> For the complete documentation index, see [llms.txt](https://www.learnros2.com/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://www.learnros2.com/probabilistic-robotics/motion-model.md).
# 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.
TODO