esa-sceva/satcom-chunk-collection huggingface dataset:
## 5 Training
### Simulation Setup  
The gazebo simulator facilitates obtain state observation as well as action execution. The Robot Operating System (ROS) is used as a bridge between the neural network and the simulator. For training, we use a 10 by 10-m walled environment. In the environment, we place multiple large cubes and cylinders, and their location is randomized at the beginning of every episode to increase the scene variance and improve the methods' generalizability. The actions are executed on simulated turtlebot 3 [41] differential drive robot models that serve as agents for our neural network. Neural network outputs are communicated to robots as ROS Twist-type messages. While the proposed method is generally sensor agnostic, we use laser readings to observe the environment for robot navigation due to the reduced number of returned sensor values and increased field of view compared to depth or RGB camera observations. A simulated Velodyne Puck sensor is placed on each robot, which records laser readings in a 180-degree field of view. This information, along with the robot's odometry and executed velocities, is used as part of the state information in (2). The goal information for calculating values of \(d\), \(cos(\theta)\) and \(sin(\theta)\) is obtained from the simulator. The goal positions, robots' starting positions and poses are also randomized at the start of each episode as not to overlap with any of the obstacles in the environment. We train the neural network with three robots sharing the environment. The simulation setup is depicted in Figure 6.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Critic Network** & \\ \hline
**Layer** & **Input Size** & **Output Size** \\ \hline
**Edge Encoder**: & & \\ Linear & 7 & 40 \\ CELU & 40 & 40 \\ Linear & 40 & 80 \\ CELU & 80 & 80 \\ Max Aggregation & (n, 80) & (1, 80) \\
**Node Encoder**: & & \\ Linear & 25 & 400 \\ CELU & 400 & 400 \\ Concatenation & 400 + 80 & 480 \\
**Q-Value Decoder**: & & \\ Linear\({}_{state}\) & 480 & 300 \\ Linear\({}_{action}\) & 2 & 300 \\ TFC & 300 + 300 & 300 \\ CELU & 300 & 300 \\ Linear & 300 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Network parameters and structure of both critic networks.
## 2 Background  
The increasing demand for energy, coupled with the need for efficient and sustainable power systems, has led to the development of smart grid technology. The power internet of things (PIoT) is an essential component in creating intelligent and efficient smart grid systems. This literature review proposal aims to explore the current state of PIoT-based smart grid demand response schemes and identify the research gaps to be addressed in the development of a novel research process. Smart grids have emerged as a promising solution to address the challenges of the energy sector, such as energy efficiency, grid resilience, and reliability. They integrate advanced technologies, such as PIoT, to optimize the generation, distribution, and consumption of electricity [13]. Demand response (DR) is a crucial component of smart grids, as it allows utilities to balance supply and demand in real-time as mentioned in [14]. DR programs typically involve time-varying pricing, direct load control, and incentive-based programs to encourage consumers to adjust their energy consumption patterns in response to grid conditions. Recent studies have shown that effective DR schemes can lead to significant cost savings, reduced peak demand, and enhanced grid stability. The power internet of things (PIoT) refers to the integration of IoT technologies into power systems, enabling seamless communication and control across various smart grid components as mentioned in [15]. PIoT provides real-time monitoring, data analysis, and decision-making capabilities, which are essential for the successful implementation of DR schemes as mentioned in [16]. In the study by [10], a novel demand response scheme that integrates plug-in electric vehicles (PEVs) and renewable energy sources in PIoT-enabled smart grids is presented. The authors in [17] develop a multi-objective optimization model that balances energy supply and demand, and demonstrate the potential of PIoT technology in facilitating efficient demand-side management, particularly in the context of increasing PEV penetration and renewable energy integration. A PIoT-based energy management system for smart homes, which effectively managed energy consumption and participated in DR programs as mentioned in [18]. A PIoT-based architecture for demand-side management in industrial parks, showing improved energy efficiency and reduced operating costs, is shown in Table 1. This study presents an integrated demand response framework for residential PIoT, focusing on the coordination and optimization of distributed energy resources and electric vehicles as mentioned in [19]. The authors propose a hierarchical control architecture to manage the power consumption of various devices and facilitate demand-side management, highlighting the potential of PIoT for improving energy efficiency and system reliability in smart grids according to [20].
