# Ski Jumping Trajectory Reconstruction Using Wearable Sensors via Extended Rauch-Tung-Striebel Smoother with State Constraints

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## Abstract

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## 1. Introduction

## 2. Data Acquisition

## 3. Methods

#### 3.1. Extended Rauch-Tung-Striebel Smoother

#### 3.1.1. Extended Kalman Filter

#### 3.1.2. Rauch-Tung-Striebel Smoother

#### 3.2. Adaption for Parameter Estimation and Soft Constraints

#### 3.2.1. Joint State and Parameter Estimation

#### 3.2.2. Constrained Filtering

#### 3.3. System Model

#### 3.3.1. Coordinate Frame Definitions

#### 3.3.2. Conversion of GPS Position Measurements to the Hill Reference Frame

#### 3.3.3. Constraint Modeling of the Geometric Shape of a Ski Jumping Hill

#### 3.3.4. Measurement Error Models

#### 3.3.5. State, Output, and Constraint Equations

## 4. Results and Discussion

#### 4.1. Setting

#### 4.2. Validation by Simulated Measurement Data

#### 4.3. Validation by Real Measurement Data

#### 4.4. Validation by Jump Length

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Take-Off and Touch-Down Time Point Detection

**Figure A1.**An illustrative example on detecting the take-off and touch-down time points by the raw measurement data.

## Appendix B. Jump Length Detected by the Video Recordings

**Figure A2.**A picture of overlaid snapshots from video recordings to illustrate the jump length detection from video recordings.

## References

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**Figure 5.**Comparison of the extended RTS smoother reconstructed trajectory (color-coded line), the generated reference trajectory (black line), and the GPS measured trajectory (purple dots) plotted on the ski jumping hill model.

**Figure 7.**Comparison of the extended RTS smoother reconstructed trajectory (color-coded line) and the GPS measured trajectory (purple solid line with dots) plotted on the ski jumping hill model (orange lines and the gray surface).

**Figure 8.**Converted GPS position measurements ${\mathit{r}}_{\mathrm{N},\phantom{\rule{4.pt}{0ex}}\mathrm{GPS}}$ (blue dots) and the extended RTS smoother reconstructed relative positions ${\mathit{r}}_{\mathrm{N}}$ (green line).

**Figure 9.**GPS velocity measurements ${\mathit{v}}_{\mathrm{O},\phantom{\rule{4.pt}{0ex}}\mathrm{GPS}}$ (blue dots) and the extended RTS smoother reconstructed velocity ${\mathit{v}}_{\mathrm{O}}$ (green line).

**Figure 10.**The extended RTS smoother estimated geometric constraints $\mathit{c}$ (green line) and pseudo measurements (blue dots).

**Figure 12.**Real magnetometer measurements ${\mathit{m}}_{\mathrm{B},\phantom{\rule{4.pt}{0ex}}\mathrm{mag}}$ (blue dots) and the extended RTS smoother estimated magnetometer measurements ${\mathit{m}}_{\mathrm{B},\phantom{\rule{4.pt}{0ex}}\mathrm{meas}}$ (green line).

**Figure 13.**The gyroscope measurements ${\mathit{\omega}}_{\mathrm{B},\phantom{\rule{4.pt}{0ex}}\mathrm{gyro}}$ (red dots).

**Figure 14.**The accelerometer measurements ${\mathit{a}}_{\mathrm{B},\phantom{\rule{4.pt}{0ex}}\mathrm{acc}}$ (red dots).

Sensor | Type | Frequency | Performance Characteristics |
---|---|---|---|

GPS logger | Qstarz BT Q1000eX | 10 Hz | Position accuracy: 3 m circular error probable (50%), velocity accuracy: 0.1 m/s. [13] |

IMU | InvenSense ICM-20600 | 100 Hz | Gyroscope: measurement range: $\pm 2000{\phantom{\rule{0.166667em}{0ex}}}^{\circ}/$s, rate noise spectral density: $\pm 0.004\phantom{\rule{0.166667em}{0ex}}{(}^{\circ}/\mathrm{s})/\sqrt{\mathrm{Hz}}$; |

Accelerometer: measurement range: ±16 g, noise spectral density: 100 $\mu \mathrm{g}/\sqrt{\mathrm{Hz}}$. [14] | |||

Magnetometer | Alps HSCDTD008A | 100 Hz | Measurement range: ±2.4 mT. [15] |

Name | Symbol | Value |
---|---|---|

The semi-major axis | $a\phantom{\rule{3.33333pt}{0ex}}$ | $6378137.0\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ |

The first eccentricity | $e\phantom{\rule{3.33333pt}{0ex}}$ | $0.0818191908426$ |

The meridian radius of curvature | ${M}_{\mu}\phantom{\rule{3.33333pt}{0ex}}$ | $a\frac{1-{e}^{2}}{{(1-{e}^{2}{sin}^{2}{\mu}_{0})}^{\frac{3}{2}}}\phantom{\rule{3.33333pt}{0ex}}$ |

