Control of rigid bodies : with applications to leader-follower spacecraft formations

Rune Schlanbusch

Control of rigid bodies

with applications to leader-follower spacecraft formations

ISBN 978-82-471-3365-1 (printed ver.) ISBN 978-82-471-3366-8 (electronic ver.) ISSN 1503-8181

NTNU Norw egian Univ er sity of Scienc e and T echnol ogy Thesis f or the degr ee of phil osophiae doct or F aculty of Inf ormation T echnol ogy, Mathematics, and El ectric al Engineering Department of Engineering Cybernetics

D oc to ra l t he se s a t N TN U , 2 01 2:4 9

Rune Schlanbusch

Control of rigid bodies

with applications to leader-follower spacecraft formations

Thesis for the degree of philosophiae doctor Trondheim, March 2012

Norwegian University of Science and Technology

Faculty of Information Technology, Mathematics, and Electrical Engineering

Department of Engineering Cybernetics

Thesis for the degree of philosophiae doctor

Faculty of Information Technology, Mathematics, and Electrical Engineering Department of Engineering Cybernetics

©Rune Schlanbusch

ISBN 978-82-471-3365-1 (printed ver.) ISBN 978-82-471-3366-8 (electronic ver.) ISSN 1503-8181

Doctoral Theses at NTNU, 2012:49

Printed by Fagtrykk, Trondheim

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R ^ef n =

⎡

⎣ − cos λ sin φ − sin λ − cos λ cos φ

− sin λ sin φ cos λ − sin λ cos φ

cos φ 0 − sin μ

⎤

⎦ ,

λ

φ

¾

¿

90 ^◦

0 ^◦

−90 ^◦

F ^ef

F ⁱ

R ⁱ ef =

⎡

⎣ cos(ω e t + α) − sin(ω e t + α) 0 sin(ω e t + α) cos(ω e t + α) 0

0 0 1

⎤

⎦ ,

t

α

x ^ef x ⁱ

t = 0

ω e = 7.292115 × 10 ⁻⁵ rad/s

z ^ef

F ⁱ

F ^s

s = f, l

!""#

R ^s i = R z (ω + ν) R x (i) R z (Ω)

=

⎡

⎣ c(ω + ν ) s(ω + ν ) 0

− s(ω + ν ) c(ω + ν ) 0

0 0 1

⎤

⎦

⎡

⎣ 1 0 0 0 ci si 0 − si ci

⎤

⎦

⎡

⎣ cΩ sΩ 0

− sΩ cΩ 0

0 0 1

⎤

⎦

=

⎡

⎣ c(ω +ν )+cΩ − cis(ω +ν)sΩ c(ω +ν)sΩ+ s(ω +ν)cicΩ s(ω+ν)si

−s(ω +ν)cΩ−cisΩc(ω +ν) −s(ω +ν)sΩ+c(ω +ν)cicΩ c(ω +ν)si

sisΩ −sicΩ ci

⎤

⎦ ,

ν

r

F c( · ) s( · )

!""#

R ⁱ s = R ⁻¹ z (Ω)R ⁻¹ x (i)R ⁻¹ z (ω + ν).

!

R ⁱ s = ( R ^s i ) ⁻¹ = ( R ^s i ) ^T

R ⁱ s = R ^T z (Ω) R ^T x (i) R ^T z (ω + ν ).

) * + !

""# ,

SO(3)

θ

k

R = cos(θ) I + sin(θ) S ( k ) + [1 − cos(θ)] kk ,

,

SO(3)

η = cos (θ/2) ∈

R

= k sin (θ/2) ∈ R ³

R = I + 2η S ( ) + 2 S ² ( ).

q = [η, ]

q ∈ S ³ = {x ∈ R ⁴ : x x = 1 }

S ³

q ₁ ⊗ q ₂ =

η ₁ η ₂ − _{1 2} η _{1 2} + η _{2 1} + S( ₁ ) ₂

.

! "

q

q ¯ =

[η, − ]

q ^c,a = ¯ q ^b,a ⊗ q ^b,c = q ^a,b ⊗ q ^b,c .

$

q q =

η ² + = 1

R = I ⇔ q = [±1, 0]

q = ±q id

!

% &'(()

C ^s a = h pv

⎡

⎣

p

r e sin ν 0

− e sin ν ^p _r 0

0 0 ^pv _h

⎤

⎦ ,

p = h ² /μ

μ

v

e

!

ν

C ^s a

det( C ^s a ) = 1 + e ² + 2e cos(ν),

! !

, % - ! *

. !

$*! $

f i = G

j = n

j =1

m i m j

r ³ _ij ( r j − r i ), i = j,

G

r ij

$

r ij := r j − r i .

% & ' !"( !"#!

d ² r i

dt ² = G

j = n

j =1

m j

r ³ _ij ( r j − r i ), i = j.

)&*!"#( +

d ² r dt ² + μ

r ³ r = 0,

r := r 2 − r 1

μ = G(m ₁ + m ₂ ),

.

m ₁ m ₂

"

/+

$ 0

" /

f ^dl f ^df

f ^al f ^af

0

¨ r l = − μ

r _l ³ r l + f dl

m l

+ f al

m l

!"#2

¨ r ^f = − μ

r _f ³ r ^f + f df

m f

+ f af

m f

.

