Time-Optimum Trajectories for Robots With Multiple End-Effectors

Exact Integrations of Polynomials and Symmetric Quadrature Formulas over Arbitrary Polyhedral Grids

In the vertical parking process, the issue of turning in place due to discontinuities in path curvature is addressed by proposing an optimal reference path planning method based on circular arcs, straight lines, and a multi-objective evaluation function. This method first analyzes the obstacle avoidance constraints between the vehicle’s outer contour and the parking space, as well as the vehicle’s kinematic constraints. The feasible driving region’s upper and lower boundaries are determined by tangent circular arcs and straight lines. Subsequently, a multi-objective evaluation function is designed, which integrates path curvature, adjustable margins at any given moment, and path length, to obtain the optimal circular arc and straight line combination within the feasible region. Finally, the path is fitted using a polynomial curve to form the optimal reference path. Simulation results demonstrate that the planned path satisfies both the continuity of path curvature and the vehicle’s kinematic constraints.

This paper presents an effective method to automatically segment and fit the edge contour curve data of a picture into straight line segments, circular arc segments or elliptical arc segments. Firstly, the curvature of each point is calculated, and the curvature change breakpoints are found to distinguish between straight lines, circular arcs and elliptical arcs; then, circular arc segments and elliptical arcs are distinguished by the change degree of curvature of adjacent points, and the automatic segmentation of contour edges is realized. Finally, the circular arc segment and elliptic arc segment are fitted by least squares method, based on which the length of the straight line, the direction angle of the circular arc, the rotation angle of the elliptic arc and other related information are calculated. The results show that our model can better identify the feature texture information of image edges.

This article presents research results of the signal-to-interference ratio dependence (S/I) on the spatial structure of the radiation field of a three-element linear sparse antenna array in communication systems with spatial signal separation. As a result of the research, it was stated that when the receiving linear antenna is located in the region of the zero order maximum of the field of the useful signal, the formation of the S/I ratio does not depend on whether the field distribution is considered on a straight line or on a circular arc. The location of the receiving antenna in the region of the first-order maximum of the field of the useful signal gives the best results in the formation of the S/I ratio if the field distribution is used on circular arc and not on a straight line.

This paper proposes an algorithm for cooperative guidance of an unmanned aerial vehicle (UAV) formation following a set of waypoints. Waypoints are linked by straight lines to form a sequence, which effectively tasks the formation to cooperatively follow a rectilinear path between these points. Unlike in earlier known papers, we propose a Fermat’s spiral rather than a circular arc as a fillet to smooth the trajectory. This is because a Fermat’s spiral has zero curvature at its origin, which helps avoid breaks in curvature observable when using straight lines and circular arcs. To make a UAV formation and to follow the path, we propose a novel guidance vector field that is direction- and magnitude-nonuniform. This is where the novelty of this paper lies, as it proposes the vector field method for following a Fermat’s spiral. Known studies used line-of-sight (LOS) guidance to that end. In that case, the UAV formation shapes itself using consensus-based coordination topology, i.e. the group has no leader. To test the algorithm for cooperative waypoint following when smoothing is provided by fillets, we utilized complete nonlinear models of two fixed-wing UAVs, each equipped with a tuned autopilot. Numerical simulation shows the proposed approach efficient.

A powerful and popular approach for estimating radial lens distortion parameters is to use the fact that lines which are straight in the scene should be imaged as straight lines under the pinhole camera model. This paper revisits this problem using the division model to parameterise the lens distortion. This turns out to have significant advantages over the more conventional parameterisation, especially for a single parameter model. In particular, we demonstrate that the locus of distorted points from a straight line is a circular arc. This allows distortion estimation to be reformulated as circle-fitting for which many algorithms are available. We compare a number of suboptimal methods offering closed-form solutions with an optimal, iterative technique which minimises a cost function on the actual image plane as opposed to existing techniques which suffer from a bias due to the fact that they optimise a geometric cost function on the undistorted image plane.

All conventional methods for calculating the trajectory of a wellbore involve assumptions. Most calculations assume that segments of the trajectory can be approximated as straight lines, polygonal lines, cylindrical helixes, circular arcs, or combination of these approximations. Conventional methods calculate the course coordinates of a well survey interval according to the predetermined shape for the trajectory. But, what does the real shape of a trajectory in each surveyed interval really look like? Why? What parameters play decisive roles in determining the true shape of a surveyed interval? Is it reasonable to assume that every survey interval has the same kind of approximate shape? Until now, the drilling industry has not found satisfactory solutions to these questions. This paper presents the results of research about the true shape of a wellbore trajectory relative to its survey stations. It also provides a universal equation for a wellbore trajectory in a surveyed interval and presents an objective approach for describing and calculating a wellbore trajectory in space. The new method does not assume that the shape of wellbore trajectory is a set of typical approximations. The study shows that the wellbore curvature and torsion at survey stations determine the shape of the wellbore trajectory at subsequent intervals, and that 3D coordinates of wellbore trajectory in a local coordinate system are linear, quadratic, and cubic functions vs. curve length, respectively. The new method yields a continuous wellbore trajectory. Extensive simulations have been carried out and are reviewed in this paper. The paper compares the results predicted by mathematical simulation using the new method with actual, observed trajectories and describes the results from other methods to show the accuracy of the new, improved method. The new method shows excellent precision in calculation and reliability. This study has also proved that the minimum curvature method and the natural curve method, the most commonly used methods, are related to the new method.

