Kinematic Equations Research Articles

Kinematic analysis applied to a solar tracker driven by a 4-bar mechanism

This paper presents the kinematic analysis of the eccentric crank-slider and crank-rocker mechanisms applied to control the 1-DOF in a solar tracker focused on improving the conversion of solar energy into electrical energy. The mechanisms for its construction and the kinematic equations that allow to know the state that keeps the movement in the working space of the tracker and its comparative analysis between them are presented. Finally, the construction to scale with 3D printing is shown to validate the functionality of each of the 4-bar systems.

Finite Tube Method for buckling analysis of tubular members using Fourier-approximation for the displacements

In this paper an efficient numerical method for the static analysis of cylindrical tubes is introduced. The method is designed for the linear buckling analysis of wind turbine support towers which are, most typically, built up from conical and/or cylindrical cans. Accordingly, the developed method uses cylindrical tube segments as elementary building blocks, along with specialized shape functions, and is named the Finite Tube Method. Within a tube segment the displacements are approximated by two-dimensional Fourier series. The curved nature of the surface is directly considered in the kinematic equations. The segments are joined and/or supported to the ground by constraint equations or by elastic links. In the current implementation internal stresses are determined in a simplified way: the circumferential stress distributions are calculated from the internal forces/moment by classic strength of material formulae, while the longitudinal distribution within each segment is quadratic. The considered internal forces/moments are: normal force, shear force, bending moment, and torsional moment. The internal forces can be arbitrarily combined. In the paper the underlying derivations are briefly summarized, then the method is demonstrated and validated by numerical examples, comparing the results to analytical and alternative numerical solutions. The authors are actively developing the method and will provide future work on utilization of the method for buckling mode identification and decomposition, as well as practical advancements to make the method a useful tool in the engineering design and analysis of wind turbine support towers.

Oriblock: The origami-blocks based on hinged dissection

Traditional origami, a well-known art of paper-folding, does not account for material thickness. Subsequently, several thickness-accommodation techniques have been developed, considering the non-negligible thickness of materials, which enables the application of origami to a wide range of mechanisms. Although these techniques prevent self-intersections and preserve kinematics after accommodating panel thickness, they are limited by their reliance on initial zero-thickness origami patterns in two dimensions. To address this issue, this study proposes a novel technique, the volume-accommodation technique in three-dimensional space, which allocates non-coplanar origami creases based on solid geometry to allow a greater variety of shape-changing motions. Typical solid geometries, such as a pyramid and a prism, are selected as examples to demonstrate the design procedures. Closure equations of the obtained oriblock are used to analyze the kinematic properties. Additionally, Bennett and Myard Linkages are derived from oriblocks based on their kinematic equivalence to strengthen their connections. As a result, this new type of origami can offer a variety of mechanisms for designers and facilitate potential applications in origami-inspired metamaterials and robotics.

Scotch yoke structure inspired piezoelectric-electromagnetic hybrid harvester for wave energy harvesting

This paper presents a piezoelectric-electromagnetic hybrid harvester for wave energy harvesting inspired by a scotch yoke structure for obtaining energy from irregular and low-frequency waves. Through simplification of the energy harvester model, the kinematic equations of the scotch yoke are established. One of the advantages of the harvester is that the wave energy can be harvested by an up-conversion mechanism, The piezoelectric transducer (PZT) and electromagnetic generator (EMG) transform the wave excitation into electricity, and by utilizing the frequency up-conversion mechanism, the PZTs’ harvesting efficiency is increased, the influencing factors of the output power of the harvester are explored by simulation and experimental studies, The results of the experiment demonstrate that this harvester could generate an ultimate output power of 286.36 mW at ultra-low frequency (1.4 Hz) operating frequency. This study offers a practical solution for harvesting low-frequency and disordered wave energy and makes significant progress in harvesting low-frequency and irregular wave energy.

Design and analysis of folding mechanism for automotive seats based on space maximization

To solve the problem of maximizing space utilization of car folding seats within limited spatial constraints, a design method for space-maximizing four-bar folding seat hinges is proposed. First, we will investigate the existing types of car folding seats, identify the mainstream configurations, and determine the optimal position of the four-bar linkage hinge point based on the constraints present within the car using theoretical analysis methods. Secondly, kinematic and dynamic models of the four-bar linkage mechanism are established, followed by the formulation of kinematic and dynamic equations. Finally, numerical methods are employed to solve these equations and obtain the kinematic and dynamic characteristics. The results indicate that the design method proposed in the article, which aims to maximize the space of the hinge four-bar folding seat, is efficient and feasible. This approach optimizes the utilization of interior space and enhances passenger comfort while accommodating the folding function of the seat within the existing vehicle environment. This method holds significant engineering implications for the design of folding seats in similar types of vehicles.

