外文翻譯----cnc加工加減速平穩(wěn)變化的方法

1、<p>　　A Generalized Approach for the</p><p>　　Acceleration and Deceleration of CNC Machine Tools</p><p>　　Jae Wook Jeon</p><p>　　Department of Control and Instrumentation Engi

2、neering</p><p>　　Sungkyunkwan University</p><p>　　300 Chunchun-Dong,Jangan-Gu,Suwon City,Korea</p><p>　　Abstract—Many techniques for the acceleration and deceleration of CNC machine

3、 tools have been proposed in order to make CNC machine tools perform given machining tasks efficiently. Since they should be calculated in a limited time, most of them are not computationally intensive. However, these pr

4、evious techniques cannot generate velocity profiles having some kinds of acceleration and deceleration characteristics though they can generate velocity profiles having various acceleration and decelerat</p><p

5、>　　I. INTRODUCTION</p><p>　　The demand for better accuracy in the manufacturing of complicated parts and the desire to increase productivity have developed CNC systems so that CNC machine tools move more

6、accurately and more quickly. Since the combined characteristics of the control and the machine tool determine the final accuracy and productivity of the CNC system, there are many factors to consider for improving these

7、quantities. One of the important factors is efficiently generating velocity profiles which have the desi</p><p>　　machine tools. One of them is generating velocity profiles by the selection of polynomial fun

8、ctions [1]. This technique can generate so many kinds of velocity profiles and furthermore can make the characteristics of deceleration be independent from that of acceleration. The major problem of this technique is com

9、putational load to increase almost exponentially with the order of polynomial in the acceleration or that in the deceleration. Due to time constraints, it is very difficult to apply these</p><p>　　In this pa

10、per, a generalized approach for the acceleration and deceleration of CNC machine tools is proposed. The proposed technique is as simple and efficient as the techniques based on a digital convolution and can generate velo

11、city profiles which have more various characteristics of acceleration and deceleration than the techniques based on a digital convolution can. That is, the proposed technique can generate velocity profiles of which the d

12、eceleration characteristics are independent from t</p><p>　　In section II, existing techniques for generating velocity profiles of CNC machine tools will be explained. In section III, it will be explained ho

13、w to generate a desired velocity profile by the proposed technique. The proposed technique and other existing techniques will be compared. In section IV, it will be shown that the proposed technique can generate some use

14、ful velocity profiles for CNC machine tools.</p><p>　　II. PREVIOUS TECHNIQUES</p><p>　　For the illustration of the previous techniques, let us consider one single-axis control system of which th

15、e maximum velocity, the maximum acceleration, and the sampling time are V max，Amax ， and Ts respectively. If this system moves the given distance S at the maximum velocity V max, then the movement time T1 in the rectangu

16、lar velocity profile will be T1 = S / V max = p Ts, (1)</p><p>　　Selecting an integer n which is the smallest integer among integers which are equal to or greater th

17、an p, the resulting rectangular velocity profile is constructed as in Fig. 1. The velocity and position equations for this profile are</p><p>　　The position increment during each sampling time is </p>

18、;<p>　　where and are the position commands at the　　and the　 sampling times respectively. That is, the position increment during every sampling time is the same. However, since no physical system can achieve

19、the above rectangular velocity profile due to impulse acceleration, the acceleration interval to increase velocity from the rest to a specified value and the deceleration interval to decrease velocity from the specified

20、value to the rest are needed.</p><p>　　II-A. Selection of Polynomial Functions</p><p>　　Given an acceleration interval Ts = naTs, a deceleration interval Td= ndTs where na= nd, and a distance S

21、, a trapezoidal velocity profile which has the linear acceleration and deceleration characteristics can be constructed as in Fig.2. If this profile has the constant velocity interval, then n=[S/(VmaxTs)] is larger than

22、na . The velocity and position equations for this profile are</p><p><b>　　And</b></p><p>　　Fig .1.A rectangular velocity profile for moving a distance S</p><p>　　Fig .2.

