Are there any advantages of doing the exponential map of an element $\in \mathfrak{se}(3)$ in order to get the translation component?

Question

I'm currently learning about Lie Groups, concretely about the special Euclidean group $SE(3)$, which in the field that I am studying (state estimation for robotics) is commonly used for representing poses, that consist of a rotation matrix $\mathbf{R}\in\mathbb{R^{3\times3}}$ and a translation vector $\mathbf{t}\in \mathbb{R^3}$: $$\mathbf{T}(\mathbf{R},\mathbf{t})=\begin{bmatrix} \mathbf{R} & \mathbf{t}\\ \mathbf{0} & 1 \end{bmatrix}$$

I understand that during the optimization of the rotation matrix, it's very important to move locally on the smooth manifold $SO(3)$ by applying additive perturbations on the tangent space and mapping them to the manifold via the exponential map. This way we are able to ensure that the estimated rotation matrix belongs to $SO(3)$.

But regarding the translational component of a pose $\in SE(3)$: Are there any advantages of applying additive perturbations on the tangent space of the manifold? I am struggling to see this, because I feel like the estimation of the translation component of the pose, could be equally well estimated without making use of the exponential map.

Answer 1

The key here is that $SE(3)$ performs translation and rotation simultaneously as a continuum. Therefore, the translation is totally dependent on the rotational component of the movement.

Think of a simple 2D robot moving on a plane. Imagine it is moving forward while rotating around $z$, so that its trajectory is a curve. The translational part of its pose after a time $\Delta t$ is clearly affected by the rotational velocity (compare it with the case in which there is no rotation).

The motion model for a robot that first translates and only then rotates can be expressed as $T(3) \times SO(3)$.

This is very well explained in the Example 7 (page 11) of Solà's paper. I recommend you to read the whole paper (dense but worth it) to fully understand the point of the example.

Are there any advantages of applying additive perturbations on the tangent space of the manifold?

Having explained the theory before, I'm aware that some authors may use $T(3) \times SO(3)$ in their optimizations. For example, compare the exponential and logarithmic maps of $SE(3)$ with the pseudo-exponential and pseudo-logarithmic maps in section 9.4.2 of Blanco's tutorial.

I'm afraid I can't really say how both approaches compare in their trade-offs between computational complexity and accuracy.