# Tracking the 6-DOF Flight Trajectory of Windborne Debris Using Stereophotogrammetry

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### Problem Formulation and Notation

${Z}_{1}$, ${Z}_{2}$ | –points at the lens positions of two cameras |

$P$ | –position of a point debris |

${P}_{1}$, ${P}_{2}$ | –projected locations of the debris on the grid wall seen from two cameras (i.e., a stereopair), respectively |

$d$ | –distance between cameras |

$l$ | –distance between camera lens and the wall along the y-direction; the wall is parallel to the z‒x plane |

${d}_{1}$, ${d}_{2}$ | –distances of points on the wall along the x-axis from the y-axis |

${h}_{1}$, ${h}_{2}$ | –heights of points on the wall along the z-axis from the x‒y plane |

${\beta}_{1}$, ${\beta}_{2}$ | –angles that ${Z}_{1}{P}_{1}$, ${Z}_{2}{P}_{2}$ rays make with the x‒y plane |

${\theta}_{1}$, ${\theta}_{2}$ | –angles between the x-axis and the projections of ${Z}_{1}{P}_{1}$, ${Z}_{2}{P}_{2}$ rays on the x‒y plane |

## 3. Debris Position and Spatial Orientation (6-DOF) in 3D Space

#### 3.1. The 2D Stereopairs’ Positions

#### 3.2. The Relationship between the 3D Spatial Position of Debris and the 2D Stereopairs’ Positions

_{.}The rationale for the scaling of an object at ${P}_{1}$ by a scale factor of $\frac{{s}_{1}}{{t}_{1}}$ along the line ${Z}_{1}{P}_{1}$ is as follows:

#### 3.3. Explicit Expressions of Debris Position in 3D Space

#### 3.4. Cartesian Position and Orientation for Thin Plate Debris

#### 3.5. Three-Degree-of-Freedom (3-DOF) Frames in Universal Coordinate System

#### 3.6. Differential Operators for 6-DOF Motion

**,**which yields the rotation about the x, y, and z axes. This is consistent with the differential operator derived from the matrix:

## 4. 6-DOF Motion Consideration

#### 4.1. Motivation

#### 4.2. The Relationship between Differential Transformations in Universal and Local Coordinate Systems

#### 4.2.1. General Transformation

#### 4.2.2. Rotational Differential Transform with Respect to Universal Coordinate System

#### 4.2.3. Differential Transform in Universal Coordinate System

**Definition**

**1.**

#### 4.2.4. Differential Transform in Local Coordinate System

- Show that the rotation differential operator $\Delta {}^{\prime}={R}^{T}\Delta R$ with respect to the local frame system, such that$$\Delta {}^{\prime}=\left[\begin{array}{c}\omega \cdot u\\ \omega \cdot v\\ \omega \cdot w\end{array}\right]\times I.$$
**Proof.**$$\Delta {}^{\prime}={R}^{T}\Delta R={R}^{T}\omega \times IR={R}^{T}\omega \times R={R}^{T}\left[\omega \times u,\omega \times v,\omega \times w\right]=\left[{R}^{T}\omega \times u,{R}^{T}\omega \times v,{R}^{T}\omega \times w\right]$$$$=\left[\left[\begin{array}{c}\omega \times u\cdot u\\ \omega \times u\cdot v\\ \omega \times u\cdot w\end{array}\right],\left[\begin{array}{c}\omega \times v\cdot u\\ \omega \times v\cdot v\\ \omega \times v\cdot w\end{array}\right],\left[\begin{array}{c}\omega \times w\cdot u\\ \omega \times w\cdot v\\ \omega \times w\cdot w\end{array}\right]\right]=\left[\begin{array}{ccc}0& -\omega \cdot w& \omega \cdot v\\ \omega \cdot w& 0& -\omega \cdot u\\ -\omega \cdot v& \omega \cdot u& 0\end{array}\right]=\left[\begin{array}{c}\omega \cdot u\\ \omega \cdot v\\ \omega \cdot w\end{array}\right]\times I.$$ - Show that the linear differential $dc{}^{\prime}={R}^{T}dc$ in the local frame simplifies to$$dc{}^{\prime}=\left[\begin{array}{c}dc\cdot u\\ dc\cdot v\\ dc\cdot w\end{array}\right].$$
**Proof.**$$dc{}^{\prime}={R}^{T}dc=\left[\begin{array}{c}{u}^{T}\\ {v}^{T}\\ {w}^{T}\end{array}\right],dc=\left[\begin{array}{c}{u}^{T}dc\\ {v}^{T}dc\\ {w}^{T}dc\end{array}\right]=\left[\begin{array}{c}u\cdot dc\\ v\cdot dc\\ w\cdot dc\end{array}\right]=\left[\begin{array}{c}dc\cdot u\\ dc\cdot v\\ dc\cdot w\end{array}\right].$$ - Differential operator with respect to local frame coordinate system.
**Theorem****1.**The frame differential transformation matrix$T{}^{\prime}$in the local coordinate system is$$T{}^{\prime}=\left[\begin{array}{cc}\Delta {}^{\prime}& dc{}^{\prime}\\ 0& 0\end{array}\right],$$**Proof.**From $T{}^{\prime}={F}^{-1}TF$ (Section 4.2.1) and $TF=\left[\begin{array}{cc}\Delta R& dc\\ 0& 0\end{array}\right]$ (Equation (34)), we get$$T{}^{\prime}=\left[\begin{array}{cc}{R}^{T}& -{R}^{T}c\\ 0& 1\end{array}\right]\left[\begin{array}{cc}\Delta R& dc\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}{R}^{T}\Delta R& {R}^{T}dc\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}\Delta {}^{\prime}& dc{}^{\prime}\\ 0& 0\end{array}\right]\phantom{\rule{0ex}{0ex}}(\mathrm{recall}\mathrm{that}\Delta {}^{\prime}={R}^{T}\Delta R\mathrm{and}dc{}^{\prime}={R}^{T}dc).$$Hence, the theorem is proved. □

## 5. 6-DOF Motion Trajectory with Velocity

#### 5.1. Velocity

#### 5.2. Experimenting with 6DOF Position and Orientation and 6DOF Motion Methods

#### 5.3. Implementation Procedure for Debris Tracking with Both Displacement and Velocity Time Histories Determined

**,**the centroids of the consecutive coordinate frames. The rotational velocity is more complex, as seen earlier. In the universal coordinate system:

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The setup of parameters for the trajectory tracking problem (adapted from [19]).

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**MDPI and ACS Style**

Sabharwal, C.L.; Guo, Y.
Tracking the 6-DOF Flight Trajectory of Windborne Debris Using Stereophotogrammetry. *Infrastructures* **2019**, *4*, 66.
https://doi.org/10.3390/infrastructures4040066

**AMA Style**

Sabharwal CL, Guo Y.
Tracking the 6-DOF Flight Trajectory of Windborne Debris Using Stereophotogrammetry. *Infrastructures*. 2019; 4(4):66.
https://doi.org/10.3390/infrastructures4040066

**Chicago/Turabian Style**

Sabharwal, Chaman Lal, and Yanlin Guo.
2019. "Tracking the 6-DOF Flight Trajectory of Windborne Debris Using Stereophotogrammetry" *Infrastructures* 4, no. 4: 66.
https://doi.org/10.3390/infrastructures4040066