Acc Decc Planning With Jerk Limitation Teza

Abstract

Motion control command of CNC controller includes path interpolation and

acceleration/deceleration planning. Smooth feed acceleration/deceleration with jerk

limitation noticeably reduces wear on mechanical parts and optimizes travel response.

Jerk limitation is an important factor when attempting to obtain high speed cutting

(HSC) on CNC machines.

This thesis imposes Blended S-Curve and Squared Sine function to provide

smooth velocity profiles for point-to-point motion. The trajectories are depended on

the acceleration and jerk constraints of machine. Overlapping one period of time has

been used for continuous motion. The proposed velocity profiles have been

implemented by two servo driver/motors and multi-axis motion control card.

Contents

Abstract...........................................................................................................................I

Contents.........................................................................................................................II

List of Figures..............................................................................................................IV

List of Tables...............................................................................................................VI

Chapter 1 Introduction..................................................................................................1

1.1 Backgrounds and Motivation.........................................................................1

1.2 Thesis Arrangement.......................................................................................2

Chapter 2 CNC Command Generation.........................................................................3

2.1 The Structure of CNC Command Generation.................................................3

2.2 Relation between Acceleration/Deceleration and Jerk...................................5

2.3 Structure of Acceleration/Deceleration..........................................................8

Chapter 3 Acceleration/Deceleration Planning...........................................................11

3.1 Blended S-Curve Velocity Profile [1]..........................................................11

3.2 Blended S-Curve velocity profile against different conditions.....................17

3.3 Squared Sine Velocity Profile......................................................................27

3.4 Squared Sine velocity profile against different conditions..........................33

3.5 Comparison of Three Velocity Profiles........................................................43

3.6 Continuous Motion.......................................................................................46

Chapter 4 Experiment and Result...............................................................................50

4.1 Configuration of Experiment.......................................................................50

4.2 Experiment Results......................................................................................51

4.2.1 Using Blended S-Curve velocity profile.....................................................51

4.2.2 Using Squared Sine velocity profile...........................................................55

4.2.3 Continuous Motion with Squared Sine velocity profile.............................59

II

Chapter 5 Conclusion and Future Work.....................................................................62

Reference.....................................................................................................................64

III

List of Figures

Figure 2-1 Structure of CNC command generation......................................................3

Figure 2-2 Velocity profile without acceleration/deceleration.....................................5

Figure 2-3 Different velocity profiles with acceleration/deceleration..........................6

Figure 2-4 Velocity profiles with jerk limitation..........................................................7

Figure 2-5 Acceleration/deceleration before interpolation...........................................8

Figure 2-6 Acceleration/deceleration after interpolation..............................................8

Figure 2-7 Acceleration/deceleration structure as a low pass filter............................10

Figure 3-1 S-Curve velocity profile............................................................................11

Figure 3-2 Blended S-Curve velocity profile..............................................................13

Figure 3-3 Blended S-Curve condition (1-1)..............................................................19





Figure 3-8 Squared Sine funciton velocity profile......................................................29

Figure 3-9 Square Sine velocity profile (Without linear section)...............................33

Figure 3-10 Squared Sine condition (1-1)..................................................................35





Figure 3-15 Comparison of three velocity profiles.....................................................44

Figure 3-16 Motion without continuity.......................................................................46

Figure 3-17 Jerk in overlapped section.......................................................................48

IV

Figure 3-18 Continuous motion under jerk constraint................................................49

Figure 4-1 PCC1620 and MSDA043A1A, MSMA042A1E.......................................50

Figure 4-2 Line interpolation with Blended S-Curve condition (1.1)........................51




Figure 4-6 Line interpolation with Squared Sine condition (1.1)...............................55




Figure 4-10 Experiment trajectory for continuous motion.........................................59

Figure 4-11 Information of continuous motion experiment.......................................59

Figure 4-12 Main motion trajectory (Experiment result)...........................................60

Figure 4-13 Connection of motion block 1 and block 2 (Experiment result).............60



V

List of Tables

Table 3-1 Time constants against different conditions (Blended S-Curve)................27

Table 3-2 Time constants against different conditions (Squared Sine)......................43

Table 3-3 Comparison of velocity profiles in total motion time................................45

Table-4-1 The specification of experiment equipments.............................................50

VI

Chapter 1 Introduction

1.1 Backgrounds and Motivation

Motion control and servo driver/motor system are the major techniques in

automation industry. Such as robotic systems, automatic packaging, material

handling, electronic equipments, all need this technique which combine mechanics

with electronics in production [1]. Computer numerical control (CNC) machine tools

play an acme role in modern automatic manufacturing system. High speed and high

accuracy machining is an important issue to high performance CNC machine tools.

While the specification and performance of machine have been fixed, promoting the

motion control technique is most effective to obtain high performances.

According to the reference command, motion control generates the motion

trajectory, and controls in the machining process. It consists of path interpolation and

acceleration/deceleration planning [2].

Path interpolation describes the desired tool path accurately, and feeds a

sequence of reference points that constitutes the desired tool path. Two conventional

interpolation algorithms are straight line interpolation and circular arc interpolation

[3]. But the work-piece shape has more and more complicated recently, in order to

satisfy the permissible accuracy, it is significant to develop interpolation algorithm

which provides smooth feed motion to high speed machining systems.

Because of the stiffness of mechanical structure and the specifications of

actuators, most machine systems have physical constraints which are limits of

acceleration and jerk [4]. If the velocity has a heavy change in high speeds, it results

in saturation of the servo driver system. It may excite system vibrations, degrade the

1

contouring accuracy, and induce wear on mechanical parts. So providing a smooth

feed acceleration and deceleration must be considered.

In this thesis, we pay attention to plan acceleration and deceleration in smooth

motion. Designing velocity profile under the constraint of mechanical system, and

achieve the accurate movement in the shortest time. In addition to this, some

automatic machines required continuous motions, it is essential to develop a method

which connects several motion blocks.

