esa-sceva/satcom-chunk-collection huggingface dataset:
# FPGA-Based Methodology for Detecting Positional Accuracy Degradation in Industrial Robots
###### Abstract
Other recent FPGA applications in robotics are agricultural robots [32], reactive robotics [33], and grasping recognition [34], among others. The works presented have demonstrated the usefulness and advantages of using FPGA processing systems in robotics problems. Considering the importance of the robot to perform tasks without losing its accuracy, some of the mentioned works that address the issue of positional degradation and fault detection require the use of external sensors such as cameras, lasers, or microphones to acquire the signals for analysis. These external sensors can work correctly in controlled environments, but in real environments, they are affected by various conditions such as sensor size, noise, illumination, temperature, and humidity This makes it difficult to implement in an industrial environment; however, a portable device of a small size equipped with the necessary components to perform signal acquisition and real-time processing presents an alternative to traditional methods of signal acquisition where data processing is performed offline. In this sense, FPGA-based digital systems are an option to perform the task of data acquisition and processing in a single, non-invasive device and have the advantages of reconfiguration and high processing speed  
In this context, the contribution of this work lies in proposing a methodology to detect positional degradation in industrial robot joints that is suitable for implementation in FPGA-based digital systems. The approach involves analyzing the current signals of their actuators to identify any anomalous behaviors that may result in end-effector deviation. For this purpose, the current signals from the robot actuators are generated synthetically by modeling the UR5 robot dynamics. The synthetic dataset contains signals with and without deviation. Then different noise levels (5 dB, 10 dB, 15 dB, and 20 dB) are added to the signals to enhance the method's robustness. Subsequently, the current signal is processed for a time-frequency analysis using the DWT. Then statistical indices are computed for the two last levels of decomposition. Finally, a neural network is trained using the indices data to classify each robot joint. The proposed method is implemented in a digital system board FPGA-based proprietary design and validated using a real UR5 robot dataset. For the synthetic case, the results show that the proposed method can classify the states of the robot's operation with and without deviation for the joints with more significant intervention in the task. The results obtained with the real-world dataset (joint five) are supported by results reported in the literature. Implementing the method in FPGA allows a quick classification of less than one second for all the joints of the robot, providing a functional system with the advantage of being a high-frequency operating system, reconfigurable, portable, and energy efficient.
## 3 Results and Discussion
### Determination of Organic Acids in Dried Sourdough  
Organic acids are the principal metabolites produced by LAB, and their action has antifungal activity due to the synergy between these organic acids, particularly lactic  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Samples**} & \multicolumn{4}{c}{**The Yield of the Weight (\%)**} \\ \cline{2-5}  & **Spray-Dry** & **Freeze-Dry** & **Dry at 40 \({}^{\circ}\)C** & **Dry at Low Humidity** \\ \hline Control & 4.0 \(\pm\)0.2 \({}^{\text{\text{\,{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{ \tiny{\tiny{\tiny{\tiny{\tiny{  } }}}}}}}}}}}}}}}\) & 42.0 \(\  
acid [46]. Lactic acid produced in the sourdoughs by the LAB fermentation have antimicrobial properties with an impact against unfavorable spoilage and decreasing the pH of the medium [47]. Table 7 summarizes the results obtained for determining the lactic acid in the dry sourdoughs and the control sample.  
The five strains produced lactic acid due to their metabolism. As expected, each sourdough had a higher lactic acid content than the control sourdough. Usually, concentrations of 26 g of lactic acid per kg of the dry product were reached. Otherwise, there were differences between each culture starter of the sourdough and the dry technique used in producing lactic acid. The highest production of acid was in spray-dry sourdoughs fermented with \(L\). _plantarum_ TN10 with 26.3 g/kg and _P. acidlactictictic_ TE4 with 20.4 g/kg, while the lowest production of lactic acid was in the sourdoughs dried at low humidity. The high concentration of lactic acid in the spray-dried samples may be due to a higher concentration of the compounds since much less powder was obtained.  
