Fast and Fault-Tolerant Passive Hyperbolic Localization Using Sensor Consensus
Abstract
:1. Introduction
- A new, fast evaluation method is introduced.
- The main contribution is the theoretical proof that ensures that the proposed algorithm always finds the global maximum of the consensus function over a finite grid.
- Finally, a comprehensive performance analysis is provided using simulations and real measurements.
2. Related Work
2.1. The TDOA Localization Problem
2.2. Consensus-Function-Based Localization
2.3. Calculation of the Consensus-Function-Based Location Estimate
3. Fast Consensus-Based Localization
3.1. Fast Search on a Finite Grid
Algorithm 1 Branch and Bound algorithm to find the maximum of the consensus function |
Input:
Branch:
|
3.2. Upper Bound of the Consensus Function
3.3. Global Convergence
4. Performance Evaluation
4.1. Simulations
- For each target position, we calculated the exact distances between the target position and the sensors.
- We calculated the exact times of arrivals as .
- We added measurement noise with normal distribution to and added additional measurement noise with normal distribution to to emulate the faulty (outlier) sensors. The number of outliers was For each target position, 100 independent measurements were created.
- Using measurements and the sensor positions , the estimated target positions were calculated by LS, COM-W, and CF.
- The tests were conducted using Matlab version R2021b on a computer with i5-8265 CPU with clock frequency of 1.6 GHz, and 24 GB of RAM.
- The LS algorithm was started from a random position within 1 m of the true position.
- The CF method was implemented in Matlab according to Algorithm 1. Apart from Matlab’s built-in vector operations, no acceleration methods (e.g., multithreading) were used.
- The final resolution of the CF method was .
4.2. Measurements
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
List of Symbols
Notations | Definition |
propagation speed | |
dimension of the search space | |
coarse grid | |
fine grid | |
source position | |
emission time | |
window length | |
location estimate | |
maximum sensor location error | |
maximum time measurement error | |
window function | |
set of points where the consensus function takes its maximum |
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ID | x (m) | y (m) | z2D (m) | z3D (m) |
---|---|---|---|---|
1 | 20 | 50 | 0 | 1.5 |
2 | 30 | 30 | 0 | 3.1 |
3 | 40 | 10 | 0 | 0.5 |
4 | 30 | 60 | 0 | 0.2 |
5 | 40 | 40 | 0 | 1.1 |
6 | 50 | 20 | 0 | 0.1 |
RMSE (m) | RMSE (m) | RMSE (m) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
