A Practical Approach for Estimating Cutter Path Motion Error of Machining Center Due to Linear Acceleration/Deceleration of Linear Segment

From the machining center (MC) user’s point of view, this paper proposes a practical approach for estimating the trajectory error of a linear interpolation cutter path produced by the linear acceleration/deceleration (Acc/Dec) processing of linear segments for a target MC. Firstly, a simple and convenient motion error model is established for describing the behavior of the servo axes. The model only contains two parameters and can be applied to any segment regardless of length. Based on the model and an assumption, in which the motion of each segment is independently performed and no interference exists between servo axes motions, a concise and efficient algorithm to simulate cutter path trajectory is developed. Comparison results of the simulated cutter path trajectories with the measured ones in verification experiments sufficiently demonstrate the effectiveness of the propose approach. Therefore, as a useful tool, the approach provides an application possibility to beforehand estimating the NC Acc/Dec motions on cutter path accuracy or judging the motion conditions for the machining purpose without performing an actual machining with the target MC.


Introduction
When precision machining a workpiece profile on MC, the cutter path is often generated by linear interpolation or circular arc interpolation for the ideal contour to be machined. In such situation, the Acc/Dec motions of interpolation segments cause some motion errors in actual cutter path, and thus the corresponding profile error occurs in the machined workpiece contour (1)(2)(3) . The shorter the segment and the higher the specified feed rate are, the larger the motion error of the cutter path becomes. Recent years, in order to achieve both accuracy and efficiency in high-speed, high-precision contour machining, some functions such as AI look-ahead control have been developed and utilized in the NC system of MC. However, since these techniques are not disclosed in detail, there are many unclear problem points in effective utilization (4,5) . Theoretically, if the control rules of Acc/Dec processing and relevant parameter values installed in the NC system are known, it should be possible to estimate and grasp the cutter path motion error caused by the cutting parameters without performing an actual contour machining (6) . However, the MC manufactures usually do not provide such detailed technical information to the users.
One author of this paper previously proposed an approach for estimating the cutter path motion error for a specific MC, based on a premise that each control axis of MC has the first-order lag characteristic and there is no interference between these axes' motions. The effectiveness of the approach has been confirmed by comparing the motion errors of simulated cutter path and the measured ones together with the machined workpiece contour errors (7,8) . However, the type of the target MC is relatively old, and a conventional exponential function operation is employed in the NC system for processing the Acc/Dec motion of cutter path segments, which is different from the mainstream method nowadays.
In this study, from the standpoint of MC users, a new motion error model of linear segment is established for a relatively new MC. The model is built on the basis of a linear Acc/Dec processing, which is currently widely utilized for NC Acc/Dec motion control of segments. Based on the Fig. 1 Cutter path consisting of n linear segments model, a simulation approach is also developed to precisely estimate the motion error of a linear interpolation cutter path. Furthermore, the effectiveness of the proposed model and simulation approach is also demonstrated by the verification experiment results.

Linear Acc/Dec Processing of Linear Segment
As shown in Fig. 1, a cutter path composed of n linear segments is specified on the XY plane of MC without any dwell at each connect point of segments. In this case, the speed command values, Fi, x and Fi, y, of the i-th segment assigned to the X and Y axis can be expressed by Eq. (1) according to the feed rate value F commanded in the NC program.
The nominal movement time of the segment, ti, is given as follows: In a case of the exponential function Acc/Dec processing to segment, Fi, x and Fi, y in Eq. (1) are used as input command to the position servo loop of the X and Y axis, respectively. However, since the commanded speed between two adjacent segments is discontinuous, the impact of Acc/Dec motion on the machine is large. In order to reduce the impact action and shorten the positioning time, a moving average type of Acc/Dec processing defined by Eq. (3) has been put into  4) and (5) practical use. This operation is also called as the linear Acc/Dec processing of segment motion control (6) .
In this equation, t and  indicate time, tA is the moving average time, usually also called the linear Acc/Dec constant, fb (t) is the speed value commanded by NC program, i.e. Fi, x and Fi, y in Eq. (1), and fa (t) is the processing result of fb (t) through the linear Acc/Dec operation and is adopted as the input command to the position servo loop of the X and Y axis.
In order to maintain the generality and conciseness of the description, in the following, Fi, x and Fi, y is described as F * , ti as tE. So, for a segment, fb (t) can be expressed as follows: Substituting Eq. (4) into Eq. (3), the corresponding fa (t) can be obtained.
As a result, the input command to the position servo loop of each driving axis can be considered to be a combination of three parts defined in the operating time intervals, [0, tA], [tA, tE] and [tE, tA+tE], respectively. Figure 2 presents the relationship between fb (t) and fa (t) to a linear segment. From this figure, the linear Acc/Dec constant tA can be treated as a command time for accelerating the speed of servo axis from 0 to the F * value or decelerating The i-th segment from F * to 0. In the case of multiple linear segments, as shown in Fig. 3, the acceleration start time of each segment is the same as the deceleration start time of the previous segment; and the segment starts to decelerate at the time tE, i.e. ti, after the acceleration started. Therefore, the speed input commanding to the position servo loop of each driving axis for whole cutter path can be generated by overlaying the command of every segment in the order of command time. From Fig. 3, it can be verified that for whole cutter path the speed input value of a driving axis is continuous and the command time tpath can be calculated by Eq. (6).
On the other hand, in this study, for a segment whose nominal movement time tE by Eq. (2) is shorter than tA, the linear Acc/Dec processing is defined by Eqs. (7) and (8), under an assumption that the displacement amount is the same as the specified value in the NC program, and the command time to movement is equal to tE +tA. Figure 4 shows the fb (t) and fa (t) of a short linear segment defined by Eqs. (7) and (8).

