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Search Results (2,884)

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25 pages, 10692 KiB  
Article
An Analytical Framework for Global Dynamic Modeling of Flexible Variable Topology Mechanisms
by Ruihai Geng, Yushu Bian, Zhihui Gao, Yize Zhao and Peng Liu
Actuators 2024, 13(12), 519; https://doi.org/10.3390/act13120519 - 15 Dec 2024
Viewed by 289
Abstract
The coupling of topology transition with flexible deformation and rigid motion presents significant challenges in the dynamic modeling of flexible variable topology mechanisms. Existing dynamics models are mostly special-purpose models for their particular cases and thus struggle to completely depict the general topology [...] Read more.
The coupling of topology transition with flexible deformation and rigid motion presents significant challenges in the dynamic modeling of flexible variable topology mechanisms. Existing dynamics models are mostly special-purpose models for their particular cases and thus struggle to completely depict the general topology transition characteristics. To address this gap, this paper proposes an analytical framework for the global dynamic modeling of flexible variable topology mechanisms, focusing on general cases. Initially, the flexible variable topology mechanisms are rigorously defined by the ordered triples and the general topology transition approaches are presented. A novel concept, the “basic flexible kinematic chain set”, is suggested, which can easily transform into the topology of each submechanism by slightly extending. Based on this concept, basic and conditional constraints are established. The continuous dynamic modeling method for each topology is developed using Jourdain’s principle and the Lagrange multiplier method. Additionally, three classes of constraints related to topology transition are identified, and their equations are formulated, elucidating the topology transition nature. Compatibility equations are proposed to define the new coordinate system for describing the deformation of flexible components after the topology transition. An impact dynamic equation is established to describe abrupt velocity change. Integrating compatibility and impact equations, a discontinuous dynamic modeling method for topology transition is developed. Finally, a flexible variable topology mechanism example is studied, and simulations and experiments are conducted to validate the proposed framework. This analytical framework is general-purpose and efficient, guiding the global dynamic modeling of various flexible variable topology mechanisms and the development of sophisticated control techniques. Full article
(This article belongs to the Section Actuators for Robotics)
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<p>Global dynamic modeling of an FVTM.</p>
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<p>Continuous dynamic modeling process of each submechanism.</p>
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<p>A kinematic analysis of the unconstrained component.</p>
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<p>Classification of constraints.</p>
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<p>A kinematic analysis of the unconstrained component.</p>
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<p>An FVTM possessing two topologies.</p>
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<p>The midpoint response of the flexible arm in a working cycle.</p>
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<p>The midpoint response of the flexible arm within the first and second working-phases: (<b>a</b>) the first working-phase; (<b>b</b>) the second working-phase.</p>
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<p>The frequency spectra of the first and second working-phases: (<b>a</b>) the first working-phase; (<b>b</b>) the second working-phase.</p>
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<p>An experimental setup possessing two topologies.</p>
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<p>Variable topology joint <math display="inline"><semantics> <mrow> <msub> <mi>J</mi> <mn>3</mn> </msub> </mrow> </semantics></math>.</p>
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<p>The midpoint response of the flexible arm in a working cycle.</p>
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<p>The midpoint response of the flexible arm within the first and second working-phases: (<b>a</b>) the first working-phase; (<b>b</b>) the second working-phase.</p>
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<p>The frequency spectra of the first and second working-phases: (<b>a</b>) the first working-phase; (<b>b</b>) the second working-phase.</p>
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15 pages, 1709 KiB  
Article
An Aircraft-Manipulator System for Virtual Flight Testing of Longitudinal Flight Dynamics
by Ademayowa A. Ishola, James F. Whidborne and Gilbert Tang
Robotics 2024, 13(12), 179; https://doi.org/10.3390/robotics13120179 - 15 Dec 2024
Viewed by 248
Abstract
A virtual flight test is the process of flying an aircraft model inside a wind tunnel in a manner that replicates free-flight. In this paper, a 3-DOF aircraft-manipulator system is proposed that can be used for longitudinal dynamics virtual flight tests. The system [...] Read more.
A virtual flight test is the process of flying an aircraft model inside a wind tunnel in a manner that replicates free-flight. In this paper, a 3-DOF aircraft-manipulator system is proposed that can be used for longitudinal dynamics virtual flight tests. The system consists of a two rotational degrees-of-freedom manipulator arm with an aircraft wind tunnel model attached to the third joint. This aircraft-manipulator system is constrained to operate for only the longitudinal motion of the aircraft. Thus, the manipulator controls the surge and heave of the aircraft whilst the pitch is free to rotate and can be actively controlled by means of an all-moving tailplane of the aircraft if required. In this initial study, a flight dynamics model of the aircraft is used to obtain dynamic response trajectories of the aircraft in free-flight. A model of the coupled aircraft-manipulator system developed using the Euler method is presented, and PID controllers are used to control the manipulator so that the aircraft follows the free-flight trajectory (with respect to the air). The inverse kinematics are used to produce the reference joint angles for the manipulator. The system is simulated in MATLAB/Simulink and a virtual flight test trajectory is compared with a free-flight test trajectory, demonstrating the potential of the proposed system for virtual flight tests. Full article
(This article belongs to the Special Issue Adaptive and Nonlinear Control of Robotics)
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Figure 1
<p>Schematic of the aircraft-manipulator system sited in the working section of an open-section wind tunnel (not to scale). The wind tunnel airspeed is shown as <math display="inline"><semantics> <msub> <mi>V</mi> <mrow> <mi>W</mi> <mi>T</mi> </mrow> </msub> </semantics></math>.</p>
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<p>A 1/12th scaled BAe Hawk aircraft.</p>
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<p>Configuration for 3-DOF aircraft longitudinal dynamics.</p>
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<p>Manipulator aircraft system configuration.</p>
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<p>Air velocity as the sum of wind tunnel velocity and aircraft velocity.</p>
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<p>Block diagram showing system control architecture for the <span class="html-italic">i</span>th loop. The block denoted as IK represents the inverse kinematics’ reference source.</p>
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<p>Short period mode pitch response of the free-flying Hawk model and the aircraft-manipulator system (AMS) model.</p>
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<p>Phugoid mode pitch response of the free-flying Hawk model and the aircraft-manipulator system (AMS) model.</p>
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<p>Short period mode altitude response of the free-flying Hawk model and the aircraft-manipulator system (AMS) model.</p>
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<p>Phugoid mode height and lateral surge position response of the free-flying Hawk model (resolved into the wind tunnel axes) and the aircraft-manipulator system (AMS) model.</p>
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<p>Phugoid mode position response of the aircraft-manipulator system (AMS) model in the wind tunnel.</p>
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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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<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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22 pages, 4715 KiB  
Article
Design and Testing of a Low-Speed, High-Frequency Straw Chopping and Returning Machine Using a Constant Breath Cam Mechanism
by Han Lin, Jin He, Guangyuan Zhong and Hanyu Yang
Agriculture 2024, 14(12), 2293; https://doi.org/10.3390/agriculture14122293 - 14 Dec 2024
Viewed by 188
Abstract
Straw incorporation offers significant advantages in agricultural crop cultivation systems. Mechanized methods constitute the predominant approach, potentially reducing yield costs and enhancing operational efficiency. The imperative to enhance the quality of straw chopping within the field is of particular significance, as suboptimal chopping [...] Read more.
