Directional control of the motion of a rolling disk by using a rotor fixed in the disk's plane

This work deals with the stabilization and control of a system which is composed of a disk rolling on a plane, and a circular rotor plate fixed in the disk's plane. The disk's motion is controlled by the above-mentioned rotor, a “tillting moment” and a pedalling moment. It is shown here that by applying a kind of inverse dynamics control, the motion of the disk is stabilized about a given angle, while simultaneously controlling its speed and direction in such a manner that the point of contact between the disk and the horizontal plane will be able, during a given time interval [0, t_f], to move from a given point. r_A to a another given point r_B, both of them fixed in the plane.     

Stabilization and control of the motion of a rolling disk by using two overhead rotors

This work deals with the stabilization and control of a system which is composed of a disk rolling on a plane, a controlled slender rod that is pivoted through its center of mass about the disk's center and two overhead rotors with their axes fixed in the upper part of the rod (see Figures 1 and 2). The rod is controlled in such a manner that it is always aligned along the line passing through the points O and C, where O denotes the center of the disk and C denotes the point of contact between the disk and the plane. The upper rotor rotates in a plane that passes through the disk's axis and the rod, whereas the lower rotor rotates in a plane that is perpendicular to the rod (see Figures 1 and 2). By using a kind of inverse dynamics control a control strategy is proposed under which the disk's inclination is stabilized about its vertical position and the disk's motion is able asymptotically to track any given smooth ground trajectory.     

Efficient Algorithms for the Riemann-Roch Problem and for Addition in the Jacobian of a Curve

We study the effective Riemann-Roch problem of computing a basis for the linear space L(D) associated with a divisor D on a plane curve C and its applications. Our presentation begins with an extension of Noether's algorithm for this problem to plane curves with singularities. Like the original, this algorithm has a worst-case complexity of Ω(n!^|D|), where n is the degree of the curve C.We next consider representations of divisors based on Chow forms. Using these, we produce a factorization-free polynomial-time algorithm for the effective Riemann-Roch problem, which improves the complexity of Noether's algorithm by an order of magnitude. We also present further improvements which, for curves of bounded degree, yield an algorithm with complexity which is linear in the size of the given divisor. Finally, we consider applications of this problem: to the arithmetic of the Jacobian of a plane curve, to the construction of rational functions on a curve with prescribed zeroes and poles, and to the construction of curves with prescribed intersections.     

Solution to a problem of guarding territory

There are two players on the plane: an invader I with maximal speed v > 1 and a defender D with maximal speed 1. It is shown that the disk of radius rv(v − 1)⁻¹ (where r denotes the ‘destruction radius’ of the defender) is the domain of largest area which can be guarded by the defender.     

Exact Minkowski Products of N Complex Disks

An exact parameterization for the boundary of the Minkowski product of N circular disks in the complex plane is derived. When N > 2, this boundary curve may be regarded as a generalization of the Cartesian oval that bounds the Minkowski product of two disks. The derivation is based on choosing a system of coordinated polar representations for the N operands, identifying sets of corresponding points with matched logarithmic Gauss map that may contribute to the Minkowski product boundary. By means of inversion in the operand circles, a geometrical characterization for their corresponding points is derived, in terms of intersections with the circles of a special coaxal system. The resulting parameterization is expressed as a product of N terms, each involving the radius of one disk, a single square root, and the sine and cosine of a common angular variable over a prescribed domain. As a special case, the N-th Minkowski power of a single disk is bounded by a higher trochoid. In certain applications, the availability of exact Minkowski products is a useful alternative to the naive bounding approximations that are customarily employed in "complex circular arithmetic."     

What can two images tell us about a third one?