23、A trapezoidal velocity profile for moving a distance S</p><p>　　In the above derivation, it is assumed that the acceleration interval is the same as the deceleration interval and the constant velocity interv

24、al is present. In the case that the acceleration interval is different from the deceleration interval or the constant velocity interval is not present, similar equations are possible to derive. This trapezoidal velocity

25、profile which has the linear acceleration and deceleration characteristics is particularly effective for controlling a machine tool havin</p><p>　　jerk quantity have been proposed [1]. Given position, veloci

26、ty, and acceleration at the initial and final positions, a class of polynomial functions for satisfying these conditions is selected. One approach is to specify a seventh-degree polynomial for each axis. Another approach

27、 is to split the entire trajectory into several trajectory segments so that different polynomials of a lower degree can be used in each trajectory segment. The most common methods are 4-3-4 trajectory, 3-5-3 trajectory,

28、</p><p>　　II-B. Digital Convolution Techniques</p><p>　　Given an acceleration interval Ta=naTs (which determines the deceleration interval Td = ndTs = naTs because two intervals cannot be

29、different in velocity profiles generated by digital convolution techniques) and a desired distance S, a trapezoidal velocity profile which has the linear acceleration and deceleration characteristics can be constructed b

30、y digital convolution techniques. The position increment during each sampling time in a trapezoidal velocity profile δPl(kTs) is constructed b</p><p>　　, 　　 　 (7)</p><p>　　where　　

31、 　 　 (8)</p><p>　　as in Fig. 3. The relationship between δP0 and δP1 is expressed as the following recursive equation [2]</p><p>　　Equation (9) provides the basic

32、 information for the trapezoidal velocity profile. Based on (9), we can design the hardware system for trapezoidal velocity profiles as in Fig. 4, where buffer registers act as delay elements [4]. A second-order velocity

33、 profile which has the parabolic acceleration and deceleration characteristics can also be constructed by digital convolution techniques. The position increment in a second-order velocity profile </p><p>　　

34、can be obtained by successive convolutions as in Fig. 5</p><p>　　Where 　 . (10)</p><p>　　The shape of second-order velocity profile can be determined by the values of m1 and m2 [2,4]

35、. The relationship between and is expressed as the following recursive equations [2].</p><p>　　Fig.3.A digital convolution for a trapezoidal velocity profile</p><p>　　Fig.4.A hardware structur

36、e for a trapezoidal velocity profile</p><p>　　Fig.5.Successive digital convolutions for a second-order velocity profile</p><p>　　Similarly, the position increment of a smooth velocity profile wh

37、ich has the high-order acceleration and deceleration characteristics can be obtained by several successive convolutions as in Fig. 6 where each sequence has the similar meaning as in (10). The shape of the smooth velocit

38、y profile can be determined by the values of ml,m2,...,mq, and q [2,4]. The relationships between and are expressed as the following recursive equations [2]:</p><p>　　The velocity profiles generat

39、ed by (13) have the property that the moving distances during the acceleration interval and during the deceleration interval are same. These distances are also same as the moving distance of trapezoidal velocity profile

40、during the acceleration interval under the same condition.</p><p>　　In order to generate an arbitrary shape velocity profile, the position increment during each sampling time in the desired velocity profile

41、can be generated from the following convolution [2,4]</p><p>　　From the appropriate of choice ai , for i = 1,2,...,m, the desired velocity profile can be obtained. The hardware system based on (14) for ge

42、nerating an arbitrary velocity profile can be designed as in Fig. 7. The acceleration and deceleration interval T, is given by </p><p>　　Ta = mTs [2,4]. In [3], a similar convolution technique which can gen

43、erate velocity </p><p>　　profiles which has several acceleration and deceleration characteristics is derived. While the computational load for generating smooth velocity profile by these digital convolution

44、techniques is much less than that for the selection of polynomial techniques, the deceleration characteristic generated by these techniques is determined from the acceleration characteristics. That is, the deceleratio

45、n characteristic cannot be made to be independent from the acceleration characteristic by using di</p><p>　　Fig.6.Successive digital convolutions for a higher-order velocity profile</p><p>　　III

46、. PROPOSED TRAJECTORY GENERATION TECHNIQUE</p><p>　　As in section II, let us consider one single-axis control system of which the maximum velocity, the maximum acceleration, and the sampling time are Vma

47、x, Amax, and Ts respectively. Given an acceleration interval Ta =naTs , a deceleration interval Td =ndTs , and a distance S , a velocity profile which has the desired characteristics of acceleration and deceleration ca

48、n be constructed by the proposed technique.</p><p>　　III-A. Linear Acceleration and Deceleration</p><p>　　In the technique selecting polynomial functions, the velocity equation which has the lin

49、ear acceleration and deceleration characteristics is calculated by (5). The coefficients of (5) may vary according to given conditions. But the ratio between the position increment during each sampling time in the accel

