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Search Results (428)

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Keywords = nonlinearity errors compensation

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21 pages, 1565 KiB  
Article
Preview-Based Optimal Control for Trajectory Tracking of Fully-Actuated Marine Vessels
by Xiaoling Liang, Jiang Wu, Hao Xie and Yanrong Lu
Mathematics 2024, 12(24), 3942; https://doi.org/10.3390/math12243942 - 14 Dec 2024
Viewed by 404
Abstract
In this paper, the problem of preview optimal control for second-order nonlinear systems for marine vessels is discussed on a fully actuated dynamic model. First, starting from a kinematic and dynamic model of a three-degrees-of-freedom (DOF) marine vessel, we derive a fully actuated [...] Read more.
In this paper, the problem of preview optimal control for second-order nonlinear systems for marine vessels is discussed on a fully actuated dynamic model. First, starting from a kinematic and dynamic model of a three-degrees-of-freedom (DOF) marine vessel, we derive a fully actuated second-order dynamic model that involves only the ship’s position and yaw angle. Subsequently, through the higher-order systems methodology, the nonlinear terms in the system were eliminated, transforming the system into a one-order parameterized linear system. Next, we designed an internal model compensator for the reference signal and constructed a new augmented error system based on this compensator. Then, using optimal control theory, we designed the optimal preview controller for the parameterized linear system and the corresponding feedback parameter matrices, which led to the preview controller for the original second-order nonlinear system. Finally, a numerical simulation indicates that the controller designed in this paper is highly effective. Full article
(This article belongs to the Special Issue Analysis and Applications of Control Systems Theory)
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Figure 1

Figure 1
<p>The coordinate frames of the target model.</p>
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<p>The block diagram for preview control design.</p>
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<p>Surge Trajectory Tracking Response.</p>
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<p>Sway trajectory tracking response.</p>
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<p>Yaw trajectory tracking response.</p>
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<p>Surge tracking error curves.</p>
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<p>Sway tracking error curves.</p>
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<p>Yaw tracking error curves.</p>
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<p>Sum of surge tracking errors.</p>
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<p>Sum of sway tracking errors.</p>
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<p>Sum of yaw tracking errors.</p>
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<p>Surge force control effort for trajectory tracking.</p>
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<p>Sway force control effort for trajectory tracking.</p>
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<p>Yaw moment control effort for trajectory tracking.</p>
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<p>Sum of surge force.</p>
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<p>Sum of sway force.</p>
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<p>Sum of yaw moment.</p>
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<p>Surge velocity tracking response.</p>
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<p>Sway velocity tracking response.</p>
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<p>Yaw angular velocity tracking response.</p>
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<p>The entire physical processes.</p>
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<p>Surge response under step input.</p>
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<p>Sway response under step input.</p>
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<p>Yaw response under step input.</p>
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18 pages, 577 KiB  
Article
Reinforcement-Learning-Based Fixed-Time Prescribed Performance Consensus Control for Stochastic Nonlinear MASs with Sensor Faults
by Zhenyou Wang, Xiaoquan Cai, Ao Luo, Hui Ma and Shengbing Xu
Sensors 2024, 24(24), 7906; https://doi.org/10.3390/s24247906 - 11 Dec 2024
Viewed by 315
Abstract
This paper proposes the fixed-time prescribed performance optimal consensus control method for stochastic nonlinear multi-agent systems with sensor faults. The consensus error converges to the prescribed performance bounds in fixed-time by an improved performance function and coordinate transformation. Due to the unknown faults [...] Read more.
This paper proposes the fixed-time prescribed performance optimal consensus control method for stochastic nonlinear multi-agent systems with sensor faults. The consensus error converges to the prescribed performance bounds in fixed-time by an improved performance function and coordinate transformation. Due to the unknown faults in sensors, the system states cannot be gained correctly; therefore, an adaptive compensation strategy is constructed based on the approximation capabilities of neural networks to solve the negative impact of sensor failures. The reinforcement-learning-based backstepping method is proposed to realize the optimal control of the system. Utilizing Lyapunov stability theory, it is shown that the designed controller enables the consensus error to converge to the prescribed performance bounds in fixed time and that all signals in the closed-loop system are bounded in probability. Finally, the simulation results prove the effectiveness of the proposed method. Full article
(This article belongs to the Section Fault Diagnosis & Sensors)
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Figure 1

Figure 1
<p>Block diagram of the overall control system.</p>
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<p>Directed communication topology graph.</p>
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<p>Schematic diagram of <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> curves [<a href="#B44-sensors-24-07906" class="html-bibr">44</a>,<a href="#B46-sensors-24-07906" class="html-bibr">46</a>].</p>
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<p>Schematic diagram of <math display="inline"><semantics> <msubsup> <mi>y</mi> <mrow> <mi>i</mi> </mrow> <mi>f</mi> </msubsup> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mi>r</mi> </msub> </semantics></math> curves.</p>
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<p>Schematic diagram of <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> </semantics></math> curvies.</p>
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<p>Schematic diagram of <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> curvies.</p>
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13 pages, 4339 KiB  
Article
FPGA Implementation for 24.576-Gbit/s Optical PAM4 Signal Transmission with MLP-Based Digital Pre-Distortion
by Sheng Hu, Tianqi Zheng, Chengzhen Bian, Xiongwei Yang, Xinda Sun, Zonghui Zhu, Yumeng Gou, Yuanxiao Meng, Jie Zhang, Jingtao Ge, Yichen Li and Kaihui Wang
Sensors 2024, 24(23), 7872; https://doi.org/10.3390/s24237872 - 9 Dec 2024
Viewed by 407
Abstract
In this work, we implemented a short-reach real-time optical communication system using MLP for pre-distortion. Lookup table (LUT) algorithms are commonly employed for pre-distortion in intensity modulation and direct detection (IM/DD) systems. However, storage limitations typically restrict the LUT pattern length to 9, [...] Read more.
In this work, we implemented a short-reach real-time optical communication system using MLP for pre-distortion. Lookup table (LUT) algorithms are commonly employed for pre-distortion in intensity modulation and direct detection (IM/DD) systems. However, storage limitations typically restrict the LUT pattern length to 9, limiting its effectiveness in compensating for nonlinear effects. A multilayer perceptron (MLP) can overcome this limitation by predicting errors and generating pre-distorted signals, thus replacing the extensive storage requirements of LUTs with minimal computational resources. The MLP-based digital pre-distortion (MLP-DPD) technique enables the creation of long-pattern LUTs for improved nonlinear compensation. In this work, an MLP-DPD scheme was implemented on a field-programmable gate array (FPGA). The FPGA was used to generate a 14.7456 GBaud pre-distorted pulse amplitude modulation 4-level (PAM4) signal. This signal was then transmitted over 20 km of standard single-mode fiber (SSMF). At the receiver, the parallel constant modulus algorithm (PCMA) was applied for signal processing. The bit error rate (BER) achieved met the 2.4 × 10−2 threshold for soft-decision forward error correction (SD-FEC), enabling a net transmission bit rate of 24.576 Gbit/s. This approach demonstrates the feasibility of using MLP-DPD for effective nonlinear compensation in high-speed optical communication systems with limited resources. Full article
(This article belongs to the Special Issue Challenges and Future Trends in Optical Communications)
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Figure 1
<p>Principle of LUT-DPD.</p>
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<p>Principle of MLP-DPD.</p>
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<p>Plot of PAA-tanh versus the original tanh when divided into different numbers of segments.</p>
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<p>Comparison of the errors of the two when divided into different segments.</p>
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<p>Principle of 64-channel parallel MLP-DPD.</p>
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<p>Experimental setup and hardware implementation.</p>
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<p>Experimental result graph and heatmap of received PAM4 signal. (<b>a</b>) Relationship between the BER curve and received optical power under BTB conditions. (<b>b</b>) BER curve after 20 km SSMF transmission.</p>
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<p>Real-time BER measurement within one hour (ROP is −1 dBm).</p>
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14 pages, 715 KiB  
Article
High-Precision Digital-to-Time Converter with High Dynamic Range for 28 nm 7-Series Xilinx FPGA and SoC Devices
by Fabio Garzetti, Nicola Lusardi, Nicola Corna, Gabriele Fiumicelli, Federico Cattaneo, Gabriele Bonanno, Andrea Costa, Enrico Ronconi and Angelo Geraci
Electronics 2024, 13(23), 4825; https://doi.org/10.3390/electronics13234825 - 6 Dec 2024
Viewed by 372
Abstract
Over the last ten years, the need for high-resolution time-domain digital signal production has grown exponentially. More than ever, applications call for a digital-to-time converter (DTC) that is extremely accurate and precise. Skew compensation and camera shutter operation represent just a few examples [...] Read more.
