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Fig. 2 Reaction wheel mathematical model

Figure 2 Reaction wheel mathematical model

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where J, , ly and I,, represent the spacecraft moment of inertia. @,, @,, and w, are the spacecraft’s angular velocities in body-fixed axes toward inertial coordinate system along x, y, and z axes, respectively. Tz,, Tay, and Tg, represent the disturbances torques, which act on the spacecraft about roll, pitch, and yaw axis respectively. t,, t,, and T, represent the torque due to the motion of the wheel on each axis. To get the space- craft’s actual attitude, which are the Euler angles roll, pitch, and yaw, the Eqs. 1, 2, and 3 shall be integrated twice. as actuators are described by Euler equations as in [28] as follows:
where J, , ly and I,, represent the spacecraft moment of inertia. @,, @,, and w, are the spacecraft’s angular velocities in body-fixed axes toward inertial coordinate system along x, y, and z axes, respectively. Tz,, Tay, and Tg, represent the disturbances torques, which act on the spacecraft about roll, pitch, and yaw axis respectively. t,, t,, and T, represent the torque due to the motion of the wheel on each axis. To get the space- craft’s actual attitude, which are the Euler angles roll, pitch, and yaw, the Eqs. 1, 2, and 3 shall be integrated twice. as actuators are described by Euler equations as in [28] as follows:
Fig. 1 Attitude control subsystem block diagram craft’s actual attitude, which are the Euler angles roll, pitch, and yaw, the Eqs. 1, 2, and
Fig. 1 Attitude control subsystem block diagram craft’s actual attitude, which are the Euler angles roll, pitch, and yaw, the Eqs. 1, 2, and
Table 1 ITHACO type-A RW main parameters In Eq. 7, b; and w; , are the bias and the weight of the ith neuron respectively. There- fore, to define the proposed RBFNN-based reaction wheel model, it is mandatory to de- termine some critical parameters. These parameters include the number of input neurons, number of hidden neurons, output layer’s neurons, and the weights of all neu- rons. In addition, other important parameters shall be tuned like the centers and the widths of the hidden neurons. Generally, the number of the problem inputs will deter- mine the number of input layer neurons [32]. Thus, the input layer of the proposed RBFNN-based reaction wheel model comprises a single neuron that represents the torque command voltage. Furthermore, the number of the output layer neurons is
Table 1 ITHACO type-A RW main parameters In Eq. 7, b; and w; , are the bias and the weight of the ith neuron respectively. There- fore, to define the proposed RBFNN-based reaction wheel model, it is mandatory to de- termine some critical parameters. These parameters include the number of input neurons, number of hidden neurons, output layer’s neurons, and the weights of all neu- rons. In addition, other important parameters shall be tuned like the centers and the widths of the hidden neurons. Generally, the number of the problem inputs will deter- mine the number of input layer neurons [32]. Thus, the input layer of the proposed RBFNN-based reaction wheel model comprises a single neuron that represents the torque command voltage. Furthermore, the number of the output layer neurons is
determined corresponding to the number of model outputs. Because the reaction wheel has only one output, which is the generated torque, thus the output layer has a single neuron. The number of hidden layer neurons has a paramount impact on the RBFNN performance. Generally speaking, the more the neurons in the hidden layer, the better the network accuracy [33]. However, the addition of hidden neurons after the right number is reached will not improve the network accuracy, but increase the computa- tional power and architectural complexity. Therefore, the optimal number of neurons in the hidden layer needs to be justified experimentally and it is based on the network designer experience [34] as will be introduced in the experimental results and discus- sions section. Furthermore, the centers, widths, and weights between the hidden neu- rons and the output layer shall be estimated. Thus, this research proposes an enhanced version of QPSO, which is EQPSO-2M to estimate the optimal values of the centers, widths, and weights. determined corresponding to the number of model outputs. Because the reaction wheel
determined corresponding to the number of model outputs. Because the reaction wheel has only one output, which is the generated torque, thus the output layer has a single neuron. The number of hidden layer neurons has a paramount impact on the RBFNN performance. Generally speaking, the more the neurons in the hidden layer, the better the network accuracy [33]. However, the addition of hidden neurons after the right number is reached will not improve the network accuracy, but increase the computa- tional power and architectural complexity. Therefore, the optimal number of neurons in the hidden layer needs to be justified experimentally and it is based on the network designer experience [34] as will be introduced in the experimental results and discus- sions section. Furthermore, the centers, widths, and weights between the hidden neu- rons and the output layer shall be estimated. Thus, this research proposes an enhanced version of QPSO, which is EQPSO-2M to estimate the optimal values of the centers, widths, and weights. determined corresponding to the number of model outputs. Because the reaction wheel