Automatic differential kinematics of serial manipulator robots through dual numbers

Dual Numbers are an extension of real numbers known for its capability of performing exact automatic differentiation of one-valued functions theoretically without error approximation. Also, Differential Kinematics of robots involves the computation of the Jacobian, which is a matrix of partial derivatives of the Forward Kinematic equations with respect to the robot’s joints. Thus, to perform the automatic calculation of the Jacobian matrix, this paper presents an extension of dual numbers composed of a scalar real part and a vector dual part, where the real part represents the angular value of the robot joint, and the dual part represents the direction of the corresponding partial derivative for each joint. The presented method was implemented in Matlab through Object Orientes Programming (OOP), and the results for calculating the Jacobian of the KUKA KR 500 robot model for 1000 random postures were subsequently compared in terms of execution time and Mean Squared Error (MSE) with other conventional methods: the geometric method, the symbolic method, and the finite difference method. The results showed a significant improvement in the computing time for calculating the Jacobian of the robotic model compared to the other methods, as well as a minimum MSE having as reference the numerical value of the symbolic calculations.

Relaxation of the kinematic dynamo equations

We compute the exact relaxation and Λ \Lambda -convex hull of the kinematic dynamo equations and show that they coincide. We also find the relaxation in the stationary case and present applications to the infinite-time limit in the kinematic dynamo problem.

Hybrid trajectory tracking control of wheeled mobile robots using predictive kinematic control and dynamic robust control

AbstractTrajectory tracking control of wheeled mobile robots (WMRs) is still a remarkable problem for many applications. In the present paper, a hybrid control is presented based on dynamic and kinematic equations of motion for wheeled mobile robots in the presence of the sum of the external disturbances and parametric uncertainty. The designed control for the WMR utilizes control and guidance to reach the reference path. In many studies, a control strategy is normally employed for WMR. However, in this study, hybrid control was used for the mentioned purpose. Akin to other studies, the kinematic control scheme here was based on the predictive control, and the dynamic control scheme was designed based on the robust control. Therefore, in this article, having introduced the kinematic model, a nonlinear predictive control was proved and designed. In the next step, a finite‐time integral type terminal sliding mode control (FITSMC) was designed based on the nonlinear dynamic model in order to automatically adjust the control gain and eliminate online disturbances and destructive chattering phenomena completely. In particular, a finite‐time disturbance observer was designed to estimate the external disturbances. The proof of the new proposed control scheme was presented using Lyapunov stability theory and numerical results. The mentioned integrated scheme, including predictive control (outer loop) and nonlinear adaptive control (inner loop), ensures the convergence and optimal tracking performance of all signals, as a result of which the tracking errors can arbitrarily converge to the origin in a finite time. In the final step, the simulation results were presented to show the effectiveness of the proposed scheme using MATLAB software, and the introduced control design was compared with a similar controller quantitatively and qualitatively.

Use of a Refined Theory for Three-Dimensional Bending Analysis of Isotropic Rectangular Thick Plates

In this paper, a refined plate theory (Alternative II theory) is presented for the three-dimensional bending analysis of an Isotropic thick plate. The theory has similarity to the first order shear deformation theory but requires no shear correction factors. The kinematics equations were developed based on the Alternative II Refined plate theory. Thereafter, using a complete three-dimensional constitutive relation, the total potential energy was developed. A governing equation and two compatibility equations were obtained by the variation of the total potential energy with respect to displacement and rotations respectively. Solving the governing and compatibility equations, a polynomial displacement function was obtained. The stiffness coefficients were then obtained using the displacement function. Thereafter, the equations for the in-plane normal and shear stresses, transverse normal and shear stresses as well as the lateral displacement were developed using the stiffness coefficients and the displacement function. Numerical values of the lateral displacement parameters were determined for a rectangular plate of aspect ratio 2.0, 1.0 and 0.5 for span to thickness ratios of 20, 10 and 7.14286. Also, numerical values of the lateral displacement and stresses were determined for a square plate for span to thickness ratios of 4, 10, 100 and 1000. The results from this work were compared with the work of previous researchers using simple percentage difference. It was observed that refined plate theories overestimate the lateral displacement of a plate. Hence, three-dimensional analysis is recommended for thick plate analysis.