Over the last ten years, the need for high-resolution time-domain digital signal production has grown exponentially. More than ever, applications call for a digital-to-time converter (DTC) that is extremely accurate and precise. Skew compensation and camera shutter operation represent just a few examples of such applications. The advantages of adopting a flexible and rapid time-to-market strategy focused on fast prototyping using programmable logic devices—such as field-programmable gate arrays (FPGAs) and system-on-chip (SoC)—have become increasingly evident. These benefits outweigh those of performance-focused yet expensive application-specific integrated circuits (ASICs). Despite the availability of various architectures, the high non-recurring engineering (NRE) costs make them unsuitable for low-volume production, especially in research or prototyping environments. To address this trend, we introduce an innovative DTC IP-Core with a resolution, also known as least significant bit (LSB), of 52 ps, compatible with all Xilinx 7-Series FPGAs and SoCs. Measurements have been performed on a low-end Artix-7 XC7A100TFTG256-2, guaranteeing a jitter lower than 50 ps r.m.s. and offering a high dynamic range up to 56 ms. With resource utilization below 1% and a dynamic power dissipation of 285 mW for our target FPGA, the design maintains excellent differential and integral nonlinearity errors (DNL/INL) of 1.19 LSB and 1.56 LSB, respectively. Full article
(This article belongs to the Special Issue Feature Papers in Circuit and Signal Processing)
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Figure 1
<p>Schematic plot of a PDL.</p>
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<p>Schematics (<b>a</b>) and waveforms (<b>b</b>) of Nutt interpolation and waveforms.</p>
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<p>(<b>a</b>) Register-transfer level (RTL) representation of the 2-bit circular buffer used to generate the CE signals. (<b>b</b>) Two BUFGCEs are used to generate the 180°-shifted clocks.</p>
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<p>Schematic (<b>a</b>) and waveforms (<b>b</b>) of the proposed dual-clock synchronous coarse logic.</p>
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<p>Overview of the proposed FPGA-based Nutt-interpolated DTC, where the dual-clock counter is used as coarse logic and the IDELAYE2 is employed as the fine interpolator.</p>
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<p>Jitter of the DTC with <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mrow> <mi>I</mi> <mi>D</mi> <mi>E</mi> <mi>L</mi> <mi>A</mi> <mi>Y</mi> <mi>C</mi> <mi>T</mi> <mi>R</mi> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math> MHz, <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>S</mi> <mi>B</mi> <mo>=</mo> <mn>52.083</mn> </mrow> </semantics></math> ps in <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>÷</mo> <mn>1023</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>S</mi> <mi>B</mi> </mrow> </semantics></math> dynamic range at 25 °C.</p>
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<p>DNL of the DTC with <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mrow> <mi>I</mi> <mi>D</mi> <mi>E</mi> <mi>L</mi> <mi>A</mi> <mi>Y</mi> <mi>C</mi> <mi>T</mi> <mi>R</mi> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math> MHz, <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>S</mi> <mi>B</mi> <mo>=</mo> <mn>52.083</mn> </mrow> </semantics></math> ps in <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>÷</mo> <mn>1023</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>S</mi> <mi>B</mi> </mrow> </semantics></math> dynamic range at 25 °C.</p>
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<p>INL of the DTC with <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mrow> <mi>I</mi> <mi>D</mi> <mi>E</mi> <mi>L</mi> <mi>A</mi> <mi>Y</mi> <mi>C</mi> <mi>T</mi> <mi>R</mi> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math> MHz, <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>S</mi> <mi>B</mi> <mo>=</mo> <mn>52.083</mn> </mrow> </semantics></math> ps in <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>÷</mo> <mn>1023</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>S</mi> <mi>B</mi> </mrow> </semantics></math> dynamic range at 25 °C.</p>
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<p>Block diagram of the measurement setup.</p>
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<p>Block diagram of IDELAYE2-based PDL-DTC used as benchmark.</p>
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<p>Jitter of the IDELAYE2-based PDL-DTC with <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mrow> <mi>I</mi> <mi>D</mi> <mi>E</mi> <mi>L</mi> <mi>A</mi> <mi>Y</mi> <mi>C</mi> <mi>T</mi> <mi>R</mi> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math> MHz, <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>S</mi> <mi>B</mi> <mo>=</mo> <mn>52.083</mn> </mrow> </semantics></math> ps in <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>÷</mo> <mn>960</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>S</mi> <mi>B</mi> </mrow> </semantics></math> dynamic range at 25 °C.</p>
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<p>DNL of the IDELAYE2-based PDL-DTC with <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mrow> <mi>I</mi> <mi>D</mi> <mi>E</mi> <mi>L</mi> <mi>A</mi> <mi>Y</mi> <mi>C</mi> <mi>T</mi> <mi>R</mi> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math> MHz, <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>S</mi> <mi>B</mi> <mo>=</mo> <mn>52</mn> </mrow> </semantics></math>.083 ps in <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>÷</mo> <mn>960</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>S</mi> <mi>B</mi> </mrow> </semantics></math> dynamic range at 25 °C.</p>
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<p>INL of the IDELAYE2-based PDL-DTC with <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mrow> <mi>I</mi> <mi>D</mi> <mi>E</mi> <mi>L</mi> <mi>A</mi> <mi>Y</mi> <mi>C</mi> <mi>T</mi> <mi>R</mi> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mn>300</mn> </mrow> </semantics></math> MHz, <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>S</mi> <mi>B</mi> <mo>=</mo> <mn>52.083</mn> </mrow> </semantics></math> ps in <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>÷</mo> <mn>960</mn> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>L</mi> <mi>S</mi> <mi>B</mi> </mrow> </semantics></math> dynamic range at 25 °C.</p>
Full article ">
30 pages, 15218 KiB  
Article
Robust Nonlinear Model Predictive Control for the Trajectory Tracking of Skid-Steer Mobile Manipulators with Wheel–Ground Interactions
by Katherine Aro, Leonardo Guevara, Miguel Torres-Torriti, Felipe Torres and Alvaro Prado
Robotics 2024, 13(12), 171; https://doi.org/10.3390/robotics13120171 - 3 Dec 2024
Viewed by 548
Abstract
This paper presents a robust control strategy for trajectory-tracking control of Skid-Steer Mobile Manipulators (SSMMs) using a Robust Nonlinear Model Predictive Control (R-NMPC) approach that minimises trajectory-tracking errors while overcoming model uncertainties and terra-mechanical disturbances. The proposed strategy is aimed at counteracting the [...] Read more.
This paper presents a robust control strategy for trajectory-tracking control of Skid-Steer Mobile Manipulators (SSMMs) using a Robust Nonlinear Model Predictive Control (R-NMPC) approach that minimises trajectory-tracking errors while overcoming model uncertainties and terra-mechanical disturbances. The proposed strategy is aimed at counteracting the effects of disturbances caused by the slip phenomena through the wheel–terrain contact and bidirectional interactions propagated by mechanical coupling between the SSMM base and arm. These interactions are modelled using a coupled nonlinear dynamic framework that integrates bounded uncertainties for the mobile base and arm joints. The model is developed based on principles of full-body energy balance and link torques. Then, a centralized control architecture integrates a nominal NMPC (disturbance-free) and ancillary controller based on Active Disturbance-Rejection Control (ADRC) to strengthen control robustness, operating the full system dynamics as a single robotic body. While the NMPC strategy is responsible for the trajectory-tracking control task, the ADRC leverages an Extended State Observer (ESO) to quantify the impact of external disturbances. Then, the ADRC is devoted to compensating for external disturbances and uncertainties stemming from the model mismatch between the nominal representation and the actual system response. Simulation and field experiments conducted on an assembled Pioneer 3P-AT base and Katana 6M180 robotic arm under terrain constraints demonstrate the effectiveness of the proposed method. Compared to non-robust controllers, the R-NMPC approach significantly reduced trajectory-tracking errors by 79.5% for mobile bases and 42.3% for robot arms. These results highlight the potential to enhance robust performance and resource efficiency in complex navigation conditions. Full article
(This article belongs to the Section Sensors and Control in Robotics)
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Figure 1

Figure 1
<p>Skid-Steer Mobile Manipulator (SSMM) model: The left figure depicts the local coordinate systems and Denavit–Hartenberg (DH) parameters for the SSMM, while the right image shows the physical robot used in this study.</p>
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<p>Scheme of the proposed Robust Nonlinear Model Predictive Control (R-NMPC) strategy.</p>
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<p>Characterization of disturbances used in tests using circular-type reference trajectory.</p>
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<p>Results of performance indexes. Tracking tests for circular trajectory considering linear and angular speed disturbances in the mobile base. The indices <math display="inline"><semantics> <mrow> <mi>C</mi> <msub> <mi>ζ</mi> <mi>b</mi> </msub> <mo>,</mo> <mi>C</mi> <msub> <mi>u</mi> <mi>b</mi> </msub> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>C</mi> <mi>t</mi> <mi>o</mi> <msub> <mi>t</mi> <mi>b</mi> </msub> </mrow> </semantics></math> correspond to the mobile base and the indices <math display="inline"><semantics> <mrow> <mi>C</mi> <msub> <mi>ζ</mi> <mi>a</mi> </msub> <mo>,</mo> <mi>C</mi> <msub> <mi>u</mi> <mi>a</mi> </msub> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>C</mi> <mi>t</mi> <mi>o</mi> <msub> <mi>t</mi> <mi>a</mi> </msub> </mrow> </semantics></math> correspond to the robotic arm of the SSMM.</p>
