robust control design
TRANSCRIPT
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A Control Approach for Thrust-Propelled
Underactuated Vehicles and its Application to VTOL
Drones
Minh-Duc Hua, Tarek Hamel, Pascal Morin, Claude Samson
To cite this version:
Minh-Duc Hua, Tarek Hamel, Pascal Morin, Claude Samson. A Control Approach for Thrust-Propelled Underactuated Vehicles and its Application to VTOL Drones. IEEE Transactionson Automatic Control, Institute of Electrical and Electronics Engineers, 2009, VOL. 54 (NO.8), pp.1837-1853.
HAL Id: hal-00415854
https://hal-unice.archives-ouvertes.fr/hal-00415854
Submitted on 11 Sep 2009
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https://hal-unice.archives-ouvertes.fr/hal-00415854https://hal-unice.archives-ouvertes.fr/hal-00415854https://hal.archives-ouvertes.fr/ -
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ap por t
de r e c he r ch e
ISSN
0249-6399
ISRN
INRIA/RR--6
453--FR+ENG
Thme NUM
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Control of Thrust-Propelled Underactuated Vehicles
Minh-Duc HUA Tarek HAMEL Pascal MORIN Claude SAMSON
N 6453
February 2008
http://hal.archives-ouvertes.fr/ -
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Unit de recherche INRIA Sophia Antipolis
2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)Tlphone : +33 4 92 38 77 77 Tlcopie : +33 4 92 38 77 65
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H2
H
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G
m
J
I {O; o, o, k o}
o o k o B {G; , , k } k
k
k
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G I x =
(x1, x2, x3)T
OG= x1
o+ x2 0+ x3k o
OG= ( o, o, k o)x
B I
R
R
, ,k I
G I
x = (x1, x2, x3)T I v = (v1, v2, v3)T
B
v = ddtOG= ( o, o, k o)x= ( , , k)v
B I B = (1, 2, 3)T
I v f= ( o, o, k o)xf= ( , , k)vf
v a G v a = v v f xa = x xf
I va = v vf B v a = ( o, o, k o)xa = ( , , k)va
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{e1, e2, e3} R3 e1 = (1, 0, 0)T e2 = (0, 1, 0)T e3 = (0, 0, 1)T
u R3 S(u)
u
S(u)v= u
v
v
R3
R3 |.|
y : [to, +) Rp c
T |y(t)| c, t T y = h(x, t) Rp
c
x = f(x, t)
(xo, to)
y(.) = h(x(., xo, to))
c
x(, xo, to)
x = f(x, t)
xo
t= to
T
k
T =Tk
{G; k } G
Fe
F = Tk + Fe
Fe
{0; k o} I
(xa, xa, , )
R
|xa|
Fe
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Fe
Fe
Fe
Fe
x
mx= T Re3+ Fe(x,x,R,, , t)
Fe
Fe
B
() :
(1) :
xmv
R
=
RvmS()v T e3+ RTFe(x,x,R,, , t)
RS()
(2) : J = S()J+ + e(x, x,R,, , t)
= (1, 2, 3)T
e
e
= 0
(2)
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G |e|
T
(1)
Fe
x
t
t Fe(x, t) t Fex(x, t)
t Fet (x, t)
x
Fe
x
c1 0, c2 > 0
|Fe(x, t)| c1+ c2|x|2 , (x, t) R2 R
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c3 0, c4 > 0
xTFe(x, t) c3|x| c4|x|3 , (x, t) R2 R
F
e
c1
v r vr d
dtv r d2dt2 v r
(1)
x = Rvv = S()v ue3+ RTe(x, t)R = RS()
e(x, t) := Fe(x, t)/m
u := T /m
3
3 = 0
3(t)
R3 || = 1 I
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RT
e3
:= RT
(; ] e3 cos=3
= 0
1 = k2(1 + 3)2
TS(Re1)
2 = k1(1 + 3)2
TS(Re2)
R = RS() = 0
(, )
xr I xr
xr
v := RT(x xr) B
(x, t) := e(x, t) xr(t)
x= Rv
v= S()v ue3+ RT(x, t)
R= RS()
x :=t0
(x(s) xr(s)) ds x := x xr
xr
x xr
v
v 0 ue3+ RT(x, t) = 0
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(x, t)
= 0
(v, R) = (0, R)
e(xr(t), t)
>0
(x, t) |(x, t)|
||
= ||e3.