According to the literature, several studies report that lactic acid's effect correlates with the inhibitory activity against fungi, as seen in [PERSON] et al. [7]. In addition, the presence of lactic acid in the sourdough can ensure its microbiological safety and prolong the final product's shelf life if the organic acid remains undissociated in the fermented product [48]. As the significant LAB metabolite, lactic acid reduces pH, and the undissociated acid collapses the electrochemical proton gradient. Studies on the effect of LAB in conjunction with fungi are complicated by considering the observed differences in sensitivity among mold species to organic acids. These differences may be related to the ability of fungi to change cellular metabolism in response to stress conditions such as high concentrations of organic acids. Specifically, organic acids generated by LAB exert their antimicrobial effect by entering cell membranes in their undissociated form, resulting in a decrease in intercellular pH and interruption of metabolic processes. [35].
# Exploring the Potential of Event Camera Imaging for Advancing Remote Pupil-Tracking Techniques
## 4 Discussion
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Test DB** & **DB Type** & **Detection Accuracy** & \begin{tabular}{c} **Tracking Accuracy** \\ **(Pupil Precision \textless{} 10 mm)** \\ \end{tabular} \\ \hline \
\begin{tabular}{c} Real Event Camera Images \\ Indoor Office (100–400 lux) \\ \end{tabular} &
\begin{tabular}{c} Large Movement \\ (Image with verifiable eye shape) \\ Various Movement \\ (minimal to large movement) \\ \end{tabular} & 98.1\% & 80.9\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance comparison of the proposed event-camera-based pupil-tracking method on different movement levels. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Sensor Type** & **Test Dataset** & \begin{tabular}{c} **Keypoint** \\ **Number** \\ \end{tabular} & **Speed** & **GFLOPs** \\ \hline \begin{tabular}{c} LAB [22] \\ Wing [23] \\ Awing [24] \\ AVS + SAN [25] \\ \end{tabular} & \begin{tabular}{c} RGB frame \\ camera \\ \end{tabular} & \
\begin{tabular}{c} Public DB \\ (WFLW [22]) \\ \end{tabular} & 98 & 41 ms (GPU) & 29.1 \\ \hline
\begin{tabular}{c} Ours \\ \end{tabular} & Event camera & In-house DB & 11 & 4 ms (CPU) & N/A \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison between previous RGB-frame-camera-based studies and the proposed method. adapt to different motion levels and lighting conditions, ultimately leading to improved generalization and performance.
## Appendix A The Quantative Example  
Consider a small matrix \(\mathbf{M^{s}}\in\mathbb{R}^{8,8}\) representing a simple wave arrival recorded by 8 receivers.  
\[\mathbf{M^{s}}=\begin{bmatrix}0&0&0&0&0&0&0&0\\ 0&2&3&2&0&0&0&0\\ 1&-2&-3&-2&1&0&0&0\\ -1&0&0&0&-1&1&0&0\\ 0&0&0&0&-1&1&0\\ 0&0&0&0&0&0&-1&1\\ 0&0&0&0&0&0&-1\\ 0&0&0&0&0&0&0\end{bmatrix}\]  
In the matrix \(\mathbf{M^{s}}\), each receiver is represented by one of the columns, and the time increases from top row to the bottom row. The source is closest to the third receiver, and the perceived waveform is simply:  
\[\mathbf{w^{s}}=\begin{bmatrix}0\\ 1\\ -1\\ 0\end{bmatrix}.\]  
Figure 10 loosely describes a toy model that can generate the matrix \(\mathbf{M^{s}}\).  
