esa-sceva/satcom-chunk-collection huggingface dataset:
## 3 Simulation Results and Analysis
### Triaxial Loading Conditions
#### 3.2.2 Failure Modes  
Figure 13 illustrates the failure modes of shale under triaxial compression at different bedding angles. As the confining pressure increases, the failure mode transitions from brittle to ductile, predominantly characterized by shear failure. At a bedding angle of 0\({}^{\circ}\), where the bedding plane is parallel to the axial force, the failure manifests as conjugate shear failure controlled by the matrix. With an increase in the confining pressure, the failure mode evolves into shear cracking. The acoustic emission signal distributions indicate that, under uniaxial compression, numerous microcracks form a complex crack network. As the confining pressure rises, the shear crack morphology becomes more defined. At a confining pressure of 22.5 MPa, a well-defined three-dimensional X-shaped conjugate shear crack emerges. At bedding angles of 30\({}^{\circ}\) and 60\({}^{\circ}\), the failure is predominantly characterized by shear slip along the bedding plane, resulting in relatively simple crack morphologies, typically a single shear plane. Due to the weak strength of the bedding plane, the compressive strength is lowest in this failure mode. For specimens with a 90\({}^{\circ}\) bedding angle, failure involves a combination of cracking along the bedding and shear cracks penetrating the specimen, forming macro-cracks typical of vertical-bedding shale; under high confining pressures, suppressing crack initiation and propagation at 90\({}^{\circ}\) bedding results in flatter crack morphologies.  
In summary, under triaxial loading conditions, the shale strength improves significantly while its degree of anisotropy decreases. As the confining pressure increases, the failure modes transition toward single-shear failure, and the crack propagation pattern shifts from disordered to more organized single-shear cracks. Overall, higher confining pressures suppress the formation and propagation of internal microcracks, thereby reducing the anisotropy of shale failure behavior.  
Figure 12: Shear strength parameters. (**a**) Shear strength parameter fitting curve; (**b**) Internal friction angle and cohesion.
## 3 Results and Discussion
### 3.1. Asphalt Binder Performance Analysis
#### 3.1.2 Tensile Properties of Asphalt Binders  
Figure 10 shows the results of the tensile strength and elongation of SSEA at 20 \({}^{\circ}\)C.  
From Figure 10, it can be seen that the tensile strength of SSEA shows a decreasing trend with the increase of asphalt content. There is a sizeable downward trend between SSEA-2 and SSEA-3, with insignificant changes between SSEA-3 and SSEA-5. This is mainly because as the asphalt content increases, the effect of resin relative to epoxy asphalt is relatively prominent in SSEA-1 and SSEA-2, and the effect of asphalt relative to epoxy asphalt is even more prominent in SSEA-3, SSEA-4, and SSEA-5. Thus, the decrease in strength is more pronounced [45, 46]. The elongation of SSEA increases with increasing bitumen dosage, which is relative to the variable change in tensile strength, with a large increase between SSEA-2 and SSEA-3. This may be due to the fact that when SSEA is subjected to external loading, the epoxy resin in the continuous phase carries most of the load, while the SBR/SBS-asphalt in the dispersed phase plays a toughening role in the overall system [47]. In addition, all five SSEA binders have higher elongation at break than BJ200. SSEA-1 and SSEA-2 have higher tensile strengths than BJ200, and SSEA-3 to SSEA-5 have tensile strengths close to BJ200.