(; ] e3 ||
cos= 3||
= 0
=||e3 = =||e3
= 0
k1
k2
k3
u = 3+ ||k1v3
1 = ||k2v2 k3||2(|| + 3)2
1
||2 TS(Re1)
2 = ||k2v1+ k3||1(|| + 3)2
1
||2 TS(Re2)
(v,) = (0, 0)
R3 (, )
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V =1
2
vTv+ 1
k21
3
||=
1
2
vTv+ 1
k2
(1
cos)
(x, t) = e(x, t) xr(t)
Fe
e
e
x xr
Iv(t) :=
t0
(x(s) xr(s)) ds+ I0
I0
h
[0, +) ,
s > 0, h(s2)s <
s
R, 0 |c|
Iv = Rv
Iv
R3 (Iv, v,) = (Iv , 0, 0)
R3 R3 (, )
h
h
Iv
c
|| < g
e
c
xr
v
x xr
x= x xr
I0 = x(0) xr(0)
Iv = x
z
x
z = x
z
x
z = 2kzz k2z(z
z) + kzhz(|x|2)x (kz > 0, z(0) = 0)
hz
z, z
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x R3 |x| (x) =x
> 0 x R3 | (x)|
(c, x)
R3
R3
|c
| 0)
=
|z| |z| |z| +z/kz
2(kz + z)
6kz(kz + z)
y:= x + z
v:= v+ RTz
:= e xr+ h(|y|2)y+ z
e
e
h
, >0
k1
k2
k3
u = 3+ ||k1v3
1 = ||k2v2 k3||2(|| + 3)2
1
||2 TS(Re1)
2 = ||k2v1+ k3
|
|1
(|| + 3)2 1
||2 T
S(Re2)
v, ,
y
c:= e e
lims+
h(s2)s > |c|
> |z| z h(|z|2)z= c
h hz
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(z, z, x, v,) = (z, 0, 0, 0, 0)
R3
R3 R3 R3 (, )
h
h
x
h
kz
kz
z
|z| x= 0 kz
|z| || kz kz
kz
|| ||
kz
u
u || || > 0
u 0
k1
k2
k3
v
: R R (0) = 0
(s)> 1k1 ,s R
u = || + ||k1(v3) ( 0)
1 = ||k2
v2 v3 2|| + 3
k3||2
(|| + 3)2 1
||2 TS(Re1)
2 = ||k2
v1 v3 1|| + 3
+
k3||1(|| + 3)2
1
||2 TS(Re2)
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v
v
(s) = k1
tanh
k1s
,
(s) = s1 +
k21s2
2
0< 1
Fe(x, x) =ca|x|x max+mge3 ca, ma
mx= T Re3+ Fe(x) m= m + ma Fe(x) = ca|x|x + mge3 (x, t)
(x, t) =Fe(x)
m xr = ca
m|x|x + mg
me3 xr
cam |x|x 0
xr mg
me3
x
(x, t) = 0
xr
>0
t |(xr(t), t)|
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x
1/||
1/(|| +3)
= 0
C1 : [0, +) [0, 1]
(s) =
sin(s
2
22 ) ,
s 1 ,
>0
u = 3+ ||k1v3
1 = ||k2v2 (|| + 3) k3||2(|| + 3)2 (||)
1
||2 TS(Re1)
2 = ||k2v1+ (|| + 3) k3||1(|| + 3)2 (||) 1||2 TS(Re2)
u
1, 2
|u| 1+ 2|x|
x
e
e
x
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(x, t)
u
: R3 R (, ) R3 R3 () 1
T
() = ()T
() := min
1,
||
d C1
(x, t) := d(t) +
M(e,d(x, t)) xr(t)
e,d(x, t) := e(x, t) d(t)
d
M
Q > 0
x(t)
|(x(t), t)| Q
u
d
e = Fe/m
c1 0, c2 > 0, c3 0, c4 > 0 (x, t) R3 R
|e,d(x, t)| c1+ c2|x|2xTe,d(x, t) c3|x| c4|x|3
s
vr> 0
P(s) := c4s3 c2vrs2 c3s c1vr
c4 > 0
(ci, vr) vr s (ci, vr) P(s) 0
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k1
