Motion Model

Recall that one of steps in Bayes Filter algorithm is to perform a prediction. The prediction is an estimate of the state transition:

$$p(x_t|u_{t-1},x_{t-1})$$

How we use the model depends on the filter algorithm. For example, with particle filter, we essentially simulate a physical event and it's a direct application of the motion model. The input is $u_{t-1}$and $x_{t-1}$, and the output is $x_t$. With particle filter and other sampling methods, we don't need to calcualte $p(x_t|u_{t-1},x_{t-1})$. For algorithms that require an explicit calculation, the inputs are $x_t$, $u_{t-1}$, and $x_1$. However, $u_{t-1}$ and $x_{t-1}$are real values while $x_t$is hypothetical.

In the book Probabilistic Robotics, the auther presents two models:

• velocity model

• odometry model

The mathematical derivation starts from assuming the instantenous movement of the robot follows a perfect circular path.

TODO