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Some Python Implementations of the Kalman Filter

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Multidimensional Kalman-Filter
==============================

#### See [Vimeo](https://vimeo.com/album/2754700/sort:preset/format:detail) for some Explanations. Or if you want to start with the basics, you might want to take a look at these Blogposts:

* [Das Kalman Filter einfach erklärt (Teil 1)](http://www.cbcity.de/das-kalman-filter-einfach-erklaert-teil-1)
* [Das Kalman Filter einfach erklärt (Teil 2)](http://www.cbcity.de/das-kalman-filter-einfach-erklaert-teil-2)
* [Das Extended Kalman Filter einfach erklärt](http://www.cbcity.de/das-extended-kalman-filter-einfach-erklaert)

Some Python Implementations of the Kalman Filter
------------------------------

![Kalman Filter Step](https://raw.githubusercontent.com/balzer82/Kalman/master/Kalman-Filter-Step.png)

### Kalman Filter with Constant Velocity Model

Situation covered: You drive with your car in a tunnel and the GPS signal is lost. Now the car has to determine, where it is in the tunnel. The only information it has, is the velocity in driving direction. The x and y component of the velocity (x˙ and y˙) can be calculated from the absolute velocity (revolutions of the wheels) and the heading of the vehicle (yaw rate sensor).

[View IPython Notebook](https://github.com/balzer82/Kalman/blob/master/Kalman-Filter-CV.ipynb?create=1) ~ [See Vimeo](https://vimeo.com/87854542)

### Kalman Filter with Constant Acceleration Model

#### in 2D

Situation covered: You have an acceleration sensor (in 2D: $\ddot x¨ and y¨) and try to calculate velocity (x˙ and y˙) as well as position (x and y) of a person holding a smartphone in his/her hand.

[View IPython Notebook](https://github.com/balzer82/Kalman/blob/master/Kalman-Filter-CA.ipynb?create=1) ~ [See Vimeo](https://vimeo.com/87854541)

Second example is the same dynamic model but this time you measure the position as well as the acceleration. Both values have to be fused together with the Kalman Filter.
Situation covered: You have an acceleration sensor (in 2D: x¨ and y¨) and a Position Sensor (e.g. GPS) and try to calculate velocity (x˙ and y˙) as well as position (x and y) of a person holding a smartphone in his/her hand.

[View IPython Notebook](https://github.com/balzer82/Kalman/blob/master/Kalman-Filter-CA-2.ipynb?create=1) ~ [See Vimeo](https://vimeo.com/87854540)

#### in 3D

Third example is in 3D space, so the state vector is 9D. This model is for ball tracking or something else in 3D space.

![Kalman 3D](https://raw.githubusercontent.com/balzer82/Kalman/master/Kalman-Filter-CA-Ball-StateEstimated.png)

[View IPython Notebook](https://github.com/balzer82/Kalman/blob/master/Kalman-Filter-CA-Ball.ipynb?create=1)

### Adaptive Kalman Filter with Constant Velocity Model

Here the Measurement Covariance Matrix R is calculated dynamically via the maximum likelihood of the acutal standard deviation of the last measurements.

[View IPython Notebook](https://github.com/balzer82/Kalman/blob/master/Adaptive-Kalman-Filter-CV.ipynb?create=1)

### Kalman Filter for Motorbike Lean Angle Estimation

Also know as the Gimbal Stabilization problem: You can measure the rotationrate, but need some validation for the correct lean angle from time to time, because simply an integration of the rotationrate adds up a lot of noise. There comes the vertical acceleration, which is a pretty good estimator for the angle in static situations. This Kalman Filter implementation fuses both together with some adaptive components.

[View IPython Notebook](https://github.com/balzer82/Kalman/blob/master/Kalman-Filter-Bike-Lean-Angle.ipynb?create=1)

## Extended Kalman Filter

![EKF Filter Step](https://raw.githubusercontent.com/balzer82/Kalman/master/Extended-Kalman-Filter-Step.png)

### Extended Kalman Filter with Constant Turn Rate and Velocity (CTRV) Model

Situation covered: You have an velocity sensor which measures the vehicle speed (v) in heading direction (ψ) and a yaw rate sensor (ψ˙) which both have to fused with the position (x & y) from a GPS sensor.

![State Vector](https://raw.githubusercontent.com/balzer82/Kalman/master/CTRV-Model.png)

[View IPython Notebook](https://github.com/balzer82/Kalman/blob/master/Extended-Kalman-Filter-CTRV.ipynb?create=1) ~ [See Vimeo](https://vimeo.com/88057157)

### Extended Kalman Filter with Constant Heading and Constant Velocity (CHCV) Model

Situation covered: You have the position (x & y) from a GPS sensor and extimating the heading direction (ψ) and the velocity (v).

[View IPython Notebook](https://github.com/balzer82/Kalman/blob/master/Extended-Kalman-Filter-CHCV.ipynb?create=1)

### Extended Kalman Filter with Constant Turn Rate and Acceleration (CTRA) Model

Situation covered: You have an acceleration and velocity sensor which measures the vehicle longitudinal acceleration and speed (v) in heading direction (ψ) and a yaw rate sensor (ψ˙) which all have to fused with the position (x & y) from a GPS sensor.

[View IPython Notebook](https://github.com/balzer82/Kalman/blob/master/Extended-Kalman-Filter-CTRA.ipynb?create=1)

## License

`CC-BY-SA2.0 Lizenz`

You are free to:

* Share — copy and redistribute the material in any medium or format for any purpose, even commercially.
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The licensor cannot revoke these freedoms as long as you follow the license terms.

Under the following terms:

* Attribution - You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
* ShareAlike - If you remix, transform, or build upon the material, you must distribute your contributions under the same license as the original.

No additional restrictions - You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.

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