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<p>Tracking tests were also considered on a circular trajectory, profiling linear and angular speed disturbances on the mobile base and the robotic arm motion. The first top row from left to right depicts the results of tracking a reference trajectory for the mobile base and robot arm. In the next two figures are shown the tracking error in the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>1</mn> </msub> </semantics></math>, and the control effort on the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>1</mn> </msub> </semantics></math>. The orange point and arrow stands for initial position and orientation of the mobile base. The second row from left to right shows the tracking error in the x-coordinate of the mobile base, the control effort for the linear displacement of the mobile base, the tracking error of the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>2</mn> </msub> </semantics></math>, the control effort of the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>2</mn> </msub> </semantics></math>. The third row from left to right presents the tracking error in the y-coordinate of the mobile base, the control effort for the angular displacement of the mobile base, tracking error of the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>3</mn> </msub> </semantics></math>, and the control effort of the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>3</mn> </msub> </semantics></math>.</p>
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<p>Disturbances used while testing the Lemniscata-type trajectory.</p>
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<p>Tracking tests on a Lemniscata trajectory considering linear and angular speed disturbances on the mobile base and the influence of the robotic arm motion. The first top row from left to right consists of the trajectory of the mobile base, the trajectory of the robotic arm, the tracking error in the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>1</mn> </msub> </semantics></math>, and the control effort of the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>1</mn> </msub> </semantics></math>. The orange point and arrow stands for initial position and orientation of the mobile base. The second row from left to right has the tracking error in the x-coordinate of the mobile base, the control effort for the linear displacement of the mobile base, the tracking error of the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>2</mn> </msub> </semantics></math>, and the control effort of the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>2</mn> </msub> </semantics></math>. The third row from left to right consists of the tracking error in the y-coordinate of the mobile base, the control effort for the angular displacement of the mobile base, tracking error of the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>3</mn> </msub> </semantics></math>, and the control effort of the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>3</mn> </msub> </semantics></math>.</p>
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<p>Performance index: Tracking tests for Lemniscata trajectory considering linear and angular speed disturbances in the mobile base. The indices <math display="inline"><semantics> <mrow> <mi>C</mi> <msub> <mi>ζ</mi> <mi>b</mi> </msub> <mo>,</mo> <mi>C</mi> <msub> <mi>u</mi> <mi>b</mi> </msub> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>C</mi> <mi>t</mi> <mi>o</mi> <msub> <mi>t</mi> <mi>b</mi> </msub> </mrow> </semantics></math> correspond to the mobile base and the indices <math display="inline"><semantics> <mrow> <mi>C</mi> <msub> <mi>ζ</mi> <mi>a</mi> </msub> <mo>,</mo> <mi>C</mi> <msub> <mi>u</mi> <mi>a</mi> </msub> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>C</mi> <mi>t</mi> <mi>o</mi> <msub> <mi>t</mi> <mi>a</mi> </msub> </mrow> </semantics></math> correspond to the robotic arm of the SSMM.</p>
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<p>Robustness tests on a circular trajectory considering linear and angular speed disturbances on the mobile base, the influence of the robotic arm motion, and parameter variation. The first top row from left to right consists of the trajectory of the mobile base, the trajectory of the robotic arm, the tracking error in the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>1</mn> </msub> </semantics></math>, and the control effort in the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>1</mn> </msub> </semantics></math>. The orange point and arrow stands for initial position and orientation of the mobile base. The second row from left to right has the tracking error in the x-coordinate of the mobile base, the control effort for the linear displacement of the mobile base, the tracking error of the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>2</mn> </msub> </semantics></math>, and the control effort of the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>2</mn> </msub> </semantics></math>. The third row from left to right consists of the tracking error in the y-coordinate of the mobile base, the control effort for the angular displacement of the mobile base, tracking error of the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>3</mn> </msub> </semantics></math>, and the control effort of the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>3</mn> </msub> </semantics></math>.</p>
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<p>Performance index: Robustness tests for circular trajectory considering linear and angular speed disturbances and parameter variations in the mobile base. The indices <math display="inline"><semantics> <mrow> <mi>C</mi> <msub> <mi>ζ</mi> <mi>b</mi> </msub> <mo>,</mo> <mi>C</mi> <msub> <mi>u</mi> <mi>b</mi> </msub> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>C</mi> <mi>t</mi> <mi>o</mi> <msub> <mi>t</mi> <mi>b</mi> </msub> </mrow> </semantics></math> correspond to the mobile base and the indices <math display="inline"><semantics> <mrow> <mi>C</mi> <msub> <mi>ζ</mi> <mi>a</mi> </msub> <mo>,</mo> <mi>C</mi> <msub> <mi>u</mi> <mi>a</mi> </msub> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>C</mi> <mi>t</mi> <mi>o</mi> <msub> <mi>t</mi> <mi>a</mi> </msub> </mrow> </semantics></math> correspond to the robotic arm of the SSMM.</p>
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<p>Robustness tests on a Lemniscata trajectory considering linear and angular speed disturbances on the mobile base, the influence of the robotic arm motion, and parameter variation. The first top row from left to right consists of the trajectory of the mobile base, the trajectory of the robotic arm, the tracking error in the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>1</mn> </msub> </semantics></math>, and the control effort in the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>1</mn> </msub> </semantics></math>. The orange point and arrow stands for initial position and orientation of the mobile base. The second row from left to right has the tracking error in the x-coordinate of the mobile base, the control effort for the linear displacement of the mobile base, the tracking error of the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>2</mn> </msub> </semantics></math>, and the control effort of the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>2</mn> </msub> </semantics></math>. The third row from left to right consists of the tracking error in the y-coordinate of the mobile base, the control effort for the angular displacement of the mobile base, tracking error of the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>3</mn> </msub> </semantics></math>, and the control effort of the joint coordinate <math display="inline"><semantics> <msub> <mi>θ</mi> <mn>3</mn> </msub> </semantics></math>.</p>
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<p>Performance index to assess robustness while tracking the Lemniscata trajectory, considering linear and angular speed disturbances and parameter variations in the mobile base. The indices <math display="inline"><semantics> <mrow> <mi>C</mi> <msub> <mi>ζ</mi> <mi>b</mi> </msub> <mo>,</mo> <mspace width="4pt"/> <mi>C</mi> <msub> <mi>u</mi> <mi>b</mi> </msub> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mspace width="4pt"/> <mi>C</mi> <mi>t</mi> <mi>o</mi> <msub> <mi>t</mi> <mi>b</mi> </msub> </mrow> </semantics></math> are associated with the mobile base, whereas <math display="inline"><semantics> <mrow> <mi>C</mi> <msub> <mi>ζ</mi> <mi>a</mi> </msub> <mo>,</mo> <mspace width="4pt"/> <mi>C</mi> <msub> <mi>u</mi> <mi>a</mi> </msub> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mspace width="4pt"/> <mi>C</mi> <mi>t</mi> <mi>o</mi> <msub> <mi>t</mi> <mi>a</mi> </msub> </mrow> </semantics></math> are associated with the arm of the SSMM.</p>
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<p>Snapshots of the SSMM during field experiments. From left to right: the Pioneer 3P-AT mobile base is mechanically coupled with a Katana 6M180 robotic arm. The middle and right images show the experimental field setup used for tracking and regulation trials with the three test controllers.</p>
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<p>Field results for the three test controllers (PID, NMPC, and R-NMPC). The top right figures show the lonitudinal and lateral trajectory-tracking errors for the mobile base, whereas the bottom right plot presents the height tracking errors for the robotic arm. The gray dashed lines indicate when the height of the reference trajectory has changed.</p>
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<p>Field results for three test controllers tracking a curved reference trajectory on terrain unevenness acting as disturbance (gray shaded area). The top right figures show the trajectory-tracking errors for the mobile base, while the bottom right figure presents the tracking errors for the robotic arm.</p>
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22 pages, 4090 KiB  
Article
Visual Servoing Using Sliding-Mode Control with Dynamic Compensation for UAVs’ Tracking of Moving Targets
by Christian P. Carvajal, Víctor H. Andaluz, José Varela-Aldás, Flavio Roberti, Carolina Del-Valle-Soto and Ricardo Carelli
Drones 2024, 8(12), 730; https://doi.org/10.3390/drones8120730 - 2 Dec 2024
Viewed by 404
Abstract
An Image-Based Visual Servoing Control (IBVS) structure for target tracking by Unmanned Aerial Vehicles (UAVs) is presented. The scheme contains two stages. The first one is a sliding-model controller (SMC) that allows one to track a target with a UAV; the control strategy [...] Read more.
