A micro Lie theory for state estimation in robotics006:D. The exponential map

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博客介绍了指数映射exp()如何将李代数的元素精确转换到群上，这一过程称为回缩。它通过解决常微分方程来实现，并且与流形上的测地线概念相关。exp()的逆操作是log()，对应于展开操作。文章还提出了通用定义，用切线增量和泰勒级数来表达指数映射，并讨论了其在几何和动力系统中的应用。

The exponential map exp() allows us to exactly transfer
elements of the Lie algebra to the group (Fig. 1), an operation
generically known as retraction. Intuitively, exp() wraps the
tangent element around the manifold following the great arc
or geodesic (as when wrapping a string around a ball, Figs. 1,
3 and 4). The inverse map is the log(), i.e., the unwrapping
operation. The exp() map arises naturally by considering the
time-derivatives of X ∈ M over the manifold, as follows.From (9) we have,

$\dot X= X v^∧ \ \ \ \ \ \ \ \ (12)$
For v constant, this is an ordinary differential equation (ODE) whose solution is $X (t) = X (0) exp(v^∧t) \ \ \ (13)$
Since X (t) and X (0) are elements of the group, then $exp(v^∧t) = X (0)^{−1}X (t)$ must be in the group too, and so exp(v∧t) maps elements v∧t of the Lie algebra to the group.This is known as the exponential map.

In order to provide a more generic definition of the exponential map, let us define the tangent increment $τ\triangleq vt∈R^m$ as velocity per time, so that we have $τ^∧ = v^∧t ∈ m$ a point in the Lie algebra. The exponential map, and its inverse the logarithmic map, can be now written as

$exp : m → M ; τ^∧→ x = exp(τ^∧) \ (14)\\ log : M → m ; x→ τ^∧ = log(x) \ \ \ (15)$
Closed forms of the exponential in multiplicative groups are obtained by writing the absolutely convergent Taylor serie

$\epsilon + τ^∧ +\frac{1}{2}{τ^∧}^2 +\frac{1}{3!}{τ^∧}^3 + · · · , (16)$

and taking advantage of the algebraic properties of the powers of τ∧ (see Ex. 4 and 5 for developments of the exponential

指数映射 exp() 允许我们准确地传递
李代数的元素到群（图1），一个操作一般称为回缩。直观地，exp() 包装了
大圆弧周围流形周围的切元素或测地线（如在球周围缠绕绳索时，图 1，3 和 4）。逆
映射是 log()，即展开手术。 exp() 映射是通过考虑
X ∈ M 在流形上的时间导数，如下。从（9）我们有，
$\dot X= X v^∧\ \ \ \ \ \ \ \ (12)$ 对于 v 常数，这是一个常微分方程 (ODE)，其解为
$X (t) = X (0) exp(v^∧t) \ \ \ (13)$
由于 X (t) 和 X (0) 是群的元素，那么 $exp(v^∧t) = X (0)^{−1}X (t)$ 也必须在群中，所以 exp(v∧t) 映射元素李代数 $v^∧t$ 到群中。这被称为指数映射。
为了提供指数映射的更通用定义，让我们将切线增量 $τ\triangleq vt∈R^m$ 定义为单位时间的速度，这样我们就有 $τ^∧ = v^∧t ∈ m$ 是李代数中的一个点。指数映射及其对数映射的逆，现在可以写成
$exp : m → M ; τ^∧→ x = exp(τ^∧) \ (14)\\ log : M → m ; x→ τ^∧ = log(x) \ \ \ (15)$
通过写出绝对收敛的泰勒级数，可以得到乘法群中指数的闭形式
$\epsilon + τ^∧ +\frac{1}{2}{τ^∧}^2 +\frac{1}{3!}{τ^∧}^3 + · · · , (16)$
并利用 τ∧ 幂的代数性质（见例 4 和例 5 以了解指数的发展)
例 4 和例 5