esa-sceva/satcom-chunk-collection huggingface dataset:
# Predictive Modeling of VO2 max Based on 20 m Shuttle Run Test for Young Healthy People
## 2 Material and Methods
### Field Assessment of VO2 max
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variable** & **Description** & **Female (_n_ = 154)**\(\bar{x}\pm sd\) & **Male (_n_ = 154)**\(\bar{x}\pm sd\) & **All (_n_ = 308)**\(\bar{x}\pm sd\) \\ \hline \(y\) & VO2 max [mL\(\cdot\)kg\({}^{-1}\).min\({}^{-1}\)] & \(44.9\pm 7.0\) & \(54.1\pm 8.1\) & \(49.5\pm 8.9\) \\ \(x_{0}\) & Gender & — & — & \(\in\{0,1\}\) * \\ \(x_{1}\) & Distance [m] & \(946.5\pm 351.9\) & \(1556.4\pm 419.7\) & \(1251.4\pm 492.8\) \\ \(x_{2}\) & HRmax [ppm] & \(189.5\pm 8.5\) & \(194.8\pm 8.4\) & \(192.1\pm 8.9\) \\ \(x_{3}\) & HRR1 [ppm] & \(159.9\pm 18.9\) & \(163.8\pm 12.9\) & \(161.9\pm 16.3\) \\ \(x_{4}\) & HRR4 [ppm] & \(125.7\pm 17.0\) & \(125.7\pm 13.5\) & \(125.7\pm 15.3\) \\ \(x_{5}\) & Age [years] & \(20.3\pm 1.2\) & \(20.8\pm 1.8\) & \(20.6\pm 1.6\) \\ \(x_{6}\) & Body weight [kg] & \(58.9\pm 9.8\) & \(75.5\pm 11.9\) & \(67.2\pm 13.7\) \\ \(x_{7}\) & Body height [cm] & \(165.0\pm 5.7\) & \(177.8\pm 6.8\) & \(171.4\pm 8.9\) \\ \(x_{8}\) & Waist\({}_{1}\) cir. [cm] & \(68.9\pm 6.8\) & \(80.7\pm 8.1\) & \(74.8\pm 9.5\) \\ \(x_{9}\) & Waist\({}_{2}\) cir. [cm] & \(75.1\pm 8.1\) & \(85.1\pm 9.3\) & \(80.1\pm 101.1\) \\ \(x_{10}\) & Hip cir. [cm] & \(95.7\pm 7.5\) & \(98.3\pm 7.5\) & \(97.0\pm 7.6\) \\ \(x_{11}\) & WHR1 & \(0.42\pm 0.04\) & \(0.45\pm 0.05\) & \(0.44\pm 0.05\) \\ \(x_{12}\) & WHRR2 & \(0.46\pm 0.05\) & \(0.48\pm 0.05\) & \(0.47\pm 0.05\) \\ \(x_{13}\) & WHR1 & \(0.72\pm 0.04\) & \(0.82\pm 0.04\) & \(0.77\pm 0.05\) \\ \(x_{14}\) & WHR2 & \(0.78\pm 0.05\) & \(0.86\pm 0.05\) & \(0.82\pm 0.06\) \\ \(x_{15}\) & BMI & \(21.6\pm 3.2\) & \(23.9\pm 3.7\) & \(22.7\pm 3.6\) \\ \(x_{16}\) & FMI & \(5.2\pm 2.3\) & \(3.9\pm 2.1\) & \(4.6\pm 2.3\) \\ \(x_{17}\) & FFMI & \(16.4\pm 1.2\) & \(20.0\pm 1.9\) & \(18.2\pm 2.4\) \\ \(x_{18}\) & BAI & \(27.2\pm 3.5\) & \(23.5\pm 3.6\) & \(25.4\pm 4.0\) \\ \(x_{19}\) & BSA & \(1.64\pm 0.14\) & \(1.93\pm 0.15\) & \(1.78\pm 0.20\) \\ \(x_{20}\) & Fat [\%] & \(23.2\pm 7.0\) & \(15.4\pm 5.9\) & \(19.3\pm 7.6\) \\ \(x_{21}\) & FFM [\%] & \(76.7\pm 7.0\) & \(84.6\pm 5.9\) & \(80.6\pm 7.6\) \\ \(x_{22}\) & TBW [\%] & \(56.1\pm 5.2\) & \(61.9\pm 4.3\) & \(59.0\pm 5.6\) \\ \hline \hline \end{tabular} HR—maximal heart rate, HRR—recovery heart rate, WHtR—waist to height ratio, WHR—waist-hip ratio, BMI—body mass index, FMI—fat mass index, FMI—fat-free mass index, BAI—body adiposity index, BSA—body surface area, Fat—body fat percentage, TFM—body fat-free percentage, TBU—total body water.  
