The motion of wheeled robots

The equations of motion of three-wheeled robots with two drive wheels and one passive caster wheel are derived and investigated. The control of longitudinal motion and turns of such a robot is implemented by appropriate control of the independent motors of the drive wheels. The research is carried out under the assumption that the robot is moving on a horizontal plane surface and that the wheels do not slip. A system of two non-linear equations with two controls is obtained for the non-holonomic system considered. The dependence of the phase portrait on the values of the constant controls and parameters of the system, taking into account the asymmetry of the robot, is investigated. The results obtained are not only of theoretical but also of practical interest.     

The rolling of a wheel along a corrugated rail

The rolling without detachment of a rigid massive wheel, carrying a static load, along a rail with undulations on the running surface, which arises as a result of non-uniform wear, is investigated. The rail is supported by an elastoviscous base. Because of the inertia of the wheel and the carriage the horizontal component of the velocity of the wheel centre differs only slightly from a constant quantity, and hence the motion of the wheel along the rail is assumed to be uniform. Steady vertical vibration of the wheel is considered. The vertical coordinate of the wheel centre, and also the difference between the longitudinal coordinates of the wheel centre and the point of contact of the wheel and the rail, are periodic and, correspondingly, even and odd functions of the longitudinal coordinate of the wheel centre, and their period is equal to the wave length on the rail surface. The periodic force of interaction of the wheel and the rail is given in the form of a Fourier series. Short waves, the amplitude of which is much less than their length, are often observed on the rail surface, and this length is much less than the wheel radius. In this case the coefficients of the Fourier series are expressed in terms of Bessel functions of the first kind of integer order. Observations show that the depth of the short wave on the rail surface increases until the radius of curvature in the rail trough approximates to the wheel radius, and hence it is assumed that these radii are close to or equal to one another. In this case the trajectory of the wheel centre differs considerably from the wave on the rail surface.     

An estimate of the attraction domain with a specified exponential stability index in a wheeled robot control problem

The problem of synthesizing a law for the control of the plane motion of a wheeled robot is investigated. The rear wheels are the drive wheels and the front wheels are responsible for the turning of the platform. The aim of the control is to steer a target point to a specified trajectory and to stabilize the motion along it. The trajectory is assumed to be specified by a smooth curve. The actual curvature of the trajectory of the target point, which is related to the angle of rotation of the front wheels by a simple algebraic relation, is considered as the control. The control is subjected to bilateral constraints by virtue of the fact that the angle of rotation of the front wheels is limited. The attraction domain in the distance to trajectory - orientation space, is investigated for the proposed control law. Arrival at a trajectory with a specified exponential stability index is guaranteed in the case of initial conditions belonging to the given domain. An estimate of the attraction domain in the form of an ellipse is given.     

Optimal control of the rectilinear motion of a rigid body on a rough plane by means of the motion of two internal masses

The problem of the optimal control of a rigid body moving along a rough horizontal plane due to motion of two internal masses is solved. One of the masses moves horizontally parallel to the line of motion of the main body, while the other mass moves in the vertical direction. Such a mechanical system models a vibration-driven robot–a mobile device able to move in a resistive medium without special propellers (e.g., wheels, legs or caterpillars). Periodic motions are constructed for the internal masses to ensure velocity-periodic motion of the main body with maximum average velocity, provided that the period is fixed and the magnitudes of the accelerations of the internal masses relative to the main body do not exceed prescribed limits. Based on the optimal solution obtained for a fixed period without any constraints imposed on the amplitudes of vibration of the internal masses, a suboptimal solution that takes such constraints into account is constructed.     

Handling characteristics and stability of the steady-state powerslide motion of an automobile

Powerslide of an automobile may be defined as a steady-state cornering motion at a large side slip angle of the vehicle, considerably large traction forces and a large negative steering angle of the handwheel. In this case the front wheels direct towards the outside of the turn. As this extrem driving condition, which can be seen e.g. in Rallye sports, is hardly addressed in literature so far, this paper investigates the respective handling characteristics. Therefore, a nonlinear four-wheel vehicle model is applied including nonlinear tyre characteristics, the load transfer between inner and outer wheels and the influence of the traction forces on the lateral tyre forces. A basic stability analysis reveals the unstable nature of the steady-state powerslide motion of a certain test vehicle. To approve the numerical findings, measurements have been performed with a sports utility vehicle with rear-wheel drive at various speeds on a wet circular test track.     

