US 2024/0202394 A1 · METHOD FOR DYNAMIC OUTPUT FORCE DISTRIBUTION OF MULTI-DEGREE-OF-FREEDOM OVER-ACTUATED MOTION STAGE

Abstract

A method for dynamic output force distribution of a multi-degree-of-freedom over-actuated motion stage includes: establishing a mathematical model G(s) of a motion stage based on a modal representation; setting a time delay interval parameter Δτ and an instruction shaper order nCS, and calculating each pulse time delay parameter τi; calculating matrices Tb and Vb; initializing q=1; defining Hq; calculating intermediate variable {tilde over (T)}jq, {tilde over (V)}, ΦHq, YHq, TΦHq, VΦHq, and βq; calculating an estimated value Ĥq of Hq; obtaining an amplitude Aikq of an instruction shaper; q=q+1, until calculation is completed for all logical axis channels; calculating α(s) by using Aikq; and obtaining a dynamic output force distribution matrix Tf(s). Zero-residue vibration suppression in all controllable flexible modalities can be realized with finite actuator redundancies.

A method for dynamic output force distribution of a multi-degree-of-freedom over-actuated motion stage, wherein the multi-degree-of-freedom over-actuated motion stage is formed of a mechanical structure and is operated in a control system that is a closed-loop feedback system that receives an input of a reference trajectory input and generates a tracking trajectory output in response to the reference trajectory input to control a motion of the multi-degree-of-freedom over-actuated motion stage, the method comprising the following steps:

step 1:
establishing a mathematical model G(s) of the multi-degree-of-freedom over-actuated motion stage based on a modal representation in accordance with the mechanical structure of the multi-degree-of-freedom over-actuated motion stage and a model identification result thereof:
G
⁡
(
s
)
[
C
C
]
[
(
s
)
(
s
)
]
[
B
B
]
(
1
)
(
s
)
[
m
⁢
s
_
⋱
m
n
R
⁢
s
_
]
⁢
s
[
ξω
⁢
s
⁢
ω
_
⋱
s
⁢
ξ
n
L
⁢
ω
n
L
⁢
s
⁢
ω
n
L
_
]
(
2
)
wherein:
B1 is a rigid modal input coupling matrix, and B2 is a flexible modal input coupling matrix;
C1 a rigid modal output coupling matrix, and C2 is a flexible modal output coupling matrix;
mq is a rigid modal quality parameter, and q is a serial number of a logical axis channel and q, , . . . , NR;
nR is a number of rigid modalities, which is equal to a total number of logical axis channels, and nL is a number of flexible modalities;
ξj is flexible modal damping, and ωj is a flexible modal frequency parameter, with a flexible modal order j, , . . . , nL;
step 2:
setting a time delay interval parameter Δτ and an instruction shaper order nCS, and calculating each pulse time delay parameter τi=Δτ(i−1) with an instruction shaper order i, , . . . , nCS;
step 3:
calculating matrices Tb and Vb:
{
T
b
=
B
T
BB
T
-
V
b
B
(
3
)
step 4:
initializing the serial number of the logical axis channel to q=1;
step 5:
defining a parameter vector Hq to be optimized in a qth column in α(s):
Hq[Aq. . . AnCSq . . . Anvq. . . AnCSnvq] (4)
wherein Aikq is an amplitude of an instruction shaper that has a serial number q of a logical axis channel, a serial number k of a redundancy, and the instruction shaper order of i;
the serial number k of the redundancy is k, , . . . , nv;
nv is a redundancy of a motion stage actuator, nv=nA−NR, and nA is the number of motion stage actuators;
step 6:
optimizing an optimization indicator
min
β
q
J
q
=

Φ
0
·
(
T
M
q
+
V
M
q
·
β
q
)

