Kinematic Equations Research Articles

Trajectory Smoothing Planning of Delta Parallel Robot Combining Cartesian and Joint Space

Delta parallel robots have been widely used in precision processing, handling, sorting, and the assembly of parts, and their high efficiency and motion stability are important indexes of their performance.Corners created by small line segments in trajectory planning cause abrupt changes in a tangential discontinuous trajectory, and the vibration and shock caused by such changes seriously affect the robot’s high-speed and high-precision performance. In this study, a trajectory-planning method combining Cartesian space and joint space is proposed. Firstly, the vector method and microelement integration method were used to establish the complete kinematic and dynamic equations of a delta parallel robot, and an inverse kinematic/dynamic model-solving program was written based on the MATLAB software R2020a. Secondly, the end-effector trajectory of the delta parallel robot was planned in Cartesian space, and the data points and inverse control points of the end effector’s trajectory were obtained using the normalization method. Finally, the data points and control points were mapped to the joint space through the inverse kinematic equation, and the fifth-order B-spline curve was adopted for quadratic trajectory planning, which allowed the high-order continuous smoothing of the trajectory planning to be realized. The simulated and experimental results showed that the trajectory-smoothing performance in continuous high-order curvature changes could be improved with the proposed method. The peak trajectory tracking error was reduced by 10.53%, 41.18%, and 44.44%, respectively, and the peak torque change of the three joints was reduced by 3.5%, 11.6%, and 1.6%, respectively.

Reentry trajectory design of a hypersonic vehicle based on reinforcement learning

In this research, we investigate control of a hypersonic vehicle (HV) following its reentry into the Earth’s atmosphere, using deep reinforcement learning (DRL) in a continuous space. Here, we incorporate the basic kinematic and force equations of motion for a vehicle in an atmospheric flight to formulate the reentry trajectory satisfying the boundary constraints and multiple mission related process constraints. The aerodynamic model of the vehicle emulates the properties of a common aero vehicle (CAV-H), while the atmospheric model of the Earth represents a standard model based on US Standard Atmosphere 1976, with significant simplification to the planetary model. In an unpowered flight, we then control the vehicle’s trajectory by perturbing its angle of attack and bank angle to achieve the desired objective, where the control problem is based on different actor-critic frameworks that utilize neural networks (NNs) as function approximators to select and evaluate the control actions in continuous state and action spaces. First, we train the model following each of the methods, that include on-policy proximal policy approximation (PPO) and off-policy twin delayed deterministic policy gradient (TD3). From the trajectory generated, we select a nominal trajectory for each algorithm that satisfies our mission requirements based on the reward model.

A two-step solution for robot-world calibration made intelligible by implementing Chasles’ motion decomposition in Ad(SE(3))

As a typical kinematic problem, an important criterion for the accuracy of robot-world calibration is whether the rotational and translational parts are calculated separately or not. Solving the kinematic equation separately increases the interest into the decomposition mode of the equation. This paper provides a decomposition mode for Ad(SE(3)) with embedded Chasles’ motion, providing a new decomposition mode for the robot-world equation. Firstly, the Lie algebra ad(se(3)) for Chasles’ rotational and translational motions are constructed and mapped exponentially to the Lie group Ad(SE(3)) separately, thus implementing the Chasles’ motion decomposition of Ad(SE(3)). Subsequently, the robot-world equation in Ad(SE(3)) is derived based on joint kinematics. A new decomposition mode for robot-world equation is proposed through the Chasles’ motion decomposition, and a two-step method based on point set matching is proposed. Finally, the effect of coordinate invariants on the calibration accuracy is discussed, and the superiority of the proposed method is verified with simulation and experimental tests.

