Control of a filament stretching device

Rangelov, K.Z.

Published: 01/01/2004

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Citation for published version (APA):Rangelov, K. Z. (2004). Control of a filament stretching device. (DCT rapporten; Vol. 2004.109). Eindhoven:Technische Universiteit Eindhoven.

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Practical Assignment

Report number DCT 2004.109 MT 04.20

Kiril Rangelov ID number :0525624 k.z.rangelov@tue.nl or kiko~rang'@yahoo.co.uk

Mechatronic Design Group Eindhoven, September 2004

Eindhoven University of Technology

Table of Contents

Experimental Setup ............................................................................................................................................ 2

The ideal uniaxial extensional flow ........................................................................................ 2

The filament stretching device ........................................................................................................ 4

System Model ........................................................................................................................................................... 6

Parameters Identification .......................................................................................................................... 8

Encoder and position Motor Constant Identi

. . Friction-Velocity Map .............................................................................................................................. 10

Gravity compensation .................................................................................................................................. 13

Carriage mass calculation ...................................................................................................................... 13 . . Equivalent inertia identification .................................................................................................. 13

Frequency domain controller loop shaping .............................................................. 14

Trajectory tracing configurations ............................................................................................. 19

Independent tracing configuration .......................................................................................... 19

Master-Slave tracing configuration ....................................................................................... 19

Initialization of the syste 21

SIMULINK Implementation layout ..................................................................................... 22

Conclusions and Recommendations .................................................................................. 23

References ....................................................................................................................................................................... 23

.................................................................................................................................................................. APPENDIX 24

Introduction

One of the great difficulties in measuring the extensional viscosity of dilute polymer solutions is the design of the extensional rheometer. In the experimental measurements of extensional viscosity, the first important constraint is a good definition of the flow during the measurements. The difficulties are associated with the need to generate flows with well-defined extensional flow field, free of a substantial shear component. Secondly the measurement of the stress and the extension rate should be done correctiy.

The existing design of a filament stretching device is used. The device is upgraded with new servo amplifiers and a new control system 'Control Desk PC-board'. Due to that, a new online control and measurement of the system should be designed in dSPACE environment.

This report describes the process sf the controller design and needed identification of the tem elements parameters as carriage friction and inertia, motors constants, the gear lo of the transition and the relation position-encoder readings.

1. Experimental Setup

I .I. The ideal uniaxial extensional flow To simplify the derivation of the extensional viscosity in the filament stretching device (FSD), ideal flow kinematics are assumed. In this experiment the filament preserves a cylindrical shape between the two plates.

The velocity field for elongational flow is described in cylindrical coordinates (r,O,z) as follows:

Figure 1.1. Uniaxial flows conditions in the filament stretching device - Left section: Ideal flow; Right section: non-ideal flow.

where, vr the velocity in the radial direction, VQ the velocity in tangential direction and v, the velocity in axial direction respectively. Following assumptions are made: The elongation rate i is taken as a function of time and it is iniform in the cylindrical filament For this cylindrical filament with length L and radius R the elongation rate 8 can be determined in terms of L or R:

The strain is a measure for the ratio of the filament length L at time t to the initial length Lo and radius Ro at time to. The Hencky strain E is defined as integrate of the elongation rate from to to t:

For a constant rate i. the length of the filament increases exponentially with the time.

The lower plate velocity is obtained by differentiating L to t:

During the experiment non ideal flow conditions are present. Non idealities of the flow are located near the plates (see Fig.1 .l right section), therefore the experiments are analyzed at the midpoint (z =1/2L) of the filament where an ideal extension flow is assumed.

The length based extension rate Eq.(l.2a) called also applied rate is equal to the rate applied to the whole fiber and does not give the actual rate in the middle of the fiber.

The radius based extension rate provides better approximation of the actual extension rate. The fiber radms R(t) and rate of decrease can be measured at the midpoint (z(t)

=1/2L(t)). The actual extension rate circa, and the actual strain &real are defined as

The initial fiber radius measured at to is designated as Ro. For extensive explanation of the uniaxial extensional flow physics see [2].

1.2. The filament stretching device In this extensional apparatus the drop is held between two disks, upper and bottom one (see Figure (1.2). The bottom disk is attached to the Carriage 1 on the right column. The carriage is belt driven by the DC motor MI. The carriage can slide over a vertical guide bar fixed to the column. On the Carriage 2 belt driven by the DC motor M2 (left) an optical devises to measure the fiber diameter are attached.