An Image-Based Visual Servoing Control (IBVS) structure for target tracking by Unmanned Aerial Vehicles (UAVs) is presented. The scheme contains two stages. The first one is a sliding-model controller (SMC) that allows one to track a target with a UAV; the control strategy is designed in the function of the image. The proposed SMC control strategy is commonly used in control systems that present high non-linearities and that are always exposed to external disturbances; these disturbances can be caused by environmental conditions or induced by the estimation of the position and/or velocity of the target to be tracked. In the second instance, a controller is placed to compensate the UAV dynamics; this is a controller that allows one to compensate the velocity errors that are produced by the dynamic effects of the UAV. In addition, the corresponding stability analysis of the sliding mode-based visual servo controller and the sliding mode dynamic compensation control is presented. The proposed control scheme employs the kinematics and dynamics of the robot by presenting a cascade control based on the same control strategy. In order to evaluate the proposed scheme for tracking moving targets, experimental tests are carried out in a semi-structured working environment with the hexarotor-type aerial robot. For detection and image processing, the Opencv C++ library is used; the data are published in an ROS topic at a frequency of 50 Hz. The robot controller is implemented in the mathematical software Matlab. Full article
(This article belongs to the Special Issue Flight Control and Collision Avoidance of UAVs)
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Figure 1
<p>UAV kinematic configuration in the workspace <math display="inline"><semantics> <mrow> <mo>{</mo> <mi mathvariant="script">F</mi> <mo>}</mo> </mrow> </semantics></math>, where <math display="inline"><semantics> <mrow> <mmultiscripts> <mi>X</mi> <mi>F</mi> <mo>′</mo> <mprescripts/> <none/> </mmultiscripts> </mrow> </semantics></math> is a line parallel to the axis-<math display="inline"><semantics> <msub> <mi>X</mi> <mi>F</mi> </msub> </semantics></math> of the fixed reference system; this allows one to identify the axis from which the UAV’s orientation <math display="inline"><semantics> <mi>ψ</mi> </semantics></math> is measured.</p>
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<p>Schematic diagram of camera configuration on the UAV. The target of interest to be followed is represented by a unicycle-type mobile robot and an aerial robot with a camera located at the bottom.</p>
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<p>Proposed cascade servo visual control scheme for target tracking based on sliding mode.</p>
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<p>Robots for experiments. At the bottom is the target located on the unicycle robot with two marks to obtain the orientation of the target, while the hexarotor with the built-in vision system can be seen at the top.</p>
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<p>Dynamic model validation of the Matrice 600PRO UAV. The red line represents the reference signal, the green line represents the actual velocity obtained from the UAV, and the black line represents the signal obtained from the dynamic model.</p>
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<p>Stroboscopic movement of the UAV during Experiment 1 in <math display="inline"><semantics> <mrow> <mo>{</mo> <mi mathvariant="script">F</mi> <mo>}</mo> </mrow> </semantics></math>: (<b>a</b>) represents the movement in the XY plane, and (<b>b</b>) represents the stroboscopic movement of the robot in the XYZ space.</p>
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<p>Target behavior in the image plane of the controller without dynamic compensation (blue line) and the controller with dynamic compensation (orange line).</p>
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<p>Secondary objective error.</p>
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<p>Stroboscopic movement executed by the UAV during the experiment.</p>
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<p>Evolution of control errors of principal tracking task and target behavior in the image plane during Experiment 2.</p>
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<p>Secondary task error evolution.</p>
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<p>Target velocity estimated by [<a href="#B30-drones-08-00730" class="html-bibr">30</a>].</p>
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<p>Reference velocities generated by DC-SMC in Experiment 2.</p>
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<p>UAV maneuverability velocity errors.</p>
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22 pages, 3505 KiB  
Article
Fixed-Time Command-Filtered Control for Nonlinear Systems with Mismatched Disturbances
by Zhiqiang Wu, Jian Zhang, Lei Xing and Liyang Sun
Mathematics 2024, 12(23), 3816; https://doi.org/10.3390/math12233816 - 2 Dec 2024
Viewed by 488
Abstract
This article concerns the issue of adaptive fuzzy command-filtered fixed-time control in the context of a category of nonlinear systems characterized by mismatched disturbances and unknown nonlinear functions. The backstepping-based disturbance observers are created to alleviate the effects of mismatched disturbances and Fuzzy [...] Read more.
This article concerns the issue of adaptive fuzzy command-filtered fixed-time control in the context of a category of nonlinear systems characterized by mismatched disturbances and unknown nonlinear functions. The backstepping-based disturbance observers are created to alleviate the effects of mismatched disturbances and Fuzzy logic systems are brought into play to model those terms that are unknown. To address the complexity explosion issue in traditional backstepping control, this paper utilizes fixed-time command filters (FTCFs) to design a novel control approach. Moreover, filtering error compensation mechanisms are developed to eliminate the errors introduced by the FTCFs. This paper derives a novel adaptive fixed-time control protocol that successfully conquers the difficulties posed by unknown nonlinear functions and mismatched disturbances. The protocol, implemented within a backstepping framework, guarantees the boundedness of all signals and tracking errors within fixed time. The efficacy of the derived control protocol is illustrated through simulation examples. Full article
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<p>Block diagram of the fixed-time control scheme.</p>
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<p>The trajectories of the states <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>x</mi> <mi>d</mi> </msub> </semantics></math>.</p>
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<p>The trajectory of the state <math display="inline"><semantics> <msub> <mi>x</mi> <mn>2</mn> </msub> </semantics></math>.</p>
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<p>The trajectory of tracking error <math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mi>d</mi> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </semantics></math>.</p>
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<p>The trajectory of control signal.</p>
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<p>The trajectories of the states <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>x</mi> <mi>d</mi> </msub> </semantics></math>.</p>
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<p>The trajectory of the state <math display="inline"><semantics> <msub> <mi>x</mi> <mn>2</mn> </msub> </semantics></math>.</p>
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<p>The trajectory of tracking error <math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mi>d</mi> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </semantics></math>.</p>
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<p>The trajectory of control signal.</p>
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<p>The curves of the states <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mi>d</mi> </msub> </semantics></math> for case 1.</p>
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<p>The curve of the tracking error for case 1.</p>
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<p>The curves of the states <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mi>d</mi> </msub> </semantics></math> for case 2.</p>
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<p>The curve of the tracking error for case 2.</p>
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<p>The curves of the states <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mi>d</mi> </msub> </semantics></math> for case 3.</p>
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<p>The curve of the tracking error for case 3.</p>
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<p>The curves of the states <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mi>d</mi> </msub> </semantics></math> for Ref. [<a href="#B49-mathematics-12-03816" class="html-bibr">49</a>].</p>
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<p>The curve of the tracking error for Ref. [<a href="#B49-mathematics-12-03816" class="html-bibr">49</a>].</p>
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<p>The curves of the states <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mi>d</mi> </msub> </semantics></math> for fixed-time controller Equation (<a href="#FD52-mathematics-12-03816" class="html-disp-formula">52</a>).</p>
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<p>The curve of the tracking error for fixed-time controller Equation (<a href="#FD52-mathematics-12-03816" class="html-disp-formula">52</a>).</p>
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<p>The curves of the states <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mi>d</mi> </msub> </semantics></math> for finite-time controller Equation (<a href="#FD74-mathematics-12-03816" class="html-disp-formula">74</a>).</p>
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<p>The curve of the tracking error for finite-time controller Equation (<a href="#FD74-mathematics-12-03816" class="html-disp-formula">74</a>).</p>
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11 pages, 4182 KiB  
Article
Identification of Intrinsic Friction and Torque Ripple for a Robotic Joint with Integrated Torque Sensors with Application to Wheel-Bearing Characterization
by Sri Harsha Turlapati, Van Pho Nguyen, Juhi Gurnani, Mohammad Zaidi Bin Ariffin, Sreekanth Kana, Alvin Hong Yee Wong, Boon Siew Han and Domenico Campolo
Sensors 2024, 24(23), 7465; https://doi.org/10.3390/s24237465 - 22 Nov 2024
Viewed by 437
Abstract
Although integrated joint torque sensors in robots dispel the need for external force/torque sensors at the wrist to measure interactions, an inherent challenge is that they also measure the robot’s intrinsic dynamics. This is especially problematic for delicate robot manipulation tasks, where interaction [...] Read more.
Although integrated joint torque sensors in robots dispel the need for external force/torque sensors at the wrist to measure interactions, an inherent challenge is that they also measure the robot’s intrinsic dynamics. This is especially problematic for delicate robot manipulation tasks, where interaction forces may be comparable to the robot intrinsic dynamics. Therefore, the intrinsic dynamics must first be experimentally estimated under no-load conditions, when the measurement only consists of torques due to the transmission of the robot actuator, before external interactions may be measured. In this work, we propose an approach for identifying and predicting the intrinsic dynamics using linear regression with non-linear radial basis functions. Then, we validate this regression on a wheel-bearing turning task, in which its friction is a measure of quality, and thus must be accurately measured. The results showed that the bearing torque measured by the joint 7 torque sensor was within an RMS error of 11% of the torque measured by the external force/torque sensor. This error is much lower than that before our proposed model in compensating the intrinsic dynamics of the robot arm. Full article
(This article belongs to the Special Issue Advances in Sensing, Control and Path Planning for Robotic Systems)
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<p>Structure of a joint in a collaborative robot with the location of torque sensor before gearbox. For a comprehensive study, see [<a href="#B5-sensors-24-07465" class="html-bibr">5</a>].</p>
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<p>Torques due to different sources in a motor. (<b>a</b>) Coulomb friction (positive rotation in green, negative rotation in red). (<b>b</b>) Viscous friction. (<b>c</b>) Torque in a cobot joint produced by inertia deployed from Equation (<a href="#FD11-sensors-24-07465" class="html-disp-formula">11</a>). (<b>d</b>) Torque in a cobot joint produced by a gearbox (harmonic drive) that is deployed from Equation (<a href="#FD12-sensors-24-07465" class="html-disp-formula">12</a>).</p>
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<p>Intrinsic dynamic compensation from torque measurements under no-load conditions. (<b>a</b>) The 1 Hz-filtered torque of freely rotating joint. (<b>b</b>) Torque vs. joint angle is repeatable and directional. (<b>c</b>) Intrinsic dynamics components in Equations (<a href="#FD7-sensors-24-07465" class="html-disp-formula">7</a>)–(<a href="#FD14-sensors-24-07465" class="html-disp-formula">14</a>). (<b>d</b>) Residuals after intrinsic dynamic compensation approach zero.</p>
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<p>The design of compliant fingerpads is key to enabling the equivalent of a power grasp (without the need for complex articulation) of the wheel-bearing by conforming to its shape and preventing slipping while turning. The robot is guided by kinesthetic teaching under high compliance to grasp the wheel-bearing and is allowed to self-adjust to minimize any misalignment between robot and wheel-bearing axes. Once settled, the end-effector joint is driven in torque to assess the wheel-bearing friction. (<b>a</b>) Grasping the bearing. (<b>b</b>) High compliance mode. (<b>c</b>) Experiment.</p>
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<p>Illustration of the sensed torques in the benchmark ATI F/T sensor (mini40), and the Kinova robot arm in the instances with and without compensating the robot intrinsic dynamics. (<b>a</b>) Single-trial observations. (<b>b</b>) Observations over three trials.</p>
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36 pages, 17182 KiB  
Article
A Fuzzy-Immune-Regulated Single-Neuron Proportional–Integral–Derivative Control System for Robust Trajectory Tracking in a Lawn-Mowing Robot
by Omer Saleem, Ahmad Hamza and Jamshed Iqbal
Computers 2024, 13(11), 301; https://doi.org/10.3390/computers13110301 - 19 Nov 2024
Viewed by 390
Abstract
This paper presents the constitution of a computationally intelligent self-adaptive steering controller for a lawn-mowing robot to yield robust trajectory tracking and disturbance rejection behavior. The conventional fixed-gain proportional–integral–derivative (PID) control procedure lacks the flexibility to deal with the environmental indeterminacies, coupling issues, [...] Read more.