By applying SMD to matrix \(\mathbf{M^{s}}\) we extract a pair of the singular vectors (\(\mathbf{u^{SMD}}\),\(\mathbf{v^{SMD}}\)), and a shift vector (\(\mathbf{s}\)). Keep in mind that when using SMD, the singular value \(\lambda^{SMD}\) is stored in right singular vector \(\mathbf{v^{SMD}}\) by multiplying the vector with it:\[\mathbf{u}^{\mathbf{SMD}}=\begin{bmatrix}0\\ 1/\sqrt{2}\\ -1/\sqrt{2}\\ 0\\ 0\\ 0\\ 0\end{bmatrix},\quad\mathbf{v}^{\mathbf{SMD}}=\begin{bmatrix}\sqrt{2}\\ 2\sqrt{2}\\ 3\sqrt{2}\\ 2\sqrt{2}\\ \sqrt{2}\\ \sqrt{2}\\ \sqrt{2}\end{bmatrix},\quad\mathbf{s}=\begin{bmatrix}1\\ 0\\ 0\\ 0\\ 1\\ 2\\ 3\\ 4\end{bmatrix}.\]  
Using the singular vectors and the shift vector we can reconstruct a matrix \(\mathbf{M}_{\mathbf{SMD}}\) to be identical to \(\mathbf{M}_{\mathbf{s}}\):  
\[\mathbf{M}^{\mathbf{SMD}}=\chi(\mathbf{u}^{\mathbf{SMD}}\mathbf{v}^{\mathbf{ SMD}}{}^{T},\mathbf{s})\]  
\[\mathbf{M}^{\mathbf{SMD}}=\begin{bmatrix}0&0&0&0&0&0&0\\ 0&2&3&2&0&0&0&0\\ 1&-2&-3&-2&1&0&0&0\\ -1&0&0&0&-1&1&0&0\\ 0&0&0&0&0&-1&1&0\\ 0&0&0&0&0&0&-1&1\\ 0&0&0&0&0&0&-1\\ 0&0&0&0&0&0&-1\\ 0&0&0&0&0&0&0\end{bmatrix}\]  
We can also apply regular SVD to the matrix \(\mathbf{M}^{\mathbf{s}}\) to extract a singular value \(\lambda\) and singular vectors \(\mathbf{u}\) and \(\mathbf{v}\):  
\[\lambda\approx 5.92,\quad\mathbf{u}\approx\begin{bmatrix}0\\ 0.68\\ -0.73\\ 0.05\\ 0\\ 0\\ 0\end{bmatrix},\quad\mathbf{v}\approx\begin{bmatrix}-0.13\\ 0.48\\ 0.72\\ 0.48\\ -0.13\\ 0.01\\ 0\\ 0\end{bmatrix},\]  
We can reconstruct a matrix \(M^{\mathbf{SVD}}\) from the extracted eigenpair with the following equation:  
\[\mathbf{M}^{\mathbf{SVD}}\approx\begin{bmatrix}0&0&0&0&0&0&0\\ -0.53&1.93&2.90&1.93&-0.53&0.03&0&0\\ 0.56&-2.06&-3.08&-2.06&0.56&-0.03&0&0\\ -0.04&0.13&0.19&0.13&-0.04&0&0&0\\ 0&0&-0.01&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\end{bmatrix}\]  
There is a significant difference between \(\mathbf{M}^{\mathbf{s}}\) and \(\mathbf{M}^{\mathbf{SVD}}\). The data from matrix \(\mathbf{M}^{\mathbf{s}}\) cannot be described by a single pair of singular vectors and a singular value obtained by applying regular SVD. However, it can be described by a pair of singular vectors coupled with a shift vector obtained by applying SMD.
## 2 Materials and Methods
### MIMO OFDM Communication and Radar Sensing
#### 2.4.2 MIMO OFDM Radar Sensing  
Consider an MIMO radar system in which there are \(N_{\rm t}\) transmit and \(N_{\rm r}\) receive antennas, with interelement spacing \(d\) and transmission wavelength \(\lambda\). The received signal at antenna \(p\) and time index \(n\) due to a moving target located at angle \(\theta\) with complex reflection coefficient \(\alpha(\theta)\) is written as  
\[y_{p}(n)=\alpha(\theta)e^{\frac{2(p-1)md}{\lambda}\sin(\theta)}e^{j2\pi f_{D}n} \sum_{q=1}^{N_{\rm t}}x_{q}(n)e^{j\frac{2(q-1)md}{\lambda}\sin(\theta)}+w_{p} (n), \tag{17}\]  
where \(\mathbf{a}_{T}(\theta)\) and \(\mathbf{a}_{R}(\theta)\) denote the transmit and receive steering vectors [17; 18].
## 4 Media Access Control
### Local Sensing Configuration
#### 4.1.2 Sensing Duration  
One of the most critical practical limitations facing the regulation of spectrum sensing is that a single antenna cannot undertake sensing and data transmission at the same time, and sensing can only take place in one of the channels [42]. As a consequence, it is often only possible to obtain partial spectral knowledge by periodically making the antenna alternate between sensing and tranceiving activities. Moreover, the length of sensing time poses a scheduling tradeoff between sensing accuracy and transmission throughput [43], as summarized in Table 2.  
