U.S. patent application number 09/805598 was filed with the patent office on 2005-06-16 for process and device for displacing a moveable unit on a base.
Invention is credited to Levine, Jean, Nguyen, Van Diep.
Application Number | 20050126892 09/805598 |
Document ID | / |
Family ID | 34681870 |
Filed Date | 2005-06-16 |
United States Patent
Application |
20050126892 |
Kind Code |
A9 |
Nguyen, Van Diep ; et
al. |
June 16, 2005 |
PROCESS AND DEVICE FOR DISPLACING A MOVEABLE UNIT ON A BASE
Abstract
According to the invention: a) a force (F) is determined which,
applied to the moveable unit (4), produces a combined effect, on
the one hand, on the moveable unit (4) so that it exactly carries
out the envisaged displacement on the base (2), especially as
regards the prescribed duration and prescribed distance of the
displacement, and, on the other hand, on the elements (MA1, MA2,
MA3, 4) brought into motion by this displacement so that all these
elements are immobile at the end of said displacement of the
moveable unit (4); and b) the force (F) thus determined is applied
to the moveable unit (4).
Inventors: |
Nguyen, Van Diep; (Nemount,
FR) ; Levine, Jean; (Paris, FR) |
Correspondence
Address: |
IRELL & MANELLA LLP
840 NEWPORT CENTER DRIVE
SUITE 400
NEWPORT BEACH
CA
92660
US
|
Prior
Publication: |
|
Document Identifier |
Publication Date |
|
US 0079198 A1 |
June 27, 2002 |
|
|
Family ID: |
34681870 |
Appl. No.: |
09/805598 |
Filed: |
March 12, 2001 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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09805598 |
Mar 12, 2001 |
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09362643 |
Jul 27, 1999 |
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6438461 |
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Current U.S.
Class: |
198/752.1 |
Current CPC
Class: |
G03F 7/70716 20130101;
B25J 9/163 20130101; B25J 9/1633 20130101 |
Class at
Publication: |
198/752.1 |
International
Class: |
B65G 027/00 |
Foreign Application Data
Date |
Code |
Application Number |
Sep 12, 2000 |
FR |
00 11584 |
Feb 23, 1999 |
FR |
99 02224 |
Claims
1. A process for displacing a moveable unit (4) on a base (2), said
moveable unit (4) being displaced linearly according to a
predetermined displacement under the action of a controllable force
(F), wherein: a) equations are defined which: illustrate a dynamic
model of a system formed by elements (2, 4, MA, MA1, MA2, MA3), of
which said moveable unit (4) is one, which are brought into motion
upon a displacement of said moveable unit (4); and comprise at
least two variables, of which the position of said moveable unit
(4) is one; b) all the variables of this system, together with said
force (F), are expressed as a function of one and the same
intermediate variable y and of a specified number of derivatives as
a function of time of this intermediate variable, said force (F)
being such that, applied to said moveable unit (4), it displaces
the latter according to said specified displacement and renders all
the elements of said system immobile at the end of said
displacement; c) the initial and final conditions of all said
variables are determined; d) the value as a function of time of
said intermediate variable is determined from the expressions for
the variables defined in step b) and said initial and final
conditions; e) the value as a function of time of said force is
calculated from the expression for the force, defined in step b)
and said value of the intermediate variable, determined in step d);
and f) the value thus calculated of said force (F) is applied to
said moveable unit (4).
2. The process as claimed in claim 1, wherein, in step a), the
following operations are carried out: the variables of the system
are denoted xi, i going from 1 to p, p being an integer greater
than or equal to 2, and the balance of the forces and of the
moments is expressed, approximating to first order if necessary, in
the so-called polynomial matrix form:A(s)X=bFwith: A(s) matrix of
size p.times.p whose elements Aij(s) are polynomials of the
variable s=d/dt; 60 X the vector ( x1 xp ) ;b the vector of
dimension p; and F the force exerted by a means of displacing the
moveable unit and in that, in step b), the following operations are
carried out: the different variables xi of said system, i going
from 1 to p, each being required to satisfy a first expression of
the form: 61 xi = j = 0 j = r pi , j y ( j ) ,the y.sup.(j) being
the derivatives of order j of the intermediate variable y, r being
a predetermined integer and the pi,j being parameters to be
determined, a second expression is obtained by putting
y.sup.(j)=s.sup.j.y: 62 xi = ( j = 0 j = r pi , j s j ) y = Pi ( s
) y ,a third expression of vectorial type is defined on the basis
of the second expressions relating to the different variables xi of
the system (S1, S2):X=P.ycomprising the vector 63 P = ( P1 Pp )
said vector P is calculated, by replacing X by the value P.y in the
following system: 64 { B T A ( s ) P ( s ) = Op - 1 bp F = j = 1 j
= p Ap , j ( s ) Pj ( s ) y in which: B.sup.T is the transpose of a
matrix B of size px(p-1) such that B.sup.Tb=Op-1; bp is the p-th
component of the vector b previously defined; and Op-1 is a zero
vector of dimension (p-1); the values of the different parameters
pi,j are deduced from the value thus calculated of the vector P;
and from these latter values are deduced the values of the
variables xi as a function of the intermediate variable y and of
its derivatives, on each occasion using the corresponding first
expression.
3. The process as claimed in claim 1, wherein, in step d), a
polynomial expression for the intermediate variable y is used to
determine the value of the latter.
4. The process as claimed in claim 3, wherein, the initial and
final conditions of the different variables of the system, together
with the expressions defined in step b), are used to determine the
parameters of the polynomial expression for the intermediate
variable y.
5. The process as claimed in claim 1 for displacing a moveable unit
(4) on a base (2) which is mounted elastically with respect to the
floor (S) and which may be subjected to linear and angular motions,
wherein the variables of the system are the linear position x of
the moveable unit, the linear position xB of the base and the
angular position .theta.z of the base, which satisfy the relations:
65 { x = y + ( rB kB + r k ) y ( 1 ) + ( mB kB + rBr kBk + J k ) y
( 2 ) + ( rBJ kBk + mBr kBk ) y ( 3 ) + mBJ kBk y ( 4 ) xB = - m kB
( J k y ( 4 ) + r k y ( 3 ) + y ( 2 ) ) z = - d m k ( m B kB y ( 4
) + rB kB y ( 3 ) + y ( 2 ) ) in which: m is the mass of the
moveable unit; mB, kB, k.theta., rB, r.theta. are respectively the
mass, the linear stiffness, the torsional stiffness, the linear
damping and the torsional damping of the base; J is the inertia of
the base with respect to a vertical axis; d is the distance between
the axis of translation of the center of mass of the moveable unit
and that of the base; and y.sup.(1), y.sup.(2), y.sup.(3) and
y.sup.(4) are respectively the first to fourth derivatives of the
variable y.
6. The process as claimed in claim 1 for displacing on a base a
moveable unit (4) on which are elastically mounted a number p of
auxiliary masses MAi, p being greater than or equal to 1, i going
from 1 to p, wherein the variables of the system are the position x
of the moveable unit (4) and the positions zi of the p auxiliary
masses MAi, which satisfy the relations: 66 { x = ( i = 1 p ( mi ki
s 2 + ri ki s + 1 ) ) y zi = ( j = 1 j i p ( mj kj s 2 + rj kj s +
1 ) ) ( ri ki s + 1 ) y in which: .PI. illustrates the product of
the associated expressions; mi, zi, ki and ri are respectively the
mass, the position, the stiffness and the damping of an auxiliary
mass MAi; mj, kj and rj are respectively the mass, the stiffness
and the damping of an auxiliary mass MAj; and s=d/dt.
