N-dimensional Box Research Articles

Determination of Workspaces and Intersections of Robot Links in a Multi-Robotic System for Trajectory Planning

One of the problems in the development of multi-robotic systems is the safe navigation of a group of robots. To solve it, the restrictions imposed by the structural elements of its agents are determined. The article presents a multi-robotic system consisting of parallel and serial robots installed on mobile platforms. The parallel robot is made based on a tripod with the ability to rotate the robot’s base relative to the horizontal axis. The analysis of its working and technological area is carried out, taking into account singularity zones. The developed algorithms for determining the workspaces are based on deterministic methods for approximating the set of solutions to systems of nonlinear inequalities. In this case, restrictions in spaces of different coordinates are presented in the form of n-dimensional boxes. Approaches to solving two problems are proposed to determine the possible intersection of links for the collaborative performance of tasks by a multi-robotic system. The first task is to determine the intersection of the links for the given positions and the relative position of the manipulators. The second is in determining the minimum distance between the technological areas of manipulators, which consist of the workspace and all possible positions of the intermediate links.

Numerical simulation of the workspace of robots with moving bases in the multi-agent system

Multi-robotic systems are used to solve many practical tasks in wide areas, including military surveillance, reconnaissance, rescue operations, warehousing, etc. Such systems can be represented as groups of intelligent robots capable of interacting with each other and jointly implementing various tasks. One of the problems in the development of such systems is the safe navigation of a group of robots. The restrictions imposed by the structural elements of its agents are determined to solve it. The article presents the structure of a multi-robotic system consisting of parallel and serial robots installed on mobile platforms. The parallel robot is made based on a tripod with the possibility of rotating the robot base about the horizontal axis. Determining the workspace boundaries of robots eliminates collisions between agents and external objects. The developed algorithms are based on deterministic methods for approximating the solutions set of nonlinear inequalities systems with restrictions in spaces of coordinates in the form of n-dimensional boxes. The relative positions of mobile platforms with manipulators, at which collisions are possible, are determined.

Localization of real algebraic hypersurfaces with applications to the enumeration of the classes of relative equilibria of a (5 + 1)-body problem

We extend classical results on the localization of zeros of real univariate polynomials to the localization of zero sets of real multivariate polynomials P, more precisely, of real algebraic hypersurfaces (assuming 0 is a regular value). Through suitable changes of variables, we may verify whether such a hypersurface P−1(0) in Rn intersects or not a given n-dimensional box Bn=Πl=1n[al,bl], and in the affirmative case, to locate with arbitrary precision the set P−1(0)∩Bn. Properties of the hypersurface such as being an analytic graph may also be deduced from our results, which include a non-differentiable, non-local version of the implicit function theorem for polynomials. Next, we apply the ideas of the first part to study the bifurcations of a one-parameter family of symmetric classes of relative equilibria of the (5+1)-body problem. The exact numbers of classes of relative equilibria are provided, and our technique allows for the localization of all relative equilibria.

An Efficient Random Algorithm for Box Constrained Weighted Maximin Dispersion Problem

The box-constrained weighted maximin dispersion problem is to find a point in an n-dimensional box such that the minimum of the weighted Euclidean distance from given m points is maximized. In this paper, we first reformulate the maximin dispersion problem as a non-convex quadratically constrained quadratic programming (QCQP) problem. We adopt the successive convex approximation (SCA) algorithm to solve the problem. Numerical results show that the proposed algorithm is efficient.

On the free path length distribution for linear motion in an n-dimensional box

We consider the distribution of free path lengths, or the distance between consecutive bounces of random particles, in an n-dimensional rectangular box. If each particle travels a distance R, then, as the free path length coincides with the distribution of the length of the intersection of a random line with the box (for a natural ensemble of random lines) and we give an explicit formula (piecewise real analytic) for the probability density function in dimension two and three.In dimension two we also consider a closely related model where each particle is allowed to bounce N times, as , and give an explicit (again piecewise real analytic) formula for its probability density function.Further, in both models we can recover the side lengths of the box from the location of the discontinuities of the probability density functions.

Inequalities for the fundamental Robin eigenvalue for the Laplacian on N-dimensional rectangular parallelepipeds

This document consists of two papers, both submitted, and supplementary material. The submitted papers are here given as Parts I and II. Part I establishes results, used in Part II, 'on functions and inverses, both positive, decreasing and convex'. Part II uses results from Part I to extablish 'inequalities for the fundamental Robin eigenvalue for the Laplacian on N-dimensional boxes'

Q-Intersection Algorithms for Constraint-Based Robust Parameter Estimation

Given a set of axis-parallel n-dimensional boxes, the q-intersection is defined as the smallest box encompassing all the points that belong to at least q boxes. Computing the q-intersection is a combinatorial problem that allows us to handle robust parameter estimation with a numerical constraint programming approach. The q-intersection can be viewed as a filtering operator for soft constraints that model measurements subject to outliers. This paper highlights the equivalence of this operator with the search of q-cliques in a graph whose boxicity is bounded by the number of variables in the constraint network. We present a computational study of the q-intersection. We also propose a fast heuristic and a sophisticated exact q-intersection algorithm. First experiments show that our exact algorithm outperforms the existing one while our heuristic performs an efficient filtering on hard problems.