/ 3 !44#

p = r f − r l = x e r + y e θ + z e h

p ¨ = ¨ r f − ¨ r l = − μ

r ³ _f r f + f df

m f

+ f af

m f

+ μ

r _l ³ r l − f dl

m l − f al

m l

,

m f p ¨ = − m f μ

r l + p (r l + p) ³ − r l

r ³ _l

+ f af + f df − m f

m l

( f al + f dl ).

r f = r l + p = (r l + x)e r + ye θ + ze h ,

¨ r f = (¨ r l + ¨ x) e r +2( ˙ r l + ˙ x) ˙ e r +(r l +x)¨ e r + ¨ y e θ +2 ˙ y e ˙ θ +y¨ e θ + ¨ z e h +2 ˙ z e ˙ h +z¨ e h .

!!"

e ˙ r = ˙ νe θ e ¨ r = ¨ νe θ − ν ˙ ² e r

e ˙ θ = − νe ˙ r e ¨ θ = −¨ νe r − ν ˙ ² e θ ,

e h

e ¨ ^h = ˙ e ^h = 0

%% &$

¨ r ^f = (¨ r l + ¨ x − 2 ˙ y ν ˙ − ν ˙ ² (r l +x) − y ν ¨ e ^r +(¨ y +2 ˙ ν ( ˙ r l + ˙ x)+ ¨ ν(r l +x) − y ν ˙ ² ) e ^θ + ¨ z e ^h .

'

r ^l = r l e ^r ,

¨ r l = ¨ r l e r + 2 ˙ r l e ˙ r + r l ¨ e r ,

)

¨ r ^l = (¨ r l − r l ν ˙ ² ) e ^r + (2 ˙ r l ν ˙ + r l ν ¨ ) e ^θ .

* #! + + +

+

p ¨ = ¨ r f − ¨ r l = (¨ x − 2 ˙ y ν ˙ − ν ˙ ² x − y ν) ¨ e r + (¨ y + 2 ˙ ν x ˙ + ¨ νx − y ν ˙ ² ) e θ + ¨ z e h .

* , #! -

!!"

m f p ¨ + C ^t ( ˙ ν) ˙ p + D ^t ( ˙ ν, ν, r ¨ f ) p + n ^t (r l , r f ) = F ^a + F ^d ,

C t ( ˙ ν ) = 2m f ν ˙

⎡

⎣ 0 − 1 0

1 0 0

0 0 0

⎤

⎦

D ^t ( ˙ ν, ν, r ¨ f ) p = m f

⎡

⎢ ⎣

μ r ³

f − ν ˙ ² − ν ¨ 0

¨ ν _r ^μ ₃

f − ν ˙ ² 0

0 0 _r ^μ ₃

f

⎤

⎥ ⎦ p

n t (r l , r f ) = m f μ

⎡

⎣

r _l r ³

f − _r ¹ 2 l

0 0

⎤

⎦

F d = f df − m f

m l f dl , F a = f af − m f

m l f al ,

R ˙ ^a b = S ω ^a a,b

R ^a b = R ^a b S ω ^b a,b

.

#$%

R ⁱ l p = r f − r l ,

& "

R ⁱ l p ¨ + 2R ⁱ l S(ω ^l i,l ) ˙ p + R ⁱ l

S ² (ω ^l i,l ) + S( ˙ ω ^l i,l )

p = ¨ r f − ¨ r l .

( )%*)' '

¨ r f − ¨ r l = − μ

r ³ _f r f + f f d

m f

+ f f a

m f

+ μ

r _l ³ r l − f ld

m l

− f la

m l

,

$% !+,

m f (¨ r f − ¨ r l ) = −m f μ

1 r ³ _f − 1

r _l ³

r _l + R ⁱ l p r _f ³

+ f f a + f f d − m f

m l (f la + f ld ).

- !$ '

m f p ¨ + C t ( ω ^l i,l ) ˙ p + D t ( ˙ ω ^l i,l , ω ^l i,l , r f ) p + n t ( r l , r f ) = F a + F d ,

C t ( ω ^l i,l ) = 2m f S ( ω ^l i,l )

D t ( ˙ ω ^l i,l , ω ^l i,l , r f ) = m f

S ² (ω ^l i,l ) + S( ˙ ω ^l i,l ) + μ r _f ³ I

n t ( r l , r f ) = μm f R ^l i

1 r ³ _f − 1

r _l ³

r l

!"

F d

F ^a

F d = R ^l i

f f d − m f

m l f ld

and F a = R ^l i

f f a − m f

m l f la

.

f

ω ⁱ i,l = S ( r ^l ) v ^l / r l r ^l ,

ω ˙ ⁱ i,l = r l r ^l S ( r ^l ) a ^l − 2 v l r ^l S ( r l ) v ^l

(r _l r l ) ² ,

ω ^l i,l = R ^l i ω ⁱ i,l

ω ˙ ^l i,l = R ^l i [S(ω ^l i,l ) + ˙ ω ⁱ i,l ]

" # $$

%

&'(

) ( ** +,-. **

p ^b = t

0 v ^b dt

/ 0!

R ⁱ f b p = r f − r l ,