Please read this note first.
$$ P (X_{t}, M | U_{t-1}, Z_{t}) $$
$X_t \rightarrow \text{Poses}; M \rightarrow \text{Map};\\ U_{t-1} \rightarrow \text{Control Inputs}; Z_t \rightarrow \text{Observations}$
Given robot's control inputs and observations, we want to estimate the probability distribution of robot's path and the map.
Over here, dark blue dots refer to the control inputs and light blue dots refer to the observations. $\mathbf{l}$ stands for landmarks and $\mathbf{x}$ stands for poses.
stop here to revise for least squares optimization.
$$ P (X_{t}| U_{t-1}) $$
question: What makes an optimisation problem above? 10 poses, 10 measurements? (9 constraints + 1 prior)
Two different kinds of constraints are necessary for pose graph SLAM. The first is odometric constraints that connect two successive states $\mathbf{x}{i}$ and $\mathbf{x}{i+1}$ via a motion model. Furthermore, in order to perform loop closing, the robot has to recognize places it already visited before. This place recognition is also a part of the front-end and provides the second type of constraint, the loop closure constraints. These constraints connect two not necessarily successive poses $\mathbf{x}{i}$ and $\mathbf{x}{j}$.
Pose Graph Representation
Odometry constraints/edges $u_i$ between $x_i$ and $x_{i+1}$: As denoted by blue 🔵 circle
$$ x_{i+1} \: \sim \: \mathcal{N}(f(x_{i}, u_{i}), \Sigma_{i}) $$
Loop closure constraints/edges $u_{ij}$ between $x_i$ and $x_j$ : As denoted by green 🎾 circle
$$ x_{j} \: \sim \: \mathcal{N}(f(x_{i}, u_{ij}), \Lambda_{ij}) $$