Differential of the Exponential Map

Transcription

1 Differential of the Exponential Map Ethan Eade May 20, 207 Introduction This document computes ɛ0 log x + ɛ x ɛ where and log are the onential mapping and its inverse in a Lie group, and x and ɛ are elements of the associated Lie algebra. 2 Definitions Let G be a Lie group, with associated Lie algebra g. Then the onential map takes algebra elements to group elements: : g G 2 x I + x + 2! x2 + 3! x The adjoint representation Adj of the group linearly transforms the onential mapping of an algebra element through left multiplication by a group element: x g 4 Y G 5 Y x Adj Y x Y 6 The adjoint operator in the algebra is the linear operator representing the Lie bracket: x, y g 7 ad x y x y y x 8

2 The adjoint operator commutes with the onential map: Adj y ad y 9 We define differentiation of a function f from algebra to group as follows: f : g G 0 f x : g g x [ ] f x ɛ0 log f x + ɛ f x 2 x ɛ In this document, we re interested in D, the differential of : D : g g 3 D x 3 Derivation of a formula for D x x x 4 This isn t a rigorous derivation the epsilon-delta proofs required for the two approximation steps are omitted, but I find it intuitively pleasing. A more rigorous approach would use theorems about integrated flows on continuous vector fields. Define F to be of x modified by an algebra element ɛ: ɛ g 5 F x, ɛ x + ɛ 6 We can also take the product of multiple smaller group elements on the same geodesic: F x, ɛ i x + ɛ Letting the number of steps go arbitrarily large, we can send accuracy: 2 F x, ɛ i x ɛ 7 0. Then we have, to arbitrary 8 Each factor of ɛ can be shifted to the left side of the product by multiplying by the adjoint an appropriate number of times: 2

3 A Adj x F x, ɛ [ A ɛ 9 ] [ x ] A2 ɛ... A ɛ 20 i [ Ai ɛ i [ Ai ɛ i ] [ x ] 2 i ] x 22 By choosing ɛ sufficiently small, the product of onentials is arbitrarily well approximated by the onential of a sum: F x, ɛ i We can use the properties of the adjoint to rewrite A : A i ɛ + O ɛ 2 x 23 for a Lie group Taking the i th power: A Adj x ad x ad x A i i ad x Thus as, the sum becomes an integral: i A i ˆ 0 i i ad x 28 t ad x dt 29 The integration can be performed on the power series of the matrix onential. i A i ˆ 0 i0 i0 3 i0 t i ad i x i! dt 30 t i+ ad i x i +! 0 3 ad i x i +! 32

4 Substituting into Eq.23: F x, ɛ i0 ad i x i +! ɛ + O ɛ 2 x Using the definition from Eq.4, D x ɛ ɛ0 log F x, ɛ x 33 ad ɛ ɛ0 i x ɛ + O ɛ 2 34 i +! i0 i0 ad i x i +! 35 4 Differential of log When x log x, we can invert the function being differentiated in Eq.4: The second term vanishes when differentiating by δ: δ f ɛ log x + ɛ x 36 ɛ log δ x x 37 D log x δ δ0 log δ x 38 In this bijective region of the function, the differential of the inverse is the inverse of the differential: ɛ δ δ 39 ɛ D log x D x 40 5 Special Cases The infinite series of Eq. 35 can be ressed in closed form in some Lie groups. 4

5 5. SO3 5.. Differential of The elements of the algebra so3 are 3 3 skew-symmetric matrices, and the adjoint representation is identical: ω R 3 4 ω 0 ω 3 ω 2 ω 3 0 ω so3 42 ω 2 ω 0 ad ω ω 43 ad 3 ω ad ω 44 Because the higher powers of ad collapse back to lower powers, we can collect terms in the series: D ω I + I + i0 i i ad ω + 2i + 2! cos ω + i0 sin i i 2i + 3! ad 2 ω 45 ω 2 46 ote that So D ω can be rewritten: ω 2 ωω T I 47 D ω I + sin cos I + ω + cos sin ω + sin ωω T I 48 ωω T 49 We label the coefficients for convenience: a sin 50 b cos 2 5 c a 2 52 D ω a I + b ω + c ωω T 53 5

6 5..2 Differential of log Recall that in the bijective region of and log, For < 2π, a closed-form inverse exists for D ω: D log ω D ω 54 D ω I 2 ω + e ω 2 55 e b 2c 2a 56 b 2 a cos 57 Depending on the value of, the more convenient of Eq. 56 or Eq.57 should be used to compute e. 5.2 SE Differential of Again, the higher powers of ad can be ressed in terms of lower powers: Collecting the terms, we have: Using the identity u, ω R ω u x se ω u ad x 6 0 ω ad 2 ω 2 x ω u + u ω 0 ω 2 62 ad 3 x 2 ad x 2 ω T 0 ω u a 2b Q ω b 3c 2 ω x + 2 ω 2 64 D x I + a ad x + c ad 2 x + ω T 0 Q ω u D ω b u + c ω u + u ω + ω T u Q ω 66 0 D ω 6

7 ω u + u ω ωu T + uω T 2 ω T u I 67...we can rewrite D x: W ω 2c I + Q ω 68 a 2b 2c I + b 3c 2 ω + 2 ωω T 2 I 69 a 2b c b I + b 3c 2 ω + 2 ωω T 70 D ω b u D x + c ωu T + uω T + ω T u W ω 7 0 D ω Differential of log A square block matrix M with the form - M A B 0 A 72...has an inverse: M A A B A 0 A 73 Thus, when < 2π, a closed form exists for D x using D ω as given by Eq.55: B b u + c D D x ω ωu T + uω T + ω T u D ω B D 0 D ω ω W ω