Information
-
Patent Grant
-
6430466
-
Patent Number
6,430,466
-
Date Filed
Monday, August 23, 199925 years ago
-
Date Issued
Tuesday, August 6, 200222 years ago
-
Inventors
-
Original Assignees
-
Examiners
Agents
- Thompson; John F.
- Breedlove; Jill M.
-
CPC
-
US Classifications
Field of Search
US
- 700 197
- 700 200
- 700 32
- 700 33
- 700 37
- 700 73
- 700 8
- 700 44
- 700 45
- 264 403
- 264 405
- 425 145
- 425 146
- 425 149
-
International Classifications
-
Abstract
The present invention provides a apparatus and method for controlling the force and velocity of a piston within the clamp assembly of a injection molding machine. The control unit comprises the following elements: a PI based cascade compensator; an estimator coupled to the clamp assembly and the cascade compensator, wherein the estimator is utilized to estimate the position of the clamp assembly piston; a velocity position controller coupled to the clamp assembly and the estimator, wherein the velocity position controller is adapted to generate a velocity error signal based on the piston velocity signal; and an auto selector coupled to the cascade compensator, wherein the auto selector is adapted to select between the velocity error signal and the force error signal to generate a valve command signal. The present invention additionally employs a method for determining the velocity profile of the piston of a clamp assembly. The method comprises the following steps: determining the maximum displacement xs of the servo valve; selecting a final velocity vs of the piston; choosing a ratio α of the piston ramp-down interval to the servo valve displacement “x” during a ramp-up interval, selecting a desired maximum piston velocity (vmax) during the ramp-up interval and a constant speed interval; and determining servo valve displacement xu from simulations or estimate from previous cycles.
Description
BACKGROUND OF THE INVENTION
The present invention relates to the regulation of a clamp assembly in a injection molding machine and, more particularly, to a closed loop system for controlling the pressure of a clamp assembly to optimize production efficiency in an injection molding machine.
Injection molding machines are used to mass produce plastic parts by automated injection of plastic into a mold that is opened and closed by a clamp assembly. At the end of an injection molding cycle, the clamp assembly is retracted, and the part is ejected. The next cycle is initiated by motion of the clamp from its retracted position to a position where the mold is nearly closed. During the cycle the mold machine should not be subjected to excessive pressure and forces generated by parts left behind in the mold, or incorrect positioning of the mold. In the event that parts are left behind nor incorrectly positioned in the mold the machine must be stopped and the piece removed. This causes production delay which impacts production efficiency.
More particularly, a typical injection molding process utilizes a injection molding machine
10
, illustrated in
FIG. 1
, wherein plastic pellets (not shown) are melted and forced into a mold
20
, by a clamp assembly
113
that opens and closes mold
20
during the mold injection cycle.
The clamp assembly
113
comprises a hydraulic cylinder which is mechanically coupled to a piston
27
. Clamp assembly
113
further comprises a stationary platen
13
which is mechanically affixed to a mold
20
, and a moving platen
12
, which is coupled to piston
27
. Piston
27
is adapted to hydraulically traverse from a clamp closed position to a clamp open position in an injection molding cycle by variable force applied to piston
27
. A hydraulic cylinder
11
is movably coupled to piston
27
. The variable force applied to piston
27
by the fluid in hydraulic cylinder
11
may be controlled by a control unit
28
.
The injection molding process comprises four successive stages. The first stage called “plastication” comprises steps wherein the plastic pellets are pushed forward from a hopper
17
through a barrel
21
towards a nozzle
14
by a rotating screw
22
while being heated in barrel
21
by electric heater bands
23
surrounding barrel
21
. The second stage called “injection,” occurs when the plastic is pushed through nozzle
14
into a mold
20
by clamp assembly
113
. The third stage called “packing,” occurs when mold
20
is packed with the molten plastic. The fourth stage called “cooling,” occurs when mold
20
is cooled to solidify the plastic part therein. After the completion of the solidification and cooling stages, piston
27
is retracted and the part is ejected. The injection molding machine
10
is thus ready for the process to be repeated in the next cycle.
It is desirable to protect the injection mold machine
10
during motion of piston
27
by “smoothing” the response of the clamp assembly cycle during clamp closure of piston
27
so as to prevent damage to the machine from plastic parts, remaining in the mold after having been jammed in mold
20
during the previous cycle. It is also desirable to control the stroke of piston
27
during the injection molding cycle so as to minimize the duration of the clamp cycle.