Straw incorporation offers significant advantages in agricultural crop cultivation systems. Mechanized methods constitute the predominant approach, potentially reducing yield costs and enhancing operational efficiency. The imperative to enhance the quality of straw chopping within the field is of particular significance, as suboptimal chopping quality can engender a cascade of issues, particularly seeding blockages. The straw chopping pass rate (CPR) is a pivotal metric for assessing the quality of straw chopping. Therefore, enhancing the CPR during the straw chopping process is necessary. This study introduces a novel maize-straw-chopping device with the ground as its supporting base. This device facilitates the rapid vertical chopping of maize straw through a constant breath cam transmission mechanism. Critical parameters were determined to optimize the performance of the chopping device by establishing mathematical models and kinematic simulation analysis methods. With the help of Rocky 2022.R2 software, the influence of the rotational velocity of the draft, tractor velocity, and blade edge angles on the CPR during the operation of the device was analyzed. The Box–Behnken test methodology was used to carry out a three-factor, three-level orthogonal rotation test to obtain the optimal working parameter combination. The results indicated that the maximum CPR value was achieved with a draft rotational velocity of 245 rpm, a tractor velocity of 3.8 km/h, and a blade edge angle of 20.75°. Finally, field validation experiments were conducted under these optimized conditions, with the average CPR of maize straw reaching an impressive 91.45%. These findings have significant implications for enhancing crop production practices. Full article
(This article belongs to the Section Agricultural Technology)
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<p>Overall structure of straw-chopping and -returning machine: (1) sliders; (2) blade; (3) plates; (4) cam; (5) drive wheels; (6) shaft; (7) sprocket.</p>
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<p>Low-speed and high-frequency working principle. The yellow represents maize straw; The blue represents plates; The green represents cam; The red represents blade.</p>
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<p>Theoretical contour lines of cams with different parameters “<span class="html-italic">n</span>”.</p>
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<p>Schematic diagram of cam mechanism movement. <span class="html-italic">Oxy</span> is a stationary coordinate system; <span class="html-italic">Ox</span><sub>1</sub><span class="html-italic">y</span><sub>1</sub> is a dynamic coordinate system; AB is the straight line where <span class="html-italic">oy</span> is located; A<sub>1</sub>B<sub>1</sub> is the straight line where <span class="html-italic">oy</span><sub>1</sub> is located; <span class="html-italic">O</span><sub>1</sub><span class="html-italic">x</span><sub>2</sub><span class="html-italic">y</span><sub>2</sub> is a moving coordinate system; <span class="html-italic">r</span><sub>0</sub> is the radius of the roller; <span class="html-italic">δ</span> is the arbitrary angular displacement through which the roller rotates; <span class="html-italic">θ</span> is the arbitrary angular displacement through which the roller rotates; <span class="html-italic">ω</span> is the angular velocity of cam rotation; <span class="html-italic">T</span> represents the coordinates of the contact point between the roller and the cam.</p>
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<p>Variation law of the pressure angle under different factors.</p>
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<p>Mechanical model of the straw chopping process.</p>
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<p>Description of a bonded sphero-cylinder model.</p>
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<p>The simulation process during operation (<b>a</b>) Front view; (<b>b</b>) top view. (1) Stubble; (2) straw; (3) chopping device; (4) blade; (5) broken straw.</p>
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<p>The simulation process during operation (<b>a</b>) Front view; (<b>b</b>) top view. (1) Stubble; (2) straw; (3) chopping device; (4) blade; (5) broken straw.</p>
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<p>Analysis of the interaction between two factors on CPR. (<b>a</b>) Influence of factor A and factor B on CPR; (<b>b</b>) influence of factor B and factor C on CPR. Different colors represent different straw CPR.</p>
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<p>Field validation experiment.</p>
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33 pages, 21111 KiB  
Review
A Review on Sheet Metal Forming Behavior in High-Strength Steels and the Use of Numerical Simulations
by Luis Fernando Folle, Tiago Nunes Lima, Matheus Passos Sarmento Santos, Bruna Callegari, Bruno Caetano dos Santos Silva, Luiz Gustavo Souza Zamorano and Rodrigo Santiago Coelho
Metals 2024, 14(12), 1428; https://doi.org/10.3390/met14121428 - 13 Dec 2024
Viewed by 376
Abstract
High-strength steels such as Dual Phase (DP), Transformation-Induced Plasticity (TRIP), and Twinning-Induced Plasticity (TWIP) steels have gained importance in automotive applications due to the potential for weight reduction and increased performance in crash tests. However, as resistance increases, there is also an increase [...] Read more.
High-strength steels such as Dual Phase (DP), Transformation-Induced Plasticity (TRIP), and Twinning-Induced Plasticity (TWIP) steels have gained importance in automotive applications due to the potential for weight reduction and increased performance in crash tests. However, as resistance increases, there is also an increase in springback due to residual stresses after the forming process. This is mainly because of the greater elastic region of these materials and other factors associated with strain hardening, such as the Bauschinger effect, that brings theory of kinematic hardening to mathematical modeling. This means that finite element software must consider these properties so that the simulation can accurately predict the behavior. Currently, this knowledge is still not widespread since it has never been used in conventional materials. Additionally, engineers and researchers use the Forming Limit Diagram (FLD) curve in their studies. However, it does not fully represent the actual failure limit of materials, especially in high-strength materials. Based on this, the Fracture Forming Limit Diagram (FFLD) curve has emerged, which proposes to resolve these limitations. Thus, this review aims to focus on how finite element methods consider all these factors in their modeling, especially when it comes to the responses of high-strength steels. Full article
(This article belongs to the Special Issue Modeling, Simulation and Experimental Studies in Metal Forming)
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<p>Advanced high-strength steels developed for automotive applications [<a href="#B20-metals-14-01428" class="html-bibr">20</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Example of springback in sheet metal bent at 90° [<a href="#B34-metals-14-01428" class="html-bibr">34</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Effect of springback on high-strength steels [<a href="#B35-metals-14-01428" class="html-bibr">35</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Springback effect after (<b>a</b>) bending in deep drawing [<a href="#B36-metals-14-01428" class="html-bibr">36</a>] and (<b>b</b>) V-bending. Reproduced with permission from Elsevier, 2024.</p>
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<p>Influence of bending and straightening on residual stresses during deep drawing of metallic sheets.</p>
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<p>(<b>a</b>) Standard geometry used for the study of springback; (<b>b</b>) measurements made on the part after bending [<a href="#B36-metals-14-01428" class="html-bibr">36</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Convergence analysis on the axial loading value (normalized) as a function of the density and the element type. Figure reproduced under Creative Commons Attribution 4.0 International License from [<a href="#B39-metals-14-01428" class="html-bibr">39</a>].</p>
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<p>The error in the axial loading estimation as a function of the CPU time, for different element types and numbers of elements. Figure reproduced under Creative Commons Attribution 4.0 International License from [<a href="#B39-metals-14-01428" class="html-bibr">39</a>].</p>
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<p>Shell element with integration points in the thickness.</p>
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<p>Strain stages in a tensile test for a conventional material.</p>
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<p>Types of localized failure that can occur through void nucleation: (<b>a</b>) Failure by localized shear plastic without necking, (<b>b</b>) Failure by localized shear plastic after necking and (<b>c</b>) Failure by void coalescence with obvious necking [<a href="#B53-metals-14-01428" class="html-bibr">53</a>]. Reproduced with permission from John Wiley and Sons, 2024.</p>
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<p>Examples of behavior under fracture of 3 metals [<a href="#B54-metals-14-01428" class="html-bibr">54</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Two types of mechanisms for void coalescence: (<b>a</b>) parallel connection between voids; (<b>b</b>) void shear connection [<a href="#B55-metals-14-01428" class="html-bibr">55</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Two failure mechanisms: necking for SPCC and SPRC and ductile fracture for other sheet metals. SPCC and SPRC are conventional carbon steels [<a href="#B56-metals-14-01428" class="html-bibr">56</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Forming Limit Diagram obtained by measuring the diffuse necking: (<b>a</b>) specimens, (<b>b</b>) fracture regions, and (<b>c</b>) plotted curve.</p>
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<p>Visualization of the circles used in the Nakajima test to obtain the main strains [<a href="#B57-metals-14-01428" class="html-bibr">57</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Location of positions where deformations can be measured. (<b>a</b>) In the necking zone. (<b>b</b>) Out of the necking zone. Figure reproduced under Creative Commons Attribution 4.0 International License from [<a href="#B58-metals-14-01428" class="html-bibr">58</a>].</p>