1.2 Thesis Arrangement

The remainder of this thesis is organized as follows. Chapter 2 deeply presents

the structure of CNC motion controller and discusses the influence of acceleration and

deceleration on motion. In chapter 3 we recommend two smooth acceleration/

deceleration algorithms with jerk limitation which are Blended S-Curve and Squared

Sine velocity profiles. These algorithms provide adjustable jerk and acceleration.

Furthermore, we can establish complete condition decisions to guarantee movement

in shortest time. Besides, continuous motion is discussed. The mentioned profiles are

simulated by MATLAB. Chapter 4 shows the experimental results obtained with

applied algorithms. Finally, the conclusions are summarized in chapter 5.

2

Chapter 2 CNC Command Generation

2.1 The Structure of CNC Command Generation

Generally, the CNC command generation can be divided into five parts, and

they can be schematized as Figure 2-1 [1].

(1) Program interpreter.

(2) Path interpolation.

(3) Acceleration/deceleration planning.

(4) Coordinate change.

(5) Detail interpolation.

MotionCommand

NC code (from

MMI or CAM)

CommandInterpretor

Line, Arc, Spline

Path Coordinate Detail Servo ControlInterpolation Change Interpolation System

Trapezoidal Position, velocity, torqueBlending S-curve command

Squared Sine

Acceleration/Deceleration

Figure 2-1 Structure of CNC command generation

3

At first, user directly sends the motion commands with MMI (man machine

interface) of the controller or the controller obtains programs of NC code made from

CAM software. These NC code contain motion information, such as motion type,

displacement, and feedrate (velocity). Program interpreter translates NC code into the

motion control command of controller. In command signal generator, path

interpolation and acceleration/deceleration planning estimate the variables like motion

time constants and specific velocity. And then, according to the real axis coordinates,

the controller does detail interpolation in every interrupt service routine. Finally,

controller sends reference command signal to servo control system and the movement

will be achieve on the reference path [1].

Motion control consists of path interpolation and acceleration/deceleration

planning.

Two basic interpolation algorithms are straight line and circular arc

interpolation. Most off-line programming stations still generate the work-piece surface

of 3D programs by describing long sequences of straight line segments. If we want to

machine a complicated contour, traditional interpolation may be hard to maintain

permissible accuracy. An alternative is to describe the contour using splines. Some

high order parametric described curves have been developed, example of these are

Hermite Curve, Bezier Curve, B-Spline, Quintic Spline, and NURBS, etc [5,6]. They

provide a more continuous feed motion compared to multiple straight line and circular

arc segments.

The purpose of path interpolation is to describe accurate moving path, but

interpolator must consider the characteristic of backward servo control system. If the

command signal changes heavily, the servo driver/motor cannot achieve

4

immediately, and it may induce vibration to make the rough motion in machining.

These will cause bad effect to the output of motors and the machining of work-piece.

So we must provide acceleration/deceleration algorithm to obtain smooth velocity

profile.

2.2 Relation between Acceleration/Deceleration and Jerk

Interpolator obtains the motion information which includes motion type,

displacement, and feedrate. Figure 2-2 shows the original curve of velocity against

time, the velocity profile without acceleration and deceleration in movement. Its

acceleration is an impulse function, actually it is impossible to create an infinite

acceleration in system. The discontinuous velocity command causes a heavy variation

that saturates the servo control system in the axes [2,7]. In order to improve the output

of motors to be more smooth and effective, we must plan the acceleration/deceleration

to the original velocity profile judging from the NC code.

F Feedrate

A Acceleration

t

Time

t

Figure 2-2 Velocity profile without acceleration/deceleration

5

Some different kinds of velocity profile are developed. Figure 2-3 (A) shows

the Trapezoidal velocity profile, and its acceleration keeps a finite value. It is helpful

to improve the output of motors. Other profiles as Exponential, Single-Cubic Spline

velocity profiles are widely adopted in mechanical industry, they show in Figure 2-3

(B) and (C) [1,2,6].

F Feedrate F

A Accelerationt

A

t

F

tA

t

t

t

J Jerk J J

Timet t t

(A) (B) (C)

Figure 2-3 Different velocity profiles with acceleration/deceleration

Position, velocity must be continuous in motion process. If the acceleration

changes discontinuous, the jerk will be the impulse function. Jerk means the first time

derivative of acceleration, also be the rate of change of acceleration. Acceleration

changes according a jump function, it is disadvantageous for machining accuracy and

service life of machines. Besides, continuity of command signal has

6

great effect in Feedforward control. When a HSC CNC machines applied

Feedforward control, discontinuous command might induce infinite Feedforward

signal to saturate the servo control system [8]. Therefore, considering the continuity

of acceleration, jerk must be limited under the constraint of machine tools.

These above-mentioned acceleration/deceleration planning are advantageous in

easily implement and the shorter motion time. But the acceleration changes

discontinuously and the jerk be infinite. Further development, 3-Cubic Spline (Figure

2-4 (A)), S-Curve velocity profile (Figure 2-4 (B)) provide the velocity profile which

has continuous acceleration and limited jerk [2,9]. Since the start of motion till the

desired feedrate has been achieved, these profiles improve the vibration of machine

obviously. In this thesis, Blended S-Curve and Squared Sine velocity profiles are

consider for better continuity [1,2,5,10,11,12].

F Feedrate F

A Accelerationt

A

t

t

t

J Jerk J

Time

t t

(A) (B)

Figure 2-4 Velocity profiles with jerk limitation

7

2.3 Structure of Acceleration/Deceleration

Judging from the sequence of acceleration/deceleration, it could be divided into

two types, acceleration/deceleration before interpolation and acceleration/deceleration

after interpolation [13].

