# 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:

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

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