ID | LS | COMW | CF | LS | COMW | CF | LS | COMW | CF | |||
1 | 0.071 | 0.103 | 0.086 | 0.077 | 0.061 | 0.090 | 0.076 | 0.068 | 0.048 | 0.084 | 0.063 | 0.060 |
2 | 0.068 | 0.092 | 0.091 | 0.077 | 0.051 | 0.073 | 0.057 | 0.057 | 0.044 | 0.079 | 0.052 | 0.059 |
3 | 0.082 | 0.092 | 0.099 | 0.090 | 0.067 | 0.076 | 0.089 | 0.072 | 0.054 | 0.078 | 0.067 | 0.063 |
4 | 0.065 | 0.114 | 0.098 | 0.067 | 0.053 | 0.083 | 0.077 | 0.060 | 0.047 | 0.082 | 0.064 | 0.059 |
5 | 0.064 | 0.083 | 0.080 | 0.065 | 0.048 | 0.086 | 0.061 | 0.064 | 0.042 | 0.078 | 0.054 | 0.062 |
6 | 0.068 | 0.072 | 0.071 | 0.073 | 0.057 | 0.071 | 0.088 | 0.064 | 0.046 | 0.074 | 0.057 | 0.059 |
mean | 0.070 | 0.092 | 0.088 | 0.075 | 0.056 | 0.080 | 0.075 | 0.064 | 0.047 | 0.079 | 0.060 | 0.060 |
RMSE (m) | RMSE (m) | RMSE (m) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
ID | LS | COMW | CF | LS | COMW | CF | LS | COMW | CF | |||
1 | 0.27 | 0.32 | 0.30 | 0.27 | 0.20 | 0.25 | 0.21 | 0.21 | 0.18 | 0.26 | 0.22 | 0.22 |
2 | 0.13 | 0.15 | 0.14 | 0.14 | 0.11 | 0.13 | 0.11 | 0.11 | 0.10 | 0.11 | 0.11 | 0.10 |
3 | 0.21 | 0.24 | 0.33 | 0.25 | 0.14 | 0.15 | 0.19 | 0.14 | 0.12 | 0.15 | 0.16 | 0.12 |
4 | 0.30 | 0.43 | 0.30 | 0.45 | 0.23 | 0.45 | 0.23 | 0.41 | 0.21 | 0.46 | 0.27 | 0.39 |
5 | 0.16 | 0.19 | 0.17 | 0.16 | 0.10 | 0.14 | 0.12 | 0.12 | 0.09 | 0.13 | 0.12 | 0.11 |
6 | 0.20 | 0.21 | 0.22 | 0.32 | 0.12 | 0.14 | 0.15 | 0.17 | 0.10 | 0.13 | 0.13 | 0.15 |
mean | 0.21 | 0.26 | 0.24 | 0.27 | 0.15 | 0.21 | 0.17 | 0.19 | 0.13 | 0.21 | 0.17 | 0.18 |
RMSE (m) | RMSE (m) | RMSE (m) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
ID | LS | COMW | CF | LS | COMW | CF | LS | COMW | CF | |||
1 | 0.071 | 3.5 | 3.5 | 0.13 | 0.061 | 2.2 | 3.6 | 0.20 | 0.048 | 1.7 | 3.4 | 0.20 |
2 | 0.068 | 3.4 | 1.9 | 0.14 | 0.051 | 1.9 | 1.2 | 0.12 | 0.044 | 3.2 | 1.1 | 0.13 |
3 | 0.082 | 3.5 | 3.8 | 0.16 | 0.067 | 2.8 | 4.8 | 0.22 | 0.054 | 4.4 | 3.1 | 0.15 |
4 | 0.065 | 2.2 | 3.8 | 0.22 | 0.053 | 1.6 | 4.0 | 0.19 | 0.047 | 1.9 | 4.8 | 0.37 |
5 | 0.064 | 4.4 | 1.5 | 0.13 | 0.048 | 2.3 | 0.8 | 0.12 | 0.042 | 2.3 | 0.7 | 0.14 |
6 | 0.068 | 2.7 | 2.9 | 0.38 | 0.057 | 2.9 | 2.1 | 0.47 | 0.046 | 1.9 | 1.5 | 0.15 |
mean | 0.070 | 3.3 | 2.9 | 0.19 | 0.056 | 2.3 | 2.8 | 0.22 | 0.046 | 2.6 | 2.4 | 0.19 |
RMSE (m) | RMSE (m) | RMSE (m) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
ID | LS | COMW | CF | LS | COMW | CF | LS | COMW | CF | |||
1 | 0.27 | 16 | 9.1 | 0.40 | 0.20 | 4.8 | 2.7 | 0.37 | 0.18 | 5.2 | 2.6 | 0.27 |
2 | 0.13 | 6.6 | 3.3 | 0.29 | 0.11 | 2.9 | 1.2 | 0.27 | 0.10 | 18.0 | 2.3 | 0.22 |