→ → Vx

TangentialMapping N x N y

Servo System

Velocity

V Acceleration/ InterpolatorDeceleration Vy

Servo System

Figure 2-5 Acceleration/deceleration before interpolation

Acceleration/deceleration before interpolation is shown in Figure 2-5. At first,

we apply the acceleration/deceleration along the desired motion path. In other words,

we plan the tangential velocity (resultant velocity) profile in the moving space. The

velocity profile is generated by the selection of polynomial function [14]. Then,

according to the geometry of respective axes and motion path, the interpolator

calculates the unit vector to map the tangent velocity profile to each axis. Finally, the

controller send reference command signal of each axis to respective servo control

systems in detail interpolator.

→ → Vx

TangentialMapping N x N y

Acceleration/ Servo System

DecelerationVelocity

V InterpolatorVy

Acceleration/ Servo SystemDeceleration

Figure 2-6 Acceleration/deceleration after interpolation

Figure 2-6 shows acceleration/deceleration after interpolation. According to the

geometry of respective axes and motion path, the interpolator first calculates the unit

vector to map the tangential velocity profile to each axis. And then, axes transmit

respective velocity profiles to the acceleration/deceleration structure. The velocity

profile is based on the digital convolution technique and digital differential analyzer,

and the velocity profile can be smoothed by this structure which acts as a low pass

filter. It shows in Figure 2-7 [14,15]. Finally, the controller sends reference command

signal of each axis to respective servo system in detail interpolator.

Although acceleration/deceleration after interpolation is advantageous in easy

implement because the acceleration/deceleration algorithm have simple structure, in

more than two blocks motion, the connection of blocks might induces contour error,

the real movement will not pass through the desired position that motion blocks

connect. Besides, the structure of low pass filter might be not easy to change [7].

Therefore, Acceleration/deceleration before interpolation is applied in this thesis.

9

F Feedrate

Time

tF

tF

t

Figure 2-7 Acceleration/deceleration structure as a low pass filter

10

Chapter 3 Acceleration/Deceleration Planning

3.1 Blended S-Curve Velocity Profile [1]

In industry, Trapezoidal, Exponential, and Single-Cubic Spline velocity profiles

are widely used because they are easy to implement and the motion time could be

shorten. But the acceleration changes with a jump function, the heavy change will

excite vibration of machine structure and reduce machining accuracy in high speed.

This will be the serious disadvantage in high speed and high accuracy machine tools.

Strengthening stiffness of machine and using high torque motor can extend the

saturated range of system, but there are not economical in automatic machine

products [1]. To solve this problem, improvement motion command is the most

effective method. Controlling rate of change of acceleration, in other words, the jerk

has to be limited. So the velocity and acceleration will be continuous. S-Curve is the

common method to achieve jerk limitation.

Feedrate

F

AccelerationT

Amax

JerkJ

max

Time

Figure 3-1 S-Curve velocity profile

11

Figure 3-1 shows the S-Curve velocity profile. S-Curve is the combination of

parabola, and named because the velocity profile looks like the English letter “S”. Its

greatest advantage is that acceleration is a continuous trapezoidal and jerk is not an

impulse function. Acceleration and jerk can be limited to desired value.

If we use complete S-Curve, the motor need greater acceleration, it means that

more torque or energy is required. In motion control, we prefer to use the energy of

motors effectively, so we can redesign this velocity profile. To combine the linear

with parabolic segment, we can limit jerk by use parabolic curve while start of the

motion and the desired feedrate will be achieved. In the middle accelerated period we

can use linear curve to obtain the maximum efficiency of motors. This combines

profile is named Blended S-Curve (also called Bell Shape) [2,6].

First we define the parameters of Blended S-Curve. And Figure 3-2 shows

Blended S-Curve and the physical meaning of parameter.

Ts : Jerk time,

Ta : Accelerate time,

To : Motion time without acceleration/deceleration,

L : Desired displacement,

F : Desired feedrate (Velocity),

J max : Desired maximum jerk,

Amax : Desired maximum acceleration,

J ref : Reference jerk

(Actual maximum jerk),

12

Aref

V1

V2

V3

: Reference acceleration

(Actual maximum acceleration),

: Velocity at end of jerk,

: Velocity at end of fixed acceleration,

: Velocity at end of acceleration

(Actual maximum feedrate).

S(t)

L

Dis

plac

emen

t

0 t1 t2 t3 t4 t5 t6 t7 Time

V(t

) V3

V2

Fee

drat

e

V1

A(t

) Time

Aref

Acc

eler

atio

n

Time

J(t) Jref

Jerk

TsTime

TaTo

Figure 3-2 Blended S-Curve velocity profile

The velocity profile can be divided to seven sections, and there have the symmetry.

Ts 0 t ≤ t1 t2 t ≤ t3 t4 t ≤ t5 t6 t ≤ t7 . (3.1)

Ta 0 t ≤ t3 t4 t ≤ t7 . (3.2)

To 0 t ≤ t4 . (3.3)

The relation between parameters can be calculated by integral theorem as follow:

Aref J ref Ts , (3.4)

V A

ref T

s , (3.5)1 2

V2 V1 Aref Ta − 2Ts , (3.6)

V3 V1 V2 Aref Ta − Ts . (3.7)

And V L . (3.8)3

To

Substitute (3.4) and (3.7) into (3.8), we get

J ref

L, (3.9)TsTa − Ts

To

Aref

L, (3.10)

Ta

− Ts

To

V1 LTs , (3.11)2Ta − Ts To

V2

2Ta

− 3Ts

L. (3.12)

2Ta

− Ts

To

Note that the above equations must satisfyTa ≥ 2Ts

. (3.13)

And

To ≥ Ta . (3.14)

Finally, Blend S-Curve equations can be derived by integral theorem and differential

theorem.