# Optimizing Field Body Fat Percentage Assessment in Professional Soccer Players
## 3 Results
\begin{table}
\begin{tabular}{c c c c c} \hline  & **Correlation** & & **Standardized** & \\  & **(\%0\% CI)** & **Bias (\(\pm\)LoA)** & **Differences (90\% CI)** & **Qualitative Assessment** \\ \hline BIA & 0.42 (0.05–0.68) & \(-\)2.30 \({}^{\ast}\) (\(\pm\)4.91) & \(-\)1.09 (0.45) & Almost certainly & 0/0/100 \\ Equations: [PERSON] (1982) & 0.78 (0.58–0.89) & \(-\)8.30 \({}^{\ast}\) (\(\pm\)2.53) & \(-\)5.65 (0.30) & Almost certainly & 0/0/100 \\ [PERSON] et al. (2006) & 0.72 (0.47–0.86) & \(-\)4.68 \({}^{\ast}\) (\(\pm\)3.02) & \(-\)2.68 (0.35) & Almost certainly & 0/0/100 \\ [PERSON] et al. (2005) & 0.79 (0.59–0.90) & \(-\)6.51 \({}^{\ast}\) (\(\pm\)2.81) & \(-\)4.11 (0.42) & Almost certainly & 0/0/100 \\ [PERSON] (1966) & 0.79 (0.59–0.90) & \(-\)4.87 \({}^{\ast}\) (\(\pm\)2.45) & \(-\)2.74 (0.23) & Almost certainly & 0/0/100 \\ [PERSON] and [PERSON] (1973) & 0.73 (0.40–0.86) & \(-\)4.73 \({}^{\ast}\) (\(\pm\)4.92) & \(-\)2.93 (0.62) & Almost certainly & 0/0/100 \\ [PERSON] et al. (2018) & 0.82 (0.64–0.91) & \(-\)6.15 \({}^{\ast}\) (\(\pm\)2.28) & \(-\)3.72 (0.27) & Almost certainly & 0/0/100 \\ [PERSON] et al. (2018) & 0.79 (0.60–0.90) & \(-\)0.30 (\(\pm\)2.56) & \(-\)0.14 (0.23) & Possibly & 1/67/32 \\ [PERSON] and [PERSON] (2000) & 0.73 (0.50–0.87) & \(-\)7.52 \({}^{\ast}\) (\(\pm\)4.36) & \(-\)5.30 (0.81) & Almost certainly & 0/0/100 \\ [PERSON] et al. (2009) & 0.78 (0.58–0.89) & \(-\)5.86 \({}^{\ast}\) (\(\pm\)2.46) & \(-\)3.47 (0.25) & Almost certainly & 0/0/100 \\ [PERSON] et al. (1980) & 0.80 (0.61–0.90) & \(-\)8.13 \({}^{\ast}\) (\(\pm\)2.46) & \(-\)5.52 (0.35) & Almost certainly & 0/0/100 \\ [PERSON] et al. (1987) & 0.78 (0.58–0.89) & \(-\)7.38 \({}^{\ast}\) (\(\pm\)2.88) & \(-\)4.89 (0.48) & Almost certainly & 0/0/100 \\ [PERSON] et al. (2005) & 0.75 (0.53–0.88) & \(-\)6.41 \({}^{\ast}\) (\(\pm\)2.80) & \(-\)3.98 (0.37) & Almost certainly & 0/0/100 \\ [PERSON] and [PERSON] (1967) & 0.84 (0.67–0.92) & \(-\)3.75 \({}^{\ast}\) (\(\pm\)2.76) & \(-\)2.10 (0.33) & Almost certainly & 0/0/100 \\ [PERSON] and [PERSON] (1974) & 0.74 (0.51–0.87) & \(-\)3.92 \({}^{\ast}\) (\(\pm\)3.52) & \(-\)1.86 (0.32) & Almost certainly & 0/0/100 \\ [PERSON] et al. (1996) & 0.81 (0.62–0.91) & \(-\)4.65 \({}^{\ast}\) (\(\pm\)3.06) & \(-\)2.74 (0.43) & Almost certainly & 0/0/100 \\ [PERSON] (1981) & 0.78 (0.58–0.89) & \(-\)3.42 \({}^{\ast}\) (\(\pm\)6.14) & \(-\)2.15 (0.65) & Almost certainly & 0/0/100 \\ [PERSON] et al. (2003) & 0.80 (0.61–0.90) & \(-\)0.81 \({}^{\ast}\) (\(\pm\)2.83) & \(-\)0.38 (0.26) & Likely & 0/11/88 \\ [PERSON] and [PERSON] (1969) & 0.75 (0.52–0.87) & \(-\)3.60 \({}^{\ast}\) (\(\pm\)3.17) & \(-\)1.98 (0.34) & Almost certainly & 0/0/100 \\ \hline \end{tabular} BIA (bioelectric impedance analysis). Significant differences between DXA percentage fat mass and other practical estimates of percentage fat mass using paired \(t\) test \({}^{\ast}\).  
\end{table} 
Table 2: Cross-sectional validity of body fat percentage measurements made by BIA and anthropometric equations using DXA as reference in elite male soccer players (\(n=21\)).
