Stability of steady motions of a mobile robot with roller-carrying wheels and a displaced centre of mass

The motion of a mobile three-wheel robotic vehicle on a horizontal surface is investigated. Passive rollers are fastened along the rim of each wheel, enabling each wheel not only to roll in the usual manner, but also to move perpendicular to its plane. Each of these wheels, as well as the ordinary wheels, is equipped with one drive, which rotates the wheel about its axis. The vehicle equipped with roller-carrying wheels can move in any direction with any orientation. The motion of the robot on a horizontal surface is studied in the case where the centre of mass of the robot deviates from the geometric centre of the triangular platform, and there is no slip at the points of contact of the rollers with the supporting surface. In the case of free motion of the robot, an additional first integral is pointed out and the exact solution found is analysed. An equation for specifying steady motions, under which a constant voltage is supplied to the DC motors that drive the wheels, is constructed. The stability of the rectilinear motion of the robot is investigated.     

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.     

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.     

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.     

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.     

Dynamically equilibrium shapes and directions of motion of ocean current rings

The dynamically equilibrium shapes of a uniform-density rotating mass of liquid (a ring) in the surface layer of a quiescent stratified ocean are determined. The examination is carried out in a plane tangential to the Earth, taking into account the vertical and horizontal projections of the angular velocity of its rotation. Exact solutions of the equations of motion of an ideal incompressibe fluid are obtained, making it possible, for a linearly stratified ocean, to determine the dynamic all equilibrium shape of the interfaces of water masses and the free boundaries of cyclonic and antocyclonic rings. These shapes comprise second-order surfaces inclined to the water level in the meridian plane, the type of surfaces depending on the governing parameters of the problem. Expressions are obtained for the angles of inclination of the principal axes. For small deviations from equilibrium, due to a difference in the gravitational forces and Archimedes forces, motion of the ring occurs, governed by the inclination of the principal axes and the nature of change (increase or reduction) in the average density of the ring, determined by the ratio of the rates of diffusion of heat and salt. The displacement along the parallel comprises geostrophic motion, for the velocity of which an analytical expression is obtained. The displacement along the meridian comprises motion over an inclined plane. An analytical expression is given that relates the change in the depth of the centre of mass of the ring to the velocity of motion along the meridian through the angle of inclination of the principal axes of the ring. This explains the motion of both types of Gulf Stream ring to the south-west and of the Oyasio ring to the north-east.     

Active vibration control of unbalanced flexible rotor–shaft systems parametrically excited due to base motion

Rotor vibrations caused by large time-varying base motion are of considerable importance as there are a good number of rotors, e.g., the ship and aircraft turbine rotors, which are often subject to excitations, as the rotor base, i.e. the vehicle, undergoes large time varying linear and angular displacements as a result of different maneuvers. Due to such motions of the base, the equations of vibratory motion of a flexible rotor–shaft relative to the base (which forms a non-inertial reference frame) contains terms due to Coriolis effect as well as inertial excitations (generally asynchronous to rotor spin) generated by different system parameters. Such equations of motion are linear but time-varying in nature, invoking the possibility of parametric instability under certain frequency–amplitude combinations of the base motion. An investigation of active vibration control of an unbalanced rotor–shaft system on moving bases is attempted in this work with electromagnetic control force provided by an actuator consisting of four electromagnetic exciters, placed on the stator in a suitable plane around the rotor–shaft. The actuator does not levitate the rotor or facilitate any bearing action, which is provided by the conventional suspension system. The equations of motion of the rotor–shaft continuum are first written with respect to the non-inertial reference frame (the moving base in this case) including the effect of rotor internal damping. A conventional model for the electromagnetic exciter is used. Numerical simulations performed on the flexible rotor–shaft modelled using beam finite elements shows that the control action is successful in avoiding the parametric instability, postponing the instability due to internal material damping and reducing the rotor response relative to the rigid base significantly, with sufficiently low demand of control current in comparison with the bias current in the actuator coils.     

