**2. Models**

The following assumptions are applied for using multi-target tracking algorithms in a cluttered environment.


Superscript *τ* denotes a target, or the index of a track that follows the target. Target *τ*'s trajectory state **x***<sup>τ</sup> <sup>k</sup>* is an *nx* × 1 state vector. In this paper, the dynamics of the targets from scan *k* to scan *k* + 1 are assumed to follow a linear dynamic model in a two-dimensional (2D) plane, such as:

$$\mathbf{x}\_{k+1}^{\tau} = \Phi\_k \mathbf{x}\_k^{\tau} + \Gamma\_k w\_{k\prime} \tag{1}$$

where **x***<sup>k</sup>* is a 6 × 1 state vector consisting of the target position, velocity, and acceleration in a 2D plane, **Φ***<sup>k</sup>* is the state propagation matrix, and **Γ***<sup>k</sup>* is the coefficient matrix of *wk*, which is a white Gaussian process noise with zero-mean and covariance matrix *diag*(*q*, *q*). The last term of (1) is white Gaussian with zero-mean and covariance matrix **Q***<sup>k</sup>* = *q***Γ***k*(**Γ***k*) , and its distribution is denoted by *N*(0, **Q***k*). The state propagation matrix **Φ***<sup>k</sup>* and the coefficient matrix for the process noise **Γ***<sup>k</sup>* follow a nearly constant velocity (NCV) model or constant turn rate (CTR) model [25].

For the NCV model, the state propagation matrix and the coefficient matrix for the process noise in (1) become:

$$
\boldsymbol{\Phi}\_{k}^{NCV} = \begin{bmatrix}
\mathbf{I}\_{2} & T\mathbf{I}\_{2} & \mathbf{O}\_{2} \\
\mathbf{O}\_{2} & \mathbf{I}\_{2} & \mathbf{O}\_{2} \\
\mathbf{O}\_{2} & \mathbf{O}\_{2} & \mathbf{O}\_{2}
\end{bmatrix} \tag{2}
$$

$$
\Gamma\_k^{NCV} = \begin{bmatrix} \frac{T^2}{2} \mathbf{I}\_2 \\ T \mathbf{I}\_2 \\ \mathbf{O}\_2 \end{bmatrix} \tag{3}
$$

where *T* is the sampling time of a discrete time interval, **I**<sup>2</sup> is a 2 × 2 identity matrix, **O**<sup>2</sup> is a 2 × 2 null matrix, and the variance of *wk* is set to be *σ*<sup>2</sup> *<sup>a</sup>* **I**<sup>2</sup> such that *wk* of the NCV model represents the acceleration uncertainty; this implies **Q***NCV <sup>k</sup>* <sup>=</sup> *<sup>σ</sup>*<sup>2</sup> *<sup>a</sup>* **Γ***NCV <sup>k</sup>* (**Γ***NCV <sup>k</sup>* ) . For the NCV model, the acceleration components of **x***<sup>k</sup>* are set to be zero.

For the CTR model, the state propagation matrix and the coefficient matrix for the process noise in (1) become:

$$\boldsymbol{\Phi}\_{k}^{CTR} = \begin{bmatrix} \mathbf{I}\_{2} & \frac{\sin(\Omega\_{k}T)}{\Omega\_{k}} \mathbf{I}\_{2} & \frac{1-\cos(\Omega\_{k}T)}{\Omega\_{k}^{2}} \mathbf{I}\_{2} \\ \mathbf{O}\_{2} & \cos(\Omega\_{k}T)\mathbf{I}\_{2} & \frac{\sin(\Omega\_{k}T)}{\Omega\_{k}} \mathbf{I}\_{2} \\ \mathbf{O}\_{2} & -\Omega\_{k}\sin(\Omega\_{k}T)\mathbf{I}\_{2} & \cos(\Omega\_{k}T)\mathbf{I}\_{2} \end{bmatrix} \tag{4}$$

$$\mathbf{I}\_k^{CTR} = \begin{bmatrix} \frac{\Omega\_k T - \sin(\Omega\_k T)}{\Omega\_k^3} \mathbf{I}\_2 \\\\ \frac{1 - \cos(\Omega\_k T)}{\Omega\_k^2} \mathbf{I}\_2 \\\\ \frac{\sin(\Omega\_k T)}{\Omega\_k} \mathbf{I}\_2 \end{bmatrix} \tag{5}$$

*wk* of the CTR model represents the uncertainty in target jerk, and its variance is *σ*<sup>2</sup> *<sup>j</sup>* **I**2; this implies **Q***CTR <sup>k</sup>* <sup>=</sup> *<sup>σ</sup>*<sup>2</sup> *<sup>j</sup>* **<sup>Γ</sup>***CTR <sup>k</sup>* (**Γ***CTR <sup>k</sup>* ) . The turn rate Ω*<sup>k</sup>* is adaptively estimated using the target acceleration and velocity estimates while tracking.

Target measurement model **z***<sup>k</sup>* is an *nz* × 1 vector, and it is expressed as:

$$\mathbf{z}\_k = \mathbf{H}\mathbf{x}\_k + v\_{k'} \tag{6}$$

where **H** is the measurement matrix denoted by:

$$\mathbf{H} = \begin{bmatrix} \mathbf{I}\_2 & \mathbf{O}\_2 & \mathbf{O}\_2 \end{bmatrix} \ . \tag{7}$$

In (6), *vk* is a white Gaussian measurement noise of the sensor with zero-mean and covariance matrix **R***k*.

The sensor obtains a set of measurements **Z***<sup>k</sup>* at each scan *k*. **z***k*,*<sup>i</sup>* is the *i*th measurement of **Z***k*, and the measurement vector of **z***k*,*<sup>i</sup>* can be expressed by:

$$\mathbf{z}\_{k,i} = \begin{bmatrix} z\_{k,i}^x & z\_{k,i}^y \end{bmatrix}^\top \quad , \tag{8}$$

where *z<sup>x</sup> <sup>k</sup>*,*<sup>i</sup>* and *z y <sup>k</sup>*,*<sup>i</sup>* represent the *x* and *y* positions in the 2D Cartesian coordinate system, respectively.