Output of MC Driving Axes for a Linear Segment and Simulation Approach of Cutter Path Trajectory
In this study, it is assumed that the position servo behavior of each driving axis can be described by a firstorder lag characteristic and no interference actions exit between these axes' motions for the target MC, same as the previous motion error model for MC cutter path (7,8) . The block diagram in Fig. 5 presents the relationship of the input and output for the position servo system of each driving axis   7) and (8) of the MC. Here T is the time constant with the position servo loop, and In(s) and Ou(s) are the input (command value) and output (response) described in s domain for the driving axis, respectively. Expressing the input and output in the time domain (t domain) as in(t) and ou(t), the following relation due to Laplace transform and inverse Laplace transform holds.
Therefore, as the input in(t) of the position servo loop, substituting the fa (t) explained in the above chapter into Eq. (9) and solving the equation, the output of the corresponding driving axis can be derived.
The speed output and displacement output,  is taken as T, and its value is given by the following equation (7) .
By using the same calculation method as above, the speed and displacement output of the driving axis for a short segment defined by Eqs. (7) and (8) can also be obtained.
Furthermore, for total tool path, the speed output and displacement output of each driving axis can be obtained by

Verification Experiment Results of MC Cutter Path Trajectory and Discussions
The target MC in this study is a vertical type of 3-axis MC produced five years ago. A numerical control unit of semi closed loop is installed with the MC. The movement ranges of the X, Y, Z axes are 600 mm, 530 mm, and 510 mm, respectively, with the least NC input increment of 1 m. The manufacturer's original NC cutting modes, R1, R2, R3, and R4, are built in the NC system. One of them must be specified during machining but R3 is the default mode. A function called as AI look-ahead control is also installed in the NC system, but there is not the detailed explanation on this function. According to the research purpose to verify the basic motion performance of the target MC and the proposed cutter path motion error model, all verification experiments of cutter path trajectory were performed in R3 mode without executing the AI look-ahead function in this study. Through checking the parameter setting values in the NC system, the servo loop gain for each driving axis is 50 sec −1 , and thus the Fig. 7 Feed rate of right-angle cutter path time constant T of the servo loop is 20 ms (9) . Moreover, the linear Acc/Dec constant tA for each axis is 48 ms. In this study, these values were adopted in the simulation calculations of cutter path trajectory.
A KGM181 type of cross grid measuring device by Heidenhain was applied to precisely measure the cutter path trajectory in a non-contact state. The resolution of position coordinates is 0.1 m. In addition, the speed value of driving axis is calculated by numerical differentiation using the obtained trajectory coordinate data and the sampling time specified in measurement.

Results of Verification Experiment for Right-Angled Corner Trajectory
A right-angled cutter path consisting of two linear segments with a length of 20 mm was measured in a commanded feed rate of 5000 mm/min. Figures 6 and 7 present the obtained results of the trajectory and feed rate. The sampling time used in the measurement is 0.1 ms. The simulation curves and the corresponding measurement curves are in good agreement with each other, as well as the experiment results under other feed rate value F. Therefore, it can be concluded that the values of the time constant T and the linear Acc/Dec constant tA used in the simulation calculation are appropriate with the target MC.