On the dynamic model and motion planning for a spherical rolling robot actuated by orthogonal internal rotors

The paper deals with the dynamics of a spherical rolling robot actuated by internal rotors that are placed on orthogonal axes. The driving principle for such a robot exploits nonholonomic constraints to propel the rolling carrier. A full mathematical model as well as its reduced version are derived, and the inverse dynamics are addressed. It is shown that if the rotors are mounted on three orthogonal axes, any feasible kinematic trajectory of the rolling robot is dynamically realizable. For the case of only two rotors the conditions of controllability and dynamic realizability are established. It is shown that in moving the robot by tracing straight lines and circles in the contact plane the dynamically realizable trajectories are not represented by the circles on the sphere, which is a feature of the kinematic model of pure rolling. The implication of this fact to motion planning is explored under a case study. It is shown there that in maneuvering the robot by tracing circles on the sphere the dynamically realizable trajectories are essentially different from those resulted from kinematic models. The dynamic motion planning problem is then formulated in the optimal control settings, and properties of the optimal trajectories are illustrated under simulation.     

Closed-form solution for the free axial-bending vibration problem of structures composed of rigid bodies and elastic beam segments

In this paper, the coupled axial-bending vibration of planar serial frame structures composed of rigid bodies and Euler-Bernoulli beam segments is considered. The corresponding mode orthogonality conditions for this kind of structures are derived. It is assumed that the mass centers of rigid bodies have both the transverse and the axial eccentricity with respect to beam neutral axes and that the mass centers are located in the plane in which the rigid bodies perform planar motion. The system responses to initial excitation in the case of distinct as well as repeated natural frequencies are considered. Theoretical considerations are accompanied by four numerical examples.     

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.     

A “retraction” method for planning the motion of a disc

A new approach to certain motion-planning problems in robotics is introduced. This approach is based on the use of a generalized Voronoi diagram, and reduces the search for a collision-free continuous motion to a search for a connected path along the edges of such a diagram. This approach yields an O(n log n) algorithm for planning an obstacle-avoiding motion of a single circular disc amid polygonal obstacles. Later papers will show that extensions of the approach can solve other motion-planning problems, including those of moving a straight line segment or several coordinated discs in the plane amid polygonal obstacles.     

Controllability and stabilization of programmed motions of an automobile-type transport robot

A non-linear model of the motion of an automobile-type transport robot (TR) with absolutely rigid wheels, a steering device and actuators based on DC motors, is considered. Such a model for TR motion is a non-holonomic electromechanical system and, if the dynamics of the actuators and the steering device (forces of elasticity and attenuation in its elements) is ignored, corresponds to the model of automobile motion devised by Lineikin [1]. Non-linear canonical transformations of the state and control space coordinates are constructed which reduce the initial equations of motion of the TR to a simpler canonical form, convenient for the analysis and synthesis of control systems for the TR. These transformations are used to find the conditions for the controllability of the TR as a controlled object. Algorithms are given for constructing programmed controls and programmed motions of the TR. Stabilizing control laws are synthesized that make the programmed motions of the TR asymptotically stable and guarantee that the transients will have preassigned properties     

Controlled rolling of a disc along a plane curve

The plane-parallel rolling of a disc along a fairly smooth curve under the action of perturbing and controlling forces and moments of forces is investigated. Models of the controlled motion corresponding to an explicit or implicit coordinate specification of the curve are constructed. The requirements imposed on the values of the sliding friction force and the normal pressure force, which ensure that the disc rolls without slipping and without loss of contact are determined. The problem of bringing the disc into the required state of motion in a time-optimal way by the action of the force of gravity and a limited axial moment of forces are considered. The situation when the reference curve contains a linear component (inclined upwards or downwards) and a periodic (wavy) component is investigated. Problems of optimal synthesis for a relatively large and relatively small controlling moment are considered.Investigations of the rolling of bodies of different geometrical shape on surfaces, both flat and curved (see 1, 2, 3, 4, etc.), are important for applications in modern machine construction, transport, mobile robot systems,5, 6, 7 etc. However, there are no results of systematic investigations of the problems of the dynamics and control of rolling of bodies on curved surfaces, including comparatively simple bodies (cylinders, discs and spheres). From the applied point of view, the problem of controlled rolling of a disc under gravitational forces on a curve containing a linear incline and periodic smooth changes in the slope is of interest as a model of the motion of a wheel along a path with a curvilinear profile. The solution of problems of the control of the motion of a disc along a closed curve, for example, inside or outside a circle, surmounting obstacles or “rolling out” from hollows of different shapes etc. is of fundamental importance.     