Synthesis of Nominal Reorientation Trajectories of a Small Satellite in Case of Failure of One Actuator

The paper devotes to the development of the approach for synthesis of reorientation trajectory of a small satellite. We consider the nanosatellite angular motion described by the kinematic quaternion equations. The aerodynamic and gravitational disturbance torques are taken into account in the angular motion model. Reorientation of a small satellite occurs from some initial position. In addition, the final angular velocity components do not exceed 0.1 °/s. The control program is given as the even Fourier series. The even Fourier series were chosen due to they can describe a complex dependency accurately. The coefficients of the even Fourier series are defined by the differential evolution algorithm. Previously, this approach has shown its efficiency for the cases of normal operation of the actuators. The paper presents the approach of synthesis of reorientation trajectory in case of failure of the actuator. The problem of reorientation reduces to optimization problem of searching of 34 coefficients of the even Fourier series that provided the achievement of the desired boundary conditions. The numerical results are given that approved the possibility of solution of the reorientation problem of small satellite in case of failure of the actuator. To compare the control programs, research was made between the cases of normal operation of the actuator and failure of the actuator. The value of the control torque differs by the order of magnitude in case of failure of the actuator. Despite this, its value is achievable for the magnetic coils.

Kinematic and optimal design of a three-wheeled holonomic omnidirectional robot platform for mobile manipulation tasks

Mobile manipulator robots are increasingly vital in industrial and service sectors, serving roles in assembly, inspection, and hazardous environments. The integration of a manipulator with a mobile robot base imposes unique demands on the vehicle's drive system. This paper introduces a three-wheel holonomic omnidirectional robot designed for mobile applications. It explores kinematic equations, electronic integration, and mechanical design for holonomic motion. This platform excels in mobility, capable of simultaneous translation and rotation, facilitating precise multidirectional motion. Its equilateral truncated triangular structure ensures stability, supporting up to 4 kg. With three omnidirectional wheels and DC motors, controlled by EMS 30A H-Bridge and Arduino Mega 2560, it offers an open platform for research and development. This robust platform aligns kinematic calculations with mathematical models, promising efficiency for diverse mobile manipulation robot applications.

Application of modified first order shear deformation theory to vibration analysis of SSSS and CCSS thick anisotropic rectangular plates

This research presents a vibration analysis of a thick anisotropic rectangular plate using modified first shear deformation theory. Modified first shear deformation theory, which is not built upon the classical plate theory, was used to develop the kinematic equations and constitutive relations of a deformed section of thick anisotropic rectangular plate from which the generalized stress equations were determined. Using the assumptions of the theory, the strain energy and external work equations were formulated, and by employing the principles of total minimum potential energy, the total potential energy functional of a thick anisotropic plate was developed. Minimization of the total potential energy functional with respect to the deflection function (w) and with respect to the shear rotations (Φx) and (Φy) respectively, resulted in the governing differential equation and the two compatibility equations of the plate. The displacement functions that satisfy the governing and compatibility equations were obtained by solving the governing and compatibility equations. From the general displacement function, the peculiar deflection equations (shape functions) were obtained for the boundary conditions considered, which are simply supported on its four edges (SSSS), clamped on two adjacent edges, and simply supported on the other two (CCSS). Using the displacement equation and the equation for rotation in the x-direction (Φx) and equation for rotation in y-direction (Φy) the direct governing and two direct compatibility equations were obtained, from which coefficients that enable the formula for calculating fundamental natural frequencies to be obtained. For the boundary condition analyzed in this work, the stiffness coefficients (kR, kQ, kRQ, kq, kRRQ , kRQQ , kNR, kNQ, kNRQ and kλ ) were computed and used in determining the fundamental natural frequency parameter values at various span to depth ratios (5,10,20, 25 and 100), aspect ratios (1 to 2 at the increment of 0.1) and angle of fibre orientation (0o, 15o, 45o). The solutions of this study were compared with those from previous researchers. The fundamental natural frequency parameter values obtained in this study were compared with the work of Reddy (1984) for 0o angle of fibre orientation at span to depth ratios of 5,10, 20, 25, 50 and 100 at an aspect ratio of 1. The percentage difference values were 6.073%, 3.197%, 1.132%, 0.788%, 0.255%, and 0.112%, respectively. These differences revealed the closeness of the results of this present study to the results of Reddy (1984). This shows that the present theory provides good and acceptable solutions to the vibration problems of thick anisotropic rectangular plates.

