# UWB-Based Human-Following System with Obstacle and Crevasse Avoidance for Polar-Exploration Robots
## 3 UWB-Based Explorer Position Estimation and Control Objectives  
This section presents the process of estimating the position of the tag using the distances obtained from the anchors mounted on the Antarctic exploration robot KAREX. Based on this estimated position, it outlines the control objectives to maintain a consistent distance between the robot and the tag. Figure 3 illustrates the relationship between the anchors and the tag relative to the robot coordinate system. In this system, (\(x_{r}\), \(y_{r}\)), (\(x_{f}\), \(y_{f}\)), (\(x_{t}\), \(y_{t}\)), (\(x_{al}\), \(y_{al}\)), and (\(x_{ar}\), \(y_{ar}\)) represent the positions of the robot, the center of the front anchors, the tag, the left anchor, and the right anchor, respectively. The distances \(d_{al}\), \(d_{ar}\), \(d_{t}\), \(d_{d}\), and \(e_{d}\) correspond to the distance between the left anchor and the tag, the right anchor and the tag, the front of the robot and the tag, the target distance to the tag, and the error distance, respectively. \(\theta_{t}\) denotes the angular position of the tag relative to (\(x_{f}\), \(y_{f}\)), while \(\theta_{al}\) and \(\theta_{ar}\) indicate the direction of the tag as seen from the left and right anchors, respectively. From Figure 3, we can determine the tag position (\(x_{t}\), \(y_{t}\)) using \(d_{al}\), \(d_{ar}\), \(p_{al}\), and \(p_{ar}\) through the process depicted in the block diagram in Figure 4. \begin{table}
\begin{tabular}{c c} \hline \hline
**Item** & **Specification** \\ \hline Dimension & 1770 mm \(\times\) 2490 mm \(\times\) 1750 mm \\ \hline Weight & 550 kg \\ \hline Minimum Operation Temperature & \(-\)50 \({}^{\circ}\)C \\ \hline Maximum Target Operating Distance & 100 km \\ \hline Payload & 100 kg \\ \hline Towing Capacity & 150\(\sim\)200 kg \\ \hline Maximum Velocity & 30 km/h \\ \hline Maximum Obstacle Clearance Height & 0.5 m \\ \hline Maximum Gap Clearance & 0.3 m \\ \hline Battery Capacity & 420 Ah \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specification of KAREX [20,21]. Figure 3: A relationship of the anchors and the tag. As shown in Figure 4, the distance information from the anchors mounted on the robot to the tag, which includes noise, is obtained. To remove this noise, the raw distance information is processed through the following Low Pass Filter (LPF). \[d_{LPF}=d_{LPF}+\alpha(d_{MEAS}-d_{LPF}) \tag{1}\]  
Here, \(d_{LPF}\) represents the result after passing through the LPF, and \(d_{MEAS}\) is the measured value. \(\alpha\) denotes the bandwidth of the LPF. The distance information between the anchor and the tag, processed through the LPF, along with \(p_{al}\) and \(p_{ar}\), is input into the tag position estimator for position estimation. In this paper, the fsolve function from the scipy.optimize library in Python [27; 28] is used to estimate the tag position \((x_{t},y_{t})\). In order to estimate the position of the tag using the fsolve library, the following algorithm is applied. As seen in Algorithm 1, it is possible to estimate the tag's position using the fsolve library, an optimization tool. The initial values (x0, y0) are set within the \(d_{d}\) range in front of the robot, which in this paper is set to (0.5, 0.5). Since the position is estimated using the distance from two anchors to one tag, two potential positions may be estimated. Therefore, only the position in front of the robot is selected. ```
1. Set anchor position: Pa(xa, ya), Pb(xb, yb)
2. Acquiring the distances between Anchors and tag: La, Lb
3. Define equations (p):
4. x, y =
5. eq1 = (x - xa)*2 + (y - ya)*2 - La*2
6. eq2 = (x - xb)*2 + (y - yb)*2 - Lb**2
7. return (eq1, eq2)
8. Set initial value of x and y: x0, y0
9. solution = fsolve(equations, (x0, y0))
```  
**Algorithm 1** Tag position estimation using fsolve library  
In this paper, the control objective is to ensure that the exploration robot follows the explorer carrying the UWB tag while maintaining a constant distance, such that \(e_{d}\to 0\) and \(\theta_{t}\to 0\).