Mathematical Model Of Kinematics Research Articles

Establishment and analysis of a unified mathematical model of kinematics and dynamics of flexible parallel mechanisms

Abstract The conventional positive solution approach for determining the forward motion of the kinematic mechanism relies primarily on the outcomes of the inverse solution of motion, which is not universally applicable to flexible parallel mechanisms. Furthermore, the process of analyzing the natural frequency of parallel mechanisms is complex, cumbersome, and challenging. More importantly, the two models are independent, and there is no unified mathematical analysis model for flexible parallel mechanisms. Therefore, this paper establishes a unified mathematical model for kinematics and dynamics analysis of flexible parallel mechanisms through an improved multi-body system transfer matrix method. Taking the flexible 3-RPU planar mechanism and the space 2-RPU (2-UPS) space mechanism as examples, respectively, the forward motion solution of the mechanism was analyzed and its working space range and resonance frequency were solved. Finally, the calculation results are compared with the simulation results, and the feasibility and accuracy of the method are proved. Through this method, we can quickly solve the motion characteristics of the mechanism without relying on the results of the inverse solution of the mechanism, which provides suggestions for the precise control of flexible parallel mechanisms and also provides corresponding reference basis for the practical application of flexible parallel mechanisms.

Mathematical modeling of seismic wave kinematics in complex media

A variational method of mathematical modeling of seismic wave kinematics has been developed at the Department of Seismometry and Geoacoustics of the Faculty of Geology of Lomonosov Moscow State University for studying the kinematics of seismic waves of different types in two-dimensional isotropic media (gradient and layered) is presented. The problem of determining the trajectories of seismic rays was solved by integrating using the Runge-Kutta method a system of differential equations with given initial conditions. The algorithm was studied in order to verify the accuracy and correctness of the solutions obtained, as well as its testing on a number of theoretical models of heterogeneous media. The developed ray tracing method was used to study the effect of the velocity gradient and the geometry of seismic boundaries on the kinematics of reflected waves in multilayer media. Based on the results of mathematical modeling of the kinematics of reflected waves, conclusions were drawn about the limits of applicability of simplified models of horizontally layered media, which often approximate complex inhomogeneous media.

Proposed solution to eliminate pitch fluctuation in case of conical screw surface machining by apex adjustment

In case of conical worm production by conventional thread grinder, the fluctuation in angular velocity causes thread error. One possible way to eliminate the pitch error at the conical screw surface machining by shaft displacement is to design a drive pin that provides a constant pitch. In the course of our research, the examination of the machining of the conical worm surfaces by apex adjustment to eliminate the pitch fluctuation problem was carried out as a generalization of studies simulated with trigonometric relations in a particular case. The investigation was continued in a constructive geometric way with the aim of exploring universal mathematical relations in explicit forms in the mathematical kinematic model instead of the point series of the driver pin creator curve calculated in a given occurrence. The developed method makes exact design of the correct threading possible as a function of angular rotation in motion transfer under the given technology conditions, and thus expands the mathematical toolbar for innovative engineering design. Here we introduce the constructive geometric path supported by the algebraic toolbar, which also required affine and projective geometric relations in developing a solution that meets freely chosen expectations.

Biologically-Inspired Learning and Adaptation of Self-Evolving Control for Networked Mobile Robots

This paper presents a biologically-inspired learning and adaptation method for self-evolving control of networked mobile robots. A Kalman filter (KF) algorithm is employed to develop a self-learning RBFNN (Radial Basis Function Neural Network), called the KF-RBFNN. The structure of the KF-RBFNN is optimally initialized by means of a modified genetic algorithm (GA) in which a Lévy flight strategy is applied. By using the derived mathematical kinematic model of the mobile robots, the proposed GA-KF-RBFNN is utilized to design a self-evolving motion control law. The control parameters of the mobile robots are self-learned and adapted via the proposed GA-KF-RBFNN. This approach is extended to address the formation control problem of networked mobile robots by using a broadcast leader-follower control strategy. The proposed pragmatic approach circumvents the communication delay problem found in traditional networked mobile robot systems where consensus graph theory and directed topology are applied. The simulation results and numerical analysis are provided to demonstrate the merits and effectiveness of the developed GA-KF-RBFNN to achieve self-evolving formation control of networked mobile robots.

Kinematic Analysis and Control Algorithm for the Ballbot

The ballbot is a dynamically stable mobile robot designed to balance on a single ball, whose dynamic stability enables improved navigability in narrow, crowded, and dynamic environments. Through its single contact point with the ground, ballbot is omnidirectional and exceptionally agile, maneuverable, and organic in motion compared to other ground vehicles. For dealing with the challenging and imperative issues about ballbot, such as balancing, yaw, position control algorithms as well as the mathematical kinematic model, a novel model employing the Lagrange Equation is derived and control algorithm based on the similar principle as the second-order inverted pendulum is proposed, which is used for tackling the balancing, yaw, and position control. A cascade fuzzy proportional derivative (PD) controller and another controller with proportional integral (PI) control and PD feedback are designed for position and speed, respectively. Yaw control is presented, including head-hold mode and head-free mode. Some experiments are carried out to validate the effectiveness of the mathematical model and control algorithm.