Results of Verification Experiment for a Path Trajectory Composed of Multiple Segments
A cutter path composed of 44 segments with a length of 10.5 mm was measured with a commanded feed rate of 5000 mm/min and the measurement sampling time is 0.1 ms. The obtained motion trajectory is shown in Fig. 8, where Point A is both the start and end of the cutter path. A part of the feed rate is presented in Fig. 9. Comparing these figures, it can be seen that each simulation curve has a good agreement with the corresponding measured one. Especially, in both the start area and the end area of the cutter path, a good overlap can be observed between the simulation curve and the measurement curve of the feed rate. Similar results were also verified from other cutter path measurements performed under different feed rate. Therefore, it can be considered that the actual motion time of the cutter path is exactly equal to that employed in the simulation calculation.

Results of Verification Experiments of Cutter Path Including Short Segments
In order to confirm the motion trajectory of a linear segment whose motion time by NC program is shorter than the linear Acc/Dec constant tA, a cutter path shown in Fig. 10 was measured under a motion condition of F=2000 mm/min. The measurement sampling time is 0.1 ms. The path contains seven X-direction segments and seven Y-direction segments. The lengths of the segment along the Y direction all are 10 mm, and the lengths of the segment along the X direction are 10 mm, 0.667 mm, 1 mm, 1.333 mm, 1.6 mm, 2 mm, and 4 mm, respectively. So, the motion time ti corresponding to each X-direction segment is 0.3 sec, 20 ms, 30 ms, 40 ms, 48 ms, 60 ms, and 0.12 sec, respectively. Figure 10 and Fig. 11 show the results of the motion trajectory and feed rate, where both the simulation and measurement curves have a good agreement with each other. The similar results are also verified from the experiments with other feed rate values. Therefore, one conclusion can be arrived that for a short linear segment, the linear Acc/Dec processing defined by Eqs. (7) and (8) Fig. 11 Feed rate of cutter path contained short segments

Results of Verification Experiment for a General Cutter Path
In order to further verify the effectiveness of the proposed motion error model and simulation approach, a general linear interpolation cutter path shown in Fig. 12 was measured under different feed rate conditions. This path is symmetrical and starts from point A, passes point B, C, D in order and stops at point E. The portion from point B to point D is consisting of 83 linear segments with a length from 1.095 mm to 3.916 mm. Figure 13 shows the cutter path trajectory in the case of sampling time of 0.2 ms and F=2000 mm/min. The thin green lines indicate the boundaries between interpolation segments (10) . Along both the simulated and measured curves,   Fig.12 a good correspondence, in either shape or size, of the local undulations around each connection point of segment can be confirmed, and the shorter the segment is, the greater the effect of Acc/Dec motion becomes. The feed rate result is v v v =+ . Small speed undulations caused by the Acc/Dec motion of segment can be observed in the area of short segments, but comparing with the measurement result of a MC employed the exponential function Acc/Dec processing (8) , it can be confirmed that the feed rate of cutter path due to the linear Acc/Dec processing is rather smoother and thus the impact action is lighter. In the results of the cutter path trajectory with other feed rate, the similar features are also observed.

Conclusions
The results of this study can be summarized as follows: (1) Based on the linear Acc/Dec processing and the characteristics of position servo system, a NC Acc/Dec motion model of linear segment has been established, which not only has a relatively simple form but also applicable to a short segment.
(2) A practical approach to simulate the motion trajectory of a linear interpolation cutter path has been proposed, under the assumptions that the motion of each segment is independent and the final trajectory can be generated by superimposing the motion of each segment in the executing time order. This approach has a simple algorithm and is easy to programing.
(3) The effectiveness of the proposed motion error model and cutter path simulation approach have been sufficiently confirmed from the comparison results of the simulated trajectories with the measured ones. As a result, the approach provides a valuable application possibility to beforehand estimating the NC Acc/Dec motions on cutter path accuracy or judging the motion conditions for the machining purpose without performing an actual machining with the target MC.
(4) In the case of the target MC, for the time constant T of the position servo loop and the linear Acc/Dec constant tA, their setting values in the NC system can be directly applied to the simulation calculation of cutter path trajectory.