\end{table} 
Table 2: Description of the variables used to construct the models. -volitional exhaustion; attainment of at least 90% of the age predicted HR\({}_{max}\) (220 beats/min minus the age of the subject in years); respiratory exchange ratio (RER) equal to or greater than 1.10; and VO\({}_{2}\) leveled off even with an increase in intensity [41; 42].
## 2 Materials and Methods
### Shear Stiffness  
Shear stiffness is the ratio of shear stress to the corresponding shear displacement prior to reaching the peak shear strength. Figure 3a shows the calculation method of the shear stiffness, and the calculation formula is [56; 57]:  
\[k_{\text{s}}=\tfrac{\Delta\tau}{\Delta u} \tag{1}\]  
where \(k_{\text{s}}\) is the shear stiffness of the fault in GPa; \(\Delta\tau\) is the increment of shear stress in MPa; and \(\Delta u\) is the increment of the corresponding shear displacement in mm.  
The shear stiffness under various stresses is different before the shear strength peaks. Therefore, it is necessary to find the elastic loading stage, that is, the linear segment of the loading curve. The shear stiffness is calculated according to the curve gradient at this stage. A program was developed to calculate the gradient of the elastic loading curve. The main idea is to start from the first data point and select one-fifth of the total number of data points for grouping. Each group of data adopts the least square method to fit the straight line. All fitted straight lines were subjected to error analysis. The gradient of straight line with the smallest error is selected as the gradient of the elastic loading curve, and this gradient corresponds to the fault shear stiffness.  
Figure 3: Shear stiffness calculation. (**a**) Typical stress-related displacement [57]. (**b**) Loading curve of direct shear test. The red line is the best-fitting line with the smallest error for the loading curve, which is fitted by the curve in the shade.  
As shown in Figure 3b, the shear test was carried out with a shear loading rate of 0.05 mm/s under constant normal stress of 5 MPa, and the surface was roughened with #60 sandpaper [29; 54]. The gradient of the fitted straight line with the smallest error is 6.53 in this test, so the shear stiffness of the fault is taken as 6.53 GPa m\({}^{-1}\).
