# Forklift Tracking: Industry 4.0 Implementation in Large-Scale Warehouses through UWB Sensor Fusion
## 3 Numerical Analysis  
The performance of the proposed UKF algorithm are here demonstrated through a numerical analysis. We consider an indoor environment, where \(N_{UIWB}=35\) UWB anchors are deployed according to a regular \(7\times 5\) grid in a 40 m \(\times\) 25 m area at the ceiling and at a height \(h=6\) m. The spacing along the \(x\)-coordinate of each anchor is 6.6 m, whereas along the \(y\)-coordinate is 6.25 m. The forklift is here considered as a single point with all of the sensor payloads (IMU, OFS, and UWB tag) placed at the same point. Therefore, \(\Delta x_{UIWB}=\Delta y_{UIWB}=0\) m. The height of the UWB tag is 2 m, so \(\Delta z=4\) m. The sampling time \(\Delta t\) is set to 10 ms. At each timestep \(k\), the algorithm exploits the measured distance between the UWB tag and the four closest UWB anchors to perform the multilateration. At the same time, the IMU and the OFS generate acceleration, angular speed, and velocity data. In accordance to the data sheet of the sensors that will be used for the experimental analysis, we set \(\sigma_{a}=16.7\) mm/s\({}^{2}\), \(\sigma_{\omega}=2.4\) mrad/s, \(r=2.3\) cm, \(\sigma_{UIWB}=0.15\) m, and \(\sigma_{\theta}=0.1\) rad. The forklift starts a \"square wave-shaped\" path from location \([x_{0},y_{0}]^{T}=[0,0]^{T}\) m with an orientation \(\theta_{0}=\pi/2\) rad, and travels with a speed of 0.75 m/s until the location \([x_{N_{c}},y_{N_{c}}]^{T}=[32.6,15]^{T}\) m after \(N_{c}=10,000\) algorithm steps. The path-length is approximately 75 m. The UKF filter is run with \(\alpha=10^{-3}\), \(\beta=2\), and \(K=0\). A top view of the performed simulated trajectory is depicted in Figure 3. The UWB anchors are the gray squared markers, and the ground truth trajectory is depicted as a continuous gray line. We represent, with a continuous blue line, the trajectory estimated  
Figure 2: RFID Smart Forklift in the Smart Warehouse, along with part of the UWB infrastructure for its localization. with the UWB system only. The output of the dead-reckoning process by using only IMU and OFS is instead depicted as a dashed red line, whereas the output of the proposed UKF-based algorithm is the pointed green line. As can be seen, the UWB system alone is not affected by drifts, but the output trajectory is not smooth; therefore, is not easy to accurately estimate the vehicle orientation. The output of the dead-reckoning process (dashed red line) is a smooth curve, but since only proprioceptive sensors are involved, it is affected by a detrimental drift, especially caused by the errors on the measured angular speed, which have a great impact on the vehicle orientation estimation. The output of the UKF (pointed green line) is a smooth trajectory without drifts, meaning that the filter converges. As expected, the trajectory of the vehicle gets further from the ground truth trajectory along the curves, but the algorithm is able to recover the track in few steps. In Figure 4, we depict the localization errors along the \(x\)-coordinate \(\epsilon_{x}\) (Figure 4a), along the \(y\)-coordinate \(\epsilon_{y}\) (Figure 4b), and the distance error \(\epsilon_{d}=\sqrt{\epsilon_{x}^{2}+\epsilon_{y}^{2}}\) (Figure 4c). From the figures, an initial transient stage is clearly observable, where the UKF must achieve the convergence, which is indeed reached in a few seconds. To better investigate the method performance, a Monte Carlo analysis with \(M=100\) test cases with similar features as the one represented in Figure 3 was conducted. The results of the cumulative distribution function of the distance error \(\epsilon_{d}\) for the different tracking methods is represented in Figure 5. As apparent, the error of the UKF is bound to 0.5 m. Figure 6, instead, shows the histogram of the orientation error \(\sigma_{\theta}\). It can be noticed that the orientation error is lower for the UKF.