Lie Groups for 2D and 3D Transformations (Ethan Eade) PDF

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SO(3) SE(3) SO(2) SE(2) Sim(3)

SO(3)

R R

−1

∈ SO(3) = RT

so(3)



0 G1 =  0 0

so(3)

3×3   0 0 −1  , G 2 =  0 −1 0

0 0 1

so(3)

  0 1 0 0 0  , G3 =  1 0 0 0

ω ω 1 G1 + ω 2 G2 + ω 3 G3

∈

∈

−1 0 0

R3 so(3)

ω ∈ so(3)

ω×



 0 −ω 3 ω 2 exp (ω × ) ≡ exp  ω 3 0 −ω 1  −ω 2 ω 1 0 1 2 1 = I + ω × + ω× + ω 3× + · · · 2! 3!

exp (ω × ) = I +

∞ X i=0

"

ω 2i+2 ω 2i+1 × × + (2i + 2)! (2i + 1)!

  ω 3× = − ω T ω · ω × θ2 ω 2i+1 ×

≡ ωT ω = (−1)i θ 2i ω ×

ω 2i+2 ×

=

(−1)i θ 2i ω 2×

#

 0 0  0

exp (ω × ) = = =

∞ X (−1)i θ 2i

!

∞ X (−1)i θ 2i

!

ω 2× (2i + 2)! i=0 i=0     1 θ2 θ4 θ2 θ4 2 + + − + · · · ω× + + · · · ω× I+ 1− 3! 4! 2! 5! 6!     1 − cos θ sin θ 2 ω× ω× + I+ θ θ2 I+

(2i + 1)!

ω× +

θ ω θ SO(3)

R

∈

so(3)

SO(3) 

tr(R) − 1 θ = arccos 2   θ · R − RT ln (R) = 2 sin θ



ln (R)

ω θ 2 sin θ

θ

X AdjX

ω

∈

so(3), R ∈SO(3)

R · exp (ω ) =

exp (AdjR · ω) · R

exp (AdjR · ω) = ω =t·v

R · exp (ω) · R−1

t∈R

t

t=0

  d  d  R · exp (t · v) · R−1 exp (AdjR · t · v) = t=0 t=0 dt dt   2     d d  = R · t=0 I + (t · v)× + O t2 · R−1 I + (AdjR · t · v)× + O t dt t=0 dt (AdjR · v)× = R · v × · R−1 = (Rv)× =⇒ AdjR

=

R

SO(3)

R3

SO(3) R ∈SO(3)

x ∈ R3

x

R

y = f (R, x) = R · x f ∂y =R ∂x

∂y ∂R

= = = = =

∂ |ω=0 (exp (ω) · R) · x ∂ω ∂ |ω=0 exp (ω) · (R · x) ∂ω ∂ |ω=0 exp (ω) · y ∂ω   G1 y G2 y G3 y −y×

x

f :G → G

G

exp (ǫ) · f (g) = f (exp (δ) · g ) ∂f ∂ǫ |δ=0 ≡ ∂δ ∂g ǫ ǫ

δ

ǫ = ∂f ∂g

≡

  −1 log f (exp (δ) · g) · f (g )   −1 ∂ log f (exp (δ) · g ) · f (g ) ∂δ

|δ=0

f (g) = g G = SO(3) R0 R2 = f (R0 ) ≡ R1 · R0 so(3) exp (ǫ) · R2 = R1 · exp (ω) · R0 ǫ

∂R2 ∂R0

ω

  −1 ∂ log (R1 · exp (ω) · R0 ) · (R1 · R0 )

ω=0

=

|ω=0 ∂ω h  i    ∂ −1 |ω=0 log exp AdjR1 · ω · R1 · R0 · (R1 · R0 ) ∂ω     ∂ |ω=0 log exp AdjR1 · ω ∂ω   ∂ |ω=0 AdjR1 · ω ∂ω AdjR1

=

R1

≡ = = =

SO(3)

so(3) Σ ∈ R3×3

R ∈SO(3)

S

ǫ S

∈

=

N (0, Σ)

exp (ǫ) · R

(R0 , Σ0 )

(R1 , Σ1 )