## 1 Introduction
A fine fiber morphology was achieved by increasing DMF concentration in the solvent mixture. The HIPS membrane demonstrated excellent performance as an air filtration medium, with a filtration efficiency up to 98.5%, as shown in filtration tests with PM\({}_{2.5}\) particles [18]. [PERSON] et al. elaborated micro-porous membranes from recycled HIPS, which presented contact angle values between 81\({}^{\circ}\) and 91\({}^{\circ}\), with surface pore sizes ranging from 150 nm to 550 nm and with macroviods in their cross-sections [19]. These membranes presented Rhodamine B (RhB) rejection values between 95 and 97% and from 85 to 87% for humic acid (HA) [20]. They exhibited a slight increase in tensile strength values as HIPS concentration increased from 20 to 35% [15]. Waste HIPS's feasibility as membranes for gas separation was also investigated [21]. The authors reported that HIPS membranes exhibited better gas transport properties than EPS and GPPS, due to their butadiene rubber content in the matrix, with a gas separation factor that was well correlated with the difference in the gas penetrant kinetic diameter. Recently, a method for the total revalorization of HIPS was investigated by [PERSON] et al. These authors employed HIPS waste as potential feedstock for styrene monomer recovery using a green fractionation process using ethyl acetate (EIOAc). This strategy consists of EIOAc addition as a pre-fractionation solvent to isolate the free PS matrix from the rubber phase, followed by thermal degradation of the isolated free PS matrix, which leads to an enhanced styrene yield, compared to the non-fractionated HIPS sample, and finally, an ethylene metathesis for the valorization of the rubber phase [22]. Here, we present the development of asymmetric sulfonated HIPS membranes from recycled HIPS (R-HIPS) for water treatment. The incorporation of sulfonic pendant groups in R-HIPS by a simple and efficient electrophilic substitution reaction was directed to enhance the hydrophilicity, morphology, permeability, and antifouling properties of HIPS membranes [23; 24]. R-HIPS sulfonated asymmetric membranes' general properties and performance in removing salts and dyes under low-pressure conditions and the effect of R-HIPS sulfonation degree on membrane fouling are discussed. Overall, this study has the potential to make a significant impact on both reducing plastic pollution and improving wastewater treatment systems through the use of recycled HIPS. With further research focused on the scalability as spiral-wound filtration modules and addressing potential challenges, sulfonated high-impact polystyrene membranes could emerge as a cost-effective and sustainable alternative for polystyrene recycling and wastewater treatment.
## 3 Results and Discussion
### Evaluation of Similarities Among Classification Methods
#### 3.3.2 Similarity Evaluation Based on Qu  
Based on Qu, the overall accuracy with NB was 47.98%, and the kappa coefficient was 0.31, indicating the highest similarity among the compared classification methods. The reclassification accuracy of the SS with NB was 36.8%, which was the highest among the compared classification methods. Additionally, the average value of reclassification accuracy for the same grade between the different classification methods was the highest at 48.4% between Qu and NB. Notably, both EI and GI reclassified the PSS from Qu as a PUS(Table 6). This is likely because EI and GI have wider weighted score ranges for the PUS than for Qu. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Classification**} & \multirow{2}{*}{**Grade**} & \multicolumn{5}{c}{**Qu**} & \multicolumn{2}{c}{**Overall**} & \multirow{2}{*}{**Kappa**} \\ \cline{3-3} \cline{5-8}  & & **SS \({}^{\gamma}\)** & **PSS** & **PUS** & **US** & **Total** & **Accuracy (\%)** & **Coefficient** \\ \hline \multirow{4}{*}{NB \({}^{2}\)} & SS & 166,718 & 0 & 0 & 0 & 166,718 & \multirow{4}{*}{47.98} & \multirow{4}{*}{0.31} \\  & PSS & 286,376 & 127,084 & 0 & 0 & 413,460 & \multirow{4}{*}{0.31} \\  & PUS & 0 & 286,818 & 356,212 & 213,096 & 856,126 & \multirow{4}{*}{0.38} \\  & US & 0 & 0 & 0 & 75,110 & 75,110 & \\  & Total & 453,094 & 413,902 & 356,212 & 288,206 & 1,511,414 & \multirow{4}{*}{0.31} \\  & (36.8) x & (30.7) & (100.0) & (26.1) & (48.4) w & & \\ \hline \multirow{4}{*}{EI} & SS & 5872 & 0 & 0 & 0 & 5872 & \multirow{4}{*}{0.21} \\  & PSS & 447,222 & 0 & 0 & 0 & 447,222 & \multirow{4}{*}{39.59} \\  & PUS & 0 & 413,902 & 356,212 & 266,691 & 1,036,805 & \multirow{4}{*}{0.21} \\  & US & 0 & 0 & 0 & 21,515 & 21,515 & \\ \cline{1-1}  & Total & 453,094 & 413,902 & 356,212 & 288,206 & 1,511,414 & \multirow{4}{*}{0.31} \\  & (36.8) x & (30.7) & (100.0) & (26.1) & (48.4) w & & \\ \hline \multirow{4}{*}{GI} & SS & 5872 & 0 & 0 & 0 & 5872 & \multirow{4}{*}{0.21} \\  & PSS & 447,222 & 0 & 0 & 0 & 447,222 & \multirow{4}{*}{39.59} \\  & PUS & 0 & 413,902 & 356,212 & 266,691 & 1,036,805 & \multirow{4}{*}{0.21} \\  & US & 0 & 0 & 0 & 21,515 & 21,515 & \\ \cline{1-1}  & Total & 453,094 & 413,902 & 356,212 & 288,206 & 1,511,414 & \multirow{4}{*}{0.31} \\  & (1.3) & (0.0) & (100.0) & (7.5) & (27.2) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Overall accuracy and kappa coefficient for the reclassification methods based on Qu using pixel counts.
