Robust control design

7/24/2019 Robust control design

1/46

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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2/46

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/


3/46


4/46

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

Minh [email protected] [email protected]

[email protected]

[email protected]


5/46


6/46

H2

H


7/46


8/46

G

m

J

I {O; o, o, k o}

o o k o B {G; , , k } k

k

k


9/46

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


10/46

{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


11/46

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)


12/46

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


13/46

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


14/46

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


15/46

(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 (, )


16/46

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


18/46

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


19/46

(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)


21/46

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


22/46

(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


23/46

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


25/46

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


26/46

= 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


27/46

= 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


28/46

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


t (s)

x(m)

x1 : x1 : x2 : x2 : x3 : .x3 : .

0 20 40 60 80 10040

30

20

10

0

10

20

30

40


t (s)

,,(o)

: : : .

= 12

= 8

0 20 40 60 80 1008

6

4

2

0

2

4

6

8


t (s)

x(m)

x1 : x2 : x3 : .

0 20 40 60 80 10020

10

0

10

20

30

40


t (s)

,,(o)

: : : .

= 6

= 1


29/46

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


30/46

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


31/46


32/46


33/46


34/46

&

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


35/46

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)


36/46

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


37/46

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


38/46

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


39/46

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


40/46

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


41/46

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))


42/46

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


43/46

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


44/46

(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

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Robust control design