## 3 Data Processing and Kinematic Analysis
### Object Tracking and Management  
To track multiple objects around the excavator, we integrated several filters to design our tracking filter composed of IMM (interacting-multiple-model), UKF (unscented Kalman filter), and JPDA (joint probabilistic data association filter) techniques. The reason for choosing the integrated filters was to handle multiple models representing different motion behaviors (IMM), estimate their non-linear states (UKF), and provide a tracking management system that allows for multitarget tracking in clutter at the stage of the data association (JPDA). The IMM-UKF-JPDA approach is computationally effective for recursive state estimation and the mode probabilities of targets. For the implementation of this filter, we consider model set \(\mathbb{M}=\left\{M_{j}\right\}_{j=1}^{r}\) with \(r=2\) meaning that the M\({}_{1}\), constant velocity (CV), and M\({}_{2}\), constant turn (CT) models are considered. These two models share a common state vector \(x=\left[p_{x},v_{x},p_{y},v_{y},p_{z},v_{z},\theta,\omega,l,w,h\right]^{T}\) consisting of bounding box center points in the \(x,y,\) and \(z\) directions \(\left(p_{x},p_{y},\)_and_\(p_{z}\right)\), their corresponding velocities  
Figure 3: Processing of point cloud: (**a**) raw point cloud; (**b**) non-ground points (green) and ground points (blue); (**c**) clustered points.  
\((v_{x},v_{y},and~{}v_{z})\), and the yaw angle \((\theta)\), turn rate \((\omega)\), and the length \((l)\), width \((w)\), and height \((h)\) of a bounding box. The discrete time-state equation for the CV model is given as:  
\[x_{k+1}=\left[\begin{array}{c}p_{x,k}+v_{x,k}Tsin\theta_{k}\\ v_{x,k}\\ p_{y,k}+v_{y,k}Tcos\theta_{k}\\ v_{y,k}\\ p_{z,k}\\ v_{z,k}\\ \theta_{k}\\ 0\\ l_{k}\\ w_{k}\\ h_{k}\end{array}\right]+u_{k}+W_{1,k}. \tag{1}\]  
and the state for the CT model is:  
\[x_{k+1}=\left[\begin{array}{c}p_{x,k}+\frac{v_{x,k}}{\omega_{k}}(cos\theta_{ k}-cos(\omega_{k}T+\theta_{k}))\\ v_{x,k}\\ p_{y,k}+\frac{v_{y,k}}{\omega_{k}}(sin(\omega_{k}T+\theta_{k})-sin\theta_{k})\\ v_{y,k}\\ p_{z,k}\\ v_{z,k}\\ \theta_{k}+T\omega_{k}\\ \omega_{k}\\ l_{k}\\ w_{k}\\ h_{k}\end{array}\right]+u_{k}+W_{2,k}. \tag{2}\]  
where \(T\) is the sampling time, \(u_{k}\) is the control input of the excavator, and both models include a white Gaussian noise \(W_{1,k}\), and \(W_{2,k}\) having a mean at zero and the covariance matrices \(Q_{1,k}\) and \(Q_{2,k}\), respectively. Such models are common for maneuvering multi-object tracking and are extensions of the models found in [21]. Among the two models, the common measurement model is given as:  
\[z_{k}=\left[\begin{array}{cccccccc}1&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&1&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0\\ \end{array}\right]\chi_{k}+~{}V_{k}. \tag{3}\]  
with the measurement noise \(V_{k}\) having the covariance matrix \(R_{k}\).  
The next step is to estimate the probability to identify the measurements according to the CV and CT models. The progress of the system among the models is operated on the Markovian model transition probability matrix.  
\[\Pi=\left(\begin{array}{ccc}p_{11}&\cdots&p_{r1}\\ \vdots&\ddots&\vdots\\ p_{1r}&\cdots&p_{rr}\end{array}\right)e\mathbb{R}^{r\times r}. \tag{4}\]where \(p_{ij}\) represents the probability of mode transition. Several studies have addressed the integration of the above filter, but this paper follows the five-step process introduced in [21; 22]. These steps include (1) interaction, (2) state prediction and measurement validation, (3) data association and tracking management, (4) mode probability update, and (5) combination. Each step can be explained in mathematical representations as follows.