## 3 Results and Discussion
### Main Volatiles from N. Caerulea Flower  
According to the results obtained from the modified HS-SPME/GC-MS analysis, the flowering period was divided to four stages: a--in half bloom; b--at the early stage of full bloom; c--at the later stage of full bloom; d--at the approach of closing (the experiment was replicated three times). Figure 3 shows the total peak areas of the VOCs under different blossom periods. The number of VOCs varied at different stages, while the total area from the GC-MS analysis at stage a was lower than that at stage b, and the total area decreased after stage c. The concentrations of volatiles at the final stage were the lowest. In general, the final-stage volatile compounds were too diluted to be smelled by humans. This observation is consistent with the human sense of smell.  
The volatile compounds from flowers are rather complex, and they are affected by different blossom periods. From the HS-SPME/GC-MS analysis, the desorption of volatiles allowed us to identify the compounds emitted by this plant. At stage b, the main components of the HS-SPME samples (concentration > 3.0%) were 6,9-heptadecadiene (40.1%), pentadecane (15.5%), 8-heptadecene (15.3%), benzyl acetate (10.4%), and benzyl alcohol (4.4%). From Figure 4, we can see that benzyl acetate, pentadecane, 6,9-heptadecadiene, and 8-heptadecene were four common major compounds under different flowering periods, while the concentration of 6,9-heptadecadiene, 8-heptadecene, and benzyl alcohol reached their highest levels at the early stage of full bloom (stage b).  
Figure 3: Total peak areas of the VOCs under different blossom periods. The box lines from upper to lower are the values of 25 th, 50 th, and 75 th percentiles. The upward and downward bars represent the maximums and minimums. The cross symbol in the box indicates average value.
# A Multi-Layer Erbium-Doped Air-Hole-Assisted Few-Mode Fiber with Ultra-Low Differential Modal Gain
###### Abstract  
The air-hole assisted few-mode fiber (AH-FMF) enables modal intensity balance and offers a profound prospect in gain equalization with the combination of adaptive ion doping. In this paper, we proposed an AH-FM-EDF with a multi-layered erbium doping profile. In AH-FM-EDF, due to the central air hole, only the first radial order modes (LP\({}_{01}\), LP\({}_{11}\), LP\({}_{21}\), and LP\({}_{31}\)) are supported, and all the modes are confined in the same high refractive index core region. The differential modal gain (DMG) is highly reduced by optimizing the erbium doping proportion in each layer. Compared with uniform doping, the DMG is reduced from 4 dB to 0.14 dB as triple-layer doping is deployed. Additionally, the proposed erbium-doped fiber performs well in gain flattening and fabrication tolerance over the whole C-band.  
mode division multiplexing (MDM); air-hole assisted; few-mode erbium-doped fiber amplifier (FM-EDFA); gain equalization +
Footnote †: journal: _photonics_
# A Novel Fast Parallel Batch Scheduling Algorithm for Solving the Independent Job Problem
## 6 Computational Experiments
### Comparison of 2-Approximation Algorithm (LIM) and CPLEX
\begin{table}
\begin{tabular}{l c c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Test Suite**} & \multicolumn{3}{c}{**CPLEX**} & \multicolumn{3}{c}{**\(C_{\textbf{max}}\)**} & \multicolumn{3}{c}{**\(C_{\textbf{max}}\)**} & \multicolumn{3}{c}{**\(T\)**} & \multicolumn{1}{c}{**97/4**} & \multicolumn{1}{c}{**5.7} \\ \cline{2-13} \multicolumn{1}{c}{**Makespan**} & **GAP (\%)** & **Run Time (s)** & **Best** & **AVG** & **Worst** & **Run Time (s)** & **Best** & **AVG** & **Worst** & **Run Time (s)** \\ \hline JIMIS1 PIK1 & 9 & 0 & 0.02 & 9 & 14.92 & 21 & 0.02 & 10 & 23.23 & 34 & 0.02 & 9.49 & 21.35 & 42.71 \\ JIMIS1 PIK1 & 8 & 0 & 0.09 & 9 & 13.02 & 22 & 0.02 & 9 & 19.06 & 36 & 0.02 & 9.63 & 21.66 & 43.32 \\ JIMIS1 PIK1 & 19 & 42.11 & 2.10 & 17 & 21.62 & 27 & 0.01 & 30 & 33.76 & 51 & 0.01 & 10.26 & 23.10 & 46.19 \\ JIMIS1 PIK1 & 10 & 40.00 & 2.26 & 15 & 19.44 & 22 & 0.01 & 24 & 32.04 & 48 & 0.01 & 10.09 & 22.27 & 45.4 \\ JIMIS1 PIK1 & 52 & 79.87 & 466.47 & 35 & 47.39 & 68 & 0.13 & 61 & 86.61 & 136 & 0.13 & 21.68 & 49.19 & 96.37 \\ JIMIS21 PIK1 & 47 & 80.85 & 25.39 & 21 & 28.84 & 40 & 0.02 & 37 & 53.53 & 106 & 0.02 & 13.57 & 30.53 & 61.07 \\ JIMIS1 PIK1 & 59 & 90.68 & 1800 & 76 & 94.57 & 129 & 0.04 & 150 & 175.85 & 255 & 0.04 & 42.73 & 96.14 & 192.92 \\ JIMIS21 PIK1 & 29 & 86.21 & 1800 & 40 & 52.75 & 108 & 0.04 & 76 & 99.02 & 180 & 0.04 & 24.22 & 54.50 & 108.99 \\ JIMIS1 PIK1 & 197 & 97.46 & 1800 & 151 & 185.53 & 255 & 0.08 & 250 & 352.33 & 496 & 0.08 & 83.29 & 187.40 & 374.81 \\ JIMIS21 PIK1 & - & - & 1800 & 78 & 105.49 & 186 & 0.08 & 146 & 203.96 & 365 & 0.08 & 47.55 & 106.99 & 213.98 \\ JIMIS1 PIK1 & 161 & 97.2 & 1800 & 213 & 283.43 & 387 & 0.12 & 37 & 535.71 & 799 & 0.12 & 126.65 & 284.96 & 569.93 \\ JIMIS21 PIK1 & - & - & 1800 & 104 & 143.73 & 283 & 0.12 & 190 & 275.35 & 560 & 0.13 & 64.65 & 145.46 & 290.93 \\ \hline \hline \end{tabular}  
* Note: (1) Column 2 is the minimum makespan 50 instances for each test suite. “ represents that CPLEX could not find a feasible solution in 1800 s. (2) Each test suite contains 50 instances. Columns 5, 6, and 7 report the best, average, and worst respectively. Columns 9, 10, and 11 report the best, average, and worst Cano, respectively. (3) Columns 13, 14, and 15 report the average, 7, 97/4, and 4, and 4,57 of 50 instances, respectively.  
\end{table} 
Table 5: Simulation results of CPLEX and the 4.5-approximation algorithm with one-job-overfull batches and a feasible schedule.
# An Analytical Framework in OFDM Wireless Networks Servicing Random or Quasi-Random Traffic
###### Abstract  
We consider the downlink of an orthogonal frequency division multiplexing (OFDM)-based cell that services calls from many service-classes. The call arrival process is random (Poisson) or quasi-random, i.e., calls are generated by an infinite or a finite number of sources, respectively. In order to determine congestion probabilities and resource utilization, we model the cell as a multirate loss model. Regarding the call admission, we consider the restricted accessibility, the bandwidth reservation (BR), and the complete sharing (CS) policies. In a system of restricted accessibility, a new call may be blocked even if resources do exist. In a BR system, subcarriers can be reserved in favor of calls of high subcarrier requirements. Finally, in a CS system, a new call is blocked due to resource unavailability. In all three policies, we show that there exist recursive formulas for the determination of the various performance measures. Based on simulation, the accuracy of the proposed formulas is found to be quite satisfactory.  
OFDMA; congestion; random; quasi-random; recursive; restricted; reservation; complete sharing +
Footnote †: journal: Applied Physics Letters  
15 November 2019; Accepted: 4 December 2019; Published: 9 December 2019
# Semi-Direct Point-Line Visual Inertial Odometry for MAVs
## 4 Experimental Results and Analysis
### Loop Closure Detection Evaluation  
The most obvious advantage of the feature-based method compared with the direct method is that the feature-based method could be used for loop closure detection, which can reduce the drift accumulated during the long-term operation. Therefore, the loop-closure detection ability of SDPL-VIO was finally evaluated to verify the integrity and reliability of the system. From the comparison results on the MH03 dataset, as shown in Figure 14, the accuracy of SDPL-VIO with loop closure detection is clearly shown to improve.  
The analysis of multiple experiments shows that SDPL-VIO can consider both speed and accuracy, which increases the operation efficiency without reducing the operation precision, and is superior to other methods in terms of comprehensive performance.