Bottom 12' disk i wrtchel

Home

Figure 1.2. Schematic picture of the filament stretching device.

The carriage 1 is considered as a master and carriage 2 as a slave. After the droplet is clamed by the disks a waiting time of approximately 1 minute is required. The master performs a desired stretching velocity (position) profile and the slave follows the master with velocity V&,= ?4 Vmmter (half of the master position). In this manner the master- slave system fulfils the experiment requirements for non ideal flow conditions of measuring the actual extension rate suggested in Eq.(1.6).

The Balance via a cantilever construction measures the force on the fiber. The left hand of the !ever is attached to the ~ p p m disk a ~ d the other lever's e! to a contra we@t ~n the balance. The contra weight forces the lever arm (upper disk) to stay at firm position during the experiment and at the same time measures the force exerted by the fluid without disturbing the fiber.

The physical system under study is mass of the carriage attached to the belt and a DC Permanent Magnet Motor drives the belt. The motor is coupled with a planetary gear transmission. On the other side of the motor shaft, an incremental encoder is attached.

upper pulley. The belt is stretched

In thk configuration, the-total bearings friction in the rotating components as the motor shaft, the gear shaft, the pulleys, and the carriage rollers in junction with the weght of the moving carriage are loading the electrical motor. The friction force will be represented by two components, dry fiiction and viscous friction. Additional dynamical load is the equivalent inertia o f the system felt by the motor. The equivalent moment of inertia is formed by the contributions of the following inertias:

9 the rotor moment of inertia, 9 the moment of inertia of the rotating gear components, 9 the moment of inertia of the belt pulleys. P the moment of inertia of the carriage rollers. 9 the masses of the carriage and the belt.

Due to the dominant contribution of the carriage mass the system equivalent inertia cam % .A&

be considered mainly caused by that mass.

lor VbIM

,r Type Peak Constant constant Inertia I s t a l l 1 Stall I I I

- A m n n n -- Torque Current Current at 25OC I Table 1.1. Data of the Master and salve motors.

Matad-- "^" I Rated 1 Rated I Max. I Volt I Torque I Rotor

Motc Rated spew ~ u u u r p~ I I 1 (Nm) 1 (A) (A) I (Vlkrpm) 1 (NmlA) 1 (gm2)

DCM 2C 30103-A2 0.47 4.6 22 I 10 0.1 1 0.06

A servo amplifier operating in current mode supplies the DC motor. The servo amplifier actually translates the input voltage u to motor current. The motor torque is proportional to the current with the coefficient of proportionality the torque constant Kt m.rn/A]. The input signal of the servo amplifier is generated by a dSPACE system. The same control system reads the TTL incremental encoder and tacho signals, as well as the data from the laser measurements and the Balance. Additional signals from the home position switches and safety limit switches are fed to the system. The sampling frequency of the dSPACE is set to 1 [kHz]. A schematic 'layo~t of the described system is given in Figure 1.3.

Motor \ , Gear

Tacho

1 ' Master

I I Servo Amplifier & Power Supply

- I system I - Host PC

Slave

I I I I

and balance data

Slave Servo Amplifier &

Power Supply

Figure 1.3. Schematic layout of the experimental setup.

Slave

2. System Model We will assume that the dry and viscous frictions on the motor side are negligible if compared with the respective friction components associated with the carriage said. To distinguish the components which belong to the carriage side we will denote them with the subscript bel t .

Due to the fact that the motor is providing the torque to drive the system we are interest in the eqiiivalent forces and inertia felt by the motor. Fclmu!as to transform. the respective forces and velocities on to the carriage (belt) side are also provided.

The system under consideration at the motor side during stretching can be modeled as

where: J the equivalent inertia of the motor-tra~smitisn-casriage combination felt

by the motor 0 the angular displacement of the motor shaft. ci, the motor angular velocity. T,,, the output torque of the motor shaft. Tfiiction the resistive friction torque. Tweight the torque generated by the weight of the carriage, positive in the

stretching direction.

Figure

Upwards

2.1. The forces and torque directions during upwards and I

constant velocity.