This paper presents the constitution of a computationally intelligent self-adaptive steering controller for a lawn-mowing robot to yield robust trajectory tracking and disturbance rejection behavior. The conventional fixed-gain proportional–integral–derivative (PID) control procedure lacks the flexibility to deal with the environmental indeterminacies, coupling issues, and intrinsic nonlinear dynamics associated with the aforementioned nonholonomic system. Hence, this article contributes to formulating a self-adaptive single-neuron PID control system that is driven by an extended Kalman filter (EKF) to ensure efficient learning and faster convergence speeds. The neural adaptive PID control formulation improves the controller’s design flexibility, which allows it to effectively attenuate the tracking errors and improve the system’s trajectory tracking accuracy. To supplement the controller’s robustness to exogenous disturbances, the adaptive PID control signal is modulated with an auxiliary fuzzy-immune system. The fuzzy-immune system imitates the automatic self-learning and self-tuning characteristics of the biological immune system to suppress bounded disturbances and parametric variations. The propositions above are verified by performing the tailored hardware in the loop experiments on a differentially driven lawn-mowing robot. The results of these experiments confirm the enhanced trajectory tracking precision and disturbance compensation ability of the prescribed control method. Full article
(This article belongs to the Special Issue Feature Papers in Computers 2024)
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<p>Position of the WMR.</p>
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<p>Baseline velocity control architecture of the WMR.</p>
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<p>Trajectory-tracking errors of the WMR.</p>
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<p>Behavior of modified error signal vs. linear error signal.</p>
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<p>Waveform of the hyperbolic tangent function.</p>
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<p>Schematic diagram of the SN-APID control scheme.</p>
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<p>Schematic of the fuzzy-immune-regulated SN-APID control scheme.</p>
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<p>Input fuzzy MFs of <math display="inline"><semantics> <mrow> <mi>u</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mover accent="true"> <mrow> <mi>u</mi> </mrow> <mo>˙</mo> </mover> </mrow> </semantics></math>.</p>
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<p>Output fuzzy MFs of <math display="inline"><semantics> <mrow> <mi>σ</mi> </mrow> </semantics></math>.</p>
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<p>Schematic diagram of the FSN-APID control architecture.</p>
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<p>Lawn mowing robot chassis used for experimental analysis.</p>
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<p>The rose-curve reference trajectory.</p>
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<p>State variations under nominal conditions.</p>
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<p>Error profile of each state variable under nominal conditions.</p>
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<p>Rose-curve tracking profiles under nominal conditions.</p>
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<p>State variations under impulsive disturbances.</p>
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<p>Error profile of each state variable under impulsive disturbances.</p>
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<p>Rose-curve tracking profiles under impulsive disturbances.</p>
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<p>State variations under step disturbances.</p>
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<p>Error profile of each state variable under step disturbances.</p>
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<p>Rose-curve tracking profiles under step conditions.</p>
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<p>State variations under randomized step disturbances.</p>
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<p>Error profile of each state variable under randomized step disturbances.</p>
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<p>Rose-curve tracking profiles under randomized step conditions.</p>
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<p>State variations under different fuzzy parameter settings of FSN-APID controller.</p>
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<p>Error profile of each state variable’s different fuzzy parameter settings of FSN-APID controller.</p>
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<p>Rose-curve tracking profiles under different fuzzy parameter settings of FSN-APID controller.</p>
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<p>Waveforms of the activation functions used for the ablation study.</p>
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<p>State variations under different activation functions of FSN-APID controller.</p>
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<p>Error profile of each state variable under different activation functions of FSN-APID controller.</p>
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<p>Rose-curve tracking profiles under different activation functions of FSN-APID controller.</p>
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<p>State variations under different terrain types.</p>
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<p>Error profile of each state variable under different terrain types.</p>
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<p>Rose-curve tracking profiles under different terrain types.</p>
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14 pages, 3643 KiB  
Article
Incremental Nonlinear Dynamics Inversion Control with Nonlinear Disturbance Observer Augmentation for Flight Dynamics
by Lamsu Kim and Jeong I. Kim
Appl. Sci. 2024, 14(22), 10615; https://doi.org/10.3390/app142210615 - 18 Nov 2024
Viewed by 590
Abstract
A flight controller formulation based on incremental nonlinear dynamics inversion (INDI) control with nonlinear disturbance observer (NDO) is proposed. INDI control is a nonlinear controller based on incremental dynamics. Aimed to attain robustness for nonlinear dynamics inversion (NDI)-based controller, incremental dynamics are derived [...] Read more.
A flight controller formulation based on incremental nonlinear dynamics inversion (INDI) control with nonlinear disturbance observer (NDO) is proposed. INDI control is a nonlinear controller based on incremental dynamics. Aimed to attain robustness for nonlinear dynamics inversion (NDI)-based controller, incremental dynamics are derived using the first-order Talyor series expansion to nonlinear systems. The incremental dynamics-based controller requires information on state derivative terms to strengthen the robustness property of the nonlinear controller. The proposed controller utilizes the first-order low-pass filter to obtain the state derivative estimate to implement incremental dynamics into the system. Because the incremental form creates uncertainty term which is an aftermath of the Taylor series expansion, the proposed controller adopts the NDO to eliminate this effect. The controller is applied to the generic transport model which was developed by NASA for simulation purposes. The proposed NDO-based INDI control underwent simulations, together with an INDI controller without disturbance observer, and showed that the developed method results in better performances, providing important advantages where it compensates the uncertainties, removes the steady-state error, and shows less oscillating longitudinal body rate response than the baseline controller, desirable for aerodynamics applications with faster system response. Full article
(This article belongs to the Section Aerospace Science and Engineering)
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<p>Inertial frame, aircraft reference frame, CG point, and corresponding <math display="inline"><semantics> <mrow> <mi>x</mi> <mspace width="1pt"/> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>y</mi> <mspace width="1pt"/> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>z</mi> </mrow> </semantics></math> basis vectors.</p>
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<p>Block diagram of incremental dynamic based on the rotational flight controller with nonlinear disturbance observer.</p>
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<p><math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mi>T</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mi>α</mi> </semantics></math>, and <math display="inline"><semantics> <mi>β</mi> </semantics></math> responses of the proposed INDI with the NDO and without the NDO.</p>
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<p>Roll rate responses of the proposed INDI with the NDO and without the NDO.</p>
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<p>Pitch rate responses of the proposed INDI with the NDO and without the NDO.</p>
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<p>Yaw rate responses of the proposed INDI with the NDO and without the NDO.</p>
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<p>Responses of the error between the roll pseudo control input and roll angular acceleration of the proposed INDI with the NDO and without the NDO.</p>
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<p>Responses of the error between the pitch pseudo control input and pitch angular acceleration of the proposed INDI with the NDO and without the NDO.</p>
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<p>Responses of the error between the yaw pseudo control input and yaw angular acceleration of the proposed INDI with the NDO and without the NDO.</p>
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34 pages, 16736 KiB  
Article
Optimized Energy Management Strategy for an Autonomous DC Microgrid Integrating PV/Wind/Battery/Diesel-Based Hybrid PSO-GA-LADRC Through SAPF
by AL-Wesabi Ibrahim, Jiazhu Xu, Abdullrahman A. Al-Shamma’a, Hassan M. Hussein Farh, Imad Aboudrar, Youssef Oubail, Fahad Alaql and Walied Alfraidi
Technologies 2024, 12(11), 226; https://doi.org/10.3390/technologies12110226 - 11 Nov 2024
Viewed by 1385
Abstract
This study focuses on microgrid systems incorporating hybrid renewable energy sources (HRESs) with battery energy storage (BES), both essential for ensuring reliable and consistent operation in off-grid standalone systems. The proposed system includes solar energy, a wind energy source with a synchronous turbine, [...] Read more.
