I am currently trying to understand the books SLAM for dummies and Simulataneous localization and mapping with the extended Kalman filter to implement slam. I have understood steps 1 and 2 SLAM for dummies. However I am having difficulty understanding Step 3: Add new landmarks to the current state (Page 40). The corresponding section in Simulataneous localization and mapping with the extended Kalman filter is 2.3.4 Landmark initialization for full observations.
Specifically, I do not understand how the SLAM specific Jacobian $J_{xr}$ (SLAM for dummies) defined below is actually derived:
I am confused about what transformation function the Jacobian $J_{xr}$ is linearizing. I notice that $J_{xr}$ is the same as Jacobian $G_R$ that linearizes the Inverse Observation model as described in section 2.3.4 of Simulataneous localization and mapping with the extended Kalman filter. Why does $J_{xr}$, contain $\Delta t$, which is the thrust applied to the robot? $\Delta t$ which belongs to the State Prediction model shouldn't be in the Inverse Observation model, no? Am I right to say that the Inverse Observation model is a function of the robot pose : [$x_{r}$, $y_{r}$, $\theta_{r}$] and the new landmark observation: [$d_{l1}$, $\phi_{l1}$] and the output is the updated state vector : [$x_{r}$, $y_{r}$, $\theta_{r}$, $x_{l1}$, $y_{l1}$]?
