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Calculating the values of K,, Ky, Ma, M,, and M, through Equation (30), V was found to be a negative semi-definite number, indicating the asymptotic stability of the closed-loop system according to the LaSalle's theorem [8].

Figure 7 Calculating the values of K,, Ky, Ma, M,, and M, through Equation (30), V was found to be a negative semi-definite number, indicating the asymptotic stability of the closed-loop system according to the LaSalle's theorem [8].

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Fig. 1. A platform with the Stewart mechanism. Where /, n, F and fare the numbers of arms, joints, and DOFs of the mechanism and the 7? joint, respectively [1]. Based on the Grubler formula, the mechanism was found to go through six DOFs.
Fig. 1. A platform with the Stewart mechanism. Where /, n, F and fare the numbers of arms, joints, and DOFs of the mechanism and the 7? joint, respectively [1]. Based on the Grubler formula, the mechanism was found to go through six DOFs.
Fig. 2. A schematic model of a Gough-Stewart
Fig. 2. A schematic model of a Gough-Stewart
The vectors Ti and Bi on the End Effector and the Base could be calculated through Equations (2) and (3), respectively. The values of the angles A; and v, are shown in Table 1.
The vectors Ti and Bi on the End Effector and the Base could be calculated through Equations (2) and (3), respectively. The values of the angles A; and v, are shown in Table 1.
Majlesi Journal of Mechatronic Systems The position of the End Effector was obtained by a position vector and a rotation matrix, with the rotation matrix defined by the roll, pitch, and yaw angles [3]. Here the rotation matrix of the End Effector was obtained from the base platform coordinate system [3] while the position vector was one from the coordinate system ‘Bxyz’ to the coordinate system ‘Txyz’. The rotation matrix and the position vector are given as Equations (4) and (5) [3].
Majlesi Journal of Mechatronic Systems The position of the End Effector was obtained by a position vector and a rotation matrix, with the rotation matrix defined by the roll, pitch, and yaw angles [3]. Here the rotation matrix of the End Effector was obtained from the base platform coordinate system [3] while the position vector was one from the coordinate system ‘Bxyz’ to the coordinate system ‘Txyz’. The rotation matrix and the position vector are given as Equations (4) and (5) [3].
Fig. 4. Demonstration of the kinematics of a Gough- Stewart platform.
Fig. 4. Demonstration of the kinematics of a Gough- Stewart platform.
Table 2. Numerical parameters of the considered Gough -Stewart platform.
Table 2. Numerical parameters of the considered Gough -Stewart platform.
Fig. 5. Subsystems related to the Adams-sub. 4. CONTROL STRATEGY Within the framework of the MRAC, the desired response characteristics are presented as a reference model on which basis the controller seeks to control the system output [7]. In this study, a second-order system was considered as the reference model for the analyzed robot. The overall transfer function of the reference model was as follows:
Fig. 5. Subsystems related to the Adams-sub. 4. CONTROL STRATEGY Within the framework of the MRAC, the desired response characteristics are presented as a reference model on which basis the controller seeks to control the system output [7]. In this study, a second-order system was considered as the reference model for the analyzed robot. The overall transfer function of the reference model was as follows:
Fig. 8. ADAMS-MATLAB co-simulation process. T This section presents the results of simulating the adaptive controller design in MATLAB software. Fig. 7 presents the diagram block of the closed-loop control under the studied Stewart platform system. For the sake of simulation, firstly, a dynamic model was built in Adams and then exported to MATLAB using the control plugin (Fig. 8). Following this section, firstly, the results of the MRAC controller were compared to those of the reference model to check for the controller performance. Secondly, an adaptive control was optimized by a genetic algorithm to demonstrate its capability.
Fig. 8. ADAMS-MATLAB co-simulation process. T This section presents the results of simulating the adaptive controller design in MATLAB software. Fig. 7 presents the diagram block of the closed-loop control under the studied Stewart platform system. For the sake of simulation, firstly, a dynamic model was built in Adams and then exported to MATLAB using the control plugin (Fig. 8). Following this section, firstly, the results of the MRAC controller were compared to those of the reference model to check for the controller performance. Secondly, an adaptive control was optimized by a genetic algorithm to demonstrate its capability.
Fig. 9. Square response for Out 1 and reference model. A comparison of the proposed system (red) with the reference model (blue) for a square input signal is shown in Fig. 9 [8], indicating a good agreement between the two, with the corresponding error shown in Fig. 10. Fig. 10 shows that the system error decreased strongly with time. The control signal exhibited regular fluctuations, as shown in Fig. 11. Fig. 12 shows that the proposed system could still reproduce the reference model appropriately in presence of white noise and disturbance. Fig.s 13 and 14 show the introduced white noise and disturbance into the system, respectively. Fig.s 9 through 12 indicate the capability of the proposed MRAC controller to respond to possible noise and disturbance signals appropriately. In the next section, the proposed methodology is optimized by a genetic algorithm to further expand the capabilities of the MRAC [9].
Fig. 9. Square response for Out 1 and reference model. A comparison of the proposed system (red) with the reference model (blue) for a square input signal is shown in Fig. 9 [8], indicating a good agreement between the two, with the corresponding error shown in Fig. 10. Fig. 10 shows that the system error decreased strongly with time. The control signal exhibited regular fluctuations, as shown in Fig. 11. Fig. 12 shows that the proposed system could still reproduce the reference model appropriately in presence of white noise and disturbance. Fig.s 13 and 14 show the introduced white noise and disturbance into the system, respectively. Fig.s 9 through 12 indicate the capability of the proposed MRAC controller to respond to possible noise and disturbance signals appropriately. In the next section, the proposed methodology is optimized by a genetic algorithm to further expand the capabilities of the MRAC [9].
Majlesi Journal of Mechatronic Systems
Majlesi Journal of Mechatronic Systems
Fig. 12. Square responses contaminated with White Noise and Disturbance for Out 1. ~ Genetic algorithm (GA) ‘refers: to a | family of evolutionary algorithms that are inspired by natural evolutionary phenomena such as inheritance, biology mutation, and Darwin's law of evolution to find an optimal formula for predicting or matching some pattern . In this work, the GA was used for optimizing the controller parameters. The main steps of the GA are
Fig. 12. Square responses contaminated with White Noise and Disturbance for Out 1. ~ Genetic algorithm (GA) ‘refers: to a | family of evolutionary algorithms that are inspired by natural evolutionary phenomena such as inheritance, biology mutation, and Darwin's law of evolution to find an optimal formula for predicting or matching some pattern . In this work, the GA was used for optimizing the controller parameters. The main steps of the GA are
Fig. 13. Added White Noise to the input.
Fig. 13. Added White Noise to the input.
Fig. 11. The control signal regenerated by MRAC.
Fig. 11. The control signal regenerated by MRAC.
Related topics:
Lyapunov Stability TheoryModel reference adaptive controlGough Stewart Platform
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