This study focuses on microgrid systems incorporating hybrid renewable energy sources (HRESs) with battery energy storage (BES), both essential for ensuring reliable and consistent operation in off-grid standalone systems. The proposed system includes solar energy, a wind energy source with a synchronous turbine, and BES. Hybrid particle swarm optimizer (PSO) and a genetic algorithm (GA) combined with active disturbance rejection control (ADRC) (PSO-GA-ADRC) are developed to regulate both the frequency and amplitude of the AC bus voltage via a load-side converter (LSC) under various operating conditions. This approach further enables efficient management of accessible generation and general consumption through a bidirectional battery-side converter (BSC). Additionally, the proposed method also enhances power quality across the AC link via mentoring the photovoltaic (PV) inverter to function as shunt active power filter (SAPF), providing the desired harmonic-current element to nonlinear local loads as well. Equipped with an extended state observer (ESO), the hybrid PSO-GA-ADRC provides efficient estimation of and compensation for disturbances such as modeling errors and parameter fluctuations, providing a stable control solution for interior voltage and current control loops. The positive results from hardware-in-the-loop (HIL) experimental results confirm the effectiveness and robustness of this control strategy in maintaining stable voltage and current in real-world scenarios. Full article
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<p>The proposed autonomous microgrid’s topology.</p>
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<p>Solar PV energy cell design.</p>
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<p>The power coefficient’s characteristics at various pitch angles (β) and tip speed ratios (λ).</p>
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<p>The BES circuit diagram with its bidirectional converter.</p>
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<p>Schematic components of (<b>a</b>) nonlinear ADRC and (<b>b</b>) linear ADRC.</p>
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<p>PSO-GA-LADRC DC-DC converter for controlling a PV system.</p>
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<p>OTC-MPPT-ADRC control for MSC.</p>
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<p>The proposed control circuit that utilizes BSC ADRC.</p>
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<p>Proposed LSC based ADRC control.</p>
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<p>Control by the SAPF ADRC using the P-Q theory methodology.</p>
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<p>Oscillating component extraction filters.</p>
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<p>The active and reactive powers transferred in the microgrid during various irradiation and wind profiles (Case 1): (<b>a</b>) radiation profile, (<b>b</b>) wind speed profile, (<b>c</b>) PV output power, (<b>d</b>) wind output power, (<b>e</b>) SAPF output power, (<b>f</b>) hybrid system output power, (<b>g</b>) active power of the load, (<b>h</b>) reactive power of the load, (<b>i</b>) battery output power, and (<b>j</b>) state of charge.</p>
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<p>The active and reactive powers transferred in the microgrid during various irradiation and wind profiles (Case 1): (<b>a</b>) radiation profile, (<b>b</b>) wind speed profile, (<b>c</b>) PV output power, (<b>d</b>) wind output power, (<b>e</b>) SAPF output power, (<b>f</b>) hybrid system output power, (<b>g</b>) active power of the load, (<b>h</b>) reactive power of the load, (<b>i</b>) battery output power, and (<b>j</b>) state of charge.</p>
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<p>The output characteristics (voltage and current) flowing in the AC link (Case 1): (<b>a</b>) AC link voltage, (<b>b</b>) AC link currents, (<b>c</b>) AC load output currents, and (<b>d</b>) injected currents in the filter.</p>
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<p>The output characteristics (voltage and current) flowing in the AC link (Case 1): (<b>a</b>) AC link voltage, (<b>b</b>) AC link currents, (<b>c</b>) AC load output currents, and (<b>d</b>) injected currents in the filter.</p>
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<p>The outcomes of the various control loops by ADRC are shown in the following order (Case 1): (<b>a</b>) SAPF DC link output voltage controls; (<b>b</b>) hybrid system DC link output voltage control; (<b>c</b>) d-axis output voltage control; (<b>d</b>) q-axis output voltage control; (<b>e</b>) d-axis output current control; (<b>f</b>) q-axis output current control; (<b>g</b>) battery current; and (<b>h</b>) controlled AC-link frequency.</p>
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<p>Active and reactive power transferred in the microgrid during constant irradiation and wind profiles (Case 2): (<b>a</b>) PV output power, (<b>b</b>) wind output power, (<b>c</b>) SAPF output power, (<b>d</b>) hybrid system output power, (<b>e</b>) active power of the load, (<b>f</b>) reactive power of the load, (<b>g</b>) battery output power, and (<b>h</b>) state of charge.</p>
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<p>The output characteristics (voltage and current) flowing in the AC link (Case 2): (<b>a</b>) AC link voltage, (<b>b</b>) AC link currents, (<b>c</b>) AC load output currents, and (<b>d</b>) injected currents in the filter.</p>
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<p>The outcomes of the various control loops by ADRC are shown in the following order (Case 2): (<b>a</b>) SAPF DC link output voltage controls; (<b>b</b>) hybrid system DC link output voltage control; (<b>c</b>) d-axis output voltage control; (<b>d</b>) q-axis output voltage control; (<b>e</b>) d-axis output current control; (<b>f</b>) q-axis output current control; (<b>g</b>) battery current; and (<b>h</b>) controlled AC-link frequency.</p>
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<p>NI PXIE-1071(HIL) experimental setup.</p>
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<p>HIL experimental results for PV output characteristics.</p>
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<p>HIL experimental results for wind output characteristics and DC bus voltage.</p>
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<p>HIL experimental results for voltage and current flowing in the AC bus.</p>
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<p>HIL experimental results for voltage and current flowing in the AC bus.</p>
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19 pages, 5088 KiB  
Article
Predefined-Time Hybrid Tracking Control for Dynamic Positioning Vessels Based on Fully Actuated Approach
by Yujie Xu, Yingjie Wang, Mingyu Fu and Hao Chen
J. Mar. Sci. Eng. 2024, 12(11), 2025; https://doi.org/10.3390/jmse12112025 - 9 Nov 2024
Viewed by 457
Abstract
This study investigates the problem of tracking the trajectory of a dynamic positioning (DP) ship under sudden surges of elevated sea states. First, the tracking problem is reformulated as an error calibration problem through the introduction of fully actuated system (FAS) approaches, thereby [...] Read more.
This study investigates the problem of tracking the trajectory of a dynamic positioning (DP) ship under sudden surges of elevated sea states. First, the tracking problem is reformulated as an error calibration problem through the introduction of fully actuated system (FAS) approaches, thereby simplifying controller design. Second, a predefined-time control term is designed to maintain the convergence time of the trajectory tracking error within a specified range; however, the upper bound of the perturbation must be estimated in advance. The high sea state during operation can result in an abrupt change in the upper bound of disturbance, thereby affecting the control accuracy and stability of the system. Therefore, a linear control matrix is developed to eliminate the system’s dependence on the estimation of the upper bound of disturbance following smooth switching, thereby achieving control decoupling and providing a conservative switching time. Additionally, a nonlinear reduced-order expansion observer (RESO) is constructed for feedforward compensation. The stability of the system is demonstrated using the Lyapunov function, indicating that the selection of appropriate poles can theoretically enhance the system’s convergence with greater control accuracy and robustness after switching. Finally, the effectiveness of the proposed method is validated through simulations and comparative experiments. Full article
(This article belongs to the Section Ocean Engineering)
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<p>Definition of the earth-fixed <math display="inline"><semantics> <mrow> <mi>O</mi> <msub> <mi>X</mi> <mn>0</mn> </msub> <msub> <mi>Y</mi> <mn>0</mn> </msub> </mrow> </semantics></math> and the body-fixed <math display="inline"><semantics> <mrow> <mi>A</mi> <mi>X</mi> <mi>Y</mi> </mrow> </semantics></math> coordinate frames.</p>
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<p>Diagram of the trajectory tracking control system of a ship.</p>
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<p>Tracking trajectories curve under different predefined and switching times.</p>
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<p>Tracking error (<math display="inline"><semantics> <mrow> <mi mathvariant="bold-italic">η</mi> <mo>−</mo> <msub> <mi mathvariant="bold-italic">η</mi> <mi mathvariant="bold-italic">d</mi> </msub> </mrow> </semantics></math>) under different predefined and switching times.</p>
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<p>Tracking rate( <math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold-italic">η</mi> <mo>˙</mo> </mover> </semantics></math> ) curve under different predefined and switching times.</p>
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<p>Surge control force <math display="inline"><semantics> <msub> <mi mathvariant="bold-italic">τ</mi> <mrow> <mi>c</mi> <mi>u</mi> </mrow> </msub> </semantics></math>, sway control force <math display="inline"><semantics> <msub> <mi mathvariant="bold-italic">τ</mi> <mrow> <mi>c</mi> <mi>v</mi> </mrow> </msub> </semantics></math>, and yaw control torque <math display="inline"><semantics> <msub> <mi mathvariant="bold-italic">τ</mi> <mrow> <mi>c</mi> <mi>r</mi> </mrow> </msub> </semantics></math> under different predefined and switching times.</p>
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<p>Time-variant external disturbances <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>d</mi> <mn>3</mn> </msub> </mrow> </semantics></math> and their estimations <math display="inline"><semantics> <mrow> <mover accent="true"> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo stretchy="false">^</mo> </mover> <mo>,</mo> <mover accent="true"> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo stretchy="false">^</mo> </mover> <mo>,</mo> <mover accent="true"> <msub> <mi>d</mi> <mn>3</mn> </msub> <mo stretchy="false">^</mo> </mover> </mrow> </semantics></math>.</p>
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<p>Comparison of tracking errors between the proposed method and the PTC method under <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>40</mn> </mrow> </semantics></math>.</p>
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<p>Comparison of thrust output between the proposed method and the PTC method under <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>40</mn> </mrow> </semantics></math>.</p>
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21 pages, 4016 KiB  
Article
Analysis of Fractional Resonant Controllers for Voltage-Controlled Applications
by Daniel Heredero-Peris, Tomàs Lledó-Ponsati, Cristian Chillón-Antón, Daniel Montesinos-Miracle and Joaquim Melendez-Frigola
Appl. Sci. 2024, 14(22), 10259; https://doi.org/10.3390/app142210259 - 7 Nov 2024
Viewed by 597
Abstract
This paper investigates the application of fractional proportional–resonant controllers within the voltage control loop of grid-forming inverters. The use of such controllers introduces an additional degree of freedom, enabling greater flexibility in manipulating frequency trajectories. This flexibility can be harnessed to improve tracking [...] Read more.
