Exponential Map

Turns an axis-times-angle vector into a rotation or pose; the log map does the reverse.

The exponential map comes from Lie group theory: it maps an element of a Lie algebra (the tangent space of the group at its identity element, which can be thought of as the linear space where 'velocities' live) onto the group itself; for matrix Lie groups, it's just the matrix exponential, exp(X) = I + X + X²/2! + …. The log map is its inverse near the identity. The most common use in robotics is with rotations: combine a unit rotation axis ω̂ and an angle θ into the 3D vector ω̂θ (exponential coordinates, also called the rotation vector), write it as the skew-symmetric matrix [ω̂]θ, and exponentiate to get the rotation matrix R — the closed-form result is Rodrigues' formula; taking the log of R recovers ω̂θ. Likewise, the exponential map from se(3) to SE(3) turns a twist into a homogeneous transformation, and underlies the product of exponentials formula. State estimation and optimization routines often perform addition and subtraction in this flat tangent space and then map back to a rotation with exp, avoiding the orthogonality-breaking errors of directly adding or subtracting rotation matrices.

ExampleWith ω̂ = (0, 0, 1) and θ = π/2, the exponential map gives the rotation matrix for a 90° rotation about the z-axis; taking the log of that matrix recovers the vector (0, 0, π/2).

Also called

Log Map, Logarithmic Map, Exp/Log

Related

Lie Group · Rotation Matrix · Axis-Angle Representation · Rodrigues' Rotation Formula · Product of Exponentials Formula · Skew-Symmetric Matrix

Sources

Solà, Deray, Atchuthan: A micro Lie theory for state estimation in robotics
Lynch & Park, Modern Robotics（Ch. 3 exponential coordinates / matrix logarithm）
Wikipedia: Exponential map (Lie theory)