4.2 Acc/Dec Control After Interpolation 125 W x (t)=(Ae −at + Bcoswt + C w sinwt) (4.47) where A = wK w 2 + a 2 ,B = −wK w 2 + a 2 ,C = awK w 2 + a 2 W x (t) in Eq. 4.47 denotes the speed of the X-axis and by integrating W x (t), we ob- tain the path radius after Exponential-type Acc/Dec control has been applied. Equa- tion 4.48a shows the result of the integration of W x (t). From Eq. 4.48a we know that after Acc/Dec time the radius of the path from Exponential-type Acc/Dec control, R  , is given by Eq. 4.48b. r = − A a e −at + 1 w  B 2 + C 2 w 2 sin(wt − θ ) (4.48a) where θ = cos −1   B 2 + C 2 w 2  R  =  B 2 + C 2 w 2 = B  1 + a 2 w 2 = −wK w 2 + a 2  1 + a 2 w 2 (4.48b) = −K a 1  1 + w 2 a 2 = −R 1  1 + w 2 a 2 As the machining error is the difference between the radius of the commanded path and the distorted path due to Exponential-type Acc/Dec control, the error is simplified as Eq. 4.49. Δ R = R −R  = R ⎛ ⎝ 1 − 1  1 = w 2 a 2 ⎞ ⎠ Δ R = R[1 −{1 − 1 2  w a  2 − 3 8  w a  4 }] (4.49) so Δ R ≈ R 2  w a  2 ≈ 1 2 τ 2 F 2 R where, 1 (1 + z) m = ∞ ∑ n=0  −m//n  z n = 1 −mz + m(m + 1) 2! z 2 − m(m + 1)(m + 2) 3! z 3 + from a binomial series. 126 4 Acceleration and Deceleration 4.2.3.4 Machining Error Summary The machining error due to the Acc/Dec control depends on the type of Acc/Dec control filter. The machining errors are summarized in Table 4.5 with respect to each type of Acc/Dec control filter. According to Table 4.5 the machining error is proportional to the square of the feedrate and the Acc/Dec time. It is also in inverse proportion to the radius of the circular path. Therefore, from this, we knowthat the higher the feedrate the longer the Acc/Dec time, the shorter the radius of the circular path and the larger the machining error. We also know that the accuracy of the S-shape-type Acc/Dec control is better than that of the alternatives. Table 4.5 Machining error due to Acc/Dec filter Control type Machining Error Remarks Linear Δ R = F 2 τ 2 24R F: Feedrate Exponential Δ R = F 2 τ 2 2R τ : Time constant S-shape Δ R = F 2 τ 2 48R R: Radius of circle 4.2.4 Block Overlap in ADCAI As mentioned in Chapter 2, the G-code system provides various instructions for con- trolling axes. Setting the block control mode is one of the G-code functions. For example, in the G-code system of the FANUC controller, there are two kinds of path control mode; exact stop mode (G61) and continuous mode (G64). In exact stop mode, the machine follows the programmed path as exactly as pos- sible, stopping at sharp corners of the path. Alternatively, in continuous mode, sharp corners of the path may be rounded slightly so that the feedrate may be kept up. Figure 4.11 shows the actual toolpath when exact stop mode is applied and Fig. 4.12 shows the actual toolpath when continuous mode is applied. Exact stop mode generally results in reduction of machined surface quality due to the stoppage of axis movement and increases machining time due to acceleration and deceleration for all blocks. In continuous mode, the tool begins the movement to the successive block before the tool reaches the end of the block. Unlike exact stop mode, this mode does not result in reduction of the surface quality and increase in machining time. In contin- 4.2 Acc/Dec Control After Interpolation 127 X Y G90 G01 G61 X50. Y20. F100 X50. Y50. Fig. 4.11 Actual path in exact stop mode uous mode, the toolpath does not pass through the programmed path as shown in Fig. 4.12. Therefore, machining error always occurs at sharp corners. The path near the corner depends on the Acc/Dec control type and, in general, the machining error is small enough so as not to reflect on machining accuracy. X Y G90 G01 G64 X50. Y20. F100 X50. Y50. Fig. 4.12 Actual path in continuous mode Figure 4.13 shows the result of X-axis interpolation and Acc/Dec control for two successive blocks. In Fig. 4.13, Block 1 and Block 2 are successive blocks and Fig. 4.13a and Fig. 4.13b show the interpolation result of Block 1 and Block 2 respec- tively. Figure 