## 3 Results and Findings from the Existing Literature
### Life Cycle Analysis (LCA)
This review reveals that the application of the life cycle assessment (LCA) and techno-economic analysis (TEA) framework directly contributes to the achievement of various SDGs. For example, the LCA aligns with objectives such as promoting good health and well-being (SDG-3), ensuring clean water and sanitation (SDG-6), facilitating affordable and clean energy (SDG-7), fostering sustainable cities and communities (SDG-11), encouraging responsible consumption and production (SDG-12), supporting climate action (SDG-13), preserving life below water (SDG-14), and protecting life on land (SDG-15). Conversely, the TEA is intertwined with the realisation of other SDGs, as illustrated in Figure 7.
# Maximum Transmit Power for UE in an LTE Small Cell Uplink
## 3 Network and Propagation Model
### Homogenous Macro/Small Cell Deployment  
This section describes the network model considered for the homogenous deployment of macro and small cells. To determine the \(P_{\text{max}}\) in small cells, which ensures the same performance when compared to the macro cell deployment, both network deployments were analyzed with similar conditions. In Figure 1, a homogenous distribution of both the macro and small cells is depicted. It is important to mention here that both the macro and small cells were considered independently, that is, the macro cells were considered first, followed by the uniformly distributed small cells. Two rings of neighboring cells were considered around the central cell. At eNodeB, omni-directional antennas were modeled to provide network coverage in each cell. In Figure 1, the central cell has a radius \(R\) and is surrounded by two rings of cells termed \"interfering rings\", at distance \(2 nR\) (\(n=1,2\)). The network size is given as \(R_{nt}=5R\). In this case, \(R=200\) m and \(1000\) m specify the cell radius, while \(R_{nt}=1000\) m and \(5000\) m denote the network range for uniformly distributed macro and small cell networks, respectively.
## 4 Conclusions  
In summary, we synthesized a series of new spirooxindole analogues based on di-substituted isatin. The anticancer activity of the compounds against three different cancer cell lines was explored. Among the analogues, the compound spirooxindole analogue **5g** had an inhibitory growth potency in HCT116 similar to that of cisplatin, but it is ca. 1.8 (in HepP2) or 2.25 (in MCF7) times more potent than the reference drug, and also showed good physicochemical parameters and lipophilicity value. Further investigation of the mechanism of action of compound **5g** is required.  
**Supplementary Materials:** The following are available online at [[http://www.mdpi.com/2076-3417/10/6/2170/s1](http://www.mdpi.com/2076-3417/10/6/2170/s1)]([http://www.mdpi.com/2076-3417/10/6/2170/s1](http://www.mdpi.com/2076-3417/10/6/2170/s1)), characterization of the synthesized compounds; protocol for anticancer activity; ROCS protocol, Figures of the NMR spectrum.  
**Author Contributions:** Conceptualization, A.B.; Data curation, F.F.E.-S., and F.A.B.; Formal analysis, F.F.E.-S., F.A.B., and A.F.M.M.R.; Funding acquisition, A.M.A.-M.; Investigation, M.S.I.; Methodology, M.S.I.; Resources, A.M.A.-M.; Supervision, Y.A.M.M.E.; Validation, A.B.; Writing--original draft, A.B. and Y.A.M.M.E.; Writing--review and editing, F.A.B., A.B., and Y.A.M.M.E. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by King Saud University, research group (No. RGP-044).  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Compounds**} & \multirow{2}{*}{**R**} & \multirow{2}{*}{**NHA**} & \multirow{2}{*}{**cLog**} & \multicolumn{3}{c}{**Breast Cancer Cells**} & \multicolumn{3}{c}{**Colon Cancer Cells**} \\ \cline{4-10}  & & & **P** & **pIC50** & & **LE** & **LipE** & **pIC50** & **LE** & **LipE** \\ \hline \multirow{2}{*}{5g} & \multirow{2}{*}{41} & \multirow{2}{*}{4.47} & \multirow{2}{*}{5.39} & \multirow{2}{*}{0.22} & \multirow{2}{*}{0.92} & \multirow{2}{*}{5.55} & \multirow{2}{*}{0.19} & \multirow{2}{*}{1.08} \\  & & & & & & & & \\ \end{tabular}
\end{table}
Table 3: _Cont._  
**Acknowledgments:** The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for providing funding to the research group (No. RGP-044). Thanks to RSUI for their technical support. [PERSON] acknowledges the OpenEye scientific software for providing academic license.  