k2
k3
0< <
u
x
(v, ) = (0, 0)
M >c1+ c2vr
M
M
M c1+ c2((ci, vr))2 |(x, t)| (x, t) R3 (, )
M
d
xr
xr = 0
e(x) =(x) =g+ ae(x)
g = (mg/m)e3
ae(x) =(ca/m)|x|x ca > 0
M
d
d =g
= g +
M(e,d)
e,d = ae
g
M
M
M < mg/m |(x, t)|
mg/m M > 0 < mg/m M (v,) = (0, 0)
R3 (, )
M
Mc1+ c2((ci, vr))
2
c1 = c3 = 0
c2 = c4 = ca
(ci, vr) = vr
M c1 + c2((ci,vr))
2
M cav2r
sup |xr(t)| c1+ c2((ci, vr))2
M >c1+ c2((ci, vr+ 2z))2
{G; k } J
J diag(J1, J1, J2)
(1) :
x
v
R
=
Rv
S()v ue3+ gRTe3+ 1mRTFae 1mLS(e3)RS()
(2) : J = S()J+ + Mae
Fae
Fae
I
Fae
Mae
L
Fae
Mae
e:= ge3+ 1
mFae 1
mLRS(e3)
(1)
Fae
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e
x
R
v
kg
ms2
J1
kgm2
J2
kgm2
m
g = 9.81 ms2
m = 3.2 kg
J
= diag(0.13, 0.13, 0.04)
1
d,3 = 0
2
d
Mae
=S()Jd JK( d)
K
k1 = 0.24
k2 = 0.08
k3 = 12.8
K = diag(20; 20; 20)
h(s) = 1+2s/2
= 1.28
= 12
(s) = k1 tanh
k1s
= 0.9
kz = 0.8
hz(s) = z1+2zs/
2z
z = 0.8
z = 0.8
= 8
d= 0
M
M= 50
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= 1
k1
k2
k3
h(0), kz, hz(0)
(z = 0, z = 0, x= 0, v= 0, R= I)
G
x(0) = (8, 5, 8)T R(0) = I3
xr = (0, 0, 0)T
xf = (4, 0, 0)T 30 s
70 s
xf = (8, 0, 0)T
e =ge3
e
g = g
m= m
z, z, z
z
G
lims+
h(s2)s > |c| c= e e
h(s2)s
10
xf = 8e1
e
x
R
T
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= 6
= 1
xr(t) = (10 cos(t/10), 10 sin(t/10), t)T
x(0) = (45, 50, 10)T R(0) =I3
0 T 1.8mg = 56.5, |i=1,2,3| 0.3T L
.c
= 6
.e
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0 20 40 60 80 1008
6
4
2
0
2
4
6
8
10Fig. a) Position tracking error
t (s)
x(m)
x1 : x2 : x3 : .
0 20 40 60 80 10040
30
20
10
0
10
20
30
40
50Fig. b) Euler angles
t (s)
,,(o)
: : : .
= 12
0 20 40 60 80 1008
6
4
2
0
2
4
6
8
10Fig. a) Position tracking error
t (s)
x(m)
x1 : x1 : x2 : x2 : x3 : .x3 : .
0 20 40 60 80 10040
30
20
10
0
10
20
30
40
50Fig. b) Euler angles
t (s)
,,(o)
: : : .
= 12
= 8
0 20 40 60 80 1008
6
4
2
0
2
4
6
8
10Fig. a) Position tracking error
t (s)
x(m)
x1 : x2 : x3 : .
0 20 40 60 80 10020
10
0
10
20
30
40
50Fig. b) Euler angles
t (s)
,,(o)
: : : .
= 6
= 1
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0 10 20 30 40 50 60 70 80 90 10010
5
0
5
10
15
t (s)
xf
(ms1
)
xf,1 : xf,2 : xf,3 : .
0 20 40 60 80 10010
5
0
5
10
15
20Fig. a) Real apparent acceleration
t (s)
e
(ms2
)
e,1 : e,2 : e,3 : .