# Can Data Assimilation Improve Short-Term Prediction of Land Surface Variables?
## 2 Materials and Methods
### Multi-Pass Land Data Assimilation Scheme  
A multi-pass land data assimilation scheme (MLDAS) has been proposed based on the Noah-MP model [4; 5] and ensemble Kalman filter (EnKF) method. Unlike other data assimilation (DA) methods, MLDAS can assimilate multisource remotely sensed or ground-measured observations in multiple passes to optimize parameters and state variables simultaneously. Currently, four passes have been built to assimilate LAI (pass I-pass II), SM (pass III), and SIF (pass IV) to update specific leaf area (SLA), leaf biomass (LFMASS), SM and maximum carboxylation rate at a reference temperature of 25 \({}^{\circ}\)C (Vcmax25) in MLDAS successfully [17]. Pass I to pass II are designed to update simulation on plant productivity, ecological behavior and plant growing rates, which are key factors for vegetation status and evapotranspiration. SLA is defined as ratio of LAI to LFMASS, so LAI can be assimilated in two passes to update these two parameters jointly. In pass I, LAI is assimilated to update the constant parameter SLA which is equivalent to BIO2 LAI in Noah-MP. In pass II, state variable LFMASS is updated. SM is updated in pass III directly. Through assimilating SIF in pass IV, we can optimize Vcmax25 to better control carbon assimilation processes and represent plant productivity [23].  
In our study, four passes were opened to jointly assimilate LAI, SM and SIF. LAI observations will be assimilated when available, and SM and SIF observations will be assimilated when available at daily and four-day intervals, respectively.
## 1 The probability of the destination profile receiving Prelearn messages (\(Pov\))  
This parameter represents the probability of the destination profile receiving Prelearn messages from the message carriers and its neighbors. The probability of the destination profile receiving Prelearn messages \(POV\) can be shown as \(POV=E_{rec}/E_{opt}\), where \(E_{rec}\) is the total number of Prelearn messages forwarded by the node, and \(E_{opt}\) is the total number of Prelearn messages obtained by the destination profile.
# Fusion of Land-Based and Satellite-Based Localization Using Constrained Weighted Least Squares
## 2 Measurement Models
### DOA Measurement Model for Land-Based Station
In practice, there will be measurement noise, so the following measurement vector can be obtained as:  
\[\mathbf{\theta}=\left[\theta_{1},\theta_{2},\dots,\theta_{N_{1}}\right]^{\mathrm{ T}}=\mathbf{\theta}^{o}+\mathbf{n}_{\mathbf{\theta}}\in\mathbb{R}^{N_{1}\times 1} \tag{5}\]  
where \(\mathbf{n}_{\mathbf{\theta}}=\left[\delta\theta_{1},\delta\theta_{2},...,\delta\theta _{N_{1}}\right]^{\mathrm{T}}\) denotes the noise vector, which follows a zero-mean Gaussian distribution, and the covariance matrix is \(\mathbf{Q}_{\mathbf{\theta}}=\mathrm{E}\left(\mathbf{n}_{\mathbf{\theta}}\mathbf{n}_{\mathbf{\theta}}^ {\mathrm{T}}\right)\). \(\triangle ABC\), based on the sine theorem, yields  
Figure 1: IVH model. \[\frac{h_{i}^{o}+R_{o}}{sin\left(\pi/2+\varphi_{i}^{o}\right)}=\frac{R_{o}}{sin \left(\pi-\left(\pi/2+\varphi_{i}^{o}+\beta_{i}\right)\right)} \tag{6}\]  