This paper investigates the application of fractional proportional–resonant controllers within the voltage control loop of grid-forming inverters. The use of such controllers introduces an additional degree of freedom, enabling greater flexibility in manipulating frequency trajectories. This flexibility can be harnessed to improve tracking error and enhance disturbance rejection, particularly in applications requiring precise voltage regulation. The paper conducts a conceptual stability analysis of ideal fractional proportional–resonant controllers using the Nyquist criterion. A tuning procedure based on robustness criteria for the proposed controller is also addressed. This tuning strategy is used to compare different controllers under the same conditions. In addition, a sensitivity analysis is provided, comparing the performance of fractional proportional–resonant controllers with traditional proportional–resonant controllers equipped with harmonic compensation. The controller’s formulation and performance are validated through simulations and tested with a 20 kVA inverter under high non-linear loads. Compared to classical control approaches, the fractional tuning parameter enhances tracking performance, reduces phase delay, and improves disturbance rejection. These improvements are achieved with a controller designed to minimise computational demands in terms of memory usage and execution time. Full article
(This article belongs to the Section Electrical, Electronics and Communications Engineering)
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<p>Voltage-controlledvoltage source converter (VC-VSC) schematic with inductive (<span class="html-italic">L</span>)–capacitive (<span class="html-italic">C</span>) filter.</p>
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<p>Open−loop (<math display="inline"><semantics> <msub> <mi>G</mi> <mi>ol</mi> </msub> </semantics></math>) bode diagrams, based on <a href="#applsci-14-10259-t002" class="html-table">Table 2</a>.</p>
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<p>Close−loop (<math display="inline"><semantics> <msub> <mi>G</mi> <mi>cl</mi> </msub> </semantics></math>) bode diagrams, based on <a href="#applsci-14-10259-t002" class="html-table">Table 2</a>.</p>
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<p>Conceptual Nyquist trajectories for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>∈</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo>]</mo> </mrow> </semantics></math> (solid lines define the Nyquist trajectory, and dashed lines describe asymptotic behaviours).</p>
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<p>Open−loop bode diagrams for ideal FPR controller, based on <a href="#applsci-14-10259-t003" class="html-table">Table 3</a>.</p>
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<p>Close−loop bode diagrams for ideal FPR controller, based on <a href="#applsci-14-10259-t003" class="html-table">Table 3</a>.</p>
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<p>Close−loop bode (<math display="inline"><semantics> <mrow> <mi>T</mi> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </semantics></math>) diagrams for the controller selection comparison.</p>
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<p>Input sensitivity (<math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>) bode diagrams for the controller selection comparison.</p>
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<p>Nyquist of <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>(</mo> <mi mathvariant="normal">I</mi> <mi>ω</mi> <mo>)</mo> </mrow> </semantics></math> ending trajectory. Grey line defines the unitary circle for discrete−time systems.</p>
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<p>Uncertainties’ superior thresholds.</p>
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<p>Time−response comparison when different references are present, being <math display="inline"><semantics> <msub> <mi>ω</mi> <mn>0</mn> </msub> </semantics></math> = 100<math display="inline"><semantics> <mi>π</mi> </semantics></math> rad/s (50 Hz). Harmonics 1 and 2.</p>
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<p>Time−response comparison when different references are present, being <math display="inline"><semantics> <msub> <mi>ω</mi> <mn>0</mn> </msub> </semantics></math> = 100<math display="inline"><semantics> <mi>π</mi> </semantics></math> rad/s (50 Hz). Harmonics 3 and 11.</p>
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<p>Time−response comparison when different harmonic disturbances are present, being <math display="inline"><semantics> <msub> <mi>ω</mi> <mn>0</mn> </msub> </semantics></math> = 100<math display="inline"><semantics> <mi>π</mi> </semantics></math> rad/s (50 Hz).</p>
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<p>VSVC power converter.</p>
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<p>Tracking voltage time response. The orange line in the right figure depicts the voltage reference on the tracking error plot.</p>
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<p>FFT for the tracking voltage time response in <a href="#applsci-14-10259-f015" class="html-fig">Figure 15</a>.</p>
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<p>Disturbance rejection time response. The orange line in the right figure depicts the disturbance on the tracking error plot.</p>
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<p>FFT for steady-state disturbance rejection in <a href="#applsci-14-10259-f017" class="html-fig">Figure 17</a>.</p>
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<p>Controller type comparison (the closer to the peak, the better the result).</p>
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23 pages, 11844 KiB  
Article
Modeling and Compensation of Stiffness-Dependent Hysteresis Coupling Behavior for Parallel Pneumatic Artificial Muscle-Driven Soft Manipulator
by Ying Zhang, Huiming Qi, Qiang Cheng, Zhi Li and Lina Hao
Appl. Sci. 2024, 14(22), 10240; https://doi.org/10.3390/app142210240 - 7 Nov 2024
Viewed by 552
Abstract
The parallel driving soft manipulator with multiple extensors and contractile pneumatic artificial muscles (PAMs) is able to operate continuously and has varying stiffness, achieving smooth movements and a fundamental trade-off between flexibility and stiffness. Owing to the hysteresis of PAMs and actuator couplings, [...] Read more.
The parallel driving soft manipulator with multiple extensors and contractile pneumatic artificial muscles (PAMs) is able to operate continuously and has varying stiffness, achieving smooth movements and a fundamental trade-off between flexibility and stiffness. Owing to the hysteresis of PAMs and actuator couplings, the manipulator outputs display coupled hysteresis behaviors with stiffness dependence, causing significant positioning errors. For precise positioning control, this paper takes the lead in proposing a comprehensive model aimed at accurately predicting the coupled hysteresis behavior with the stiffness dependence of the soft manipulator. The model consists of an inherent hysteresis submodule, an actuator coupling submodule, and a stiffness-dependent submodule in series. The asymmetrical hysteresis nonlinearity of the PAM is established by the generalized Prandtl–Ishlinskii model in the inherent hysteresis submodule. The serial actuator coupling submodule is dedicated to modeling the actuator couplings, and the stiffness-dependent submodule is implemented with a fuzzy neural network to characterize the stiffness dependence and other system nonlinearities. In addition, an inverse compensator on the basis of the proposed model is conducted. Experiments demonstrate that this model possesses high accuracy and good generalization, and its compensator is effective in decoupling and mitigating hysteresis coupling of the manipulator. The proposed model and control methods significantly improve the positioning accuracy of the pneumatic soft manipulator. Full article
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<p>PAM-driven soft manipulators. (<b>a</b>) OCTARM continuum manipulator; (<b>b</b>) BIONIC HANDLING ASSISTANT; (<b>c</b>) a cylindrical manipulator; (<b>d</b>) continuum robot with parallel cable and pneumatic artificial muscles; (<b>e</b>) seven parallel PAMs-based manipulator with variable stiffness; (<b>f</b>) four parallel PAMs-based manipulators with variable stiffness.</p>
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<p>Experimental system for coupled input–output response test of the soft manipulator.</p>
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<p>The pneumatic soft manipulator. (<b>a</b>) Actual photograph of the manipulator. (<b>b</b>) Front view of the manipulator.</p>
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<p>Coupled hysteresis effects shown in the manipulator system. (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>e</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>f</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>g</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>h</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>i</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>j</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mtext> </mtext> </mrow> </semantics></math>; (<b>k</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mtext> </mtext> </mrow> </semantics></math>; (<b>l</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> </mrow> </semantics></math>.</p>
Full article ">Figure 4 Cont.
<p>Coupled hysteresis effects shown in the manipulator system. (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>e</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>f</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>g</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>h</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>i</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>j</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mtext> </mtext> </mrow> </semantics></math>; (<b>k</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mtext> </mtext> </mrow> </semantics></math>; (<b>l</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> </mrow> </semantics></math>.</p>
Full article ">Figure 5
<p>Stiffness effects on coupled hysteresis output of the manipulator system. (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math>.</p>
Full article ">Figure 6
<p>Stiffness-dependent and coupled hysteresis model.</p>
Full article ">Figure 7
<p>Structure of the stiffness-dependent submodule.</p>
Full article ">Figure 8
<p>Decoupled inverse hysteresis compensator.</p>
Full article ">Figure 9
<p>Identification results and errors of the inherent hysteresis model. (<b>a</b>) Identification results for <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>GPI</mi> </mrow> <mn>1</mn> </msub> </mrow> </semantics></math>; (<b>b</b>) identification results for <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>GPI</mi> </mrow> <mn>2</mn> </msub> </mrow> </semantics></math>; (<b>c</b>) identification results for <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>GPI</mi> </mrow> <mn>3</mn> </msub> </mrow> </semantics></math>.</p>
Full article ">Figure 10
<p>The prediction results of the SDCHM and TSFNN model with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> <mi>bar</mi> </mrow> </semantics></math>. (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math>; (<b>e</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math>; (<b>f</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math>; (<b>g</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math>; (<b>h</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math>; (<b>i</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math>; (<b>j</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math>; (<b>k</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math>; (<b>l</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math>.</p>
Full article ">Figure 10 Cont.