Downwards

downwards motion wi

where Km is the motor constant [N.m/V] and is,, the gear transmission ratio. If fact the Km is formed by the product of the motor torque constant Kt [N.mlA] and the constant of the servo amplifier KSe,, [AN]. 'in order to hi the ni~ior i o~q ie aiid ~iigda velocity to the respective fmce 2cd longitudinal velocity exerted from the belt on the carriage following relations are used:

One of the possible ways to describe the resistive friction torque as combination of dry friction Fc and viscous fiiction b coefficients is presented below.

Coulomb + Viscous Friction

-10 -8 -6 -4 -2 0 2 4 6 8 10 Angular Velocity Thitadot [rads]

Figure 2.2. Coulomb-Viscous Friction Model.

The presented model is one of the simplest. For more precise description of the friction phenomenon a combination of the coulomb, viscous, static and Stribeck fnction is used. For more details see [I] and [3]. The system dynamics described by Eq.(2.1) now can be rewritten as

3. Parameters Identification In order to identify the friction model parameters and unknown setup parameters a series of experiments are planed and performed on the real system.

3.1. Encoder and position identification Due to the absent of information of the encoder it was necessary to find the relation between number of pulses per rotor revolution and the respective longitudinal displacement. The following experiment supplied with the needed data.

The cover of the encoder was removed and a special position marker was aligned to the distinct mark on the encoder disk (the encoder disk was never touched by any object or hand!). The two marks where aligned with respect to each other. The displayed encoder reading was assigned for initial. Four rotations of the rotor in one direction where performed. The motion is initiated by hand without any reversing of the rotation direction! On each complete turn the value of the encoder readings are recorded. Simultaneously the longitudinal displacement of the carriage for the same turns is measured with the precise caliper with respect to the special marker on the guide bar. The processed measurement data and computed values are given in Table 3.1. The values of the known system components are *resented in ~able3.2.

22.25 22.40

500 22.35 22.35

average 500 average 22.35

1 0.04474 [mmlencoder line] 22.35 [encoder lineslmm] 22.35 [mmlrotor turn] RDuuev = 23.69*10J [m]

3.2. Motor Constant Identification

Table 3.2. Encoder, Tacho and gear data.

A servo amplifier in current mode supplies the DC motor. The dSPACE commands a control voltage of the amplifier. In the loop dSPACE-amplifier-motor the commanded voltage is translated to motor torque. We assume a linear relation between the motor torque and the motor current with coefficient of proportionality the toque constant Kt [N.mnT]. We also assume proportionality between the servo ampiifier input voitage and the output current with coefficient the servo-gain K,,, [AIV].

Tacho

Ktacho= 10V/1000 rpm

In order to identify the motor constant K, [N.mN] for the given set of power supply- amplifier-motor the following experiment is set up. The motor shaft is locked and an Amper-meter is connected in series with the amplifier and the motor. Various voltage

Reduction Gear ratio

iWr= 6,66 : 1

levels are commanded by the dSPACE and the respective measured current values are recorded and averaged. In this manner the generated still torque (current times the Kt) is mapped to the commanded voltage. The collected experimental data are presented in Table 3.3.

As a result we adopt the same motor constant for the master and the slave motors K,= 0.5157 m.m/V].

T&he 3.3 Ide~tificatior, of the servo constmt, experimental data. [ dSPACE / Motor 1 Ks,, 1

K,,, average value 5.1572 1

3.3. Friction-Velocity Map After close inspection of Eq.(2.6) we can conclude that, at the given constant angular speed ( 03 = 0) the motor generates the amount of torque needed to overcome the resistive fiction forces and the torque caused by the weight of the carriage.

Eq.(3.1) reveals that if we can measure the drive torque generated by the motor at various pairs of constant velocities (upwards and downwards), we are able to reconstruct fiiction- velocity map and thus separate and identify the viscous and coulomb fiction components, and the equivalent mass felt by the motor.

u(t). K,,, = T,,,,,, - F, - sign(w(t)) - b - w(t) for w = const

h=O (3.1)

During the downward motion the motor torque reads

When the motor performs upward motion the torque is

u, ( t ) Km = +Tweight + - s i g n ( u ( t ) ) + b . ~ ( t )

By subtracting Eq.(2.8) and Eq.(2.9) we obtain the friction components

And by summing the same equations we obtain Twight.