Algorithm 1: Pseudocode for the proposed EQPSO-2M algorithm where r, and r2 represent two different arrays of uniform distribution random numbers. The two consecutive mutations will help the particles to explore extensively different re- gions of the search space to find the best positions. Thus, this is to enhance the convergence speed of the QPSO and to avoid premature convergence. Applying the diversity control and the two successive single dimension Gaussian mutations will avoid the premature conver- gence that may occur in the conventional QPSO. Moreover, these two processes can en- hance the convergence speed of QPSO and prevent the algorithm from trapping in local minima. The pseudocode for the proposed EQPSO-2M is shown in Algorithm 1 as below:
Algorithm 1: Pseudocode for the proposed EQPSO-2M algorithm where r, and r2 represent two different arrays of uniform distribution random numbers. The two consecutive mutations will help the particles to explore extensively different re- gions of the search space to find the best positions. Thus, this is to enhance the convergence speed of the QPSO and to avoid premature convergence. Applying the diversity control and the two successive single dimension Gaussian mutations will avoid the premature conver- gence that may occur in the conventional QPSO. Moreover, these two processes can en- hance the convergence speed of QPSO and prevent the algorithm from trapping in local minima. The pseudocode for the proposed EQPSO-2M is shown in Algorithm 1 as below:
Then again, decreasing the number of particles to a specific limit might cause the
Then again, decreasing the number of particles to a specific limit might cause the
Table 2 QPSO-trained RBFNN performance at different hidden layer neurons where NSR addresses the number of successful runs and TNR is the total number of runs, which are 20 runs in runs in all the experiments. Besides, it is considered for the single run to be effective at the end of 3000 iterations in a manner that MSE ~6.5E-7. The results acquired from these trials are given in Table 3 and Fig. 6. As per the results in Table 3, it can be observed that the QPSO algorithm with 20 particles has a 50% success rate. The STD of the best fitness value for the four cases is around something similar. It can be observed from Fig. 6 that the four cases have approximately the same average fitness values. Although, QPSO with 50 particles has a slightly fast convergence speed, but expanding the population size will increment the computational time and the calculation intricacy. Consequently, we chose the population size to be 20 particles. The results acquired from these trials are given in Table 3 and Fig. 6. As per the results
Table 2 QPSO-trained RBFNN performance at different hidden layer neurons where NSR addresses the number of successful runs and TNR is the total number of runs, which are 20 runs in runs in all the experiments. Besides, it is considered for the single run to be effective at the end of 3000 iterations in a manner that MSE ~6.5E-7. The results acquired from these trials are given in Table 3 and Fig. 6. As per the results in Table 3, it can be observed that the QPSO algorithm with 20 particles has a 50% success rate. The STD of the best fitness value for the four cases is around something similar. It can be observed from Fig. 6 that the four cases have approximately the same average fitness values. Although, QPSO with 50 particles has a slightly fast convergence speed, but expanding the population size will increment the computational time and the calculation intricacy. Consequently, we chose the population size to be 20 particles. The results acquired from these trials are given in Table 3 and Fig. 6. As per the results
The SR is computed as follows:
The SR is computed as follows:
Table 3 Effect of swarm size change on QPSO-RBFNN performance
Table 3 Effect of swarm size change on QPSO-RBFNN performance
Fig. 6 Convergence speed of QPSO algorithm at different population size
Fig. 6 Convergence speed of QPSO algorithm at different population size
Table 4 AGPSO coefficients updating strategies iments and have the most exceedingly awful mean best fitness. The convergence speed is a significant factor that can be utilized to assess optimization problems. Therefore, to additionally assess the performance of the pro- posed EQPSO-2M strategy, its convergence speed is compared with different algo- rithms. Figure 7 shows the average convergence curves that are plotted in a logarithmic scale for the proposed EQPSO-2M and different algorithms. As can be seen from Fig. 7, that the average fitness of the EQPSO-2 M algorithm is the minimum. In addition, the EQPSO-2M algorithm shows the best convergence speed compared with different peers. It converges to an average fitness value of ~ 6.5E-07 at around 300 iterations. Be- sides, EQPSO-1M has the convergence speed after the proposed algorithm. Thanks for applying the diversity control and the single mutation. Based on the convergence speed, the second-best result is obtained by the PSO-In algorithm. This improvement in the results of the PSO-In ensures strong relation between the PSO convergence behavior and the control parameters