A Mathematical Model for an Acceleration of a Rotating Object about a Fixed Axis

Kinematics of a rotating object about a fixed axis is well described in science of classical mechanics. All fundamental publications and textbooks contain a mathematical model for an acceleration of a rotating object about a fixed axis that is used for computing of forces acting on an object. Practice of usage for computing an acceleration a rotating object did not show some claims to the known equation. Nevertheless, the detailed analysis of the mathematical model for kinematics of a rotation of an object with acceleration demonstrates its incorrectness. The value of the actual acceleration of the rotating object is less than computed by the known equation. This is reason that running mechanisms are operating without fails because the value of acting forces is bigger than actual one. It means, the component of mechanisms is stronger and heavier than necessary. This paper addresses an acceleration analysis of a rotating object about a fixed axis and presents the mathematical model for computing an acceleration that can be used in engineering.

Kinematic and dynamic analysis of the Gantry-Tau, a 3-DoF translational parallel manipulator

• A closed-from mathematical model for dynamics of the Gantry-Tau is derived. • Kinematics of the Gantry-Tau is formulated and singular conditions are discussed. • Simulation studies are carried out to assess the performance of the proposed models. This paper deals with the mathematical modeling of kinematics and dynamics of the 3-degrees-of-freedom Gantry-Tau manipulator. Compared to many other parallel robots, Gantry-Tau offers a large accessible workspace and high stiffness. The kinematics of Gantry-Tau is presented which includes inverse kinematics formulation for the position, velocity and acceleration of the mechanism. Also, based on the obtained Jacobin matrices, singular configurations of the robot are studied. Afterwards, the equations of the inverse dynamic model of the Gantry-Tau are obtained through two different methods, i.e., virtual work and Newton–Euler. Finally, a case study is performed to verify the correctness of the derived models and investigate their computational efficiency.

Математическая модель кинематики и динамики древовидного исполнительного механизма шестиногого шагаюшего робота

Математическая модель кинематики и динамики древовидного исполнительного механизма шестиногого шагаюшего робота

Математическая модель кинематики и динамики древовидного исполнительного механизма активного экзоскелета

Математическая модель кинематики и динамики древовидного исполнительного механизма активного экзоскелета

Crawler robot kinematics modeling by using a two-wheeled approach

The aim of this paper is to analyze the kinematics of a small crawler robot. A mathematical model of kinematics based on a two-wheeled approach is proposed. This model is experimentally verified using vision-based motion measurements of a crawler vehicle equipped with encoders and a remote control system. It is assumed that the vehicle moves along a few curves with different angular speeds of its wheels. Based on the model of motion and the values of these speeds, numerical simulations are investigated. The results obtained from numerical and experimental validation are presented and discussed. The comparison delivered some important conclusions.

The kinematic analysis of wheeled chassis of type 8K4P

The current level of scientific and technological progress determines the widespread robots and robotic systems in all areas of human activity - from the entertainment to industry. Robotic systems are also used in construction, geological exploration, the repair utilities, etc. It should be noted that the design of such systems takes into account the increased requirements for size, terrain, autonomous functioning (mobility) and the range of tasks. Mobile robotics plays a particularly important role in the performance of specific tasks, including dangerous to life or health (liquidation of various accidents, consequences of emergencies, working in dangerous environments). Complex robotic systems are also used for planetary studies. Therefore, scientific and applied research which related to the improvement of existing robotic systems or developing new are relevant and have obvious practical importance. The mathematical model for kinematics’ investigation of eight-wheeled chassis of the class 8K4P is presented. The chassis can move along the surface of an arbitrary profile. Wheel chassis have a number of advantages over other types of chassis, such as simple mechanisms, high speed, easy to control, energy efficiency, etc. Compared to most common wheel chassis with a rocker-bogie [2] and based on rotational wheel units [4, 5] the benefits 8K4P include simplicity of design, high terrain crossing capacity in both directions of motion [1] and the possibility of overcoming various obstacles, including commensurate with dimensions of chassis. Concrete numerical results of kinematic analysis are shown for modelling of chassis’s movement on obstacles «stairway». The advantage of developed program is the ability to take into account various requirements, including restrictions on the relative angles of sections, restrictions on the maximum value of linear/angular velocity, taking into account the possibility of changing the position of the center of mass of the mechanical system, etc. The criterion which can be used to forecast occurrence of undesirable phenomenon such as breakaway of wheels from a base surface is offered also. The specified criterion defines a relative deviation of a trajectory of movement of the center of weights of mechanical system from a straight line. As the result of performed numerical experiments geometrical parameters of the chassis which satisfy minimal value of the criterion have been obtained.

