Assume we have a robot that moves and observes landmarks on his path. The path is composed of $k$ poses $s_1 \: ... s_k$ where $s_i = [x,\ y,\ \theta]$. From the sensors, we read the motion measurements as a sequence of gaussian random variables $\mathcal{N}(\boldsymbol{\mu}^1_2, \boldsymbol{\Sigma}^1_2) \ ... \ \mathcal{N}(\boldsymbol{\mu}^{k-1}_k, \boldsymbol{\Sigma}^{k-1}_k)$, where $\boldsymbol{\mu}^{i-1}_i = [\Delta x, \ \Delta y, \ \Delta\theta]$ is the measurement of the motion from $s_{i-1}$ to $s_i$ and $\Sigma^{i-1}_{i}$ is the covariance of that measurement.
At pose $s_1$, the robot measures a landmark $l$, generating the measurement $\phi = \mathcal{N}(l^{(1)}, \Sigma_{l^{(1)}})$ where $l^{(1)} = [x, y]$ and represents the landmark position in the reference frame of $s_1$.
We now transform the landmark measurement $l^{(1)}$ into the reference frame of the last pose $s_k$ with the homogeneous transformation \begin{equation} l^{(k)} = M^{k}_1 \ \ l^{(1)} \end{equation} where $M^k_1$ is the transformation from $s_k$ to $s_1$ given by \begin{equation} M^k_1 = \prod_{i=1}^{k-1}\Big(M_{i+1}^{i}\Big)^{-1} \end{equation} and $M_{i+1}^i$ is the homogeneous matrix correspondent to the motion measurement $\boldsymbol{\mu}_i$.
I am interested in understanding what will the covariance of $l^{(k)}$ be. My intuition is that since components of $l^{(1)}$ do not appear in $M^1_k$, then $l^{k}$ is just a linear transformation of $l^{(1)}$ and the covariance $\Sigma_{l^{(k)}}$ will be \begin{equation} \Sigma_{l^{(k)}} = \Sigma^1_k \ \Sigma_{l^{(i)}} \ \Sigma^1_k \end{equation} where $\Sigma^1_k$ is the covariance associated with the transformation $M^1_k$, and I cannot directly compute it, but only approximate it with an EKF-like algorithm. Am I correct?
