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Software Control of the Movement of a Differential Drive Robot for Different Friction Models
Abstract
Introduction . Designing motion control systems for mobile robots requires the construction of mathematical models. Researchers have repeatedly addressed this topic. In particular, works have been published on the calculations of multiphysical processes, modeling the movement of various types of wheels under certain conditions. In addition, the dynamics of deformable contacting bodies during sliding, rolling and rotation, issues of autonomy and controllability of mobile robots were considered. Note, however, that the dynamics and positioning accuracy of wheeled robots is largely determined by friction. The literature does not present studies on the dynamics of a robot with a differential drive taking into account the interrelationships of sliding, spinning and rolling friction effects based on the theory of multicomponent friction. Research in this area can reveal new dynamic effects. Based on the data obtained in this way, it is possible to improve the accuracy of positioning in building mathematical control models. The presented work aims at investigating the movement of an automatic device with a differential drive taking into account three contact models: nonholonomic, Coulomb friction, and multicomponent models.
Materials and Methods . The scheme of a two-wheeled robot with differential drive and continuous movement on the support surface was adopted as the basic one. The movement of the device was provided through software control. The dynamics was described in the form of Appel equations. Mathematical models were used for calculations, taking into account friction in different ways. Coordination of the actions of the mechanism was formed at a dynamic level. The control actions were the moments of the wheel motors. When visualizing the models under study, the built-in numerical methods of the Wolfram Mathematica system were used with a minimum accuracy of 10 ⁻⁶ .
Results . When building a mathematical model, the equations for the angular velocities of the wheels were determined. The authors took into account the presence of a contact site and derived the equations of dynamics of a differential drive robot. The elements of the system were force and moment projections, indicators of platform spin, masses, angular accelerations, and inertia of the wheels. It was shown how control actions were formed within the framework of nonholonomic mechanics. The model of engines that created a moment of control on the driving wheels was described. The solution was derived as the relationship between the inductance of the conductors of electric motors and the operation of the power supply. Three models describing the dynamics of a differential drive robot were examined in detail. The first model was nonholonomic. The second and third included a system of equations for the dynamics of a differential drive robot for a general case with a contact platform. At the same time, in the second model, the switching time in the engine was ignored and the Coulomb friction was involved. In the third model, a parameter to determine the speed of transients in the engine was introduced, and Pade decomposition was involved. This was a model with multicomponent friction. The calculation results were shown in the form of graphs. On them, the studied models were visualized in the form of curves of different colors. Comparison of the graphs showed in which cases, after the completion of transients, the control provided the required accuracy. These were models 1 and 2. In model 3, the software control generated an error in the angular velocity of rotation of the platform. This error could not be predicted within the framework of the 1st and 2nd models. In all the systems considered, the sliding speed of the wheels in the transverse direction dropped to zero. The condition of continuous motion of the support wheel was obtained and validated.
Discussion and Conclusion . Software control is acceptable in models that do not take into account wheel friction during simultaneous sliding, spinning and rolling (general case of spatial motion). However, it is important to consider the relationship between these processes and multicomponent friction. This is required for the robot to perform program movements more accurately. It was established that software control in a model that takes into account the friction of spinning and rolling caused deviations from the program values of the angular velocity of the platform. The results obtained can be used in the building of a control system with predictive models.
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The implementations of the theory of multicomponent dry friction [1-19] for analyze the dynamics of some robotic systems, such as a butterfly robot [16-18, 20] or a humanoid robot is proposed. Since the main controlled element of these systems is a spherical, elastic composite shell, it is required to calculate the distribution of normal contact stresses inside the contact spot. The contact pressure distribution for such elements is constructed using the S. A. Ambartsumyan’s equation for a transversally isotropic spherical shell. This equation is modified by introducing the averaged contact pressure and normal displacements for the shell. The construction of the resolving integral equation for the contact pressure is based on the principle of superposition and the method of Green's functions. For this, the corresponding Green's function is constructed, which is the normal displacement of the shell as a solution to the problem of the effect of concentrated pressure. Green's function as well as the contact pressure, it is sought in the form of series expansions in Legendre polynomials, taking into account additional relations for the reduced contact pressure and normal displacements. Using the Green's function, an integral equation solving the problem is constructed. As a result, the problem is reduced to determining the expansion coefficients in a series of the reduced contact pressure. Restricting ourselves to a finite number of terms in the series of expansions, using the discretization of the contact area and the properties of Legendre polynomials, the problem is reduced to solving a system of algebraic equations for the expansion coefficients for the reduced pressure. After that, from the additional relation, the coefficients of the required expansion of the contact pressure in a series in Legendre polynomials are determined. To describe the conditions of shell contact with the surface, the theory of multicomponent anisotropic dry friction is used, taking into account the combined kinematics of shell motion (simultaneous sliding, rotation and rolling). The coefficients of the dry friction model can be calculated using simple explicit formulas [1-19] based on numerical experiments.