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<p>Comparison between FLD (black and red) and FFLD (blue) curves: (<b>a</b>) in the space of major and minor principal strains; (<b>b</b>) in the space of stress triaxiality and equivalent strain to failure [<a href="#B76-metals-14-01428" class="html-bibr">76</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Global strain status, experimental FLC, and damage: (<b>a</b>) results at the integration points located on the negative surface; (<b>b</b>) results at the integration points located on the positive surface [<a href="#B76-metals-14-01428" class="html-bibr">76</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Comparison of the (<b>a</b>) experiment and (<b>b</b>) simulation [<a href="#B76-metals-14-01428" class="html-bibr">76</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Comparison between results obtained in a (<b>a</b>) simulation using the FLC curve and (<b>b</b>) a simulation using the MMC fracture criterion of an automotive front rail made from DP780 steel [<a href="#B73-metals-14-01428" class="html-bibr">73</a>]. Reproduced with permission from SAE international, 2024.</p>
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<p>Fracture strain versus stress triaxiality and Lode angle simulated. Figure reproduced under Creative Commons Attribution 3.0 International License from [<a href="#B88-metals-14-01428" class="html-bibr">88</a>].</p>
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<p>Mathematical adjustment curves for tensile testing. Figure reproduced under Creative Commons Attribution 4.0 International License from [<a href="#B92-metals-14-01428" class="html-bibr">92</a>].</p>
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<p>Comparison of tensile test data with Bulge test. Figure reproduced under Creative Commons Attribution 3.0 International License from [<a href="#B93-metals-14-01428" class="html-bibr">93</a>].</p>
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<p>Three main directions of anisotropy measurements in the sheet rolling Direction.</p>
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<p>Compressed disk specimens of AA2090-T3 using different lubricants and at different thickness strains (ε<sub>z</sub>) [<a href="#B80-metals-14-01428" class="html-bibr">80</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Strains measured in samples after the disk compression test [<a href="#B80-metals-14-01428" class="html-bibr">80</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Calculation of the biaxial anisotropy coefficient. Figure reproduced under Creative Commons Attribution 4.0 International License from [<a href="#B103-metals-14-01428" class="html-bibr">103</a>].</p>
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<p>Comparison of true stress–strain curves determined by uniaxial tension and Bulge test for 3 different steels [<a href="#B104-metals-14-01428" class="html-bibr">104</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Schematic representation of the Bauschinger effect.</p>
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<p>Typical tensile–compression test [<a href="#B104-metals-14-01428" class="html-bibr">104</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Stress–strain curves of the tensile test under loading–unloading–reloading condition for determination of Young’s modulus of elasticity at different pre-strains [<a href="#B104-metals-14-01428" class="html-bibr">104</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Young’s modulus change with plastic strain.</p>
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<p>Scheme of a tribological system in sheet forming. Figure reproduced under Creative Commons Attribution 4.0 International License from [<a href="#B116-metals-14-01428" class="html-bibr">116</a>].</p>
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<p>The 3D images and mean roughness Rz values of bionic structures (<b>a</b>) St1, (<b>b</b>) St2, (<b>c</b>) St3, (<b>d</b>) St4 (<b>e</b>) St5, and (<b>f</b>) flat reference surface [<a href="#B118-metals-14-01428" class="html-bibr">118</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Friction coefficient results obtained [<a href="#B118-metals-14-01428" class="html-bibr">118</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Coefficient of friction under different contact pressures [<a href="#B120-metals-14-01428" class="html-bibr">120</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Friction variation with working temperature: (<b>a</b>) Variation curves of the friction coefficient with time under different temperatures; (<b>b</b>) Experimental friction coefficients at different temperatures. Figure reproduced under Creative Commons Attribution 4.0 International License from [<a href="#B121-metals-14-01428" class="html-bibr">121</a>].</p>
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<p>Results of the coefficient of friction as a function of sliding speed and contact pressure. Figure reproduced under Creative Commons Attribution 4.0 International License from [<a href="#B122-metals-14-01428" class="html-bibr">122</a>].</p>
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<p>Comparison of springback for different mesh sizes in finite element simulation [<a href="#B38-metals-14-01428" class="html-bibr">38</a>]. Reproduced with permission from Springer Nature, 2024.</p>
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<p>Comparison of springback for different time steps in finite element simulation [<a href="#B38-metals-14-01428" class="html-bibr">38</a>]. Reproduced with permission from Springer Nature, 2024.</p>
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<p>The comparison of calculated springback between isotropic and kinematic hardening mode [<a href="#B38-metals-14-01428" class="html-bibr">38</a>]. Reproduced with permission from Springer Nature, 2024.</p>
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<p>Finite element simulation results for DP980 in a springback test: (<b>a</b>) θ<sub>1</sub>, (<b>b</b>) θ<sub>2</sub>, and (<b>c</b>) sidewall radius ρ [<a href="#B36-metals-14-01428" class="html-bibr">36</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Finite element simulation results for TWIP980 in a springback test: (<b>a</b>) θ<sub>1</sub>, (<b>b</b>) θ<sub>2</sub>, and (<b>c</b>) sidewall radius ρ [<a href="#B36-metals-14-01428" class="html-bibr">36</a>]. Reproduced with permission from Elsevier, 2024.</p>
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<p>Numerical simulation results for models with constant and variable friction [<a href="#B120-metals-14-01428" class="html-bibr">120</a>]. Reproduced with permission from Elsevier, 2024.</p>
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22 pages, 4429 KiB  
Article
Deep Reinforcement Learning-Based Robotic Puncturing Path Planning of Flexible Needle
by Jun Lin, Zhiqiang Huang, Tengliang Zhu, Jiewu Leng and Kai Huang
Processes 2024, 12(12), 2852; https://doi.org/10.3390/pr12122852 - 12 Dec 2024
Viewed by 326
Abstract
The path planning of flexible needles in robotic puncturing presents challenges such as limited model accuracy and poor real-time performance, which affect both efficiency and accuracy in complex medical scenarios. To address these issues, this paper proposes a deep reinforcement learning-based path planning [...] Read more.
The path planning of flexible needles in robotic puncturing presents challenges such as limited model accuracy and poor real-time performance, which affect both efficiency and accuracy in complex medical scenarios. To address these issues, this paper proposes a deep reinforcement learning-based path planning method for flexible needles in robotic puncturing. Firstly, we introduce a unicycle model to describe needle motion and design a hierarchical model to simulate layered tissue interactions with the needle. The forces exerted by tissues at different positions on the flexible needle are considered, achieving a combination of kinematic and mechanical models. Secondly, a deep reinforcement learning framework is built, integrating obstacle avoidance and target attraction to optimize path planning. The design of state features, the action space, and the reward function is tailored to enhance the model’s decision-making capabilities. Moreover, we incorporate a retraction mechanism to bolster the system’s adaptability and robustness in the dynamic context of surgical procedures. Finally, laparotomy simulation results validate the proposed method’s effectiveness and generalizability, demonstrating its superiority over current state-of-the-art techniques in robotic puncturing. Full article
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<p>The trajectory of flexible needle movement under the interaction force between the needle and tissue.</p>
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<p>The design of a simplified human tissue layering model. This model constructs three layers of puncture spaces based on the density differences in internal tissues in the human body. Each layer of space represents a portion of tissues that are adjacent to each other in the internal space of the human body and have little density difference. When the puncture needle passes through these spaces, the radius of movement in each layer will be different due to the density difference.</p>
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<p>Deep reinforcement learning-based robotic puncturing path planning of flexible needle.</p>
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<p>Model of obstacle repulsion and lesion target gravity.</p>
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<p>A flowchart of the DQN-based path planning of puncturing.</p>
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<p>(<b>a</b>) Reward convergence curve for different learning rates; (<b>b</b>) reward convergence curve for different discount factors; (<b>c</b>) reward convergence curve for different batch sizes; (<b>d</b>) reward convergence curve for different target Q-network update frequencies.</p>
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<p>Reward graph of the DQN-based PPFNP.</p>
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<p>Reward convergence plot for different space sizes.</p>
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<p>Reward convergence plots for different number of agents.</p>
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17 pages, 4021 KiB  
Article
Neural Network Identification-Based Model Predictive Heading Control for Wave Gliders
by Peng Jin, Baolin Zhang and Yun Zhang
J. Mar. Sci. Eng. 2024, 12(12), 2279; https://doi.org/10.3390/jmse12122279 - 11 Dec 2024
Viewed by 352
Abstract
This paper deals with the neural network identification-based model predictive heading control problem in a wave glider. First, based on a kinematic model of the wave glider subjected to external disturbance and system uncertainty, a state space model of the wave glider is [...] Read more.