The displacement equations are

1 3J

ref t

6

1

A

t −

t 2

Vt

−t st 2 ref 1 1 1 1

1 t −

t 31

A t 2

t −

t st − J − t V

6 2ref 2 2 2 2 2

St V3 t − t3 st 3

1

− Jref

t −

t43 V

t

−

t4 st

4

6 3

1

A

t −

t 2 V

t −

t st −2 ref 5 2 5 5

1 t − t6

1

Aref t

− t6

V1 t − t6 st 6 Jref

3

−2

6 2

The velocity equations are

1 2J

ref t

2

t − t

A Vref 1 1

1 t −

t 2 A

t −

t V− J2 ref 2 ref 2 2

Vt V3

1− J ref t − 4 2 V

2 3

t − t V− Aref 5 2

1− Aref t − t6

V1

2

2 J

ref t −

t

6

0 t ≤ t1 Ts

t1 t ≤ t2 Ta − Ts

t2 t ≤ t3 T

t3 t ≤ t4 To . (3.15)

t4 t ≤ t5 To Ts

t5 t ≤ t6 To Ta − Ts

t6 t ≤ t7 To Ta

0 t ≤ t1 Ts


t2 t ≤ t3 T

t3 t ≤ t4 To . (3.16)

t4 t ≤ t5 To Ts


t6 t ≤ t7 To Ta

15

The acceleration equations are

J t 0 t ≤ t Tref 1 s

A t t ≤ t T− T

ref 1 2 a s

− J

ref

t − t

2 A t

2 t ≤ t

3 T

ref

At 0 t 3 t ≤ t

4 T . (3.17)

o

t4 t ≤ t5 To Ts− Jref t − t4

− A t5

t ≤ t6

T T − Tref o a s

Jref

t − t

6

− A t6

t ≤ t7

T Tref o a

And the jerk equations are

Jref 0 t ≤ t1 Ts

0 t t ≤ t

2

T− T

1 a s

t2 t ≤ t3 T− J

ref

Jt 0 t3

t ≤ t

4 T . (3.18)

o

t4 t ≤ t5 To Ts− Jref

0 t5

t ≤ t

6

T T −To a s

J

ref

t6

t ≤ t

7

T To a

As above-mentioned planning, the displacement, velocity, acceleration and jerk

can be calculated by giving the time constants Ts , Ta , and To . But it can not indicate

whether the motion command exceed the real performance and physical limit of

machine tools or not [1,6,16]. If the real motion can not achieve the desired command,

saturated motor current will induce exceeding torque to vibrate machine

16

structure. Therefore, we have to use movement, feedrate, maximum acceleration, and

maximum jerk as designed parameter, estimate the shortest time constant satisfied

above physical constraints.

3.2 Blended S-Curve velocity profile against different conditions

Figure 3-1 shows that F 1 A T and A J T .2 2max max max

According to above two equations, we will obtain

2

F Amax . (3.19)

J max

It means that if the displacement is long enough, the feedrate F can be achieved with

the maximum acceleration and the maximum jerk. And judging from the user given

feedrate, we can discuss the changes of Blended S-Curve against different conditions.

If we know the actual maximum jerk J ref , the actual maximum acceleration

Aref , the actual maximum feedrate V3 , and the desired displacement L in motion,

according to (3.4), (3.7), (3.8), the following equations can be obtained.

The jerk time is

T A

ref . (3.20)s

J ref

The accelerate time is

T V3 T . (3.21)a A

ref

s

And the motion time without acceleration/deceleration is

T L . (3.22)

o

V3

If we decide the desired maximum acceleration Amax , maximum jerk J max , and

feedrate F in motion process, according to (3.19), there will be two kind of condition

in motion.

Condition (1): F ≥ Amax 2

J max

It means that the user given feedrate can be achieved by using

maximum acceleration and maximum jerk. But we must consider that

displacement may be not long enough to achieve our desired maximum

acceleration or feedrate. So we divide condition (1) to three conditions

further.

(1-1): The desired feedrate can be achieved, and the desired maximum

acceleration can be achieved.

According to (3.20), (3.21), (3.22),

Ts

Amax

, (3.23)J max

T F Ts

, (3.24)a A

max

T L . (3.25)o F

And the motion time without acceleration/deceleration must be equal

or greater than the accelerate time, (3.14) must be satisfied.

L ≥ F A

max . (3.26)AF J

maxmax

And the above equation can be

L ≥F 2

A F

max

. (3.27)A Jmaxmax

18

If we assume L is 20 mm , F is 50 mm s , Amax is 500 mm s 2 ,

and J max is 20000 mm s3 , the result will be shown as Figure 3-3.

This is the complete Bell Shape.

Figure 3-3 Blended S-Curve condition (1-1)

(1-2): The desired feedrate cannot be achieved, and the desired maximum


Ts A

max, (3.28)J

max

T V3 Ts

, (3.29)a A

max

T L . (3.30)o

V3

In this condition, the motion time without acceleration/deceleration

equal to the accelerate time.19

To Ta . (3.31)

Thus L V3 A

max . (3.32)V A J

max3 max


− A 2 A 4 4A Jmax

2 LV3

max max max

. (3.33)2J max

Then we obtain the time constants

Ts

Amax

, (3.34)J max

− A 2 A 4 4A Jmax

2 LT

a max max max

Ts , (3.35)2J max

Amax

To Ta . (3.36)

The accelerate time must be equal or greater than double jerk time, the

maximum acceleration can achieve.

− A

max 2 A

max 4 4Amax J max 2 L≥

A(3.37)2J A J max .

max maxmax

It can be simplify to

3

L ≥ 2Amax . (3.38)

2

J max

In condition (1-2), the displacement will be

F 2 A F L ≥

2A 3 max max . (3.39)

Amax

J max

J max

2

If we assume L is 5 mm , F is 50 mm s , A is 500 mm s 2 ,max


20



acceleration cannot be achieved.

T A

ref ,s J

max

T V

3 T ,a A

refs

T L .o

V3

The accelerate time must be equal to double jerk time.

V

Aref

.3

A Jmaxref

The above equation can be

A 2

V3 ref .

J max

(3.40)

(3.41)

(3.42)

(3.43)

(3.44)

21


to accelerate time.