# A New Real-Time Detection and Tracking Method in Videos for Small Target Traffic Signs
## 3 Results
### Effectiveness Analyses
#### 3.2.2 Comparisons of YOLOv3 Detection Results before and after Improvements  
The precision, recall, and speed of YOLOv3 and Improved YOLOv3 are shown in Table 2. An additional output layer is added, resulting in a slight increase in computing cost, but the detection results, which are made up of precision and recall, have the greatest impact on sensitivity. Besides, the loss curve and P-R curve of YOLOv3 and improved YOLOv3 are shown in Figure 6.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Data Augmentation** & **TP** & **FP** & **FN** & **P** & **R** \\ \hline - & 1524 & 586 & 461 & 0.72 & 0.77 \\ Image corruption & 2474 & 591 & 592 & 0.81 & 0.81 \\ \hline (P: Precision, R: Recall, “-” Means No Data Augmentation is Used). & & & & \\ \end{tabular}
\end{table}
Table 1: Test Results of YOLOv3 on TT100K after Applying Data Augmentation Technology.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Detection Algorithm** & **P** & **R** & **mAP** & **Speed** & **FPS** \\ \hline YOLOv3 & 0.81 & 0.81 & 0.704 & 0.0269s & 37.94 \\ Ours & 0.91 & 0.90 & 0.8476 & 0.0323s & 24.22 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparisons of YOLOv3 Detection Results before and after Improvements ([EMAIL_ADDRESS]).  
In order to further verify the efficiency of the model for small targets and to ensure the detection ability of other scale targets, the Microsoft COCO benchmark [44] evaluation metrics are adopted. According to the size of traffic signs, Ref. [44] attributes the pixel value of object length and width between 0 and 32 to small objects, the pixel value of target length and width between 32 and 96 to medium objects, and the pixel value of target length and width between 96 and 400 to large objects. YOLOv3 and the improved YOLOv3 are tested on the TT100K, which has performed data augmentation. As suggested in Table 3, meanwhile, the method enhances the detection ability of YOLOv3 for small targets and slightly enhances the detection effect for medium targets and large targets.
# Automatic Recognition of Mexican Sign Language Using a Depth Camera and Recurrent Neural Networks
## 5 Discussion  
After comparing the three most common architectures for sequence classification, we found that the architecture based on LSTM performed the best when the inputs were noisy. On the other hand, GRU performed better when the inputs were not noisy. We also demonstrate that training with Gaussian noise augmentation to the inputs makes the system more robust and can generalize better.  
