# An Action Classification Method for Forklift Monitoring in Industry 4.0 Scenarios
## 3 Materials and Methods
### Signal Model  
The phase-based action classification method proposed here enables a smart-gate operation with a single antenna to determine the moving direction of the forklift crossing the gate. When the reader interrogates a tag, the latter backscatters the impinging signal, thus enabling the reader to measure a phase proportional to the distance between the reader and the tag. When the tagged forklift crosses the gate, the reader antenna performs several queries of the moving tag and measures the phase of the signal at different time steps \(t_{n}\), being \(n\in\{0,...,N_{R}-1\}\) and \(N_{R}\) the number of successful queries of the tag  
Figure 6: RFID tags placed on the forklift. (**a**) Tag placed on the upright, and (**b**) tag placed on the forklift top. Figure 7: Photocells installed at one of the UHF-RFID Smart Gates. during the vehicle crossing. To be more precise, the phase of the signal measured by the reader at time \(t_{n}\) can be resumed as:  
\[\phi(t_{n})=\text{mod}\Big{(}\frac{-4\pi d(t_{n})}{\lambda}+\phi_{0}(\theta_{R}, \psi_{R},\theta_{T},\psi_{T},t_{n})+\phi_{m}(t_{n})\Big{)}_{2\pi} \tag{1}\]  
where \(d(t_{n})\) is the distance between the tag and the reader at time \(t_{n}\), \(\lambda\) is the carrier wavelength in free-space, \(\phi_{0}(\theta_{R},\psi_{R},\theta_{T},\psi_{T},t_{n})\) is the phase bias caused by reader and tag antennas and by the electrical circuitry, where \(\theta_{R}\) and \(\psi_{R}\) are the elevation and azimuth angle at time \(t_{n}\), respectively, at the reader antenna side, and \(\theta_{T}\) and \(\psi_{T}\) are the elevation and azimuth angle at time \(t_{n}\), respectively, at the tag antenna side. \(\phi_{m}(t_{n})\) is the contribution to the phase caused by multipath phenomena at time \(t_{n}\). The distance \(d(t_{n})\) is defined as:  
\[d(t_{n})=\|\mathbf{P_{ant}}-\mathbf{P_{tag}}(t_{n})\| \tag{2}\]  
where \(\mathbf{P_{ant}}\) is the vector \([x_{ant},y_{ant},z_{ant}]^{T}\in\mathbb{R}^{3}\) of the reader antenna location, and \(\mathbf{P_{tag}}(t_{n})\) is the vector \([x_{tag}(t_{n}),y_{tag}(t_{n}),z_{tag}(t_{n})]^{T}\in\mathbb{R}^{3}\) of the tag trajectory sample at time \(t_{n}\). The value of \(\phi_{0}(\theta_{R},\psi_{R},\theta_{T},\psi_{T},t_{n})\) is defined as:  
\[\phi_{0}(\theta_{R},\psi_{R},\theta_{T},\psi_{T},t_{n})=\phi_{TX}(\theta_{R}, \psi_{R},t_{n})+\phi_{RX}(\theta_{R},\psi_{R},t_{n})+\phi_{tag}(\theta_{T}, \psi_{T},t_{n}) \tag{3}\]  
where \(\phi_{TX}\) and \(\phi_{RX}\) are the phase offsets caused by the transmitting and receiving circuitry of the reader, and \(\phi_{tag}\) is a phase offset that depends on the tag itself and may be different even among tags of the same model. The \(\phi_{0}(\theta_{R},\psi_{R},\theta_{T},\psi_{T},t_{n})\) term is almost constant over consecutive tag query responses within the reader antenna's main beam, and it will be indicated in the rest of the paper as \(\phi_{0}\). To overcome the problem of the phase \(2\pi-\)ambiguity, we can perform phase unwrapping [40]:  
\[\phi^{u}(t_{n})=\frac{-4\pi d(t_{n})}{\lambda}+\phi_{0}+\phi_{m}(t_{n}) \tag{4}\]  
To correctly execute the phase unwrapping, consecutive phase samples must not differ more than \(\pi\). If we consider the value of \(\phi_{m}(t_{n})-\phi_{m}(t_{n-1})\approx 0\), meaning that the phase difference caused by the multipath between consecutive time steps is negligible, only the condition \(d(t_{n})-d(t_{n-1})<\lambda/4\) must be satisfied. This fact is a direct consequence of the Nyquist-Shannon Sampling Theorem, which states the condition for which a signal is sampled without aliasing. Further considerations on the topic applied to the RFID field can be found in [41; 42]. As it will be discussed later, a relatively high forklift speed or a poor RFID reader sampling rate may both lead to errors during the phase unwrapping process and, therefore, to classification errors. Now, for the sake of simplicity, the value of \(\phi^{u}(t_{n})\) is normalized by the first sample acquired at \(n=0\).