7. The process as claimed in claim 1 for displacing a moveable unit
(4) on a base (2) which is mounted elastically with respect to the
floor (S) and on which is elastically mounted an auxiliary mass
(MA), wherein the variables of the system are the positions x, xB
and zA respectively of the moveable unit (4), of the base (2) and
of the auxiliary mass (MA), which satisfy the relations: 67 { x = [
( mAs 2 + rAs + kA ) ( mBs 2 + ( rA + rB ) s + ( kA + kB ) ) - (
rAs + kA ) 2 ] y xB = - My ( 2 ) zA = - M ( rAy ( 3 ) + kAy ( 2 ) )
in which: M, mB and mA are the masses respectively of the moveable
unit (4), of the base (2) and of the auxiliary mass (MA); rA and rB
are the dampings respectively of the auxiliary mass (MA) and of the
base (2); kA and kB are the stiffnesses respectively of the
auxiliary mass (MA) and of the base (2); and s=d/dt.
8. The process as claimed in claim 1 for displacing on a base
mounted elastically with respect to the floor, a moveable unit on
which is elastically mounted an auxiliary mass, wherein the
variables of the system are the positions x, xB and zC respectively
of the moveable unit, of the base and of the auxiliary mass, which
satisfy the relations: 68 { x = [ ( mCs 2 + rCs + kC ) ( mBs 2 +
rBs + kB ] y xB = [ ( mCs 2 + rCs + kC ) ( Ms 2 + rCs + kC ) - (
rCs + kC ) 2 ] y zC = ( rCs + kC ) ( mBs 2 + rBs + kB ) y in which:
M, mB and mC are the masses respectively of the moveable unit, of
the base and of the auxiliary mass; rB and rC are the dampings
respectively of the base and of the auxiliary mass; kB and kC are
the stiffnesses respectively of the base and of the auxiliary mass;
and s=d/dt.
9. A device comprising: a base (2); a moveable unit (4) which may
be displaced linearly on said base (2); and a controllable actuator
(5) able to apply a force (F) to said moveable unit (4) with a view
to its displacement on said base (2), wherein it furthermore
comprises means (6) which implement steps a) to e) of the process
specified under claim 1, so as to calculate a force (F) which may
be applied to said moveable unit (4), and which determine a control
command and transmit it to said actuator (5) so that it applies the
force (F) thus calculated to said moveable unit (4).
Description
BACKGROUND OF THE INVENTION
[0001] 1. Field of the Invention
[0002] The present invention relates to a process and a device for
displacing a moveable unit on a base.
[0003] Said device is of the type comprising a controllable
actuator, for example an electric motor, intended to give rise to a
linear displacement of the moveable unit on the base, as well as a
system which is formed of a plurality of elements which are brought
into motion upon the displacement of said moveable unit.
[0004] Within the context of the present invention, said system
exhibits at least two different motions and comprises as elements
which may be brought into motion, in particular:
[0005] said base which can be mounted elastically with respect to
the floor, especially so as to isolate it from vibrations
originating from said floor; and/or
[0006] one or more auxiliary masses, for example measurement
supports and/or loads, which are tied elastically to the base;
and/or
[0007] one or more auxiliary masses, for example likewise
measurement supports and/or loads, which are tied elastically to
the moveable unit.
[0008] When the moveable unit is set into motion, said elements of
the system begin to move. However, especially by reason of the
aforesaid elastic link, these elements still continue to move when
the displacement of the moveable unit has terminated and when the
latter comes to a stop.
[0009] Such a continuance of the motions of said system is
generally undesirable, since it may entail numerous drawbacks. In
particular, it may disturb measurements, especially positioning
measurements, which are made on the moveable unit or on these
elements.
[0010] Also, an object of the present invention is to control the
moveable unit in such a way that all the moving elements of said
system, for example the base and/or auxiliary masses, are
stationary at the end of the displacement of the moveable unit.
[0011] As regards said base, if it is mounted elastically with
respect to the floor, it is known that, when the moveable unit is
set into motion, during the acceleration and deceleration phases,
it is subjected to the reaction of the force applied to the
moveable unit by the actuator. This reaction load excites the base
which then oscillates on its supports. This disturbs the relative
positioning of the moveable unit with respect to the base, and
greatly impedes the accuracy of the device.
[0012] This relative position error persists after the end of the
displacement of the moveable unit and disappears only after the
stabilization (which takes place much later) of the base.
[0013] Various solutions for remedying this drawback are known.
Some of these solutions make provision in particular:
[0014] to immobilize the base during the acceleration and
deceleration phases via a disabling system, for example an
electromagnetic disabling system, which is mounted in parallel with
the elastic supports. However, this known solution prevents the
supports from isolating the base from the vibrations originating
from the floor during said acceleration and deceleration
phases;
[0015] to cancel the effect produced by the force developed by the
actuator, by making provision for an additional actuator which is
arranged between the base and the floor and which develops an
additional force of the same amplitude but oppositely directed;
or
[0016] to displace an additional moveable unit on the base
according to a similar displacement, but oppositely directed, with
respect to the displacement of the moveable unit, so as to cancel
the inertia effects.
[0017] However, none of these known solutions is satisfactory,
since their effectivenesses are restricted and since they all
require supplementary means (disabling system, additional actuator,
additional moveable unit) which increase in particular the
complexity, the cost and the bulkiness of the device.
[0018] Moreover, above all, these solutions implement an action
which acts only on the base and not on the other elements of the
system which, for their part, continue to move when the moveable
unit is stationary.
[0019] The object of the present invention is to remedy these
drawbacks. It relates to a process for displacing, in an extremely
accurate manner and at restricted cost, a moveable unit on a base
mounted for example on the floor, whilst bringing all the motions
to which this displacement gives rise to a stop at the end of the
displacement, said moveable unit being displaced linearly according
to a displacement which is predetermined in terms of distance and
time, under the action of a controllable force.
[0020] Accordingly, said process is noteworthy according to the
invention in that:
[0021] a) equations are defined which:
[0022] illustrate a dynamic model of a system formed by elements,
of which said moveable unit is one, which are brought into motion
upon a displacement of said moveable unit; and
[0023] comprise at least two variables, of which the position of
said moveable unit is one;
[0024] b) all the variables of this system, together with said
force, are expressed as a function of one and the same intermediate
variable y and of a specified number of derivatives as a function
of time of this intermediate variable, said force being such that,
applied to said moveable unit, it displaces the latter according to
said specified displacement and renders all the elements of said
system immobile at the end of said displacement;
[0025] c) the initial and final conditions of all said variables
are determined;
[0026] d) the value as a function of time of said intermediate
variable is determined from the expressions for the variables
defined in step b) and said initial and final conditions;
[0027] e) the value as a function of time of said force is
calculated from the expression for the force, defined in step b)
and said value of the intermediate variable, determined in step d);
and
[0028] f) the value thus calculated of said force is applied to
said moveable unit.
[0029] Thus, the force applied to the moveable unit enables the
latter to carry out the predetermined displacement envisaged,
especially in terms of time and distance, whilst rendering the
elements brought into motion by this displacement immobile at the
end of the displacement so that they do not oscillate and, in
particular, do not disturb the relative positioning between
themselves and the moveable unit.
[0030] It will be noted moreover that, by reason of this combined
control of said moveable unit and of said moving elements, one
obtains an extremely accurate displacement of the moveable unit in
a reference frame independent of the base and tied for example to
the floor.