This paper deals with the neural network identification-based model predictive heading control problem in a wave glider. First, based on a kinematic model of the wave glider subjected to external disturbance and system uncertainty, a state space model of the wave glider is established. Then, a neural network identification-based model predictive heading controller (NNI-MPHC) is designed for the wave glider. The heading controller mainly includes three components: a model predictive controller, a neural network-based model identifier, and a linear reduced-order extended state observer. Third, a design algorithm of the NNI-MPHC is presented. The algorithm is demonstrated through simulation, where the results show the following: (i) The designed NNI-MPHC is remarkably capable of guaranteeing the tracing effects of the wave glider. (ii) Comparing the NNI-MPHC and existing heading controllers, the former is better than the latter in terms of tracking accuracy and rapidity and robustness to model uncertainty and/or external disturbances. Full article
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<p>The overall structure of a wave glider.</p>
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<p>Motion coordinate system for a wave glider [<a href="#B30-jmse-12-02279" class="html-bibr">30</a>].</p>
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<p>The overall structure of NNI-MPHC.</p>
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<p>The overall structure of Hopfield network.</p>
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<p>External disturbance added in wave glider.</p>
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<p>Case I—Identified systematic parameters versus time.</p>
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<p>Case II—Identified systematic parameters versus time.</p>
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<p>Case III—Identified systematic parameters versus time.</p>
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<p>Case I—Yaw angle, tracking error, and speed of glider with different heading controllers.</p>
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<p>Case II—Yaw angle, tracking error, and speed of glider with different heading controllers.</p>
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<p>Case III − Yaw angle, tracking error, and speed of glider with different heading controllers.</p>
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17 pages, 4573 KiB  
Article
Study on Trajectory Optimization for a Flexible Parallel Robot in Tomato Packaging
by Tianci Guo, Jiangbo Li, Yizhi Zhang, Letian Cai and Qicheng Li
Agriculture 2024, 14(12), 2274; https://doi.org/10.3390/agriculture14122274 (registering DOI) - 11 Dec 2024
Viewed by 344
Abstract
Currently, flexible robots, exemplified by parallel robots, play a crucial role in the automated packaging of agricultural products due to their rapid, accurate, and stable characteristics. This research systematically explores trajectory planning strategies for parallel robots in the high-speed tomato-grabbing process. Kinematic analysis [...] Read more.
Currently, flexible robots, exemplified by parallel robots, play a crucial role in the automated packaging of agricultural products due to their rapid, accurate, and stable characteristics. This research systematically explores trajectory planning strategies for parallel robots in the high-speed tomato-grabbing process. Kinematic analysis of the parallel robot was conducted using geometric methods, deriving the coordinates of each joint at various postures, resulting in a kinematic forward solution model and corresponding equations, which were verified with data. To address the drawbacks of the point-to-point “portal” trajectory in tomato grabbing, a 3-5-5-3 polynomial interpolation method in joint space was proposed to optimize the path, enhancing trajectory smoothness. To improve the efficiency of the tomato packaging process, a hybrid algorithm combining particle swarm optimization (PSO) and genetic algorithms (GA) was developed to optimize the operation time of the parallel robot. Compared to traditional PSO, the proposed algorithm exhibits better global convergence and is less likely to fall into local optima, thereby ensuring a smoother and more efficient path in the robot-grabbing tomato process and providing technical support for automated tomato packaging. Full article
(This article belongs to the Section Agricultural Technology)
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<p>Schematic diagram and actual figure of the parallel robot. (<b>a</b>) Schematic diagram of the parallel robot structure; (<b>b</b>) Actual parallel robot.</p>
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<p>Traditional “portal” type trajectory.</p>
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<p>A 3-5-5-3 polynomial interpolation joint angle curve.</p>
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<p>A 3-5-5-3 polynomial interpolation joint angular velocity curve.</p>
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<p>A 3-5-5-3 polynomial interpolation joint angular acceleration curve.</p>
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<p>Optimal time planning related curves for joint 1. (<b>a</b>) Angle curve post-optimization of joint 1. (<b>b</b>) Angular velocity curve post-optimization of joint 1. (<b>c</b>) Angular acceleration curve post-optimization of joint 1. (<b>d</b>) Fitness during the optimization of joint 1.</p>
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<p>Optimal time planning related curves for joint 2. (<b>a</b>) Angle curve post-optimization of joint 2. (<b>b</b>) Angular velocity curve post-optimization of joint 2. (<b>c</b>) Angular acceleration curve post-optimization of joint 2. (<b>d</b>) Fitness during the optimization of joint 2.</p>
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<p>Optimal time planning related curves for joint 2. (<b>a</b>) Angle curve post-optimization of joint 2. (<b>b</b>) Angular velocity curve post-optimization of joint 2. (<b>c</b>) Angular acceleration curve post-optimization of joint 2. (<b>d</b>) Fitness during the optimization of joint 2.</p>
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<p>Optimal time planning related curves for joint 3. (<b>a</b>) Angle curve post-optimization of joint 3. (<b>b</b>) Angular velocity curve post-optimization of joint 3. (<b>c</b>) Angular acceleration curve post-optimization of joint 3. (<b>d</b>) Fitness during the optimization of joint 3.</p>
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<p>Robot experiment platform.</p>
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<p>A 3D modeling drawing of a packaging robot.</p>
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<p>A 100-time tomato-boxing test results.</p>
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21 pages, 19119 KiB  
Article
Caterpillar-Inspired Multi-Gait Generation Method for Series-Parallel Hybrid Segmented Robot
by Mingyuan Dou, Ning He, Jianhua Yang, Lile He, Jiaxuan Chen and Yaojiumin Zhang
Biomimetics 2024, 9(12), 754; https://doi.org/10.3390/biomimetics9120754 - 11 Dec 2024
Viewed by 402
Abstract
The body structures and motion stability of worm-like and snake-like robots have garnered significant research interest. Recently, innovative serial–parallel hybrid segmented robots have emerged as a fundamental platform for a wide range of motion modes. To address the hyper-redundancy characteristics of these hybrid [...] Read more.
