# On the Functional and Extra-Functional Properties of IMU Fusion Algorithms for Body-Worn Smart Sensors

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Background and Methods

#### 3.1. Used Algorithms

- Complementary Filter

- Mahony Filter

- Madgwick filter

- Kalman Filter

#### 3.2. Quaternion Representation

#### 3.3. Data Formats

#### 3.3.1. Single Precision Floating-Point

#### 3.3.2. Fixed-Point Numbers

#### 3.4. Used Hardware

- The SAM D20 contains a single cycle hardware multiplier for 32-bit integer numbers, which means that addition and multiplication take the same time.
- The SAM D20 does not have hardware support for floating-point numbers. All floating-point operations have to be emulated in software resulting in higher execution time and power consumption.

#### 3.5. Analysis of Extra-Functional Properties

#### 3.5.1. Code Size

#### 3.5.2. Computational Effort

#### 3.6. Analysis of Functional Properties

#### Filter and Measurement Parameters

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#### 3.7. Statistical Analysis

## 4. Results: Extra-Functional Properties

#### 4.1. Code Size

#### Explanation of the Size Differences

#### 4.2. Execution Time

#### 4.3. Summary for the Extra-Functional Properties

## 5. Results: Functional Properties

#### 5.1. General Comparison of the Fusion Results

#### 5.1.1. Quantization Errors

#### 5.2. Statistical Analysis

#### 5.2.1. Results Grouped by Data Format

#### 5.2.2. Results Grouped by Movement Speed

#### 5.3. Analysis of External Influences

#### 5.3.1. Movement Speed

#### 5.3.2. User Interaction

#### 5.3.3. Other Factors

- Precision of the image analysis. This factor is influenced by the resolution of the camera, the frame rate of the camera, and the precision of the used image analysis algorithm.
- Cross correlation of the data from reference and sensor fusion. When the timestamps of the data do not fit precisely, there will be an error added to the whole measurement.

## 6. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AHRS | Attitude and Heading Reference System |

BLE | Bluetooth Low Energy |

MEMS | Micro-Electro-Mechanical Systems |

$\mathsf{\mu}$C | Microcontroller |

IMU | Inertial Measurement Unit |

SiP | System in Package |

ROM | Read Only Memory |

SRAM | Static Random-Access Memory |

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**Figure 1.**Measurement setup for the assessment of the computation time of the sensor fusion algorithms.

**Figure 2.**Data flow of the measurement setup. Steps involving the camera data are colored orange. Steps involving the sensor fusion result are colored blue. The resulting error is colored in gray.

**Figure 6.**Box-Plot of the execution times for the 32-bit floating-point implementations. The boxes show the deviations from the median execution time in milliseconds.

**Figure 10.**Comparison of the fusion out of the Kalman filter for the three data formats. (

**A**) output of the Kalman filter with 32-bit floating-point data; (

**B**) output of the Kalman filter with 32-bit fixed-point data; (

**C**) output of the Kalman filter with 16-bit fixed-point data; (

**D**) output of the Mahony filter with 16-bit fixed-point data.

**Figure 11.**Estimated influence of rotation speed, external influences, and used algorithm on the result of the sensor fusion grouped by data format.

**Figure 12.**Estimated influence of data format, external influences, and used algorithm on the result of the sensor fusion grouped by rotation speed.

**Figure 14.**Comparison between accuracy with which the user conducted a predefined movement and error from the sensor fusion output.

Data Format | Kalman | Madgwick | Mahony | Complementary |
---|---|---|---|---|

32-bit Floating-Point | 3.963 ms | 1.142 ms | 0.758 ms | 0.782 ms |

32-bit Fixed-Point | 1.923 ms | 0.560 ms | 0.350 ms | 0.382 ms |

16-bit Fixed-Point | 0.621 ms | 0.166 ms | 0.121 ms | 0.123 ms |

**Table 2.**Average error in degree measured for the four sensor fusion algorithms using the three data formats.

Data Format | Kalman | Madgwick | Mahony | Complementary |
---|---|---|---|---|

32-bit Floating-Point | 1.557 | 1.657 | 1.588 | 1.489 |

32-bit Fixed-Point | 1.562 | 1.603 | 1.557 | 1.478 |

16-bit Fixed-Point | 2.429 | 1.722 | 1.626 | 1.539 |

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**MDPI and ACS Style**

Büscher, N.; Gis, D.; Kühn, V.; Haubelt, C.
On the Functional and Extra-Functional Properties of IMU Fusion Algorithms for Body-Worn Smart Sensors. *Sensors* **2021**, *21*, 2747.
https://doi.org/10.3390/s21082747

**AMA Style**

Büscher N, Gis D, Kühn V, Haubelt C.
On the Functional and Extra-Functional Properties of IMU Fusion Algorithms for Body-Worn Smart Sensors. *Sensors*. 2021; 21(8):2747.
https://doi.org/10.3390/s21082747

**Chicago/Turabian Style**

Büscher, Nils, Daniel Gis, Volker Kühn, and Christian Haubelt.
2021. "On the Functional and Extra-Functional Properties of IMU Fusion Algorithms for Body-Worn Smart Sensors" *Sensors* 21, no. 8: 2747.
https://doi.org/10.3390/s21082747