A number of studies have been carried out to explore the tuning of sensing duration for the detection of certain predefined spectral events [26, 44, 45, 46, 47, 48]. The approach presented in [44] establishes an analytical model of the relationship between sensing time and detection accuracy by employing a maximum _a posteriori_ (MAP) energy detector. Based on the model, the optimal sensing and transmission time is derived. The work is subject to two main limitations: first, it assumes the prior knowledge of the characteristics of the target spectral event; second, it does not specifically address the tradeoff of network throughput in the context of periodic spectrum sensing. The iterative algorithm proposed in [47] finds the sensing period that maximizes the throughput. However it does not optimize the detection performance but rather requires the sensing accuracy as a constraint for the algorithm. Such a limitation is also found with the algorithm proposed in [26]. By associating the probability mass function (PMF) of detection accuracy with sensing period and related overhead, the algorithm is able to find the sensing time that meets the detection requirement with minimized overhead.
## 1 Introduction
In this work, we propose a low-power and lightweight RD/SD mechanism, denoted as DiRPL, which builds upon control features already available in RPL protocol, namely Destination-Oriented Directed Acyclic Graph (DODAG) Information Object (DIO)/Destination Advertisement Object (DAO) messages exchange. Unlike other RD/SD mechanisms, DiRPL is particularly suitable for constrained and duty-cycled SOs operating in 6 LoWPAN multi-hop networks and avoids the introduction ofredundant transmissions. It is targeted to Contiki OS-enabled SOs but, being based on an RPL-oriented approach, may be extended to other OSs, e.g., RIOT OS [7]. An experimental performance evaluation is carried out in the Cooja simulator, to evaluate the feasibility and efficiency, in terms of delay and energy consumption, of the proposed DiRPL RD/SD mechanism. The rest of this paper is organized as follows. In Section 2, an overview of related works on SD and RD mechanisms for IoT systems is provided. In Section 3, a detailed description of the building blocks of the proposed architecture is given. Implementation details of the proposed architecture and experimental performance results are presented in Section 4. A brief discussion on the proposed RPL-based RD/SD mechanism, denoted as DiRPL, is done in Section 5. Finally, conclusions are drawn in Section 6.
## 5 Simulation Results
### Metrics  
We use the message delivery ratio \(M_{s}\), the average end-to-end delay \(M_{d}\), and the overhead \(M_{o}\) as metrics to evaluate performance. The metrics can be expressed as  
\[M_{s}=\frac{m}{M}*100\%, \tag{22}\]  
\[M_{d}=\frac{1}{m}\sum_{k=1}^{m}\left[\sum_{i=1}^{n}\left(T_{S_{i}}+T_{P_{i}}+T ^{\prime}{}_{P_{i}}+T_{Q_{i}}\right)+\sum_{i=1}^{n-1}T_{C_{i}}\right]_{k}, \tag{23}\]  
\[M_{o}=\left(H-\sum_{i=1}^{m}F_{i}\right)/\sum_{i=1}^{m}F_{i}, \tag{24}\]where \(m\) represents the number of messages received by the DN, \(M\) represents the message generated by the SN, \(H\) represents the number of packets, and \(F_{i}\) represents the number of successfully transmitted packets.
# Inter-System Differencing between GPS and BDS for Medium-Baseline RTK Positioning
## 4 Experiments of Medium-Baseline RTK Positioning  
In this section, we mainly test the accuracy and reliability of combined GPS and BDS RTK positioning with the developed inter-system differencing model and classical differencing model. Two medium baselines CUT0-PERT and TG00-TGT0 were tested. The details of the two baselines can be seen in Table 4. The data of the baseline CUT0-PERT were downloaded from Multi-GNSS Experiment and Pilot Project (MGEX). The data of the baseline TG00-TG00 were collected in Tianjin, China.  
The numbers of the visible GPS and BDS satellites for the two baselines with a cut-off elevation of \(10^{\circ}\) are given in Figure 6. We can see with combined GPS and BDS, there are no less than 14 satellites available at all times for both the two baselines. In order to test the positioning performance with different satellite visibility, both of the positioning experiments below are carried out with 7-12 and all satellites, which are chosen in descending order of elevation.
