1 Derivation of Zero Vibration (ZV) input shaper University of Technology System and Control...
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Toyohashi University of Technology System and Control Laboratory 1 Derivation of Z ero V ibration (ZV) input shaper Denote G is a second order system, ω and ζ are natural frequency and damping ratio of G respectively. The time response of G to an impulse, whose amplitude and time location are A i and t i , is given as y i (t)= " A i ω p 1 - ζ 2 e -ζω(t-t i ) # sin ω p 1 - ζ 2 (t - t i ) . (1) Based on the superposition law for linear system, the response of system G to a series of n impulses is given as follows y Σ (t)= n X i=1 " A i ω p 1 - ζ 2 e -ζω(t-t i ) # sin ω p 1 - ζ 2 (t - t i ) . (2) By using following trigonometric identity n X i=1 B i sin(ωt +Φ i )= A Σ sin(ωt + ψ), where A Σ = v u u t n X i=1 B i cos Φ i ! 2 + n X i=1 B i sin Φ i ! 2 , (3) we can obtain the magnitude of y Σ (t) as follows A Σ (t)= ωe -ζωt p 1 - ζ 2 v u u t " n X i=1 A i e ζωt i sin ω p 1 - ζ 2 t i # 2 + " n X i=1 A i e ζωt i cos ω p 1 - ζ 2 t i # 2 (4) The vibration percentage of magnitude response between the impulse sequence and an unity impulse which is applied at t = 0 is given as V (ω,ζ )= A Σ A ↑ = ωe -ζωt p 1 - ζ 2 r h ∑ n i=1 A i e ζωt i sin ω p 1 - ζ 2 t i i 2 + h ∑ n i=1 A i e ζωt i cos ω p 1 - ζ 2 t i i 2 ωe -ζωt p 1 - ζ 2 Δ = q V 2 1 (ζ,ω)+ V 2 2 (ζ,ω), (5) where V 1 (ω,ζ )= n X i=1 A i e ζωt i sin(ω p 1 - ζ 2 t i ) and V 2 (ω,ζ )= n X i=1 A i e ζωt i cos(ω p 1 - ζ 2 t i ). Note that the ratio V (ω,ζ ) represents the vibration percentage between input shaping to that without input shaping. The ZV input shaper can be obtained by letting V (ω,ζ ) = 0, which yields in ( V 1 (ω,ζ ) = 0 V 2 (ω,ζ ) = 0 ⇒ ( ∑ n i=1 A i e ζωt i sin(ω p 1 - ζ 2 t i ) = 0 ∑ n i=1 A i e ζωt i cos(ω p 1 - ζ 2 t i ) = 0 (6) Without lost of generality, assume there has two impulses in the sequence and the first impulse is applied at t = 0 (i.e. t 1 = 0), hence A 2 e ζωt 2 sin ω p 1 - ζ 2 t 2 =0 A 1 + A 2 e ζωt 2 cos ω p 1 - ζ 2 t 2 =0 (7) (8) From (7), we have sin(ω p 1 - ζ 2 t 2 ) = 0, therefore ω p 1 - ζ 2 t 2 = kπ (k ∈ Z + ) thus t 2 = kπ ω √ 1-ζ 2 . The minimum time of t 2 is chosen as t 2 = π ω √ 1-ζ 2 . Substituting t 2 into (8), we have A 1 = A 2 e ζπ √ 1-ζ 2 . In practice, in order to preserve the response of original command, following constraint should be hold 2 X i=1 A i =1. (9) 1
Transcript of 1 Derivation of Zero Vibration (ZV) input shaper University of Technology System and Control...
Toyohashi University of Technology System and Control Laboratory
1 Derivation of Zero Vibration (ZV) input shaper
Denote G is a second order system, ω and ζ are natural frequency and damping ratio of G respectively.The time response of G to an impulse, whose amplitude and time location are Ai and ti, is given as
yi(t) =
[Aiω√1− ζ2
e−ζω(t−ti)
]sin(ω√
1− ζ2(t− ti)). (1)
Based on the superposition law for linear system, the response of system G to a series of n impulses isgiven as follows
yΣ(t) =
n∑i=1
[Aiω√1− ζ2
e−ζω(t−ti)
]sin(ω√
1− ζ2(t− ti)). (2)
By using following trigonometric identity
n∑i=1
Bi sin(ωt+ Φi) = AΣ sin(ωt+ ψ), where AΣ =
√√√√( n∑i=1
Bi cos Φi
)2
+
(n∑i=1
Bi sin Φi
)2
, (3)
we can obtain the magnitude of yΣ(t) as follows
AΣ(t) =ωe−ζωt√
1− ζ2
√√√√[ n∑i=1
Aieζωti sin(ω√
1− ζ2ti
)]2
+
[n∑i=1
Aieζωti cos(ω√
1− ζ2ti
)]2
(4)
The vibration percentage of magnitude response between the impulse sequence and an unity impulsewhich is applied at t = 0 is given as
V (ω, ζ) =AΣ
A↑
=
ωe−ζωt√1− ζ2
√[∑ni=1Aie
ζωti sin(ω√
1− ζ2ti
)]2+[∑n
i=1Aieζωti cos
(ω√
1− ζ2ti
)]2
ωe−ζωt√1− ζ2
∆=√V 2
1 (ζ, ω) + V 22 (ζ, ω), (5)
where V1(ω, ζ) =
n∑i=1
Aieζωti sin(ω
√1− ζ2ti) and V2(ω, ζ) =
n∑i=1
Aieζωti cos(ω
√1− ζ2ti). Note that the
ratio V (ω, ζ) represents the vibration percentage between input shaping to that without input shaping.