[0031] It will be noted that the implementation of the process in
accordance with the invention is not limited to a displacement
along a single axis, but can also be applied to displacements along
several axes which can be regarded as independent.
[0032] Advantageously, in step a), the following operations are
carried out: the variables of the system are denoted xi, i going
from 1 to p, p being an integer greater than or equal to 2, and the
balance of the forces and of the moments is expressed,
approximating to first order if necessary, in the so-called
polynomial matrix form:
A(s)X=bF
[0033] with:
[0034] A(s) matrix of size p.times.p whose elements Aij(s) are
polynomials of the variable s=d/dt;
[0035] X the vector 1 ( x1 xp ) ;
[0036] b the vector of dimension p; and
[0037] F the force exerted by the motor.
[0038] Advantageously, in step b), the following operations are
carried out:
[0039] the different variables xi of said system, i going from 1 to
p, each being required to satisfy a first expression of the form: 2
xi = j = 0 j = r pi , j y ( j ) ,
[0040] the y.sup.(j) being the derivatives of order j of the
intermediate variable y, r being a predetermined integer and the
pi, j being parameters to be determined, a second expression is
obtained by putting y.sup.(j)=s.sup.j.y: 3 xi = ( j = 0 j = r pi ,
j s j ) y = Pi ( s ) y ,
[0041] a third expression of vectorial type is defined on the basis
of the second expressions relating to the different variables xi of
the system:
[0042] comprising the vector 4 X = P y P = ( P1 Pp )
[0043] said vector P is calculated, by replacing X by the value P.y
in the following system: 5 { B T A ( s ) P ( s ) = Op - 1 bp F = j
= i j = p Ap , j ( s ) Pj ( s ) y
[0044] in which:
[0045] B.sup.T is the transpose of a matrix B of size px(p-1), such
that B.sup.Tb=Op-1;
[0046] bp is the p-th component of the vector b previously defined;
and
[0047] Op-1 is a zero vector of dimension (p-1);
[0048] the values of the different parameters pi,j are deduced from
the value thus calculated of the vector P; and
[0049] from these latter values are deduced the values of the
variables xi as a function of the intermediate variable y and of
its derivatives, on each occasion using the corresponding first
expression.
[0050] Thus, a fast and general method of calculation is obtained
for calculating the relations between the variables of the system
and said intermediate variable, in the form of linear combinations
of the latter and of its derivatives with respect to time.
[0051] Advantageously, in step d), a polynomial expression for the
intermediate variable y is used to determine the value of the
latter.
[0052] In this case, preferably, the initial and final conditions
of the different variables of the system, together with the
expressions defined in step b), are used to determine the
parameters of this polynomial expression.
[0053] In a first embodiment, for displacing a moveable unit on a
base which is mounted elastically with respect to the floor and
which may be subjected to linear and angular motions,
advantageously, the variables of the system are the linear position
x of the moveable unit, the linear position xB of the base and the
angular position .theta.z of the base, which satisfy the relations:
6 { x = y + ( r B k B + r k ) y ( 1 ) + ( m B k B + r B r k B k + J
k ) y ( 2 ) + ( r B J k B k + m B r k B k ) y ( 3 ) + m B J k B k y
( 4 ) x B = - m k B ( J k y ( 4 ) + r k y ( 3 ) + y ( 2 ) ) z = - d
m k ( m B k B y ( 4 ) + r B k B y ( 3 ) + y ( 2 ) )
[0054] in which:
[0055] m is the mass of the moveable unit;
[0056] mB, kB, k.theta., rB, r.theta. are respectively the mass,
the linear stiffness, the torsional stiffness, the linear damping
and the torsional damping of the base;
[0057] J is the inertia of the base with respect to a vertical
axis;
[0058] d is the distance between the axis of translation of the
center of mass of the moveable unit and that of the base; and
[0059] y.sup.(1), y.sup.(2), y.sup.(3) and y.sup.(4) are
respectively the first to fourth derivatives of the variable y.
[0060] This first embodiment makes it possible to remedy the
aforesaid drawbacks (inaccurate displacement, etc) related to the
setting of the base into oscillation during the displacement of the
moveable unit.
[0061] In a second embodiment, for displacing on a base a moveable
unit on which are elastically mounted a number p of auxiliary
masses MAi, p being greater than or equal to 1, i going from 1 to
p, advantageously, the variables of the system are the position x
of the moveable unit and the (linear) positions zi of the p
auxiliary masses MAi, which satisfy the relations: 7 { x = ( i = 1
p ( m i ki s 2 + ri ki s + 1 ) ) y zi = ( j = 1 j i p ( m i kj s 2
+ ri kj s + 1 ) ) ( r i k i s + 1 ) y
[0062] in which:
[0063] .PI. illustrates the product of the associated
expressions;
[0064] mi, zi, ki and ri are respectively the mass, the position,
the stiffness and the damping of an auxiliary mass MAi;
[0065] mj, kj and rj are respectively the mass, the stiffness and
the damping of an auxiliary mass MAj; and
[0066] s=d/dt.
[0067] In a third embodiment, for displacing a moveable unit on a
base which is mounted elastically with respect to the floor and on
which is elastically mounted an auxiliary mass, advantageously, the
variables of the system are the positions x, xB and zA respectively
of the moveable unit, of the base and of the auxiliary mass, which
satisfy the relations: 8 { x = [ ( m A s 2 + r A s + k A ) ( m B s
2 + ( r A + r B ) s + ( k A + k B ) ) - ( r A s + k A ) 2 ] y x B =
- M y ( 2 ) z A = - M ( r A y ( 3 ) + k A y ( 2 ) )
[0068] in which:
[0069] M, mB and mA are the masses respectively of the moveable
unit, of the base and of the auxiliary mass;
[0070] rA and rB are the dampings respectively of the auxiliary
mass and of the base;
[0071] kA and kB are the stiffnesses respectively of the auxiliary
mass and of the base; and
[0072] s=d/dt.
[0073] In a fourth embodiment, for displacing on a base mounted
elastically with respect to the floor, a moveable unit on which is
elastically mounted an auxiliary mass, advantageously, the
variables of the system are the positions x, xB and zC respectively
of the moveable unit, of the base and of the auxiliary mass, which
satisfy the relations: 9 { x = [ ( m C s 2 + r C s + k C ) ( m B s
2 + r B s + k B ) ] y x B = [ ( m C s 2 + r C s + k C ) ( M s 2 + r
C s + k C ) - ( r C s + k C ) 2 ] y z C = ( r C s + k C ) ( m B s 2
+ r B s + k B ) y
[0074] in which:
[0075] M, mB and mC are the masses respectively of the moveable
unit, of the base and of the auxiliary mass;
[0076] rB and rC are the dampings respectively of the base and of
the auxiliary mass;
[0077] kB and kC are the stiffnesses respectively of the base and
of the auxiliary mass; and
[0078] s=d/dt.
[0079] The present invention also relates to a device of the type
comprising:
[0080] a base mounted directly or indirectly on the floor;
[0081] a moveable unit which may be displaced linearly on said
base; and
[0082] a controllable actuator able to apply a force to said
moveable unit with a view to its displacement on said base.
[0083] According to the invention, said device is noteworthy in
that it furthermore comprises means, for example a calculator:
[0084] which implement steps a) to e) of the aforesaid process, so
as to calculate a force which, applied to said moveable unit, makes
it possible to obtain the combined effect or control indicated
above; and
[0085] which determine a control command and transmit it to said
actuator so that it applies the force thus calculated to said
moveable unit, during a displacement.