The motion in a perfect fluid of a body containing a moving point mass

The motion of a system (a rigid body, symmetrical about three mutually perpendicular planes, plus a point mass situated inside the body) in an unbounded volume of a perfect fluid, which executes vortex-free motion and is at rest at infinity, is considered. The motion of the body occurs due to displacement of the point mass with respect to the body. Two cases are investigated: (a) there are no external forces, and (b) the system moves in a uniform gravity field. An analytical investigation of the dynamic equations under conditions when the point performs a specified plane periodic motion inside the body showed that in case (a) the system can be displaced as far as desired from the initial position. In case (b) it is proved that, due to the permanent addition of energy of the corresponding relative motion of the point, the body may float upwards. On the other hand, if the velocity of relative motion of the point is limited, the body will sink. The results of numerical calculations, when the point mass performs random walks along the sides of a plane square grid rigidly connected with the body, are presented.     

Dynamic modeling and simulations of the wave glider

Wave glider is a new wave-powered autonomous marine vehicle, which is composed of a submerged glider connected to a surface floater via a tether. Such an advanced vessel is designed to harvest energy in ocean surface waves to generate forward thrust. Equipped with solar panel and battery as well as some dedicated sensors, the wave glider is able to achieve long duration missions via sea-side control. In this paper, a 4-DOF (degree-of-freedom) mathematical model of the wave glider is established using Newton–Euler approach. The second-order wave drift force on the horizontal plane and the first order wave force on the vertical direction are considered. The hydrodynamic parameters were calculated using the potential flow theory and empirical formula. Furthermore, motion simulation of the wave glider with respect to the sensitivity analysis to some key environmental factors and the heading control ability is conducted. The simulation results are presented and discussed in detail, which provides theoretical guidance and reference for wave glider design.     

On final motions of a Chaplygin ball on a rough plane

A heavy balanced nonhomogeneous ball moving on a rough horizontal plane is considered. The classical model of a “marble” body means a single point of contact, where sliding is impossible. We suggest that the contact forces be described by Coulomb’s law and show that in the final motion there is no sliding. Another, relatively new, contact model is the “rubber” ball: there is no sliding and no spinning. We treat this situation by applying a local Coulomb law within a small contact area. It is proved that the final motion of a ball with such friction is the motion of the “rubber” ball.     

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.     

Informational Sets in a Model Problem of Homing

A linear pursuit problem in the plane under incomplete pursuer information about the evader is investigated. At discrete time instants, the pursuer measures with errors the angle of vision to the evader, the angular velocity of the line of sight, and the relative distance. Other combinations of measurable parameters are possible (for example, angle of vision and relative distance or angle of vision only). The measurements errors obey certain geometric constraints. The initial uncertainties on the evader coordinates and velocities are given in advance. Having a resource of impulse control, the pursuer tries to minimize the miss distance. The evader control is bounded in modulus.The problem is formulated as an auxiliary differential game. Here, the notion of informational set is central. The informational set is the totality of pointwise phase states consistent with the history of the observation-control process. The informational set depends on the current measurements; it changes in time and plays the role of a generalized state, which is used for constructing the pursuer control.A control method designed for the linear pursuit problem is used in the planar problem of a vehicle homing toward a dangerous space object. The nonlinear dynamics is described by the Kepler equations. Nonlinear terms of the equations in relative coordinates are small and are replaced by an uncertain vector parameter, which is bounded in modulus and is regarded as an evader control. As a result, we obtain the mentioned control problem in the plane.The final part of the paper is devoted to the simulation of a space vehicle homing toward a dangerous space object. In testing the control method developed, two variants are considered: random measurement errors and game method of constructing the measurements; the latter is also described in the paper.     

The influence of friction forces on the dynamics of a two-link mobile robot

Controlled periodic motions of a planar two-link robot in a horizontal plane when there is dry friction are considered. The two-link is controlled by means of an internal torque applied to the joint connecting the links. The dynamics of the two-link, taking into account the influence of friction forces and the constrained nature of the control torque, is analysed assuming that the angle between the links is small. The conventional locomotion algorithm of a two-link is modified to ensure rectilinear displacement of the two-link. The influence of various geometrical and mechanical parameters of the system on the average rate of locomotion and on the power consumption during the motion of the two-link robot in a plane is investigated.     

An Invariant Measure and the Probability of a Fall in the Problem of an Inhomogeneous Disk Rolling on a Plane

This paper addresses the problem of an inhomogeneous disk rolling on a horizontal plane. This problem is considered within the framework of a nonholonomic model in which there is no slipping and no spinning at the point of contact (the projection of the angular velocity of the disk onto the normal to the plane is zero). The configuration space of the system of interest contains singular submanifolds which correspond to the fall of the disk and in which the equations of motion have a singularity. Using the theory of normal hyperbolic manifolds, it is proved that the measure of trajectories leading to the fall of the disk is zero.