The ZV input shaper can be obtained by letting V (ω, ζ) = 0, which yields in{V1(ω, ζ) = 0
V2(ω, ζ) = 0⇒
{ ∑ni=1Aie
ζωti sin(ω√
1− ζ2ti) = 0∑ni=1Aie
ζωti cos(ω√
1− ζ2ti) = 0(6)
Without lost of generality, assume there has two impulses in the sequence and the first impulse is appliedat t = 0 (i.e. t1 = 0), hence
A2eζωt2 sin
(ω√
1− ζ2t2
)= 0
A1 +A2eζωt2 cos
(ω√
1− ζ2t2
)= 0
(7)
(8)
From (7), we have sin(ω√
1− ζ2t2) = 0, therefore ω√
1− ζ2t2 = kπ (k ∈ Z+) thus t2 = kπ
ω√
1−ζ2. The
minimum time of t2 is chosen as t2 = π
ω√
1−ζ2. Substituting t2 into (8), we have A1 = A2e
ζπ√1−ζ2 . In
practice, in order to preserve the response of original command, following constraint should be hold
2∑i=1
Ai = 1. (9)
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Toyohashi University of Technology System and Control Laboratory
Therefore, A1 +A2 = 1
A1 = A2expζπ√1− ζ2
⇒
A1 =
1
1 +K
A2 =K
1 +K,
(10)
where K = e
−ζπ√1− ζ2
. In conclusion, ZV input shaper is given asAiti
=
1
1 +K
K
1 +K
0π
ω√
1− ζ2
. (11)
2 Derivation of Zero Vibration Derivative (ZVD) input shaper
In order to improve the robustness of ZV input shaper with respect to the uncertainty of natural frequency,following constraint should be added
dV
dω
∣∣∣∣ω=ωn
= 0, (12)
where ωn is modeled natural frequency whilst ω denotes the actual natural frequency of the system. We
have V (ω, ζ) =√V 2
1 (ω, ζ) + V 22 (ω, ζ), therefore
dV
dω=
(∂V
∂V1
dV1
dω+∂V
∂V2
dV2
dω
), (13)
By choosingdV1
dω
∣∣∣∣ω=ωn
= 0 anddV2
dω
∣∣∣∣ω=ωn
= 0, (13) can be fulfilled. We have
V1 =n∑i=1
Aieζωti sin
(ω√
1− ζ2ti
)⇒ dV1
dω=
n∑i=1
Aitieζωti
[ζ sin
(ω√
1− ζ2ti
)+√
1− ζ2 cos(ω√
1− ζ2ti
)]. (14)
By using following identity
a sinx+ b cosx = c sin(x+ φ), where c =√a2 + b2 and φ = atan(b, a), (15)
the (14) can be further written as
dV1
dω=
n∑i=1
Aitieζωti sin
[ω√
1− ζ2ti + atan
(√1− ζ2
ζ
)]. (16)
In the same manner, it can be obtained that
dV2
dω=
n∑i=1
Aitieζωti cos
[ω√
1− ζ2ti + atan
(√1− ζ2
ζ
)]. (17)
It is recognized that two equations (16) and (17) are added, therefore it needs two more unknowns. Thiscan be satisfied if one more impulse is appended, thus ZVD input shaper will have 3 impulses in thesequence, and is governed by
V1(ωn, ζ) = 0
V2(ωn, ζ) = 0
dV1
dω
∣∣∣∣ω=ωn
= 0
dV2
dω
∣∣∣∣ω=ωn
= 0.
⇒
∑3i=1Aie
ζωnti sin(ωn√
1− ζ2ti
)= 0
∑3i=1Aie
ζωnti cos(ωn√
1− ζ2ti
)= 0
∑3i=1Aitie
ζωnti sin
[ωn√
1− ζ2ti + atan
(√1− ζ2
ζ
)]= 0
∑3i=1Aitie
ζωnti cos
[ωn√
1− ζ2ti + atan
(√1− ζ2
ζ
)]= 0.
