Lie Groups and Lie Algebras

一些在SLAM中经常用到的关于李群李代数方面的知识

Special Orthogonal Group SO(3) S O ( 3 ) and so(3) s o ( 3 )

The group SO(3) S O ( 3 ) forms a smooth manifold, and its tangent space (at the identity) denoted as so(3) s o ( 3 ) .

SO(3)so(3)={R∈R3×3∣RR⊺=I,det(R)=1}={ϕ∈R3,Φ=ϕ∧∈R3×3} S O ( 3 ) = { R ∈ R 3 × 3 ∣ R R ⊺ = I , det ( R ) = 1 } s o ( 3 ) = { ϕ ∈ R 3 , Φ = ϕ ∧ ∈ R 3 × 3 }

where

ϕ∧=⎡⎣⎢0ϕ3−ϕ2−ϕ30ϕ1ϕ2−ϕ10⎤⎦⎥ ϕ ∧ = [ 0 − ϕ 3 ϕ 2 ϕ 3 0 − ϕ 1 − ϕ 2 ϕ 1 0 ]

• SO(3) S O ( 3 ) : Rotation Matrix R R
• so(3) s o ( 3 ) : Angle-Axis ϕ=θa ϕ = θ a
• Connection: Rodrigues’s Formula

Exp Map

so(3)→SO(3) s o ( 3 ) → S O ( 3 )

R=exp(ϕ∧)=exp(θa∧)=cosθI+(1−sinθ)aa⊺+sinθa∧ R = exp ⁡ ( ϕ ∧ ) = exp ⁡ ( θ a ∧ ) = cos ⁡ θ I + ( 1 − sin ⁡ θ ) a a ⊺ + sin ⁡ θ a ∧

Introduce Exp(ϕ)=exp(ϕ∧) E x p ( ϕ ) = exp ⁡ ( ϕ ∧ ) , then the abode equation can written as:

R=Exp(ϕ)=Exp(θa)=cosθI+(1−sinθ)aa⊺+sinθa∧ R = E x p ( ϕ ) = E x p ( θ a ) = cos ⁡ θ I + ( 1 − sin ⁡ θ ) a a ⊺ + sin ⁡ θ a ∧

Log Map

SO(3)→so(3) S O ( 3 ) → s o ( 3 )

θRaϕ=arccos(tr(R)−12)=a=θa θ = arccos ⁡ ( t r ( R ) − 1 2 ) R a = a ϕ = θ a

BCH Formula

Exp(ϕ+Δϕ)≈Exp(ϕ)Exp(Jr(ϕ)Δϕ)=Exp(Jl(ϕ)Δϕ)Exp(ϕ) E x p ( ϕ + Δ ϕ ) ≈ E x p ( ϕ ) E x p ( J r ( ϕ ) Δ ϕ ) = E x p ( J l ( ϕ ) Δ ϕ ) E x p ( ϕ )

where

Jl(ϕ)=Jr(−ϕ)=sinθθI+(1−sinθθ)aa⊺+(1−cosθθ)a∧ J l ( ϕ ) = J r ( − ϕ ) = sin ⁡ θ θ I + ( 1 − sin ⁡ θ θ ) a a ⊺ + ( 1 − cos ⁡ θ θ ) a ∧

Log(Exp(ϕ)Exp(δϕ))≈Log(Exp(δϕ)Exp(ϕ))≈ϕ+J−1r(ϕ)δϕJ−1l(ϕ)δϕ+ϕ L o g ( E x p ( ϕ ) E x p ( δ ϕ ) ) ≈ ϕ + J r − 1 ( ϕ ) δ ϕ L o g ( E x p ( δ ϕ ) E x p ( ϕ ) ) ≈ J l − 1 ( ϕ ) δ ϕ + ϕ

Property

RExp(ϕ)R⊺=⇔Exp(ϕ)R=exp(Rϕ∧R⊺)=Exp(Rϕ)RExp(R⊺ϕ) R E x p ( ϕ ) R ⊺ = exp ⁡ ( R ϕ ∧ R ⊺ ) = E x p ( R ϕ ) ⇔ E x p ( ϕ ) R = R E x p ( R ⊺ ϕ )

Special Euclidean Group SE(3) S E ( 3 ) and se(3) s e ( 3 )

SE(3)se(3)={T=[R0⊺p1]∈R4×4∣R∈SO(3),p∈R3}={ξ=[ρϕ]∈R6,ρ∈R3,ϕ∈so(3),ξ∧=[ϕ∧0⊺ρ0]∈R4×4} S E ( 3 ) = { T = [ R p 0 ⊺ 1 ] ∈ R 4 × 4 ∣ R ∈ S O ( 3 ) , p ∈ R 3 } s e ( 3 ) = { ξ = [ ρ ϕ ] ∈ R 6 , ρ ∈ R 3 , ϕ ∈ s o ( 3 ) , ξ ∧ = [ ϕ ∧ ρ 0 ⊺ 0 ] ∈ R 4 × 4 }

• SE(3) S E ( 3 ) : Pose (Rotation R R + Translation p p ), Matrix
• se(3) s e ( 3 ) : Angle Axis ϕ ϕ + Translation ρ ρ , Vector

Property

T−1=[R⊺0⊺−R⊺p1] T − 1 = [ R ⊺ − R ⊺ p 0 ⊺ 1 ]

Gaussian Random Variables

R˜T˜=RExp(Δϕ)=[RExp(Δϕ)0⊺p+RΔp1] R ~ = R E x p ( Δ ϕ ) T ~ = [ R E x p ( Δ ϕ ) p + R Δ p 0 ⊺ 1 ]

where R R and p p is the noise-free rotation and translation respectively, and Δϕ Δ ϕ and Δp Δ p is the perturbations (Gaussian white noise)