Theory and Design of CNC Systems Part 5 potx

However, in the case when a shortblock and a normal block are successive, since the speed profile can be generatedwith an identical equation regardless of the commanded feedrate of the t

W x (t) = (Ae −at + Bcoswt + C w sin wt) (4.47)

where A= w2wK + a2,B = w −wK2+ a2,C = w awK2+ a2

Wx (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 integraEqua-tion 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+w1



B2+C w22sin(wt −θ) (4.48a)whereθ = cos −1

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/Deccontrol filter The machining errors are summarized in Table 4.5 with respect to eachtype of Acc/Dec control filter

According to Table 4.5 the machining error is proportional to the square of thefeedrate and the Acc/Dec time It is also in inverse proportion to the radius of thecircular path Therefore, from this, we know that the higher the feedrate the longer theAcc/Dec time, the shorter the radius of the circular path and the larger the machiningerror We also know that the accuracy of the S-shape-type Acc/Dec control is betterthan that of the alternatives

Table 4.5 Machining error due to Acc/Dec filter

Control type Machining Error Remarks

Linear ΔR = F2 τ 2

24R F: Feedrate Exponential ΔR = F2 τ 2

2R τ: Time constant

S-shape ΔR = F2 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 trolling axes Setting the block control mode is one of the G-code functions Forexample, in the G-code system of the FANUC controller, there are two kinds of pathcontrol mode; exact stop mode (G61) and continuous mode (G64)

con-In exact stop mode, the machine follows the programmed path as exactly as sible, stopping at sharp corners of the path Alternatively, in continuous mode, sharpcorners 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 4.12shows the actual toolpath when continuous mode is applied

pos-Exact stop mode generally results in reduction of machined surface quality due

to the stoppage of axis movement and increases machining time due to accelerationand deceleration for all blocks

In continuous mode, the tool begins the movement to the successive block beforethe tool reaches the end of the block Unlike exact stop mode, this mode does notresult in reduction of the surface quality and increase in machining time In contin-

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 inFig 4.12 Therefore, machining error always occurs at sharp corners The path nearthe 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

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 andFig 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 controlfor 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 arecontinuously 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 withthe 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 atthe corner between two success blocks join The speed is accelerated or deceleratedconsidering the difference in the feedrate of the two blocks

Time Block 1

Pulse

Time

Pulse

Acc/Dec control

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 beforeexecuting rough interpolation Also unlike ADCAI-type NCK, where Acc/Dec con-trol is carried out separately for individual axes, ADCBI-type NCK carries out theAcc/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 machiningspeed of machine tools is getting faster Therefore, ADCBI is essential to implement

the high-speed machining functions that have become a typical machine-tool tion and consequently, the latest machine tools provide ADCAI as a basic function.

Fig 4.15 ADCBI-type NCK flowchart

Figure 4.15 shows the flowchart for the overall procedure of the ADCBI-typeNCK 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 speedprofile 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 tion time for rough interpolation, and the commanded feedrate are considered whengenerating a speed profile For convenience, let us suppose that Acc/Dec control is

itera-applied to a linear path, the length of the linear path is L(mm), the allowable celeration is A(mm/s2), the allowable deceleration is D(mm/s2), the iteration time

ac-130 4 Acceleration and Deceleration

Blockinformation

Block end position

Block start position

Fig 4.16 Linear path Acc/Dec control

for rough interpolation isτ(s), and the commanded feedrate from a part program is

F(mm/s2)

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 notinclude the constant-speed zone Equation 4.50 is the condition that a normal blockshould satisfy If Eq 4.50 is not satisfied then the block is a short block

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.

(a) Normal block (b) Short block

Fig 4.17 Speed profiles

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, which is

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 aftersubtracting the length needed for acceleration and deceleration by the commandedfeedrate, as given by Eq 4.53

Li= V i2+12A −V i2where, V i is the velocity of the ith interval and V0= 0

Li is the displacement for the ith sampling time.

NA=TτA .

132 4 Acceleration and Deceleration

In the constant speed range the commanded feedrate is F and the tool movesτ×F

every iteration time for interpolation In the deceleration interval, the length throughwhich the tool moves every iteration time for interpolation can be calculated using

Eq 4.56

L i= V i2−V 2D i2+1where, Vi is the velocity of the ith interval and V0= F

L i is the displacement for the ith sampling time.

N D=TτD .