<p>The prediction results of the SDCHM and TSFNN model with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> <mi>bar</mi> </mrow> </semantics></math>. (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math>; (<b>e</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math>; (<b>f</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math>; (<b>g</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math>; (<b>h</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math>; (<b>i</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> <mo>,</mo> <mtext> </mtext> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math>; (<b>j</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math>; (<b>k</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math>; (<b>l</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <mi>p</mi> <mo>*</mo> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math>.</p>
Full article ">Figure 11
<p>RMSE of the SDCHM and TSFNN model corresponding to different pressure in the extensor PAM. (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> <mtext> </mtext> <mi>bar</mi> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1.5</mn> <mtext> </mtext> <mi>bar</mi> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1.7</mn> <mtext> </mtext> <mi>bar</mi> </mrow> </semantics></math>.</p>
Full article ">Figure 11 Cont.
<p>RMSE of the SDCHM and TSFNN model corresponding to different pressure in the extensor PAM. (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> <mtext> </mtext> <mi>bar</mi> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1.5</mn> <mtext> </mtext> <mi>bar</mi> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1.7</mn> <mtext> </mtext> <mi>bar</mi> </mrow> </semantics></math>.</p>
Full article ">Figure 12
<p>Inverse compensation results. (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.7</mn> <mrow> <mtext> </mtext> <mi>bar</mi> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mrow> <mn>1</mn> <mi>d</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.7</mn> <mrow> <mtext> </mtext> <mi>bar</mi> </mrow> </mrow> </semantics></math>, …; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.7</mn> <mrow> <mtext> </mtext> <mi>bar</mi> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mrow> <mn>1</mn> <mi>d</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1.5</mn> <mrow> <mtext> </mtext> <mi>bar</mi> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math>; (<b>e</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1.5</mn> <mrow> <mtext> </mtext> <mi>bar</mi> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math>; (<b>f</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1.5</mn> <mtext> </mtext> <mi>bar</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math>; (<b>g</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>2.3</mn> <mrow> <mtext> </mtext> <mi>bar</mi> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mrow> <mn>3</mn> <mi>d</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math>; (<b>h</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>2.3</mn> <mrow> <mtext> </mtext> <mi>bar</mi> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mrow> <mn>3</mn> <mi>d</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math>; (<b>i</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>2.3</mn> <mrow> <mtext> </mtext> <mi>bar</mi> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mrow> <mn>3</mn> <mi>d</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math>.</p>
Full article ">Figure 12 Cont.
<p>Inverse compensation results. (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.7</mn> <mrow> <mtext> </mtext> <mi>bar</mi> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mrow> <mn>1</mn> <mi>d</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.7</mn> <mrow> <mtext> </mtext> <mi>bar</mi> </mrow> </mrow> </semantics></math>, …; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.7</mn> <mrow> <mtext> </mtext> <mi>bar</mi> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mrow> <mn>1</mn> <mi>d</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1.5</mn> <mrow> <mtext> </mtext> <mi>bar</mi> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math>; (<b>e</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1.5</mn> <mrow> <mtext> </mtext> <mi>bar</mi> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math>; (<b>f</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1.5</mn> <mtext> </mtext> <mi>bar</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math>; (<b>g</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>2.3</mn> <mrow> <mtext> </mtext> <mi>bar</mi> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mrow> <mn>3</mn> <mi>d</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> </mrow> </semantics></math>; (<b>h</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>2.3</mn> <mrow> <mtext> </mtext> <mi>bar</mi> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mrow> <mn>3</mn> <mi>d</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> </mrow> </semantics></math>; (<b>i</b>) <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>2.3</mn> <mrow> <mtext> </mtext> <mi>bar</mi> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mrow> <mn>3</mn> <mi>d</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> </mrow> </semantics></math>.</p>
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<p>Compensation control results with the SDIHC compensator with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.6</mn> <mi>bar</mi> </mrow> </semantics></math>. (<b>a</b>) PAM I; (<b>b</b>) PAM II; (<b>c</b>) PAM III.</p>
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29 pages, 4318 KiB  
Article
Adaptive Integral Sliding Mode Control with Chattering Elimination Considering the Actuator Faults and External Disturbances for Trajectory Tracking of 4Y Octocopter Aircraft
by Samir Zeghlache, Hilal Rahali, Ali Djerioui, Hemza Mekki, Loutfi Benyettou and Mohamed Fouad Benkhoris
Processes 2024, 12(11), 2431; https://doi.org/10.3390/pr12112431 - 4 Nov 2024
Viewed by 816
Abstract
This paper presents a control strategy for a 4Y octocopter aircraft that is influenced by multiple actuator faults and external disturbances. The approach relies on a disturbance observer, adaptive type-2 fuzzy sliding mode control scheme, and type-1 fuzzy inference system. The proposed control [...] Read more.
This paper presents a control strategy for a 4Y octocopter aircraft that is influenced by multiple actuator faults and external disturbances. The approach relies on a disturbance observer, adaptive type-2 fuzzy sliding mode control scheme, and type-1 fuzzy inference system. The proposed control approach is distinct from other tactics for controlling unmanned aerial vehicles because it can simultaneously compensate for actuator faults and external disturbances. The suggested control technique incorporates adaptive control parameters in both continuous and discontinuous control components. This enables the production of appropriate control signals to manage actuator faults and parametric uncertainties without relying only on the robust discontinuous control approach of sliding mode control. Additionally, a type-1 fuzzy logic system is used to build a fuzzy hitting control law to eliminate the occurrence of chattering phenomena on the integral sliding mode control. In addition, in order to keep the discontinuous control gain in sliding mode control at a small value, a nonlinear disturbance observer is constructed and integrated to mitigate the influence of external disturbances. Moreover, stability analysis of the proposed control method using Lyapunov theory showcases its potential to uphold system tracking performance and minimize tracking errors under specified conditions. The simulation results demonstrate that the proposed control strategy can significantly reduce the chattering effect and provide accurate trajectory tracking in the presence of actuator faults. Furthermore, the efficacy of the recommended control strategy is shown by comparative simulation results of 4Y octocopter under different failing and uncertain settings. Full article
(This article belongs to the Special Issue Fuzzy Control System: Design and Applications)
Show Figures

Figure 1

Figure 1
<p>The 4Y octocopter aircraft configuration [<a href="#B16-processes-12-02431" class="html-bibr">16</a>], (<b>a</b>) Real 4Y octocopter (<b>b</b>) 4Y octocopter configuration.</p>
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<p>Controlling the motion of the 4Y octocopter aircraft utilizing virtual control.</p>
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<p>Overview of the developed fault-tolerant control.</p>
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<p>Architecture of a type-2 fuzzy logic system.</p>
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<p>Interval type-2 Gaussian membership functions for the antecedent sets.</p>
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<p>The membership functions of input variables <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mi>i</mi> </msub> </mrow> </semantics></math> and output <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <mo> </mo> <msub> <mi>u</mi> <mi>i</mi> </msub> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>6</mn> </mrow> </semantics></math>.</p>
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<p>Block diagram of the proposed control algorithm.</p>
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<p>(<span class="html-italic">x</span>, <span class="html-italic">y</span>, <span class="html-italic">z</span>) positions and yaw angle (<span class="html-italic">ψ</span>) outputs of the 4Y octocopter aircraft in the presence of actuator faults and external disturbances (Scenario 1)). (<b>a</b>) Evolution of x real vs. x desired (<b>b</b>) Evolution of x real vs. y desired (<b>c</b>) Evolution of z real vs. z desired (<b>d</b>) Evolution of ksi real vs. ksi desired.</p>
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<p>Roll and pitch angles (<span class="html-italic">φ</span>, <span class="html-italic">θ</span>) of the 4Y octocopter aircraft in the presence of actuator faults and external disturbances (Scenario 1).(<b>a</b>) Evolution of phi angle (<b>b</b>) Evolution of theta angle.</p>
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<p>The input forces of the 4Y octocopter aircraft (Scenario 1). (<b>a</b>) Evolution of input F1 (<b>b</b>) Evolution of input F2 (<b>c</b>) Evolution of input F3 (<b>d</b>) Evolution of input F4 (<b>e</b>) Evolution of input F5 (<b>f</b>) Evolution of input F6 (<b>g</b>) Evolution of input F7 (<b>h</b>) Evolution of input F8.</p>
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<p>3D position tracking result of the 4Y octocopter aircraft (Scenario 1).</p>
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<p>3D position tracking result of the 4Y octocopter aircraft (Scenario 1). (<b>a</b>) Evolution of x real vs. x desired (<b>b</b>) Evolution of x real vs. y desired (<b>c</b>) Evolution of z real vs. z desired (<b>d</b>) Evolution of ksi real vs. ksi desired.</p>
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<p>Roll and pitch angles (<span class="html-italic">φ</span>, <span class="html-italic">θ</span>) of the 4Y octocopter aircraft in the presence of actuator faults and external disturbances (Scenario 2). (<b>a</b>) Evolution of phi angle (<b>b</b>) Evolution of th et a angle.</p>
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<p>The input forces of the 4Y octocopter aircraft (Scenario 2). (<b>a</b>) Evolution of input F1 (<b>b</b>) Evolution of input F2 (<b>c</b>) Evolution of input F3 (<b>d</b>) Evolution of input F4.</p>
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<p>3D position tracking result of the 4Y octocopter aircraft (Scenario 2).</p>
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