selection. The acquired results demonstrate that the other algorithms (ELSPO, AGPSO, QPSO, MPSO, and TAPSO) show less performance and they have a slow convergence to the optimal minimum value. These algorithms have a delay in converging to the minimum best values due to stagnation conditions. The main reason for the high convergence rate of the proposed EQPSO-2M is that the utilization of the two successive mutations mechanism helps the swarm in each iter- ation to explore the search space extensively near the personal best position to find more best positions. In addition, the utilization of diversity control improves the diver- sity of the particles. Consequently, it reveals a better performance and more efficacy in jumping out from local minima in case of stagnation, and hence obtaining more high- optimization problems. Therefore, to additionally assess the performance of the pro-
Table 4 AGPSO coefficients updating strategies iments and have the most exceedingly awful mean best fitness. The convergence speed is a significant factor that can be utilized to assess optimization problems. Therefore, to additionally assess the performance of the pro- posed EQPSO-2M strategy, its convergence speed is compared with different algo- rithms. Figure 7 shows the average convergence curves that are plotted in a logarithmic scale for the proposed EQPSO-2M and different algorithms. As can be seen from Fig. 7, that the average fitness of the EQPSO-2 M algorithm is the minimum. In addition, the EQPSO-2M algorithm shows the best convergence speed compared with different peers. It converges to an average fitness value of ~ 6.5E-07 at around 300 iterations. Be- sides, EQPSO-1M has the convergence speed after the proposed algorithm. Thanks for applying the diversity control and the single mutation. Based on the convergence speed, the second-best result is obtained by the PSO-In algorithm. This improvement in the results of the PSO-In ensures strong relation between the PSO convergence behavior and the control parameters selection. The acquired results demonstrate that the other algorithms (ELSPO, AGPSO, QPSO, MPSO, and TAPSO) show less performance and they have a slow convergence to the optimal minimum value. These algorithms have a delay in converging to the minimum best values due to stagnation conditions. The main reason for the high convergence rate of the proposed EQPSO-2M is that the utilization of the two successive mutations mechanism helps the swarm in each iter- ation to explore the search space extensively near the personal best position to find more best positions. In addition, the utilization of diversity control improves the diver- sity of the particles. Consequently, it reveals a better performance and more efficacy in jumping out from local minima in case of stagnation, and hence obtaining more high- optimization problems. Therefore, to additionally assess the performance of the pro-
Table 5 Comparison results between EQPSO-2M and other algorithms
Table 5 Comparison results between EQPSO-2M and other algorithms
Fig. 7 Average fitness curves for different algorithms
Fig. 7 Average fitness curves for different algorithms
Table 6 p-values of Wilcoxon rank sum test comparison between the results of EQPSO-2M and other algorithms
Table 6 p-values of Wilcoxon rank sum test comparison between the results of EQPSO-2M and other algorithms
Fig. 8 Output of reaction wheel model compared to the actual output at 10° (a) Roll output, (b) Pitch output, (¢) Yaw output
Fig. 8 Output of reaction wheel model compared to the actual output at 10° (a) Roll output, (b) Pitch output, (¢) Yaw output
Fig. 9 Performance of the developed RW models measured at tilting angle from — 90° to 90° roll, pitch, and yaw algorithm. Statistical benchmark rules dependent on the SR, convergence speed, and This research proposes another modeling scheme for the spacecraft reaction wheel util- izing RBFNN with an enhanced version of QPSO. In light of the principles of the diver- sity control and mutation strategy, EQPSO-2M is proposed to ameliorate the RBFNN parameters. In this way, the estimation of the RBFNN parameters is demonstrated as an optimization problem that was settled in terms of the EQPSO-2M algorithm. Add- itionally, the performance of the proposed algorithm was compared with other strat- egies like ELPSO, AGPSO, PSO-In, MPSO, TAPSO, and the conventional QPSO algorithm. Statistical benchmark rules dependent on the SR, convergence speed, and
Fig. 9 Performance of the developed RW models measured at tilting angle from — 90° to 90° roll, pitch, and yaw algorithm. Statistical benchmark rules dependent on the SR, convergence speed, and This research proposes another modeling scheme for the spacecraft reaction wheel util- izing RBFNN with an enhanced version of QPSO. In light of the principles of the diver- sity control and mutation strategy, EQPSO-2M is proposed to ameliorate the RBFNN parameters. In this way, the estimation of the RBFNN parameters is demonstrated as an optimization problem that was settled in terms of the EQPSO-2M algorithm. Add- itionally, the performance of the proposed algorithm was compared with other strat- egies like ELPSO, AGPSO, PSO-In, MPSO, TAPSO, and the conventional QPSO algorithm. Statistical benchmark rules dependent on the SR, convergence speed, and
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