R0

R1

  (R1 , Σ1 ) ◦ (R0 , Σ0 ) = R1 · R0 , Σ1 + R1 · Σ0 · R1T

(Rc , Σc )

Σc

= =

v Rc

−1  −1 Σ0 + Σ1−1

−1

Σ0 − Σ0 (Σ0 + Σ1 )

Σ0

≡ R1 ⊖ R0   = ln R1 · R−1 0   = exp Σc · Σ1−1 · v · R0

(R0 , Σ0 ) (R1 , Σ1 )

v

K −1

K ≡ Σ0 (Σ0 + Σ1 )

R0 ⊕ (K · v)

Rc

=

Σc

= exp (K · v) · R0 = (I − K) · Σ0 Σ0

Σ1

SE(3)

R

∈

C

= R

SO(3), t ∈ R3   R t ∈ SE(3) 0 1 t

SO(3)

C1 , C2 C1 · C2

C −1 1

∈

SE(3)     R 2 t2 R 1 t1 = · 0 1 0 1   R 1 R 2 R 1 t2 + t1 = 0 1  T  Tt R1 −R 1 = 0 1

T z w ∈ RP3 (λx ≃ x ∀λ ∈ R)   R t C ·x = ·x 0 1    T R x y z + wt = w 

x =

w=1

x

y

x w = 0,

x se(3)

4×4

so(3) 

0 0 G1 =  0 0 

0  0 G4 =   0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 −1 0 0

se(3)

 1 0  , 0  0



0  0 G2 =   0 0

  0 0  0 0   , G5 =   −1 0  0 0

0 0 0 0

0 0 0 0

 0 1  , 0  0



0  0 G3 =   0 0

  0 1 0 0 1 0 0 0   , G6 =  0 0 0 0  0 0 0 0



u ω

T

u1 G1 + u2 G2 + u3 G3 + ω 1 G4 + ω 2 G5 + ω 3 G6 

u ω

se(3)

δ=





T

∈

−1 0 0 0

0 0 0 0

 0 0  , 1  0

 0 0 0 0   0 0  0 0

R6 se(3)

∈ se(3)

SE(3)

∈

se(3)   ω× u exp (δ) = exp 0 0  2   1 ω× ω× u = I+ + 0 0 0 2!

u ω

∈

0 0 0 0

ω×u 0



1 + 3!



ω 3× 0

2 ω× u 0



+ ···

SO(3)

exp

V



ω× 0

u 0



=

V

=

 exp (ω × ) Vu 0 1 1 1 2 I + ω × + ω× + · · · 3! 2!



"

# 2i+1 ω 2i+2 ω× × = I+ + (2i + 2)! (2i + 3)! i=0 ! ! ∞ ∞ X X (−1)i θ 2i (−1)i θ 2i = I+ ω 2× ω× + (2i + 3)! (2i + 2)! i=0 i=0 ∞ X

V

V

= =

   1 θ2 θ2 θ4 θ4 1 − + + − + · · · ω× + + · · · ω 2× 2! 4! 5! 3! 6! 7!     1 − cos θ θ − sin θ 2 I+ ω× ω× + θ3 θ2

I+



u, ω

∈

θ

=

A = B

=

C

=

R

=

R3 √ ωT ω sin θ θ 1 − cos θ θ2 1−A θ2 2 I + Aω × + Bω×

V = I + Bω × + Cω×2     u R Vu exp = 0 1 ω A B V

C

θ2

1 1 V−1 = I − ω × + 2 2 θ ln()

  A 2 ω× 1− 2B

SE(3) u = V−1 · t

ln(R)

SE(3)

SO(3) 

R t δ= u ω ∈ se(3), C = 0 1 C · exp (δ ) = exp (AdjC · δ) · C T





∈SE(3)

exp (AdjC · δ) = AdjC · δ

C · exp (δ) · C −1 ! 6 X = C· δ i Gi · C −1

i=1  Ru + t × Rω = Rω   R t× R = ∈ R6×6 0 R



=⇒ AdjC

C =



R 0

t 1



x ∈ R3

∈SE(3)

y

=

f (C, x) =

=

R·x+t

x



R

t

f ∂y =R ∂x SO(3)



·



x 1

 x

∂y ∂C



=



=

C ∂C ∂C0

SO(3)