0 20 40 60 80 10010
5
0
5
10
15
20Fig. b) Estimated apparent acceleration
t (s)
e
(ms2
)
e,1 : e,2 : e,3 :.
e
e
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20 10 0 10 20 30 40 5020
10
0
10
20
30
40
50
60Fig. a) Projected trajectories on a horizontal plane
x1(m)
x2
(m
)
ref. trajectory : act.trajectory:
20 10 0 10 20 30 40 500
20
40
60
80
100
120Fig. b) Projected trajectories on a vertical plane
x1(m)
x3(
m)
ref. trajectory: act.trajectory:
0 20 40 60 80 10010
0
10
20
30
40
50Fig. c) Position tracking error
t (s)
x1,2,3
(m
) x1 :
x2 : x3 : .
0 20 40 60 80 100100
80
60
40
20
0
20
40
60Fig. d) Euler angles
t (s)
,,(o)
: : : .
0 20 40 60 80 10025
30
35
40
45
50
55
60Fig. e) Thrust control input
t (s)
T=
mu(N)
0 20 40 60 80 1002.5
2
1.5
1
0.5
0
0.5
1
1.5
2
2.5Fig. f) Torque control inputs
t (s)
1,2,3
(N.m
)
1 : 2 : 3 :.
= 6
= 1
e
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&
V = 1 3 = 1 cos
V R = RS()
|| = 1
V =
1 2 2
1
+
TS(Re2)TS(Re1)
1, 2
V = k 21 +
22
(1 + 3)2 = k
1 31 + 3
= kV1 + 3
kV2
0
V
= 0
tan2(/2) = 21 +
22
(|| + 3)2 =
|| 3|| + 3
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V
V
V = vT (
ue3+ ) +
1
||k2 1 2
21 +
1
||2
TS(Re2)TS(Re
1)
= v3(u + 3) + 1||k2
1 2 2
1
+
1
||2TS(Re2)
TS(Re1)
+ ||k2
v1v2
u
1
2
V = ||k1v23k3k2
21 + 22
(|| + 3)2 = ||k1v23
k3k2
tan2(/2)
v V
v3
V
v
v3
ddttan2(/2)
v
v = RT(x xr)
x
e
u
x
x= uRe3+ e(x, t)
e(x, t) =e
x(x, t)x +
et
(x, t),
x
x
e
>0
|(t)| ,t
d
dt(1 cos) = k2(1v1+ 2v2) k3 || 3|| + 3 = k2(1v1+ 2v2) k3tan
2(/2)
v
1 > 0
|| > 1 = ddt
(1 cos)< 0
= min{1, |(0)|} >0
tan(/2)
1/(|| + 3)
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v, , ,
1
2
, v , , u v
ddttan
2(/2)
1/(||+3)
V
v3
1
2
v1
v2
1,2
(1, 2)T
d
dt
1,2|| =a(t) + b(t)
a(t) := 3
k2v1 k31(|| + 3)2
k2v2 k32(||
+ 3
)2
, b(t) := 1
||
3+ 1
||2 TS(Re3)
21
a(t)
a(t)
v, ,
1/(|| + 3) b(t)
3
3
1,2
b(t)
ddt
1,2||
1,2 || + 3 > 0
v1, v2
(v, ) = (0, 0)
v= S()v ue3+ RT RTh(|Iv|2)Iv+ RTc
f : s h(s2)s f1 f
h
f(0) = 0
cTIv |c||Iv | |Iv|0
f(s) ds +
|c|0
f1(s) ds
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V
V =1
2vTv+
1
k2
1 3||
+
|Iv|0
f(s) ds cTIv+ |c|0
f1(s) ds
V
v
V
Iv
V
Iv 2V
I2v> 0
h
Iv = Rv
V = vT (ue3+ ) + 1||k2
1 2 2
