Tracking the 6-DOF Flight Trajectory of Windborne Debris Using Stereophotogrammetry
Abstract
:1. Introduction
2. Background
Problem Formulation and Notation
, | –points at the lens positions of two cameras |
–position of a point debris | |
, | –projected locations of the debris on the grid wall seen from two cameras (i.e., a stereopair), respectively |
–distance between cameras | |
–distance between camera lens and the wall along the y-direction; the wall is parallel to the z‒x plane | |
, | –distances of points on the wall along the x-axis from the y-axis |
, | –heights of points on the wall along the z-axis from the x‒y plane |
, | –angles that , rays make with the x‒y plane |
, | –angles between the x-axis and the projections of , 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
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
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
4.2.4. Differential Transform in Local Coordinate System
- Show that the rotation differential operator with respect to the local frame system, such thatProof.
- Show that the linear differential in the local frame simplifies toProof.
- Differential operator with respect to local frame coordinate system.Theorem 1.The frame differential transformation matrixin the local coordinate system isProof.From (Section 4.2.1) and (Equation (34)), we getHence, 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
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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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
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 StyleSabharwal, 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