L 2A

ref. (3.45)V J

max3

ThusLJ max

2Aref

. (3.46)A 2

J max

ref


A 3 J max2 L . (3.47)

ref 2


Ts 3L

, (3.48)2J max

Ta 2Ts , (3.49)

To Ta . (3.50)

The displacement is shorter than condition (1-2), so

3

L ≤ 2Amax . (3.51)

J max

2

If we assume L is 0.3 mm , F is 50 mm s , A is 500 mm s 2 ,max


22


Condition (2): F Amax 2

J max

It means that the user given feedrate can be achieved without

maximum acceleration. We also consider the displacement long

enough or not to achieve the user desired feedrate. So we divide

condition (2) to two conditions.



T A

ref , (3.52)s J

max

T F Ts

, (3.53)a A

ref

T L . (3.54)o F

23


F A

ref.

A Jmaxref


Aref

FJ max .

The time constants are

Ts JF

,max

Ta 2Ts ,

To FL

.

(3.55)

(3.56)

(3.57)

(3.58)

(3.59)

The motion time without acceleration/deceleration must be equal or

greater than the accelerate time.

L ≥

2 F . (3.60)

FJ

max

The displacement may be

L ≥ 2

F 3

. (3.61)J

max



24




This condition is similar to condition (1-3), so

Ts 3L

, (3.62)2J max

Ta 2Ts , (3.63)

To Ta . (3.64)

The displacement is shorter than condition (2-1).

L 2

F 3

. (3.65)J

max



25


Table 3-1 shows each condition and its corresponding time constants.

2

(1) F ≥

Amax

J max

(1-1) L ≥

F 2

A F F 2

A F

L ≥2A 3

(1-3) L ≤2A 3

max (1-2) max max max

Amax

J max

Amax

J max

J max

2 J max

2

Ts

Amax

Amax

3L

J max

J max

2J max

T F Ts

− A

max2 A

max 4 4Amax J max 2 L T 2Ta A

max2J

max A

max

ss

ToL

Ta TaF

26

2

(2) F

Amax

J max

(2-1) L ≥ 2

F 3

(2-2) L 2

F 3

J max

J max

TsF

3

LJ

max2J

max

Ta 2Ts 2Ts

ToL

TaF

Table 3-1 Time constants against different conditions (Blended S-Curve)

Judging from the given feedrate and displacement, we can calculate the shortest

time to form the Blended S-Curve velocity profile which under the acceleration and

jerk constraint.

3.3 Squared Sine Velocity Profile

In this section, we will present another acceleration/deceleration planning with

jerk limitation, and it is Squared Sine function velocity profile [11].

If we want the continuous jerk profile, using cubic polynomial is one direct way.

But the parameter of cubic polynomial which is one time derivative of jerk is more

difficult for us to experience and design. The most advantage of Squared Sine function

is the jerk profile not only can be limited but also can be continuous, and we do not

need to design any other parameter difficult to set. It is smoother than Blended S-

Curve, and more better to reduce vibration of machine structure.

27

Similar to Blended S-Curve, we define the parameters of Squared Sine function

at first. And Figure 3-8 shows Squared Sine function and the physical meaning of

parameter.

Ts : Jerk time,

Ta : Accelerate time,

To : Motion time without acceleration/deceleration,

L : Desired displacement,

F : Desired feedrate (Velocity),

J max : Desired maximum jerk,

Amax : Desired maximum acceleration,

J ref : Reference jerk

(Actual maximum jerk),

Aref : Reference acceleration

(Actual maximum acceleration),

V1 : Velocity at end of jerk,

V2 : Velocity at end of fixed acceleration,

V3 : Velocity at end of acceleration

(Actual maximum feedrate).

28

S(t)

L

Dis

plac

emen

t

0 t1 t2 t3 t4 t5 t6 t7 Time

V(t

) V3

V2

Fee

drat

e

V1

A(t

) Time

Aref

Acc

eler

atio

n

Time

J(t) Jref

Jerk

TsTime

TaTo

Figure 3-8 Squared Sine funciton velocity profile

The velocity profile can be divided to seven sections, and there have the symmetry.

Ts 0 t ≤ t1 t2 t ≤ t3 t4 t ≤ t5 t6 t ≤ t7 . (3.66)

Ta 0 t ≤ t3 t4 t ≤ t7 . (3.67)

To 0 t ≤ t4 . (3.68)

The acceleration equations can be defined as

29

2

ktAref

sin

Aref

Aref cos2 k t − t2

At 0

sin2 k t − t4

− A

ref

− Aref

− A

cos2 k t −

t 6ref

Where k π .2T

s

And J πA

ref kA .ref ref

2Ts

0 t ≤ t1 Ts

t1 t ≤ t2 Ta −Ts

t2 t ≤ t3 T

t3 t ≤ t4 To . (3.69)

t4 t ≤ t5 To Ts

t5 t ≤ t6 To Ta −

Ts t6 t ≤ t7 To Ta

(3.70)

(3.71)

Squared Sine equations can be derived by integral theorem and differential theorem.

The velocity equations are

Aref sin2kt

t −2 2k

A t− t V

ref 1 1

Aref

sin2kt −

t 2

t − t2 V2

2 2k

Vt V3

Aref sin2kt

− t

4− t − t4 − V3

2 2k

− Aref t − t5 V2

Aref sin2kt

− t

6

− t − t6 V12 2k

0 t ≤ t1 Ts


t2 t ≤ t3 T

t3 t ≤ t4 To . (3.72)

t4 t ≤ t5 To Ts


t6 t ≤ t7 To Ta

30

The displacement equations are

Aref 2 cos2kt

Aref

0 t ≤ t1 Tst −

4 2k2

8k2

Aref t −t12 V1t−t1 S(t1) t1 t ≤ t2 Ta −Ts2

cos2kt −t

Aref 2

Aref

t −t2− 2

V2t −t2 St2 t2 t ≤ t3 T

4 2k2

8k2

St V

t

−t St t t ≤ t T

3 3 3 3 4 o

Aref cos2kt

−t

A

ref2

− t −t4 4

V3t −t4 S(t4) t4 t ≤ t5 To Ts4 2k

28k

2

A− ref t −t 2 Vt −t S(t ) t t ≤ t T T −T

2 5 2 5 5 5 6 o a s

cos2kt −t

Aref 2

Aref

− t −t6 − 6

V1t −t6− St6 t6 t ≤ t7 To Ta

4 2k2

8k2

. (3.73)