Figure 8: Precision recall curves for features combinations.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Features**} & \multicolumn{5}{c}{**Testing Noise**} \\ \cline{2-7}  & **No-Noise** & **10 cm** & **20 cm** & **30 cm** & **40 cm** & **50 cm** \\ \hline All features & **96.44** & 64.44 & 63.55 & 61.77 & 63.55 & 62.44 \\ Hands-only & 96.0 & **69.11** & **69.33** & **68.66** & **68.44** & 66.0 \\ Face-only & 3.55 & 5.11 & 5.55 & 4.22 & 5.77 & 4.0 \\ Body-only & 71.55 & 8.66 & 10.44 & 10.22 & 10.66 & 10.0 \\ Face + Body & 63.55 & 12.88 & 12.44 & 12.88 & 10.88 & 13.77 \\ Hands + Face & 96.22 & 56.88 & 58.44 & 58.0 & 58.22 & 58.88 \\ Hands + Body & 92.0 & 65.55 & 68.22 & 65.33 & 66.22 & **67.33** \\ \hline \hline \end{tabular}
\end{table}
Table 8: Model accuracy when varying the input features. Every row reports the results of a model training with a specific set of features. For example, the row names _All features_ used hands, body, and face keypoints. The best result on each column is highlighted in bold.  
One ablation study demonstrated that a more extensive network was not better, and the optimal size was a 2-layer network with noise augmentation and dropout.  
The second ablation study showed that the hands' keypoints are the most important features for classification with a 96% accuracy, followed by body features with 72% accuracy. Facial keypoints alone are not good features, with an accuracy of 4%; however, when combined with the hand keypoints, the accuracy increased to 96.2%. The best model was the one mixing the three sets of keypoints with an accuracy of 96.44%, proving our hypothesis that the facial and body features play a role on the sign language recognition. This second ablation study also showed that adding Gaussian noise to the facial keypoints affects model accuracy. The hands-only model had an accuracy of 69%, and a model using hands plus face had a lower accuracy of 57%.
# Robust Transceiver Design for IRS-Assisted Cascaded MIMO Communication Systems
## 3 Achievable Rate (Full-Duplex Relay)
### Algorithm Description
We have

\[\mathbf{H}_{R\uparrow}\mathbf{\Theta}\mathbf{H}_{IR}= \left(\sum_{r_{1}=1}^{K_{r}}\sum_{r_{2}=1}^{M}h_{RIr_{1},r_{2}} \left|r_{1}\right\rangle\!\!\left\langle r_{2}\right|\right)\left(\sum_{r_{3} =1}^{M}\theta_{r_{3},r_{3}}\left|r_{3}\right\rangle\!\!\left\langle r_{3} \right|\right)\left(\sum_{r_{4}=1}^{M}\!\!\sum_{r_{5}=1}^{K_{t}}h_{IRr_{4},r_ {5}}\left|r_{4}\right\rangle\!\!\left\langle r_{5}\right|\right) \tag{17}\]3 \[\mathbf{H}_{R\uparrow}\mathbf{\Theta}\mathbf{H}_{IR}= \left(\sum_{r_{1}=1}^{K_{r}}\sum_{r_{2}=1}^{M}h_{RIr_{1},r_{2}} \left|r_{1}\right\rangle\!\!\left\langle r_{2}\right|\right)\left(\sum_{r_{3} =1}^{M}\theta_{r_{3},r_{3}}\left|r_{3}\right\rangle\!\!\left\langle r_{3} \right|\right)\left(\sum_{r_{4}=1}^{M}\!\!\sum_{r_{5}=1}^{K_{t}}h_{IRr_{4},r_ {5}}\left|r_{4}\right\rangle\!\!\left\langle r_{5}\right|\right) \tag{17}\]4 (102) \[\mathbf{H}_{R\uparrow}\mathbf{\Theta}\mathbf{H}_{IR}= \left(\sum_{r_{1}=1}^{K_{r}}\sum_{r_{2}=1}^{M}h_{RIr_{1},r_{2}} \left|r_{1}\right\rangle\!\!\left\langle r_{2}\right|\right)\left(\sum_{r_{3} =1}^{M}\theta_{r_{3},r_{3}}\left|r_{3}\right\rangle\!\!\left\langle r_{3} \right|\right)\left(\sum_{r_{4}=1}^{M}\!\!\sum_{r_{5}=1}^{K_{t}}h_{IRr_{4},r_ {5}}\left|r_{4}\right\rangle\!\!\left\langle r_{5}\right|\right) \tag{17}\]5

