Target Localization and Sensor Movement Trajectory Planning with Bearing-Only Measurements in Three Dimensional Space
Abstract
:1. Introduction
2. Bearing-Only Localization in 3D Space
3. Bias Compensation Estimator for Localization in 3D Space
3.1. Bias Compensation Localization in 2D Space
3.2. Bias Compensation Method in Z-Axis
3.3. BC-WIV Estimator in 3D Space
4. Sensor Trajectory Design
4.1. Constraints
4.2. Sensor Trajectory Planning Strategies
4.3. Optimal Solution Region
- When the sensor motion area and the target safety area are separated, define the boundary part of the sensor motion area inside as surface 1 in Figure 3a. The feasible region is the sensor motion area;
- When the sensor motion area intersects with the target safety area, the feasible region is the sensor motion area where the part inside the target safety area is excluded, and the surface is where part of the target safety boundary is inside the sensor motion area. According to the position of , it can be divided into three sub-cases: (i) when is outside the sensor motion area and is not contained by the target safety area, define the part of the sensor motion boundary inside with the section inside the target safety area replaced by the section of the target safety boundary inside the sensor movement area, such as surface 2 in Figure 3b. E and F (E and F are symmetrically distributed) are the points where the two spheres intersect; (ii) when the is outside the sensor motion area and if is contained by the target safety area, define the part of the target safety boundary inside the sensor motion area as surface 3 in Figure 3c; (iii) when the is inside the sensor motion area, define the part of the target safety boundary inside the sensor motion area as surface 4 in Figure 3d;
- when the sensor motion area contains the target safety area, the feasible region is the sensor motion area where the interior section of the target safety area is excluded. The target safety boundary is defined as surface 5 in Figure 3e.
4.4. Analytical Derivation for the Global Maximizer
4.5. Single Sensor Trajectory Design Procedure
- Generate through the maximization of (69) or (72). The maximizer of (69) or (72) can be produced in an analytical way described in Section 4.4;
- Gather the measurement information , , and from the sensor at ;
- Use localization technique to generate ;
- .
5. Simulation
5.1. Example 1
5.2. Example 2
6. Conclusions and Future Works
7. Patents
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
FIM | Fisher information matrix |
AOA | Angle of arrival |
2D | Two dimensional |
3D | Three dimensional |
MSE | Mean square error |
IV | Instrumental variable |
CRLB | Cramér-Rao lower bound |
UAV | Unmanned aerial vehicle |
BC | Bias compensation |
BC-WIV | Bias compensation weighted instrumental variable |
PLE | Pseudo-linear estimator |
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Observation Point | |||
---|---|---|---|
k = 10 | 98/100 | 96/100 | 3/100 |
k = 15 | 99/100 | 96/100 | 2/100 |
Trajectory 1 | Trajectory 2 | Trajectory 3 | Trajectory 4 | |
---|---|---|---|---|
mean of | 91.16 | 90.58 | 89.62 | 90.49 |
mean of | 93.42 | 92.86 | 88.85 | 90.99 |
Trajectory 1 | Trajectory 2 | Trajectory 3 | Trajectory 4 | |
---|---|---|---|---|
runtime(s) | 0.5558 | 0.5240 | 0.6996 | 0.5899 |
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Zou, Y.; Gao, B.; Tang, X.; Yu, L. Target Localization and Sensor Movement Trajectory Planning with Bearing-Only Measurements in Three Dimensional Space. Appl. Sci. 2022, 12, 6739. https://doi.org/10.3390/app12136739
Zou Y, Gao B, Tang X, Yu L. Target Localization and Sensor Movement Trajectory Planning with Bearing-Only Measurements in Three Dimensional Space. Applied Sciences. 2022; 12(13):6739. https://doi.org/10.3390/app12136739
Chicago/Turabian StyleZou, Yiqun, Bilu Gao, Xiafei Tang, and Lingli Yu. 2022. "Target Localization and Sensor Movement Trajectory Planning with Bearing-Only Measurements in Three Dimensional Space" Applied Sciences 12, no. 13: 6739. https://doi.org/10.3390/app12136739
APA StyleZou, Y., Gao, B., Tang, X., & Yu, L. (2022). Target Localization and Sensor Movement Trajectory Planning with Bearing-Only Measurements in Three Dimensional Space. Applied Sciences, 12(13), 6739. https://doi.org/10.3390/app12136739