50、eration interval is fixed. That is, </p><p>　　in Fig. 8. Similarly, the ratio between each position increment in the deceleration interval is fixed. Therefore, the velocity profile which has the linear accel

51、eration and deceleration characteristics can be calculated as the following:</p><p>　　(i) The velocity after the acceleration Vm is determined as </p><p>　　Fig.7.The hardwar

52、e structure of arbitrary acceleration and deceleration</p><p>　　where (16)</p><p>　　(ii) Then the position increment during each sampling time is calculated as</p>

53、<p><b>　　if ,</b></p><p><b>　　if , </b></p><p>　　According to the acceleration and deceleration interval, the coefficients</p><p>　　and in (17)an

54、d (18) can be calculated and be </p><p>　　stored in advance. Therefore a velocity profile which has the linear acceleration and deceleration characteristics is able to be efficiently calculated for a given

55、distance.</p><p>　　III-B. Arbitrary Acceleration and Deceleration</p><p>　　Let us consider a velocity profile V(t) which has file acceleration characteristic and the deceleration characteristic

56、represented by and respectively,</p><p>　　where both of and are differential on and are continuous on , and Tc is the time to start deceleration and Vm is the velocity after acceleration. Then the

57、 position increment during each sampling time in the acceleration interval can be represented as</p><p>　　and it can be written as </p><p>　　where is the coefficients which can be calculated fr

58、om fa(u) and na=Ta /Ts and be stored. Similarly, the position increments during each sampling time in the deceleration interval can be represented as</p><p>　　Fig.8.The position increment during each sampli

59、ng time</p><p>　　in the acceleration interval of a trapezoidal velocity</p><p>　　Where dγk is the coefficients which can be calculated from fd(u) and nd = Td / Ts and be stored. The area Sa

60、 under fa during the acceleration interval and the area Sd under　fd during the deceleration interval can be represented as</p><p>　　where can be calculated from and , and be stored. Similarly, can be

61、calculated from and and be stored. In the technique selecting polynomial functions, the velocity equation which has the acceleration and deceleration characteristics represented by and respectively is calculated by app

62、ropriate polynomial functions. The coefficients of the polynomial function may vary according to given conditions. But the ratio between the position increment during each sampling time in the acceleration inte</p>

63、<p>　　(i) For an acceleration interval Ta =naTs and a deceleration interval Td =ndTs, the　corresponding coefficients and are retrieved.</p><p>　　(ii) For mo

64、ving a distance S, the velocity after the acceleration Vm is determined as</p><p>　　where </p><p>　　(iii) Then the position increment during each sam

65、pling time is calculated as</p><p><b>　　if </b></p><p><b>　　if </b></p><p>　　Since the coefficients and can be calculated and be stored in advance acc

66、ording to the acceleration and deceleration intervals, file velocity profile which has arbitrary acceleration and deceleration characteristics is able to be efficiently calculated for a given distance.</p><p&g

67、t;　　In digital convolution techniques as in (14) and Fig. 7, the shape of a velocity profile is determined by the values of ai for i = 1,2,...,m where m determines the acceleration and deceleration intervals which are sa

68、me na = nd = m. This means that the values of ai determine the above coefficients and where na = nd. Therefore, the coefficients in velocity profiles generated by digital convolution techniques cannot be made to be

69、independent from the coefficients . The deceleration characteris</p><p>　　IV. GENERATION OF SOME VELOCITY PROFILES</p><p>　　While a number of velocity profiles can be generated by the digital co

70、nvolution techniques [2-6], some velocity profiles cannot be generated by them. Velocity profiles</p><p>　　Fig.9.The position increment during each sampling time</p><p>　　in the acceleration int

71、erval of a velocity profile</p><p>　　which cannot be generated by the digital convolution techniques are illustrated in Fig. 10 where the acceleration characteristic is represented by and the deceleration c

72、haracteristics is represented by ．Let us consider one single-axis linear motion control system of which the maximum velocity, the sampling time, the deceleration interval, and the coupling ratio are Vmax = 3000rpm, Ts=

73、 2msec , Td = 40Ts = 80msec , and P=10 tums/in respectively. Given linear distances L1= 0.5 in. and L2 = 1 i</p><p>　　V. CONCLUSION</p><p>　　A generalized approach for the acceleration and dec