The body structures and motion stability of worm-like and snake-like robots have garnered significant research interest. Recently, innovative serial–parallel hybrid segmented robots have emerged as a fundamental platform for a wide range of motion modes. To address the hyper-redundancy characteristics of these hybrid structures, we propose a novel caterpillar-inspired Stable Segment Update (SSU) gait generation approach, establishing a unified framework for multi-segment robot gait generation. Drawing inspiration from the locomotion of natural caterpillars, the segments are modeled as rigid bodies with six degrees of freedom (DOF). The SSU gait generation method is specifically designed to parameterize caterpillar-like gaits. An inverse kinematics solution is derived by analyzing the forward kinematics and identifying the minimum lifting segment, framing the problem as a single-segment end-effector tracking task. Three distinct parameter sets are introduced within the SSU method to account for the stability of robot motion. These parameters, represented as discrete hump waves, are intended to improve motion efficiency during locomotion. Furthermore, the trajectories for each swinging segment are determined through kinematic analysis. Experimental results validate the effectiveness of the proposed SSU multi-gait generation method, demonstrating the successful traversal of gaps and rough terrain. Full article
(This article belongs to the Section Locomotion and Bioinspired Robotics)
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<p>Natural caterpillar locomotion pattern. (<b>a</b>) Natural caterpillar locomotion sequence (the red dashed line represents stable segment; the yellow dashed line represents swinging segment). (<b>b</b>) Schematic diagram of natural caterpillar segments.</p>
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<p>Nine-state of one segment motion trajectory.</p>
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<p>The hump formed on the natural caterpillar locomotion in a single segment 9-state trajectory. The illustration of the hump formed in the SSU method (red segment ((<b>3</b>)–(<b>6</b>) left) is the segment that is about to enter the swinging phase during the stance phase; red segment (right) is the segment that has ended the swinging phase during the stance phase. The yellow segment is the swinging segment in the swinging phase).</p>
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<p>Footfall-pattern diagram of nature caterpillar gait.</p>
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<p>Robot mechanism and variables. (<b>a</b>) 3-RSR. (<b>b</b>) 4-3-RSR.</p>
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<p>The kinematics analysis of 4-3-RSR robot SSU parameters <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>min</mi> </mrow> </msub> </mrow> </semantics></math>. In the 3-RSR parallel mechanism, (<b>a</b>) the relationship of the distal plate center in axis <math display="inline"><semantics> <mi>X</mi> </semantics></math> coordinate component and base angle <math display="inline"><semantics> <mrow> <msub> <mi>θ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>θ</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>θ</mi> <mn>3</mn> </msub> </mrow> </semantics></math>; (<b>b</b>) the relationship of the pitch angle of the distal plate and base angle <math display="inline"><semantics> <mrow> <msub> <mi>θ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>θ</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>θ</mi> <mn>3</mn> </msub> </mrow> </semantics></math>; (<b>c</b>) the relationship of the pitch angle of the distal plate and distal plate center in axis <math display="inline"><semantics> <mi>X</mi> </semantics></math> coordinate component. (<b>d</b>) The 2-3-RSR mechanism and variables.</p>
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<p>The robot posture when <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>=</mo> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </mrow> </semantics></math>.</p>
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<p>SSU gait generation flowchart.</p>
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<p>The SSU swinging segment trajectory. (<b>a</b>) The gaits sequence for <math display="inline"><semantics> <mrow> <mfenced> <mrow> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> <mi>th</mi> </mrow> </semantics></math> segment trajectory <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>s</mi> <mo stretchy="false">^</mo> </mover> <mrow> <msub> <mi>c</mi> <mn>1</mn> </msub> </mrow> </msub> <mfenced> <mi>t</mi> </mfenced> </mrow> </semantics></math>. (<b>b</b>) The gaits sequence for <math display="inline"><semantics> <mrow> <mfenced> <mrow> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> <mi>th</mi> </mrow> </semantics></math> segment trajectory <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>s</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi>c</mi> <mn>2</mn> </mrow> </msub> <mfenced> <mi>t</mi> </mfenced> </mrow> </semantics></math>. (<b>c</b>) The trajectory of <math display="inline"><semantics> <mrow> <mfenced> <mrow> <mi>m</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfenced> <mi>th</mi> </mrow> </semantics></math> segment progressive. (<b>d</b>) The compensate trajectory <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>s</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi>c</mi> <mn>1</mn> </mrow> </msub> <mfenced> <mi>t</mi> </mfenced> </mrow> </semantics></math> of <math display="inline"><semantics> <mrow> <mfenced> <mrow> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> <mi>th</mi> </mrow> </semantics></math> segment. (<b>e</b>) The compensate trajectory <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>s</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi>c</mi> <mn>2</mn> </mrow> </msub> <mfenced> <mi>t</mi> </mfenced> </mrow> </semantics></math> of <math display="inline"><semantics> <mrow> <mfenced> <mrow> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> <mi>th</mi> </mrow> </semantics></math> segment progressive.</p>
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<p>Three gaits pattern of 4-3-RSR robot. (<b>a</b>) The 1-1-1-1-1 gait, (<b>b</b>) 1-1-2-1 gait, and (<b>c</b>) 1-2-2 gait. Footfall-pattern diagram of the (<b>d</b>) 1-1-1-1-1 gait, (<b>e</b>) 1-1-2-1 gait, and (<b>f</b>) 1-2-2 gait.</p>
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<p>The 4-3-RSR robot. (<b>a</b>) Three rotary joints replace the sphere joint. (<b>b</b>) The 4-3-RSR robot press plate (left) and main view (right).</p>
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<p>Joint trajectories of 4-3-RSR robot. (<b>a</b>) The 1-1-1-1-1 gait, (<b>b</b>) 1-1-2-1 gait, and (<b>c</b>) 1-2-2 gait, where (1) (2) (3) (4) illuminate the 1st, 2nd, 3rd, and 4th 3-RSR parallel mechanism joint trajectories.</p>
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<p>Three gaits experiment of the 4-3-RSR robot. (<b>a</b>) The 1-1-1-1-1 gait, (<b>b</b>) 1-1-2-1 gait, and (<b>c</b>) 1-2-2 gait. (The red dotted line represents the stable segment, and the yellow dotted line represents the swinging segment).</p>
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<p>Locomotion of the 4-3-RSR robot rectilinear gait.</p>
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<p>The 1-1-1-1-1-1 gait crossing gaps.</p>
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<p>The 1-1-1-1-1-1 gait on roughness terrain.</p>
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21 pages, 2964 KiB  
Article
Prediction of Drivers’ Red-Light Running Behaviour in Connected Vehicle Environments Using Deep Recurrent Neural Networks
by Md Mostafizur Rahman Komol, Mohammed Elhenawy, Jack Pinnow, Mahmoud Masoud, Andry Rakotonirainy, Sebastien Glaser, Merle Wood and David Alderson
Mach. Learn. Knowl. Extr. 2024, 6(4), 2855-2875; https://doi.org/10.3390/make6040136 - 11 Dec 2024
Viewed by 415
Abstract
Red-light running at signalised intersections poses a significant safety risk, necessitating advanced predictive technologies to predict red-light violation behaviour, especially for advanced red-light warning (ARLW) systems. This research leverages Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU) models to forecast the red-light [...] Read more.
Red-light running at signalised intersections poses a significant safety risk, necessitating advanced predictive technologies to predict red-light violation behaviour, especially for advanced red-light warning (ARLW) systems. This research leverages Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU) models to forecast the red-light running and stopping behaviours of drivers in connected vehicles. We utilised data from the Ipswich Connected Vehicle Pilot (ICVP) in Queensland, Australia, which gathered naturalistic driving data from 355 connected vehicles at 29 signalised intersections. These vehicles broadcast Cooperative Awareness Messages (CAM) within the Cooperative Intelligent Transport Systems (C-ITS), providing kinematic inputs such as vehicle speed, speed limits, longitudinal and lateral accelerations, and yaw rate. These variables were monitored at 100-millisecond intervals for durations from 1 to 4 s before reaching various distances from the stop line. Our results indicate that the LSTM model outperforms the GRU in predicting both red-light running and stopping behaviours with high accuracy. However, the pre-trained GRU model performs better in predicting red-light running specifically, making it valuable in applications requiring early violation prediction. Implementing these models can enhance red-light violation countermeasures, such as dynamic all-red extension (DARE), decreasing the likelihood of severe collisions and enhancing road users’ safety. Full article
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<p>Red-light running traffic violation and right-angle collision [<a href="#B10-make-06-00136" class="html-bibr">10</a>].</p>
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<p>Flowchart of red-light running behaviour prediction methodology.</p>
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<p>(<b>a</b>) Vehicle passing intersection during yellow signal which then turns red when the vehicle enters the conflict zone. (<b>b</b>) Vehicle turning during green signal but the straight signal is red [<a href="#B40-make-06-00136" class="html-bibr">40</a>].</p>
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<p>The windowing of datasets with different traffic monitoring times at different distances before the stop line [<a href="#B20-make-06-00136" class="html-bibr">20</a>].</p>
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<p>Transfer learning with pre-trained LSTM model to predict red-light running behaviour.</p>
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<p>The comparison of LSTM and GRU models’ prediction accuracy before and after data upsampling.</p>
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<p>The comparison of LSTM and GRU models’ prediction accuracy before and after data upsampling.</p>
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<p>The comparison of LSTM and GRU models’ performance measures.</p>
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14 pages, 5971 KiB  
Article
Flight Altitude and Sensor Angle Affect Unmanned Aerial System Cotton Plant Height Assessments
by Oluwatola Adedeji, Alwaseela Abdalla, Bishnu Ghimire, Glen Ritchie and Wenxuan Guo
Drones 2024, 8(12), 746; https://doi.org/10.3390/drones8120746 - 10 Dec 2024
Viewed by 374
Abstract
Plant height is a critical biophysical trait indicative of plant growth and developmental conditions and is valuable for biomass estimation and crop yield prediction. This study examined the effects of flight altitude and camera angle in quantifying cotton plant height using unmanned aerial [...] Read more.