(18)
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Toyohashi University of Technology System and Control Laboratory
Without lost of generality, assume t1 = 0 and A1 = 1 (normalization will be carried out later), hence wehave
A2eζωnt2 sin
(ωn√
1− ζ2t2
)+A3e
ζωnt3 sin(ωn√
1− ζ2t3
)= 0
1 +A2eζωnt2 cos
(ωn√
1− ζ2t2
)+A3e
ζωnt3 cos(ωn√
1− ζ2t3
)= 0
A2t2eζωnt2 sin
[ωn√
1− ζ2t2 + atan
(√1− ζ2
ζ
)]+A3t3e
ζωnt3 sin
[ωn√
1− ζ2t3 + atan
(√1− ζ2
ζ
)]= 0
A2t2eζωnt2 cos
[ωn√
1− ζ2t2 + atan
(√1− ζ2
ζ
)]+A3t3e
ζωnt3 cos
[ωn√
1− ζ2t3 + atan
(√1− ζ2
ζ
)]= 0.
(19)
(20)
(21)
(22)
Square sum of (19) and (20) yields
A22e
2ζωnt2 +A23e
2ζωnt3 + 2A2A3eζωnt2eζωnt3 cos
[ωn√
1− ζ2 (t3 − t2)]
= 1. (23)
By choosing
cos[ωn√
1− ζ2(t3 − t2)]
= −1, (24)
we obtain
A2eζωnt2 −A3e
ζωnt3 = 1
⇒ −A3 +A2e−ζωn(t3−t2) = e−ζωnt3 . (25)
From (24), we have ωn√
1− ζ2(t3− t2) = π thus t3− t2 =π
ωn√
1− ζ2. Substituting t3 = t2 +
π
ωn√
1− ζ2
into (19) and also multiple e−ζωnt3 both sides, we have
A2e−ζ(t3−t2)ωn sin(t2ωn
√1− ζ2) +A3 sin
[(t2 +
π
ωn√
1− ζ2
)ωn√
1− ζ2
]= 0
⇒ A2e−ζ(t3−t2)ωn sin(t2ωn
√1− ζ2) +A3 sin
(t2ωn
√1− ζ2 + π
)= 0
⇒(A2e
−ζ(t3−t2)ωn −A3
)sin(t2ωn
√1− ζ2
)= 0
⇒ e−ζωnt3 sin(t2ωn√
1− ζ2) = 0
⇒ sin(t2ωn
√1− ζ2
)= 0. (26)
From (26), it can be obtained that t2ωn√
1− ζ2 = π thus t2 =π
ωn√
1− ζ2, which also results in
t3 =2π
ωn√
1− ζ2.
Now, square sum of (21) and (22) yields
A22t
22e
2ζωnt2 +A33t
23e
2ζωnt3 + 2A2A3t2t3eζωnt2eζωnt3 cos
(ωn√
1− ζ2(t3 − t2))
= 0. (27)
But cos[ωn√
1− ζ2(t3 − t2)]
= −1 (as in (24)), thus(A2t2e
ζωnt2 −A3t3eζωnt3
)2= 0. (28)
The amplitude of A2 and A3 can be solved by adjoining (25) and (28) A2t2eζωnt2 = A3t3e
ζωnt3
A3 = A2e−ζωn(t3−t2) − e−ζωnt3 .
(29)
Substituting t2 =π
ωn√
1− ζ2and t3 =
2π
ωn√
1− ζ2into (29) and solving A2, A3. The result is easily
derived as A2 = 2e
−ζπ√1− ζ2
A3 = e
−2ζπ√1− ζ2
,
, or
A2 = 2K
A3 = K2,, where K = e
−ζπ√1− ζ2
.
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Toyohashi University of Technology System and Control Laboratory
Normalizing A1, A2, and A3 such that
3∑i=1
Ai = 1, the ZVD input shaper is given as follows
Aiti
=
1
K2 + 2K + 1
2K
K2 + 2K + 1
K2
K2 + 2K + 1
0π
ωn√
1− ζ2
2π
ωn√
1− ζ2.
(30)
3 Derivation of multi-mode input shaper
Suppose the system has two modes, e.g. multi-body system with two distinct natural frequencies ω1,ω2, and damping ratios ζ1, ζ2. In this case, input shaping sequence can be obtained by convolving twoimpulse sequences designed for each mode. The procedure is illustrated in Fig. 1, where “ * ” notationdenotes the convolution operator.
1
11KK
1
11 K
time
2
21KK
2
11 K
time0 1T 0 2T
time
1A
2A3A
4A
1T 2T 1 2T T0
* (a) (b) (c)
Figure 1: (a) ZV impulse sequence for the first mode, (b) ZV impulse sequence for the second mode, (c)ZV impulse sequence for the multi-mode.
In Fig. 1, the convolution process gives
A1 =1
(1 +K1)(1 +K2)
A2 =K2
(1 +K1)(1 +K2)
A3 =K1
(1 +K1)(1 +K2)
A4 =K1K2
(1 +K1)(1 +K2)
. (31)
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