esa-sceva/satcom-chunk-collection huggingface dataset:
# Cooperative Jamming-Based Physical-Layer Group Secret and Private Key Generation
## 4 Performance Analysis
### The SK/PK Upper Bound
#### 4.2.1 Upper Bound on SK Capacity  
For (24), its first term can be calculated as  
\[I\left(\tilde{Y}_{A};\tilde{Y}_{B},\tilde{Y}_{C},\tilde{Y}_{R}^ {(1)},\tilde{Y}_{R}^{(2)}\right) \tag{26}\] \[=h\left(\tilde{Y}_{A}\right)+h\left(\tilde{Y}_{B},\tilde{Y}_{C}, \tilde{Y}_{R}^{(1)},\tilde{Y}_{R}^{(2)}\right)-h\left(\tilde{Y}_{A};\tilde{Y} _{B},\tilde{Y}_{C},\tilde{Y}_{R}^{(1)},\tilde{Y}_{R}^{(2)}\right)\] \[=\log_{2}\frac{\left(p_{A}+\sigma_{1}^{2}\right)\left|\mathbf{C} _{\tilde{Y}_{B}\tilde{Y}_{C}\tilde{Y}_{R}^{(1)}\tilde{Y}_{R}^{(2)}}\right|}{ \left|\mathbf{C}_{\tilde{Y}_{A}\tilde{Y}_{B}\tilde{Y}_{C}\tilde{Y}_{R}^{(1)} \tilde{Y}_{R}^{(2)}}\right|},\]  
where the covariance matrices \(\mathbf{C}_{\tilde{Y}_{B}\tilde{Y}_{C}\tilde{Y}_{R}^{(1)}\tilde{Y}_{R}^{(2)}}\) and \(\mathbf{C}_{\tilde{Y}_{A}\tilde{Y}_{B}\tilde{Y}_{C}\tilde{Y}_{R}^{(1)}\tilde {Y}_{R}^{(2)}}\) can be given by  
\[\mathbf{C}_{\tilde{Y}_{B}\tilde{Y}_{C}\tilde{Y}_{R}^{(1)}\tilde{Y}_{R}^{(2)}} =\left[\begin{array}{cccc}p_{B}+\sigma_{1}^{2}&0&p_{B}&0\\ 0&p_{C}+\sigma_{1}^{2}&0&p_{C}\\ p_{B}&0&p_{A}+p_{B}+\sigma_{2}^{2}&p_{A}\\ 0&p_{C}&p_{A}&p_{A}+p_{C}+\sigma_{3}^{2}\end{array}\right], \tag{27}\]  
\[\mathbf{C}_{\tilde{Y}_{A}\tilde{Y}_{B}\tilde{Y}_{C}\tilde{Y}_{R}^{(1)}\tilde{Y }_{R}^{(2)}} =\left[\begin{array}{cccc}p_{A}+\sigma_{1}^{2}&0&0&p_{A}&p_{A}\\ 0&p_{B}+\sigma_{1}^{2}&0&p_{B}&0\\ 0&0&p_{C}+\sigma_{1}^{2}&0&p_{C}\\ p_{A}&p_{B}&0&p_{A}+p_{B}+\sigma_{2}^{2}&p_{A}\\ p_{A}&0&p_{C}&p_{A}&p_{A}+p_{C}+\sigma_{3}^{2}\end{array}\right], \tag{28}\]  
respectively.  
