4.1.2. Nonlinear Dynamic Model

For the demonstration of the nonlinear tracking scenario, we use a constant turn model with 5-D state vector *xk* = *px*, *py*, *p*˙ *<sup>x</sup>*, *p*˙ *<sup>y</sup>*, *ω* .*T* , where *ω* is the object's turn rate. The transition density is given as follows:

$$f\_{+}(\mathfrak{x}\_{+}|\mathfrak{x}) = \mathcal{N}(\mathfrak{x}\_{+}; F(\omega)\mathfrak{x}, Q)$$

$$\text{where}\,F\left(\begin{bmatrix}p\_{x}\\p\_{x}\\p\_{y}\\p\_{y}\\\omega\end{bmatrix}\right)=\begin{bmatrix}1&\frac{\sin(\omega\Lambda)}{\omega}&0&-\frac{1-\cos(\omega\Lambda)}{\omega}&0\\0&\cos(\omega\Lambda)&0&-\sin(\omega\Lambda)&0\\0&\frac{1-\cos(\omega\Lambda)}{\omega}&1&\frac{\sin(\omega\Lambda)}{\omega}&0\\0&\sin(\omega\Lambda)&0&\cos(\omega\Lambda)&0\\0&0&0&0&1\end{bmatrix},\newlineQ\_{\mathbb{S}=}\begin{bmatrix}\sigma\_{\omega}^{2}G\overline{G}^{T}&0\\0&\sigma\_{v}^{2}\end{bmatrix}\text{ and }G=\begin{bmatrix}\Lambda^{2}/2&0\\\Lambda&0\\0&\Lambda^{2}/2\\0&\Lambda\end{bmatrix}.$$

In this experiment, we set *σω* = *π*/180 rad/s and *σ<sup>v</sup>* = 5 m/s. The observation model is given as the bearing and range detection of the 2D vector *zk* = [*θ*,*r*] *<sup>T</sup>* with *σθ* = *π*/90 rad and *σ<sup>r</sup>* = 5 m.

The surveillance region is the half disc of the radius 2000 m with *K* = 100 time steps and Δ = 1. The ground truth for this experiment is given in Figure 5. The surviving probability is set to *pS* = 0.99 and the detection probability is *pD* = 0.95. Clutter rate is set to 66 false alarms per scan. The expected birth states are *m*(1) *<sup>B</sup>* = [−1500, 0, 250, 0, *π*/180] *<sup>T</sup>*, *m*(2) *<sup>B</sup>* = [−250, 0, 1000, 0, *π*/180] *T*, *m*(3) *<sup>B</sup>* = [250, 0, 750, 0, *π*/180] *<sup>T</sup>*, and *m*(4) *<sup>B</sup>* = [1000, 0, 1500, 0, *π*/180] *<sup>T</sup>* with *rB* = 0.02, and the birth covariance is *PB* = *diag*([50, 50, 50, 50, *π*/30]). The number of hypotheses is also capped at 20, 000 components. The smoothing interval is the entire tracking sequence from *k* = 1 to *k* = *K*. We also set the track pruning threshold for the smoother to 3 time steps in this experiment.

For this scenario, we also test the performance of the tracker over 100 Monte Carlo runs. The means of OSPA error and OSPA<sup>2</sup> error of the estimates are plotted in Figures 6 and 7, respectively, while the set cardinality is shown in Figure 8.

**Figure 5.** Ground truth for nonlinear dynamic scenario (circle: track start position, triangle: track end position).

**Figure 6.** OSPA error for nonlinear dynamic scenario.