Autonomous longitudinal separation control of aircraft in random wind fields based on MPC

Aiming at the problem of autonomous longitudinal separation under random disturbance of flight path, the random factor of high-altitude wind often leads to poor robustness of longitudinal separation between two aircraft. This paper proposes an autonomous longitudinal separation control method for aircraft based on model predictive control. Firstly, the nonlinear kinematic differential equation of the wind field difference between the two aircraft and the longitudinal separation is established, and the linear time-varying prediction model is derived. The longitudinal separation and path deviation distance of the two aircraft are selected as the optimization objectives, the high-altitude wind is the random disturbance, and the true airspeed and yaw angle of the leading aircraft are measurable. Terminal equality constraints are added to the air safety and aircraft performance constraints to maintain the stability of the system. To verify the effectiveness of the algorithm, within the prescribed simulation time of 120 seconds, this paper sets three groups of different expected intervals 12, 13, 14km. Through the designed MPC controller, by controlling the true airspeed and yaw angle of the trailing aircraft within the rolling time domain period, the interval curve between the two aircraft is relatively smooth and is always not less than the minimum safety interval of 10 km. It stabilizes at the expected target interval at the 74th, 90th and 118th seconds respectively, and begins to return to the route at the 58th, 74th and 95th seconds, and finally returns to the center line of the route; two groups of wind field control groups are set up, one group is the forecast wind intensity increased by 2 times, and the other group is the turbulent wind disturbance intensity increased by 8 times. Both can smoothly and stably establish the expected interval of 12 km at the 61st and 72nd seconds respectively.

An energy perspective on Galileo’s kinematic paradox

The connection between kinematics and work–energy principles is often not emphasized enough for students to appreciate fully. Highlighting the relationship between the kinematic equations and the conservation of total mechanical energy is crucial. We revisit the solution to Galileo’s kinematic paradox from an energy perspective. Galileo stated that the time it takes for an object to travel from the top to the bottom point of a vertical circle is the same regardless of its path. While mathematical proofs of this statement using kinematic approaches are well-documented, this paper explores the problem from an energy standpoint, using experimental evidence to elucidate the relationship between kinematics and energy.

The effect of disc deformation on the dynamic response of multi-disc systems: A tilting-impact model originating from the wet clutch

Analyzing the dynamic response of a wet multi-disc clutch is crucial for relevant fault diagnosis research. However, characterizing the tilting-impact phenomenon between the friction discs remains ambiguous. This paper presents a tilting-impact model of a multi-disc system under dry conditions and investigates the effect of disc deformation on the system dynamics. Firstly, Euler angles and Euler kinematic differential equations are introduced to describe the spatial position and orientation of the spinning-tilting disc. The aligning centrifugal moment, spline friction force, and tilting-impact forces for both flat and deformed discs are derived. A bench test is conducted to validate the model by comparing simulation and experimental signals at various rotation speeds. The results indicate that the occurrence of tilting-impact phenomena in dry conditions requires external disturbances, and the discs eventually stop tilting-impacting once the disturbance subsides, and the system with deformed discs experiences longer recovery times. Furthermore, the tilting-impact forces in systems with deformed discs are smaller than those with flat discs only.

Design and Analysis of a Planar Six-Bar Crank-Driven Leg Mechanism for Walking Robots

