# Exercise - Sensorfusion and Localization (1D-Kalman Filter)

[TODO]

## Requirements

### Knowledge

To complete this exercise notebook, you should possess knowledge about the following topics:

Kalman Filter 1D

• Notes by Christian Herta [HER18]

### Python Modules

import scipy.stats
import numpy as np
from matplotlib import pyplot as plt

%matplotlib inline

## Exercises

### Exercise - Sensorfusion

We have three sensor which measure a distance.

We get the following values from the measurements:

• Sensor 1:$50.1$ cm
• Sensor 2:$49.3$ cm
• Sensor 3:$49.7$ cm

From the calibration of the sensors we have the following estimate of the standard deviations for the sensors (in the range 30-90 cm):

• Standard deviation of sensor 1:$0.8$ cm
• Standard deviation of sensor 2:$1.2$ cm
• Standard deviation of sensor 3:$0.9$ cm

Assume that the sensors are prefectly calibrated such we have no systematic error (bias).

Remark: If you are interesed how to estimate the error of an sensor, then read for an easy explaination https://amloceanographic.com/blog/sensor-accuracy/

• What is the distance if we combine of all three measurements?

• Also give an error estimate of the result (the true value should be in at least 0.95 of the Gaussian area), e.g. as$88\pm 4$m (How is this related to the standard deviation?)

Solve the exercise

1. with pen & paper
2. implement it with numpy
3. plot the differnet Gaussians.

Note: Take care for the significant digits you use.

sigma123 * 2

### Exercise - 1D Kalman Filter

A robot moves with a velocity of about$3$ m/s. We assume that if the robot moves in a time$\Delta t=2$s we have an standard deviation of the moved distance of$0.8$m.

Each 2 seconds we measure the position with an standard deviation$\sigma$ of$1.2$ m.

We get the following measurements:

-$z(t=0 SINGLESINGLE ) = -2$m (inital$\sigma=0$) -$z(t=2 SINGLESINGLE ) = 3.4$ m -$z(t=4 SINGLESINGLE )$ = no measurement -$z(t=6 SINGLESINGLE )= 16.3$ m

What is the predicted position before the measurement at$t=8$s.

In the state space we use just the position of the robot (1D). Solve the exercise

1. with pen & paper
2. implement it with numpy

## Literature

### Notebook License (CC-BY-SA 4.0)

The following license applies to the complete notebook, including code cells. It does however not apply to any referenced external media (e.g., images).

Exercises - Sensorfusion and Localization (1D-Kalman Filter)
by Christian Herta