74、eleration of CNC machine tools has been proposed. Given a machining task in a CNC system, the appropriate acceleration characteristic, the deceleration characteristic, the acceleration interval, and the deceleration inte

75、rval can be determined. According to the acceleration characteristic, the deceleration characteristic, the acceleration interval, and the deceleration interval, some coefficients are calculated and are stored. By using t

76、hese coef</p><p>　　Fig.10-a.A velocity profile for L1=0.5 in .=40,and =40</p><p>　　Fig.10-b.A velocity profile for L2=1 in . =40,and =40</p><p>　　Fig.10-c.A velocity profile for L1=

77、0.5 in . =50,and =40</p><p>　　Fig.10-d.A velocity profile for L2=1 in . =50 , and =40</p><p>　　REFERENCES</p><p>　　1. K. S. Fu, R. C. Gonzalez, and C. S. G. Lee, Robotics: Control

78、, Sensing,Vision and Intelligence, McGraw-Hill, 1987.</p><p>　　2. D. I. Kim, J. W. Jeon, and S. Kim, "Software acceleration/deceleration methods for industrial robots and CNC machine tools," Mechat

79、ronics, Vol. 4, No. 1, pp. 37-53, 1994.</p><p>　　3. D. S. Khalsa, "High Performance Motion Control Trajectory Commands Based on The Convolution Integral and Digital Filtering" Proceedings of Intell

80、igent Motion, pp. 54-61, Oct. 1990.</p><p>　　4. United States Patent, Patent Number 4,555,758, Nov. 26, 1985.</p><p>　　5. Masory O. and Korea Y., "Reference-Word Circular Interpolators for

81、CNC Systems," Trans. of ASME, J. Eng. Ind., vol 104, pp. 400-405, 1982.</p><p>　　6. Koren Y., Computer Control of Manufacturing ,Systems, McGraw-Hill Inc. 1988.</p><p>　　CNC加工加減速平穩(wěn)變化的方法<

82、/p><p>　　Jac Wook Jcon </p><p>　　Sungkyunkwan大學(xué) 控制和儀器控制工程系</p><p>　　300 Chunchun-Dong、 Jangan-Gu 、 Suwon市,韓國</p><p>　　摘要 ： 為了使CNC機(jī)床有效率地完成預(yù)定的加工任務(wù)，人們提出了許多數(shù)控機(jī)床加減速的方法。 因為機(jī)床要在限

83、定時間完成任務(wù), 而這些方法并不能準(zhǔn)確的計算。然而，這些早先的技術(shù)不能夠形成某些加減速度特性的速度分布圖，盡管他們能形成很多種加減速度特性的速度分布圖。這篇文章提出了一個早先的技術(shù)并不能做到的形成速度分布圖的一般方法，這些。根據(jù)需要達(dá)到的加減速特性，計算出各個系數(shù)并存儲，對CNC系統(tǒng)設(shè)定一個移動距離，一個加速度時間和一個減速度時間，運(yùn)用這些系數(shù)就可以形成所需求的加減速特性的速度分布圖。下面將詳細(xì)說明用這個技術(shù)形成典型的速度分布圖的方法。

84、</p><p><b>　　一、引言</b></p><p>　　生產(chǎn)復(fù)雜零件的精確度要求和提高生產(chǎn)力的要求改進(jìn)了CNC系統(tǒng)，使得CNC機(jī)床運(yùn)作的快速精確。因為控制器的組合特性和工作母機(jī)決定了CNC系統(tǒng)的最終精確度和生產(chǎn)率，所以有許多因素需要考慮。其中一個重要因素就是如何有效的形成給定加工任務(wù)所需求的加減速特性的速度分布圖。許多研究者提出了CNC機(jī)床形成速度分布圖的

85、技術(shù)。其中一個就是通過多項式功能選擇來完成速度分布圖[1]。這項技術(shù)能夠形成很多種速度分布圖，此外還將減速度特性從加速度中獨立出來。這項技術(shù)的主要問題是計算負(fù)荷將隨加減速多項式的需求成指數(shù)增長。因為時間的限制，用這些技術(shù)控制CNC系統(tǒng)非常困難。之前其他形成速度分布圖的其他技術(shù)都是基于數(shù)字回旋 [2-6]。這些技術(shù)比多項式功能選擇技術(shù)更加有效，用硬件也更加容易實現(xiàn)。但是，在這些技術(shù)形成的速度分布圖過程中，加速度時間總是和減速時間相等，而且

