# 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](http://www.probabilistic-robotics.org/), 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.

<figure><img src="/files/fLnUycUNnGyLzk967C9f" alt=""><figcaption></figcaption></figure>

TODO

<figure><img src="/files/EyQvEiIcTKJWuhKuRoin" alt="" width="158"><figcaption></figcaption></figure>


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