BRIEF SUMMARY OF THE INVENTION
The present invention provides an apparatus and method for controlling the force and velocity of a piston within the clamp assembly of a injection molding machine. The control unit comprises the following elements: a cascade compensator, wherein the cascade compensator is adapted to generate a force error signal having lead and lag compensation based on the desired force minus the actual force of the piston; an estimator coupled to the clamp assembly, wherein the estimator is utilized to estimate the position of the servo valve and to generate a corresponding actual force signal; a velocity and position controller coupled to the clamp assembly and the estimator, wherein the velocity and position controller is adapted to generate a velocity error signal based on the piston velocity signal; and an auto selector coupled to the cascade compensator, wherein the auto selector is adapted to select between the velocity error signal and the force error signal so as to generate the valve position command signal.
The present invention additionally employs a method for determining the velocity profile of the piston of a clamp assembly, wherein the velocity profile having a ramp-up interval, a constant speed interval, ramp-down interval; and a final touchdown interval. The method comprises the following steps: determining the maximum displacement x
f
of the servo valve; selecting a final touch down velocity v
s
of the piston; choosing a ratio α of the piston ramp-down interval to the servo valve displacement “x
v
” during the ramp-up interval, selecting a desired maximum piston velocity (v
max
) during the ramp-up interval and constant speed interval; and determining servo valve displacement x
u
from simulations or estimate from previous cycles.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1
is a prior art illustration of an injection molding machine;
FIG. 2
is a functional block diagram of the piston portion of the clamp assembly of
FIG. 1
;
FIG. 3
is a functional block diagram of the clamp assembly and control unit of the present invention;
FIG. 4
is a schematic block diagram of a typical velocity position controller;
FIG. 5
is a graphical illustration of the piston velocity profile of the velocity position controller of
FIG. 8
;
FIGS. 6A and 6B
are process flow block diagrams of the velocity position controller of
FIG. 5
;
FIG. 7A
is a graphical illustration of a piston velocity and servo valve position profile of the present invention which employs a linear response to the servo valve error signal;
FIG. 7B
is a graphical illustration of a piston velocity and servo valve position profile of the present invention which employs a quadratic response to the servo valve error signal; and
FIG. 8
is a schematic block diagram of the velocity position controller of the present invention.
DETAILED DESCRIPTION OF THE INVENTION
Referring now to
FIGS. 1 and 2
, in which like elements have like reference numbers, a prior art illustration of an injection molding machine
10
is shown. Control unit
28
generates a servo valve position command signal
122
that acts to control the velocity of piston
27
enabling piston to traverse from a clamp open position to a clamp closed position. More particularly, piston assembly
50
comprises servo valve
52
which is actuated by control unit
28
to cause servo valve
52
to be displaced a distance “x
v
” from a null position
53
during normal operation. This displacement regulates the net pressure (P
2
inlet pressure minus P
1
return pressure) of hydraulic fluid on piston
27
so as to cause piston
27
to traverse from the clamp open position to the clamp closed position at a desired velocity.
The present invention is employed to estimate and regulate the force of clamp assembly
113
on the mold/plastic part as it is forged during the clamp cycle. The force exerted by piston
27
on mold
20
may be estimated by equation (1) listed below:
F
=(
P
3
A
2
−P
4
A
4
)−
Ma
equation 1
where “F” is the force exerted by piston 27 on the mold, P
3
is the hydraulic pressure on a first surface of piston
27
, P
4
is the hydraulic pressure on a second surface of piston
27
, A
2
is the first surface area of piston
27
, A
4
is the second surface area of piston
27
, “M” is the mass of mold,
20
and “a” is the acceleration of piston
27
.
