# Covariance of Gaussian after sequence of homogeneous transformations

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 $$$$l^{(k)} = M^{k}_1 \ \ l^{(1)}$$$$ where $$M^k_1$$ is the transformation from $$s_k$$ to $$s_1$$ given by $$$$M^k_1 = \prod_{i=1}^{k-1}\Big(M_{i+1}^{i}\Big)^{-1}$$$$ and $$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 $$$$\Sigma_{l^{(k)}} = \Sigma^1_k \ \Sigma_{l^{(i)}} \ \Sigma^1_k$$$$ where $$\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?