4.13c and Fig. 4.13d show the results of Linear Type Acc/Dec control for Block 1 and Block 2. If we combine the result of interpolation and Acc/Dec con- trol for the two blocks with respect to time, we obtain the time–pulse graph shown in Fig. 4.14. In continuous mode, the end result of Block 1 and the beginning of Block 2 are continuously connected. The connected interpolation pulse train is input continu- ously to the Acc/Dec controller and the Acc/Dec controller performs Acc/Dec con- trol without considering blocks. Figure 4.14 shows the result of Linear-type Acc/Dec 128 4 Acceleration and Deceleration Time (a) The interpolation result of Block 1 Pulse Time (b) The interpolation result of Block 2 Pulse Time (c) The result of Acc/Dec control for Block 1 Pulse Time (d) The result of Acc/Dec control for Block 2 Pulse Acc/Dec control Acc/Dec control Fig. 4.13 X-axis interpolation and Acc/Dec control control for two successive blocks. The time–pulse graph in Fig. 4.14 is identical with the summation of the two time–pulse graphs in Fig. 4.13b and Fig. 4.13d. As shown in Fig. 4.14, in Continuous Mode, reduction of speed does not occur at the corner between two success blocks join. The speed is accelerated or decelerated considering the difference in the feedrate of the two blocks. Time Block 1 Pulse Time Pulse Acc/Dec control Block 2 Block 1 Block 2 Fig. 4.14 Time–pulse graph for two successive blocks 4.3 Acc/Dec Control Before Interpolation Unlike ADCAI-type NCK, ADCBI-type NCK generates the speed profile before executing rough interpolation. Also unlike ADCAI-type NCK, where Acc/Dec con- trol is carried out separately for individual axes, ADCBI-type NCK carries out the Acc/Dec control for the programmed path itself. Therefore, theoretically, ADCBI- type NCK does not result in machining error. As mentioned in Section 4.2.3, ADCAI generates machining error in proportion to the feedrate and this has become a serious problem considering that the machining speed of machine tools is getting faster. Therefore, ADCBI is essential to implement 4.3 Acc/Dec Control Before Interpolation 129 the high-speed machining functions that have become a typical machine-tool func- tion and consequently, the latest machine tools provide ADCAI as a basic function. Part program Interpreter Acceleration/Deceleration Rough interpolation Mapping to each axis Fine interpolation Position control Fig. 4.15 ADCBI-type NCK flowchart Figure 4.15 shows the flowchart for the overall procedure of the ADCBI-type NCK. Figure 4.16 shows the sequence of executing Acc/Dec control and rough in- terpolation and the output at each stage. The Acc/Dec Controller calculates the speed profile considering acceleration and deceleration. The rough interpolator then gener- ates the interpolated points considering tool displacement and the remaining length of the programmed path for every iteration time instant based on the speed profile. 4.3.1 Speed-profile Generation In ADCBI, the path length, the allowable acceleration and deceleration, the itera- tion time for rough interpolation, and the commanded feedrate are considered when generating a speed profile. For convenience, let us suppose that Acc/Dec control is applied to a linear path, the length of the linear path is L(mm), the allowable ac- celeration is A(mm/s 2 ), the allowable deceleration is D(mm/s 2 ), the iteration time 130 4 Acceleration and Deceleration Block information Block end position Block start position Velocity Time F Interpret Acceleration/Deceleration Rough interpolation Fig. 4.16 Linear path Acc/Dec control for rough interpolation is τ (s), and the commanded feedrate from a part program is F(mm/s 2 ). In order to generate a speed profile, it is necessary to check if the linear path is a normal block or a short block. The normal block includes an acceleration zone, constant-speed zone, and deceleration zone, while the zone, or short block, does not include the constant-speed zone. Equation 4.50 is the condition that a normal block