Plant height is a critical biophysical trait indicative of plant growth and developmental conditions and is valuable for biomass estimation and crop yield prediction. This study examined the effects of flight altitude and camera angle in quantifying cotton plant height using unmanned aerial system (UAS) imagery. This study was conducted in a field with a sub-surface irrigation system in Lubbock, Texas, between 2022 and 2023. Images using the DJI Phantom 4 RTKs were collected at two altitudes (40 m and 80 m) and three sensor angles (45°, 60°, and 90°) at different growth stages. The resulting images depicted six scenarios of UAS altitudes and camera angles. The derived plant height was subsequently calculated as the vertical difference between the apical region of the plant and the ground elevation. Linear regression compared UAS-derived heights to manual measurements from 96 plots. Lower altitudes (40 m) outperformed higher altitudes (80 m) across all dates. For the early season (4 July 2023), the 40 m altitude had r2 = 0.82–0.86 and RMSE = 2.02–2.16 cm compared to 80 m (r2 = 0.66–0.68, RMSE = 7.52–8.76 cm). Oblique angles (45°) yielded higher accuracy than nadir (90°) images, especially in the late season (24 October 2022) results (r2 = 0.96, RMSE = 2.95 cm vs. r2 = 0.92, RMSE = 3.54 cm). These findings guide optimal UAS parameters for plant height measurement. Full article
(This article belongs to the Special Issue Advances of UAV Remote Sensing for Plant Phenology)
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<p>Study site on a research farm in Lubbock County, Texas, in 2022 and 2023.</p>
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<p>DJI Phantom 4 RTKs and GNSS mobile station for acquiring RGB images in a research field in Lubbock, Texas, 2022. (<b>a</b>) DJI Phantom 4 RTKs UAS platform (<b>Left</b>), (<b>b</b>) Phantom 4 UAS controller (<b>middle</b>), and (<b>c</b>) D-RTKs 2 High-Precision GNSS Mobile Station (<b>right</b>) (source: <a href="https://www.dji.com" target="_blank">https://www.dji.com</a>, accessed on 5 January 2024.).</p>
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<p>Image acquisitions at two flight altitudes (40 m and 80 m) and three camera angles (45°, 60°, and 90°) using a UAS in a cotton field in Lubbock, Texas.</p>
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<p>Workflow for processing unmanned aerial system (UAS) images to estimate plant height.</p>
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<p>Boxplot of plant height measurements in a research field in Lubbock, Texas, on 4 July and 2 August 2023 and 28 August and 24 October 2022.</p>
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<p>Errors in UAS-derived cotton plant height at two UAS flight altitudes and three camera angles on (<b>a</b>) 4 July 2023, (<b>b</b>) 2 August 2023, (<b>c</b>) 28 August 2022, and (<b>d</b>) 24 October 2022.</p>
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<p>Interactions between flight altitude and camera angle for errors in plant heights derived from UAS image on (<b>a</b>) 4 July 2023, (<b>b</b>) 2 August 2023, (<b>c</b>) 28 August 2022, and (<b>d</b>) 24 October 2022.</p>
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<p>Tukey’s post hoc test for different camera angles (45°, 60°, 90°) at different flight altitudes for errors in plant heights derived from UAS images on (<b>a</b>) 4 July 2023, (<b>b</b>) 2 August 2023, (<b>c</b>) 28 August 2022, and (<b>d</b>) 24 October 2022. Significance levels: * <span class="html-italic">p</span> &lt; 0.05, ** <span class="html-italic">p</span> &lt; 0.01, and *** <span class="html-italic">p</span> &lt; 0.001, and n.s. represents non-significant results.</p>
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<p>Relationship between measured plant height and UAS-derived plant height from different UAS altitudes and angles in a research field in Lubbock, Texas. (<b>a</b>) 4 July 2023, (<b>b</b>) 2 August 2023, (<b>c</b>) 28 August 2022, and (<b>d</b>) 24 October 2022.</p>
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<p>Relationship between measured and UAS-derived plant heights using 30% test data for a flight altitude of 40 m and a camera angle of 45° for 4 July and 2 August 2023 and 28 August and 24 October 2022.</p>
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33 pages, 12166 KiB  
Article
Dual Approach to Large-Scale Seismic Vulnerability Assessment of Churches Through Representative Archetypes
by Giorgia Cianchino, Maria Giovanna Masciotta, Gianfranco De Matteis and Giuseppe Brando
Heritage 2024, 7(12), 6998-7030; https://doi.org/10.3390/heritage7120324 - 10 Dec 2024
Viewed by 440
Abstract
In this paper, the seismic vulnerability of churches is assessed using different methods characterized by different levels of complexity depending on the accuracy of the results to be achieved. Specifically, we compare the two main types of methodologies applied in the literature, namely, [...] Read more.
In this paper, the seismic vulnerability of churches is assessed using different methods characterized by different levels of complexity depending on the accuracy of the results to be achieved. Specifically, we compare the two main types of methodologies applied in the literature, namely, empirical and analytical methods. Empirical methods assess seismic vulnerability based on engineering judgements. In this study, these evaluations were carried out through an automatic tool, the MACHRO form, which was introduced in the past by the authors with the purpose of making evaluations as objective as possible. Analytical methods evaluate the vulnerability of a stock of churches through linear and nonlinear kinematic analyses performed for the most vulnerable macro-elements, which are treated by means of mechanical models. When the number of churches in the stock is huge, this type of evaluation might prove unfeasible. For this reason, churches are grouped into a manageable number of archetypes in order to be analyzed. The above-described methodologies were applied to a relevant number of churches, aiming to appraise discrepancies in terms of results and highlight advantages and drawbacks of their application. Full article
(This article belongs to the Special Issue Risk Analysis and Preservation Strategies of Architectural Heritage)
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<p>Identification of the sample of churches analyzed with reference to the main shocks of the 2016 earthquake sequence.</p>
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<p>The stock of analyzed churches: photographic overview.</p>
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<p>The stock of analyzed churches: photographic overview.</p>
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<p>The stock of analyzed churches: photographic overview.</p>
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<p>Excerpt of the MaCHRO form structure with close-ups of the significant sections.</p>
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<p>Frequencies of (<b>a</b>) ages of construction and (<b>b</b>) states of maintenance for the entire sample of churches.</p>
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<p>Frequencies of (<b>a</b>) tie rods, (<b>b</b>) retaining elements and (<b>c</b>) corner connections in the investigated churches.</p>
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<p>Fragility curves of the studied churches obtained through the application of the empirical method proposed by De Matteis et al. [<a href="#B12-heritage-07-00324" class="html-bibr">12</a>].</p>
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<p>The ideal reference models for the three identified archetype churches.</p>
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<p>Collapse mechanisms activated in the analyzed sample of churches following the 2016 earthquake.</p>
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<p>Kinematic mechanisms considered for the “façade” and “lateral wall” macro-elements.</p>
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<p>(<b>a</b>) Capacity curve schematization and (<b>b</b>) capacity curves of the identified archetypes.</p>
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<p>Generic capacity curve with indication of damage levels (D0 to D5) and associated damage thresholds (TDs).</p>
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<p>Performance Point definition (red point) through superposition of capacity curve and demand spectrum for Archetype #1.</p>
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<p>Construction of fragility curves for the façade macro-elements of the archetype churches, where points are interpolated by a lognormal function.</p>
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<p>Analytical fragility curves of archetype churches (grey lines) versus empirical fragility curves with Iv = 0.51 (black lines).</p>
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<p>Analytical fragility curves (dotted lines) versus empirical fragility curves (continuous lines) for Archetype #1.</p>
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21 pages, 7204 KiB  
Technical Note
A Method for Developing a Digital Terrain Model of the Coastal Zone Based on Topobathymetric Data from Remote Sensors
by Mariusz Specht and Marta Wiśniewska
Remote Sens. 2024, 16(24), 4626; https://doi.org/10.3390/rs16244626 - 10 Dec 2024
Viewed by 366
Abstract
This technical note aims to present a method for developing a Digital Terrain Model (DTM) of the coastal zone based on topobathymetric data from remote sensors. This research was conducted in the waterbody adjacent to the Vistula Śmiała River mouth in Gdańsk, which [...] Read more.
