02.1 Lie Algebra

• when optimizing estimations

• -> we want that certain constraints still hold (i.e. orthogonality of rotation matrix… RRT = I)

• => difficlt to simultaneously optimize and enforce constraints

• => use Lie algebra to find constraing free optimization strategy

• algebraic structure of

  • one set

    • and

  • one operator

G = (A,*) (* is operator, not multipication…)

conditions it should satisfy:

• closure

  • ∀ a,b ∈ A; a*b ∈ A

• associative law

  • ∀ a,b,c ∈ A;

    • (a*b)*c = a*(b*c)

• identity element

  • ∃ a0 ∈ A

  • so that

  • ∀ a ∈ A

    • a0 * a = a * a0 = a

• inverse

  • ∀ a ∈ A,

  • ∃ a^-1 ∈ A

  • so that

    • a*a^-1 = a0

    • -> element times inverse of element equals identity element

• invertible n*n matrix (set)

• matrix multiplication (operator)

• Rotation matrix group

  • -> here, SO(2) (2-dim) and SO(3) (3-dim) is most common

• rotation matrices (set)

• matrix multiplicatoin (operator)

• unity element: identity matrix

• identitiy element: inverse

  • R * R^-1 = I

• group with continuous properties

• -> e.g. SO(n), SE(n) in real space…

• can be multiplied

• but not aded

• => affects derivative computation!!! (-> and in turn optimization with gradient descent…)

• need derivatives for optimizatoin

• -> map Lie gorup in other space

• there, we perform addition and multiplication

• can be rewritten as vector

• transpose equal to negative of itself

• a^ = A

• A (reversed hat) = a

-> 6 dim. vector consisting of translatoin part and rotation(so(3)) part

=> BUT!!! translatoin part not the same as in translation matrix!

• i.e. similar to SE(3) we have a rotation and translation part

• Here, the so(3) part (I.e. rotation in Lie group) is the phi while the translation part is the p

• where theta is the norm of the vector

• and n is the unit vector relating to the direction…

• Of the 6D rotation vector phi

• SO(3) to so(3)

But basically solve for rotation angle and rotation axis separately… (norm and unit vector…)

=> axis n is eigenvector corresponding to the matrix R's eigenvalue of 1…

• Lie group (SO(3) and SE(3) have no addition)

• => map from Lie group to lie algebra

• -> perform addition

• -> map back

• no

• -> we need some approximation (BCH approximation)

The derivative of the R matrix (Lie group)

Lie group:

• group with continuous properties

Lie algebra:

• stuff we map our Lie groups into to perform constraint free optimisation