[0086] Thus, over and above the aforesaid advantages, the device in
accordance with the invention does not require any additional
mechanical means, thereby reducing its cost and its bulkiness and
simplifying its embodiment, with respect to the known and aforesaid
devices.
[0087] The figures of the appended drawing will elucidate the
manner in which the invention may be embodied. In these figures,
identical references designate similar elements.
[0088] FIGS. 1 and 2 respectively illustrate two different
embodiments of the device in accordance with the invention.
[0089] FIGS. 3 to 7 represent graphs which illustrate the
variations over time of variables of the system, for a first
embodiment of the device in accordance with the invention.
[0090] FIGS. 8 to 13 represent graphs which illustrate the
variations over time of variables of the system, for a second
embodiment of the device in accordance with the invention.
[0091] The device 1 in accordance with the invention and
represented diagrammatically in FIGS. 1 and 2, according to two
different embodiments, is intended for displacing a moveable unit
4, for example a moveable carriage, on a base 2, in particular a
test bench.
[0092] This device 1 can for example be applied to fast XY tables
used in microelectronics, to machine tools, to conveyors, to
robots, etc.
[0093] In a known manner, said device 1 comprises, in addition to
the base 2 and to the moveable unit 4:
[0094] supports 3, of known type, arranged between the base 2 and
the floor S;
[0095] means (not represented), for example a rail, fixed on the
base 2 and enabling the moveable unit 4 to be displaced linearly on
said base 2; and
[0096] a controllable actuator 5, preferably an electric motor,
able to apply a force F to said moveable unit 4 with a view to its
displacement on the base 2.
[0097] Within the context of the present invention, the device 1
comprises a system S1, S2 which is formed of various elements
specified hereinbelow and variables according to the embodiment
contemplated, which are brought into motion upon the displacement
of the moveable unit 4.
[0098] According to the invention, said device 1 is improved in
such a way as to obtain directly at the end of a displacement of
the moveable unit 4:
[0099] accurate positioning of the latter in a reference frame (not
represented), independent of the moveable unit 4 and of the base 2
and tied for example to the floor; and
[0100] immobilization of all the moving elements of said system S1,
S2.
[0101] To do this, the device 1 moreover comprises, according to
the invention, calculation means 6 which calculate a particular
force F, which is intended to be transmitted in the form of a
control command to the actuator 5, as illustrated by a link 7, and
which is such that, applied to said moveable unit 4, it produces a
combined effect (and hence combined control):
[0102] on the one hand, on the moveable unit 4 so that it exactly
carries out the envisaged displacement, especially as regards the
prescribed duration and prescribed distance of displacement;
and
[0103] on the other hand, on said system S1, S2 so that all its
moving elements are immobile at the end of the displacement of the
moveable unit 4.
[0104] Accordingly, said calculation means 6 implement the process
in accordance with the invention, according to which:
[0105] a) equations are defined which:
[0106] illustrate a dynamic model of said system (for example S1 or
S2) formed by the different elements, of which said moveable unit 4
is one, which are brought into motion upon a displacement of said
moveable unit 4; and
[0107] comprise at least three variables, of which the position of
said moveable unit 4 is one;
[0108] b) all the variables of this system, together with said
force F, are expressed as a function of one and the same
intermediate variable y and of a specified number of derivatives as
a function of time of this intermediate variable, said force F
being required to be such that, applied to said moveable unit 4, it
displaces the latter according to said specified displacement and
renders all the elements of said system immobile at the end of said
displacement;
[0109] c) the initial and final conditions of all said variables
are determined;
[0110] d) the value as a function of time of said intermediate
variable is determined from the expressions for the variables
defined in step b) and said initial and final conditions; and
[0111] e) the value of said force is calculated from the expression
for the force, defined in step b) and said value of the
intermediate variable, determined in step d).
[0112] Thus, by virtue of the invention, the force F applied to the
moveable unit 4 enables the latter to carry out the predetermined
displacement envisaged, especially in terms of time and distance,
whilst rendering the elements (specified hereinbelow) which are
brought into motion by this displacement immobile at the end of the
displacement so that they do not oscillate and, in particular, do
not disturb the relative positioning between themselves and the
moveable unit 4.
[0113] It will be noted moreover that, by reason of this combined
effect or control of said moveable unit 4 and of said moving
elements, one obtains an extremely accurate displacement of the
moveable unit 4 in a reference frame independent of the base 2 and
tied for example to the floor S.
[0114] Of course, the implementation of the present invention is
not limited to a displacement along a single axis, but can also be
applied to displacements along several axes which can be regarded
as independent.
[0115] According to the invention, in step d), a polynomial
expression for the intermediate variable y is used to determine the
value of the latter, and the initial and final conditions of the
different variables of the system, together with the expressions
defined in step b) are used to determine the parameters of this
polynomial expression.
[0116] The process in accordance with the invention will now be
described in respect of four different systems (of moving
elements).
[0117] In a first embodiment (not represented), the supports 3 are
of elastic type and make it possible to isolate the base 2 from the
vibrations originating from said floor S. The natural frequency of
the base 2 on said elastic supports 3 is generally a few Hertz.
Furthermore, in addition to the translational motion of the
moveable unit 4 controlled by the force F, an angular motion is
created between the base 2 and the moveable unit 4. Specifically,
in this case, the axis of the moveable unit 4 does not pass through
its center of mass, the force produced by the actuator 5 creates a
moment about the vertical axis. The rail is assumed to be slightly
flexible and thus allows the moveable unit 4 small rotational
motions about the vertical axis, which corresponds to the aforesaid
relative angular motion between the base 2 and the moveable unit
4.
[0118] Consequently, in this first embodiment, to displace the
moveable unit 4 on the base 2 which is mounted elastically with
respect to the floor and which may be subjected to a (relative)
angular motion, the variables of the system are the linear position
x of the moveable unit 4, the linear position xB of the base 2 and
the angular position .theta.z of the base 2, which satisfy the
relations: 10 { x = y + ( r B k B + r k ) y ( 1 ) + ( m B k B + r B
r k B k + J k ) y ( 2 ) + ( r B J k B k + m B r k B k ) y ( 3 ) + m
B J k B k y ( 4 ) x B = - m k B ( J k y ( 4 ) + r k y ( 3 ) + y ( 2
) ) z = - d m k ( m B k B y ( 4 ) + r B k B y ( 3 ) + y ( 2 ) )
[0119] in which
[0120] m is the mass of the moveable unit 4;
[0121] mB, kB, k.theta., rB, r.theta. are respectively the mass,
the linear stiffness, the torsional stiffness, the linear damping
and the torsional damping of the base 2;
[0122] J is the inertia of the base 2 with respect to a vertical
axis;
[0123] d is the distance between the axis of translation of the
center of mass of the moveable unit 4 and that of the base 2;
and
[0124] y.sup.(1), y.sup.(2), y.sup.(3) and y.sup.(4) are
respectively the first to fourth derivatives of the variable y.
[0125] Specifically, in this first embodiment, the balance of the
forces and of the moments, the angle .theta.z being approximated to
first order, may be written: 11 { mx ( 2 ) = F mBxB ( 2 ) = - F -
kBxB - rBxB ( 1 ) J z ( 2 ) = - dF - k z - r z ( 1 ) ( 1 )
[0126] It will be noted that, within the context of the present
invention, .alpha..sup.(.beta.) is the derivative of order .beta.
with respect to time of the parameter .alpha., regardless of
.alpha.. Thus, for example, x.sup.(1) is the first derivative of x
with respect to time.