This paper discusses the problem of predicting image features in an image from image features in two other images and the epipolar geometry between the three images. We adopt the most general camera model of perspective projection and show that a point can be predicted in the third image as a bilinear function of its images in the first two cameras, that the tangents to three corresponding curves are related by a trilinear function, and that the curvature of a curve in the third image is a linear function of the curvatures at the corresponding points in the other two images. Our analysis relies heavily on the use of the fundamental matrix which has been recently introduced (Faugeras et al, 1992) and on the properties of a special plane which we call the trifocal plane. Though the trinocular geometry of points and lines has been very recently addressed, our use of the differential properties of curves for prediction is unique.We thus completely solve the following problem: given two views of an object, predict what a third view would look like. The problem and its solution bear upon several areas of computer vision, stereo, motion analysis, and model-based object recognition. Our answer is quite general since it assumes the general perspective projection model for image formation and requires only the knowledge of the epipolar geometry for the triple of views. We show that in the special case of orthographic projection our results for points reduce to those of Ullman and Basri (Ullman and Basri, 1991). We demonstrate on synthetic as well as on real data the applicability of our theory.     

The Geometry of D-Decomposition of a Two-Dimensional Plane of Arbitrary Coefficients of the Characteristic Polynomial of a Discrete System

The D-decomposition of a two-dimensional plane of arbitrary coefficients of a discrete characteristic polynomial is studied. The geometry of the boundary curve is analyzed and its common properties with a linear discrete system of any order are described. Example are given.     

Motion of a disk in contact with a parametric 2D curve and Painlevé’s paradox

Multibody System Dynamics - This paper addresses the modeling and simulation of a homogeneous disk that undergoes plane motion constrained to be in permanent contact with and in the same plane as a...     

Perturbation Analysis and Simulation Study of the Effects of Phase on the Classical Hydrogen Atom Interacting with Circularly Polarized Electromagnetic Radiation

The classical hydrogen atom is examined for the situation where a circularly polarized electromagnetic plane wave acts on a classical charged point particle in a near-circular orbit about an infinitely massive nucleus, with the plane wave normally incident to the plane of the orbit. The effect of the phase of the polarized wave in relation to the velocity vector of the classical electron is examined in detail by carrying out a perturbation analysis and then comparing results using simulation methods. By expanding the variational parts of the radius and angular velocity about their average values, simpler nonlinear differential equations of motion are obtained that still retain the key features of the oscillating amplitude, namely, the gradual increase of the envelope of the oscillating amplitude and the point of rapid orbital decay. Also, as shown here, these key features carry over nicely to conventional quantities of interest such as energy and angular momentum. The phase is shown here to have both subtle yet very significant effects on the quasistability of the orbital motion. A far wider range of phase conditions are found to provide stability than might intuitively be expected, with the time to orbital decay, t _d , varying by orders of magnitude for any plane wave with an amplitude A above a critical value, A _c .     

Simulation of a pad slider flying on a wavy surface

This paper describes following-up characteristic of a pad slider to a wavy surface. The pad slider comprises a leading pad and a trailing pad. We use a sine wave to replace an actual wavy surface. The response of the pad slider due to the sine wave is analyzed by computer simulation. The response of the pad slider is evaluated from variation in slider flying height (FH). A variation gain that is a ratio of FH fluctuation and wave amplitude is defined to explain the following-up characteristic of slider. To analyze the following-up characteristic, different wavelengths are used as parameter to calculate the variation gain. Because pads on the air-bearing surface of the slider are able to occur at two pressure peaks on the leading edge and trailing edge that enable a pitching motion of the slider, the slider can follow the disk waviness with a wavelength that is shorter than the slider length. The variation gain is 0.33 for the wavelength that is equal to the slider length. However, the slider cannot follow the disk waviness with a wavelength that is shorter than the pad length. The variation gain is greater than 1 for the wavelength 0.2 mm. To study the relationship between the variation gain and the dynamics of slider, a transient response simulation is carried out in order to investigate a natural frequency of slider. We set a projection on a plane surface. When the slider is flying over the projection, we can obtain a flying height response curve and a pitch response curve. The natural frequency of flying height and pitch angle can be known by FFT. The transient response of slider in pitch mode is compared with the variation gain. The simulation results make clear the fact that the following-up characteristic has correlation with the dynamics of the pitch model and the phase difference between a locus of head and a wave of disk surface.     