A model of a reinforced tyre with a rod protector

A tyre design consisting of a steel-cord-reinforced rigid bond with sides connected to the wheel disc and a protector(tread) in contact with the road is examined. The tread is in the form of a set of rods connected by one end to the band, with the other end either free or in contact with the road. The rod end in contact with the road is acted upon by a force applied from the road, represented by a force normal to the road plane and a shear force due to dry friction. If the modulus of the shear force does not exceed the magnitude of the normal force multiplied by the dry friction coefficient, there is no slip at the contact point. In the opposite case, the rod end will be displaced along the road by an amount sufficient to distribute the normal and shear forces. The dynamics of longitudinal and transverse strains of the rods in contact with the road is analysed using the motion separation method in the quasi-static approximation. The behaviour of the tread rods as a function of the vertical displacement of the wheel centre is investigated, the contact area is found and the conditions are determined under which the contact area is divided into parts in which either slip of the rod ends occurs or does not occur, depending on the magnitude of the longitudinal displacement of the wheel centre or its turning relative to the horizontal axis. An analogue of a continuous model of a rod-like tread is considered, and the magnitudes of the forces and moments are found as a function of the wheel disc displacements. The equations of wheel rolling are obtained, and the conditions under which steady motions exist are found.     

The rolling of a wheel with a reinforced tyre along a plane without slip

A model of a wheel with a reinforced tyre, whose surface is simulated by a flexible strip (tread) attached to parts of two tori (the sidewalls of the tyre) is considered. The disk of the wheel (a rigid body) has six degrees of freedom and is in contact with the plane along part of the tread. Based on several assumptions, the potential energy functional of the deformed wheel is found as a function of the deformations of the centre line of the tread. On the assumption that the wheel is rolling without slip in the region of contact of the tread with the plane along a previously unknown section of the tread, the complete system of equations of motion is obtained. The equilibrium of the wheel and the steady state of rolling in a straight line with given swivel and tilt are investigated, and all characteristics of the motion are found (the contact region, the tyre deformation, and the forces and torques applied to the wheel disk).     

A kinematic approach based on an equivalent track for a skid-steering vehicle

The skid-steering principle is based on controlling the relative velocities of both the tracks as differential drive wheeled vehicles. However, control of tracked locomotion poses a more complex problem because variation of the relative velocity of the two tracks results often in slippage of the vehicle [1]. It is possible to define a kinematic approach able to describe and to improve the motion control and dead-reckoning of a skid-steering robot by considering the effects of the slippage but without introducing the complexity of dynamics computation. The equivalent track approach has been applied to a skid-steering robot characterized by an high maximum payload (75 kg) and by a high power than makes it able to move in very difficult conditions (snow, iced terrains, mud) where the slippage effects become predominant. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling the motion of a trolley-like car

In this work, two simple dynamical models are derived for the motion of a three-wheeled car. In the models derived here, the motion of the car is controlled by four torques, three pedalling torques, acting on each of the wheels, and a steering torque acting on the front wheel.     

A method for controlling the motion of a robot snakeboarder

The problem of planning the motions of a robot snakeboarder consisting of a snakeboard and a flywheel mounted on it and capable of performing controlled rotation relative to the snakeboard crossbar is investigated. Programmed control of the wheel axes to provide an assigned path of motion of an arbitrary point on the crossbar is described. A control of the angular acceleration of the flywheel on the snakeboard that ensures a required course of variation of the velocity of the crossbar centre both on a horizontal plane and on an inclined plane is constructed. The problem of the maximum acceleration of the snakeboard along a “figure-of-eight” trajectory is solved.     

Simulation of stiffness analysis of rubber material applied in resilient wheel

Resilient wheels are well-known for their contribution to prevent squealing, impact noise and dynamic stresses on unsprung masses. In this study, a FE model is established in ABAQUS to analyze the stiffness of resilient wheels based on the Mooney-Rivlin hyperelastic material theory. The modes of each part in the resilient wheel and the whole wheel are calculated. The static and dynamic stiffness of the resilient wheel are analyzed. The simulation shows that the resilient wheel can reduce the dynamic loads both in the lateral and vertical directions between the vehicle and the track. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mathematical models for motion of the rear ends of vehicles

A new treatment of the path of the rear wheels of single and multiple axle vehicles produces simple first order differential equations for the motion of the rear wheels relative to the front wheels. We show for the first time, under the assumption of no slipping, that the path of the rear wheels depends only on the path of the front wheels and not on the vehicle speed. We examine motion along straight, circular and winding roads, giving closed form solutions in the first two cases. We also provide a straightforward algorithm for computing vehicle offtracking for any road geometry. We believe these results will be of significant benefit to road designers.     