where \(\beta_{i}=arcsin\left(\frac{||\mathbf{u}-\mathbf{g}^{\prime}||_{2}}{2R_{o}}\right)\). (6) can be converted into  
\[\varphi_{i}^{o}=arctan\left(\frac{\left(R_{o}+h_{i}^{o}\right)cos(\beta_{i})-R _{o}}{\left(R_{o}+h_{i}^{o}\right)sin(\beta_{i})}\right),1\leq i\leq N_{1} \tag{7}\]  
Based on (7), the true value vector of the elevation angle is expressed as \(\varphi^{o}=\left[\varphi_{1}^{o},\varphi_{2}^{o},\ldots,\varphi_{N_{1}}^{o} \right]^{\mathrm{T}}\), and further, the measurement vector is given as  
\[\mathbf{\varphi}=\left[\varphi_{1},\varphi_{2},\ldots,\varphi_{N_{1}}\right]^{ \mathrm{T}}=\mathbf{\varphi}^{o}+\mathbf{n_{\varphi}}\in\mathbb{R}^{N_{1}\times 1} \tag{8}\]  
\(\mathbf{n_{\varphi}}=\left[\delta\varphi_{1},\delta\varphi_{2},\ldots,\delta \varphi_{N_{1}}\right]^{\mathrm{T}}\) denotes the elevation noise vector, and the covariance matrix of \(\mathbf{n_{\varphi}}\) is \(\mathbf{Q_{\varphi}}\). Combining (5) and (8) gives the DOA measurement model for land-based stations as  
\[\mathbf{z}=\left[\mathbf{\theta}^{\mathrm{T}},\mathbf{\varphi}^{\mathrm{T}}\right]^{ \mathrm{T}}=\mathbf{z}^{o}+\mathbf{n}_{a} \tag{9}\]  
where \(\mathbf{z}^{o}=\left[\left(\mathbf{\theta}^{o}\right)^{\mathrm{T}},\left(\mathbf{\varphi} ^{o}\right)^{\mathrm{T}}\right]^{\mathrm{T}}\) and \(\mathbf{n}_{a}=\left[\mathbf{n}_{\mathbf{\theta}}^{\mathrm{T}},\mathbf{n}_{\mathbf{\varphi}}^{ \mathrm{T}}\right]^{\mathrm{T}}\). The covariance matrix of \(\mathbf{n}_{a}\) is \(\mathbf{Q}_{a}=blkdiag\Big{\{}\mathbf{Q_{\varphi}},\mathbf{Q_{\varphi}}\Big{\}}\).
## 4 Results
### Sponsors  
Figure 3 shows Trial sponsors. Pharmaceutical companies funded 45.24% of trials and 57.14% were funded by a Public Entity. There was no difference between GA and GB (\(p\) = 0.2753).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Investigated Drugs** & **GA, \(n\) = 53** & **\%** & **GB, \(n\) = 73** & **\%** & **Total** & \(p\)**-Value** \\ \hline
**Androgen deprivation** & 5 & 9.43 & 11 & 15.07 & 16 & 0.4237 \\
**therapy** & 6 & 11.32 & 19 & 26.03 & 25 & 0.0449 \\ \hline
**Enzalutamide** & 14 & 26.42 & 27 & 36.99 & 41 & 0.2503 \\ \hline
**Abiaterone** & 13 & 24.53 & 32 & 43.84 & 45 & 0.0377 \\ \hline
**Prednisone** & 9 & 16.98 & 32 & 43.84 & 41 & 0.0019 \\ \hline
**Apalutamide** & 11 & 20.75 & 16 & 21.92 & 27 & 1.0000 \\ \hline
**Bicalutamide** & 5 & 9.43 & 7 & 9.59 & 12 & 1.0000 \\ \hline
**Leuprolide** & 13 & 24.53 & 13 & 17.81 & 26 & 0.3802 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Most frequently investigated drugs. ADT: see specific section for ADT.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Molecular Targets** & **GA, \(n\) = 53** & **\%** & **GB, \(n\) = 73** & **\%** & **Total** & **\%** & \(p\)**-Value** \\ \hline AR & 26 & 49.06 & 54 & 73.97 & 80 & 63.49 & 0.0051 \\ \hline
**CYP17** & 12 & 22.64 & 27 & 36.99 & 39 & 30.95 & 0.1181 \\ \hline
**GnRH** & 13 & 24.53 & 12 & 16.44 & 25 & 19.84 & 0.2691 \\ \hline
**LHRH** & 12 & 22.64 & 9 & 12.33 & 21 & 16.67 & 0.1497 \\ \hline \hline \end{tabular}
\begin{tabular}{c} AR = androgen receptor; CYP17 = Cytochrome P450 17 alpha-hydroxylase/17,20-lyase; GnRH = gonadotropin-releasing hormone; LHRH = luteinizing hormone-releasing hormone. \\ \end{tabular}
\end{table}
Table 7: Distribution of most frequent molecular targets in the study groups.