A dual‐band microstrip antenna using a circular ring and a concentric disk

A dual‐band microstrip antenna using a differentially driven annular ring and a single‐ended concentric circular disk is reported. The ring is designed in the TM₁₂ mode at a lower frequency f_l and the disk in the TM₁₁ mode at a higher frequency f_h. It is found that the mutual loading perturbs the resonant frequencies and thus the outer and inner radii of the ring and the radius of the disk calculated from the cavity models need to be slightly adjusted. More importantly, it is also found that the mutual loading affects the gains and may reduce the gain of the disk antenna substantially. The cavity‐model design formulas to determine the ring and disk dimensions are modified to account for their mutual loading between the ring and disk. It is shown that the differentially driven ring has an improved current distribution, a better impedance matching, and an enhanced impedance bandwidth. Moreover, a higher gain is obtained for the disk. A design sample for f_l = 2.45 GHz and f_h = 5.3 GHz on an FR4 substrate was fabricated. Simulated and measured results agree satisfactorily. © 2016 Wiley Periodicals, Inc. Int J RF and Microwave CAE 26:268–276, 2016.     

Linear-time algorithms for problems on planar graphs with fixed disk dimension

The disk dimension of a planar graph G is the least number k for which G embeds in the plane minus k open disks, with every vertex on the boundary of some disk. Useful properties of graphs with a given disk dimension are derived, leading to an algorithm to obtain an outerplanar subgraph of a graph with disk dimension k by removing at most 2k−2 vertices. This reduction is used to obtain linear-time exact and approximation algorithms on graphs with fixed disk dimension. In particular, a linear-time approximation algorithm is presented for the pathwidth problem.     

3D Video Recorder: a System for Recording and Playing Free‐Viewpoint Video†

We present the 3D Video Recorder, a system capable of recording, processing, and playing three‐dimensional video from multiple points of view. We first record 2D video streams from several synchronized digital video cameras and store pre‐processed images to disk. An off‐line processing stage converts these images into a time‐varying 3D hierarchical point‐based data structure and stores this 3D video to disk. We show how we can trade‐off 3D video quality with processing performance and devise efficient compression and coding schemes for our novel 3D video representation. A typical sequence is encoded at less than 7 Mbps at a frame rate of 8.5 frames per second. The 3D video player decodes and renders 3D videos from hard‐disk in real‐time, providing interaction features known from common video cassette recorders, like variable‐speed forward and reverse, and slow motion. 3D video playback can be enhanced with novel 3D video effects such as freeze‐and‐rotate and arbitrary scaling. The player builds upon point‐based rendering techniques and is thus capable of rendering high‐quality images in real‐time. Finally, we demonstrate the 3D Video Recorder on multiple real‐life video sequences. ACM CSS: I.3.2 Computer Graphics—Graphics Systems, I.3.5 Computer Graphics—Computational Geometry and Object Modelling, I.3.7 Computer Graphics—Three‐Dimensional Graphics and Realism     

Rolling on a smooth biparametric surface

In this paper we present a method to calculate the rolling of a rigid convex object on a smooth biparametric surface where a single contact point is maintained during the animation. The object's motion is computed using a prediction-correction schema. The prediction computes the motion of the object rolling on the tangent plane at the current contact point. The next contact point is obtained by projecting the predicted point onto the curved surface. A correction is made according to the local surface curvature. An algorithm which calculates the initial contact point between the surface and the object is given. The rolling of a ball is presented as an example.     

The μ-basis of a planar rational curve—properties and computation

A moving line L(x,y;t)=0 is a family of lines with one parameter t in a plane. A moving line L(x,y;t)=0 is said to follow a rational curve P(t) if the point P(t₀) is on the line L(x,y;t₀)=0 for any parameter value t₀. A μ-basis of a rational curve P(t) is a pair of lowest degree moving lines that constitute a basis of the module formed by all the moving lines following P(t), which is the syzygy module of P(t). The study of moving lines, especially the μ-basis, has recently led to an efficient method, called the moving line method, for computing the implicit equation of a rational curve [3 and 6]. In this paper, we present properties and equivalent definitions of a μ-basis of a planar rational curve. Several of these properties and definitions are new, and they help to clarify an earlier definition of the μ-basis [3]. Furthermore, based on some of these newly established properties, an efficient algorithm is presented to compute a μ-basis of a planar rational curve. This algorithm applies vector elimination to the moving line module of P(t), and has O(n²) time complexity, where n is the degree of P(t). We show that the new algorithm is more efficient than the fastest previous algorithm [7].     

