# Intelligence-Aware Batch Processing for TMA with Bearings-Only Measurements

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## Abstract

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## 1. Introduction

- We develop a novel MLE approach to carry out batch target-tracking estimation based on noisy bearing-only measurements, which incorporates as inequality constraints additional information in terms of sets where the target’s trajectory is assumed to be contained and other sets which have empty intersection with the target’s trajectory;
- We characterize the CRLB associated to the constrained problem by considering a generalized set-valued Jacobian matrix of the constraints function and by resorting to nonsmooth theory;
- We provide a heuristic way to select the ownship’s trajectory based on the although coarse-grained available information regarding the target’s trajectory.

## 2. Materials and Methods

#### 2.1. Problem Statement

#### 2.2. Maximum Likelihood Estimation without Additional Information

#### 2.3. Maximum Likelihood Estimation with Additional Information

#### 2.4. Computational Approach to Solve MLE Problems

#### 2.5. CRLB of the Estimate in the Unconstrained Case

#### 2.6. CRLB for Constrained MLE

#### 2.7. Ownship Trajectory Selection Based on Artificial Potential Fields

#### Ownship Trajectory

## 3. Experimental Analysis

- No information: the ownship does not rely on the additional information and selects the trajectory in Equation (19) with $\gamma =0\begin{array}{c}\hfill \mathrm{rad}\end{array}$.
- Unconstrained, APF direction: the ownship does not rely on the additional information for the computations but selects the trajectory according to the proposed APF approach; in other words, it selects the trajectory in Equation (19) with $\gamma =0.749\begin{array}{c}\hfill \mathrm{rad}\end{array}$.
- Constrained, no APF direction: the ownship relies on the additional information for the computations but selects the trajectory in Equation (19) with $\gamma =0\begin{array}{c}\hfill \mathrm{rad}\end{array}$.
- Proposed Approach: the ownship follows the trajectory in Equation (19) with $\gamma =0.749\begin{array}{c}\hfill \mathrm{rad}\end{array}$ and, further to that, actively relies on the additional information during the computation of the MLE solution, of the experimental covariance matrix and CRLB.

^{®}i7 quad-core @ 2.27 GHz. For each execution of MIDACO-SOLVER, we set the number of evaluated solutions to ${10}^{6}$. All other MIDACO-SOLVER parameters were used by their default values, which especially means that a feasibility accuracy of $0.001$ was used for all individual constraints. In all cases, we compute the MLE solution with MIDACO-SOLVER, and we consider 100 MLE solutions for each operational scenario.

- a standard extended Kalman filter (EKF) (e.g., see [44] and references therein), where we approximate the nonlinear output function $h(\xb7)$ by its Jacobian matrix at each time instant;
- the pseudolinear Kalman filter (PL-KF) [45], where the nonlinear and noisy output $z\left(k\right)=\begin{array}{c}\hfill \mathrm{atan}2({y}_{t}\left(k\right)-{y}_{o}\left(k\right),{x}_{t}\left(k\right)-{x}_{o}\left(k\right))\end{array}+w\left(k\right)$ is approximated by$$\tilde{z}\left(k\right)=\left[\begin{array}{c}sin\left(z\right(k\left)\right)\\ -cos\left(z\right(k\left)\right)\end{array}\right]\underset{M}{\underbrace{\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\end{array}\right]}}{\widehat{\mathit{x}}}_{k|k}+\eta \left(k\right),$$$$\eta \left(k\right)\sim \mathcal{N}\left(\right)open="("\; close=")">0,{R}_{k}$$$${R}_{k}\approx {\parallel {\widehat{\mathit{d}}}_{k|k-1}\parallel}^{2}{\sigma}^{2}$$$${\widehat{\mathit{d}}}_{k|k-1}=M{\widehat{\mathit{x}}}_{k|k-1}-\left[\begin{array}{c}{x}_{o}\left(k\right)\\ {y}_{o}\left(k\right)\end{array}\right],$$${\widehat{\mathit{x}}}_{k|k-1}$ being the vector collecting the prediction of the target’s states at time k;
- a statistical linearization extended Kalman filter (SL-EKF) [46] where, instead of the Jacobian of the output function $h(\xb7)$, we approximate the nonlinear measurement function $y\left(k\right)=h(\mathit{x},k)+w\left(k\right)$ via the linear approximation ${H}^{*}\mathit{x}$, with$${H}^{*}=\underset{H\in {\mathbb{R}}^{2\times 6}}{argmin}{\parallel y\left(k\right)-H{\widehat{\mathit{x}}}_{k|k}\parallel}^{2}.$$
- the shifted Rayleigh filter (SRF) [47], where $z\left(k\right)$ is approximated by$$z\left(k\right)\approx \Pi \left(\right)open="("\; close=")">M\mathit{x}\left(k\right)+\mathit{u}\left(k\right)+\omega \left(k\right)$$$$\omega \left(k\right)\sim \mathcal{N}\left(\right)open="("\; close=")">{\mathbf{0}}_{2},{\sigma}^{2}E\left(\right)open="["\; close="]">{\parallel M\mathit{x}\left(k\right)+\mathit{u}\left(k\right)\parallel}^{2}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}z\left(1\right),\dots ,z\left(k\right).$$

