From source code for Factor2Poses2D residuals calculated:
+$$ +\begin{equation} +\overline{r} = R_i^T\cdot(\overline{x}_{t+1} - \overline{x}_t) - \overline{z}_t, +\end{equation} +$$$$ +\overline{x}_t = (x_t, y_t,\theta_t)^T +$$where:
+$$ +\begin{equation} +R_i^T = \left[ +\begin{array}{cc} +cos(\theta_t) & sin(\theta_t)\\ +-sin(\theta_t) & cos(\theta_t)\\ +\end{array} +\right] \in \mathbb{R}^{2 \times 2} +\end{equation} +$$and +$$ +\begin{equation} +\dfrac{dR_i^T}{d\theta_t} = \left[ +\begin{array}{cc} +-sin(\theta_t) & cos(\theta_t)\\ +-cos(\theta_t) & -sin(\theta_t)\\ +\end{array} +\right] \in \mathbb{R}^{2 \times 2} +\end{equation} +$$
+and
+$$ +\delta x = x_{t+1} - x_{t}\\ +\delta y = y_{t+1} - y_{t}\\ +$$ +Thus, for residuals have the following derivatives:
+$$ +\boxed{ +\dfrac{d \overline{r}}{d \overline{x}_t} = \left[ +\begin{array}{ccc} +-R_i^T & | & \dfrac{dR_i^T}{d\theta_t} \left(\begin{array}{c}\delta x\\ \delta y\\ \end{array}\right)\\ +\hline\\ +\begin{array}{cc} 0&0\\ \end{array} & | &-1\\ +\end{array} +\right] \in \mathbb{R}^{3 \times 3} +} +$$$$ +\boxed{ +\dfrac{d \overline{r}}{d\overline{z}_t} = -I \in \mathbb{R}^{3 \times 3} +} +$$$$ +\boxed{ +\dfrac{d^2\overline{r}}{d\overline{x}_t d\overline{z}_t} = 0 \in \mathbb{R}^{3 \times 3\times 3} +} +$$ +