G1 y

· · · G6 y 

−y×

I



≡

C1 · C0 ∂ = [C1 · exp (δ) · C0 ] ∂δ = AdjC1

SO(3)

SO(2)

∈ SO(2) = RT

R R so(2)

−1

so(2)

2×2

G=



0 1

−1 0



so(2)

θ θG θ ∈ so(2)

θ×

∈

∈

R so(2) θG

exp (θ× ) ≡ = =

 −θ 0 1 1 2 I + θ× + θ × + θ×3 + · · · 2! 3!    1 −θ 2 0 −θ + I+ θ 0 0 2! exp



0 θ

0 −θ 2



1 + 3!

sin θ exp (θ× ) =



cos θ sin θ

− sin θ cos θ



SO(2)

R ∈ ln (R) = θ =

so(2)

SO(2) arctan (R21 , R11 )

SO(2)

SE(2)

SE(3)

SE(2)

R

∈

C

=

cos θ

∈ SO(2)

θ

SO(2), t ∈ R2   R t ∈ SE(2) 0 1



0 −θ 3

θ3 0



R

∈

SE(2)     R 2 t2 R 1 t1 = · 0 1 0 1   R 1 R 2 R 1 t2 + t1 = 0 1   T Tt −R R1 1 = 0 1

C1 , C2 C1 · C2

C −1 1

x = C ·x = = w=1

t



x



R



y

R 0

t 1 

x

w 

T

∈ RP2

(λx ≃ x ∀λ ∈ R)

·x

y w

T

+ wt



x w = 0,

x se(2)

3×3 

0 G1 =  0 0

0 0 0

  1 0 0  , G2 =  0 0 0

0 0 0

  0 0 1  , G3 =  1 0 0

se(2)



u1

u2

θ

T

u1 G1 + u2 G2 + θG3 

u θ

T

∈ se(2)

∈

∈

R3 se(2)

−1 0 0

 0 0  0

se(2)

δ=





∈

se(2)   θ× u exp (δ) = exp 0 0  2   1 θ× θ× u = I+ + 0 0 0 2!

u θ

SE(2)

θ× u 0



+

1 3!



3 θ× 0

θ 2× u 0



SO(2)

exp



θ× 0

=

V

u 0



=

V

=

∞ X

i=0

2i θ× = 2i+1 θ× =

"



 exp (θ× ) Vu 0 1 1 2 1 I + θ× + θ × + · · · 3! 2!

θ 2i+1 θ×2i × + (2i + 1)! (2i + 2)!

#



 1 0 0 1   0 −1 (−1)i θ 2i+1 · 1 0 i 2i

(−1) θ ·

V

V

 ∞ X (−1)i θ 2i

   θ 1 1 0 0 + · · 0 1 1 (2i + 1)! (2i + 2)! i=0 !  !   ∞ ∞ X X (−1)i θ 2i (−1)i θ 2i+1 1 0 = + · · 0 1 (2i + 1)! (2i + 2)! i=0 i=0

=

−1 0



0 −1 1 0



+ ···

     θ θ2 θ4 1 0 − + + + ··· · V = 1− 0 1 2! 3! 5!        1 − cos θ sin θ 0 1 0 + · · = 1 0 1 θ θ   1 sin θ −(1 − cos θ ) = · 1 − cos θ sin θ θ

θ5 θ3 + + ··· 4! 6!  −1 0

   0 −1 · 1 0

V ln() Vu = t

SE(2) u

θ = ln(R)

sin θ θ 1 − cos θ B ≡ θ   1 A B −1 V = −B A A2 + B 2     −1 R t V ·t ln ∈ se(2) = θ 0 1 A ≡

SE(2)

δ=

θ



T

∈

se(2), C =

AdjC · δ

=

C·

u θ



3 X

δ i Gi

i=1

Ru + θ =  =⇒ AdjC



R =  0



 θ

t2 −t1 1



R 0 !

t 1

t2 −t1

 

∈SE(2)

· C −1





 ∈ R3×3

Sim(3)