- (1)
- a scenario where the average initial condition ${\widehat{x}}_{0|0}$ is drawn from a zero-mean Gaussian variable with standard deviation equal to $\mathbf{\psi}$, while the initial covariance ${\Sigma}_{0|0}$ is equal to the square of $\mathbf{\psi}$, i.e.,$${\widehat{x}}_{0|0}\sim \mathcal{N}\left(\right)open="("\; close=")">{\mathbf{0}}_{6},\begin{array}{c}\hfill \mathrm{diag}\end{array}{\left(\mathbf{\psi}\right)}^{2}$$
- (2)
- a scenario where the average initial condition is drawn from a Gaussian variable with a mean equal to $\mathbf{\psi}$ and standard deviation equal to $\mathbf{\psi}$, while the initial covariance is equal to the square of $\mathbf{\psi}$, i.e.,$${\widehat{x}}_{0|0}\sim \mathcal{N}\left(\right)open="("\; close=")">\mathbf{\psi},\begin{array}{c}\hfill \mathrm{diag}\end{array}{\left(\mathbf{\psi}\right)}^{2}$$
- (3)
- a scenario where the average initial condition is exactly $\mathbf{\psi}$, while the initial covariance is equal to the square of $\mathbf{\psi}$, i.e.,$${\widehat{x}}_{0|0}=\mathbf{\psi},\phantom{\rule{1.em}{0ex}}{\Sigma}_{0|0}=\begin{array}{c}\hfill \mathrm{diag}\end{array}{\left(\mathbf{\psi}\right)}^{2}.$$

## 4. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

APF | Artificial potential fields |

CRLB | Cramér–Rao lower bound |

DOAJ | Directory of open access journals |

EKF | Extended Kalman filter |

FIM | Fisher information matrix |

MAP | Maximum a posteriori |

MDPI | Multidisciplinary Digital Publishing Institute |

MIDACO-SOLVER | Mixed integer distributed ant colony optimization solver |

MLE | Maximum likelihood estimation |

PL-KF | Pseudo-linear Kalman filter |

SL-EKF | Statistical linearization extended Kalman filter |

SRF | Shifted Rayleigh filter |

TMA | Target motion analysis |

## Appendix A

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**Figure 1.**Example of the proposed approach for selecting the ownship’s trajectory. In this example, we consider two sets, ${\mathcal{X}}_{1}$ and ${\mathcal{X}}_{2}$, and one setm ${\mathcal{Y}}_{1}$, represented by the circles shown with a solid line and by a dotted line, respectively. The points ${\mathit{x}}_{1}$ and ${\mathit{x}}_{2}$ and the point ${\mathit{y}}_{1}$ (i.e., the centers of the circles) are shown with an x mark and with a cross, respectively. In this example, we choose ${\alpha}_{1}$ and ${\alpha}_{2}$ equal to the area of ${\mathcal{X}}_{1}$ and ${\mathcal{X}}_{2}$, respectively, while ${\beta}_{1}$ is the reciprocal of the area of ${\mathcal{Y}}_{1}$. The resulting direction for the ownship is shown with an arrow (the initial position for the ownship is given by the starting endpoint of the arrow).

**Figure 2.**Scenario considered in the experimental analysis. We assume additional information is available, in that the target’s trajectory is known to be confined in the intersection of the two solid circles and to lie outside the dotted circle. The red (shorter) arrow represents the target’s trajectory. The APF direction is shown with a black (longer) arrow.

**Figure 4.**Sampling of 100 trajectories (in cyan) based on the average solution found via the proposed approach and on the experimental covariance matrix.

**Figure 5.**Sampling of 100 trajectories (cyan) based on the average solution found via the proposed approach and on the CRLB covariance matrix.

**Figure 6.**Sampling of 100 trajectories (cyan) based on the best solution found via the unconstrained, APF direction approach and on the CRLB covariance matrix.

**Figure 7.**Zoomed version of a portion of Figure 6.

**Figure 8.**Sampling of 100 trajectories (cyan) based on the best solution found via for the constrained case, but without relying on the APF, and on the experimental covariance matrix.