## 3 Improved PSO-RBFNN Model
### Improved PSO Algorithm
#### 3.1.2 The Addition of Local Best Information  
In PSO, information sharing among individuals is critical in finding the optimal solution. At each iteration, each particle is navigated by the personal best information and the global best information. As the iteration increases, all particles tend to approach the global best and thus the diversity of the population is decreased, leading to the \"premature phenomenon\". To overcome this, the local best information obtained at each iteration is added to the velocity updating in Equation (4) to increase the diversity during the optimization process.  
In our improved PSO, velocities of particles are updated as:  
\[\mathbf{v}_{t}^{k+1}=w\mathbf{v}_{t}^{k}+c_{1}r_{1}(\mathbf{pbest}_{t}^{k}-\mathbf{x}_{t}^{k})+ c_{2}r_{2}[\mu(\mathbf{gbest}^{k}-\mathbf{x}_{t}^{k})+(1-1/k)(1-\mu)(\mathbf{lbest}^{k}-\mathbf{x}_{t}^ {k})] \tag{9}\]  
where \(\mathbf{lbest}^{k}\) is the local best position at \(k\)th iteration which is expressed as:  
\[\mathbf{lbest}^{k}=\operatorname*{argmax}_{\mathbf{x}}(f(\mathbf{x}_{t}^{k})) \tag{10}\]  
\(\mu\in(0,1)\) is a constant used to adjust the proportion of information sharing between \(\mathbf{lbest}\) and \(\mathbf{gbest}\). That is, PSO tends to be premature with a very large value of \(\mu\), while it will not converge with a too small value of \(\mu\); The factor of \((1-1/k)\) is used to increase the weight of \(\mathbf{lbest}\) as the iteration increases.  
In brief, by introducing the time-varying learning factors and local best information, the improved PSO may ensure good exploration and exploitation capabilities even at the later stage of the optimization process. Figure 4 shows the schematic optimization process of our improved PSO.  
Figure 3: Curves of time-varying learning factors: (**a**) with different values of \(\alpha\) when \(c=2\); (**b**) with different values of \(c\) when \(\alpha=3\).
## 3 Fuzzy MPC Algorithm with Advanced Generation of Prediction
### Generation of the Free Response  
The basic idea of the free response generation for an MPC algorithm is to use a non-linear process model assuming that the control signals do not change over the entire prediction horizon--a classic free response; see for example, Reference [36]. In the case of the proposed algorithm, future control increments calculated in the algorithm in the previous iterations are also used when the free response is calculated. Thus, more information is exploited to obtain the improved prediction. Assume that future control values are decomposed as follows:  
\[u_{k+i|k}^{m}=\tilde{u}_{k+i|k}^{m}+u_{k+i|k-1}^{m}\, \tag{24}\]  
where \(\tilde{u}_{k+i|k}^{m}\) can be interpreted as a corrections of the control signal \(u_{k+i|k-1}^{m}\) obtained in the last \((k-1)\)st iteration. Similarly, future control increments will be given by the formula:  
\[\Delta u_{k+i|k}^{m}=\Delta\tilde{u}_{k+i|k}^{m}+\Delta u_{k+i|k-1}^{m}. \tag{25}\]Assume \[\mathbf{u}_{k+i|k-1}=\left[u^{1}_{k+i|k-1},\ldots,u^{n_{u}}_{k+i|k-1}\right]^{T}\,\] (26) In this case, using the nonlinear model, the free response is iteratively derived as follows:  