Likewise, the computation of the second term in (24) is given by  
\[I\left(\tilde{Y}_{A},\tilde{Y}_{R}^{(1)},\tilde{Y}_{R}^{(2)}; \tilde{Y}_{B},\tilde{Y}_{C}\right) =h\left(\tilde{Y}_{A},\tilde{Y}_{R}^{(1)},\tilde{Y}_{R}^{(2)} \right)+h\left(\tilde{Y}_{B},\tilde{Y}_{C}\right)-h\left(\tilde{Y}_{A},\tilde{Y }_{B},\tilde{Y}_{C},\tilde{Y}_{R}^{(1)},\tilde{Y}_{R}^{(2)}\right) \tag{29}\] \[=\log_{2}\frac{\left|\mathbf{C}_{\tilde{Y}_{A}\tilde{Y}_{R}^{(1)} \tilde{Y}_{R}^{(2)}}\right|\left|\mathbf{C}_{\tilde{Y}_{B}\tilde{Y}_{C}} \right|}{\left|\mathbf{C}_{\tilde{Y}_{A}\tilde{Y}_{B}\tilde{Y}_{C}\tilde{Y}_{R}^ {(1)}\tilde{Y}_{R}^{(2)}}\right|},\]where \(\mathbf{C}_{\widehat{Y}_{A}\widehat{Y}_{R}^{(1)}\widehat{Y}_{R}^{(2)}}\) and \(\mathbf{C}_{\widehat{Y}_{A}\widehat{Y}_{R}\widehat{Y}_{C}\widehat{Y}_{R}^{(1)} \widehat{Y}_{R}^{(2)}}\) are given in (23) and (28), respectively, and \(\mathbf{C}_{\widehat{Y}_{R}\widehat{Y}_{C}}\) can be expressed as  
\[\mathbf{C}_{\widehat{Y}_{R}\widehat{Y}_{C}}=\left[\begin{array}{cc}p_{B}+ \sigma_{1}^{2}&0\\ 0&p_{C}+\sigma_{1}^{2}\end{array}\right]. \tag{30}\]
## 3 Y-Branch Coupler
### Structure of the Y-Branch Coupler  
Figure 3 shows the structure of the Y-branch coupler, including the input part and branch part. There are two identical and parallel straight structures in the input part. Width of each straight structure is \(W\) and gap between two straight structures is \(G\). In the branch part, two straight structures merge into one and length of this part is \(L_{\text{b}}\). Height of the Y-branch coupler along the \(y\) direction is 250 nm, which is consistent with the height of the silicon nitride (SiN) core.  
To get the maximum coupling efficiency, the SMF should be aligned to the Y-branch coupler center and the two should be as close as possible. Coupling efficiency of the Y-branch coupler \(\eta_{y}\) can be calculated as follow:  
\[\eta_{y}=\eta_{y1}\cdot\eta_{y2} \tag{1}\]  
where \(\eta_{y1}\) denotes the matching degree between the mode in the SMF and mode in the input part, \(\eta_{y2}\) denotes the transmission efficiency of the branch part.
## 3 Results and Discussion
### Cost Parameters Variation  
A preliminary study of the robustness of the solution in terms of the number of iterations and the computational cost was carried out. In all conducted trials, the algorithm finds a solution to the planning problem posed, using as few as 100 iterations. It is noteworthy that the cost of the optimal trajectory obtained varies depending on the weights assigned to each parameter of the cost function. Decreasing the percentage assigned to security (from 80% to 60%), as shown in Figure 8a, highlights the difference in the cost value obtained within the same dataset, where the selected optimal trajectories may or may not vary according to the adjustment of these weights. On the other hand, the obtained results do not significantly vary based on the number of iterations employed in the search for the optimal trajectory. While increasing the number of iterations leads to a greater number of outliers, the values of the median, upper quartile, and lower quartile remain fairly constant, as illustrated in Figure 8b for the case of weights assigned corresponding to 60% in security, 20% effort, and 20% entry angle convenience.  
Figure 6: Path-planning results for the RRT algorithm at 100 iterations. Optimal trajectory (blue) calculated from the cost function detailed in Section 2.2, and the shortest path (dashed magenta) and straight line between start and goal (red): (**a**) Three-dimensional visualization. (**b**) Coronal plane. (**c**) Sagittal plane.  
Figure 7: Path-planning results for RRT algorithm at 100 iterations on each potential entry point (black asterisks). Optimal trajectory (blue) calculated from the cost function detailed in Section 2.2 and shortest path (dashed magenta): (**a**) Three-dimensional visualization. (**b**) Coronal plane. (**c**) Sagittal plane.