[0127] The calculation of the intermediate variable y is achieved
by putting 12 s = t ,
[0128] x=P(s)y, xB=PB(s)y, .theta.z=P.theta.(s)y and by rewriting
the system (1) with this notation: 13 { ms 2 P ( s ) y = F ( mBs 2
+ rBs + kB ) PB ( s ) y = - F ( Js 2 + r s + k ) P ( s ) y = -
dF
[0129] i.e.: 14 ( mBs 2 + rBs + kB ) PB ( s ) = 1 d ( Js 2 + r s +
k ) P ( s ) = - ms 2 P ( s )
[0130] and hence: 15 { P ( s ) = ( mB kB s 2 + rB kB s + 1 ) ( J k
s 2 + r k s + 1 ) PB ( s ) = - m kB s 2 ( J k s 2 + r k s + 1 ) P (
s ) = - d m k s 2 ( mB kB s 2 + rB kB s + 1 )
[0131] From these expressions, we immediately deduce: 16 x = ( mB
kB s 2 + rB kB s + 1 ) ( J k s 2 + r k s + 1 ) y { x = y + ( rB kB
+ r k ) y ( 1 ) + ( mB kB + rBr kBk + J k ) y ( 2 ) + ( rBJ kBk +
mBr kBk ) y ( 3 ) + mBJ kBk y ( 4 ) xB = - m kB ( J k y ( 4 ) + r k
y ( 3 ) + y ( 2 ) ) z = - d m k ( mB kB y ( 4 ) + rB kB y ( 3 ) + y
( 2 ) ) ( 2 )
[0132] The expression for y as a function of x, x.sup.(1), xB,
xB.sup.(1), .theta.z and .theta.z.sup.(1) is obtained by inversion.
However, this formula is not necessary in order to plan the
trajectories of x, xB and .theta.z. Specifically, since we want a
stop-stop displacement of the moveable unit 4 between x0 at the
instant t0 and x1 at the instant t1, with
x.sup.(1)(t0)=0=x.sup.(1)(t1) and xB(t0)=0=xB(t1),
xB.sup.(1)(t0)=0=xB.sup- .(1)(t1) and .theta.z(t0)=0=.theta.z(t1),
.theta.z.sup.(1)(t0)=0=.theta.z.- sup.(1)(t1), with in addition
F(t0)=0=F(t1),
[0133] we deduce therefrom through the aforesaid expressions (2)
that y(t0)=x0,y(t1)=x1 and
y.sup.(1)(ti)=y.sup.(2)(ti)=y.sup.(3)(ti)=y.sup.(4)(ti)=y.sup.(5)(ti)=y.su-
p.(6)(ti)=0, i=0.1
[0134] i.e. 14 initial and final conditions.
[0135] It is sufficient to choose y as a polynomial with respect to
time of the form: 17 y ( t ) = x0 + ( x1 - x0 ) ( ( t ) ) i = 0 ai
( ( t ) ) i ( 3 )
[0136] with 18 ( t ) = t - t0 t1 - t0
[0137] and .alpha..gtoreq.7 and .beta..gtoreq.6. The coefficients
a0, . . . , a.beta. are then obtained, according to standard
methods, by solving a linear system.
[0138] The reference trajectory sought for the displacement of the
moveable unit 4 is then given by expressions (2) with y(t) given by
expression (3).
[0139] Moreover, the force F as a function of time to be applied to
the means 5 is obtained by integrating the value of y obtained via
expression (3) in the expression F(t)=M.x.sup.(2)(t).
[0140] In this first embodiment, we obtain: 19 F ( t ) = M [ y 2 +
( rB kB + r k ) y ( 3 ) + ( mB kB + rBr kBk + J k ) y ( 4 ) + ( rBJ
+ mBr kBk ) y ( 5 ) + mBJ kBk y ( 6 ) ] (3A)
[0141] with y(t) given by expression (3).
[0142] Thus, since by virtue of the device 1 the base 2 is
immobilized at the end of the displacement, it does not disturb the
positioning of the moveable unit 4 in the aforesaid reference frame
so that said moveable unit 4 is positioned in a stable manner as
soon as its displacement ends. Moreover, since its displacement is
carried out in an accurate manner, its positioning corresponds
exactly in said reference frame to the sought-after
positioning.
[0143] Represented in FIGS. 3 to 7 are the values respectively of
said variables y (in meters m), x (in meters m), xB (in meters m),
.theta.z (in radians rd) and F (in Newtons N) as a function of time
t (in seconds s) for a particular exemplary embodiment, for
which:
[0144] m=40 kg;
[0145] mB=800 kg;
[0146] kB=mB(5.2.pi.).sup.2 corresponding to a natural frequency of
5 Hz;
[0147] rB=0.3{square root}{square root over (kBmB)} corresponding
to a normalized damping of 0.3;
[0148] J=120 Nm corresponding to the inertia of the moveable unit
4;
[0149] k.theta.=J(10.2.pi.).sup.2 corresponding to a natural
rotational frequency of 10 Hz;
[0150] r.multidot.=0.03{square root}{square root over (k.theta.J)}
corresponding to a normalized rotational damping of 0.3;
[0151] d=0.01 m corresponding to the off-centering of the moveable
unit 4;
[0152] t1-t0=0.4 s; and
[0153] x1-x0=25 mm.
[0154] The moveable unit 4 is displaced from the position x0 at
rest (x0.sup.(1)=0) at the instant t0, to the position x1 at rest
(x1.sup.(1)=0) at the instant t1. It is therefore displaced over a
distance of 25 mm in 0.4 s. To obtain this displacement, as well as
the immobilization (at the end of said displacement) of the various
motions to which the displacement gives rise, the force F
represented in FIG. 7 should be applied to said moveable unit 4.
This force is given by expression (3A) with y given by (3) for
.alpha.=7 and .beta.=6. In this case, the coefficients a0 up to a6
are given by a0=1716, a1=-9009, a2=20020, a3=-24024, a4=16380,
a5=-6006, a6=924.
[0155] In a second embodiment represented in FIG. 1, the system S1
comprises, in addition to the moveable unit 4, a number p of
auxiliary masses MAi, p being greater than or equal to 1, i going
from 1 to p, which are linked respectively by elastic links e1 to
ep of standard type, in particular springs, to said moveable unit
4. In the example represented, p=3.
[0156] In this case, the variables of the system are the position x
of the moveable unit 4 and the positions zi of the p auxiliary
masses MAi, which satisfy the relations: 20 { x = ( i = 1 p ( mi ki
s 2 + ri ki s + 1 ) ) y zi = ( j = 1 p j i ( mj kj s 2 + rj kj s +
1 ) ) ( ri ki s + 1 ) y ( 4 )
[0157] in which:
[0158] .PI. illustrates the product of the associated
expressions;
[0159] mi, zi, ki and ri are respectively the mass, the position,
the stiffness and the damping of an auxiliary mass MAi;
[0160] mj, kj and rj are respectively the mass, the stiffness and
the damping of an auxiliary mass MAj; and -s=d/dt.
[0161] Specifically, the dynamic model of the system S1 may be
written: 21 { Mx ( 2 ) = F + i = 1 p ( ki ( zi - x ) + ri ( zi ( 1
) - x ( 1 ) ) ) Mizi ( 2 ) = ki ( x - zi ) + ri ( x ( 1 ) - zi ( 1
) ) , i = 1 , p . ( 5 )
[0162] As in the foregoing, we wish to find laws of motion which
ensure the desired displacement of the moveable unit 4, the
auxiliary masses MAi (for example measurement devices and/or loads)
being immobilized as soon as the moveable unit 4 stops.