Development and verification of a mathematical model to quantify the joint spaces of the elbow.

The purpose of the study was a development and verification of a model to quantify elbow joint spaces focusing on posterolateral rotatory instability. The model was tested for feasibility and applied to open and arthroscopic surgeries. A mathematical kinematic model was created, consisting of two segments representing the humerus and radius and ulna. Model calculations of the joint opening were compared with direct measurements on a mechanical model with attached marker arrays, and compared with values determined by a digital sliding caliper in order to evaluate the mathematical kinematic rigid body model of the elbow joint. Joint angles predicted by the mathematical model were compared with values determined by a two axis goniometer. The evaluation showed significant accordance (intercorrelation coefficient ≥0.8). The highest difference was 1.54° (root mean square: 0.46°) for all movement conditions. Ligament transaction at the open surgery situation affected an increase in joint opening (maximum joint opening: 10.9 mm±0.3). Distances in the ulnohumeral joint space (ligaments intact) under arthroscopic surgery situation varied between 4.0 and 6.8 mm (mean: 5.2 mm±1.0). Thus a possibility is created to achieve an objective assessment of joint space opening to provide a more complete description of the elbow kinematics using this approach.

Structural Parameter Identification of Articulated Arm Coordinate Measuring Machines

Precise structural parameter identification of a robotic articulated arm coordinate measuring machine (AACMM) is essential for improving its measuring accuracy, particularly in robotic applications. This paper presents a constructive parameter identification approach for robotic AACMMs. We first develop a mathematical kinematic model of the AACMM based on the Denavit-Hartenberg (DH) approach established for robotic systems. This model is further calibrated and verified via the practical test data. Based on the difference between the calculated coordinates of the AACMM probe via the kinematic model and the given reference coordinates, a parameter identification approach is proposed to estimate the structural parameters in terms of the test data set. The Jacobian matrix is further analyzed to determine the solvability of the identification model. It shows that there are two coupling parameters, which can be removed in the regressor. Finally, a parameter identification algorithm taking the least-square solution of the identification model as the structural parameters by using the obtained poses data is suggested. Practical experiments based on a robotic AACMM test rig are carried out, and the results reveal the effectiveness and robustness of the proposed identification approach.

Design and Implementation of Robot Arm Control Using LabVIEW and ARM Controller

The design, analysis and implementation of a robot arm system, which is expressed towards its performance with an analytical model by using LabVIEW and embedded system tools was presented in this article. Mathematical modeling of kinematics plays an essential role in design and implementation of a robot arm control. The projected work was focused to control the end effector of the robot arm to achieve any accessible point in an amorphous region using LabVIEW, ARM (Advanced RISC (Reduced Instruction Set Computing) Machine) microcontroller and Dexter ER2 robotic arm. LabVIEW uses analytical method to design the inverse kinematic model of the robot arm. The inverse kinematic model and robot arm control was implemented using LabVIEW and ARM microcontroller. LabVIEW uses the parallel communication to send the joint angles of the robot arm to the ARM. ARM microcontroller uses five PWM (Pulse Width Modulation) signals in order to control the robot arm, which was geared up with servo motors. Robot arm was controlled manually through the LabVIEW GUI (Graphical User Interface) controls. The present paper discussed about the mechanical configuration, analytical modeling, software and hardware of the above said work.

Simulation Methodology for Mechatronic Applications using Multilanguage Techniques

This paper presents a simulation methodology proposal, applying a multilanguage technique to simulate a mechatronic system. Firstly, a way to interact between two programming languages is presented, by taking advantage of each language to facilitate the development of a complex simulation tool. Later, the mechanical structure is obtained in VRML format. Next, special software is used to compute the mathematical model for kinematics and dynamic behavior. The results are showed by graphical simulation with Graphic Libraries support. Finally, the methodology is applied to flexible manipulator with one degree of freedom as study case, and the graphical simulation is presented.

Mathematical Modeling of Kinematics of Central Asian Plates

A two-dimensional geomechanical model is developed for numerical modeling of the displacement, strain, and stress fields of the lithosphere of Central Asia. This model considers the basic structural elements, i.e., plates, microplates, and tectonic faults. For the fault edges, the relative displacements caused by the Indo-Eurasian collision at the velocity of Hindustan motion 55 mm/yr are calculated by means of finite-element method. It is shown that the velocity of opening of different sections of the Baikal rift ranges from 1.2 to 4.8 mm/yr.