should satisfy. If Eq. 4.50 is not satisfied then the block is a short block. F 2 2A + F 2 2D ≥ L (4.50) In the case of a normal block, we can obtain a speed profile like that shown in Fig. 4.17a. In the case of a short block, we can obtain a speed profile like that shown in Fig. 4.17b. In the case of a short block, the length of the path is shorter than the length needed for the actual speed to reach the commanded feedrate F from zero speed and return back to zero speed. It is therefore impossible for the actual speed to reach the commanded feedrate, F. Velocity (mm/s ) Time (ms) 2 F Velocity (mm/s ) Time (ms) 2 F (a) Normal block (b) Short block Fig. 4.17 Speed profiles 4.3 Acc/Dec Control Before Interpolation 131 After checking whether the path is a normal block or a short block using Eq. 4.50, the speed profile is generated according to the path type. In the case of a normal block, the acceleration time T A that is spent to reach the commanded feedrate F from 0(mm/sec), is computed by Eq. 4.51 and the deceleration time T D ,whichis spent to reach 0(mm/s) from the commanded feedrate F is computed using Eq. 4.52. The constant speed time T C is calculated by dividing the length of the path after subtracting the length needed for acceleration and deceleration by the commanded feedrate, as given by Eq. 4.53. T A = F A (4.51) T D = F D (4.52) T C = L − F 2 2A − F 2 2D F (4.53) In the case of a short block, the length of the block is obtained by integrating the speed profile shown in Fig. 4.17b with respect to time. If the maximum reachable speed for the short block is F  , acceleration time T A , deceleration time T D ,andF  are calculated using Eq. 4.54. T A = F  A T D = F  D (4.54) L = F  ×(T A + T D ) 2 From the above equations, it is possible to generate a speed profile for both normal blocks and short blocks. Also, based on the generated speed profile, the interpolation for a linear path can be carried out. In the ADCBI-type NCK, the rough interpolator calculates the interpolated point through which the tool should go for every constant iteration time for interpolation, τ . In the acceleration range, the length that the tool should move every iteration time for interpolation can be calculated using Eq. 4.55. V i+1 = V i + τ ·A,(i = 0,1,2, ,N A ) (4.55) L i = V 2 i+1 −V 2 i 2A where, V i is the velocity of the ith interval and V 0 = 0 L i is the displacement for the ith sampling time. N A = T A τ . 132 4 Acceleration and Deceleration In the constant speed range the commanded feedrate is F and the tool moves τ ×F every iteration time for interpolation. In the deceleration interval, the length through which the tool moves every iteration time for interpolation can be calculated using Eq. 4.56. V i+1 = V i − τ ·D,(i = 0,1,2, ,N D ) (4.56) L i = V 2 i −V 2 i+1 2D where, V i is the velocity of the ith interval and V 0 = F L i is the displacement for the ith sampling time. N D = T D τ . It is possible to calculate the interpolated point by projecting the displacement through which the tool moves in every iteration time for interpolation onto the pro- grammed path. 4.3.2 Block Overlap Control Hardly ever is only one linear block or one circular block used for actual machining. In general, because an NC program consists of multiple linear blocks and circular blocks, it is true that direct usage of the above-mentioned equations for generating speed profile and interpolating is impossible. In ADCAI, interpolation and Acc/Dec control are applied to the individual block and it is not necessary to consider the connection of blocks. However, in ADCBI, because the speed at the beginning and the end of a block should be considered when generating a speed profile, the previous and the successive blocks should be considered when generating a speed profile and interpolating. In the next sections, all possible cases for connection relationships that can occur between two successive blocks in actual machining will be addressed. The equations for generating a speed profile for each case will be described. 