Generally, the hydraulic fluid flow rate “Q” is related to the difference in pressure on both sides of servo valve
52
and the area of the servo valve opening according to the general mathematical relationship illustrated in equation 2
where C
d
is a fluid pressure coefficient, “ρ” is the fluid density, “A” is the area of the orifice through with the fluid passes, and ΔP is the pressure differential across the orifice. Additionally, the compressibility effect in the hydraulic fluid is defined by equation 3 illustrated below,
where “V” is the volume of the hydraulic fluid, β is the bulk modulus, and dP/dt is the derivative of the pressure with respect to time. From these three equations the following first order, non-linear differential equations relating to pressures, velocity and valve opening were derived,
F=P
3
A
2
−P
4
A
4
−Ma
equation 4
where “F” is force exerted by piston
27
on mold
20
and “a” is the acceleration of the piston.
where dp
3
/dt is the derivative of pressure on the first surface A
2
of piston
27
with respect to time, and Q
4
is the flow rate of the hydraulic fluid into the left chamber, V
1
is the initial volume of the fluid in the left chamber, “x” is displacement of the piston from initial position and “v” is the velocity of the piston
where dp
4
/dt is the derivative of the pressure on the second surface A
4
of piston
27
with respect to time, Q
2
is the flow rate of the hydraulic fluid returning from piston
27
to servo valve
52
and V
2
is the initial volume of the fluid in contact with area A
4
. Simulations were conducted using equations 4-6 having the following limitations. The maximum velocity of piston
27
did not exceed about 2,000 millimeters/second. The maximum displacement “x
v
” of servo valve
52
was not be greater than about 650 millimeters. The above limitations were employed using a mold mass “M” having values in a range from about 1800 kg to about 5400 kg.
From the simulations performed using equations 2 through 7, control algorithms were developed to regulate the force exerted by piston
27
on mold
20
and the stroke duration of piston
27
, as further described below.
Referring now to
FIG. 3
, which illustrates a functional block diagram of the clamp assembly controller
100
and to
FIGS. 4 and 8
, showing further detail of a velocity position controller
115
, in which like elements have like reference numbers.
Clamp assembly controller
100
, comprises desired force limit
118
, summer
117
, cascade compensator
111
, auto selector
112
, clamp assembly
113
, estimator
114
, and velocity position controller
115
. Clamp assembly controller
100
optionally may further comprise feedback compensator
116
.
Desired force limit “F”
118
provides a maximum force “F” that piston
27
may generate. This force limit is established by the maximum force that the piston assembly
50
may generate so as not to cause excessive force on mold
20
. The maximum force “F” is selectable and depends on the size of the machine and mold.
Force summer
117
generates a difference signal, referred to as the force difference signal, between the desired force “F” and actual force signal
121
produced by clamp assembly
113
. Alternatively, force summer
117
generates force difference signal that is the difference between the desired force signal “F” and the force feedback signal generated by feedback compensator
116
.
Cascade comparator
111
is a lead/lag controller, having proportional and integral gains, that provides lead and lag compensation to the force difference signal generated by force summer
117
. The proportional and integral gains values are chosen to provide damped response without oscillations, within a specified bandwidth. The gains are selected through conventional compensation techniques and simulations.
Velocity and position controller
115
generates a piston velocity signal having a piston velocity profile, illustrated in
FIG. 5
, in correspondence with the servo valve displacement “x
v
” as represented by piston velocity signal
120
. Piston velocity profile
200
(
FIG. 5
) is a graphical illustration of an exemplary piston velocity waveform
210
. A maximum velocity (v
max
) is the maximum mechanical velocity of piston
27
. A final velocity (v
f
) is the final estimated velocity of piston
27
. A ramp up section
212
, having a servo valve displacement interval extending from x
v
=0 to x
v
=x
u
, is defined as the interval where the piston velocity increases from about zero to about 99 percent of v
max
. A constant speed interval
214
, having a servo valve displacement interval extending from x
u
to x
d
, is the phase where the piston velocity is substantially constant, typically about 99% of v
max
. A ramp down interval
216
, having a servo valve displacement interval extending from x
d
to x
s
, is the phase in which the piston velocity is reduced to v
f
, typically much less than v
max
. In one exemplary embodiment v
max
may be about 2000 meters/second and v
f
may be about 40 meters/second. A touchdown interval
218
, having a servo valve displacement interval extending from x
s
to x
f
, is the phase where the piston velocity is constant at about v
f
. Each of the intervals x
u
, x
d
, and x
s
are selected based on the velocity and position control process described below.
Auto selector
112
selects a force control function
123
or a velocity control function
124
to control clamp assembly
113
. The force control function
123
is represented by the signal generated by cascade compensator
111
and the velocity control function
124
is represented by the output of the velocity and position controller
115
(e). Auto selector
112
selects the minimum of the two control functions as valve command
122
. Servo valve command signal
122
is coupled to clamp assembly
113
.