A four-dimensional model of dry friction in the interaction of a solid wheel and a horizontal rough surface is investigated. It is assumed that there is no separation between the wheel and the horizontal surface. The movement of the body occurs in conditions of combined dynamics, when in addition to the sliding movement, the body participates in spinning and rolling. The equation of motion of the wheel is compiled using the Appel equation.
The resulting model of sliding, spinning, and rolling friction is given for the case where the contact area is a circle. The cumbersome integral expressions were replaced by fractional-linear Pade approximations. Pade approximations accurately describe the behavior of the components of the friction model. A mathematical model is proposed that describes the simultaneous sliding, spinning and rolling of a solid wheel. The dependences of the parallel and perpendicular components of the friction force and the torque of the spinning friction were ploted with respect to the parameter that characterizes the movement of the wheel. Comparisons of the integral friction model and the model based on Pade approximations are presented. The results of the comparison showed a qualitative correspondence of the models. After obtaining the equation of motion, the simulation of motion at a constant control torque of the wheel is carried out. The graphs allow you to match the logical behavior of the wheel movement.
In this paper, the modelling of a reaction wheel bicycle robot (RWBR) is identified from a second-order mathematical model which is similar to an inverted pendulum, and an adaptive integral terminal sliding mode (AITSM) control scheme is developed for balancing purpose of the RWBR. The proposed AITSM control scheme can not only stabilize the bicycle robot and reject external disturbances generated by uncertainties and unmodelled dynamics, but also eliminate the need of the required bound information in the control law via the designed adaptive laws. The experimental results verify the excellent performance of the proposed control scheme in terms of strong robustness, fast error convergence in comparison with other control schemes.
This paper investigates the influence of the material properties on the deceleration dynamics of a deformable cylinder rolling with slipping on a half-space of the same material. The interaction of the cylinder and the half-space is described by the 2D quasistatic contact problem of viscoelasticity (Goryacheva: J Appl Math Mech 37(5):877–885, 1973; Contact mechanics in tribology. Kluwer, Dordrecht 1998) which includes as limiting cases the absolutely rigid and elastic materials. Full dynamical analysis of the problem including the phase portrait, the dependence of the deceleration distance on the mechanical properties of the contacting bodies and on the friction coefficient is provided. The qualitative features of deceleration are justified by asymptotic analysis.
A mobility mechanism for robots to be used in tight spaces shared with people requires it to have a small footprint, to move omnidirectionally, as well as to be highly maneuverable. However, currently there exist few such mobility mechanisms that satisfy all these conditions well. Here we introduce Omnidirectional Balancing Unicycle Robot (OmBURo), a novel unicycle robot with active omnidirectional wheel. The effect is that the unicycle robot can drive in both longitudinal and lateral directions simultaneously. Thus, it can dynamically balance itself based on the principle of dual-axis wheeled inverted pendulum. This letter discloses the early development of this novel unicycle robot involving the overall design, modeling, and control, as well as presents some preliminary results including station keeping and path following. With its very compact structure and agile mobility, it might be the ideal locomotion mechanism for robots to be used in human environments in the future.
The study of the dynamical behavior of a system becomes really complex when friction forces are introduced in calculus since they directly depend on the values of the normal reactions and on the motion of the system, but the motion, at its turn, depends upon the forces from the system. The analysis is more difficult when dry friction, characterized by inequalities, is present between the parts of the system. When inequalities are involved in dynamical equilibrium equations, theoretically simultaneous equilibrium states may occur. In scientific literature there are famous examples supporting this statement. The present paper presents the dynamical study of a simple system, namely a ball obliged to roll in a groove with flat walls.
It is proved experimentally that the motion of the ball is a planar one. The dynamical analysis of the system with planar motion leads to an undeterminate system of equations and therefore the spatial approach of the system is required. After writing the equations of the spatial motion of the ball, an incompatible system of equations is obtained.