It is possible to calculate the interpolated point by projecting the displacementthrough 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 circularblocks, it is true that direct usage of the above-mentioned equations for generatingspeed profile and interpolating is impossible In ADCAI, interpolation and Acc/Deccontrol are applied to the individual block and it is not necessary to consider theconnection of blocks However, in ADCBI, because the speed at the beginning andthe end of a block should be considered when generating a speed profile, the previousand the successive blocks should be considered when generating a speed profile andinterpolating

In the next sections, all possible cases for connection relationships that can occurbetween two successive blocks in actual machining will be addressed The equationsfor 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 blockand the block without constant speed interval as a short block From the way in whichtwo 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 shortblock and a normal block are successive, since the speed profile can be generatedwith an identical equation regardless of the commanded feedrate of the two blocks,

a method to calculate the speed profile when the commanded feedrate of the twoblocks is identical will be described Therefore, the way in which two blocks areconnected 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

t

(b) Normal block → Normal block (Speed : high → low) F

t

(c) Normal block → Normal block (Speed : low → high)

F

t

(d) Short block → Normal block (Constant speed) F

t N1 N2 (e) Normal block → Short block (Constant speed)

F

t N1 N2

(f) Short block → Short block (Constant speed) F

t N1 N2

(g) Short block → Short block (Speed : high → low)

F

t N1 N2

(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 inSection 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 C1is computed by Eq 4.57

134 4 Acceleration and Deceleration

T C1= L1− F

2

2A

where, L1is the displacement of block N1

In Block N2, because at the beginning of the block the tool is moving with 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 C2is computed

where, L2is the displacement of block N2

When two successive blocks have the same feedrate, the speed profile for theacceleration 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-mentioned equa-tions, it is possible to generate the speed profile for two successive normal blockswith the same feedrate as in Fig 4.19

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 N1 and N2 are F1and

F2, respectively, and F1is higher than F2, the speed at the corner is defined as F2.This is done in order to avoid abnormal machining status such as tool breakage due

to the high speed In Block N1, acceleration time T A1 is computed by Eq 4.59 and

deceleration time T D1is computed by Eq 4.60

In Block N1, the speed profile for the acceleration interval is obtained by Eq 4.55

and the speed profile for the deceleration interval is obtained by Eq 4.56 where the

speed at the beginning of deceleration is F1, and the speed at the end of deceleration

is F2 The constant speed time of Block N1 is calculated by Eq 4.61.

the beginning of deceleration is F2and the speed at the end of deceleration is 0 The

constant speed time of Block N2 is calculated by Eq 4.63.

pro-4.3.2.4 Normal Block (Low Speed)/Normal Block (High Speed)

Figure 4.18c shows the case where two normal blocks with different feedrate aresuccessive and the speed of the first block is smaller than that of the second block

In this case, the smaller speed between the two block speeds is defined as the speed

at the corner as shown in Fig 4.18b

If the commanded feedrate of Block N1 is F1 and the commanded feedrate of

Block N2 is F2, the speed at the corner is defined as F1 In Block N1, acceleration time T A1 is computed by Eq 4.64, but it is not necessary to calculate deceleration

because the speed at the end position is F1and so it is not necessary to decelerate

The constant-speed time T C1is computed by Eq 4.65

136 4 Acceleration and Deceleration

In Block N1, the speed profile of the acceleration interval can be obtained by

Eq 4.55 and the speed at constant speed interval is held at the commanded feedrate

F1

In Block N2, because the feedrate is lower than the commanded feedrate of Block N2, F2, at the end of Block N1, the speed at the beginning of the Block N2 is not

changed and only deceleration is required at the end of the block The

decelera-tion time T D2 is calculated by Eq 4.66 and the constant speed time is calculated by

Eq 4.67 The speed profile of the deceleration interval can be obtained by Eq 4.56

where the speed at the beginning of deceleration is F2and the speed at the end ofdeceleration is 0

pro-4.3.2.5 Short Block/Normal Block with Identical Speed

Figure 4.18d shows the case where a short block precedes a normal block and thefeedrate of the two blocks is identical In order to generate a speed profile, firstlythe speed at the connection point of the two blocks should be calculated Unlike the

Fig 4.21 Speed profiles for normal blocks with F1lower than F2

case where two normal blocks are connected, because it is impossible to reach at thecommanded feedrate on a short block, it is first necessary to consider the maximumreachable speed on the short block Equation 4.64 is used for calculating this

The speed F from Eq 4.68 is defined as the corner speed and the speed of the

beginning of Block N2 The time spent to reach F  from 0 in a short block, T A1, iscomputed by Eq 4.69 and the time spent to reach the commanded feedrate of Block

N2 from F  , T A2, 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

pro-138 4 Acceleration and Deceleration

Fig 4.22 Speed profiles for short block/normal block with identical feedrates

4.3.2.6 Short Block/Normal Block with Different Speed

In the case where a short block precedes a normal block are continued and the manded feedrate of the two blocks are different from each other It is possible togenerate a speed profile by the same method as mentioned in Section 4.3.2.5 Thereason is that it is impossible to reach the commanded feedrate in a short block and

com-the corner speed is decided based only on com-the length of com-the short block, L1.