('down ( t ) + ' u p ( t)) ' Km = ' Twe&t (3.5)

nonlinear friction pheno almost impossible to maintain wide range of constant angular velocities in open-loop system. To insure constant angular velocities a close-loop experiment is employed. The PD velocity controller is implemented. The values of the proportional and derivative gains are Kp =10 and I(d = 0.04 respectively. The derivative is filtered by a low-pass fitter with cut-off frequency of 200 [Hz]. The desired velocity in rad/s is set and the controller output voltage is recorded and averaged. The controller voltage is assumed proportional to the toque generated by the motor. The collected experimental data for the master and slave carriages are presented in Table 3.4. Graphical representation of the experimental data is shown in Figure 3.1.

Table 3.4. Friction-velocitv maD data for both the master and the slave. . A

Identified friction parameters b = 7*1oW4 [N.m.s/radl. F, = 0.1014 TN.ml

Master

I

Identified friction parameters b = 5* 1 o4 [N.m.s/radl, F, = 0.1 372 rr\r.ml

Slave Twaght

[Nm] 0.16361

W

[radls] 0.01

Tfrietron

[Nm] 0.11338

W

[radls] 0.01

Tmotor up

[Nm] 0.44688

Tmotor down

[Nm] 0.05297

Tmotor down

[Nm] 0.05023

Tfrlcnon

[Nm] 0.19696

Tmotor up

[Nm] 0.27698

Twezght

[Nm] 0.24993

Master and Slave friction-velocity maps, expeimental data 0.45

0 20 40 60 80 100 motor shaft velocity [radls]

Figure 3.1. Friction-velocity maps for the angular velocity range 0.01+100 [rads].

Close inspection of Figure 3.1 reveals pre-sliding region at low velocities 0.01tl [radls], and sliding region at velocity range ls100 [radls]

Figure 3.2. Friction-velocity maps and fitted curves for the master and the slave for the angular

Masterfrictiowvdodty map, expeimRltal and fitted curves

velocity range 1+100 [rads].

Slavefriction-velodty map, expeimental and fitted wrves

The identified friction parameters will be used for friction compensation to improve the controller performance.

3.4. Gravity compensation In order to free the controller of the duty to compensate for the Tweight a separate constant value can be added to the control voltage. The value of that constant voltage to be feed forward can be estimated from the already identified parameters as follows.

- weight .master ' weight .master - = 0.3335 ; K m

- weight .slave 'weight slave - = 0.4905 ;

K m

3.5. Carriage mass calculation The toque Tweight felt by the motor can be easily transformed to a belt force if Eq.(2.3) applied. As it shown in Figure 2.1 the Fweigh.belt equals the marriage times the gravitational constant. The mass of the cakiage can be found as

- Tweight.slave . igear mcarriage.slave - = 7.25 [kg]

Rpulley . g

3.6. Equivalent inertia identification After we have calculated the carriage mass it is possible to identify the equivalent inertia felt by the motor. We can obtain the-value of the load equivalent inertia as

When the respective values are substituted in Eq.(3.10) following equivalent inertias are calculated:

The experiment data of the estimated equivalent inertia in both rotational directions are given in Table 4.3. value assliined io represent the iiisrtia of the meter retor arid the other associated with it rotating components is JmofoY = 8.7039*10-~ [kg.m2]. The total inertia of all components in the master loop, respective in the slave loop is:

4. Frequency domain controller loop shaping The controller design objectives are: obtaining stable close-loop system whit an as higher as possible bandqvidth, with 30 [deg] phase margin and 6 [dB] gain margin. To analyze the close-loop and its stability margins we need information on the linear dynamics of the system under study. We will obtain the Frequency Response Function (FRF) of the system by measuring the sensitivity FRF Sow) of the stable close-loop system with a known controller Cli,). The system is excited by the band-limited white noise signal Was is it shown in Figure 4.1 and by measuring the signal Uwe can write for the sensitivity

Figure 4.1. Measurement of the system sensitivity FRF and controller FRF identification.

The frequency response of the system is reconstructed by substituting Eq(4.1) in Eq.(4.2).

To avoid the uncertainty due tg,,the transform of the continuous controller in discrete implementation a separate standalone replica of the controller is excited by the same W signal and the controller output signal Upd is recorded and afterwards used in FRF reconstruction instead of the numerical derived controller C(,) in Eq.(4.2).