* First, the nonlinear model (12) is used to obtain \[\widehat{\mathbf{y}}_{k+1}=\mathbf{f}(\mathbf{y}_{k},\mathbf{y}_{k-1},\ldots,\mathbf{y}_{k-n_{a}+1}, \mathbf{u}_{k|k-1},\mathbf{u}_{k-1},\ldots,\mathbf{u}_{k-n_{b}+1})\ ;\] (27)
* Next, the values \(\widehat{\mathbf{y}}_{k+1}\) are used when calculating the output values \(\widehat{\mathbf{y}}_{k+2}\) for the next sampling instant: \[\widehat{\mathbf{y}}_{k+2}=\mathbf{f}(\widehat{\mathbf{y}}_{k+1},\mathbf{y}_{k},\ldots,\mathbf{y}_ {k-n_{a}+2},\mathbf{u}_{k+1|k-1},\mathbf{u}_{k|k-1},\mathbf{u}_{k-1},\ldots,\mathbf{u}_{k-n_{b} +2})\ ;\] (28)
* For the \(i\)th iteration, using the values \(\widehat{\mathbf{y}}_{k+1},\ldots,\widehat{\mathbf{y}}_{k+i-1}\) one obtains: \[\widehat{\mathbf{y}}_{k+i}=\mathbf{f}(\widehat{\mathbf{y}}_{k+i-1},\widehat{\mathbf{y}}_{k+i- 2},\ldots,\mathbf{y}_{k-n_{a}+i},\mathbf{u}_{k+i-1|k-1},\mathbf{u}_{k+i-2|k-1},\ldots,\mathbf{ u}_{k-n_{b}+i})\.\] (29)
* After taking into account the estimation of unmeasured disturbances \(\mathbf{d}_{k}=\mathbf{y}_{k}-\widehat{\mathbf{y}}_{k|k-1}\), containing also influence of modeling errors, the final form of the formula describing the elements of the free response is obtained: \[\widetilde{\mathbf{y}}_{k+i\tilde{k}}=\widehat{\mathbf{y}}_{k+i}+\mathbf{d}_{k}\.\] (30)
## 6 Total Longitudinal OAM at the Focus  
Utilizing (9) and (15), we obtain the total longitudinal OAM at the field focus (1) (averaged over the beam cross section):  
\[\begin{array}{l}\hat{L}_{z,\alpha}=\int\limits_{0}^{\infty}\int\limits_{0}^ {2\pi}L_{z,\alpha}rdrd\varphi\\ =\frac{1}{1+\alpha^{2}}\int\limits_{0}^{\infty}J\int\limits_{0}^{2\pi}rdrd \varphi\Big{[}n(1+\alpha^{2})I_{0,n}^{2}+\alpha^{2}(n-2)I_{2,n-2}^{2}+(n+2)I _{2,n+2}^{2}+\\ \quad\quad+2(n+1)I_{1,n+1}^{2}+2\alpha^{2}(n-1)I_{1,n-1}^{2}\\ \quad\quad+2\alpha\cos(2\varphi)((n-1)I_{0,n}I_{2,n-2}+(n+1)I_{0,n}I_{2,n+2}-2 nI_{1,n-1}I_{1,n+1})]=\\ =nW_{0}+\frac{\alpha^{2}(n-2)}{1+\alpha^{2}}W_{2}+\frac{(n+2)}{1+\alpha^{2}}W_ {2}+\frac{2(n+1)}{1+\alpha^{2}}W_{1}+\frac{2\alpha^{2}(n-1)}{1+\alpha^{2}}W_ {1}\\ =nW_{0}+\Big{(}n+2\frac{1-\alpha^{2}}{1+\alpha^{2}}\Big{)}W_{2}+2\Big{(}n+ \frac{1-\alpha^{2}}{1+\alpha^{2}}\Big{)}W_{1}=nW+2\frac{1-\alpha^{2}}{1+ \alpha^{2}}(W_{2}+W_{1}).\end{array} \tag{19}\]  
From (10) and (19) follows the expression for the sum of the total longitudinal SAM and OAM at the field focus (1):  
\[\begin{array}{l}\hat{S}_{z,\alpha}+\hat{L}_{z,\alpha}=\left(n+\frac{1- \alpha^{2}}{1+\alpha^{2}}\right)W.\end{array} \tag{20}\]  
It follows from (18) that the total longitudinal OAM in the initial plane is equal to the expression:  
\[\begin{array}{l}\hat{L}_{z,0,\alpha}=\int\limits_{0}^{\infty}\int\limits_{0 }^{2\pi}L_{z,0,\alpha}rdrd\varphi=nW.\end{array} \tag{21}\]  
In the initial plane, it follows from (11) and (21) that the sum of the SAM and OAM is equal to the expression:  
\[\begin{array}{l}\hat{S}_{z,0,\alpha}+\hat{L}_{z,0,\alpha}=\left(n+\frac{1- \alpha^{2}}{1+\alpha^{2}}\right)W.\end{array} \tag{22}\]The total longitudinal SAM and OAM for the field (1) is seen from comparison of (20) and (22) to remain unchanged during focusing. By varying the parameter \(\alpha\) from 1 to 0, the angular momentum of the field (1) at the focus can be changed in the range:  
\[nW\leq\hat{S}_{z,\alpha}+\hat{L}_{z,\alpha}\leq(n+1)W. \tag{23}\]