[0163] Accordingly, the intermediate variable y is calculated by
the same approach as earlier and the trajectory of the moveable
unit 4 is planned by way thereof.
[0164] The intermediate variable y being required to satisfy
x=P(s)y, zi=Pi(s)y, i=1, . . . , p, with 22 s = t ,
[0165] we must have, substituting these relations into the system
(5):
(mis.sup.2+ris+ki)Pi=(ris+ki)P, i=1, . . . , p
[0166] From this expression, we immediately derive: 23 P ( s ) = (
i = 1 p ( mi ki s 2 + ri ki s + 1 ) ) , Pi = ( j = 1 j i p ( mj kj
s 2 + rj kj s + 1 ) ) ( ri ki s + 1 ) ,
[0167] thereby proving the aforesaid formulae (4).
[0168] In this case, it may be demonstrated that the force F to be
applied satisfies the relation: 24 F ( t ) = [ ( Ms 2 + ( j = 1 p
rj ) s + ( j = 1 p kj ) ) i = 1 p ( mi ki s 2 + ri ki s + 1 ) - i =
1 p ( ris + ki ) j = 1 j i p ( mj kj s 2 + rj kj s + 1 ) ] y .
[0169] The aforesaid formulae are verified and specified
hereinbelow for two and three auxiliary masses MAi
respectively.
[0170] In the case of two auxiliary masses (p=2), the model may be
written: 25 { Mx ( 2 ) = F - k1 ( x - z1 ) - r1 ( x ( 1 ) - z1 ( 1
) ) - k2 ( x - z2 ) - r2 ( x ( 1 ) - z2 ( 1 ) ) m1z1 ( 2 ) = k1 ( x
- z1 ) + r1 ( x ( 1 ) - z1 ( 1 ) ) m2z2 ( 2 ) = k2 ( x - z2 ) + r2
( x ( 1 ) - z2 ( 1 ) )
[0171] From this we immediately deduce: 26 { x = ( m1 k1 s 2 + r1
k1 s + 1 ) ( m2 k2 s 2 + r2 k2 s + 1 ) y z1 = ( r1 k1 s + 1 ) ( m2
k2 s 2 + r2 k2 s + 1 ) y z2 = ( r2 k2 s + 1 ) ( m1 k1 s 2 + r1 k1 s
+ 1 ) y ( 6 )
[0172] i.e, putting 27 mi ki = Ti 2
[0173] and 28 ri ki = 2 DiTi , i = 1.2 : 29 { x = y + 2 ( D1T1 +
D2T2 ) y ( 1 ) + ( Ti 2 + T2 2 + 4 D1D2T2 ) y ( 2 ) + 2 ( D1T1T2 2
+ D2T2T1 2 ) y ( 3 ) + ( T1 2 T2 2 ) y ( 4 ) z1 = y + 2 ( D1T1 +
D2T2 ) y ( 1 ) + ( T2 2 + 4 D1D2T1T2 ) y ( 2 ) + ( 2 D1T1T2 2 ) y (
3 ) z2 = y + 2 ( D1T1 + D2T2 ) y ( 1 ) + ( T1 2 + 4 D1D2T1T2 ) y (
2 ) + ( 2 D2T2T1 2 ) y ( 3 ) .
[0174] The expression for y, or more precisely the expressions for
y, y.sup.(1), y.sup.(2), y.sup.(3), y.sup.(4) and y.sup.(5) are
deduced therefrom by inverting the system obtained on the basis of
x, z1, z2, x.sup.(1), z1.sup.(1), z2.sup.(1).
[0175] We deduce therefrom that, to perform a displacement from x0
at the instant t0 to x1 at the instant t1, with the auxiliary
masses at rest at t0 and t1, it is sufficient to construct a
reference trajectory for y with the initial and final
conditions
[0176] y(t0)=x0, y(t1)=x1 and all the derivatives
y.sup.(k)(t0)=y.sup.(k)(- t1)=0, k varying from 1 to 6 or more if
necessary, and to deduce therefrom the reference trajectories of
the main and auxiliary masses, as well as of the force F to be
applied to the motor.
[0177] In this case, the force F satisfies the relation: 30 F ( t )
= [ ( Ms 2 + ( r1 + r2 ) s + ( k1 + k2 ) ) ( m1 k1 s 2 + r1 k1 s +
1 ) ( m2 k2 s 2 + r2 k2 s + 1 ) - ( r1s + k1 ) ( m2 k2 s 2 + r2 k2
s + 1 ) - ( r2s + k2 ) ( m1 k1 s 2 + r1 k1 s + 1 ) ] y .
[0178] Furthermore, the model for three auxiliary masses MAi (p=3)
[see FIG. 1], may be written, as earlier: 31 { Mx ( 2 ) = F - k1 (
x - z1 ) - r1 ( x ( 1 ) - z1 ( 1 ) ) - k2 ( x - z2 ) - r2 ( x ( 1 )
- z2 ( 1 ) ) - k3 ( x - z3 ) - r3 ( x ( 1 ) - z3 ( 1 ) ) m1z1 ( 2 )
= k1 ( x - z1 ) + r1 ( x ( 1 ) - z1 ( 1 ) ) m2z2 ( 2 ) = k2 ( x -
z2 ) + r2 ( x ( 1 ) - z2 ( 1 ) ) m3z3 ( 2 ) = k3 ( x - z3 ) + r3 (
x ( 1 ) - z3 ( 1 ) )
[0179] From this we immediately deduce: 32 { x = ( m1 k1 s 2 + r1
k1 s + 1 ) ( m2 k2 s 2 + r2 k2 s + 1 ) ( m3 k3 s 2 + r3 k3 s + 1 )
y z1 = ( r1 k1 s + 1 ) ( m2 k2 s 2 + r2 k2 s + 1 ) ( m3 k3 s 2 + r3
k3 s + 1 ) y z2 = ( r2 k2 s + 1 ) ( m1 k1 s 2 + r1 k1 s + 1 ) ( m3
k3 s 2 + r3 k3 s + 1 ) y z3 = ( r3 k3 s + 1 ) ( m1 k1 s 2 + r1 k1 s
+ 1 ) ( m2 k2 s 2 + r2 k2 s + 1 ) y ( 7 )
[0180] We proceed as earlier in order to determine the values as a
function of time of the different variables and in particular of
the force F, the latter satisfying the expression: 33 F ( t ) = [ (
Ms 2 + ( r1 + r2 + r3 ) s + ( k1 + k2 + k3 ) ) . ( m1 k1 s 2 + r1
k1 s + 1 ) ( m2 k2 s 2 + r2 k2 s + 1 ) ( m3 k3 s 2 + r3 k3 s + 1 )
- ( r1s + k1 ) ( m2 k2 s 2 + r2 k2 s + 1 ) ( m3 k3 s 2 + r3 k3 s +
1 ) - ( r2s + k2 ) ( m1 k1 s 2 + r1 k1 s + 1 ) ( m3 k3 s 2 + r3 k3
s + 1 ) - ( r3s + k3 ) ( m1 k1 s 2 + r1 k1 s + 1 ) ( m2 k2 s 2 + r2
k2 s + 1 ) ] y .