Clamp assembly
113
operates as described above to open and close mold
20
. Clamp assembly
113
generates a piston velocity signal
120
in correspondence with displacement “x
v
” of the servo valve
52
.
Estimator
114
operates to measure the actual force generated by piston
27
. Alternatively, estimator
114
operates to estimate the force of piston
27
utilizing equation 4, where the acceleration of piston
27
is calculated using piston velocity signal
120
(based on approximate differentiation), P
3
, and P
4
. If piston velocity is not directly measured, it can be estimated from the piston position signal. Estimator
114
generates actual force signal
121
in correspondence with the measured or calculated force of piston
27
.
In another alternative embodiment of the present invention, feedback compensator
116
may be employed in control
100
to provide lead/lag compensation to actual force signal
121
. The compensator may be designed to cancel sensor lags and add damping to the closed loop system.
Desired final position
119
provides a final position x
f
set point for piston
27
. The final position x
f
is determined based on the desired position for clamp closing. Summer
125
generates a position error signal, referred to as the position error signal (e), which is the difference between the desired final position and the actual position generated by the clamp assembly
113
. Desired velocity schedule
154
is a schedule of position error signal (e) versus the desired piston velocity, as illustrated in
FIGS. 7A and 7B
. Desired velocity schedule
154
generates a desired velocity signal (v). Between zero and small error e
f
, i.e., near final position, the desired velocity is the touch down velocity, v
f
. The final velocity is established by the maximum necessary velocity of piston
27
in touchdown interval
218
(FIG.
5
). This final velocity must be low enough to provide safe mold closing and high enough to minimize the cycle time of clamp assembly
113
. In an exemplary embodiment the final velocity is in a range from about 10 to about 50 meters/second. At very high position error, i.e., near the initial position, the desired velocity is the maximum velocity, v
max
.
In between v
max
and v
f
, the profile of desired velocity verses error depends on the designer's choice. Here, consideration is given to two choices: 1) constant deceleration (dt) with respect to time, and 2) constant deceleration (d
x
) with respect to position. If the deceleration is constant with respect to position and is equal to d
x
, then v=v
f
+d
x
(e−e
f
) from e=e
f
onwards, until “v” becomes v
max
. If the deceleration is constant with respect to time and is equal to dt, then v={square root over (v
f
2
+2+L d
t
+L (e−e
f
+L ))} from e=e
f
onwards, until “v” becomes v
max
.
The profile of velocity schedule
154
is selected to minimize the cycle time of clamp assembly
113
while at the same time insuring system stability.
Servo valve displacement look-up table
156
may be employed to provide a valve drive signal which corresponds with the desired velocity, as output by
154
. Servo valve limiter
158
limits the valve command to reasonable, physical limits—these may be the same as saturation limits for servo valve
52
.
FIG. 8
provides further detail of an alternative velocity and position controller
115
of the present invention, illustrated in FIG.
3
. The present position and velocity controller
115
is adapted to employ a closed loop around the piston force, velocity, and position. Velocity and position controller
115
, described in
FIG. 4
, further comprises a lead/lag controller
157
which is coupled to limiter
158
. Clamp assembly
113
generates an actual force signal
159
, a piston position signal
120
, and a piston velocity signal
161
. The force signal
159
is coupled to controller
157
. Piston position signal
120
is coupled to a sample and hold block
152
and the piston velocity signal
161
is coupled to a summer
126
via a sample & hold
152
. Piston and velocity controller
115
is adapted to control the piston position based on piston force, velocity, and position. Each component operates as described above to regulate piston force, velocity, and position to limit the piston force and the maximize the cycle time of the piston.
The following steps, illustrated in
FIGS. 6A and 6B
, identify a process
220
that may be performed by a computer or circuit to determine the velocity profile of piston
27
, identified in
FIGS. 6A and 6B
.
Step
222
. Determine the total servo valve displacement x
f
from the initial to the final position x
v
=0 to x
v
=x
f
when mold
20
is closing.
Step
224
. Select a servo valve displacement (x
s
) for mold closing position for the beginning of touchdown phase x
s
at v
f
. The closer x
s
is to x
f
the less the cycle time. For example, x
s
has a value that is about 99 percent of x
f
.