1
+
1
||2TS(Re2)
TS(Re1)
V = ||k1v23k3k2
21 + 22
(
|
|+ 3)
2
(Iv, v,)
(Iv , 0, 0)
h
Iv
h(|Iv |2)Iv = c
Iv
Iv
a(t) := RTh(|Iv|2)Iv+ RTc , b(t) := S()v ue3+ RT
(Iv, v,) = (Iv 0, 0)
V
v
v= S()v ue3+ RTz+ RT(e xr)
v= S()v ue3 RTh(|y|2)y+ RT+ RTc
V =1
2vTv+
1
k2
1 3||
+
|y|0
f(s) ds cTy+ |c|0
f1(s) ds
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f
V
v
y
y= Rv
V = ||k1v23k3
k2
21
+ 22
(|| + 3)2
z, z
z
y
v
z
z
y= x + z
v= v+ RTz
x
v
hz
z(3)
z
h(|z|2)z = c
(y, v, )
(z, 0, 0)
(y, v,)
y:= y z, z := z z, w:= z,gz(y, z) := hz(
|y
z
|2)(y
z) + hz(
|z
|2)z
gz(y, z)
y
x= y z= y z z = ww = 2kzw k2z z+ k2z(
(z+ z) z) kzhz(|z|2)z+ kzgz(y, z)
Z= F(Z) + G(y, Z)
Z:= (z, w)T
F(Z) :=
w
2kzw k2z z+ k2z(
(z+ z) z) kzhz(|z|2)z
,
G(y, Z) := (0, kzgz(y, z))T
Z = 0 Z =
F(Z)
U= 1
2kz
|z|20
hz(s) ds+1
2|z|2 +1
2
z+ wkz2
Z= F(Z)
U= hz(|z|2)|z|2 kz
|z|2 +z+
w
kz
2
zT + ( (z+ z) z)T
z+ w
kz
hz(|z|2)|z|2 kz |z|2 kzz+ wkz
2 + 2kz|z| z+ wkz
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z
z
z > 0
hz(|z|2)> z z, w > 0
U
z|z|2
w z+ w
kz2
z
w
U
Z= 0 Z=F(Z)
Z = 0 Z = F(Z)
Z
G(y, Z)
Z
(z, z)
(z, 0)
z
y:= z+ x
x
v := v+ RTz
z
v
(z, z, x, v, ) = (z, 0, 0, 0, 0)
(v, y,) = (0, 0, 0)
Z = 0
Z= F(Z)
G
y = 0
(v, y,, Z) = (0, 0, 0, 0)
(z, z, x, v,) = (z, 0, 0, 0, 0)
u
u
V = vT (ue3+ ) + 1||k2
1 2 2
1
+
1
||2TS(Re2)
TS(Re1)
= v3(u + ||) v3(|| 3)
+ 1
||k2
1 2 21 + 1||2
TS(Re2)TS(Re1)
+ ||k2v1v2
= v3(u + ||)
+ 1
||k2
1 2 2
1
+
1
||2TS(Re2)
TS(Re1)
+ ||k2
v1v2
||k2v3|| + 3
12
V = ||k1(v3)v3 k3k2
21 + 22
(|| + 3)2
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x
V = 12 |x|2 |x|
V = xT(uRe3+ e(x, t))
V |x||u| + c3m
|x| c4m
|x|3
|x|
1+ (2+c3m
)|x| c4m
|x|2
x
12
2+c3m
+1
2
2+c3m
2
+41c4
m
e
x
u
x
e
e,d
M c1+ c2((ci, vr))2 (x, t)
vTRT(e,d(x, t) M(e,d(x, t))) 0
|e,d(x, t)| < M
M
e,d(x, t) =
M(e,d(x, t))
|e,d(x, t)| M |x| > (ci, vr)
M
(e,d(x, t)) 1 Rv= x xr
vTRT(e,d(x, t) M(e,d(x, t))) = (1 (e,d(x, t)))(x xr)Te,d(x, t) (1 (e,d(x, t)))(xe,d(x, t) + vr|e,d(x, t)|) (1 (e,d(x, t)))(c4|x|3 c2vr|x|2 c3|x| c1vr)
|x| > (ci, vr)
c4|x
|3
c2vr
|x
|2
c3
|x
| c1vr
0
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1, 2
e
e
x
d(t)
e,d(x, t)
M
(x, t)
u
1,2
3