2
2
by using a method of least squares to calculate a parameter vector βq in Ĥq:
βq(VMqTTVMq)−(VMqTTVMq) (5)
wherein
Φ
0
=
[
1
/
ω
1
2
⋱
1
/
ω
n
L
2
]
⁢
T
M
q
=
[
T
~
1
q
+
V
~
V
⁢
1
·
T
Φ
H
q
·
Y
H
q
⋮
T
~
n
L
q
+
V
~
Vn
L
·
T
Φ
H
q
·
Y
H
q
]
⁢
V
M
q
=
[
V
~
V
⁢
1
·
V
Φ
H
q
⋮
V
~
Vn
L
·
V
Φ
H
q
]
⁢
V
~
Vj
=
[
V
~
j
⁢
1
,
…
︸
n
CS
V
~
j
⁢
2
,
…
︸
n
CS
…
V
~
jn
V
,
…
︸
n
CS
]
(
6
)
{
T
Φ
H
q
=
Φ
H
qT
(
Φ
H
q
·
Φ
H
qT
)
-
1
V
Φ
H
q
=
ker
⁡
(
Φ
H
q
)
(
7
)
Φ
H
q
=
[
φ
111
C
…
φ
n
CS
⁢
11
C
…
φ
11
⁢
n
V
C
…
φ
n
CS
⁢
1
⁢
n
V
C
⋮
⋮
⋮
⋮
φ
1
⁢
n
L
⁢
1
C
…
φ
n
CS
⁢
n
L
⁢
1
C
…
φ
1
⁢
n
L
⁢
n
V
C
…
φ
n
CS
⁢
n
L
⁢
n
V
C
φ
111
S
…
φ
n
CS
⁢
11
S
…
φ
11
⁢
n
V
S
…
φ
n
CS
⁢
1
⁢
n
V
S
⋮
⋮
⋮
⋮
φ
1
⁢
n
L
⁢
1
S
…
φ
n
CS
⁢
n
L
⁢
1
S
…
φ
1
⁢
n
L
⁢
n
V
S
…
φ
n
CS
⁢
n
L
⁢
n
V
S
]
(
8
)
{
φ
ijk
C
=
V
~
jk
·
e
ξ
j
⁢
ω
j
⁢
τ
i
⁢
cos
⁡
(
ω
dj
·
τ
i
)
φ
ijk
S
=
V
~
jk
·
e
ξ
j
⁢
ω
j
⁢
τ
i
⁢
sin
⁡
(
ω
dj
·
τ
i
)
ω
dj
=
ω
j
⁢
1
-
ξ
j
2
(
9
)
Y
H
q
=
[
-
T
~
q
…
-
T
~
j
q
…
-
T
~
n
L
q
…
]
(
10
)
{tilde over (T)}jq bjTbq is a qth column of Tb, and bj is a jth column of B2;
{tilde over (V)}jbjVb, {tilde over (V)}jk is a kth element of {tilde over (V)}j;
step 7:
calculating an estimated value Ĥq of Hq:
ĤqTqH·YHqVqH·βq·YHq (11);
step 8:
correspondingly obtaining the amplitude Aikq of the instruction shaper according to Hq defined in step 5 and Ĥq defined in step 7;
step 9:
letting q=q+1, determining whether q is greater than the total number of the logical axis channels;
if yes, proceeding to step 10;
and if no, skipping to step 5;
step 10:
calculating α(s) by using Aikq:
α
⁡
(
s
)
=
[
∑
i
=
1
n
CS
A
i
11
⁢
e
-
τ
i
⁢
s
∑
i
=
1
n
CS
A
i
12
⁢
e
-
τ
i
⁢
s
…
∑
i
=
1
n
CS
A
i
1
⁢
n
R
⁢
e
-
τ
i
⁢
s
⋮
⋮
⋱
⋮
∑
i
=
1
n
CS
A
i
n
V
⁢
1
⁢
e
-
τ
i
⁢
s
∑
i
=
1
n
CS
A
i
n
V
⁢
2
⁢
e
-
τ
i
⁢
s
…
∑
i
=
1
n
CS
A
i
n
V
⁢
n
R
⁢
e
-
τ
i
⁢
s
]
;
(
12
)
step 11:
obtaining a dynamic output force distribution matrix T(s):
Tf(s)TbVb·α(s) (13).

685 more elements

Method for dynamic output force distribution of multi-degree-of-freedom over-actuated motion stage

Life of this patent

Method for dynamic output force distribution of multi-degree-of-freedom over-actuated motion stage

Life of this patent

Abstract

Claims1