[0181] Represented in FIGS. 8 to 13 are the values respectively of
the variables y, x, z1, z2, z3 and F as a function of time t for a
particular example of the embodiment of FIG. 1, z1 to z3 being the
displacements of the auxiliary masses MA1, MA2 and MA3
respectively. The variables y, x, z1, z2 and z3 are expressed in
meters (m) and the force F in Newtons (N).
[0182] This example is such that:
[0183] M=5 kg;
[0184] m1=0.1 kg;
[0185] m2=0.01 kg;
[0186] m3=0.5 kg;
[0187] k1=m1(5.2.pi.).sup.2, k2=m2(4.2.pi.).sup.2,
k3=m3(6.2.pi.).sup.2, corresponding to natural frequencies of 5, 4
and 6 Hz respectively;
[0188] r1=0.3{square root}{square root over (k1m1)}, r2=0.2{square
root}{square root over (k2m2)}, r3=0.15{square root}{square root
over (k3m3)}; corresponding to normalized dampings of 0.3, 0.2 and
0.15 respectively;
[0189] t1-t0=0.34 s; and
[0190] x1-x0=40 mm.
[0191] Additionally, in a third embodiment represented in FIG. 2,
the system S2 comprises the moveable unit 4, the base 2 which is
mounted elastically with respect to the floor S and an auxiliary
mass MA which is linked by way of an elastic link eA of standard
type to said base 2.
[0192] In this case, the variables of the system are the positions
x, xB and zA of the moveable unit 4, of the base B and of the
auxiliary mass MA, which satisfy the relations: 34 { x = [ ( mAs 2
+ rAs + kA ) ( mBs 2 + ( rA + rB ) s + ( kA + kB ) ) - ( rAs + kA )
2 ] y xB = - My ( 2 ) zA = - M ( rAy ( 3 ) + kAy ( 2 ) ) ( 8 )
[0193] in which:
[0194] M, mB and mA are the masses respectively of the moveable
unit 4, of the base 2 and of the auxiliary mass MA;
[0195] rA and rB are the dampings respectively of the auxiliary
mass MA and of the base 2;
[0196] kA and kB are the stiffnesses respectively of the auxiliary
mass MA and of the base 2; and
[0197] s=d/dt.
[0198] Specifically, the dynamic model of the system S2 may be
written: 35 { Mx ( 2 ) = F mBxB ( 2 ) = - F - kBxB - rBxB ( 1 ) - k
( xB - zA ) - rA ( xB ( 1 ) - zA ( 1 ) ) mzA ( 2 ) = kA ( xB - zA )
+ rA ( xB ( 1 ) - zA ( 1 ) ) ( 9 )
[0199] The intermediate variable must satisfy: x=P(s)y, xB=PB(s)y
and zA=Pz(s)y with 36 s = t .
[0200] Substituting these expressions into (9), we obtain: 37 { F =
Ms 2 P ( s ) y ( mBs 2 + ( rA + rB ) s + ( kA + kB ) ) PB ( s ) = -
Ms 2 P ( s ) + ( rAs + kA ) Pz ( s ) ( mAs 2 + rAs + kA ) Pz ( s )
= ( rAs + kA ) PB ( s ) .
[0201] On eliminating Pz from the last equation, it follows
that:
[(mAs.sup.2+rAs+kA)(mBs.sup.2+(rA+rB)s+(kA+kB))-(rAs+kA).sup.2]PB=-(mAs.su-
p.2+rAs+kA)Ms.sup.2P
[0202] from which we derive: 38 { P = ( mAs 2 + rAs + kA ) ( mBs 2
+ ( rA + rB ) s + ( kA + kB ) ) - ( rAs + kA ) 2 PB = - Ms 2 Pz = -
Ms 2 ( rAs + kA )
[0203] thus making it possible to obtain the aforesaid expressions
(8).
[0204] The values as a function of time of the different variables,
and in particular the force F, are then obtained as before.
[0205] In this case, said force F satisfies the expression:
F(t)=M[(mAs.sup.2+rAs+kA)(mBs.sup.2+(rA+rB)s+(kA+kB))-(rAs+kA).sup.2]y.sup-
.(2).
[0206] In a fourth and last embodiment (not represented), the
system is formed of the moveable unit 4, of the base 2 and of an
auxiliary mass MC which is tied elastically to said moveable unit
4.
[0207] In this case, the variables of the system are the positions
x, xB and zC respectively of the moveable unit 4, of the base 2 and
of the auxiliary mass MC, which satisfy the relations: 39 { x = [ (
mCs 2 + rCs + kC ) ( mBs 2 + rBs + kB ] y xB = [ ( mCs 2 + rCs + kC
) ( Ms 2 + rCs + kC ) - ( rCs + kC ) 2 ] y zC = ( rCs + kC ) ( mBs
2 + rBs + kB ) y
[0208] in which:
[0209] M, mB and mC are the masses respectively of the moveable
unit 4, of the base 2 and of the auxiliary mass MC;
[0210] rB and rC are the dampings respectively of the base 2 and of
the auxiliary mass MC;
[0211] kB and kC are the stiffnesses respectively of the base 2 and
of the auxiliary mass MC; and
[0212] s=d/dt.
[0213] Specifically, the dynamic model of this system may be
written: 40 { Mx ( 2 ) = F - kC ( x - zC ) - rC ( x ( 1 ) - zC ( 1
) ) mBxB ( 2 ) = - F - kBxB - rBxB ( 1 ) mCzC ( 2 ) = kC ( x - zC )
+ rC ( x ( 1 ) - zC ( 1 ) ) ( 10 )
[0214] By using, as in the foregoing, the polynomial representation
of the variable 41 s = t ,
[0215] the system (10) becomes: 42 { ( Ms 2 + rCs + kC ) x = F + (
rCs + kC ) zC ( mBs 2 + rBs + kB ) xB = - F ( mCs 2 + rCs + kC ) zC
= ( rCs + kC ) x ,
[0216] which, together with the expressions for each of the
variables as a function of the intermediate variable (and of its
derivatives) x=P(s)y, xB=PB(s)y, zC=Pz(s)y, finally gives: 43 { P =
( mCs 2 + rCs + kC ) ( mBs 2 + rBs + kB ) PB = ( mCs 2 + rCs + kC )
( Ms 2 + rCs + kC ) - ( rCs + kC ) 2 Pz = ( rCs + kC ) ( mBs 2 +
rBs + kB )
[0217] The construction of the reference trajectories of y, and
then of x, xB, zC and F is done as indicated earlier.
[0218] In this case, the force F satisfies:
F(t)=-(mBs.sup.2+rBs+kB)[(mCs.sup.2+rCs+kC)(Ms.sup.2+rCs+kC)-(rCs+kC).sup.-
2]y.
[0219] A method in accordance with the invention will now be
described which makes it possible to determine in a general and
fast manner the expressions defined in the aforesaid step b) of the
process in accordance with the invention, for linear systems of the
form: 44 j = 1 p Ai , j ( s ) xj = biF , i = 1 , , p ( 11 )
[0220] where the Ai,j(s) are polynomials of the variable s, which,
in the case of coupled mechanical systems, are of degree less than
or equal to 2 and where one at least of the coefficients bi is
non-zero. F is the control input which, in the above examples, is
the force produced by the actuator 5.