Droplet absorption and spreading into thin layers of polymer hydrogels

From biological tissues to microactuators and absorption of solvents into layers of paint, macroscopically non-porous materials with the capacity to swell when in contact with a solvent are ubiquitous. In these systems, owing to strong solid–fluid interactions, chemically driven flows can yield large geometric changes. We study experimentally and theoretically the canonical problem of the swelling of a thin hydrogel layer by a single water drop. Using a bespoke experimental set-up, we observe fast absorption leading to a radially spreading axisymmetric blister. We use a fully three-dimensional linear poroelastic framework with nonlinear kinematic equations to obtain governing equations, which we then reduce with thin-layer scalings to a one-dimensional nonlinear diffusion equation for the evolution of the blister geometry. In the limits of large and small deformations, the evolution of the blister characteristic height and radius are self-similar, following power laws in time. Our experimental measurements show that the evolution of the blister is broadly within the theoretical predictions in the large and small deformation regimes. In the general intermediate deformation regime, the measurements are well captured by our reduced one-dimensional diffusion model, which does not require the sophisticated and computationally expensive numerical approaches necessary for the original two-dimensional nonlinear coupled transport problem. By adapting modelling techniques from the fluid dynamics of thin porous elastic layers to a polymer swelling problem, our modelling framework extends the range of polymer swelling problems that can be treated with semi-analytical methods. Moreover, our detailed experimental data can serve as a test case for future nonlinear poroelastic frameworks of swelling polymer materials.

Gait analysis of biped robot based on inverted pendulum model

The spring inverted pendulum model was used to analyze the motion of the biped robots under different motion states and the effective model parameters were obtained. In this paper, the biped robot was simplified to an inverted pendulum model and the kinematics equations of the robot under different motion states were obtained. The kinematics equations were solved by MATLAB, and the analysis was used to get the right sports condition during motion process. The corresponding simulation of the inverted pendulum model was carried out based on MATLAB, a series of data such as the trajectory of mass point of the robot and the variation of the ground reaction force during motion process were obtained. By analyzing the results, the model parameters and conditions of the biped robots to keep stable in different motion states were obtained.

Para- and Transcellular Transport Kinetics of Nanoparticles across Lymphatic Endothelial Cells.

Lymphatic vessels have received significant attention as drug delivery targets, as they shuttle materials from peripheral tissues to the lymph nodes, where adaptive immunity is formed. Delivery of immune modulatory materials to the lymph nodes via lymphatic vessels has been shown to enhance their efficacy and also improve the bioavailability of drugs when delivered to intestinal lymphatic vessels. In this study, we generated a three-compartment model of a lymphatic vessel with a set of kinematic differential equations to describe the transport of nanoparticles from the surrounding tissues into lymphatic vessels. We used previously published data and collected additional experimental parameters, including the transport efficiency of nanoparticles over time, and also examined how nanoparticle formulation affected the cellular transport mechanisms using small molecule inhibitors. These experimental data were incorporated into a system of kinematic differential equations, and nonlinear, least-squares curve fitting algorithms were employed to extrapolate transport coefficients within our model. The subsequent computational framework produced some of the first parameters to describe transport kinetics across lymphatic endothelial cells and allowed for the quantitative analysis of the driving mechanisms of transport into lymphatic vessels. Our model indicates that transcellular mechanisms, such as micro- and macropinocytosis, drive transport into lymphatics. This information is crucial to further design strategies that will modulate lymphatic transport for drug delivery, particularly in diseases like lymphedema, where normal lymphatic functions are impaired.

