1.3 Parameter Perturbations

Above, the operator $\ominus$ was defined for computing differences in the observation space, in such a way that $\mathbf{z}_{i}\ominus\mathbf{h}_{i}(\mathbf{x})$ is a vector in $\mathbb{R}^{m}$ even when $\mathbf{z}_{i}$ and $\mathbf{h}_{i}(\mathbf{x})$ are not represented as vectors. Similarly, we define the operator $\oplus$ for ``adding'' a perturbation to our parameters. The parameter space $X$ could be a vector space like $\mathbb{R}^{n}$, or instead some other manifold with $n$ degrees of freedom. For pose estimation, $X=\mathrm{SE}(3)$ and $n=6$.

Consider a parameter perturbation vector $\boldsymbol{\delta}\in\mathbb{R}^{n}$. Then for $\mathbf{x}\in X$, we have

$$\oplus : X\times\mathbb{R}^{n}\rightarrow X$$ (12)

When $X=\mathbb{R}^{n}$, this is just standard vector addition:

$$\mathbf{x}\oplus\boldsymbol{\delta}\equiv\mathbf{x}+\boldsymbol{\delta}$$ (13)

When $X=G$ for a Lie group $G$, the perturbation is expressed as left multiplication in the group:

$$\mathbf{x}\oplus\boldsymbol{\delta}\equiv\exp\left(\boldsymbol{\delta}\right)\cdot\mathbf{x}$$ (14)

Thus in Lie groups, using Eqs. :autorefequation8 and :autorefequation14, our intuition for $\oplus$ and $\ominus$ holds:

$$\left(\mathbf{x}\oplus\boldsymbol{\delta}\right)\ominus\mathbf{x} = \ln\left(\exp\left(\boldsymbol{\delta}\right)\cdot\mathbf{x}\cdot\mathbf{x}^{-1}\right)$$ (15)

$$= \ln\left(\exp\left(\boldsymbol{\delta}\right)\right)$$ (16)

$$= \boldsymbol{\delta}$$ (17)

In manifolds without the exponential map, the perturbation can be computed as an update that might violate the manifold constraints, followed by a projection back onto the manifold.