## 5 Simulation Experiments, Verification, and Comparative Analysis
The proposed architecture is verified through detailed simulation experiments. Nevertheless, the authors intend to develop a small-scale prototype to experimentally validate the proposed architecture in the future. Figure 9: Response of the proposed architecture when the PV array is under the influence of the Figure 7b shading case, with sub-graphs: (**a**) \(P_{pv}\) curves, (**b**) \(I_{pv}\) curves, (**c**) \(V_{pv}\) curves, (**d**) \(D\) signals of buck converters, and (**e**) DC-link voltage controlled by the centralized MPPT inverter.
## Instrument Description
### Offline Signal Processor
#### 2.6.2 Interferometric Correlator  
For every millisecond \(T_{\mathrm{ms}}\), the complex raw samples from SPIR's front ends are combined to obtain the up and down-looking beamformed signals as:  
\[z_{\mathrm{UP}}^{T_{\mathrm{ms}}}[t] = \sum_{i=1}^{8}s_{i}[t]e^{-j\beta_{i}^{T_{\mathrm{ms}}}} \tag{3}\] \[z_{\mathrm{DOWN}}^{T_{\mathrm{ms}}}[t] = \sum_{i=9}^{16}s_{i}[t]e^{-j\beta_{i}^{T_{\mathrm{ms}}}} \tag{4}\]  
where \(i\) denotes the antenna element number, \(s_{i}\) is its corresponding complex signal component for the time series of \(t\in[T_{\mathrm{ms}}\), \(T_{\mathrm{ms}}+1)\) ms, and \(\beta_{i}^{T_{\mathrm{ms}}}\) is its beamformer phase at millisecond \(T_{\mathrm{ms}}\).  
The next step consists of cross-correlating both signals at the output of the beamformers in the frequency domain by using FFTs. We transform the time-domain signal to the frequency domain and compute the complex conjugate product of their spectra. Before returning the result to the time domain, we apply a comb-like filter \(H\left[f\right]\) to remove interfering tones. The bandwidth is set according to the nominal values for each type of signal (GPS/Galileo at L1/L5). Therefore, for every millisecond \(T_{ms}\) and set of \(\beta_{i}^{T_{ms}}\), we obtain the complex waveform:  
\[w^{T_{ms}}[\tau]=\mathcal{F}^{-1}\{Z_{\mathrm{DOWN}}^{T_{ms}}[f]\cdot Z_{ \mathrm{UP}}^{T_{ms}*}[f]\cdot H[f]\} \tag{5}\]  
where \(Z_{\mathrm{UP}}^{T_{ms}}[f]\) and \(Z_{\mathrm{DOWN}}^{T_{ms}}[f]\) are the discrete Fourier transforms (\(\mathcal{F}\{\}\)) of \(z_{\mathrm{UP}}^{T_{ms}}[t]\) and \(z_{\mathrm{DOWN}}^{T_{ms}}[t]\) respectively, and \(H[f]\) is the filter applied.  
Finally, coherent (complex sum) and non-coherent (power averaging) integrations are performed in order to reduce thermal noise and speckle. The coherent and non-coherent integration intervals are programmable. Before each integration process, the waveform series are aligned with respect to the first sample (retracking), based on a precise altimetric delay model. This is done in order to compensate for the movement of the aircraft in the vertical direction during the integration period. Not doing so would result in smeared correlation waveforms along the horizontal delay axis. In order to avoid the use of interpolation methods between samples, this step is also done in the frequency domain. We use the Fourier transform property, by which a delay in the temporal domain is equivalent to a phasor multiplication in the frequency domain [15].
## 3 Results and Discussion
### Comparative LCA of Hydrogen Production: Biogas Reforming vs. Agricultural Residue
#### 3.2.1 Global Warming Potential  
One of the most crucial impact categories analyzed is global warming potential (GWP), measured in kg CO\({}_{2}\) equivalent per kg of hydrogen produced. As depicted in Table 5, the GHG emissions from biogas reforming (5.0471 kg CO\({}_{2}\) eq) are significantly higher than those from gasification (1.2987 kg CO\({}_{2}\) eq), primarily due to reduced reliance on fossil-based electricity and methane emissions. This indicates that biogas reforming generates 4.89 times more CO\({}_{2}\) emissions, reinforcing the environmental advantage of gasification in mitigating climate change. Figures 5 and 6 further highlight these differences, showing the substantial contribution of process emissions in biogas reforming to overall environmental burdens. These findings align with the work of previous researchers [30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42], who reported that thermochemical gasification of biomass results in substantially lower GHG emissions than reforming-based hydrogen production due to the reduced reliance on fossil carbon sources.