where

\[\mathbf{H}_{R\uparrow}\mathbf{\Theta}\mathbf{H}_{IR}= \left(\sum_{r_{1}=1}^{K_{r}}\sum_{r_{2}=1}^{M}h_{RIr_{1},r_{2}} \left|r_{1}\right\rangle\!\!\left\langle r_{2}\right|\right)\left(\sum_{r_{3} =1}^{M}\theta_{r_{3},r_{3}}\left|r_{3}\right\rangle\!\!\left\langle r_{3} \right|\right)\left(\sum_{r_{4}=1}^{M}\!\!\sum_{r_{5}=1}^{K_{t}}h_{IRr_{4},r_ {5}}\left|r_{4}\right\rangle\!\!\left\langle r_{5}\right|\right) \tag{17}\]6

and

\[\mathbf{H}_{R\uparrow}\mathbf{\Theta}\mathbf{H}_{IR}= \left(\sum_{r_{1}=1}^{K_{r}}\sum_{r_{2}=1}^{M}h_{RIr_{1},r_{2}} \left|r_{1}\right\rangle\!\!\left\langle r_{2}\right|\right)\left(\sum_{r_{3} =1}^{M}\theta_{r_{3},r_{3}}\left|r_{3}\right\rangle\!\!\left\langle r_{3} \right|\right)\left(\sum_{r_{4}=1}^{M}\!\!\sum_{r_{5}=1}^{K_{t}}h_{IRr_{4},r_ {5}}\left|r_{4}\right\rangle\!\!\left\langle r_{5}\right|\right) \tag{17}\]7

respectively. Next, we show \(g(P_{r})=R_{\text{sr}}^{\text{FD}}(P_{r})-R_{\text{rd}}^{\text{FD}}(P_{r})\) is a monotonically decreasing function of \(P_{r}\). It is sufficient to show \(\frac{dR_{\text{rd}}^{\text{FD}}}{dP_{r}}\leq 0\) and \(\frac{dR_{\text{rd}}^{\text{FD}}}{dP_{r}}>0\).
## 5 Fuel-Consumption Assessment Results
### Drivers' Advice  
Figure 11 depicts the Uyanik measurement of relevant CAN-bus and IMU signals corresponding to D1 (green) and D11 (red) during five uninterrupted minutes of the route. DS evaluation times from 8 s to 292 s were considered for the performance analysis, as can be seen in Figure 12. It can be observed that the longer the evaluation time, the lower the number of involved DS classes for each evaluation period. For many drivers, this number of classes reached a local minimum at a length of 100 s, while overshooting for longer times until the low-pass filtering effects of a very long evaluation time happened, which would eliminate the details that are needed for a correct DS evaluation. For that reason and, according to [58], the corresponding driving behavior was evaluated every 100 s in order to be correctly assessed and to provide useful advice to the drivers, with the last 100 s stretch being test data (i.e., unseen by the system). Taking into account the evolution of the DS during each 100 s stretch, the cluster with the maximum percentage was selected. D11 and D1 were both classified into a single cluster during the whole segment of the trip: D11 drove according to the Very low cluster, and D1's driving style was mostly in the Medium cluster. GT-Suite simulations of fuel consumption are also displayed. As can be seen, the average fuel consumption of D11 is much lower than D1, as expected. Again, the measured RPMs are lower for D11 than for D1, and the same is the case for the variance of the positive XACC, as in the cluster distribution of Figure 10. On the other hand, measured speeds are not significantly different, proving that fuel consumption under similar conditions has more to do with the car handling itself than with speed. In consequence, the eco-driving system would provide the following advice: D11: \"Keep driving style\"; D1: \"Lower RPM/Operate gas softly\".  