The jerk equations are

2kAref sinkt coskt 0 t ≤ t1 Ts

0 t t ≤ t

2 T

− T

1 a s

sink t − t

cosk t − t − 2kA

2 2t

2

t ≤ t

3T

ref

Jt 0 t3

t ≤ t

4 T . (3.74)

o

− 2kAref sink t − t4 cosk t − t4 t

4 t ≤ t5 To Ts

0 t5

t ≤ t

6 T T − T

o a s

2kA sink t − cosk t − 6 6 t6 t ≤ 7 T T

tref o a

As Blended S-Curve, we substitute (3.71) and (3.7) into (3.8), we will derive

31

J ref π L ,

2 TsTa − Ts To

Aref L .

Ta − Ts To

And

V t 1 V Ts A

ref sin2kT V1

Ts − s ,

2 2k

V2 V t 2 V Ta − Ts Aref Ta − 2Ts V1 .

V V V2

L3 1

To

(3.75)

(3.76)

(3.77)

(3.78)

(3.79)

Note that the above equations must satisfy (3.13), (3.14).

For the same reason as Blended S-Curve, we instead that using time constants

Ts , Ta , To of using movement, feedrate, maximum acceleration, and maximum jerk

as designed parameter, avoid the motion command exceed the real performance and

physical limit of machine tools.

32

3.4 Squared Sine velocity profile against different conditions

Feedrate

F

AccelerationT

Amax

J max

Jerk

Time

Figure 3-9 Square Sine velocity profile (Without linear section)

Figure 3-9 shows that F 1 A T and A J T .

2 max max max πAccording to above two equations, we will obtain

2

F π A

max . (3.80)

2J

max

It means that if the displacement is long enough, the feedrate F can be achieved with

the maximum acceleration and the maximum jerk. And judging from the user given

feedrate, we can discuss the changes of Squared Sine against different conditions.

If we know the actual maximum jerk J ref , the actual maximum acceleration

Aref , the actual maximum feedrate V3 , and the desired displacement L in motion,

33

according to (3.71), (3.78), (3.79), the following equations can be obtained.

The jerk time is

T π Aref . (3.81)s

2 J

ref

The accelerate time is

T V3 T . (3.82)a A

ref

s

And the motion time without acceleration/deceleration is

T L . (3.83)o

V3

If we decide the desired maximum acceleration Amax , maximum jerk J max , and

feedrate F in motion process, according to (3.80), there will be two kind of condition

in motion.

Condition (1): F ≥ π

Amax 2

2 J

max

It means that the user given feedrate can be achieved by using

maximum acceleration and maximum jerk. But we must consider that

displacement may be not long enough to achieve our desired maximum

acceleration or feedrate. So we divide condition (1) to three conditions

further.



According to (3.81), (3.82), (3.83).

Ts πA

max , (3.84)2

J

max

T F Ts

, (3.85)a A

max

34

T L . (3.86)o F


or greater than the accelerate time, (3.14) must be satisfied.

L ≥ F π A

max . (3.87)AF 2 J

maxmax


L ≥F 2 π A F

2max

. (3.88)A Jmaxmax



Figure 3-10 Squared Sine condition (1-1)


acceleration can be achieved.35

Ts πA

max , (3.89)2

J

max

T V3 Ts

, (3.90)a A

max

T L . (3.91)o

V3

In this condition, the motion time without acceleration/deceleration

equal to the accelerate time.

To Ta . (3.92)

Thus L V3 π A

max . (3.93)AV

3

2 Jmaxmax


− πA 2 π 2 A 4 16A Jmax

2 LV3

max max max

, (3.94)4J max

then we obtain the time constants

Ts

πA

max

, (3.95)2 J

max

− πA 2 π 2 A 4 16A Jmax

2 L

Ta max max max

Ts , (3.96)4J max

Amax

To Ta . (3.97)

The accelerate time must be equal or greater than double jerk time, the

maximum acceleration can achieve.

− πAmax2 π 2 Amax

4 16Amax J max 2 L≥

π A(3.98)4J A 2 J max .

max maxmax


L ≥

π 2 A 3

. (3.98)max

2J

max

2

In condition (1-2), the displacement will be

36

F 2 π A F L ≥π 2 A 3

max max . (3.100)2

Amax 2

J max 2

J max

If we assume L is 5 mm , F is 50 mm s , Amax is 500 mm s 2 ,





T π Aref ,s 2

J

max

T V3 T ,a A

refs

T L .o

V3

The accelerate time must be equal to double jerk time.37

(3.101)

(3.102)

(3.103)

Vπ

Aref

. (3.104)3

A 2 Jmaxref

The above equation can be

V3 πA

ref 2 . (3.105)2

J

max


to accelerate time.

L πA

ref. (3.106)V J

max3

Thus2 LJ max π

Aref

. (3.107)π A 2

J max

ref


A 3 2J 2 L. (3.108)max

ref

π 2


Ts 3 πL , (3.109)

4J max

Ta 2Ts , (3.110)

To Ta . (3.111)

The displacement is shorter than condition (1-2), so

L ≤

π 2 A 3

. (3.112)max

2J

max

2



38


Condition (2): F π

Amax 2

2 J

max

It means that the user given feedrate can be achieved without

maximum acceleration. We also consider the displacement long

enough or not to achieve the user desired feedrate. So we divide

condition (2) to two conditions.



T πA

ref , (3.113)s 2

J

max

T F Ts

, (3.114)a A

ref

T L . (3.115)o F

39


F πA

ref.

A 2 Jmaxref


A 2FJ

max .ref π

The time constants are

Ts πF

,2J max

Ta 2Ts ,

To FL

.