4.3.2.1 Classification of Continuous Blocks In Section 4.3.1, we defined the block with constant speed interval as a normal block and the block without constant speed interval as a short block. From the way in which two blocks are connected it is possible to classify pairs of blocks into twelve types depending on the type of block (e.g. normal block and short block) and the difference of commanded feedrate between the two blocks. However, in the case when a short block and a normal block are successive, since the speed profile can be generated with an identical equation regardless of the commanded feedrate of the two blocks, 4.3 Acc/Dec Control Before Interpolation 133 a method to calculate the speed profile when the commanded feedrate of the two blocks is identical will be described. Therefore, the way in which two blocks are connected can be classified into eight types, as shown in Fig. 4.18. For convenience, it is supposed that the direction of two successive blocks is identical. F tN1 N2 (a) Normal block → Normal block (Constant speed) F tN1 N2 (b) Normal block → Normal block (Speed : high → low) F tN1 N2 (c) Normal block → Normal block (Speed : low → high) F tN1 N2 (d) Short block → Normal block (Constant speed) F tN1 N2 (e) Normal block → Short block (Constant speed) F tN1N2 (f) Short block → Short block (Constant speed) F tN1N2 (g) Short block → Short block (Speed : high → low) F tN1N2 (h) Short block → Short block (Speed : low → high) Fig. 4.18 Speed profiles for identical blocks 4.3.2.2 Normal Block/Normal Block, Identical Speed As shown in Fig. 4.18a, if two blocks with an identical feedrate F are successive, it is possible to generate the successive speed profile by the methods mentioned in Section 4.3.1. Because in Block N1, deceleration is not necessary, the acceleration time T A1 is computed by Eq. 4.51 and the constant-speed time T C1 is computed by Eq. 4.57. 134 4 Acceleration and Deceleration T C1 = L 1 − F 2 2A F (4.57) where, L 1 is the displacement of block N1 In Block N2, because at the beginning of the block the tool is moving with feed- rate F, acceleration is not required and only deceleration is necessary. The decelera- tion time T D2 is computed by Eq. 4.52 and the constant-speed time T C2 is computed by Eq. 4.58. T C2 = L 2 − F 2 2D F (4.58) where, L 2 is the displacement of block N2 When two successive blocks have the same feedrate, the speed profile for the acceleration interval can be obtained based on Eq. 4.55. The speed profile for the de- celeration interval can be obtained by Eq. 4.56. Based on the above-mentionedequa- tions, it is possible to generate the speed profile for two successive normal blocks with the same feedrate as in Fig. 4.19. TT TT F N1 N2 Time A1 C1 C2 D2 Velocity Fig. 4.19 Speed profiles for identical blocks 4.3.2.3 Normal Block (High Speed)/Normal Block (Low Speed) In the case when two normal blocks with different feedrates are successive as shown in Fig. 4.18b, the lower of the two blocks’ speeds is defined as the speed at the corner. For example, if the commanded feedrates of Block N1andN2areF 1 and F 2 , respectively, and F 1 is higher than F 2 , the speed at the corner is defined as F 2 . This is done in order to avoid abnormal machining status such as tool breakage due [...]... the length of the blocks, while Fig 4.40 shows the speed profile when both the length of blocks and the angle between blocks are considered From Fig 4.40 we can see that, due to the Cartesian maximum allowable acceleration, the speed decreases at the corner where the arc and the line join 25 20 Y-axis (mm) 15 10 5 0 -5 -10 - 25 -20 - 15 -10 -5 0 X-axis (mm) Fig 4.38 Look-ahead path 5 10 15 155 Feedrate... spent to reach the commanded feedrate of Block N2 from F , TA2 , is computed by Eq 4.70 Further, the speed profile of the acceleration interval in Block N1 can be obtained by Eq 4.69 and Eq 4 .55 and the speed profile of acceleration interval in Block N2 can be obtained by Eq 4.70 and Eq 4 .55 where