# Meaning of the equation of graphical model

The paper Topology-based Representations for Motion Planning and Generalisation in Dynamic Environments with Interactions by Ivan et.al., says on page 10 that the Approximate Inference Control (AICO) framework translates the robot dynamics to the graphical model by the following equation:

What does p(x0:T,u0:T) mean? I feel that p means 'prior of' some uncertain quantity, but I'm not sure about this.

• Maybe you should be more specific in your question. $p(x_{0:T},u_{0:T})$ normally means the joint probability distribution of all states and control inputs. Apr 1, 2016 at 7:46

It is not a prior. It is the posterior of trajectory and controls. Basically that equation gives you the motion model of the robot similar to the motion models described in Sebastian Thrun's Probabilistic Robotics. $x_{0:t}$ is a set of poses i.e., trajectory and $u_{0:t}$ is a set of controls, i.e., Motor commands for the joints. The posterior here provides a likelihood estimate of the robot's pose after a control command. From what I understand after skimming through the rather obscure paper and another paper Robot Trajectory Optimization using Approximate Inference that the LHS is used to compute a likelihood for the next pose, $x_{t+1}$ if a control action $u_t$ is chosen with the knowledge of $x_t$. The LHS can be viewed as described in the Background section below. From the latter paper, the estimation of state $x_{t+1}$ requires some confidence on the knowledge of current state $x_t$ which is given by the covariance matrix $A$ and mean $a$.
$p(x_{0:t},u_{0:t})$ is called a state transition probability, it can also be written as $p(x_{t}|x_{0:t-1}, u_{0:t})$ since it can be assumed that $x_t$ is a complete state or here just the pose of the robot, i.e. it is a summary of all that happened in the past poses after executing commands in the set $u_{1:t}$. What this results in is that $x_t$ is conditional stochastically on $x_{t-1}$ and can be estimated by executing the command $u_t$. Which is what the motion model does for you and it is shown as $p(x_t|u_t, x_{t-1})$. It says that the state $x_t$ depends on $u_t$ and can be computed as long as $x_{t-1}$ is known.