The system sensitivity measurements data are collected when the carriage is commanded to move with constant velocity and the level of the added white nose is adjusted to avoid zero velocity during the experiment. The measure master and slave systems FRF are presented in Figure 4.2, Figure 4.3, and Figure 4.4, Figure4.5 respectively.

Figure 4.2. Sensitivity FRF and Frequency Response Function of the master system.

Figure 4.3. Controller FRF and open-loop FRF of the master system. , , % -

Figure 4.4. Sensitivity FRF and Frequency Response Function of the slave system.

Figure 4.5. Controller FRF and open-loop FRF of the slave system.

The observed anti-resonance resonance pattern in the plant FWF is due to the belt connecting the carriage and"t.e drive:

Several attempts to design a proper controller were made. A combination of LeadLag, low-pass and Notch were tried. The controllers' design was done in "Do It Easy Toll" toolbox for Mathlab. After the implementation of the designed controllers of the Master and Slave in the "Control Desk", the system appears to be unstable and starts to oscillate in up or down movement.

To further investigate the reason for that system behavior measurements of the FRF were made at various velocities and locations of the carriages in upwards and downwards motion. The analysis of the data revealed that the frequency of the resonance picks depends on the direction of the carriage motion and at the same time depends on the carriage position on the guide bar. The following four figures illustrate that phenomenon.. : . .

Figure 4.6 Master FRF at the upper half of the bar, left - downward motion; right - upward motion;

Figure 4.8 Slave FRF at the upper half of the bar, left - downward motion; right - upward motion;

Figure 4.9 Slave FRF at the lower half of the bar, left - downward motion; right - upward motion;

The final decision was made to apply a PID controller with mass and friction feedforward. To add extra phase a leadllag was added to the controller. The controller feedforward parameters were tuned online by applying a third order profile of fore and aft potion. When the controller was tested with the exponent third order profile function in fore direction (downwards motion) error different parameters were obtained. Doe to the desired exponential motion profile the last settings values were set as controller's settings. The adopted parameter values are given in Table 4.1 and the system performance is shown in Figure 4.10. The measmements cf the fiber will be made dur i~g the extefision region. The tracing emrs &iring extensim (zcce!eraticlr) do not exceed + 50 pm, and the errors increase in the deceleration.

Table 4.1 Controller settings

Figure 4.10. System performance at exponential motion profile with controller parameters given in Table 4.1.

KP Kn

Master 10.02 0.04

Slave 13.34 0.08

5. Trajectory tracing configurations The tracing of the prescribed trajectory by the master and the slave carriages was made in two different configurations namely independent and master-slave .

5.1. Independent tracing configuration h the independent configuration the master md slave carriages are tracing the prescribed trajectory iadepende~t from each ~ther. The layxt ef the implemented control is shovm in Figure 5.1. In this configuration the deviations of the master and slave position with respect to the followed path will cause measurement of the fiber diameter not always at the middle point during the stretching.

Figure 5.1. The layout of the implemented Independent tracing configuration.

5.2. Master-Slave tracing configuration In contrast with the independent tracing the Master-Slave configuration links the slave carriage with the master carriage. In that configuration (See Figure 5.2) the Slave (the slave carriage) uses as a reference the position trajectory of the Master (the master carriage) instead of the prescribed trajectory. The benefit of such configuration is that the Slave will follow the Master path despite of the Maser error and measurements of the fiber diameter will take place in the midpoint length during entire stretching. To verify that a validation experiment is made by increasing the Master error with respect to the prescribed trajectory (same trajectory as in Figure 4.10) and observing the Slave error with respect to the Master path. The results of the experiment are presented in Figure 5.3.

The Master-Slave configuration requires additional changes in the initialization procedure used by the Independent configuration. In details the initialization procedure is discussed in section 6.

Figure 5.2. The layout of the implemented Master-Slave tracing configuration.

6. Initialization of the system Of great importance is the correct initialization of the master slave system in the first moment after the power is turned on. The encoders of the master and slave carriages are initialized with the respect value which corresponds to a proximally their current position in rest. This initialization is not accurate.