Step
226
. Choose a ratio α of the servo valve displacement “x
v
” during ramp-down interval
216
to the servo valve displacement “x
v
” during ramp-up interval
212
, according to equation 7 illustrated below.
x
u
is the displacement reached by the piston, by the time “v” reaches about 99% of v
max
. x
d
is the displacement where the ramp down phase begins. Note that x
d
>=x
u
. A numerical value for α of about 1.0 or slightly higher/lower may be chosen. Values of α much less than 1.0 should not be selected because they result in mold injection system instability during a rapid ramp down phase. Very high values of α lead to larger cycle time.
Step
228
. Select a piston maximum velocity (v
max
) during the ramp-up interval
212
and constant speed interval
214
. The desired piston velocity depends on the machine and mold size.
Step
230
. Determine x
u
from simulations or estimate from previous cycles. It is noted that x
u
is approximately proportional to mold mass “M”. X
u
can also be estimated from previous cycles. X
u
should be chosen according to the mathematical relationship
so that x
d
is greater than or equal to x
u
. In the event that x
u
does not satisfy equation 8, v
max
may be reduced to satisfy the relationship between x
u
and x
s
in equation 8.
Step
236
. Begin ramp up phase with v
max
.
Step
232
. If
choose x
d
=x
s
−αx
u
. If
choose x
d
=x
u
.
Step
233
. If at
v<99% of v
max
, it implies that x
u
>x
s
/(α+1). Then, change v
max
, x
u
, and x
d
as follows:
v
max
=v at x=x
s
/(α+1), x
u
=x
s
/(α+1), and set x
d
=x
u
.
Step
234
. Select the piston deceleration profile during ramp-down interval
218
according to a linear profile as further described below. For the linear deceleration profile the slope (dx) is estimated, according to equation 9, based on linear trajectory from x
d
to x
s
. The linear deceleration profile is illustrated in FIG.
7
A.
Alternatively, a quadratic profile is a more accurate estimate of the deceleration profile of piston
27
. When the quadratic deceleration profile is utilized the profile is estimated according to equation 10. The quadratic deceleration profile is illustrated in FIG.
7
B.
It should be noted that steps
222
through
230
may be executed on the “fly”, that is, during a mold machine cycle after piston
27
has complete the ramp-up interval
212
of it trajectory or off line.
It will be apparent to those skilled in the art that, while the invention has been illustrated and described herein in accordance with the patent statutes, modifications and changes may be made in the disclosed embodiments without departing from the true spirit and scope of the invention. It is, therefore, to be understood that the appended claims are intended to cover all such modifications and changes as fall within the true spirit of the invention.
Claims
- 1. A control unit for controlling the force and velocity of a piston on a mold within the clamp assembly of a injection molding machine, based on a servo valve command signal which acts to operate a servo valve, said control unit comprising:a cascade compensator, wherein said cascade compensator is adapted to generate the valve command signal, based on comparison of a desired force and an estimated actual force of the piston on the mold; an estimator coupled to the clamp assembly, wherein said estimator is adapted to estimate an actual force using cylinder pressure and a piston position signal; a position and velocity controller being coupled to the clamp assembly and coupled to said estimator, wherein said position and velocity controller is adapted to generate the valve command signal based on desired position and velocity profiles; and an auto selector being coupled to said cascade compensator, wherein said auto selector is adapted to select between a velocity control function signal and a force control function signal so as to generate the servo valve command signal.
- 2. The control unit as recited in claim 1, further comprising a desired force limit, wherein said desired force limit is adapted to generate a desired force limit signal.
- 3. The control unit as recited in claim 1, further comprising a feedback compensator being coupled to said estimator, wherein said feedback compensator is adapted to generate a feedback signal in correspondence with said estimated actual force signal.
- 4. The control unit as recited in claim 1, wherein said position and velocity controller further comprises:a desired velocity scheduler adapted to generate a velocity request signal based on a position error signal (e); and a servo valve look-up table being coupled to said desired velocity scheduler, wherein said servo valve look-up table is adapted to generate the servo valve command signal in correspondence with said velocity request signal.
- 5. The control unit as recited in claim 4, wherein said desired velocity scheduler is adapted to generate said velocity schedule signal based on said position error signal (e).
- 6. The control unit as recited in claim 5, further comprising a lead/lag controller coupled to said clamp assembly and said estimator, wherein said lead/lag controller provides a closed loop response between the desired velocity and actual velocity.