Flow-induced vibration analysis in a pump-turbine runner under transient operating conditions

The dynamic characteristics of the pumped storage unit (pump-turbine runner) make it highly susceptible to vibrations. Previous studies seldom addressed the clearance variation due to runner vibration, largely because of two challenges: the integration of the moving grid with the clearance and governing equations, and the intricacy of factoring in the entire shaft system's influence on vibration. By employing user-defined functions (UDF), kinematic equations for the pump-turbine runner's rotation and translation were formulated and integrated with computational fluid dynamics results. This paper introduces a computational method to simulate clearance-induced vibration displacement. The study examines the impact of clearance vibration displacement on the pump-turbine's transient flow field and runner vibrations under various operating conditions. Notably, deviations from the rated load condition led to an expansion-contraction trajectory of the runner axis, culminating in chaotic behaviour in the "S" characteristic zone. The correlation between runner vibrations and pressure fluctuations strengthened with the clearance displacement model, especially in narrow labyrinth seal clearances. Simulations also provided a refined estimation of the secondary rotational speed increase during load rejection. The method's reliability is confirmed through field test data comparison, offering a fresh perspective to identify the vibration characteristics of the pump-turbine runner.

SH Waves in a Half-Space Model of Functionally Gradient Piezoelectric Semiconductors with Initial Stresses

In terms of piezoelectric class (6 mm) crystal semiconductor, the propagation of shear surface waves on the half-space surface under initial stresses is studied. The initial stresses and material parameters are hypothesized to exponentially vary along direction only. Besides, the velocity equations of SH waves can be obtained based on the boundary conditions and the kinematic equations of the initially stressed graded piezoelectric semiconductor material, as well as the traction-free boundary conditions. Based on the numerical results, we explore how the semiconductivity, initial stresses as well as material gradient index impact the dispersive curves.

Design and modeling of a planar-to-spatial tendon-driven continuum manipulator subjected to uncertain forces

Continuum Manipulators (CMs) have gained popularity as a promising solution for exploring unstructured environments due to their dexterity, compliance, and redundancy. Studies on CMs have focused individually on planar and spatial configurations, ranging from single to multi-section constructions. Planar CMs are easier to design, have simpler kinematics equations and possesses less number of tendons with reduced dexterity, redundancy, and reach. Whereas spatial CMs, offer a larger workspace with improved dexterity and redundancy while computationally expensive. This paper introduces a novel two-section tendon-driven continuum manipulator (TDCM) design that uses two planar sections to manipulate the distal end of the manipulator in three-dimensional space to handle external interaction from the environment. The proposed solution offers improved performance in terms of handling payloads and tip deflection under external interactions. This design is intended for agricultural applications such as vegetable harvesting and requires larger workspace and fewer actuators. We demonstrate the effectiveness of our design through kinematic and static analysis. The kineto-static model estimates the profile of the proposed TDCM under external interaction acting at the manipulator's distal end. Finally, the proposed design is experimentally validated by an in-housed prototype of an two-section TDCM, actuated by two tendons/section.

A flat-foldable equiangular spiral folding pattern inspired by sunflowers for deployable structures

Flasher origami pattern has been widely utilized to improve the stowage efficiency of deployable structures. Nevertheless, flasher origami cannot be folded fully flat, and they still have great potential for optimization in terms of storage volume and folding creases. In this paper, a flat foldable equiangular spiral folding pattern inspired by the sunflower disk is introduced. Then, a parametric design method for this equiangular spiral crease diagram is introduced in detail. Subsequently, a kinematic model of the equiangular spiral folding pattern is established based on the kinematic equivalence between rigid origami and spherical linkages. A simulation of the developed model demonstrates that the equiangular spiral folding pattern can be folded flat. Using the folded ratio as an evaluation index, the calculated results and experiments show that the equiangular spiral crease pattern can yield fewer creases and improve stowage efficiency in comparison to flasher origami pattern. Equiangular spiral folding pattern can save a considerable amount of space and provide a new approach to spatially deployable structures.

