The Acceleration/deceleration Control Algorithm Based on Trapezoid-curve Jerk in Cnc Machining

In this study, we propose the acceleration/deceleration control algorithm based on trapezoid-curve jerk in CNC machining. In aviation and mould and die industry, it is much significant to achieve high accuracy CNC machining on complex profile parts. The unsmooth Acceleration/Deceleration (ab. Acc/Dec) control in feed movement is one of the main reasons to bring about machine tools impact and vibration in practical machining. After analyzing the CNC machine tools dynamic model, an Acc/Dec control algorithm based on trapezoid-curve jerk is put forward in order to avoid step change in jerk curve in the study; Moreover, the motion profile smooth control approach based on continuous jerk is developed in details to decrease machine tools impact according to various kinematics constraint conditions, such as the maximum acceleration, the maximum jerk, the machining program segment displacement, the instruction feedrate and so on; Finally, the developed Acc/Dec approach and the traditional linear Acc/Dec approach are compared in the CNC experimental table. The results reveal that the developed approach can achieve more smooth and flexible motion profile, which is helpful to minish machine tools impact and enhance parts machining surface quality.


INTRODUCTION
It is important to ensure smooth movement and reduce machine tools impact in high precision CNC machining.However, in practical machining the impact and vibration on machine tools worsen cutting process and destroy machining quality.It has been reported that the unsmooth Acc/Dec control in feed movement is one of the main reasons to bring about machine tools impact and vibration (Chen et al., 2006).
Consequently, several researchers have developed various Acc/Dec control approach in CNC machining.The jerk profile in the linear Acc/Dec, exponent Acc/Dec approach and the S-shape curve Acc/Dec approach is discontinuous, which leads to flexible impact on machine tools (Shi et al., 2007).The excessive jerk still exists due to abrupt change in acceleration profile (Xu et al., 2008), which will cause shock to the machine as well as deteriorate the surface accuracy (Zhang et al., 2009).Accordingly, Chi-Wei et al. (2003) presents optimal S-curve velocity profiles of PTP motion generation for a multiple-axis machine subject to individual axis' maximum velocity, acceleration and jerk constraints; Jiang et al. (2011) introduces the feedrate control algorithm based on the trigonometric acceleration-deceleration and machine dynamics conditions; Sheng-Jung et al. (2009) proposes a parametric interpolator composed of a look-ahead stage and a real-time sampling stage for jerk-limited acceleration planning based on constant jerk (Sung-Ho and Min-Yang, 2004) presents a new recursive trajectory generation method to produce exact feedrate trajectory generation through jerk-limited acceleration profiles for the parametric curves (Yih-Fang et al., 2010) Chang introduces a look-ahead linear jerk filter to ensure the smooth and accurate motion of a tool with a linear change in jerk during real-time machining; Wan et al. (2009) Daping presents a feedrate scheduling and jerk control interpolation algorithm to seek the optimal feedrate by evaluating the tool path ahead.
The CNC Acc/Dec control algorithms mentioned above can reduce impact on machine tools a certain extent.However, the step change in jerk curve will produce flexible impact on machine tools unavoidably.Accordingly, in order to achieve high precision CNC machining, an Acc/Dec approach based on trapezoidcurve jerk is put forward in the study firstly; Secondly, the motion profile smooth control algorithm based on continuous jerk is developed in details to decrease machine tools impact according to various kinematics constraint conditions.
This study proposes the acceleration/deceleration control algorithm based on trapezoid-curve jerk in CNC machining.In aviation and mould and die industry, it is much significant to achieve high accuracy CNC machining on complex profile parts.The unsmooth Acceleration/Deceleration (ab.Acc/Dec) control in feed movement is one of the main reasons to bring about machine tools impact and vibration in practical machining.After analyzing the CNC machine tools dynamic model, an Acc/Dec control algorithm based on trapezoid-curve jerk is put forward in order to avoid step change in jerk curve in the study; Moreover, the motion profile smooth control approach based on continuous jerk is developed in details to decrease machine tools impact according to various kinematics constraint conditions, such as the maximum acceleration, the maximum jerk, the machining program segment displacement, the instruction feedrate and so on; Finally, the developed Acc/Dec approach and the traditional linear Acc/Dec approach are compared in the CNC experimental table.The results reveal that the developed approach can achieve more smooth and flexible motion profile, which is helpful to minish machine tools impact and enhance parts machining surface quality.

