*3.4. Update of Track State and Existence Probability*

Using the validated measurements selected by track *τ*, the posterior trajectory state of track *τ* and the posterior target existence probability are calculated.

Let *β<sup>τ</sup> <sup>k</sup>*,*<sup>i</sup>* denote the data association probability that is conditioned on the target existence event *χτ <sup>k</sup>* . The data association probability *<sup>β</sup><sup>τ</sup> <sup>k</sup>*,*<sup>i</sup>* is expressed for the event *<sup>χ</sup><sup>τ</sup> k*,*i* , which indicates that the *i*th validated measurement of track *τ* is a target measurement, and the event *χ<sup>τ</sup> <sup>k</sup>*,0, which indicates that all the validated measurements of track *τ* are regarded as clutter measurements.

$$\mathcal{B}\_{k,i}^{\tau} = P\left\{\chi\_{k,i}^{\tau}|\chi\_{k'}^{\tau}\mathbf{Z}^{k}\right\} = \frac{P\_{\rm D}P\_{\rm G}}{\Lambda\_{k}^{\tau}}\frac{p\_{k,i}^{\tau}}{\bar{\rho}\_{k,i}^{\tau}}\,\mathrm{}\tag{19}$$

$$\beta\_{k,0}^{\tau} = P\left\{\chi\_{k,0}^{\tau}|\chi\_{k'}^{\tau} \mathbf{Z}^{k}\right\} = \frac{1 - P\_D P\_G}{\Lambda\_k^{\tau}},\tag{20}$$

where Λ*<sup>τ</sup> <sup>k</sup>* is the measurement likelihood ratio of track *τ* such that:

$$
\Lambda\_k^\pi = 1 - P\_D P\_G + P\_D P\_G \sum\_{i=1}^{m\_k^\pi} \frac{p\_{k,i}^\pi}{\bar{\rho}\_{k,i}^\pi}. \tag{21}
$$

The posterior trajectory state of track *τ* is calculated by the total probability theorem [31] such as:

$$p(\mathbf{x}\_k^\tau | \boldsymbol{\chi}\_k^\tau, \mathbf{Z}^k) = \sum\_{i=0}^{m\_k^\tau} p(\mathbf{x}\_k^\tau | \boldsymbol{\chi}\_{k'}^\tau, \boldsymbol{\chi}\_{k,i'}^\tau \mathbf{Z}^k) P\left\{ \boldsymbol{\chi}\_{k,i}^\tau | \boldsymbol{\chi}\_{k'}^\tau, \mathbf{Z}^k \right\},\tag{22}$$

where *p*(**x***<sup>τ</sup> <sup>k</sup>* <sup>|</sup>*χ<sup>τ</sup> <sup>k</sup>* , *<sup>χ</sup><sup>τ</sup> k*,*i* , **Z***k*) is a single Gaussian distribution as a posterior probability density function for the target trajectory state conditioned on the facts that target *τ* exists and **z***k*,*<sup>i</sup>* is the target measurement.

$$p(\mathbf{x}\_{k}^{\tau}|\chi\_{k'}^{\tau},\chi\_{k,i'}^{\tau},\mathbf{Z}^{k}) = N(\mathbf{x}\_{k'}^{\tau};\mathbf{\hat{x}}\_{k|k,i'}^{\tau},\mathbf{P}\_{k|k,i}^{\tau})\_{\prime} \tag{23}$$

where the conditional mean **x**ˆ *<sup>τ</sup> <sup>k</sup>*|*k*,*<sup>i</sup>* and covariance **<sup>P</sup>***<sup>τ</sup> <sup>k</sup>*|*k*,*<sup>i</sup>* satisfy:

$$\mathbf{\hat{x}}\_{k|k,i}^{\tau} = \begin{cases} \mathbf{\hat{x}}\_{k|k-1}^{\tau} + \mathbf{K}\_{k|k-1}^{\tau} (\mathbf{z}\_{k,i} - \mathbf{H} \mathbf{\hat{x}}\_{k|k-1}^{\tau}) & i > 0 \\ \mathbf{\hat{x}}\_{k|k-1}^{\tau} & i = 0 \end{cases} \tag{24}$$

$$\mathbf{P}\_{k|k,i}^{\tau} = \begin{cases} (\mathbf{I}\_6 - \mathbf{K}\_{k|k-1}^{\tau}\mathbf{H})\mathbf{P}\_{k|k-1}^{\tau} & i > 0\\ \mathbf{P}\_{k|k-1}^{\tau} & i = 0. \end{cases} \tag{25}$$

where **<sup>I</sup>***<sup>n</sup>* denote an *<sup>n</sup>* <sup>×</sup> *<sup>n</sup>* identity matrix, and the Kalman gain **<sup>K</sup>***<sup>τ</sup> <sup>k</sup>*|*k*−<sup>1</sup> is expressed by:

$$\mathbf{K}\_{k|k-1}^{\tau} = \mathbf{P}\_{k|k-1}^{\tau} \mathbf{H}^{\top} \left(\mathbf{S}\_{k|k-1}^{\tau}\right)^{-1}.\tag{26}$$

Using the data association probabilities for the validated measurements, the updated track state estimates are obtained in the form of a Gaussian mixture such as:

$$\mathfrak{X}\_{k|k}^{\tau} = \sum\_{i=0}^{m\_k^{\tau}} \beta\_{k,i}^{\tau} \mathfrak{X}\_{k|k,i}^{\tau} \tag{27}$$

$$\mathbf{P}\_{k|k}^{\tau} = \sum\_{i=0}^{m\_k^{\tau}} \beta\_{k,i}^{\tau} \left( \mathbf{P}\_{k|k,i}^{\tau} + \hat{\mathbf{x}}\_{k|k,i}^{\tau} (\hat{\mathbf{x}}\_{k|k,i}^{\tau})^{\top} \right) - \hat{\mathbf{x}}\_{k|k}^{\tau} (\hat{\mathbf{x}}\_{k|k}^{\tau})^{\top}. \tag{28}$$

The posterior target existence probability is used as a track score for track management including confirmation and termination. It is obtained by using the prior target existence probability and the measurement likelihood ratio such as [10]:

$$P\left\{\chi\_k^{\mathsf{T}}|\mathbf{Z}^k\right\} = \frac{P\left\{\chi\_k^{\mathsf{T}}|\mathbf{Z}^{k-1}\right\}\Lambda\_k^{\mathsf{T}}}{1 - (1 - \Lambda\_k^{\mathsf{T}})P\left\{\chi\_k^{\mathsf{T}}|\mathbf{Z}^{k-1}\right\}}.\tag{29}$$