Research and Simulation of Kinematics and Dynamics of Tracked Support Equipment Based on Multi-Body Dynamics

Aiming at addressing the problems of low production efficiency and low safety factors in coal mines, this study designed a new type of support equipment and verified its theoretical reliability through the analysis of driving theory. The analysis was carried out through kinematics and dynamics, the position coordinates of the prototype at different times were analyzed using the RPY angles, and the spatial coordinate system of the prototype was established. Then, the position coordinates at different times could be solved by establishing the plane kinematics equations of the prototype. A 14-degree-of-freedom dynamic system model was established to reflect the longitudinal dynamic characteristics of the traveling mechanism of the prototype, and the differential equations of the traveling mechanism were established via the Lagrange dynamic equation method. Finally, the dynamic stability of the prototype under different working conditions was studied by using multi-body dynamic simulation technology. The research results show that the prototype meets the design requirements, has good reliability, realizes the mechanization, automation and high-efficiency production of coal mines, and realizes safe underground production. The research results provide effective solutions for ensuring the sustainable development of coal resources and the safe production of coal mines.

The structural design and simulation of the tubing handling manipulator

The size of the oil pipes in the factory is relatively large, making them inconvenient to handle. There are few existing oil pipe handling devices, and a 2P2R four degree of freedom manipulator device has been innovatively designed to complete the oil pipe handling work. First, the three-dimensional model of the manipulator was established using SolidWorks software, and then the improved D-H method was used to carry out the kinematics modeling of the manipulator system, and the forward and Inverse kinematics equations were derived. Then, the MATLAB software was used to carry out the kinematics analysis of the manipulator, and the manipulator workspace was calculated. The movement trajectory, as well as the displacement, speed, and acceleration curves under the trajectory were obtained using the fifth order Polynomial interpolation method, Finally, the 3D model was imported into Adams software, and an Adams virtual prototype was established for dynamic analysis. The force and torque curves of each joint were obtained, laying the foundation for further design and research in the future.

COLOR AND SIZE SORTING SYSTEM WITH AN ESP32-BASED ROBOT ARM

Robotics has seen an explosive increase in usage in the past few decades owing to its abundance of benefits and automation capabilities. The use of robotics is especially prominent in industries where the demand for precise and automatic robots in order to increase production efficiency is high. One of the robots commonly used in these industries is the arm robot model, which is a machine designed specifically to mimic some of the human arm’s capabilities in order to execute certain tasks over and over again, either autonomously or not. This paper will discuss the design of a robotic arm with five degrees of freedom capable of picking up and placing objects with three different colors and sorting the objects based on their size, starting from the largest to the smallest. The inclusion of an additional degree of freedom allows the robot to adjust the rotation of its grip to properly grab rotated objects. With an ESP32-CAM acting as its eyes, the robot is aided by a Python program run remotely on a separate Laptop in order to analyze video stream uploaded by the camera to determine the coordinates, color, size, and rotation of objects placed within the Detection Zone. The implementation of inverse kinematics equations run on an ESP32 microcontroller allows the robot to reach arbitrary coordinates specified by the Python program. Testing revealed that the resulting robot is able to perform the tasks it was designed to, namely recognizing and sorting colored boxes of different colors and sizes, with an accuracy 100% in recognition tasks and an accuracy of 91.5% in sorting tasks.

Design and motion planning of a 7-DOF assembly robot with heavy load in spacecraft module