4.3.2.7 Normal Block/Short Block with Identical Speed

Figure 4.18e shows the case where a normal block precedes a short block and thefeedrate of the two blocks is identical As mentioned in Section 4.3.2.5, the speed

at the connection point of the two blocks should be calculated based on the length

of the short block in order to generate a speed profile In this case, because a shortblock is executed after a normal block, the start speed of the short block that makesthe speed at the end of the block zero should be calculated Equation 4.72 is used for

calculating the start speed of Block N2, F 

The speed F  from Eq 4.72 is defined as the corner speed and the speed at the

end of Block N1 the acceleration time of Block N1, T A1, is computed by Eq 4.73

and the deceleration time of Block B1, T D1, is computed by Eq 4.74 Further, the

constant speed time, T C1, is calculated by Eq 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 N2can be obtained by Eq 4.74 and Eq 4.56, where the initial speed at the deceleration

interval, V0, is F1and the end speed of the deceleration interval is F 

Fig 4.23 Speed profile for Normal block/Short block with F1larger than F2

There is only a deceleration interval in Block N2 The deceleration time, T D2,can be obtained by Eq 4.76 Figure 4.23 shows the speed profile generated from theabove-mentioned equations

4.3.2.8 Normal Block/Short Block with Different Speed

When a normal block precedes a short block, the commanded feedrate of two blockscan be different from each other In this case, it is possible to generate a speed pro-file by a method similar to that of a normal block and short block with the samecommanded feedrate, described in Section 4.3.2.7 This is because the corner speed

is decided based on the length of the short block, L2, regardless of its commanded

speed

140 4 Acceleration and Deceleration

Therefore, for the case that is described in this section, the corner speed F  iscomputed by Eq 4.72 and the speed profile is generated in the same way as themethod mentioned in Section 4.3.2.7

4.3.2.9 Short Block/Short Block with Identical Speed

Figure 4.18f shows the case where two short blocks with identical feedrate are nected In this case, in order to generate a speed profile, it is first necessary to calcu-

con-late the maximum feasible speed at the corner, F  The end speed of Block N1, F1, is

computed by Eq 4.77 and the start speed of Block N2, F2, is computed by Eq 4.78

The smaller of the two speeds F 

1and F 

2is selected as the corner speed F and it

is possible to calculate the maximum speed, F max , based on F  If F is the same as

F2

max − F 2

F max2 − F 2 2A +F max2

If F  is F2 , the acceleration time of Block N1, T A1, is calculated by Eq 4.81 and

the deceleration time, T D1, is calculated by Eq 4.82 In addition, the speed profilecan be obtained by Eq 4.55 and Eq 4.56 where the initial speed of the deceleration

interval, V0, is F max and the end speed of the deceleration interval is F  Also, the

deceleration time of Block N2, T D2, is calculated by Eq 4.76 and the speed profile

of Block N2 can be obtained by Eq 4.56 where the initial speed of deceleration, V0,

is F and the end speed of deceleration is zero

Figure 4.24 shows the speed profile generated from the above-mentioned

equa-tions in the case that F  is F 

2 In the case that F  is F 

1, it is possible to generate aspeed profile in a similar way

4.3.2.10 Short Block (High Speed)/Short Block (Low Speed)

Figure 4.18g shows the case where two short blocks with different feedrates areconnected As mentioned in Section 4.3.2.6, the corner speed of two short blocks isdecided by the length of the short blocks regardless of the commanded feedrate ofthe blocks Therefore, the speed profile for the case mentioned in this section can beidentically obtained by the method of the case in Section 4.3.2.9

4.3.2.11 Short Block (Low Speed)/Short Block (High Speed)

Figure 4.18h shows the case where two short blocks are connected and the speed ofthe first block is smaller than that of the second block Although the speed of the twoblocks is different, the method to obtain the speed profile is identical with that of thecase mentioned in Section 4.3.2.9 because the corner speed of the two short blocks

is decided by the length of the short blocks regardless of the commanded feedrate ofthe blocks

During circular-path machining, the speed of each axis is continually changing.Therefore it is necessary to reduce the speed (feedrate) compared with the linearpath The change of the axis speed results in mechanical shock and, especially at thetransition point from a circular path to a linear path, large mechanical shock occurs.The mechanical shock is proportional to the acceleration The acceleration is propor-tional to the square of the feedrate and is inversely proportional to the radius of thecircular path Therefore, it is necessary to restrict the maximum allowable accelera-tion for a circular path The allowable speed for a circular path is obtained as below

4.3.2.12 Overlap Between a Linear and a Circular Profile