## 4 Results
### Association of Remote Sensing Observations with Ground Data  
After exploring and analyzing remote sensing data, on-ground characteristics, type of vehicle, and type of fuel, a correlation was suggested based on the literature survey [24] in Figure 6. The purpose of suggesting a correlation with pollutants was to develop a key to transform the results obtained from this study into surveillance policies and action. In Pakistan, motorcycles and cars use petrol, rikshaws use liquefied petroleum gas (LPG), and buses and trucks are powered by diesel, generally with little variation. As can be seen from Figure 6, the high CO was the result of faulty rikshaws and cars; similarly, high NO\({}_{2}\) was due to trucks and buses. Similarly, in Ferozpur, SO\({}_{2}\) was high, suggesting that buses must be restricted to enter the city only partially, or must be checked for maintenance. The brick kilns, which were high contributors of SO\({}_{2}\), must be converted to a better technology, such  
Figure 4: The percentage contribution of each site on the corresponding pollutant.  
as contemporary brick kilns. The comparative pollution contribution of different type of brick kilns have been explained in Figure 7[40].  
Figure 5: The top contributor of specific pollutants within various economic activities. The top contributor of (**a**) CO in Roads (**b**) CO in Brick kilns (**c**) CO in Industrial Estates (**d**) NO\({}_{2}\) in Roads (**e**) NO\({}_{2}\) in Brick kilns (**f**) NO\({}_{2}\) in Industrial Estates (**g**) SO\({}_{2}\) in Roads (**h**) SO\({}_{2}\) in Brick kilns (**i**) SO\({}_{2}\) in Industrial Estates (**j**) HCHO in Roads (**k**) HCHO in Brick kilns (**l**) HCHO in Industrial Estates (**m**) CH\({}_{4}\) in Roads (**n**) CH\({}_{4}\) in Brick kilns (**o**) CH\({}_{4}\) in Industrial Estates (**p**) AOD in Roads (**q**) AOD in Brick kilns (**r**) AOD in Industrial Estates.
## 3 Materials and Methods
### Determination of VOCs
#### 3.4.2 HS-SPME-GC-MS  
HS-SPME-GC-MS measurements were included to help with identification of compounds detected by PTR-ToF-MS. At the end of fermentation (after 14 days), samples were removed from the PTR-ToF-MS autosampler sample tray and transferred to a GC-MS autosampler sample tray held at either 25 or 35 \({}^{\circ}\)C. A SPME fibre (Divinylbenzene/Carboxen/P polydimethylsiloxane (DVB/CAR/PDMS) 2 cm, 50/30 \(\upmu\)m thickness) was exposed to the headspace of the sample for 40 min at either 25 or 35 \({}^{\circ}\)C. VOCs were desorbed from the SPME fibre at 250 \({}^{\circ}\)C for 5 min in the injector of the GC in splitless mode, and helium was used as the carrier gas at a flow rate of 2 mL/min. Volatiles were separated using a capillary column (InnoWax 30 m/0.32 mm/0.5 \(\upmu\)m). The oven temperature program was set 40 \({}^{\circ}\)C held for 1 min, and then increased to 250 \({}^{\circ}\)C at 5 \({}^{\circ}\)C/min and held for 2 min. MS was performed with an ion source temperature of 200 \({}^{\circ}\)C and an electron ionisation energy of 70 eV over the mass range of \(m/z\) 33-350.  
According to the retention time (RT) of n-alkane series (C7-C30) obtained under the same conditions, retention indices (RI) of the detected VOCs were computed. By comparing RI calculated and from the NIST library (NIST14, version 2.2, National Institute of Standards and Technology), the VOCs were identified.
## 3 Spatio-Temporal Distribution Characteristics of PMGIM
### PMGIM Distribution Law of 182602 Working Face
#### 3.3.3 Characteristics of Agglomeration Law  
Prior to mining in the working face, PMGIM dynamics adhere to three major patterns: sporadic, clustered, and staged. Sporadic microseismic activity suggests a non-uniform distribution closely related to regional stress variations. For instance, the initiation and cessation of grouting activities significantly impact microseismic frequency, facilitating real-time monitoring of grouting effects. Clustered microseismic events are concentrated in geologically fragile areas, such as fracture zones, indicating tectonic stress anomalies and potential water conductivity risks. The staged patterns of microseismic energy release progress through three phases: \"low frequency low energy-high frequency low energy-high frequency high energy\", consistent with the grouting process. Those patterns aid in predicting dangerous periods in the working face, particularly during the \"high frequency high energy\" stage, when vigilance for water gushing risks is imperative.
