As drone technology advances, Unmanned Aerial Vehicles (UAVs) have become critical military assets requiring effective countermeasures. Interception missions demand precise and smooth trajectory predictions to minimize energy consumption and prevent command oscillations in autonomous interceptors. Traditional Extended Kalman Filter (EKF) approaches exhibit significant fluctuations when tracking highly maneuverable UAVs, compromising interception success rates. We address this limitation through a novel EKF-RS algorithm that integrates dual smoothing techniques for superior trajectory prediction.
Theoretical Foundations
Our approach combines three core methodologies:
Extended Kalman Filter
EKF handles nonlinear systems through first-order Taylor approximations. For drone technology applications, we model UAV dynamics using Constant Velocity (CV) and Constant Acceleration (CA) models:
$$ \mathbf{x}_t = \mathbf{F}\mathbf{x}_{t-1} + \mathbf{w}_{t-1} $$
$$ \mathbf{z}_t = \mathbf{H}\mathbf{x}_t + \mathbf{v}_t $$
Where state vector $\mathbf{x}_t$ contains position, velocity, and acceleration components. CV and CA transition matrices are:
$$ \mathbf{F}_{\text{CV}} = \begin{bmatrix}
1 & 0 & T & 0 & 0 & 0 \\
0 & 1 & 0 & T & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1
\end{bmatrix} \quad
\mathbf{F}_{\text{CA}} = \begin{bmatrix}
1 & 0 & T & 0 & \frac{T^2}{2} & 0 \\
0 & 1 & 0 & T & 0 & \frac{T^2}{2} \\
0 & 0 & 1 & 0 & T & 0 \\
0 & 0 & 0 & 1 & 0 & T \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1
\end{bmatrix} $$
RTS Smoothing
Rauch-Tung-Striebel smoothing refines EKF estimates through backward passes:
$$ \mathbf{G}_{t-1} = \mathbf{P}_{t-1}\mathbf{F}^T(\mathbf{P}_t^-)^{-1} $$
$$ \hat{\mathbf{x}}_{t-1}^s = \hat{\mathbf{x}}_{t-1} + \mathbf{G}_{t-1}(\hat{\mathbf{x}}_t^s – \hat{\mathbf{x}}_t^-) $$
$$ \mathbf{P}_{t-1}^s = \mathbf{P}_{t-1} + \mathbf{G}_{t-1}(\mathbf{P}_t^s – \mathbf{P}_t^-)\mathbf{G}_{t-1}^T $$
Savitzky-Golay Filtering
SG filtering preserves kinematic trends using local polynomial fitting:
$$ \hat{y}_i = \sum_{j=-m}^{m} c_j y_{i+j} $$
Where $2m+1$ defines the smoothing window and $c_j$ are convolution coefficients.
Algorithm Design
Our EKF-RS framework addresses EKF’s volatility through dual-smoothing fusion within sliding windows:
| Module | Function | Parameters |
|---|---|---|
| Sliding Window | Maintains L-step history | L=5 (default) |
| EKF Core | State estimation | Q=0.1, R=0.03 |
| RTS Smoother | Trajectory smoothing | Fixed-interval |
| SG Filter | Trend preservation | 2nd-order polynomial |
| Fusion Engine | Optimal weighting | Adaptive λ |
The fusion process minimizes this objective function:
$$ J = \frac{1}{N} \sum_{k=1}^{N} \left[ \lambda \| \hat{\mathbf{x}}_k^f – \hat{\mathbf{x}}_k^{\text{SG}} \|^2 + (1-\lambda) \| \hat{\mathbf{x}}_k^f – \hat{\mathbf{x}}_{k-1}^f \|^2 \right] $$
Producing optimal smoothed estimates:
$$ \hat{\mathbf{x}}^f = \mathbf{A}\hat{\mathbf{x}}^s + \mathbf{B}\hat{\mathbf{x}}^{\text{SG}} \quad \text{where} \quad \mathbf{A} + \mathbf{B} = \mathbf{I} $$
Simulation Analysis
We validated our approach using simulated UAV trajectories under CV and CA models:
| Algorithm | CV RMSE (m) | CA RMSE (m) | Smoothness Index |
|---|---|---|---|
| EKF | 4.32 ± 0.87 | 5.91 ± 1.24 | 0.38 ± 0.12 |
| EKF-SG | 3.15 ± 0.62 | 4.83 ± 1.05 | 0.21 ± 0.08 |
| EKF-RS | 1.89 ± 0.41 | 2.57 ± 0.73 | 0.09 ± 0.03 |
Critical observations from trajectory analysis:
- Smoothness Enhancement: EKF-RS reduced trajectory fluctuations by 76.3% versus EKF
- Error Reduction: 56.2% lower RMSE in CA scenarios compared to EKF-SG
- Trend Preservation: Maintained kinematic consistency during acceleration phases
Interception Implications
This drone technology advancement significantly impacts Unmanned Aerial Vehicle interception:
$$ \Delta E_{\text{intercept}} \propto \frac{1}{\text{Smoothness Index}} \times \text{RMSE} $$
Implementation benefits include:
- 35-40% reduction in interceptor energy consumption
- Command oscillation amplitude decreased by 68%
- Real-time operation at 20Hz update rates
Conclusion
Our EKF-RS algorithm demonstrates significant improvements in UAV trajectory prediction by achieving 56.2% higher accuracy and 76.3% smoother outputs than conventional EKF. The dual-smoothing approach effectively balances precision and smoothness through adaptive fusion of RTS and SG techniques. Future work will integrate neural networks for dynamic weight optimization across diverse drone technology scenarios. This advancement in Unmanned Aerial Vehicle tracking directly enhances interception system effectiveness while reducing operational energy requirements.