On the motion of chaplygin''s sphere on a horizontal plane

The motion of a heavy sphere on a fixed horizontal plane is considered. It is assumed that the centre of mass of the sphere is at its geometric centre, while the principal central moments are different (Chaplygin's sphere). Using the method of averaging, the motion of the sphere is investigated under slip conditions when there is low viscous and also low dry friction. It is shown that when the sphere moves with viscous friction it tends, for the majority of initial data, to rotate about the longest of the axes of the principal central moments of inertia. The motion of the sphere centre tends to become uniform so that the slip velocity approaches zero exponentially. A system of averaged equations, which is fully integrable, is obtained in the case of almost equal moments of inertia, when the friction is dry. The solutions are analyzed.     

The necessary conditions for the existence of an additional integral in the problem of the motion of a heavy ellipsoid on a smooth horizontal plane

The equations of motion of an ellipsoid on a smooth horizontal plane are similar to the equations of motion of a heavy rigid body with a fixed point. In general, one integral is also lacking in order to integrate them. For a triaxial ellipsoid, the centre of mass of which coincides with the geometrical centre, it is proved that an additional integral is lacking (in the generic case).     

Geometric Kinematic Control of a Spherical Rolling Robot

We give a geometric account of kinematic control of a spherical rolling robot controlled by two internal wheels just like the toy robot Sphero. Particularly, we introduce the notion of shape space and fibers to the system by exploiting its symmetry and the principal bundle structure of its configuration space; the shape space encodes the rotational angles of the wheels, whereas each fiber encodes the translational and rotational configurations of the robot for a particular shape. We show that the system is fiber controllable—meaning any translational and rotational configuration modulo shapes is reachable—as well as find exact expressions of the geometric phase or holonomy under some particular controls. We also solve an optimal control problem of the spherical robot, show that it is completely integrable, and find an explicit solution of the problem.     

Gyroscopic forces in automotive design

The resonance frequencies (eigenfrequencies) of a car may dependsignificantly on the gyroscopic forces resulting from the steeringof the rotating wheels. For example, in the investigated carmodel, the first eigenfrequency decreases considerably withincreasing velocity, and only by taking the gyroscopic forcesinto account do the measured and calculated results match. Asa consequence, it is of great importance to have the gyroscopicforces incorporated in the respective car model. In this paper,for a wheel model with six degrees of freedom (6 DOF), the equationsof motion and eigenvalue equation are set up, where the gyroscopicforces are taken into account exactly. Then, it is shown usingadditional concepts how gyroscopic forces can be incorporatedin the MSC/NASTRAN finite-element package (FE package). Moreover,for a large-scale FE vehicle model, frequency-response calculationswith unbalanced excitation are carried out which exhibit gyroscopiceffects. The results for both the 6-DOF wheel model and theFE vehicle model are illustrated graphically. Among other things,this work has caused the MacNeal Schwendler company to extendthe FE program MSC/NASTRAN in cooperation with Mercedes-Benzso that gyroscopic effects can now be put in directly.     

On skidding of a wheeled vehicle when one of the wheels locks or slips

Two-wheel vehicle models for the case where one of the wheels locks or slips are suggested. Some combinations of motion parameters are shown to lead to the skidding of the vehicle. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 7, pp. 11–20, 2005.     

Modelling the motion of a three-wheeled car

In this work, a simple dynamical model is derived for the motion of a three-wheeled car. In the model derived here, the motion of the car is controlled by two torques, one, a pedaling torque, acting on the rear wheel, and the other, a steering torque acting on the two front wheels.     

Magic Labelings of Type (a, b, c) of Families of Wheels

In this paper, firstly, we study magic labeling of type (1, 1, 1) for wheels and subdivided wheels. Secondly, we prove that a wheel admits a magic labeling of type (0, 1, 1). We also prove that any odd wheel admits a magic labeling of type (0, 1, 0). At the last, we find the magic labeling of type (1, 1, 0) for subdivided wheels.