The radius of curvature in the prime vertical | ${N}_{\mu}\phantom{\rule{3.33333pt}{0ex}}$ | $\frac{a}{\sqrt{1-{e}^{2}{sin}^{2}{\mu}_{0}}}\phantom{\rule{3.33333pt}{0ex}}$ |

**Table 3.**Diagonal elements of the estimated initial states error covariance matrix ${\overline{\mathit{P}}}_{0}$.

System States | Symbol | Estimated Variance |
---|---|---|

Position | ${\mathit{r}}_{\mathrm{N},0}\phantom{\rule{3.33333pt}{0ex}}$ | $\mathrm{diag}{\left(\left[5\phantom{\rule{3.33333pt}{0ex}}5\phantom{\rule{3.33333pt}{0ex}}15\right]\right)}^{2}\phantom{\rule{3.33333pt}{0ex}}\left[{\mathrm{m}}^{2}\right]$ |

Velocity | ${\mathit{v}}_{\mathrm{B},0}\phantom{\rule{3.33333pt}{0ex}}$ | $\mathrm{diag}{\left(\left[1\phantom{\rule{3.33333pt}{0ex}}1\phantom{\rule{3.33333pt}{0ex}}3\right]\right)}^{2}\phantom{\rule{3.33333pt}{0ex}}\left[{\left(\mathrm{m}/\mathrm{s}\right)}^{2}\right]$ |

Attitude quaternions | ${\mathit{q}}_{\mathrm{BO},0}\phantom{\rule{3.33333pt}{0ex}}$ | ${\left(0.1{\mathit{I}}_{4}\right)}^{2}\phantom{\rule{3.33333pt}{0ex}}[-]$ |

Gyroscope bias | $\mathrm{\Delta}{\mathit{\omega}}_{0}\phantom{\rule{3.33333pt}{0ex}}$ | ${\left(0.0873{\mathit{I}}_{3}\right)}^{2}\phantom{\rule{3.33333pt}{0ex}}\left[{\left(\mathrm{rad}/\mathrm{s}\right)}^{2}\right]$ |

Accelerometer bias | $\mathrm{\Delta}{\mathit{a}}_{0}\phantom{\rule{3.33333pt}{0ex}}$ | ${\left(0.2{\mathit{I}}_{3}\right)}^{2}\phantom{\rule{3.33333pt}{0ex}}\left[{\left({\mathrm{m}/\mathrm{s}}^{2}\right)}^{2}\right]$ |

Magnetometer bias | $\mathrm{\Delta}{\mathit{m}}_{0}\phantom{\rule{3.33333pt}{0ex}}$ | ${\left(5{\mathit{I}}_{3}\right)}^{2}\phantom{\rule{3.33333pt}{0ex}}\left[{\left(\mu \mathrm{T}\right)}^{2}\right]$ |

Magnetometer scaling error | $\mathrm{\Delta}{\mathit{S}}_{\mathrm{m},0}\phantom{\rule{3.33333pt}{0ex}}$ | ${\left(0.1{\mathit{I}}_{3}\right)}^{2}\phantom{\rule{3.33333pt}{0ex}}[-]$ |

System Inputs | Symbol | Estimated Variance |
---|---|---|

Gyroscope | ${\mathit{\omega}}_{\mathrm{B},\phantom{\rule{4.pt}{0ex}}\mathrm{gyro}}\phantom{\rule{3.33333pt}{0ex}}$ | ${\left(0.0175{\mathit{I}}_{3}\right)}^{2}\phantom{\rule{3.33333pt}{0ex}}\left[{\left(\mathrm{rad}/\mathrm{s}\right)}^{2}\right]$ |

Accelerometer | ${\mathit{a}}_{\mathrm{B},\phantom{\rule{4.pt}{0ex}}\mathrm{acc}}\phantom{\rule{3.33333pt}{0ex}}$ | ${\left(0.1{\mathit{I}}_{3}\right)}^{2}\phantom{\rule{3.33333pt}{0ex}}\left[{\left({\mathrm{m}/\mathrm{s}}^{2}\right)}^{2}\right]$ |

System Measurements | Symbol | Estimated Variance |
---|---|---|

GPS position | ${\mathit{r}}_{\mathrm{N},\phantom{\rule{4.pt}{0ex}}\mathrm{GPS}}\phantom{\rule{3.33333pt}{0ex}}$ | $\mathrm{diag}{\left(\left[3\phantom{\rule{3.33333pt}{0ex}}3\phantom{\rule{3.33333pt}{0ex}}5\right]\right)}^{2}\phantom{\rule{3.33333pt}{0ex}}\left[{\mathrm{m}}^{2}\right]$ |

GPS velocity | ${\mathit{v}}_{\mathrm{O},\phantom{\rule{4.pt}{0ex}}\mathrm{GPS}}\phantom{\rule{3.33333pt}{0ex}}$ | $\mathrm{diag}{\left(\left[0.1\phantom{\rule{3.33333pt}{0ex}}0.1\phantom{\rule{3.33333pt}{0ex}}0.3\right]\right)}^{2}\phantom{\rule{3.33333pt}{0ex}}\left[{\left(\mathrm{m}/\mathrm{s}\right)}^{2}\right]$ |