Summary An explicit solution to the general 3D point-to-target problem based on the minimum curvature method has been sought for more than four decades. The general case involves the trajectory's start and target points connected by two circular arcs joined by a straight line with the position and direction defined at both ends. It is known that the solutions are multivalued, and efficient iterative schemes to find the principal root have been established. This construction is an essential component of all major trajectory construction packages. However, convergence issues have been reported in cases where the intermediate tangent section is either small or vanishes and rigorous mathematical conditions under which solutions are both possible and are guaranteed to converge have not been published. An implicit expression has now been determined that enables all the roots to be identified and permits either exact or polynomial-type solution methods to be used. Most historical attempts at solving the problem have been purely algebraic, but a geometric interpretation of related problems has been attempted, showing that a single circular arc and a tangent section can be encapsulated in the surface of a horn torus. These ideas have now been extended, revealing that the solution to the general 3D point-to-target problem can be represented as a 10th-order self-intersecting geometric surface, characterized by the trajectory's start and end points, the radii of the two arcs, and the length of the tangent section. An outline of the solution's derivation is provided in the paper together with complete details of the general expression and its various degenerate forms so that readers can implement the algorithms for practical application. Most of the degenerate conditions reduce the order of the governing equation. Full details of the critical and degenerate conditions are also provided, and together these indicate the most convenient solution method for each case. In the presence of a tangent section, the principal root is still most easily obtained using an iterative scheme, but the mathematical constraints are now known. It is also shown that all other cases degenerate to quadratic forms that can be solved using conventional methods. It is shown how the general expression for the general point-to-target problem can be modified to give the known solutions to the 3D landing problem and how the example in the published works on this subject is much simplified by the geometric, rather than algebraic treatment.

This paper presents a method for designing stationary cams with constant speed roller followers. A stationary cam is immovable, but it guides its follower to produce a desired motion with the follower being driven by other links. In order to avoid infinite jerk of the follower motion, the pitch curve of a stationary cam should consist of pieces of transition curve, which provide gradual change in curvature and are interposed between straight lines and circular arcs. The general parametric equations of transition curves are derived and the cam design process is demonstrated. This method is applicable for designing the guideways of tool magazine in automatic tool-changing system of the machine center.

Analytical solution for groundwater recharge on a hill

This column will publish short (from just a few paragraphs to ten or so pages), lively and intriguing computer-related mathematics vignettes. These vignettes or snapshots should illustrate ways in which computer environments have transformed the practice of mathematics or mathematics pedagogy. They could also include puzzles or brain-teasers involving the use of computers or computational theory. Snapshots are subject to peer review. This issue’s snapshot explores some generalizations of the definition of geometric reflection. Dynamic geometry tools can facilitate generalizations such as those obtained by relaxing the requirement that the reflection be through a straight line. The author compares the families of curves obtained by reflecting thru circular arcs with the curves generated in response to a physical problem proposed by Wittgenstein. He suggests that the strategy of generalizing definitions is a good avenue for bringing students quickly to the activity of doing mathematics.

The logarithmic spiral bevel gear is a new type of spiral bevel gear and has received great attention in the industry for its excellent engineering characteristics. The objective of this study is to develop an accurate geometric model for the logarithmic spiral bevel gear. Based on the gear tooth flank formation mechanism, a conical logarithmic spiral curve with a constant spiral angle was constructed as the tooth trace curve. The profiles for the exterior and interior transverse of the tooth were built with an accurate involutes curve, with transition being circular arcs and straight lines. The first tooth model was established by sweeping the tooth profile accurately along the tooth trace curve, and the rest of the teeth were created by an array operation. The accurate three-dimensional model of logarithmic spiral bevel gear was finally obtained by means of a Boolean addition operation among all of the gear teeth and the root cone. An experimental study was carried out for such a gear whose number of teeth was 37, with modules being 4.5 mm, normal pressure angle being 20° and spiral angle being 35°. A DMU 40 monoBLOCK five-axis computer numerical control milling machine tool was used to produce the prototype, and a Zeiss CONTURA G3 three-dimensional coordinate instrument was employed to measure the exterior transverse addendum diameter. The linear error between the theoretical model value and the measured average value was 0.0027 mm, indicating the effectiveness and practicability of this modeling method, which provides an accurate geometric model for design and for subsequent tasks like computer-aided engineering analysis and manufacturing.

We consider the reflection of an ultrashort pulse from a motionless extended object having cylindrical symmetry. The reflected pulse is described using the nonstationary Kirchhoff integral. The parameters of body boundaries shaped as circular arcs, straight lines, parabolas, and splines are reconstructed using neural networks. The obtained results allow one to perform spatial visualization of the object. The limitations and possibilities of the further development of the method are presented.

This paper explores the design of excitation current profiles for a type of three-phase magnetically coupled linear variable reluctance motor. A model relating force production to air gap geometry is derived using magnetic circuit analysis under an assumption that flux flows across the air gap through tubular paths bounded by straight lines and circular arcs. Constrained optimization techniques are used for two purposes: first, to determine the motor's continuous force limit corresponding to some given continuous current limit; second, to determine currents that satisfy the current limits and produce a desired force with minimum power dissipation. The optimization results indicate that a 6-wire connection is preferable to a 3-wire connection, and that wide teeth are preferable to narrow teeth, if the goal is to maximize the range of ripple-free force.

Kurvengetriebe mit umlaufender Doppelkurve zur erzeugung von Bahnkurven mit vorgeschriebener geschwindigkeit