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.     

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.     

Asymptotics of Maxwell Time in the Plate-Ball Problem

The problem on rolling of a sphere on a plane without slipping or twisting is considered. One should roll the sphere from one contact configuration to another so that the length of the curve traced by the contact point in the plane is the shortest possible. The asymptotics of Maxwell time for rolling of the sphere along small amplitude sinusoids is studied. A two-sided estimate for this asymptotics is obtained.     

Isoperimetric inequalities and the gap between the first and second eigenvalues of an Euclidean domain

In this paper we introduce a weighted Cheeger constant and show that the gap between the first two eigenvalues of a Riemannian manifold given Dirichlet conditions can be bounded from below in terms of this constant. When the Riemannian manifold is a bounded Euclidean domain satisfying an interior rolling sphere condition we give an estimate on the weighted Cheeger constant in terms of the rolling sphere radius, volume, a bound on the principal curvatures of the boundary and the dimension. This yields a lower bound on the nontrivial gap for Euclidean domains. S-Y. Cheng’s research partially supported by the CUHK direct grant A/C # 220600260. K. Oden’s research partially supported by the Department of Education Graduate Fellowship     

Rolling of a non-homogeneous ball over a sphere without slipping and twisting

Consider the problem of rolling a dynamically asymmetric balanced ball (the Chaplygin ball) over a sphere. Suppose that the contact point has zero velocity and the projection of the angular velocity to the normal vector of the sphere equals zero. This model of rolling differs from the classical one. It can be realized, in some approximation, if the ball is rubber coated and the sphere is absolutely rough. Recently, J. Koiller and K. Ehlers pointed out the measure and the Hamiltonian structure for this problem. Using this structure we construct an isomorphism between this problem and the problem of the motion of a point on a sphere in some potential field. The integrable cases are found.      

Dynamics-Based Motion Planning for a Pendulum-Actuated Spherical Rolling Robot

This paper deals with the dynamics and motion planning for a spherical rolling robot with a pendulum actuated by two motors. First, kinematic and dynamic models for the rolling robot are introduced. In general, not all feasible kinematic trajectories of the rolling carrier are dynamically realizable. A notable exception is when the contact trajectories on the sphere and on the plane are geodesic lines. Based on this consideration, a motion planning strategy for complete reconfiguration of the rolling robot is proposed. The strategy consists of two trivial movements and a nontrivial maneuver that is based on tracing multiple spherical triangles. To compute the sizes and the number of triangles, a reachability diagram is constructed. To define the control torques realizing the rest-to-rest motion along the geodesic lines, a geometric phase-based approach has been employed and tested under simulation. Compared with the minimum effort optimal control, the proposed technique is less computationally expensive while providing similar system performance, and thus it is more suitable for real-time applications.     

Chaplygin ball over a fixed sphere: an explicit integration

We consider a nonholonomic system describing the rolling of a dynamically nonsymmetric sphere over a fixed sphere without slipping. The system generalizes the classical nonholonomic Chaplygin sphere problem and it is shown to be integrable for one special ratio of radii of the spheres. After a time reparameterization the system becomes a Hamiltonian one and admits a separation of variables and reduction to Abel-Jacobi quadratures. The separating variables that we found appear to be a non-trivial generalization of ellipsoidal (spheroconic) coordinates on the Poisson sphere, which can be useful in other integrable problems. Using the quadratures we also perform an explicit integration of the problem in theta-functions of the new time.      

Optimal Recovery of Functions on the Sphere on a Sobolev Spaces with a Gaussian Measure in the Average Case Setting

In this paper, we study optimal recovery(reconstruction) of functions on the sphere in the average case setting. We obtain the asymptotic orders of average sampling numbers of a Sobolev space on the sphere with a Gaussian measure in the Ld-1q(S) metric for 1 ≤ q ≤∞, and show that some worst-case asymptotically optimal algorithms are also asymptotically optimal in the average case setting in the Ldq(S-1)metric for 1 ≤ q ≤∞.     

Modelling and simulation of a sphere rolling on the inside of a rough vertical cylinder

A classical problem of nonholonomic system dynamics—the motion of a sphere on the inside of a rough vertical cylinder—is extended to rolling friction. The case study is modelled in independent coordinates. Due to the nonholonomic constraints imposed on the sphere, the governing equations arise as a set of differential-algebraic equations. The results of numerical simulations show the transition of the sphere from a sinusoid path on the vertical cylinder surface to a fall with slip. The physics of the ‘paradoxical’ motion is explained in detail.     

Computing the minimum distance between two Bézier curves

A sweeping sphere clipping method is presented for computing the minimum distance between two Bézier curves. The sweeping sphere is constructed by rolling a sphere with its center point along a curve. The initial radius of the sweeping sphere can be set as the minimum distance between an end point and the other curve. The nearest point on a curve must be contained in the sweeping sphere along the other curve, and all of the parts outside the sweeping sphere can be eliminated. A simple sufficient condition when the nearest point is one of the two end points of a curve is provided, which turns the curve/curve case into a point/curve case and leads to higher efficiency. Examples are shown to illustrate efficiency and robustness of the new method.     