Currently, there is great demand for automated assembly of equipment inside spacecraft module, which would require an assembly robot with small size, large workspace, high load capacity, and high flexibility. To meet this demand, a redundant seven-degree-of-freedom (7-DOF) assembly robot with three prismatic joints and four rotating joints is proposed in this study. First, a kinematic model of the assembly robot is established, and the positional forward kinematics are derived. Then, the positional inverse kinematic equations are derived by using the fixed joint method and the coordinate inverse transformation method for the characteristics of the robot with redundant degrees of freedom. The inverse solution optimization model of the assembly robot is designed based on genetic algorithm, and B-spline interpolation algorithm is used for joint trajectory planning. Next, NSGA-2 algorithm is used to construct the trajectory optimization model, and the time-pulsation multi-objective Pareto optimal frontier is obtained. Finally, the joint simulation model of mechatronics and the prototype experiment are established to verify the effectiveness of the optimization results. Analysis shows that the proposed assembly robot has better load-bearing capacity and dynamic characteristics than similar existing assembly robots. Furthermore, the segmented trajectory optimization strategy for the assembly robot using inverse solution optimization, interpolation algorithm, and NSGA-2 based multi-objective optimization gets each motion joint to realize the trajectory tracking function quickly and stably, and the trajectory tracking error is small enough to meet the requirements of end positioning accuracy.

Kinematics in metric-affine geometry

In a given geometry, the kinematics of a congruence of curves is described by a set of three quantities called expansion, rotation, and shear. The equations governing the evolution of these quantities are referred to as kinematic equations. In this paper, the kinematics of congruence of curves in a metric-affine geometry are analysed. Without assuming an underlying theory of gravity, we derive a generalised form of the evolution equations for expansion, namely, Raychaudhuri equation (timelike congruences) and Sachs optical equation (null congruences). The evolution equations for rotation and shear of both timelike and null congruences are also derived. Generalising the deviation equation, we find that torsion and non-metricity contribute to a relative acceleration between neighbouring curves. We briefly discuss the interpretation of the expansion scalars and derive an equation governing angular diameter distances. The effects of torsion and non-metricity on the distances are found to be dependent on which curves are chosen as photon trajectories. We also show that the rotation of a hypersurface orthogonal congruence (timelike or null) is a purely non-Riemannian feature.

Nonlinear Multi-Object Differential Game Simulation Model in LabVIEW

This article presents the synthesis of a nonlinear multi-object differential game model in relation to the process of safe ship control in collision situations at sea. Nonlinear dynamic equations of a target ship and linear kinematic equations of passing ships were used to formulate the game state equations. The model of such a differential game was developed using LabVIEW 2022 version software. This was then subjected to simulation tests using the example of a navigational situation in which the target ship passed three encountered ships at a safe distance under the conditions of non-cooperation of ships, their cooperation, and optimal non-game control. The results of the computer simulation are presented in the form of ship trajectories and time courses of individual game control variables. The distinguishing feature of the model built in LabVIEW software is the ability to conduct research in online mode, where the user has the opportunity to track the impact of changes in the model parameters on the course of the differential game simulation on an ongoing basis. Further refinements of the simulation model should concern the larger number of ships and test the sensitivity of the game control quality to inaccuracies in the measured state variables and to changes in the parameters of the ship’s dynamics.

DEVELOPMENT OF A METHOD FOR CALCULATING LOADS ON IMPLANTS AND PROSTHESES USED FOR THE HUMAN SKELETON

The study proposes a novel computational approach for customizing sustainable knee disarticulation prostheses, aimed at improving the quality of life for users. A specialized calculation technique for assessing the loads and moments on the prosthesis was formulated, leveraging MATLAB for solving kinematic equations, Solidworks for motion analysis, and ANSYS Workbench for material and static analysis. The integration of these tools enabled the validation of the design and analytical outcomes. The kinematic solutions accounted for individual and prosthesis weights, analyzing linear and angular dynamics over a motion range pertinent to the prosthetic leg's function. Static analysis was executed to determine maximum force impact on the prosthesis. The study's results were conducive to identifying the most suitable prosthesis characteristics for individuals aged 20 to 80, with a height of 160-190 cm and a weight of 80-120 kg. The prosthetic design promoted ease of movement in activities requiring a range of motion, such as running and jumping. The prosthesis adapted swiftly to body movements, achieving readiness in approximately three seconds. The research underscores the importance of interdisciplinary collaboration between engineers and medical professionals to optimize the anatomical and kinematic aspects of prosthesis design.