A KalmanBased Compensation Strategy for Platoons Subject to Data Loss: Numerical and Empirical Study
Abstract
:1. Introduction
 We propose a new strategy to estimate and replace the predecessortransmitted data in case of loss. This strategy includes a stage where a linear extrapolation is carried out to estimate data from two vehicles ahead, as well as a stage based on the Kalman filter with intermittent observations [19,20], to estimate the predecessor vehicle’s state.
 The performance of the proposed strategy is analyzed numerically for two cases for the data loss probability; a case with a constant data loss probability and another case where the transmission success is dependent on the intervehicle distance. As part of this analysis, we also compare the performance of our proposal with a simpler strategy based purely on linear extrapolation [9]. Our numerical results show that although both strategies can achieve string stability for the means and variances of both the tracking and estimation errors, the Kalman filteringbased approach produces better performance compared to the linear extrapolation strategy. Moreover, our proposed strategy is capable of achieving string stability for channels with higher dataloss probability values than those attainable using the extrapolation strategy.
 We also implement these strategies on the experimental platform PLTOON [31,32], which is a lowcost platform with scale vehicles around 20 cm long that can is suitable for platooning studies. We show that both strategies can achieve tracking errors with stringstable performance, and although the comparison of these strategies does not demonstrate improvements, mainly due to sensor noise levels, the performance of the platoon is slightly better in terms of the variance when using the Kalman filter strategy.
2. Platooning Problem Description
2.1. Platooning Setup with Ideal Communication
2.2. Platooning with Lossy Communication
2.2.1. Channel Model for Lossy Communications
Constant Probability
DistanceDependence Probability
2.2.2. Problems with Lossy Communication
2.3. Platoon String Stability
2.4. PLTOON Implementation Considerations
3. Compensation Strategies
3.1. Linear Extrapolation
3.2. Strategy Based on the Intermittent Kalman Filter
3.3. Strategies for Relative Distance Model
3.3.1. Linear Extrapolation
Algorithm 1 Data replacement using linear extrapolation (holding previous distance) 

3.3.2. Strategy Based on the Intermittent Kalman Filter
Algorithm 2 Data replacement using strategy based on intermittent Kalman filter 

4. Simulation Results
4.1. Constant Transmission Probability
4.2. DistanceDependent Transmission Probability
5. Experimental Results
5.1. Experiment 1
5.2. Experiment 2
5.3. Discussion about Noise Level
5.4. Discussion about Other Potential Issues in Real Scenarios
 In practice, heterogeneous platoons are expected to be more common than homogeneous ones. This implies that the models of the vehicles ${G}_{i}$ are generally different, which is beyond our framework setup. One way to deal with this is to design stabilizing controllers ${C}_{i}$ such that the closedloop system T is common to all vehicles in the platoon, if possible. Another approach is to extend our setup to heterogeneous platoons by considering a different model of the preceding vehicle in our derivations.
 Our approach requires estimating the previous vehicle information, which implies having an accurate model of such a vehicle. Model uncertainty may yield a poor estimation. To deal with this issue, collaborative systems identification algorithms can be included in order to reduce model uncertainty [36], as well as the inclusion of robust estimation techniques [37].
 In our setup, we assume linear models. In practice, general models are expected to be nonlinear; hence, our setup cannot be straightforwardly applied. One solution is to use feedbacklinearization techniques [38] or obtain a closedloop linear model. Another alternative is to extend our approach by considering an estimator for nonlinear systems such as the unscented Kalman filter (UKF) with intermittent observations [39].
 The intervehicle communication channels can suffer from different types of phenomena that can affect the performance of the platoon beyond data loss. These phenomena include random delays, fading, signaltonoise ratio limitations, and cyber attacks, among others. Our approach could be extended to include tools from networked control systems theory to reduce the effect of such communication issues on the control performance (see, e.g., [40,41,42,43,44]).
6. Conclusions and Future Work
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Vehicle  KalmanBased  Extrapolation 

1  0.5425  0.5654 
2  0.4367  0.4387 
3  0.5049  0.5721 
4  0.4244  0.5144 
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Villenas, F.I.; Vargas, F.J.; Peters, A.A. A KalmanBased Compensation Strategy for Platoons Subject to Data Loss: Numerical and Empirical Study. Mathematics 2023, 11, 1228. https://doi.org/10.3390/math11051228
Villenas FI, Vargas FJ, Peters AA. A KalmanBased Compensation Strategy for Platoons Subject to Data Loss: Numerical and Empirical Study. Mathematics. 2023; 11(5):1228. https://doi.org/10.3390/math11051228
Chicago/Turabian StyleVillenas, Felipe I., Francisco J. Vargas, and Andrés A. Peters. 2023. "A KalmanBased Compensation Strategy for Platoons Subject to Data Loss: Numerical and Empirical Study" Mathematics 11, no. 5: 1228. https://doi.org/10.3390/math11051228