v= S()v ue3+ RT(x, t) + RT(e,d(x, t) M(e,d(x, t)))
V
V =v3(u + 3) + vTRT(e,d M(e,d))
+ 1||k2
1 2 2
1
+ 1||2
TS(Re2)TS(Re1)
+ ||k2 v1v2
u, 1, 2
V = ||k1v23 (|| + 3)k3k2
21 + 22
(|| + 3)2+ vTRT(e,d M(e,d))
+(1 (||))
||3k2 (1TS(Re2)+ 2
TS(Re1))
M(e,d(xr(t), t)) = e,d(xr(t), t)
M c1 + c2v2r
M > c1 + c2v2r
M(e,d(x, t)) = e,d(x, t)
x
xr
(0, )
(||) = 1 xr
V = ||k1v23 (|| + 3)k3k2
21 + 22
(|| + 3)2
M c1 + c2((ci, vr))2 |(x, t)| ,(x, t)
(||) = 1,(x, t)
V = ||k1v23 (|| + 3)k3k2
21 + 22
(
|
|+ 3)
2 + vTRT(e,d M(e,d))
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V ||k1v23 (|| + 3)k3k2
21 + 22
(|| + 3)2
d
dt
||
= 1
||S()
1||2 RTS()
d
dt
1 3||
=
1
||
1 2 2
1
+
1
||2TS(Re2)
TS(Re1)
|| =R
T ||
d
dt || = S()RT
|| + RT d
dt ||
= S() || + RT d
dt
||
= 1
||S()+ RT d
dt
||
d
dt
||
=(||2I3 T)
||3 = S()2
||3
RT d
dt || =
1
||3RTS()2=
1
||3S()RTS()
d
dt
1 3||
= 1||e
T3S()
1||2 R
TS()
= 1||2 1 0
1||2 S()R
T
= 1
||
1 22
1
+
1
||3
1 2TS(Re2)
TS(Re1)
1 + 3
||
3
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x(t)
x= a(t) + b(t)
a(t)
limt+
x(t) =c
limt+
b(t) = 0
c
limt+ x(t) = 0
b= 0
z = 0, z = 0, x = 0
v = 0
R = I3
u = g
= 0
Fe = mg e3
ge3+ (h(0) + kzhz(0))x + h(0)z 2kzz
w:= z (s)
s |s=0= 1
u g+ (h(0) + kzhz(0))x3+ gk1v3+ h(0)z3+ (gk1 2kz)w31 k34g (h(0) + kzhz(0))x2 gk2v2 k3h(0)4g z2 (gk2 k3kz2g )w2 k34 (eT2RTe3)2 k34g (h(0) + kzhz(0))x1+ gk2v1+ k3h(0)4g z1+ (gk2 k3kz2g )w1+ k34 (eT1RTe3)3 0
1 := eT1RTe3
2 := eT2RTe3
RT =S()RT 3 0 R I3
1
2, 2
1
z = ww = 2kzw+ kzhz(0)xx = vv = (g1, g2, (h(0) + kzhz(0))x3 gk1v3 h(0)z3 (gk1 2kz)w3)T1 = k34g (h(0) + kzhz(0))x1 gk2v1 k3h(0)4g z1 (gk2 k3kz2g )w1 k34 12 = k34g (h(0) + kzhz(0))x2 gk2v2 k3h(0)4g z2 (gk2 k3kz2g ))w2 k34 2
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(3) :
z3 = w3w3 = 2kzw3+ kzhz(0)x3x3 = v3v3 = (h(0) + kzhz(0))x3 gk1v3 h(0)z3 (gk1 2kz)w3
(i) :
zi = wiwi = 2kzwi+ kzhz(0)xixi = vivi = gii = k34g (h(0) + kzhz(0))xi gk2vi k3h(0)4g zi (gk2 k3kz2g )wi k34 i
(i= 1, 2)
P3() = 4 + (2kz+ gk1)
3 + (h(0) + 2gkzk1+ kzhz(0))2
+ (2kzh(0) + gkzhz(0)k1) + kzhz(0)h(0)
Pi() = 5
+
2kz+
k34
4
+
g2
k2+
kzk32
3
+
2g2
kzk2+
h(0)k34 +
kzhz(0)k34
2
+
g2kzhz(0)k2+
kzh(0)k32
+
kzhz(0)h(0)k34
h(0) = 2gk1kz 4k2z , k2 =k1k3
4g , kz