## 7 Feature Extraction Approaches
### Subspace Learning-Based Approaches
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Year**} & \multirow{2}{*}{**Paper**} & \multirow{2}{*}{**Method**} & \multicolumn{3}{c}{**Used Datasets**} & \multirow{2}{*}{**Evaluation Protocol**} & \multirow{2}{*}{**Acc. (\%)**} \\ \cline{5-6}  & & & **Name** & & **\#IDs.** & & **\#Imgs.** & \\ \hline
2003 & [PERSON] et al. [50] & Fisherpalms & Private & 300 & 3000 &
\begin{tabular}{c} 6 Imgs/sub (Rand) for training, \\ remaining 4 Imgs/sub for testing \\ \end{tabular} & 99.75 \\ \hline
2005 & [PERSON] et al. [2] & WFDA & Private & 150 & 900 &
\begin{tabular}{c} 3 Imgs/sub for training, \\ remaining 3 Imgs/sub for testing \\ \end{tabular} & 98.64 \\ \hline
2007 & [PERSON] et al. [51] & 2 DLDPP & PolyU & 100 & 600 &
\begin{tabular}{c} First session for training, second \\ session for testing \\ \end{tabular} & 84.67 \\ \hline
2008 & Pan and & I2 DLPPG & Private-1 & 40 & 400 & \
\begin{tabular}{c} 5 Imgs/sub for training (Rand), \\ remaining for testing \\ \end{tabular} & 99.50 \\ \cline{2-6}  & [PERSON] [52] & Private-2 & 346 & 1730 &
\begin{tabular}{c} 5-fold cross-validation \\ \end{tabular} & 95.77 \\ \hline
2011 & [PERSON] and [PERSON] [53] & W2D\(-\)DLPP & PolyU & 100 & 600 &
\begin{tabular}{c} 4 Imgs/sub for training (Rand, \\ 20 iterations), remaining for \\ testing \\ \end{tabular} & 94.90 \\ \hline
2018 & [PERSON] et al. [54] & SR & \
\begin{tabular}{c} PolyU-MS \\ \end{tabular} & 500 & 6000 &
\begin{tabular}{c} 4 Imgs/sub for training, \\ remaining for testing (Rand, \\ 10 iterations) \\ \end{tabular} & 99.24 \\ \hline
2019 & [PERSON] et al. [55] & RSM & \begin{tabular}{c} PolyU \\ \end{tabular} & 374 & 3740 & \begin{tabular}{c} 99.96 \\ remaining for testing \\ \end{tabular} \\ \cline{2-6}  & \begin{tabular}{c} [PERSON] et al. \\ \end{tabular} & RSM & \begin{tabular}{c} PolyU-MS \\ \end{tabular} & 500 & 6000 & \begin{tabular}{c} First 4 Imgs/sub for training, \\ remaining for testing \\ \end{tabular} & 99.15 \\ \cline{2-6}  & \
\begin{tabular}{c} Private \\ \end{tabular} & 400 & 8000 &
\begin{tabular}{c} 94.50 \\ \end{tabular} \\ \hline
2021 & [PERSON] et al. [56] & SF2 DDLPP & PolyU & 100 & 600 &
\begin{tabular}{c} 50\% for training (Rand), 50\% for testing \\ \end{tabular} & 95.18 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: A comparative overview of subspace learning-based approaches (#ID: number of identities, #Imgs: number of images, Acc: valid accuracy).