The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere

In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of various rigid bodies on a plane and a sphere. It is shown that a hierarchy of possible types of dynamical behavior arises depending on the body’s surface geometry and mass distribution. New integrable cases and cases of existence of an invariant measure are found. In addition, these systems are used to illustrate that the existence of several nontrivial involutions in reversible dissipative systems leads to quasi-Hamiltonian behavior.     

Approximate solution of a control problem on the basis of the nilpotent approximation

We consider the control problem for deterministic systems described by ordinary differential equations with linear controls. On the basis of the nilpotent approximation method, we construct an algorithm for finding an approximate solution of the control problem for three-dimensional nonlinear systems with two linear controls. The algorithm was implemented in Maple and tested in examples including the control of a mobile robot on a plane and the attitude control of a sphere rolling on a plane.     

A Solution to the Weber Location Problem on the Sphere

In this paper we present an algorithm for the solution of the minisum location problem on the sphere. The algorithm is finite for the solution within a given accuracy of the optimal solution.     

A variational crime on the tangent bundle of the sphere

Summary. We study here the finite element approximation of the vector Laplace-Beltrami Equation on the sphere . Because of the lack of a smooth parametrization of the whole sphere (the so-called “poles problem”), we construct a finite element basis using two different coordinate systems, thus avoiding the introduction of artificial poles. One of the difficulties when discretizing the Laplace operator on the sphere, is then to recover the optimal order error. This is achieved here by a suitable perturbation of the vector field basis, locally, near the matching region of the coordinate systems. Received July 16, 1998 / Published online December 6, 1999     

A Sharp Inequality on the Exponentiation of Functions on the Sphere

In this paper we show a new inequality that generalizes to the unit sphere the Lebedev-Milin inequality of the exponentiation of functions on the unit circle. It may also be regarded as the counterpart on the sphere of the second inequality in the Szegö limit theorem on the Toeplitz determinants on the circle. On the other hand, this inequality is also a variant of several classical inequalities of Moser-Trudinger type on the sphere. The inequality incorporates the deviation of the center of mass from the origin into the optimal inequality of Aubin for functions with mass centered at the origin, and improves Onofri's inequality with the contribution of the shifting of the mass center explicitly expressed. © 2021 Wiley Periodicals LLC.     

The dynamical analysis of a uniparametric family of three-point optimal eighth-order multiple-root finders under the Möbius conjugacy map on the Riemann sphere

Numerical Algorithms - In this paper, we present dynamical viewpoints under the Möbius conjugacy map on the Riemann sphere for a uniparametric family of optimal eighth-order multiple-root...     

Closed-Loop Control of the Motion of a Sphere Rolling on a Moving Horizontal Plane

A uniform sphere is rolling without slipping on a horizontal plane. The motion of the sphere is controlled via the control of the acceleration of the plane. At the time t=0, the sphere and the plane are stationary and the center of the sphere is located at a point A in the plane. Given a time interval [0, t _f], the problem dealt with here is: Find a closed-loop strategy for the acceleration of the moving plane such that, at the time t=t _f, the plane and the sphere will be nearly at rest and the center of the sphere will be in a given neighborhood of the origin. By introducing the concept of path controllability, a closed-loop strategy for the solution of the above-mentioned problem is proposed and its efficiency is demonstrated by solving numerically some examples.     

A formula for the reactions of ideal constraints

A formula for determining the reactions of Lagrange-ideal constraints when using arbitrary linear quasi-velocities is obtained. An example of rolling of a sphere along a pair of skew lines is considered with different assumptions regarding whether stick or slip occurs at the points of contact.     

The model of dry friction in the problem of the rolling of rigid bodies

The Contensou model of combined dry friction [1] is considered. The problem of integrating the shear stresses over the contact area is solved in terms of elementary functions, unlike the solution in [1], reduced to elliptic quadratures. The problem of the rolling of a homogeneous sphere over a plane with dry friction is investigated.     

The distribution of points on the sphere and corresponding cubature formulae

In applications, for instance in optics and astrophysics, thereis a need for high-accuracy integration formulae for functionson the sphere. To construct better formulae than previouslyused, almost equidistantly spaced nodes on the sphere and weightsbelonging to these nodes are required. This problem is closelyrelated to an optimal dispersion problem on the sphere and tothe theories of spherical designs and multivariate Gauss quadratureformulae. We propose a two-stage algorithm to compute optimal point locationson the unit sphere and an appropriate algorithm to calculatethe corresponding weights of the cubature formulae. Points aswell as weights are computed to high accuracy. These algorithmscan be extended to other integration problems. Numerical examplesshow that the constructed formulae yield impressively smallintegration errors of up to 10^-12.