[0221] Accordingly, according to the invention, in step b), the
following operations are carried out:
[0222] the different variables xi of said system (for example S1 or
S2), i going from 1 to p, p being an integer greater than or equal
to 2, each being required to satisfy a first expression of the
form: 45 xi = j = 0 j = r pi , j y ( j ) ,
[0223] the y.sup.(j) being the derivatives of order j of the
intermediate variable y, r being a predetermined integer and the
pi, j being parameters to be determined, a second expression is
obtained by putting y.sup.(j)=s.sup.(j).y: 46 xi = ( j = 0 j = r pi
, j s j ) y = Pi ( s ) y
[0224] a third expression of vectorial type is defined on the basis
of the second expressions relating to the different variables xi of
the system: 47 X = P y { P = ( P1 P P ) X = ( x1 xp )
[0225] comprising the vectors
[0226] said vector P is calculated, replacing X by the value P.y in
the following expressions: 48 { B T A ( s ) P ( s ) = Op - 1 bp F =
( j = 1 j = p Ap , j ( s ) Pj ( s ) Y )
[0227] in which:
[0228] B.sup.T is the transpose of a matrix B of size px(p 1) and
of rank p-1, such that B.sup.Tb=Op-1;
[0229] bp is the p-th component of the vector b; and
[0230] Op-1 is a zero vector of dimension (p-1);
[0231] the values of the different parameters pi,j are deduced from
the value thus calculated of the vector P; and
[0232] from these latter values are deduced the values of the
variables xi as a function of the intermediate variable y and of
its derivatives, on each occasion using the corresponding first
expression.
[0233] The aforesaid method is now justified.
[0234] Let us denote by A(s) the matrix of size pxp whose
coefficients are the polynomials Ai, j(s), i, j=1, . . . , p, i.e.:
49 A ( s ) = ( A1 , 1 ( s ) Ap , 1 ( s ) A1 , p ( s ) Ap , p ( s )
) , X = ( x1 xp ) and b = ( b1 bp )
[0235] Without loss of generality, it can be assumed that the rank
of A(s) is equal to p (otherwise, the system is written together
with its redundant equations and it is sufficient to eliminate the
dependent equations) and that bp.noteq.0. There then exists a
matrix B of size px(p-1) and of rank p-1such that:
B.sup.Tb=0p-1
[0236] where T represents transposition and 0p-1 the vector of
dimension p-1, all of whose components are zero. The system (11)
premultiplied by B.sup.T then becomes: 50 B T A ( s ) X = Op - 1 ,
bpF = j = 1 p Ap , j .times. j . ( 12 )
[0237] As indicated earlier, an intermediate variable y is
characterized in that all the components of the vector X can be
expressed as a function of y and of a finite number of its
derivatives. For a controllable linear system, such an output
always exists and the components of X can be found in the form of
linear combinations of y and of its derivatives, i.e.: 51 xi = j =
0 r pi , jy ( j )
[0238] where y.sup.(j) is the derivative of order j of y with
respect to time and where the pi,j are real numbers which are not
all zero, or alternatively: 52 xi = ( j = 0 r pi , js j ) y = Pi (
s ) y , i = 1 , , p .
[0239] We shall calculate the vector 53 P ( s ) = ( P1 ( s ) Pp ( s
) ) ,
[0240] by replacing X by its value P(s)y in (12): 54 bpF = j = 1 p
Ap , j ( s ) Pj ( s ) y . ( 13 )
[0241] Consequently, P belongs to the kernel of the matrix
B.sup.TA(s) of dimension 1, since B is of rank p-1 and A(s) of rank
p. To calculate P, let us denote by A1(s), . . . , Ap(s) the
columns of the matrix A(s) and (s) the matrix of size
(p-1).times.(p-1) defined by:
(s)=(A2(s), . . . , Ap(s)).
[0242] Let us also denote by P(s) the vector of dimension p-1
defined by: 55 P ^ ( s ) = ( P2 ( s ) Pp ( s ) ) .
[0243] Let us rewrite (13) in the form
B.sup.TA1(s)P1(s)+B.sup.T(s){circum- flex over (P)}(s)=0p-1 or
alternatively B.sup.T(s){circumflex over
(P)}(s)=-B.sup.TA1(s)P1(s). Since the matrix B.sup.T(s) is
invertible, we have:
{circumflex over (P)}(s)=-(B.sup.T(s)).sup.-1B.sup.TA1(s)P1(s)
[0244] i.e.: 56 P ^ ( s ) = - 1 det ( B T A ^ ( s ) ) ( co ( B T A
^ ( s ) ) ) T B T A1 ( s ) P1 ( s ) ( 14 )
[0245] where co(B.sup.T(s)) is the matrix of the cofactors of
B.sup.T(s).
[0246] From this we immediately deduce that it is sufficient to
choose: 57 { P1 ( s ) = det ( B T A ^ ( s ) ) P ^ ( s ) = - ( co (
B T A ^ ( s ) ) ) T B T A1 ( s ) ( 15 )
[0247] this completing the calculation of the vector P(s).
[0248] It will be observed that if the Ai,j(s) are polynomials of
degree less than or equal to m, the degree of each of the
components of P is less than or equal to mp. Specifically, in this
case, the degree of the determinant det (B.sup.T(s)) is less than
or equal to (p-1)m and the degree of each of the rows of (co
(B.sup.T(s))).sup.TB.sup.T A1(s), using the fact that the degree of
a product of polynomials is less than or equal to the sum of the
degrees, is less than or equal to (p-1)m+m=pm, hence the aforesaid
result.
[0249] In all the examples presented earlier, which model
mechanical subsystems, we have m=2.
[0250] It may easily be verified that this general method yields
the same calculations for P as in each of the examples already
presented hereinabove.
[0251] We shall return to certain of the examples dealt with
earlier and show how the calculation of the variable y makes it
possible to achieve passive isolation of the elastic modes.
[0252] In all these examples, the trajectories are generated on the
basis of polynomial trajectories of the intermediate value y, which
are obtained through interpolation of the initial and final
conditions. Furthermore, we are interested only in the particular
case where the system is at rest at the initial and final instants,
thereby making it possible to establish simple and standard
formulae which depend only on the degree of the polynomial.
[0253] In the simplest case, where the initial and final
derivatives of y are zero up to order 4, the sought-after
polynomial is of degree 9: 58 { y ( t0 ) = y0 y ( 1 ) ( t0 ) = 0 y
( 2 ) ( t0 ) = 0 y ( 3 ) ( t0 ) = 0 y ( 4 ) ( t0 ) = 0 y ( t1 ) =
y1 y ( 1 ) ( t1 ) = 0 y ( 2 ) ( t1 ) = 0 y ( 3 ) ( t1 ) = 0 y ( 4 )
( t1 ) = 0
[0254] which gives:
y(t)=y0+(y1-y0).sigma..sup.5(126-420.sigma.+540.sigma..sup.2-315.sigma..su-
p.3+70.sigma..sup.4), 59 = ( t - t0 t1 - t0 ) ( 16 )
[0255] If we ask for a polynomial such that the initial and final
derivatives are zero up to order 5, the sought-after polynomial is
of degree 11:
y(t)=y0+(y1-y0).sigma..sup.6(462-1980.sigma.+3465.sigma..sup.2-3080.sigma.-
.sup.3+1386.sigma..sup.4-252.sigma..sup.5)
[0256] still with .sigma. defined as in (16).
[0257] If we ask for a polynomial such that the initial and final
derivatives are zero up to order 6, the sought-after polynomial is
of degree 13:
y(t)=y0+(y1-y0).sigma..sup.7(1716-9009.sigma.+20020.sigma..sup.2-24024.sig-
ma..sup.3+16380.sigma..sup.4-6006.sigma..sup.5+924.sigma..sup.6).
* * * * *