86、加速度特性決定了減速度特性。因此，這樣有些速度分布圖是不能用這些方法實現(xiàn)的。</p><p>　　這篇論文提出數(shù)控機(jī)床加減速的一般途徑。這種技術(shù)的難易程度和數(shù)字回旋相當(dāng)，并且能形成比它更多種加減速特性。換句話說，這種技術(shù)能夠產(chǎn)生減速度特性獨立于加速度特性的速度分布圖。首先，根據(jù)加速特性、減速特性、加速時間，減速時間計算出一些系數(shù)并存儲。然后給定需要移動距離，通過計算經(jīng)過每個采樣時間的位移增量就產(chǎn)生了所需特性的速度

87、分布圖。每個采樣時間的位移增量可以通過</p><p>　　將存儲的系數(shù)和采樣時間與加速后速度的乘積相乘得到。</p><p>　　在第二部分中，將會對數(shù)控機(jī)床產(chǎn)生速度分布圖的現(xiàn)有技術(shù)進(jìn)行說明。第三部分將說明怎樣運(yùn)用這個技術(shù)產(chǎn)生所需求的速度分布圖。這種技術(shù)將和現(xiàn)有的技術(shù)做個比較。在第四部分，將會說明這種技術(shù)能夠產(chǎn)生一些對CNC機(jī)床有用的速度分布圖。</p><p>

88、<b>　　二、 先前的技術(shù)</b></p><p>　　為了說明的技術(shù)，就以單軸控制系統(tǒng)為例，它的最大速度、最大加速度、采樣時間分別是Vmax, Amax, 和Ts。如果系統(tǒng)以最大速度Vmax移動給定的距離S，則矩形速度分布圖中的運(yùn)動時間 T1 將是 </p><p><b>　　(1)</b></p><p

89、>　　選取一個大于等于p最小整數(shù)n，則形成如圖１的矩形速度分布圖。圖形的速度和位移方程如下：</p><p>　　每個采樣時間的位移增量是</p><p>　　, (4)</p><p>　　其中　和　　分別是采樣點為　 和 位移。也就是說，每個采樣時間的位置增量相同。然而沒有一個物理系統(tǒng)可以通過推進(jìn)加速度來完成上面所講的矩形速度分布圖，

90、需要加速時間內(nèi)將速度由零升到一個特定階段而減速時間內(nèi)將當(dāng)前速度降為零。</p><p>　　1.多項式函數(shù)的選擇</p><p>　　給定加速度時間 , 減速時間 這里 距離為S，</p><p>　　則形成如圖２所示的具有線性加減速度特性的梯形速度分布圖。如果有恒速度時段，那么</p><p>　　。速度和位置方程如下：<

91、/p><p><b>　　另外</b></p><p>　　圖１　移動距離?。印〉木匦嗡俣确植紙D</p><p>　　圖２ 移動距離?。印〉奶菪嗡俣确植紙D</p><p>　　根據(jù)上文所說，假定加速度時間和減速度時間相等,當(dāng)前速度恒定。在加減速度不等，現(xiàn)有速度不是恒定的情況下，就會得到一些類似的方程。這種有線性加減速度特性的

92、梯形速度分布圖特別適用于控制剛性強(qiáng)的機(jī)床。然而，梯形圖連接點的大的進(jìn)給可能將鐵屑聚集還會可能影響機(jī)床的剛性。因為平滑的速度分布圖有更高的要求加減速度特性，在頂點處不會產(chǎn)生大的進(jìn)給，這些速度分布圖常被應(yīng)用在數(shù)控機(jī)床上[1-6]，于是提出了一些選擇多項式函數(shù)來產(chǎn)生速度分布圖的技術(shù)方法[1]。給定位置，速度以及在起始和終了位置的加速度，就選擇了一類滿足這些條件的多項式函數(shù)。一個方法就是對每個軸列入一個第7次多項式。另一個方法就是將一個完整的軌

93、跡分成一些軌跡片段，這樣不同多相式的低次數(shù)就可以用于軌跡片段。最普通的方法就是4-3-4分段法，3-5-3分段法還有5-立方分段法。用選擇多項式函數(shù)的方法產(chǎn)生速度分布圖的主要問題就是計算的負(fù)荷。產(chǎn)生一個平滑的速度分布圖所需要的時間在增加次數(shù)后成指數(shù)的增長。因此，在CNC系統(tǒng)的控制中，用這些技術(shù)產(chǎn)生任意平滑的速度分布圖幾乎是不可能的。</p><p><b>　　2.數(shù)字回旋技術(shù)</b><