## 3 GaN HEMT Device on MMIC Die
### Device Description's Model  
For the thermal analysis of a multi-fingers AlGaN/GaN HEMT die, Figure 4 shows the dimension geometry, with its geometrical parameters listed in Table 2[40]. The epitaxial layers were grown by MOCVD on the semi insulating SiC substrates. In its vertical direction, the substrate is a silicium carbide SiC of 350 \(\upmu\)m in thickness, while the die attach layer is a Sn-3.5 Ag of 50 \(\upmu\)m in thickness. The thickness of metallization layout (Au), AlGaN (-23% of Aluminum), and the GaN, are selected to be 2, 0.03, and 2 \(\upmu\)m, respectively. The die is attached to a Cu-tungsten micro channel heat-sink for efficient thermal management. The device is passivated by SiO2 layer, which is omitted in thermal simulations.  
Figure 4: 3D cross section view of the AlGaN/GaN HEMT die.  
Figure 3: Model described of a GaN MMIC chip. (**a**) Schematic diagram of a 12.5 W AlGaN/GaN MMIC die; (**b**) 1.56 W power amplifier HEMT consisting of 8 gate fingers structure; (**c**) 50 W SSPA (4 \(\times\) 12.5 W MMIC die). * is mentioned in the (a) showing one chip. In the (C), 4 chips of (a) is presented.  
In GaN devices, the total heat generated is equal to \(I_{dS}V_{dS}\), which is the power in the device, leading to large quantities of heat generation.  
During RF operation under high-voltage, the current flows from the drain fingers to the source fingers and is modulated by the voltages applied to the gate fingers. Therefore, localized self-heating increases the channel temperature and happens in a region near the gate contact inside the most majority multi-finger HEMT structure.
## 2 Theoretical Background
### Limitations of Existing Approaches  
Broadly speaking, there are two major shortcomings of existing quantitative multivariate approaches that are apparent in technology adoption research. The first is caused by the use of existing and pretested measurement scales that rarely take into account extreme events, many of which are hard to detect and predict by definition. Such measurement approaches regularly force the full range of opinions into a standardized scheme, often by using Likert-type items, in which (dis)agreements are frequently expressed on five-, six-, or seven-point scales. Furthermore, when it comes to statistical analysis, outliers often pose a nuisance that distort the main findings. They can either be eliminated (i.e., ignored) or avoided by using robust statistics. Moreover, the amount of complexity that can be simultaneously accounted for in a quantitative and theory-testing manner is limited, which is especially true in case a whole model is evaluated as such (\"model fit\").  
For example, the highly popular TAM unfolds its full tautological power in cases where decision-makers are asked about their perceived usability, usefulness, or intention to accept a technology in a cross-sectional survey. Quite obviously, those who perceive a technology to be useful and easy to use are likely to accept it (and vice versa), leading to those significant statistical relationships academic researchers crave. As of the time of writing this paper, I am not aware of any single study that has plausibly demonstrated an insignificant or even negative relationship between the two constructs of \"ease of use\" and \"usefulness\" and the intention to accept a technology. However, those outliers, extreme cases, and unlikely events that are never supposed to happen are swept under the carpet by this approach. Alas, they do happen, and they frequently lead to market turmoil, which can cause harm to individuals, organizations, and even nation states. [PERSON] labels such extremely rare events as black swans and concludes that they are potentially unpredictable [48]. In the case of Bitcoin, numerous such events can be identified in the past, including a high volatility caused by events as diverse as [PERSON]'s Twitter posts [49], China's crackdown on cryptocurrencies in general [50], or unforeseeable market actionsof major Bitcoin investors, who are aptly titled \"whales\" [51]. [PERSON] suggests that we deal with such events through the creation of robustness, but scenario technique (which is used synonymously with scenario analysis in this paper) might be a proper tool to identify such rare occurrences since outliers are considered seriously and independent of majority mainstream opinions. Furthermore, it can be used to create bundles of factors and further develop them into elaborate descriptions of potential future states. In this regard, previous research has shown that the simultaneous consideration of the effects of numerous contingency factors can deliver some important insights into organizational behavior and performance from different managerial angles [52].