the speed at the beginning of acceleration is F and the speed at the end of acceleration is F2 TA1 = F A (4.69)... acceleration time of Block N1, TA1 , is calculated by Eq 4.81 and the deceleration time, TD1 , is calculated by Eq 4.82 In addition, the speed profile can be obtained by Eq 4 .55 and Eq 4 .56 where the initial speed of the deceleration interval, V0 , is Fmax and the end speed of the deceleration interval is F Also, the deceleration time of Block N2, TD2 , is calculated by Eq 4.76 and the speed profile of Block... used instead of the method mentioned in Section 4.3.3 For convenience of explanation, we define the first block as N1 and the next block as N2 We define that the start point and the end point of N1 are (XS1 ,YS1 , ZS1 ) and (XE1 ,YE1 , ZE1 ), respectively and the start point and the end point of N2 are (XS2 ,YS2 , ZS2 ) and (XE2 ,YE2 , ZE2 ), respectively In this case, the speed of blocks N1 and N2 in the... the performance of CNC machine tools Machining accuracy depends on the ability to follow the trajectory of the controller As mentioned in Chapter 3, the accuracy of the machining 146 4 Acceleration and Deceleration trajectory is inversely proportional to the feedrate and sudden changes of feedrate result in reduction of accuracy of the CNC equipment In the ADCBI type of NCK, the accuracy of machining... 80 60 40 20 0 0 5 10 Time(s) 15 20 25 Fig 4.30 Speed profile for circular profile speed of the current block zero and it is possible to control the speed of successive blocks depending on the commanded feedrate and the length 4.4.1 Look-Ahead Algorithm A Look Ahead algorithm calculates the start speed and the end speed of each block based on the remaining length of the successive blocks and the maximum... compute the speed of the blocks to be looked at ahead For example, assume that there is a part program that has paths shown in Fig 4.38, and assume that the programmed feedrate of the paths is 2000 mm/min, the acceleration time is 200 msec, and the maximum allowable acceleration is 200000 mm/min2 154 4 Acceleration and Deceleration 400 Velocity (mm/min) 350 300 250 200 150 100 50 0 1 2 3 4 5 Time (s) 6... TD1 = F1 − F D F2 1 L1 − 2A − TC1 = F1 (4.74) 2 F1 −F 2 2D (4. 75) The speed profile of the acceleration interval of Block N1 can be obtained by Eq 4.73 and Eq 4 .55 The speed profile of the deceleration interval of Block N2 can be obtained by Eq 4.74 and Eq 4 .56 , where the initial speed at the deceleration interval, V0 , is F1 and the end speed of the deceleration interval is F TD2 = F D (4.76) Velocity... Look-ahead (mm/min) 4 .5 Summary 250 0 2000 150 0 1000 50 0 0 0 0.2 0.4 0.6 0.8 1 1.2 Time (s) 1.4 1.6 1.8 2 Fig 4.39 Look-ahead speed profile 1 Feedrate after Look-ahead and Comer control (mm/min) 250 0 2000 150 0 1000 50 0 0 0 0 .5 1 Time (s) 1 .5 2 Fig 4.40 Look ahead speed profile 2 4 .5 Summary In ADCBI, the speed profile for two successive blocks is generated by considering the type of the current block (e.g... appropriate control algorithms and the instructions generated are sent to the drivers As mentioned above, a fully functional CNC system consists of various control modules that provide many functions Practically, however, the majority of CNC systems include control systems that consist of the interpolation module of the middle layer and the servo and spindle control modules of the lower layer In this chapter, . profile of the acceleration interval in Block N1 can be obtained by Eq. 4.69 and Eq. 4 .55 and the speed profile of acceleration interval in Block N2 can be obtained by Eq. 4.70 and Eq. 4 .55 where. acceleration and deceleration by the commanded feedrate, as given by Eq. 4 .53 . T A = F A (4 .51 ) T D = F D (4 .52 ) T C = L − F 2 2A − F 2 2D F (4 .53 ) In the case of a short block, the length of the block. by Eq. 4.73 and Eq. 4 .55 . The speed profile of the deceleration interval of Block N2 can be obtained by Eq. 4.74 and Eq. 4 .56 , where the initial speed at the deceleration interval, V 0 ,isF 1 and the