(3.116)

(3.117)

(3.118)

(3.119)

(3.120)

The motion time without acceleration/deceleration must be equal or

greater than the accelerate time.

L ≥ 2πF . (3.121)

FJ

max

The displacement may be

L ≥2πF 3

(3.122).

J max



40




This condition is similar to condition (1-3), so

Ts 3 πL , (3.123)4J

max

Ta 2Ts , (3.124)

To Ta . (3.125)

The displacement is shorter than condition (2-1).

L 2πF 3. (3.126)

J max



41


Table 3-2 shows each condition and its corresponding time constants.

π2

(1) F ≥

Amax

2J

max

(1-1)(1-3)

F 2 π A F L ≥

π 2 A 3

(1-2) F 2 π Amax Fmax max

π 2 3

L ≥ A

max 2 J max 2 J

max

2

L ≤

Amax

2

Amax 2

J max 2

J max

Ts

π A

maxπ

A

max

3

πL

2 J

max2

J

max4J

max

Ta F Ts − πAmax2 π 2 Amax

4 16Amax J max2 L T

s2Ts

Amax

4J max

Amax

ToL

Ta TaF

42

π2

(2) F

Amax

2J

max

(2-1) L ≥ 2πF 3(2-2) L 2πF 3

J max

J max

TsπF

3

πL2J

max4J

max

Ta 2Ts 2Ts

ToL

TaF

Table 3-2 Time constants against different conditions (Squared Sine)

Judging from the given feedrate and displacement, we can calculate the shortest

time to form the Squared Sine velocity profile which under the constraints of

acceleration and jerk.

3.5 Comparison of Three Velocity Profiles

The above-mentioned two acceleration/deceleration planning were commonly

adopted in industrial motion control systems to achieve the jerk limitation. The

displacement, velocity, acceleration can be continuous and jerk can be limited. Now

we assume a motion command, displacement is 20 mm , feedrate is 50 mm s ,

maximum acceleration is 500 mm s 2 , maximum jerk is 20000 mm s3 , and then we

implement the acceleration/deceleration by Trapezoidal velocity profile, Blended S-

Curve profile and Squared Sine velocity profile. The results are shown in Figure 3-15.

43

Figure 3-15 Comparison of three velocity profiles

As it can be seen, the velocity and acceleration profiles with jerk limitation

(Blended S-Curve, Squared Sine) are smoother. This also can be seen by comparing

the frequency content of acceleration profiles for the three trajectories, which directly

relate to motor current, and actuation torque delivered by the drivers to the

44

mechanical structure. The Trapezoidal velocity profile produces high frequency

harmonics in acceleration, whereas the signal contents of jerk limited profile are

mainly in the low frequency range, making the trajectory also easier to track by the

limited bandwidth of the servo controller [5]. Blended S-Curve also can improve the

servo lag of position command, it had been compared to Trapezoidal and Exponential

velocity profile [7]. Attaching to ideal Feedforward controller, the contour error

almost can be reduced completely [8].

If the constraints of a motion control system, such as maximum acceleration and

maximum jerk have been decided, we can compare the total motion time (To Ta ) of

these three velocity profiles. Table 3-3 shows that different acceleration/ deceleration

types and the corresponding total motion time.

Type of acceleration/deceleration Total motion time

TrapezoidalL

V3

V3

Aref

Blended S-CurveL

V

Aref

V3

3

J ref

Aref

Squared SineL V

π Aref

V3

3

2 J

refA

ref

Table 3-3 Comparison of velocity profiles in total motion time

The total motion time of Squared Sine is the longest in these three

acceleration/deceleration profiles, and it is also the smoothest.

45

3.6 Continuous Motion

In automation industry, the point-to-point motion trajectory is usually used.

Sometimes, we required shortening the total machining time and the machining

accuracy also can be permitted, the continuous motion trajectory is considered. In

general, the end position of one forward motion block is the start position of one

backward motion block, and the velocity, acceleration and jerk will be zero at the turn

point in machining trajectory. It is shown in Figure 3-16.

Figure 3-16 Motion without continuity

Continuous motion means that as several single motion blocks connect to each

other, the motion trajectory will close to the desired connected positions but not pass

through them. The trajectory only passes through the start and end position of the

motion curve which combined with several single motion blocks. The velocity and

46

acceleration will be more smooth and each motion block will not need to accelerate at

rest. The purposed continuous motion method is polynomial addition which is to

overlap velocity profile of two connected motion blocks for one period of time [2].

Complete Blended S-Curve and Squared Sine velocity profiles are consist of

seven polynomial sections. We can use the property to plan the continuous motion.

We consider overlapping the last polynomial section of forward motion profile with

the first polynomial section of backward motion profile.

Equation (3.15) shows the jerk equations of Blended S-Curve, the jerk in the

first section and last section are the same. If we add the jerk function of these two

sections, the feedrate and acceleration may not exceed the desired value, but the jerk

value will be double of the desired maximum jerk. Therefore, we can not plan the

continuous motion to Blended S-Curve velocity profile by polynomial addition.

Equation (3.32) shows the jerk of Squared Sine velocity profile, similar to

Blended S-Curve, the jerk function in the first section and the last section are planned

the same. The jerk is not a jump function in these two sections, thus we can try to add

the jerk function of Squared Sine velocity profile under the jerk constraint.

47

Jerk

Block 1 Block 2

Tctime

J(t)

Tc t

Figure 3-17 Jerk in overlapped section

Figure 3-17 shows the jerk in overlapped section, this is the extreme condition

because these two motion blocks move in the same direction. We assume the

combined time is Tc here. Because the maximum jerk we set is fixed and the jerk

function in single block is symmetric, the jerk equation in the overlapped section can

be assumed as

J (t) 2kAmax sinktcoskt 2kAmax sinkTc − tcosk Tc − t

kAmax sin2kt sin2kTc − t. (3.127)

Then we differentiate J t to make sure that the time maximum jerk exist.

dJ t 2k 2 A cos2kt − cos2 kT − t 0 . (3.128)dt max c

The maximum jerk occurs at t Tc .2

The maximum jerk in combined section must be under the desired jerk constraint in

motion.