# Numerical Study of Endwall Modification with Micro-Scale Ribs in a Turbine Cascade
## 3 Results and Discussions
### Heat Transfer Characteristic
From a comprehensive perspective, the tilted and vertical rib showed an obvious advantage on film cooling with little difference in the heat transfer characteristic (less than 1%), among which the vertical rib endwall with an s value of 8 mm presented the best increment (26.9%) in \(\eta\). The horizontal rib exhibited not only a decrease in \(\eta\) but also an increase in \(h\), which indicated that the rib layout should consider the flow direction of the mainstream. In addition, the heat transfer of the tilted rib presented a slight increase with 0.9% and 2.9%.
# A Back-to-Back Gap Waveguide-Based Packaging Structure for E-Band Radio Frequency Front-End
## 2 Unit Design of the Packaging Structure
### Hybrid Coupler and Bandpass Filter Design
It can be seen that when \(w\) is a constant and \(l\) increases, the resonant frequency of the resonator decreases accordingly. In addition, due to the fact that the sidewalls of the resonant cavity created by the EBG can be equivalent to ideal electrical walls, it has properties similar to those of rectangular waveguide resonators. Therefore, the initial value of its resonant frequency  
Figure 6: Simulated results of the designed coupler: (**a**) power distribution unbalance and output-port phase difference; (**b**) return loss and isolation levels. Figure 7: Structure and dimensions of the BPF (number 1–5 represent five resonators). can also be estimated according to Formula (4), and then the resonant frequency can be accurately designed by adjusting the dimensions. \[f_{101}=\frac{c}{2}\sqrt{{(\frac{1}{w})}^{2}+{(\frac{1}{l})}^{2}} \tag{4}\]  
Additionally, based on the model, and with copper used as the metal boundary for analysis, the simulated unloaded quality factor of the resonator is around 1800. When considering a passband range of 71-76 GHz/81-86 GHz with a return loss of \(-18\) dB, the coupling coefficients are as follows: \(m_{\mathrm{S1}}=0.066/0.058\), \(m_{\mathrm{12}}=m_{\mathrm{45}}=0.0562/0.05\), \(m_{\mathrm{23}}=m_{\mathrm{34}}=0.0421/0.037\), and \(Q_{\mathrm{ex}}=15.63/17.76\) (where \(m_{\mathrm{ij}}\) denotes the coupling coefficients between the \(i\)th and \(j\)th resonators) [23]. The external quality factor (\(Q_{\mathrm{ex}}\)) can be adjusted by varying the heights of the ridges used for achieving electromagnetic coupling between adjacent resonators (as shown in Figure 9), and the coupling coefficient can be adjusted by changing the heights of the ridges. In addition, as the heights of the input/output-coupled ridges increase, the value of \(Q_{\mathrm{ex}}\) also increases. After optimizing the coupling structures, the final dimensions and simulated results are shown in Table 1 and Figure 10, in which both bandpass filters exhibit insertion losses below 0.4 dB and return losses exceeding \(-18\) dB while demonstrating excellent out-of-band rejection. Figure 8: Structure of the resonator and the resonant frequency versus the resonator length. Figure 9: The effect of ridge height on the coupling coefficient and external quality factor: (**a**) effect of \(h\)t on the coupling coefficient; (**b**) effect of \(h\)i on the external quality factor.