**Figure 9.**Sampling of 100 trajectories (cyan) based on the best solution found via for the constrained case, but without relying on the APF, and on the CRLB covariance matrix.

**Figure 10.**Comparison of the proposed approach against EKF, PLKF, SRF and SL-EKF, considering a scenario where the methods compared with the proposed one are initialized with ${\widehat{x}}_{0|0}\sim \mathcal{N}\left(\right)open="("\; close=")">{\mathbf{0}}_{6},\begin{array}{c}\hfill \mathrm{diag}\end{array}{\left(\mathbf{\psi}\right)}^{2}$ and ${\Sigma}_{0|0}=\begin{array}{c}\hfill \mathrm{diag}\end{array}{\left(\mathbf{\psi}\right)}^{2}$.

**Figure 11.**Comparison of the proposed approach against EKF, PLKF, SRF and SL-EKF, considering a scenario where the methods compared with the proposed one are initialized with ${\widehat{x}}_{0|0}\sim \mathcal{N}\left(\right)open="("\; close=")">\mathbf{\psi},\begin{array}{c}\hfill \mathrm{diag}\end{array}{\left(\mathbf{\psi}\right)}^{2}$ and ${\Sigma}_{0|0}=\begin{array}{c}\hfill \mathrm{diag}\end{array}{\left(\mathbf{\psi}\right)}^{2}$.

**Figure 12.**Comparison of the proposed approach against EKF, PLKF, SRF and SL-EKF, considering a scenario where the methods compared with the proposed one are initialized with ${\widehat{x}}_{0|0}=\mathbf{\psi}$ and ${\Sigma}_{0|0}=\begin{array}{c}\hfill \mathrm{diag}\end{array}{\left(\mathbf{\psi}\right)}^{2}$.

**Table 1.**Parameters describing the sets ${\mathcal{X}}_{1}$,${\mathcal{X}}_{2}$ and ${\mathcal{Y}}_{1}$ considered in the experimental analysis.

Set | Centroid | Radius |
---|---|---|

${\mathcal{X}}_{1}$ | ${\mathit{x}}_{1}={[{10}^{4},5\times {10}^{4}]}^{T}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ | ${\rho}_{{\mathcal{X}}_{1}}=5\times {10}^{4}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ |

${\mathcal{X}}_{2}$ | ${\mathit{x}}_{2}={[2\times {10}^{4},6\times {10}^{4}]}^{T}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ | ${\rho}_{{\mathcal{X}}_{2}}=4.5\times {10}^{4}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ |

${\mathcal{Y}}_{1}$ | ${\mathit{y}}_{1}={[4\times {10}^{4},8\times {10}^{4}]}^{T}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ | ${\rho}_{{\mathcal{Y}}_{1}}=2.5\times {10}^{4}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ |

${\mathit{x}}_{\mathit{t}0}\phantom{\rule{0.166667em}{0ex}}\mathbf{m}$ | ${\mathit{y}}_{\mathit{t}0}\phantom{\rule{0.166667em}{0ex}}\mathbf{m}$ | ${\dot{\mathit{x}}}_{\mathit{t}0}\phantom{\rule{0.166667em}{0ex}}\mathbf{m}/\mathbf{s}$ | ${\dot{\mathit{y}}}_{\mathit{t}0}\phantom{\rule{0.166667em}{0ex}}\mathbf{m}/\mathbf{s}$ | ${\ddot{\mathit{x}}}_{\mathit{t}}\phantom{\rule{0.166667em}{0ex}}\mathbf{m}/\mathbf{s}$ | ${\ddot{\mathit{y}}}_{\mathit{t}}\phantom{\rule{0.166667em}{0ex}}\mathbf{m}/\mathbf{s}$ | rel. pos. err. | rel. vel. err. | abs. acc. err. | |
---|---|---|---|---|---|---|---|---|---|

Ground truth | $3.000\times {10}^{4}$ | $3.000\times {10}^{4}$ | $8.333$ | $7.778$ | $0.000$ | $0.000$ | - | - | - |

No information, average | $3.933\times {10}^{4}$ | $3.848\times {10}^{4}$ | $-5.048$ | $-4.123$ | $3.437\times {10}^{-3}$ | $2.303\times {10}^{-3}$ | $4.202\times {10}^{-1}$ | $2.218$ | $4.137\times {10}^{-3}$ |

Unconstrained, APF direction, average | $3.796\times {10}^{4}$ | $3.733\times {10}^{3}$ | $-4.714$ | $-4.143$ | $4.780\times {10}^{-3}$ | $3.958\times {10}^{-3}$ | $3.605\times {10}^{-1}$ | $2.191$ | $6.206\times {10}^{-3}$ |