In sum, most drivers' classifiable behaviors vary among different clusters and, with the Very low consumption class being the ideal one, indications should be addressed to drivers to modify their DS if they fall into the other four classes at any moment of the ongoing ride. The eco-driving system provides advice to the driver according to a user-configurable time interval.  
\begin{table}
\begin{tabular}{l l c} \hline
**Current Cluster** & **Target Cluster** & **Reduction (\%)** \\ \hline Low & Very low & 9.5 \\ Medium & Low & 31.5 \\ High & Medium & 18.1 \\ \hline \end{tabular}
\end{table}
Table 9: Expected fuel-consumption reduction between contiguous clusters.  
Figure 11: Uyanik measurement of relevant CAN-bus and IMU signals corresponding to D1 (green) and D11 (red) during five uninterrupted minutes of the route. The driving behavior was evaluated every 100 s, and the cluster with the maximum percentage was selected. Both D11 and D1 were classified into a single cluster during the whole segment of the trip: D11 drives according to the Very low cluster, and D1’s driving style is mostly into the Medium cluster. GT-Suite simulations of fuel consumption are also displayed.
# Enhanced Trajectory Tracking via Disturbance-Observer-Based Modified Sliding Mode Control
## 2 Control Design
### ESO-Based PIDSMC
#### 2.2.1 System Stability  
Stability is an important feature of control algorithms. The control stability is analyzed using the Lyapunov stability theorem:  
\[\hat{s}\cdot\hat{s}\leq 0, \tag{38}\]  
by using Equation (29) in above relation:  
\[\hat{s}\cdot\dot{\hat{s}}\leq|\hat{s}|\bigg{(}\dot{\hat{x}}_{2}+\bigg{(}\frac {l_{1}}{\epsilon_{0}}\bigg{)}\widetilde{x}_{1}-\ddot{x}_{d}+c_{1}\Big{(}\dot{ \hat{x}}_{1}-\dot{x}_{d}\Big{)}+c_{2}\cdot\!\hat{e}\bigg{)}\leq 0, \tag{39}\]  
by using Equation (7) in (39):  
\[\hat{s}\cdot\dot{\hat{s}}\leq|\hat{s}|\bigg{(}\dot{x}_{3}+\bigg{(}\frac{l_{2} }{\epsilon_{0}  
after using the control input Equation (35) in (41):  
\[\begin{split}\hat{s}\dot{\hat{s}}\leq|\hat{s}|\Big{(}\hat{x}_{3}+ \Big{(}\frac{l_{1}}{\epsilon_{o}}+\frac{c_{1}l_{1}}{\epsilon_{o}}\Big{)}\widetilde {x}_{1}-K^{\prime}\cdot\text{sat}(\hat{s})-\Big{(}\frac{l_{1}}{\epsilon_{o}}+ \frac{c_{1}l_{1}}{\epsilon_{o}}\Big{)}\widetilde{x}_{1}+\ddot{x}_{d}-c_{1} \big{(}\hat{x}_{2}-\dot{x}_{d}\big{)}-\\ c_{2}\dot{e}-\dot{x}_{3}+\Big{(}\frac{l_{1}}{\epsilon_{o}}\Big{)} \widetilde{x}_{2}-\ddot{x}_{d}+c_{1}\big{(}\hat{x}_{2}-\dot{x}_{d}\big{)}+c_ {2}\dot{e}\Big{)}\leq 0,\end{split} \tag{42}\]  
after the cancelation, Equation (42) can be written as:  
\[\hat{s}\dot{\hat{s}}\leq|\hat{s}|\bigg{(}-K^{\prime}\cdot\text{sat}(\hat{s})+ \bigg{(}\frac{l_{1}}{\epsilon_{o}}\bigg{)}\widetilde{x}_{2}\bigg{)}\leq 0, \tag{43}\]  
in order to enforce the stability, the magnitude of gain \(K^{\prime}\) should satisfy the following condition (44):  
\[K^{\prime}>\frac{l_{1}}{\epsilon_{0}}\cdot\Big{|}\widetilde{x}_{2}\Big{|}. \tag{44}\]