## 4 Fractional-Order Controller and Fractional-Order Disturbance Observer Design
### Longitudinal and Latitudinal Controllers Design  
Define the state variables \(x_{j},\ j=3,4,5,6\), as \(x_{3}=\left[x\ y\right]^{T}\), \(x_{4}=\left[\dot{x}\ \dot{y}\right]^{T}\), \(x_{5}=\left[\dot{\varphi}\ \dot{\theta}\right]^{T}\), and \(x_{6}=\left[\dot{\varphi}\ \dot{\theta}\right]^{T}\) from x, y, roll, and pitch axes. The state space model can be expressed as follows:  
\[\dot{x}_{3}=x_{4}, \tag{50}\] \[\dot{x}_{4}=R_{4}(\dot{\varphi},u_{1})T_{4}(x_{5})+f_{44},\] (51) \[\dot{x}_{5}=x_{6}\] \[\dot{x}_{6}=f_{6}+g_{6}u_{6}+\tau_{d6},\]  
where \(R_{4}(\dot{\varphi},u_{1})=\frac{1}{\pi} \begin{bmatrix}\cos\psi&\sin\psi\\ \sin\psi&-\cos\psi\end{bmatrix} u_{1}\), \(T_{4}(x_{5})=\left[\cos\phi\sin\theta\sin\phi\right]^{T}\), \(f_{44}=\left[f_{dx}\ f_{dy}\right]^{T}\), \(g_{6}=\)diag\(\left(1/J_{x}\,\ l/J_{y}\right)\), \(u_{6}=\left[u_{\phi}\ \dot{u}_{\theta}\right]^{T}\), and \(f_{6}=\left[(J_{y}-J_{z})\dot{\theta}\dot{\psi}/J_{x}+J_{r}\dot{\theta}\Omega _{r}/J_{x}\ \ \ \left(J_{z}-J_{x}\right)\dot{\psi}\dot{\phi}/J_{y}\right.\)\(\left.-J_{r}\dot{\phi}\dot{\Omega}_{r}/J_{y}\right]^{T}\), \(\tau_{d6}=\left[\tau_{d\phi}\ \tau_{d6}\right]^{T}\). The tracking errors are defined as \(e_{3}=x_{3}-x_{3d}\), \(e_{4}=x_{4}-\zeta_{31}\), \(e_{5}=x_{5}-x_{5d}\), \(e_{6}=x_{6}-\zeta_{51}\). We introduce the fractional-order filtered error vectors as  
\[\sigma_{j}=D^{\gamma_{1}}e_{j}+D^{\gamma_{2}-1}k_{j}^{2}(b_{j1}e_{j}+b_{2}sig^ {1/2}(\varepsilon_{j}))-\xi_{j\mu}i=1,2j=3,4,5,6, \tag{52}\]  
where \(k_{j}>0\), \(b_{j1}>0\), and \(b_{j2}>0\), \(j=3,4,5,6\), are constants. In the sliding mode surfaces, the compensating signal dynamics \(\xi_{ji}\), \(j=3,4,5,6\), \(i=1,2\), are expressed as  
\[D^{\overline{7}_{1}}\xi_{j1}=-\beta_{j1}\xi_{j1}+\xi_{j2}+\zeta_{j1}-\alpha_{k} -\lambda_{j1}sig^{1/2}(\xi_{j1}), \tag{53}\]  
\[D^{\overline{7}_{1}}\xi_{j2}=-\beta_{j2}\xi_{j2}-\xi_{j1}-\lambda_{j2}sig^{1/2} (\xi_{j2}),\ j=3,4,5,6, \tag{54}\]  
where \(\beta_{j1}>0\), \(\beta_{j2}>0\), \(\lambda_{j1}>0\), and \(\lambda_{j2}>0\), \(j=3,4,5,6\), are constants. Furthermore, the output of the tracking differentiator \(\zeta_{j1}\) is obtained from the following tracking differentiator described in Theorem 4:  
\[ \begin{array}{c}D^{\gamma_{1}}\xi_{k1}=\zeta_{k2},\\ D^{\gamma_{1}}\xi_{k2}=-\Lambda_{k}^{2}[\zeta_{k1}\sinh(r_{k1}\varphi_{k1}(\zeta_{ k1}-\alpha_{k}))+c_{k2}\sinh(r_{k2}\varphi_{k2}(\zeta_{k2})/\Lambda_{k})],k=3,5, \end{array}  \tag{55}\]  
where \(\varphi_{k1}(\xi_{k1}-\alpha_{k})=\epsilon_{k}sig^{1/2}(\zeta_{k1}-\alpha_{k})+(1- \epsilon_{k})sig^{(2+\
u_{k})/2}(\zeta_{k1}-\alpha_{k})\), \(\varphi_{k2}(\zeta_{k2})=sig^{1/2}(\zeta_{k2})\), \(0<\epsilon_{k}<1\), and \(0<\
u_{k}<1\), \(k=3,5\), are constants.