AN ACC/DEC CONTROL APPROACH BASED ON TRAPEZOID-CURVE JERK
The CNC machine tools dynamic model: Suppose the CNC machine tools motion components mass be m then the relational expression among the servo-actuator driving force F, the cutting force F c , the friction force F f and the feeding acceleration a can be obtained: Differentiate the Eq. ( 1): where, the jerk j describes the relation between the machine tools response speed and the movement smooth characteristic in theory (Leng et al., 2008).

An Acc/Dec control approach based on trapezoidcurve jerk (a new approach):
A new Acc/Dec control approach based on trapezoid-curve jerk is put forward in the study, where the jerk curve adopts linear Acc/Dec which is first-order continuous.The displacement S, feedrate V and acceleration a curves can be obtained when integrating j curve in turn: (3) As shown in Fig. 1, a whole motion process includes seven segments: AB is acceleration speed-up portion, BC is even speed-up portion, CD is decrease speed-up portion, DE is uniform speed portion, EF is acceleration slowing-down portion, FG is even slowing-down portion and GH is decrease slowingdown portion.
Let the slope coefficient of j curve be l, which is computed through machining experimentations.The q 0 , q 1 , q 2 , q 3 , and q 4 value are various in Eq. ( 3), (4), ( 5) and ( 6).Consequently, the expression of S, V and a are unlike in each segment shown in Fig. 1.
For example, When ∈ 0, , the C curve is speedup: , the j curve is uniform-speed: As shown in Fig. 1, the jerk j is first-order continuous, which can obtain smooth movement, but make the Acc/Dec process more complicated at the same time.Accordingly, the jerk process algorithm in the introduced Acc/Dec approach should be simplified rationally in order to meet the demand of real-time property in the study.Set an example with AB segment (acceleration speed-up portion) to tell how to simplify operation: According to the maximum permissible jerk j max and the slope coefficient l of j curve, the time when j accelerates from 0 to j max can be obtained: Suppose the CNC system interpolation period be T, then the required period numbers when j accelerates from 0 to j max is as follows: In order to simplify operation, round-off number to n 1 : So the practical jerk in each interpolation period when j accelerates from 0 to j max can be obtained: where, K = [0, 1,…, n j ].In like manner, the practical jerk in each interpolation period when j decelerates from j max to 0, i.e., ∈ , In AB segment, the needed time t q when a accelerates from 0 to a max can be obtained: Then the time when j keeps to j max according to Eq. ( 7) and ( 10) can be obtained: The algorithms mentioned above simplify the calculation of a and j and reduce the Acc/Dec process operation of AB segment (acceleration speed-up portion).Furthermore, the algorithms are appropriate for the other movement segments.With the simplified algorithms, the developed Acc/Dec approach accords with the demand of 4 ms interpolation period completely.

MOTION PROFILE SMOOTH PROCESS ALGORITHM BASED ON THE DEVELOPED ACC/DEC APPROACH
All of the motion profile portions shown in Fig. 1 don't always exist in a certain machining program segment.Consequently, the machining program segment Acc/Dec control should be done according to various kinematics constraint conditions, such as the permissible maximum acceleration a max , the maximum permissible jerk j max , the machining program segment displacement S and the instruction feedrate V c .
Above all, the time that a gets to a max and the time that V gets to V c should be estimated in CNC interpolation pretreatment.Suppose the time when V accelerates from 0 to V c be v t , the time when a accelerates from 0 to a max and decelerates from a max to 0 be t a , it can be concluded: The even speed-up portion is needed for velocity to achieve V c in speed-up process if t v >t a ; The even speedup portion is unnecessary for velocity to arrive at V c in speed-up process if t v >t a .The motion profile smooth process algorithm is researched in details according to different conditions.