Doe to the gravity and the large mass the both carriages are located at the end of the guide-bars. For the measure of safety on both ends of each guide-bar a magnetic NC contactless limit-switches are fitted (see Figure 1.2). The four switches are connected in series. When the magnet mounted on the carriage is in the range of one of the four switches the power supply of the motors is shutdown and need to be turned on manually. The carriages ones passed the switches continue their motion due to the inertial forces or gravity until stop by hitting the end shock absorbers (one up and one down for each bar). To prevent carriage to reach the limit-switches under the gravity force a support made of soft material was added. The soft supports under the carriages provide them with the

the r~zpport deformation the kitial position is not accurate.

guide bars before the upper limit-switches so called home sensors are firmly attached to the bar. The home sensors are of the same type as the limit- switches and are triggered by different magnets mount on the carriages. The zero position of the system is considered with the two carriages positioned at the upper end of the guide bars. The position coordinates of the home sensors with respect to the zero position is known. The coordinates of the home sensors can be used to correct the actual carriage position.

To improve the accuracy of the position initialization following procedure is applied. The carriages approximate initial coordinates are loaded in the encoders. The carriages are command to move upwards with low constant velocity. When a carriage approaches its home sensor the sensor switch is triggered (switch is open) and sends a signal. This sensor signal activates index search of the respective encoder, and when the index is found the coordinate of the home sensor is assigned as position of the encoder. This procedure is active only at the very first time a carriage passes trough its home sensor.

The respective values of the used encoder positions for the current setup configuration are given in Table 6.1. If the configuration is changed the respective coordinate values must be corrected!

Table 6.1. Coordinate values used in the current setup for the encoder initialization. 1 Initial position I Position when home sensor and 1

I Slave I 19445 I 5200 I Master

Master-Slave tracing configuration requires modification of the described procedure. The modification consists of switching the Slave position input from the trajectory position

[encoder lines] 34643

Index found [encoder lines] 1000

(see Figure 5.1) to the Master encoder position readings (see Figure 5.2) after the initialization

7. SlMULlNK Implementation layout The layout of the entire system build in mathlab-SIMULINK is presented in Figure 7.1. This layout is used to create a dSPACE real time control implementation. In the real time implementation the operator has the following options:

9 Start the Initialization procedure (Indicators for accomplished master and slave initialization).

9 Manual position of the Master-Slave system (maser and slave carriage position displays (with respect to the zero position) [m], and error displays in [m]).

9 Access to the settings of the exponential third order Trajectory generator. 9 Possibility to import third order motion profiles (data for the acceleration, velocity

and position as a matlab workspace variables) k To observe the plots of carriages positions, fiber diameter, applied strain, actual

strain, and the fiber force.

8. Conclusions and Recommendations The concept of the Master-Slave tracing coniigwation would be preferred instead of Independent configuration (see section 5).

For fbrther improvement of the racing accwacy of the system a better controller may be designed and implemented. The resonant frequencies of the master and slave carriages might be influenced by adjusting the respective belt tension. The change in the belt tension most probably will result in different friction parameters.

More advanced friction compensation can also contribute to the system accwacy.

A better exponential trajectory generator will additionally reduce the tracing error especially in the deceleration section of the motion profile.

9. References

[I] Hensen, Ronnie Controlled Mechanical Systems with Friction, PhD. Thesis, Eindhoven University of Technology, 2000, ISBN 90-3 86-2693-2

[2] Macosko, Christopher Rheology: principles, measurements, and applications, VCH Publishers, Inc. 1994, ISBN 1-5608 1-579-5

[3] Putra, Devi Control of Limit Cycling in Frictional Mechanical Systems, PhD. Thesis, Eindhoven University of Technology, 2004, ISBN 90-386-2636-3

10. APPENDIX

10.1. Exponential third order trajectory generator The design of the path generator is based on the Eq. 1.4 and Eq. 1.5. The operator sets the experiment values for the desired strain rate 2 , the initial length Lo [m] and radius Ro [m] of the filament (fiber) and the trajectory generatof is ready to output the path profile. The generztor nezds also the setting for the !irnitztior,s of the red s e tq as:

> maximum possible stretching length- position limit > maximum possible acceleration- acceleration limit > maximum possible velocity- velocity limit > maximum possible deceleration- deceleration

When one of the specified limits values is reached (except of the deceleration) the enerator decelerates and

Figure 10.1. Layout of the exponential third order trajectory generator.