48

T T TJ c kA sin 2k c sin 2k T − c ≤ J max . (3.129)

2max

2c

2

2kAmax sinkTc ≤ kAmax , and π 1It can be simplify to ≤ .sin

2T sT

c

2

Finally we obtain the overlapped time T ≤1 T .

c 3 s

Figure 3-18 shows continuous motion in Squared Sine velocity profile, and we take

1T

c as

3 T

s .

Figure 3-18 Continuous motion under jerk constraint

49

Chapter 4 Experiment and Result

4.1 Configuration of Experiment

We use a PC to calculate the command with motion control, and then transmit

position command to servo driver/motors by PC-Based multi-axis motion control card.

Two servo motors are used to simulate two axes motion here. Encoders installed on

servo motors are used to measure real position, and decoder circuit is installed on

motion control card. C language is used to implement purposed algorithm with

Borland Turbo C++ 3.0. Figure 4-1 shows the motion control card and AC servo

driver/motor, Table 4-1 lists the experiment equipments [17-19].

Figure 4-1 PCC1620 and MSDA043A1A, MSMA042A1E

PC Intel Pentium Ⅱ 233 MHz , 128 MB RAM

Motion control card Pou Yuen Tech（PCC1620）

AC servo driver Panasonic（MSDA043A1A）

AC servo motor Panasonic（MSMA042A1E）

Table-4-1 The specification of experiment equipments

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4.2 Experiment Results

In the experiments, the interrupt service routine time of motion control is 3ms .

4.2.1 Using Blended S-Curve velocity profile

For the experiment, we use the line interpolation to generate a motion trajectory.

Different displacements and feedrates are set to test different conditions in motion.

Figure 4-2 shows the motion information and the parameters are shown below.

It corresponds to the Blended S-Curve condition (1.1) presented at last chapter.

Maximum jerk 0.02 pulse ms3

Maximum acceleration 0.5 pulse ms2

Feedrate 50 pulse ms

Displacement：： pulseX 40000,Y 30000

Figure 4-2 Line interpolation with Blended S-Curve condition (1.1)

51






Displacement X

：4800 ,Y：3600 pulse




52




Displacement

：400 ：300 pulse

X ,Y







Displacement

：8000

,Y：6000 pulse

X

53


Because the time constants of Blended S-Curve condition (2.2) is the same to

condition (1.3), Figure 4-4 also shows the motion information and its parameters are

shown below. It corresponds to the Blended S-Curve condition (2.2) presented at last

chapter.




Displacement

：400 ：300 pulse

X ,Y

54

4.2.2 Using Squared Sine velocity profile

For the experiment, we use the line interpolation to generate a motion trajectory.

Different displacements and feeerates are set to test different condition in motion.

Figure 4-6 shows the motion information and its parameters are shown below.

It corresponds to the Squared Sine condition (1.1) presented at last chapter.




Displacement：： pulseX 40000,Y 30000

Figure 4-6 Line interpolation with Squared Sine condition (1.1)

55






Displacement X

：4800 ,Y：3600 pulse




56




Displacement

：1200

,Y：900 pulse

X






Feedrate 18 pulse msDisplacement X ,Y

pulse：12000

：9000

57


Because the time constants of Squared Sine condition (2.2) is the same to

condition (1.3), Figure 4-8 also shows the motion information and its parameters are

shown below. It corresponds to the Squared Sine condition (2.2) presented at last

chapter.




Displacement

：1200

,Y：900 pulse

X

58

4.2.3 Continuous Motion with Squared Sine velocity profile

We take a rhombus as the experiment trajectory for continuous motion and it

shows in Figure 4-10.

Y-axis (0,120000)

(-80000,60000) (80000,60000)

(0,0) X-axis

Figure 4-10 Experiment trajectory for continuous motion

We decide maximum jerk is 0.02 pulse ms3 , maximum acceleration is 0.5

pulse ms 2 , and feedrate is 80 pulse ms . We decide overlapped time is 1

3 Ts , and the

experiment result is shown in Figure 4-11.

Figure 4-11 Information of continuous motion experiment59

The experiment trajectories are shown in Figure 4-12, 4-13, 4-14, 4-15.

Figure 4-12 Main motion trajectory (Experiment result)

Figure 4-13 Connection of motion block 1 and block 2 (Experiment result)

60



61

Chapter 5 Conclusion and Future Work

In this thesis we mainly impose acceleration/deceleration planning with jerk

limitation to generate a smooth motion trajectory which could avoid the saturation of

the motion control system effectively.

Using acceleration/deceleration before interpolation, Blended S-Curve velocity

profile provides the adjustable jerk and acceleration of motion, and in the middle

process of acceleration, linear velocity profile can obtain maximum efficiency of

motor. Besides, Squared Sine velocity profile has the continuity of jerk, it is more

advantageous to reduce vibration of machine and maintain the machining accuracy.

And it is convenient for planning the continuous motion under jerk constraint of

motion.

While the maximum acceleration and jerk already be decided, we calculate the

shortest motion time to form a smooth profile against different desired displacement

and feedrate both to two purposed velocity profile. As we calculate, the jerk time of

Squared Sine is approximately 1.57 times to the jerk time of Blended S-Curve. It is

more significant to develop an acceleration/deceleration algorithm that has continuous

jerk and its motion time can be shorter as possible.

In continuous motion, the purposed overlapped time 13 Ts is considered to the

extreme condition that the motion directions of two blocks are the same. Further we

can lengthen overlapped time by determining the motion direction of block. Such as

two connected blocks in opposite direction, we can overlap the complete jerk time and

it will not exceed desired constraint of jerk and acceleration.

The dynamic of system can be considered after adding load (ex. ball screw) to

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servo driver/motors. It is important to research the vibration and jerk induced from

real output of system.

63

Acc Decc Planning With Jerk Limitation Teza

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