SE(3

R

∈

T

=

T1 , T2 T1 · T2

T 1−1

SO(3), t ∈ R3 , s ∈ R   R t ∈ Sim(3) 0 s−1

∈

Sim(3)     R 2 t2 R 1 t1 · = −1 −1 0 s2 0 s1   R1 R2 R1 t2 + s2−1t1 = 0 (s1 · s2 )−1  T  R1 −s1 RT1 t = 0 s1 s

x = T ·x = = ≃



x

y



T w ∈ RP3  t ·x −1

R 0 s   R x

z

y z s−1 w

  s R x

y

T z

w

+ wt T

(λx ≃ x ∀λ ∈ R)



+ wt

!

w=1 sim(3)

se(3)



0  0  G7 =  0 0

 0 0 0 0 0 0   0 0 0  0 0 −1

sim(3) 

T u ω λ

u1 G1 + u2 G2 + u3 G3 + ω 1 G4 + ω 2 G5 + ω 3 G6 + λG7 



 u ω λ

∈

R7 sim(3)

T u ω λ ∈ sim(3)

sim(3)

δ=

∈

Sim(3)

∈

sim(3)   ω× u exp (δ) = exp 0 −λ  2   1 ω× ω× u = I+ + 0 λ 0 2!



ω × u − λu λ2



exp (ω × ) 0

Vu exp (−λ)

1 3!

+

3 ω× 0

ω 2× u − λω × u + λ2 u −λ3

se(3)

exp



ω× 0

u 0



=

V

=



n k ∞ X X (−λ) ω n−k × (n + 1)! n=0



k=0

k ∞ X ∞ X ω×n−k (−λ) (n + 1)!

=

k=0 n=k

k ∞ X ∞ j X ω × (−λ) (j + k + 1)! j=0

=

k=0

θ2 = ω T ω ω×

V

=

=

!

"

# 2i+1 ω 2i+2 ω× × + (−λ) I+ (2i + k + 2)! (2i + k + 3)! k=0 i=0 ! ! ! ∞ X ∞ X ∞ ∞ ∞ k k k X X X (−1)i θ 2i (−λ) (−1)i θ 2i (−λ) (−λ) ω 2× ω× + I+ (2i + k + 3)! (2i + k + 2)! (k + 1)! i=0 i=0

∞ k X (−λ) (k + 1)! k=0

k=0

∞ X

k=0

k

∞ X

k=0



+ ···

V

=

A = B C

= =

AI + Bω × + Cω 2× 1 − exp (−λ) λ ∞ ∞ X k X (−1)i θ 2i (−λ) (2i + k + 2)! k=0 i=0 ∞ X ∞ X k=0 i=0

k

(−1)i θ 2i (−λ) (2i + k + 3)!

B

B

= =

∞ X ∞ k X (−1)i θ 2i (−λ)

k=0 i=0 " ∞ X i=0

= =

=

=

(2i + k + 2)! ∞ X

k

(−λ) (−1) θ (2i + k + 2)! k=0 i 2i

#

" # ∞ ∞ X (−1)i θ 2i X (−λ)2i+k (2i + k + 2)! λ2i i=0 k=0 # " ∞ ∞ X (−1)i θ 2i X (−λ)m (m + 2)! λ2i m=2i i=0 #! " ∞ ∞ 2i−1 X (−λ)m X (−1)i θ 2i X (−λ)m − (m + 2)! λ2i (m + 2)! i=0 m=0 m=0 #! " ! 2i−1 ∞ ∞ ∞ X X (−1)i θ 2i X (−λ)m (−1)i θ 2i X (−λ)m − (m + 2)! λ2i λ2i m=0 (m + 2)! m=0 i=0 i=0

L ≡ B

∞ X (−1)i θ 2i λ2i i=0

"

∞ m X (−λ) (m + 2)! m=0

#!

! 2i−1 ∞ X (−1)i θ 2i X (−λ)m = L− λ2i m=0 (m + 2)! i=0 " #! ∞ i−1 2p 2p+1 X (−1)i θ 2i X (−λ) (−λ) = L− + (2p + 3)! λ2i p=0 (2p + 2)! i=0  ! i−1 ∞ X λ 1 (−1)i θ 2i X 2p − (−λ) = L− (2p + 2)! (2p + 3)! λ2i p=0 i=0    ∞ X (−1)i θ 2i 1 λ 2p (−λ) − = L− λ2i (2p + 2)! (2p + 3)! p...