Modelling and control of the motion of a disk rolling on a spherical dome

This work deals with the modelling and control of the motion of a disk rolling without slipping on a rigid spherical dome. It is assumed here that the motion of the disk is controlled by a tilting moment, a directional moment, and a pedalling moment. First, a mathematical model of the motion of the disk rolling on the dome is derived. Then, by using a kind of an inverse control transformation, a control strategy is proposed under which the motion of the disk is stabilized and is able asymptotically to track any smooth trajectory which is located on the spherical dome.     

Modelling of the motion of a disk rolling on a smooth rigid surface

This letter deals with the modelling of the motion of a disk rolling without slipping on a rigid surface.     

Stabilization and control of the motion of a rolling disk

This work deals with the stabilization and control of the motion of a disk rolling on the horizontal plane. It is assumed here that the motion of the disk is controlled by a tilting moment, a directional moment, and a pedalling moment. By using a kind of an inverse control transformation, a control strategy is proposed under which the motion of the disk is stabilized and is able asymptotically to track any given smooth ground trajectory.     

Feasible strategies for the control of a disk rolling on a moving horizontal plane

Let (X,Y,Z) be an inertial coordinate system and suppose that a horizontal plane is moving in a uniform velocity parallel to the (X,Y)-plane. A disk is rolling on the moving plane. Given two points A and B fixed in the (X,Y)-plane. Open-loop strategies are computed, for rolling the disk, on the moving plane, from A to B, during a given time interval [0, _f] and subject to state and control constraints.     

Modelling of the controlled motion ofa point on a plane

A solution of the problem of feedback control of the motion of a point on a plane is presented. The equations of the controlprogramme (the objective) are set up as a system of differential equations with a given set of singular trajectories in the domain of admissible positions of the controlled point, as well as a given topological structure of the partition into trajectories. These equations define the vector field of velocities of the programmed motions of the point and are used to find the corresponding control forces.     

The nearly horizontally rolling of a thick disk on a rough plane

In the paper the analytical solution of the partially linearized equations of motion of nearly horizontally rolling of a thick rigid disk on a perfectly rough horizontal plane under the action of gravity is given in terms of Whittaker functions. The solution is used to obtain the asymptotic solutions for a very small inclination angle, the study of unilateral contact between a disk and a plane and the study of disk colliding motion.      

The rolling motion of a truncated ball without slipping and spinning on a plane

This paper is concerned with the dynamics of a top in the form of a truncated ball as it moves without slipping and spinning on a horizontal plane about a vertical. Such a system is described by differential equations with a discontinuous right-hand side. Equations describing the system dynamics are obtained and a reduction to quadratures is performed. A bifurcation analysis of the system is made and all possible types of the top’s motion depending on the system parameters and initial conditions are defined. The system dynamics in absolute space is examined. It is shown that, except for some special cases, the trajectories of motion are bounded.     

Modelling the motion of a three-link manipulator based on a plane with a time-dependent inclination

This work deals with the modelling of a three-link manipulator mounted on a plane with a time-dependent inclination. Two cases are considered.

• (i)The plane is part of a rigid body.
• (ii)The plane is in a moored ship.

     

Dynamics of a rolling and sliding disk in a plane. Asymptotic solutions,stability and numerical simulations

We present a qualitative analysis of the dynamics of a rolling and sliding disk in a horizontal plane. It is based on using three classes of asymptotic solutions: straight-line rolling, spinning about a vertical diameter and tumbling solutions. Their linear stability analysis is given and it is complemented with computer simulations of solutions starting in the vicinity of the asymptotic solutions. The results on asymptotic solutions and their linear stability apply also to an annulus and to a hoop.     

The interconnection of sliding and rolling in the problem of the motion of a homogeneous sphere on a rough horizontal plane

The motion of a homogeneous sphere on a rough horizontal plane when the angular velocities of the twisting and spinning of the sphere are equal to zero at the initial instant is considered. It is proved that, for any initial conditions, the angular velocity of the rolling of the sphere and the sliding velocity vanish after the same finite time. It is shown that the sliding and rolling are interconnected and, in particular, that the rolling of a sphere without sliding is impossible.     

Modelling the motion of a disk Rolling on a curve in R: The case where the motion is driven by using two overhead rotors

This work deals with the modelling of the motion of a disk rolling without slipping ona regular curve in R³. The disk's motion is driven by a pedalling torque and by using two overhead rotors.     

The motion of a cylinder-rod system on a horizontal plane: Path controllability and closed-loop control

This work deals with the guidance and control of a system which is composed of a rolling cylinder and a controlled slender rod that is pivoted, through its center of mass, about the cylinder's center. Given a finite time interval [0, t_f], and let P₁ and P₂ be two points in the (X, Y)-plane. The problem dealt with here is to find a closed-loop control law for the cylinder-rod system such that:

• 1.(i) the cylinder center will move from P₁ to P₂ during [0, t_f] and will come to rest at P₂, and
• 2.(ii) the rod will rotate from an angle ψ21 to ψ22 during [0, t_f] and will stop to rotate at t = t_f.

By introducing the concept of path controllability, a closed-loop control law for the solution of the above-posed problem is proposed and its efficiency is demonstrated by solving numerically some examples.     

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.     

On a conditioned Brownian motion and a maximum principle on the disk

Bounds for the 3G-expression∫ ^Ω G(x,z)G(z,y)d,z/G(x,y) play a fundamental role in potential theory. Here,G(x,y) is the Green function for the Laplace problem with zero dirichlet boundary conditions on Ω. The 3G-formula equals , the expected lifetime for a Brownian motion starting in that is killed on exiting ω and conditioned to converge to and to be stopped at . Although it was shown by probabilistic methods for bounded (simply connected) 2d-domains that ifx ε δΩ, then the supremum ofy \at E _x ^y is assumed for somey at the boundary, the analogous question remained open forx in the interior. Here we are able to give an answer in the case thatB ⊂ ℝ is the unit disk. The dependence of this quantity on the positions ofx andy is investigated, and it is shown that indeed E _x ^y (\gt_\om) is maximized on by opposite boundary points. The result also gives an answer to a number of questions related to the best constant for the positivity-preserving property of some elliptic systems. In particular, it confirms a, relationE _x ^y (\gt_\om) with a ‘sum of inverse eigenvalues’ that was conjectured recently by Kawohl and Sweers.     

Covering the plane with copies of a convex disk

We prove that for every convex disk in the plane there exists a double-lattice covering of the plane with copies of with density ≤ 1.2281772 . This improves the best previously known upper bound ≤ 8/(3+2\sqrt{3}) 1.2376043 , due to Kuperberg, but it is still far from the conjectured value . <lsiheader> <onlinepub>7 August, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>20n2p251.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>no <sectionname> </lsiheader> Received June 1, 1996, and in revised form January 24, 1997.     

The steady rolling of a disc on a rough plane

The well-known problem of the rolling without slipping of a heavy circular disc along a horizontal plane is considered. The steady motions of a disc for which the angle between the plane of the disc and the supporting plane (the angle of nutation) is constant are investigated. The problem of the range of variation of the angle of nutation within which the given motions are stable, irrespective of the values of the constants of the two linear first integrals or, in other variables, irrespective of the angular velocities of precession and proper rotation, is investigated. It is shown that this range is wider than was established earlier in [1].     

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.     

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.     

Diffraction of a plane wave by a circular disk

In this paper we solve the problem of diffraction of a normally incident plane wave by a circular disk. We treat both the hard and soft disk. In each case we obtain the solution as a series which converges when the product of the wave number and the radius of the disk is large. Our construction leads directly to asymptotic approximations to the solution for large wave number.