- 7. The control unit as recited in claim 6, wherein a ramp up velocity of the servo valve has an upper limit of vmax, wherein the final touch down velocity of the servo valve is vf based on error signal (e), and wherein a ramp down deceleration of the servo valve is constant with respect to time and position.
- 8. The control unit as recited in claim 7, wherein the ramp down velocity of the servo valve is constant with respect to time.
- 9. The control unit as recited in claim 7, wherein the ramp down velocity of the servo valve is constant with respect to position.
- 10. A method for determining the velocity profile of the piston of a clamp assembly, wherein the piston velocity is controlled by the operation of a servo valve within the clamp assembly, and wherein the displacement of the servo valve is controllable, the velocity profile having a ramp-up interval, a constant speed interval, ramp-down interval; and a touchdown interval, said method comprising the following steps:determining a maximum displacement xs of the servo valve; selecting a final velocity of the piston (vf); choosing a ratio α of a piston ramp-down interval to the servo valve displacement “x” during the ramp-up interval, according to equation α=(xs-xd)xuwhere xu is a servo valve displacement and xd is a displacement where a ramp down interval begins;selecting a desired maximum piston velocity (vmax) during the ramp-up interval and constant speed interval; and determining the servo valve displacement xu from simulations or estimate from previous cycles according to a mathematical relationship xu≤xs(α+1).
- 11. The method as recited in claim 10, further comprising the step of choosing a servo valve displacement of xd=xs−αxu wherein xu<xs(α+1).
- 12. The method as recited in claim 11, further comprising the step of choosing a servo valve displacement of xd=xu wherein xu=xs(a+1).
- 13. The method as recited in claim 12, further comprising the step of selecting a piston deceleration profile during the ramp-down interval according to a linear and a quadratic velocity profile.
- 14. The method as recited in claim 13, further comprising the step of determining said linear velocity profile having a slope (dx) according to the equation dx=(vmax-vf)α xubased on the piston having a linear velocity trajectory from xd to xs.
- 15. The method as recited in claim 12, further comprising the step of selecting the piston deceleration profile during the ramp-down interval according to a quadratic velocity profile.
- 16. The method as recited in claim 15, further comprising the step of determining said quadratic velocity profile (dt) according to the equation dt=(vmax2-vf2)2α xu.
- 17. A control unit for controlling the force and velocity of a piston on a mold within the clamp assembly of a injection molding machine, based on a servo valve command signal which acts to operate a servo valve, said control unit comprising:a desired force limit, wherein said desired force limit is adapted to generate a desired force limit signal; a cascade compensator connected to the desired force limit, wherein said cascade compensator is adapted to generate the valve command signal, based on comparison of the desired force and an estimated actual force of the piston on the mold; an estimator coupled to the clamp assembly, wherein said estimator is adapted to estimate an actual force using cylinder pressure and a piston position signal; a position and velocity controller being coupled to the clamp assembly and coupled to said estimator, wherein said position and velocity controller is adapted to generate the valve command signal based on desired position and velocity profiles; an auto selector being coupled to said cascade compensator, wherein said auto selector is adapted to select between a velocity control function signal and a force control function signal so as to generate the servo valve command signal; and a feedback compensator being coupled to said estimator, wherein said feedback compensator is adapted to generate a feedback signal in correspondence with said estimated actual force signal.
- 18. A method for determining the velocity profile of the piston of a clamp assembly, wherein the piston velocity is controlled by the operation of a servo valve within the clamp assembly, and wherein the displacement of the servo valve is controllable, the velocity profile having a ramp-up interval, a constant speed interval, ramp-down interval; and a touchdown interval, said method comprising the steps of:determining a maximum displacement xs of the servo valve; selecting a final velocity of the piston (vf); choosing a ratio α of a piston ramp-down interval to the servo valve displacement “x” during the ramp-up interval, according to equation α=(xs−xd)/xu where xu is a servo valve displacement and xd is a displacement where a ramp down interval begins; selecting a desired maximum piston velocity (vmax) during the ramp-up interval and constant speed interval; determining the servo valve displacement xu from simulations or estimate from previous cycles according to a mathematical relationship xu≤xs(α+1);andselecting a piston deceleration profile during the ramp-down interval according to a linear velocity profile.
- 19. The method as recited in claim 18 wherein the step of selecting the piston deceleration profile during the ramp-down interval also selects the piston deceleration according to a quadratic velocity profile.
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