This study presents the design and a thorough analysis of a six-bar crank-driven leg mechanism integrated with a skew pantograph, developed for walking robots. The mechanism’s dimensions were optimized using a rigorous dimensional synthesis process in GIM software (version 2024). Subsequently, a detailed kinematic analysis was performed in GIM to simulate the leg’s motion trajectory, velocity, and acceleration. In parallel, kinematic equations were formulated using the vector loop method, implemented in MATLAB (version R2013-b), and compared with the GIM results for validation, demonstrating the strong agreement between both tools. These results confirm the mechanism’s ability to generate a compact, high-lift foot trajectory while maintaining system stability and energy efficiency. An inverse dynamic analysis was carried out to determine the actuator’s driving torque, ensuring efficient operation under expected load conditions. Furthermore, topology optimization conducted in SOLIDWORKS (version 2021) significantly reduced the weight of the ground-contacting link while preserving its structural integrity. A subsequent stress analysis validated the mechanical viability of the optimized design, supporting its feasibility for real-world implementation. This research provides a robust foundation for the development of a functional prototype. Its potential applications include mobile robots for sectors such as agriculture and all-terrain vehicles, where efficient, reliable, and adaptive locomotion is crucial. The proposed mechanism strikes an optimal balance between mechanical simplicity, cost-effectiveness, and high performance, making it well-suited for challenging operational environments.

Kinematic modeling and control of orthogonal omnidirectional wheeled two-dimensional conveyor platform

This paper proposes an orthogonal omnidirectional wheeled two-dimensional conveyor platform and the inverse kinematic equations based on the cell model and the omnidirectional wheel model are derived, respectively. The two models use the way of the circumcircle of the cargo box and the dot product method to obtain the cell or omnidirectional wheel directly contacted under the cargo box, respectively. This paper uses YOLOV8_OBB and Strong SORT target tracking algorithms to assign tracking IDs to the cargo box and output the box position information. A feedback mechanism for faulty omnidirectional wheels or faulty cells is proposed, where the faulty area is classified as impassable during path planning and the A* path planning algorithm is used for path planning to avoid the faulty area. A visual feedback-based path tracking is adopted to enable timely correction when the cargo box deviates from the established path. The experimental results show that the omnidirectional wheel-based model improves the path-tracking accuracy by 44.52% and reduces the average power consumption by 2.6 times compared to the cell-based model.

Sloshing in a tank with elastic side walls and a membrane cover

The coupled problem of liquid sloshing in a tank with a membrane cover and elastic side walls is investigated. The velocity potential theory is used for the flow and the linear elastic theory for the cover and side walls. For the former, the vertical mode expansion is used, in which all the roots of the dispersion relationship are first found. The expansion automatically satisfies the governing equation, the tank bottom, and the tank cover conditions. The deflections of the side walls are expanded into a cosine series together with a four term polynomial, which is vital for the procedure to succeed. These expansions are then matched through the dynamic and kinematic equations of the plates, and the problem is completed by imposing the edge conditions of the plate and membrane. Through the combined matrix equation, the natural frequencies of the system are obtained. In addition, the nature of the dispersion relationship of the membrane is analyzed. Explicit solutions are obtained for some special cases, and the link with the free surface sloshing is established. Extensive numerical results are provided, and their physics is analyzed.

A guidance of solid element application in predicting the ultimate strength of flat plates in compression

With the increasing demand in ocean engineering field for the ultimate limit state (ULS) analysis of thick plates, the solid (3D) element has been more frequently used in the ULS analysis. Compared to 2D element, 3D element can better consider the transverse shearing effect during the analysis which usually occurs on thick plates, due to its reliable reflection of geometrical characteristics and kinematic equations of the structures. Besides, 3D element can also facilitate the introduction of influencing factors into the ULS analysis, such as the erosions, cracks and residual stress. This may improve the simulation precision of ULS analysis of thick plate models. Until now, there is a lack of 3D element modelling technique in the ULS analysis of flat/stiffened plates. The present study aims to provide helpful information on 3D elements in the ULS analysis of flat plates under axial compression load. A total of 350 plate scenarios were adopted in the parametric study to consider the effects of 3D element shape (αyz). It is found that the element shape significantly influences the ULS analysis of flat plates, where planar-like 3D element is not recommended. An empirical formula for determining the optimal 3D element shape of the finite element (FE) model is proposed based on the parametric results. A guidance for implementing the 3D elements is then documented, which may help engineers further understand the ultimate strength characteristic of very thick flat plates.