Constrained, no APF direction, average | $3.204\times {10}^{4}$ | $3.193\times {10}^{4}$ | $7.161$ | $6.884$ | $2.543\times {10}^{-4}$ | $4.987\times {10}^{-5}$ | $9.384\times {10}^{-2}$ | $1.815\times {10}^{-1}$ | $2.591\times {10}^{-4}$ |

Proposed Approach, average | $3.040\times {10}^{4}$ | $3.035\times {10}^{4}$ | $7.912$ | $7.382$ | $2.652\times {10}^{-4}$ | $2.598\times {10}^{-4}$ | $1.767\times {10}^{-2}$ | $7.169\times {10}^{-2}$ | $3.713\times {10}^{-4}$ |

No information, best | $3.028\times {10}^{4}$ | $3.024\times {10}^{4}$ | $8.045$ | $7.507$ | $1.989\times {10}^{-4}$ | $1.969\times {10}^{-4}$ | $1.243\times {10}^{-2}$ | $4.902\times {10}^{-2}$ | $2.789\times {10}^{-4}$ |

Unconstrained, APF direction, best | $3.704\times {10}^{4}$ | $3.640\times {10}^{4}$ | $0.686$ | $0.618$ | $3.754\times {10}^{-3}$ | $3.560\times {10}^{-3}$ | $3.171\times {10}^{-1}$ | $1.299$ | $5.173\times {10}^{-3}$ |

Constrained, no APF direction, best | $2.811\times {10}^{4}$ | $2.806\times {10}^{4}$ | $10.081$ | $7.382$ | $-1.460\times {10}^{-3}$ | $-2.202\times {10}^{-3}$ | $9.023\times {10}^{-2}$ | $3.412\times {10}^{-1}$ | $2.642\times {10}^{-3}$ |

Proposed Approach, best | $3.026\times {10}^{4}$ | $3.023\times {10}^{4}$ | $8.070$ | $7.531$ | $1.835\times {10}^{-4}$ | $1.821\times {10}^{-4}$ | $1.179\times {10}^{-2}$ | $4.471\times {10}^{-2}$ | $2.586\times {10}^{-4}$ |

**Table 3.**Euclidean norm of the parameter estimation via MIDACO-SOLVER for the different operational scenarios.

Experimental Covariance ${\parallel \xb7\parallel}_{2}$ | Experimental Covariance ${\parallel \xb7\parallel}_{2}$ | CRLB ${\parallel \xb7\parallel}_{2}$ | |
---|---|---|---|

(Solutions with Errors ≤ 50th Percentile) | |||

No information | $3.851\times {10}^{8}$ | $3.743\times {10}^{6}$ | $7.713\times {10}^{5}$ |

Unconstrained, APF direction | $5.666\times {10}^{8}$ | $4.144\times {10}^{6}$ | $7.713\times {10}^{5}$ |

Constrained, no APF direction | $3.696\times {10}^{7}$ | $5.410\times {10}^{6}$ | $1.454\times {10}^{3}$ |

Proposed Approach | $4.784\times {10}^{5}$ | $6.600\times {10}^{4}$ | $1.873\times {10}^{3}$ |

**Table 4.**Average and standard deviation over 100 trials of the computational time (in seconds) for the computation of the MLE solution via MIDACO-SOLVER for the different operational scenarios.

Time (Average) $\mathbf{s}$ | Time (Standard Deviation) $\mathbf{s}$ | |
---|---|---|

No information | $58.790$ | $3.235$ |

Unconstrained, APF direction | $59.124$ | $3.049$ |

Constrained, no APF direction | $85.926$ | $21.231$ |

Proposed Approach | $86.230$ | $22.395$ |

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## Share and Cite

**MDPI and ACS Style**

Oliva, G.; Farina, A.; Setola, R.
Intelligence-Aware Batch Processing for TMA with Bearings-Only Measurements. *Sensors* **2021**, *21*, 7211.
https://doi.org/10.3390/s21217211

**AMA Style**

Oliva G, Farina A, Setola R.
Intelligence-Aware Batch Processing for TMA with Bearings-Only Measurements. *Sensors*. 2021; 21(21):7211.
https://doi.org/10.3390/s21217211

**Chicago/Turabian Style**

Oliva, Gabriele, Alfonso Farina, and Roberto Setola.
2021. "Intelligence-Aware Batch Processing for TMA with Bearings-Only Measurements" *Sensors* 21, no. 21: 7211.
https://doi.org/10.3390/s21217211