This technical note aims to present a method for developing a Digital Terrain Model (DTM) of the coastal zone based on topobathymetric data from remote sensors. This research was conducted in the waterbody adjacent to the Vistula Śmiała River mouth in Gdańsk, which is characterised by dynamic changes in its seabed topography. Bathymetric and topographic measurements were conducted using an Unmanned Aerial Vehicle (UAV) and two hydrographic methods (a Single-Beam Echo Sounder (SBES) and a manual survey using a Global Navigation Satellite System (GNSS) Real-Time Kinematic (RTK) receiver). The result of this research was the development of a topobathymetric chart based on data recorded by the above-mentioned sensors. It should be emphasised that bathymetric data for the shallow waterbody (less than 1 m deep) were obtained based on high-resolution photos taken by a UAV. They were processed using the “Depth Prediction” plug-in based on the Support Vector Regression (SVR) algorithm, which was implemented in the QGIS software as part of the INNOBAT project. This plug-in allowed us to generate a dense cloud of depth points for a shallow waterbody. Research has shown that the developed DTM of the coastal zone based on topobathymetric data from remote sensors is characterised by high accuracy of 0.248 m (p = 0.95) and high coverage of the seabed with measurements. Based on the research conducted, it should be concluded that the proposed method for developing a DTM of the coastal zone based on topobathymetric data from remote sensors allows the accuracy requirements provided in the International Hydrographic Organization (IHO) Special Order (depth error ≤ 0.25 m (p = 0.95)) to be met in shallow waterbodies. Full article
(This article belongs to the Special Issue Remote Sensing: 15th Anniversary)
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<p>The location of bathymetric and topographic measurements carried out at the Vistula Śmiała River mouth in Gdańsk.</p>
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<p>The location of depth points recorded by an SBES integrated with a GNSS RTK receiver and designed sounding profiles in the study area.</p>
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<p>Flight trajectory of the UAV using the LiDAR system in the study area.</p>
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<p>The distribution of GCPs and UAV flights in the study area.</p>
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<p>A visualisation of the integrated data derived from a total of three mutually independent instruments (GNSS RTK receiver, LiDAR system, SBES).</p>
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<p>A view of georeferenced photos based on the entered GCPs (<b>a</b>) and a point cloud (<b>b</b>).</p>
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<p>The “Depth Prediction” plug-in window (<b>a</b>) and the depth points obtained based on photos (<b>b</b>).</p>
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<p>A bathymetric and topographic DTM of the Vistula Śmiała River mouth in Gdańsk.</p>
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<p>A diagram showing the development of the DTM of the coastal zone based on bathymetric and topographic data integration.</p>
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<p>The location of underwater GCPs that were used to assess the accuracy of the generated DTM of the coastal zone based on bathymetric and topographic data integration.</p>
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21 pages, 5349 KiB  
Article
Post-Processing Kalman Filter Application for Improving Cooperative Awareness Messages’ Position Data Accuracy
by Maximilian Bauder, Robin Langer, Tibor Kubjatko and Hans-Georg Schweiger
Sensors 2024, 24(24), 7892; https://doi.org/10.3390/s24247892 - 10 Dec 2024
Viewed by 441
Abstract
Cooperative intelligent transportation systems continuously send self-referenced data about their current status in the Cooperative Awareness Message (CAM). Each CAM contains the current position of the vehicle based on GPS accuracy, which can have inaccuracies in the meter range. However, a high accuracy [...] Read more.
Cooperative intelligent transportation systems continuously send self-referenced data about their current status in the Cooperative Awareness Message (CAM). Each CAM contains the current position of the vehicle based on GPS accuracy, which can have inaccuracies in the meter range. However, a high accuracy of the position data is crucial for many applications, such as electronic toll collection or the reconstruction of traffic accidents. Kalman filters are already frequently used today to increase the accuracy of position data. The problem with applying the Kalman filter to the position data within the Cooperative Awareness Message is the low temporal resolution (max. 10 Hz) and the non-equidistant time steps between the messages. In addition, the filter can only be applied to the data retrospectively. To solve these problems, an Extended Kalman Filter and an Unscented Kalman Filter were designed and investigated in this work. The Kalman filters were implemented with two kinematic models. Subsequently, driving tests were conducted with two V2X vehicles to investigate and compare the influence on the accuracy of the position data. To address the problem of non-equidistant time steps, an iterative adjustment of the Process Noise Covariance Matrix Q and the introduction of additional interpolation points to equidistance the received messages were investigated. The results show that without one of these approaches, it is impossible to design a generally valid filter to improve the position accuracy of the CAM position data retrospectively. The introduction of interpolation points did not lead to a significant improvement in the results. With the Q matrix adaptation, an Unscented Kalman Filter could be created that improves the longitudinal position accuracy of the two vehicles under investigation by up to 80% (0.54 m) and the lateral position accuracy by up to 72% (0.18 m). The work thus contributes to improving the positioning accuracy of CAM data for applications that receive only these data retrospectively. Full article
(This article belongs to the Special Issue Sensors and Systems for Automotive and Road Safety (Volume 2))
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<p>General methodological approach to investigate the Extended Kalman and Unscented Kalman Filter to improve the accuracy of the position information of the Cooperative Awareness Message.</p>
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<p>Extension of the methodological approach by temporarily adding additional CAM interpolation points to generate an equidistant time interval.</p>
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<p>Top view of the CARISSMA outdoor test site with the trajectory of the test drives using the Golf 8 as an example.</p>
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<p>Schematic representation of the Kalman filter function according to [<a href="#B33-sensors-24-07892" class="html-bibr">33</a>].</p>
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<p>Presentation of the position information results after applying the Kalman filters. Top: Initialization of the R-matrix with heading in rad and yaw rate in rad/s. Bottom: Initialization of the R matrix with heading in ° and yaw rate in °/s.</p>
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<p>Result of the <math display="inline"><semantics> <mrow> <mi>ω</mi> </mrow> </semantics></math>-analysis: progression of median, mean, and standard deviation via omega. In green: Filtered position data (EKF and UKF) of the ID.3 with CTRA model. In blue: Filtered position data (EKF and UKF) of the Golf 8 with CTRA model. (SD = standard deviation).</p>
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<p>Result of the position accuracies as a boxplot diagram after applying the various Kalman filters (EKF and UKF) with the respective kinematic models. The median is an orange line, and the mean value is a green triangle.</p>
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<p>Result of the position accuracies as a boxplot diagram after applying the various Kalman filters (EKF and UKF) with the respective kinematic models and the <math display="inline"><semantics> <mrow> <mi mathvariant="bold-italic">Q</mi> </mrow> </semantics></math> matrix adjustment. The median is an orange line, and the mean value is a green triangle.</p>
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<p>Result of the position accuracies as a boxplot diagram after applying the various Kalman filters (EKF and UKF) with the respective kinematic models and after equidistancing to <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>t</mi> <mo>=</mo> <mn>0.1</mn> <mo> </mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math>. The median is shown as an orange line and the mean value as a green triangle.</p>
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<p>Result of the position accuracies as a boxplot diagram after applying the various Kalman filters (EKF and UKF) with the respective kinematic models and after equidistancing to <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>t</mi> <mo>=</mo> <mn>0.01</mn> <mo> </mo> <mi mathvariant="normal">s</mi> </mrow> </semantics></math>. The median is shown as an orange line and the mean value as a green triangle.</p>
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<p>Plot of the position points of the driven trajectory according to the original CAM data (black), the ADMA reference data (green), and the position data after applying the UKF with SSA model and <math display="inline"><semantics> <mrow> <mi mathvariant="bold-italic">Q</mi> </mrow> </semantics></math>-matrix adjustment (red).</p>
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23 pages, 3192 KiB  
Article
Design Optimization of a Parallel–Serial Manipulator Considering Stiffness Criteria
by Anton Antonov
Robotics 2024, 13(12), 176; https://doi.org/10.3390/robotics13120176 - 10 Dec 2024
Viewed by 363
Abstract
In this paper, we analyze stiffness and perform geometrical optimization of a parallel–serial manipulator with five degrees of freedom (5-DOF). The manipulator includes a 3-DOF redundantly actuated planar parallel mechanism, whose stiffness determines the stiffness of the whole mechanical system. First, we establish [...] Read more.