94、;/p><p>　　給定加速度時間 Ta=naTs （它決定了減速度時間 Td = ndTs = naTs ，因為數(shù)字回旋技術(shù)產(chǎn)生的速度分布圖的加減速時間不可能不同）給定距離 S， 通過數(shù)字式回旋技術(shù)就會構(gòu)造出一個有線性加減速特性的梯形速度分布圖。公式（4）中顯示梯形分布圖中每個采樣時間的關(guān)系，梯形速度分布圖中每個采樣時間里的位置增量 由位置增量 的反饋構(gòu)成，次序 H(kTs ) 由以下得到：</p

95、><p>　　, (7)</p><p>　　這里 (8)</p><p>　　圖3顯示δP0 和 δP1 的關(guān)系可以表達(dá)成如下遞歸方程[2]：</p><p>　　方程(9)提供給梯形速度分布圖基礎(chǔ)信息。根據(jù)方程（

96、9），我們可以為梯形速度分布圖設(shè)計硬件系統(tǒng)如圖４，緩沖寄存器充當(dāng)延時元件［４］。有假想加減速度特性的2級速度分布圖也能夠用數(shù)字回旋技術(shù)構(gòu)造。二次分布圖的位置增量 可從連續(xù)的反饋后得到，如圖5。</p><p>　　這里 　　 (10)</p><p>　　二級速度分布圖的輪廓由m1 和 m2 的值決定[4]。 和 的關(guān)系可以

97、由下面的遞歸方程[2]表示：</p><p>　　圖3 梯形速度分布圖的數(shù)字反饋</p><p>　　圖 4 梯形速度分布圖的硬件組成</p><p>　　圖 5 第2次速度分布圖的連續(xù)數(shù)字反饋</p><p>　　同樣的，具有高速加減速特征的平滑速度分布圖的位置增量可由幾次回旋得到例圖6。每次回旋都有相似的意義。平滑速度分布圖的輪廓可以由,

98、決定，q[2，4]。 和 的關(guān)系可以用下面的遞歸方程表示[2]：</p><p>　　由公式（13）產(chǎn)生的速度分布圖有這樣的特性：加速度和減速度時間內(nèi)的位移相等。在相同條件下也和梯形速度分布圖在相同加速度時間內(nèi)的位移相等。</p><p>　　為了形成任意形速度分布圖，所需的速度分布圖中每段采樣時間的位置增量 </p><p>　　都可以由以后的回饋產(chǎn)生[2，4

99、]：</p><p>　　選擇合適的 ，,就會獲得所需的速度分布圖。根據(jù)公式（14）設(shè)計的為產(chǎn)生任意速度分布圖的硬件系統(tǒng)可設(shè)計成如圖 7 所示。加速度和減速時間 由公式 得出。在[3]中，可得到有多次加減速的速度分布圖，這種變化圖可由一種相似的反饋技術(shù)得到。而用反饋技術(shù)產(chǎn)生平滑速度分布圖的計算負(fù)荷則比用多項式技術(shù)少的多，這個方法產(chǎn)生的減速度特性由加速度特性決定。換句話說，用數(shù)字反饋技術(shù)產(chǎn)生的減速度特性不可能

100、獨立于加速度特性。因此，一些對CNC機(jī)床有益的速度分布圖就不能用這些反饋技術(shù)形成。</p><p>　　圖6 更高次速度分布圖的連續(xù)數(shù)字反饋</p><p>　　三、關(guān)于軌跡形成技術(shù)</p><p>　　如第二部分所說，讓我們考慮下單軸控制系統(tǒng)，它的最大速度、最大加速度和采樣時間分別是 ,, 和 。給定加速度時間 ，減速時間 ,距離為 S ，具有所需加減速特性

101、的速度分布圖就能用這個技術(shù)形成。</p><p>　　1.線性加速度、減速度</p><p>　　在選擇多項式函數(shù)的技術(shù)，有線性加、減速特性的速度方程可以由（5）計算出。（5）式的系數(shù)根據(jù)條件的不同而不同。但是加速度時間內(nèi)每個采樣時間內(nèi)的位置增量的比率是固定的。也就是 如圖8。同樣的，減速時間內(nèi)位置增量的比率也是固定的。因此，具有線性加減速特性的速度分布圖可以由以下公式計算：</