Алгоритмы группового управления подводными подвижными объектами

The problem of formation control is relevant when searching for objects, surveying a given area, and group monitoring. In the underwater environment, this problem is complicated by restrictions on the frequency of receiving navigation data and the speed of information exchange between underwater robots. The purpose of this article is to develop and study group control algorithms that ensure the given formation and the maintenance of this formation during movement. The article provides an overview of the results in the field of group control of underwater vehicles, provides a mathematical model of the object, navigation system, underwater environment and communication system. The equations of kinematics and dynamics of a solid body in three-dimensional space, supplemented by equations of actuators, are used as a mathematical model. Algorithms for the distribution of moving objects in formation and nonlinear control algorithms for moving in a line are proposed. When constructing the devices, it is assumed that there is a group leader who transmits his coordinates to the rest of the underwater robots. Motion control is synthesized by the method of positional trajectory control in the form of a function of external coordinates. A study was carried out using numerical methods, during which the process of the formation and movement along the trajectory described by straight line segments was studied. The influence of errors in the navigation system and the frequency of data updates on the error of maintaining a given position by a separate underwater robot is investigated.

Attitude determination for multirotor aerial vehicles using a prescribed‐time super‐twisting algorithm

AbstractThis paper is concerned with the prescribed‐time robust attitude determination (AD) of multirotor aerial vehicles (MAVs) using vector measurements from the local magnetic field and local gravity. To address this problem, we first introduce a novel modified super‐twisting algorithm endowed with the prescribed‐time convergence property. The state of the proposed algorithm is governed by an unbounded time‐varying gain up to the prescribed settling time (PST) and by a function after that. Therefore, after the PST, the new algorithm coincides with the conventional super‐twisting, thus showing robust stability at the origin. This prescribed‐time super‐twisting algorithm (PTSTA) is then applied to the formulation of a three‐stage gyro‐free attitude determination method for MAVs. In the first stage, the classical QUEST algorithm is used to compute a Wahba‐optimal attitude estimate from the vector measurements. In the second stage, the PTSTA is employed in the formulation of a robust state estimator that provides estimates of the attitude Gibbs vector and its rate. Finally, in the third stage, these state estimates as well as the attitude kinematic equation are immediately used to compute the MAV angular velocity. The proposed robust prescribed‐time gyro‐free AD method is evaluated numerically, showing invariance with respect to disturbance and model uncertainty.

An Algorithm for Strapdown Airborne Gravity Disturbance Vector Measurement Based on High-Precision Navigation and EGM2008.

Attitude errors, accelerometer bias, the gravity disturbance vector, and their coupling are the primary factors obstructing strapdown airborne vector gravimetry. This paper takes the geocentric inertial frame as a reference and solves the kinematic equations of its motion and its errors of the body frame and local geographic frame in the Lie group, respectively; the attitude accuracy is improved through a high-precision navigation algorithm. The constant accelerometer bias is estimated through Kalman filtering and is deducted from the accelerometer output to eliminate its influence. Based on the EGM2008 model, the low-frequency components of the gravity disturbance vector are corrected. The gravity disturbance vectors after model data fusion were low-pass filtered to obtain the ultimate results. This method was applied to flight experimental data in the South China Sea, and a gravity anomaly accuracy of better than 0.5 mGal, a northward gravity disturbance accuracy of 0.85 mGal, and an eastward gravity disturbance accuracy of 4.0 mGal were obtained, with a spatial resolution of approximately 4.8 km.

Optimization of high-speed angular contact ball bearing for aircraft gearbox utilizing an evolutionary multi-objective algorithm, NSGA-III

We develop a program for predicting bearing performance, including the maximum contact stress, bearing power loss, minimum lubricating film thickness, static safety factor, and dynamic life, and an optimal design for high-speed angular contact ball bearings in aircraft gearboxes using optimization algorithms. A program is developed to predict bearing performance by calculating the loads and displacements of individual balls based on the bearing’s kinematic and static equilibrium equations. Next, an optimization program is developed with non-dominated sorting genetic algorithm III (NSGA-III), wherein calculated bearing performance indicators serve as constraints or objective functions. The performance of individuals within the final Pareto front is evaluated using the inter-criteria correlation (CRITIC) and weighted product methods. The optimization performances of NSGA-III and NSGA-II are compared based on their hypervolume indicators. A solution of the kinematic and static equilibrium equations under the load cases and duty cycles of real aircraft gearboxes is obtained, enhancing the prediction accuracy. Furthermore, optimization, including bearing performance metrics rarely considered in previous studies, is performed. Optimal specifications superior to reference bearings employed in aircraft gearboxes are obtained, enhancing all three objective functions or specific objective functions. The optimization performance of NSGA-III is confirmed to surpass that of conventional NSGA-II.