# Experimental Study on the Shear Strength of Silt Treated by Xanthan Gum during the Wetting Process
## 3 Results and Discussion
### Direct Shear Test
#### 3.4.3 Relationship between Strength and Water Content  
Figure 17 shows the shear strength-displacement relation curves of the pure silt and XG-silt specimens with different water contents under vertical stress of 100 kPa. The strength of the silt specimens gradually increased with decreasing water content, as did the strength of the XG-silt specimens.
## 3 Results and Discussion
### Effectiveness Analysis  
Two different tests were used to assess IVEE performance. Both were the same and instructed the participants to visualize data, subset data, merge data, and answer knowledge-related questions in R and VR environments. Knowledge-related questions instructed the participants to visually report certain data values. This section of the chapter focuses on various comparisons between study conditions (R and VR) using parametric and non-parametric tests. After completing the tests and the efficacy measures questionnaires, the final questionnaire asked participants to report their preferences regarding both tools.
## 7 Direct Cross-Coupling Reactions between 2-Bromo-1,3,4-Oxadiazoles and Organoboron Compounds  
Cross-coupling reactions of halogen-containing 1,3,4-oxadiazoles with phenylboronic derivatives have appeared in the literature only twice. Both reactions are classified as Suzuki cross-coupling reactions. In 2004, [PERSON] and [PERSON] described an effective method for the coupling of 2-bromo-5-phenyl-1,3,4-oxadiazole with derivatives of boronic acid in the presence of Pd(PPh)\({}_{4}\) and Na\({}_{2}\)CO\({}_{3}\) as a base. The transformations gave 2,5-diaryl-1,3,4-oxadiazoles in high yields (Scheme 11; 46a,b; 85-93%) [85]. Another article from 2016 by [PERSON] and co-workers presented the Suzuki cross-coupling of the pinacol ester of 4-aminophenylboronic acid to 2-bromo-5-(4-chlorophenyl)-1,3,4-oxadiazole derivatives. This was also performed under the action of Pd(PPh)\({}_{4}\) and Na\({}_{2}\)CO\({}_{3}\). Despite the presence of two potential coupling centers (C(sp\({}^{2}\))-Br, C(sp\({}^{2}\))-Cl), the reaction occurred selectively at the C2 carbon atom in the 1,3,4-oxadiazole ring (Scheme 11; 47; 96%) [46].
## 6 Conclusions  
We have reviewed the current state-of-the-art of OPS technologies using coherently modulated PDM-QPSK/PDM-16 QAM optical packets with special attention on the digital coherent burst-mode receiver and the electro-optical PLZT optical switch subsystems and have characterized a complete 2 \(\times\) 2 OPS node with a control plane capable of resolve packet contention by dropping or buffering. We have proposed and experimentally demonstrated a novel DCBMRX that makes use of the Stokes parameters to promptly estimate the SOP. The estimator makes use of the SVD decomposition and an estimator of size 16 is enough to guarantee a packet payload at the FEC limit of 10\({}^{-3}\) and this may reduce the overall complexity of the receiver. We have also experimentally demonstrated the suitability of PLZT optical switches as a subsystem for the switching of high-order QAM modulated optical packets. Using a transmitter with a two-bit DAC made of discrete components to generate PDM-16 QAM modulated optical packets, we demonstrated a switching capacity of over  
Figure 16: BER of 64 wavelength channels (**a**) back to back (**b**) after dropping packets (marked as DROP in Figure 15) and (**c**) after the packet forwarding process (marked as FORWARD in Figure 15).  
10.24 Tb/s/port. We used the PLZT optical switch as a building block for a full 2 \(\times\) 2 OPS node with packet contention resolution by packet dropping or packet buffering in FDLs. We first evaluated this optical switch in a back-to-back to demonstrate the node operation with packet buffering at 10.24 Tb/s/port. Finally, we increased the switching capacity to 12.8 Tb/s/port and transmitted the OPS signal thought two spans of 50 km of dispersion compensated SMF, owing to an improved packet transmitter consisting of eight-bit AWGs, which did not require a packet carver.  
The authors would like to thank [PERSON] and [PERSON] for their technical support during the experimental work.  