3 | 0.21 | 7.2 | 6.5 | 0.38 | 0.14 | 3.9 | 4.5 | 0.24 | 0.12 | 2.2 | 3.8 | 0.25 |
4 | 0.30 | 5.9 | 7.7 | 0.88 | 0.23 | 6.5 | 3.5 | 0.45 | 0.21 | 10.1 | 5.7 | 0.81 |
5 | 0.16 | 5.7 | 3.7 | 0.19 | 0.10 | 5.2 | 2.3 | 0.44 | 0.09 | 4.3 | 0.95 | 0.17 |
6 | 0.20 | 9.8 | 5.4 | 0.84 | 0.12 | 2.6 | 3.1 | 0.25 | 0.10 | 4.7 | 1.7 | 0.37 |
mean | 0.21 | 8.5 | 6.0 | 0.50 | 0.15 | 4.3 | 2.9 | 0.34 | 0.13 | 7.4 | 2.8 | 0.35 |
Mean Execution Time, 2D (ms) | Mean Execution Time, 3D (ms) | |||||
---|---|---|---|---|---|---|
LS | COM-W | CF | LS | COM-W | CF | |
15 | 3.9 | 34.8 | 18.8 | 5.0 | 95.6 | 89.0 |
25 | 3.7 | 164.8 | 19.8 | 4.7 | 814.7 | 85.8 |
35 | 3.6 | 439.5 | 20.8 | 4.7 | 3225.7 | 83.1 |
mean | 3.7 | 213.1 | 19.8 | 4.8 | 1378.7 | 86.0 |
RMSE-xy (m) | RMSE-xyz (m) | Execution Times (ms) | |
---|---|---|---|
LS | 0.24 | 0.62 | 4.2 |
COM-W | 0.19 | 0.47 | 4.3 |
CF | 0.15 | 0.41 | 11.1 |
ID | x (m) | y (m) | z (m) |
---|---|---|---|
1 | 36.34 | 67.55 | 3.55 |
2 | 30.30 | 66.42 | −0.30 |
3,4 | 31.94 | 57.34 | −0.30 |
5,6,7,8 | 28.93 | 45.45 | 7.30 |
9 | 25.85 | 40.44 | −0.20 |
10 | 33.37 | 48.24 | −0.25 |
ID | Position Estimation Error (m) | Run-Time (ms) | N | Cw | ||||
---|---|---|---|---|---|---|---|---|
LS | COM-W | CF | LS | COM-W | CF | |||
1 | 14.22 | 10.04 | 0.80 | 105 | 1404 | 109 | 29 | 25 |
2 | 2.34 | 2.74 | 1.28 | 24 | 11 | 350 | 10 | 9 |
3 | 1.50 | 2.00 | 1.18 | 6 | 69 | 52 | 26 | 23 |
4 | 4.65 | 3.91 | 0.89 | 6 | 2261 | 92 | 34 | 25 |
5 | 12.41 | 8.79 | 0.34 | 8 | 2789 | 60 | 38 | 23 |
6 | 3.29 | 6.13 | 0.14 | 6 | 292 | 141 | 21 | 19 |
7 | 13.00 | 12.51 | 0.50 | 4 | 215 | 490 | 20 | 16 |
8 | 3.54 | 3.37 | 0.77 | 5 | 67 | 125 | 15 | 14 |
9 | 7.82 | 11.37 | 1.06 | 6 | 1159 | 34 | 30 | 21 |
10 | 4.82 | 5.32 | 1.94 | 6 | 1224 | 79 | 30 | 22 |
average | 6.76 | 6.62 | 0.89 | 18 | 949 | 153 |
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Simon, G.; Zachár, G. Fast and Fault-Tolerant Passive Hyperbolic Localization Using Sensor Consensus. Sensors 2024, 24, 2891. https://doi.org/10.3390/s24092891
Simon G, Zachár G. Fast and Fault-Tolerant Passive Hyperbolic Localization Using Sensor Consensus. Sensors. 2024; 24(9):2891. https://doi.org/10.3390/s24092891
Chicago/Turabian StyleSimon, Gyula, and Gergely Zachár. 2024. "Fast and Fault-Tolerant Passive Hyperbolic Localization Using Sensor Consensus" Sensors 24, no. 9: 2891. https://doi.org/10.3390/s24092891
APA StyleSimon, G., & Zachár, G. (2024). Fast and Fault-Tolerant Passive Hyperbolic Localization Using Sensor Consensus. Sensors, 24(9), 2891. https://doi.org/10.3390/s24092891