The motion profile smooth process algorithm if t v >t a :
The even speed-up portion is needed for the velocity to arrive at the instruction feedrate V c if  How to compute ?As shown in Fig. 2, the motion profile doesn't include uniform speed portion (DE segment).In other words, the velocity adds from 0 in the motion program segment, arrives at V c on point D, decreases to 0 in succession on the terminal point.The key to compute displacement S AH from point A to point H is to obtain even speed-up time t BC .The even speed-up time t BC is equal to the even slowing-down time t FG owing to symmetrical characteristic.Because: where suppose the instantaneous velocity on point D be V D . Since, where, The unknown BC t can be concluded according to Eq. ( 13), ( 14) and ( 15), then S 1 will be readily obtained: can be calculated in like manner.
The motion profile smooth process algorithm if t v >t a : The even speed-up portion isn't needed for the velocity to arrive at the instruction feedrate V c when t v >t a .The machining program segments can be divided into two instances according to various displacement: If S is long enough (suppose ), the machine tools instantaneous velocity can arrive at instruction feedrate; If S is short enough (suppose ), the machine tools instantaneous velocity can't arrive at instruction feedrate.
As shown in Fig. 4, compute the speed-up time of acceleration with jerk j max firstly: Then S 3 can be concluded: The machining program segments can be divided into two instances owing to S 3 .

 S S 3
As shown in Fig. 5, the machining program segment motion profile includes five portions only and Because the machining program segment displacement is much less, the acceleration can't get to maximum and velocity can't get to instruction value, the motion profile includes four portions only and doesn't involve uniform speed segment, even speed-up segment and even slowing-down segment.

ACC/DEC CONTROL EXPERIMENTAL ON THE CNC TABLE
The CNC experimental table hardware structure is shown in Fig. 6.The CNC controller is made up of personal computer and DSP movement control card SEED-DEC2812.And the Acc/Dec control, Fig. 6: The CNC table hardware structure Fig. 7: Adopting the developed Acc/Dec approach interpolation, position control are carried out on SEED-DEC2812.Especially, the operating system (RTOS)-UC/OS-II is introduced to SEED-DEC2812 to meet the demands of CNC system multitask and real-time property.Both the interpolation period and sampling period of the CNC table are 4 ms:  Adopt the developed Acc/Dec approach based on trapezoid-curve jerk and the main kinematics parameters are defined as follows: Let the X axis table move forward 15 mm and then countermove 15 mm.The displacement curve, velocity curve and acceleration curve is shown in Fig. 7.The displacement curve, velocity curve and acceleration curve is shown in Fig. 8. Compare the developed Acc/Dec approach and the linear Acc/Dec approach through velocity and acceleration curve: as shown in Fig. 9, the data1 curve represents the developed Acc/Dec approach, while the data 2 curve represents the linear Acc/Dec approach.

EXPERIMENTATION RESULTS
It can be seen from Fig. 7, 8 and 9 that the developed Acc/Dec approach can achieve smoother and flexible motion profile, but need more run time.In particular, the run time when adopting the introduced Acc/Dec approach is 2.26 s in the experimentation, while the run time when adopting the linear Acc/Dec approach is 2.19 s.

CONCLUSION
It is important to ensure smooth movement and reduce machine tools impact in high precision CNC machining.The CNC machine tools dynamic model is set up to analyze the cause of machine impact in the study, then a Acc/Dec approach based on trapezoidcurve jerk is put forward which can ensure that the jerk is first-order continuous; Moreover, the motion profile smooth control algorithm based on continuous jerk is developed in details according to various kinematics constraint conditions; Finally, the experimentation results reveal that the developed approach can achieve more smooth and flexible motion profile, which is helpful to reduce machine tools impact and enhance parts machining surface quality.In sum, the developed approach avoids flexible impact on machine tools produced by step change jerk and achieves smooth motion profile, which is significant to high accuracy CNC machining.