**Conflicts of Interest:** The authors declare no conflict of interest.
## 4 Biomedical Applications of ML-Assisted SWIR Techniques
### Quantitative Imaging and Prognosis
Furthermore, a DNN was employed to accurately estimate fundamental parameters encompassing the count of scatterers within a resolution volume, lateral and axial resolution, as well as signal-to-noise ratio (SNR) by analyzing local speckle patterns within the OCT images [104]. These networks can also find utility in calibrating OCT systems for exceptionally precise measurementsof the attenuation coefficient, exemplifying their potential to enhance the precision and versatility of OCT techniques. Theoretical models developed for PT-OCT signals demonstrate that PT-OCT signals are influenced by multiple parameters, necessitating consideration of the interplay between them for accurate quantification of titers of molecules of interest [16; 29]. Recent applications of SVM models to PT-OCT data successfully classified phantoms based on their lipid content irrespective of their depth within the sample [71]. This study can have a significant impact on personalized medicine by enhancing diagnostic capabilities and improving patient outcomes through a better understanding of lipid-related aspects of various diseases. OCE is another modality of OCT that can provide mechanical-related properties of tissues. A DNN was applied on obtained mechanical properties data from phantoms mimicking tissues with different mechanical elasticity to extract mechanical properties [105]. This advancement in elastic property estimation using DNN from OCE data holds the promise of enhancing personalized medicine by providing clinicians with real-time, non-invasive tools for assessing tissue characteristics and tailoring treatments to individual patient needs.
# Spacing Allocation Method for Vehicular Platoon: A Cooperative Game Theory Approach
## 4 Methodology Based on Cooperative Game Model for Platoon
### Motion Controller  
In this section, a more general control law which is suitable for most of the spacing policy will be proposed. To describe the movement of the platoon, the dual-integrator model is used which is shown in Equation (21).  
\[\begin{array}{l}V_{i}=\dot{X}_{i}\\ \dot{A}_{i}=\dot{V}_{i}\end{array} \tag{21}\]  
where \(X_{i}\), \(V_{i}\) and \(A_{i}\) represent the position, velocity and acceleration of vehicle \(i\) respectively.  
Assuming that the \(\Delta i\) denote the actual relative position between vehicle \(i-1\) and vehicle \(i\), and the actual space is given by Equation (22).  
\[\Delta_{i}=\dot{X}_{i-1}-\dot{X}_{i} \tag{22}\]  
Existing spacing policy can be written as the form of Equation (23), which is a function of the relative position and relative speed between host vehicle and nearest neighbors.  
\[R_{ds}=R\Big{(}\Delta_{i},\Delta_{i+1},\dot{\Delta}_{i},\dot{\Delta}_{i+1} \Big{)} \tag{23}\]  
Notice that we can get the dynamic model of the following error with Equation (24).  
\[\begin{array}{l}e=\Delta_{i}-R_{i}\\ \dot{e}=\dot{\Delta}_{i}-\dot{R}_{i}\end{array} \tag{24}\]  
where \(\dot{R}_{i}=\frac{\partial R}{\partial\dot{\Delta}_{i}}\dot{\Delta}_{i}+\frac {\partial R}{\partial\dot{\Delta}_{i+1}}\dot{\Delta}_{i+1}+\frac{\partial R}{ \partial\dot{\Delta}_{i}}\ddot{\Delta}_{i}+\frac{\partial R}{\partial\dot{ \Delta}_{i+1}}\ddot{\Delta}_{i+1}\).  
Let \(\dot{e}=~{}-\eta e\), and the control law can be obtained in Equation (25).  
\[\ddot{\Delta}_{i}=\frac{-\eta[\Delta_{i}-R]+\dot{\Delta}_{i}-\left(\frac{ \partial R}{\partial\dot{\Delta}_{i}}\dot{\Delta}_{i}+\frac{\partial R}{ \partial\dot{\Delta}_{i+1}}\dot{\Delta}_{i+1}+\frac{\partial R}{\partial\dot {\Delta}_{i+1}}\ddot{\Delta}_{i+1}\right)}{\partial R/\partial\dot{\Delta}_{i}} \tag{25}\]  
where \(\eta\) is a positive constant.