. 36, No. 4, (993), pp (6) ISO 3-, Test code for machine tools - Part : Geometric accuracy of machines operating under no-load or finishing conditions, (996). (7) Lei, W. T. and Hsu, Y. Y., Accuracy test of five-axis CNC machine tool with 3D probe-ball, Part : Design and modeling, International Journal of Machine Tools and Manufacture, vol. 4, (), pp (8) Utsumi, K. Kosugi, T., Saito, A. and Tsutsumi, M., Measurement method of geometric accuracy of five-axis controlled machining centers (Measurement by a master ball and a displacement sensor), Japan Society of Mechanical Engineers, Series C, vol. 73, No. 79, (6), pp (9) Sakamoto, S. and Inasaki, I., Identification of alignment errors in five-axis machining centers, 34

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