An efficient algorithm for one-step planar compliant motion planning with uncertainty

Uncertainty in the execution of robot motion plans must be accounted for in the geometric computations from which plans are obtained, especially in the case where position sensing is inaccurate. We give anO(n ² logn) algorithm to find a single commanded motion direction that will guarantee a successful motion in the plane from a specified start to a specified goal whenever such a one-step motion is possible. The plans account for uncertainty in the start position and in robot control, and anticipate that the robot may stick on or slide along obstacle surfaces with which it comes in contact. This bound improves on the best previous bound by a quadratic factor, and is achieved in part by a new analysis of the geometric complexity of the backprojection of the goal as a function of commanded motion direction.A preliminary version of this paper appeared in theProceedings of the ACM Symposium on Computational Geometry, Saarbrücken, June 1989. This paper describes research done in the Computer Science Robotics Laboratory at Cornell University. Support for our robotics research is provided in part by the National Science Foundation under Grant IRI-8802390 and by a Presidential Young Investigator award, and in part by the Mathematical Sciences Institute.     

Radical parametrizations of algebraic curves by adjoint curves

We present algorithms for parametrizing by radicals an irreducible curve, not necessarily plane, when the genus is less than or equal to 4 and the curve is defined over an algebraically closed field of characteristic zero. In addition, we also present an algorithm for parametrizing by radicals any irreducible plane curve of degree d having at least a point of multiplicity d−r, with 1≤r≤4 and, as a consequence, every irreducible plane curve of degree d≤5 and every irreducible singular plane curve of degree 6.     

Point-to-point and collision avoidance control of the motion of an autonomous bicycle

This work deals with the stabilization and collision avoidance control of a riderless bicycle (see Figure 1). It is assumed here that the bicycle is controlled by a pedalling torque, a directional torque, and by a rotor mounted on the crossbar that generates a tilting torque.Given two points r_A, r_B, and a circular obstacle in the horizontal plane. Also, given a time interval [0, t_f], where 0 < t_f < ∞. It is shown here that by applying a kind of inverse dynamics control, the motion of the bicycle is stabilized while simultaneously controlling its speed and direction in such a manner that the point of contact between the bicycle's rear wheel and the horizontal plane will be able, during [0, t_f], to move from r_A to r_B without hitting the obstacle.     

Shortest paths for line segments

We study the problem of shortest paths for a line segment in the plane. As a measure of the distance traversed by a path, we take the average curve length of the orbits of prescribed points on the line segment. This problem is nontrivial even in free space (i.e., in the absence of obstacles). We characterize all shortest paths of the line segment moving in free space under the measured ₂, the average orbit length of the two endpoints.The problem ofd ₂ optimal motion has been solved by Gurevich and also by Dubovitskij, who calls it Ulam's problem. Unlike previous solutions, our basic tool is Cauchy's surface-area formula. This new approach is relatively elementary, and yields new insights.This work was partially supported by the ESPRIT II Basic Research Actions Program of the EC under Contract No. 3075 (project ALCOM) and by the Deutsche Forschungsgemeinschaft Grant Ot 64/5-3. Chee Yap acknowledges support from the Deutsche Forschungsgemeinschaft, and partial support from NSF Grants DCR-8401898 and DCR-8401633.     

Position control of a flexible gantry robot arm using smart material actuators

This article presents new feedback actuators that achieve accurate position control of a flexible gantry robot arm. Translational motion in the plane is generated by two dc motors and controlled by applying electric fields to electro‐rheological (ER) clutch actuators. On the other hand, during control action of translational motion, a flexible arm attached to the moving part produces undesirable oscillations due to its inherent flexibility. Oscillations are actively suppressed by employing feedback voltage to the piezoceramic actuator attached to the surface of the flexible arm. Consequently, an accurate position control at the end‐point of the flexible arm can be achieved. To accomplish this control goal, governing equations of the proposed system are derived and written as transfer functions. Transfer functions are used in design of a set of robust H_∞ controllers. Electric fields to be applied to ER clutch and control voltage for the piezoceramic actuator are determined via H_∞ methodology which is incorporated with classical loop shaping design technique. To evaluate effectiveness of the proposed control system, experiments for both regulating and tracking controls are undertaken. ©1999 John Wiley & Sons, Inc.