In this paper, we analyze stiffness and perform geometrical optimization of a parallel–serial manipulator with five degrees of freedom (5-DOF). The manipulator includes a 3-DOF redundantly actuated planar parallel mechanism, whose stiffness determines the stiffness of the whole mechanical system. First, we establish the kinematic and stiffness models of the mechanism and define its stiffness matrix. Two components of this matrix and the inverse of its condition number are chosen as stiffness indices. Next, we introduce an original two-step procedure for workspace analysis. In the first step, the chord method is used to find the workspace boundary. In the second step, the workspace is sampled inside the boundary by solving the point-in-polygon problem. After that, we derive stiffness maps and compute the average stiffness indices for various combinations of design variables. The number of these variables is reduced to two geometrical parameters, simplifying the representation and interpretation of the obtained results. Finally, we formulate the multi-objective design optimization problem, whose main goal is to maximize the lateral stiffness of the mechanism. We solve this problem using a hierarchical (ε-constraint) method. As a result, the lateral stiffness with optimized geometrical parameters increases by 54.1% compared with the initial design. Full article
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Figure 1
<p>Organization of this paper.</p>
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<p>Considered manipulator: (<b>a</b>) computer model; (<b>b</b>) prototype during the operation.</p>
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<p>The parallel mechanism: (<b>a</b>) general configuration; (<b>b</b>) lowest configuration.</p>
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<p>Kinematic parameters of the mechanism.</p>
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<p>Determining the starting point on the workspace boundary.</p>
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<p>Computing the workspace boundary using the chord method: (<b>a</b>) finding point <math display="inline"><semantics> <msub> <mi>U</mi> <mrow> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </semantics></math> by a circular search at point <math display="inline"><semantics> <msub> <mi>U</mi> <mi>j</mi> </msub> </semantics></math>; (<b>b</b>) angle <math display="inline"><semantics> <mi>ψ</mi> </semantics></math> depending on the <math display="inline"><semantics> <mi>σ</mi> </semantics></math> value.</p>
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<p>Sampling rectangle <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mn>1</mn> </msub> <msub> <mi>V</mi> <mn>2</mn> </msub> <msub> <mi>V</mi> <mn>3</mn> </msub> <msub> <mi>V</mi> <mn>4</mn> </msub> </mrow> </semantics></math> that envelops workspace boundary <math display="inline"><semantics> <mrow> <msub> <mi>U</mi> <mn>1</mn> </msub> <msub> <mi>U</mi> <mn>2</mn> </msub> <mo>…</mo> <msub> <mi>U</mi> <mi>n</mi> </msub> </mrow> </semantics></math>. The blue and red dots are examples of samples inside and outside the boundary.</p>
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<p>Numbering of polygon vertices and edges: (<b>a</b>) the numbers near the vertices inside the parentheses correspond to values <math display="inline"><semantics> <msub> <mi>a</mi> <mi>j</mi> </msub> </semantics></math>, while the blue numbers near the edges are values <math display="inline"><semantics> <msub> <mi>b</mi> <mi>j</mi> </msub> </semantics></math>; (<b>b</b>) edge <math display="inline"><semantics> <mrow> <msub> <mi>U</mi> <mn>1</mn> </msub> <msub> <mi>U</mi> <mn>2</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>(</mo> <mn>1</mn> <mo>)</mo> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math> and edge <math display="inline"><semantics> <mrow> <msub> <mi>U</mi> <mn>3</mn> </msub> <msub> <mi>U</mi> <mn>4</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>b</mi> <mn>3</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>(</mo> <mn>3</mn> <mo>)</mo> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math> are not counted in Algorithm 1, as they do not match the if statements of the algorithm.</p>
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<p>Workspace boundary for various end-effector orientations (<math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>255</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>505</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>): (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>=</mo> <msup> <mn>10</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>=</mo> <msup> <mn>20</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>. The squares correspond to the boundary points.</p>
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<p>Workspace sampling for various parameters (<math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>): (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>255</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>505</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>325</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>425</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>345</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>575</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>455</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>575</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>.</p>
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<p>Stiffness maps of <math display="inline"><semantics> <msub> <mi>k</mi> <mi>y</mi> </msub> </semantics></math> for various parameters (<math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>): (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>255</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>505</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>325</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>425</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>345</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>575</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>455</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>575</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>.</p>
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<p>Stiffness maps of <math display="inline"><semantics> <msub> <mi>k</mi> <mi>z</mi> </msub> </semantics></math> for various parameters (<math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>): (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>255</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>505</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>325</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>425</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>345</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>575</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>455</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>575</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>.</p>
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<p>Stiffness map of <math display="inline"><semantics> <msup> <mi>k</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math> for various parameters (<math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>): (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>255</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>505</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>325</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>425</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>345</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>575</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>455</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>575</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>.</p>
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<p>Stiffness maps for the case when the adjacent branches coincide (<math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>255</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>): (<b>a</b>) lateral stiffness <math display="inline"><semantics> <msub> <mi>k</mi> <mi>y</mi> </msub> </semantics></math>; (<b>b</b>) vertical stiffness <math display="inline"><semantics> <msub> <mi>k</mi> <mi>z</mi> </msub> </semantics></math>; (<b>c</b>) overall stiffness <math display="inline"><semantics> <msup> <mi>k</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>.</p>
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<p>The values of the performance metrics for various parameters <math display="inline"><semantics> <msub> <mi>y</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mn>2</mn> </msub> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>=</mo> <msup> <mn>0</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>): (<b>a</b>) average lateral stiffness <math display="inline"><semantics> <msub> <mover accent="true"> <mi>k</mi> <mo stretchy="false">¯</mo> </mover> <mi>y</mi> </msub> </semantics></math>; (<b>b</b>) average vertical stiffness <math display="inline"><semantics> <msub> <mover accent="true"> <mi>k</mi> <mo stretchy="false">¯</mo> </mover> <mi>z</mi> </msub> </semantics></math>; (<b>c</b>) average overall stiffness <math display="inline"><semantics> <msup> <mover accent="true"> <mi>k</mi> <mo stretchy="false">¯</mo> </mover> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>; (<b>d</b>) workspace area <span class="html-italic">W</span>. The white circles correspond to the initial design with <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>255</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>505</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>; the white squares correspond to the optimized design with <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>345</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>575</mn> <mspace width="0.166667em"/> <mi>mm</mi> </mrow> </semantics></math>.</p>
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