Magnetometer | ${\mathit{m}}_{\mathrm{B},\phantom{\rule{4.pt}{0ex}}\mathrm{mag}}\phantom{\rule{3.33333pt}{0ex}}$ | ${\left(5{\mathit{I}}_{3}\right)}^{2}\phantom{\rule{3.33333pt}{0ex}}\left[{\left(\mu \mathrm{T}\right)}^{2}\right]$ |

Quaternions constraint | ${c}_{\mathrm{q}}\phantom{\rule{3.33333pt}{0ex}}$ | ${0.01}^{2}\phantom{\rule{3.33333pt}{0ex}}[-]$ |

In-run vertical constraint | ${c}_{\mathrm{V},\phantom{\rule{4.pt}{0ex}}\mathrm{IR}}\phantom{\rule{3.33333pt}{0ex}}$ | ${0.1}^{2}\phantom{\rule{3.33333pt}{0ex}}\left[{\mathrm{m}}^{2}\right]$ |

In-run horizontal constraint | ${c}_{\mathrm{H},\phantom{\rule{4.pt}{0ex}}\mathrm{IR}}\phantom{\rule{3.33333pt}{0ex}}$ | ${0.1}^{2}\phantom{\rule{3.33333pt}{0ex}}\left[{\mathrm{m}}^{2}\right]$ |

Land area vertical constraint | ${c}_{\mathrm{V},\phantom{\rule{4.pt}{0ex}}\mathrm{LA}}\phantom{\rule{3.33333pt}{0ex}}$ | ${0.1}^{2}\phantom{\rule{3.33333pt}{0ex}}\left[{\mathrm{m}}^{2}\right]$ |

Variables | $\overline{\mathbf{\Delta}{\mathit{x}}_{\mathit{N}}}$ | $\overline{\mathbf{\Delta}{\mathit{y}}_{\mathit{N}}}$ | $\overline{\mathbf{\Delta}{\mathit{z}}_{\mathit{N}}}$ | $\overline{\mathbf{\Delta}{\mathit{v}}_{\mathit{O},\mathit{x}}}$ | $\overline{\mathbf{\Delta}{\mathit{v}}_{\mathit{O},\mathit{y}}}$ | $\overline{\mathbf{\Delta}{\mathit{v}}_{\mathit{O},\mathit{z}}}$ | $\overline{\mathbf{\Delta}\mathit{\varphi}}$ | $\overline{\mathbf{\Delta}\mathit{\theta}}$ | $\overline{\mathbf{\Delta}\mathit{\psi}}$ |
---|---|---|---|---|---|---|---|---|---|

RMS Error: | 0.159 m | 0.123 m | 0.109 m | 0.021 m/s | 0.026 m/s | 0.023 m/s | 0.584 ${}^{\circ}$ | 0.319 ${}^{\circ}$ | 0.759 ${}^{\circ}$ |

**Table 7.**Comparison between jump length obtained from the trajectory reconstruction results (proposed method) and video recordings (reference).

Jump Length: | From Trajectory Reconstruction ${\mathit{L}}_{\mathit{TR}}$ | From Video Recording ${\mathit{L}}_{\mathit{VR}}$ | Error $\mathbf{\Delta}\mathit{L}$ |
---|---|---|---|

Jump No. 1 | 91.6 m | 90.0 m | 1.6 m |

Jump No. 2 | 85.8 m | 85.0 m | 0.8 m |

Jump No. 3 | 98.6 m | 97.5 m | 1.1 m |

Jump No. 4 | 69.5 m | 70.0 m | -0.5 m |

Jump No. 5 | 86.0 m | 85.5 m | 0.5 m |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fang, X.; Grüter, B.; Piprek, P.; Bessone, V.; Petrat, J.; Holzapfel, F.
Ski Jumping Trajectory Reconstruction Using Wearable Sensors via Extended Rauch-Tung-Striebel Smoother with State Constraints. *Sensors* **2020**, *20*, 1995.
https://doi.org/10.3390/s20071995

**AMA Style**

Fang X, Grüter B, Piprek P, Bessone V, Petrat J, Holzapfel F.
Ski Jumping Trajectory Reconstruction Using Wearable Sensors via Extended Rauch-Tung-Striebel Smoother with State Constraints. *Sensors*. 2020; 20(7):1995.
https://doi.org/10.3390/s20071995

**Chicago/Turabian Style**

Fang, Xiang, Benedikt Grüter, Patrick Piprek, Veronica Bessone, Johannes Petrat, and Florian Holzapfel.
2020. "Ski Jumping Trajectory Reconstruction Using Wearable Sensors via Extended Rauch-Tung-Striebel Smoother with State Constraints" *Sensors* 20, no. 7: 1995.
https://doi.org/10.3390/s20071995