[PERSON] and [PERSON] conceived and designed the experiments; [PERSON] performed the experiment described in Section 2; [PERSON] and [PERSON] performed the experiments described in Sections 3-5; all authors were involved in the discussion of the results; [PERSON] and [PERSON] wrote the manuscript and prepared the figures; and all authors reviewed the manuscript and approved it for publication.  
The authors declare no conflict of interest.
## 2 Dynamic Model of a Robot Manipulator and Fault  
The dynamics of an n-degree-of-freedom (DOF) robot manipulator are defined as:  
\[\mathbf{M}(q)\ddot{q}+\mathbf{C}(q,\dot{q})\dot{q}+\mathbf{G}(q)+\mathbf{F}_{ f}(\dot{q})=\tau \tag{1}\]  
where \(\ddot{q},\dot{q},q\in\Re^{n}\) are the vectors for joint acceleration, velocity, and position, respectively. \(\mathbf{M}(q)\in\Re^{n\times n}\), \(\mathbf{C}(q,\dot{q})\in\Re^{n\times n}\), and \(\mathbf{G}(q)\in\Re^{n}\) represent the inertia matrix, the centripetal and Coriolis matrix, and the vector of gravitation force, respectively. \(\mathbf{F}_{f}(\dot{q})\in\Re^{n}\) is the vector of friction term which includes a viscous friction and a dynamic friction, and \(\tau\in\Re^{n}\) is the vector of torque at the joints.  
In practice, the dynamic model of a robot is not known exactly, so Equation (1) can be written as:  
\[(\mathbf{M}(q)+\Delta M(q))\ddot{q}+(\mathbf{C}(q,\dot{q})+\Delta C(q,\dot{q}) )\dot{q}+(\mathbf{G}(q)+\Delta G(q))+(\mathbf{F}_{f}(\dot{q})+\Delta F_{f}( \dot{q}))+\delta=\tau \tag{2}\]  
where \(\Delta M,\Delta C,\Delta G\), and \(\Delta F\) are unknown dynamic uncertainties, and \(\delta\) is an unknown external disturbance. \(\mathbf{M}(q)\), \(\mathbf{C}(q,\dot{q})\), \(\mathbf{G}(q)\) and \(\mathbf{F}(\dot{q})\) are estimates of \(\mathbf{M}(q),\mathbf{C}(q,\dot{q}),\mathbf{G}(q)\) and \(\mathbf{F}(\dot{q})\). Thus, Equation (2) can be rewritten as:  
\[\mathbf{M}(q)\ddot{q}+\mathbf{C}(q,\dot{q})\dot{q}+\mathbf{G}(q)+\mathbf{F}_{ f}(\dot{q})+\Psi(q,\dot{q},\ddot{q},t)=\tau \tag{3}\]  
where \(\Psi(q,\dot{q},\ddot{q},t)=\Delta M\ddot{q}+\Delta C\dot{q}+\Delta G+\Delta F+\delta\).  
In general, actuator faults can be divided into two types: bias faults and gain faults. In a robot manipulator, these are known as loss-of-effectiveness and lock-in-place faults. In practice, both kinds of actuator faults commonly occur. Therefore, the total torque including both kinds of actuator faults can be comprehensively described as:  
\[\tau^{t}=(I-\boldsymbol{\rho}(t))\tau+\mathbf{f}(t)\ (t>t_{f}) \tag{4}\]  
where \(\mathbf{f}(t)=diag(f_{i})\in\Re^{n}\) denotes a bounded signal. \(\boldsymbol{\rho}(t)=diag(p_{i}(t))\in\Re^{n\times n}\), \(0\leq p_{i}(t)<1\)\((i=1,2,\ldots,n)\), which is unknown, denotes the remaining control rate. \(I\in\Re^{n\times n}\) is the identity matrix, and \(t_{f}\) is the time of occurrence of each fault.  
Substituting Equation (4) into Equation (3), the dynamics model of an n-degree-of-freedom robot manipulator with actuator faults can be written as:  
\[\mathbf{M}(q)\ddot{q}+\mathbf{C}(q,\dot{q})\dot{q}+\mathbf{G}(q)+\mathbf{F}_{f}( \dot{q})+\psi(q,\dot{q},\ddot{q},t)=(I-\rho(t))\tau+\mathbf{f}(t) \tag{5}\]