We present a multibody simulator being used for compliant humanoid robot modelling and report our reasoning for choosing the settings of the simulator’s key features. First, we provide a study on how the numerical integration speed and accuracy depend on the coordinate representation of the multibody system. This choice is particularly critical for mechanisms with long serial chains (e.g. legs and arms). Our second contribution is a full electromechanical model of the inner dynamics of the compliant actuators embedded in the COMAN robot, since joints’ compliance is needed for the robot safety and energy efficiency. Third, we discuss the different approaches for modelling contacts and selecting an appropriate contact library. The recommended solution is to couple our simulator with an open-source contact library offering both accurate and fast contact modelling. The simulator performances are assessed by two different tasks involving contacts: a bimanual manipulation task and a squatting tasks. The former shows reliability of the simulator. For the latter, we report a comparison between the robot behaviour as predicted by our simulation environment, and the real one.
This review presents a collection of the solved dynamic problems taking into account the normal and shear stress distributions in the contact region due to the deformation of contacting bodies. The considered dynamic problems differ in the number of degrees of freedom and the type of relative displacements of contacting bodies (2-D models with rolling and sliding, 3-D models with sliding and spinning, 3-D models with sliding, rolling and spinning). The contact mechanics solutions are used in formulation of the dynamic problems, which are studied based on the analytical or semi-analytical approach. The effects of the mechanical properties of the contacting bodies and the contact conditions on the dynamics are analyzed and discussed.
Our goal is to build up an efficient computer model of the omni-vehicle comprising several omni-wheels (each one carries several freely rotating rollers on its periphery); the vehicle rolls without control on a horizontal plane. For simulating contact interaction we use the holonomic non-ideal constraints instead of ideal non-holonomic ones. Although there exist papers on the dynamical models of the vehicle and its control, e. g. [1, 2], here we study how the algorithm for computing the contact forces between a particular roller and the plane influences a description of the vehicle dynamics. To simulate dynamics we build up our dynamical models using Modelica language of object-oriented modeling. Earlier we proposed the contact tracking algorithms [3] for the point-contact dry friction regularized by linear saturation function. Using numerical simulations of the models developed we compare the vehicle dynamics for three particular contact models: (a) ideal non-holonomic contact “with impacts”, (b) model of viscous friction with the large coefficient of viscosity, and (c) one with the regularized dry friction concentrating mostly on the tangent forces description.
The dynamics of a symmetrical vehicle with omniwheels, moving along a fixed, absolutely rough horizontal plane, is considered, making the following assumptions: the mass of each roller is nonzero, there is a point contact between the rollers and the plane, and there is no slip. The equations of motion composed with the use of the Maxima symbolic computation system, contain additional terms, proportional to the axial moment of inertia of the roller and depending on angles of rotation of the wheels. The mass of the rollers is taken into account in those phases of motion when there is no change of rollers at the contact. The mass of rollers is considered to be negligible when wheels change from one roller to another. It is shown that a set of motions, existing in the inertialess model (i.e., the model that does not take into account mass of rollers), disappears, as well as its linear first integral. The main types of motion for a symmetrical three-wheeled vehicle, obtained by a numerical integration of equations of motion, are compared with results obtained on the basis of the inertialess model.
The analytic solution of the problem of forced vibrations of a rigid body with cylindrical surface on a horizontal foundation is given. It is assumed that the dry friction force acts at the point of contact between the cylindrical surface of the body and the foundation and the foundation moves by a harmonic law in the horizontal direction perpendicularly to the cylindrical surface element. The averaging method is used to determine the forced vibration mode near the natural frequency of the body vibrations on the fixed foundation. The results are presented as amplitude-frequency and phase-frequency characteristics.
Kinematics is the branch of mechanics that studies the motion of a body, or a system of bodies, without considering its mass or the forces acting on it.
The dry friction model accounting the combined kinematics is constructed using the differential Coulomb's law formulation for a small contact spot element. Integrating over the contact area results the exact dynamically coupled model that can be approximated by the one containing linear and quadratic forms of sliding and spinning velocities. The example of the real tire is studied using the finite element solution for the quasi-static contact pressure distribution; its analytic approximation allows one to compute the dry friction model coefficients. The effect of the contact pressure distribution on the friction spin is shown.
О моделях шины, учитывающих как деформированное состояние, так и эффекты сухого трения в области контакта. Компьютерные исследования и моделирование
- А А Киреенков
- С И Жаворонок
- Д В Нуштаев
Киреенков А.А., Жаворонок С.И., Нуштаев Д.В. О моделях шины, учитывающих как деформированное
состояние, так и эффекты сухого трения в области контакта. Компьютерные исследования и моделирование.
2021;13(1):163-173. https://doi.org/10.20537/2076-7633-2021-13-1-163-173