Mesh-independent a priori bounds for nonlinear elliptic finite difference boundary value problems

In this paper we prove mesh independent a priori L∞-bounds for positive solutions of the finite difference boundary value problem−Δhu=f(x,u)in Ωh,u=0on ∂Ωh, where Δh is the finite difference Laplacian and Ωh is a discretized n-dimensional box. On the one hand this completes a result of [9] on the asymptotic symmetry of solutions of finite difference boundary value problems. On the other hand it is a finite difference version of a critical exponent problem studied in [10]. Two main results are given: one for dimension n=1 and one for the higher dimensional case n≥2. The methods of proof differ substantially in these two cases. In the 1-dimensional case our method resembles ode-techniques. In the higher dimensional case the growth rate of the nonlinearity has to be bounded by an exponent p<nn−1 where we believe that nn−1 plays the role of a critical exponent. Our method in this case is based on the use of the discrete Hardy–Sobolev inequality as in [3] and on Moser's iteration method. We point out that our a priori bounds are (in principal) explicit.

Conservation laws of generalized billiards that are polynomial in momenta

This paper deals with dynamics particles moving on a Euclidean n-dimensional torus or in an n-dimensional parallelepiped box in a force field whose potential is proportional to the characteristic function of the region D with a regular boundary. After reaching this region, the trajectory of the particle is refracted according to the law which resembles the Snell -Descartes law from geometrical optics. When the energies are small, the particle does not reach the region D and elastically bounces off its boundary. In this case, we obtain a dynamical system of billiard type (which was intensely studied with respect to strictly convex regions). In addition, the paper discusses the problem of the existence of nontrivial first integrals that are polynomials in momenta with summable coefficients and are functionally independent with the energy integral. Conditions for the geometry of the boundary of the region D under which the problem does not admit nontrivial polynomial first integrals are found. Examples of nonconvex regions are given; for these regions the corresponding dynamical system is obviously nonergodic for fixed energy values (including small ones), however, it does not admit polynomial conservation laws independent of the energy integral.

Tilings With $n$-Dimensional Chairs and Their Applications to Asymmetric Codes

An <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional chair consists of an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional box from which a smaller <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional box is removed. A tiling of an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional chair has two nice applications in some memories using asymmetric codes. The first one is in the design of codes that correct asymmetric errors with limited magnitude. The second one is in the design of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> cells <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -ary write-once memory codes. We show an equivalence between the design of a tiling with an integer lattice and the design of a tiling from a generalization of splitting (or of Sidon sequences). A tiling of an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional chair can define a perfect code for correcting asymmetric errors with limited magnitude. We present constructions for such tilings and prove cases where perfect codes for these type of errors do not exist.

Analysis of nonlinear electrical circuits using bernstein polynomials

In electrical circuit analysis, it is often necessary to find the set of all direct current (d.c.) operating points (either voltages or currents) of nonlinear circuits. In general, these nonlinear equations are often represented as polynomial systems. In this paper, we address the problem of finding the solutions of nonlinear electrical circuits, which are modeled as systems of n polynomial equations contained in an n-dimensional box. Branch and Bound algorithms based on interval methods can give guaranteed enclosures for the solution. However, because of repeated evaluations of the function values, these methods tend to become slower. Branch and Bound algorithm based on Bernstein coefficients can be used to solve the systems of polynomial equations. This avoids the repeated evaluation of function values, but maintains more or less the same number of iterations as that of interval branch and bound methods. We propose an algorithm for obtaining the solution of polynomial systems, which includes a pruning step using Bernstein Krawczyk operator and a Bernstein Coefficient Contraction algorithm to obtain Bernstein coefficients of the new domain. We solved three circuit analysis problems using our proposed algorithm. We compared the performance of our proposed algorithm with INTLAB based solver and found that our proposed algorithm is more efficient and fast.

A solution to one of Knuth's permutation problems

We answer a problem posed recently by Knuth: an n-dimensional box, with edges lying on the positive coordinate axes and generic edge lengths W 1 < W 2 < ⋯ < W n , is dissected into n! pieces along the planes x i = x j . We describe which pieces have the same volume, and show that there are C n distinct volumes, where C n denotes the nth Catalan number.

Higher-Dimensional Box Integrals

Herein, with the aid of substantial symbolic computation, we solve previously open problems in the theory of n-dimensional box integrals Bn (s) := ∊ [0, 1] n . In particular, we resolve an elusive integral called K 5 that previously acted as a “blockade” against closed-form evaluation in n = 5 dimensions. In consequence, we now know that Bn (integer) can be given a closed form for n = 1, 2, 3, 4, 5. We also find the general residue at the pole at s = −n, this leading to new relations and definite integrals; for example, we are able to give the first nontrivial closed forms for six-dimensional box integrals and to show hyperclosure of B 6 (even). The Clausen function and its generalizations play a central role in these higher-dimensional evaluations. Our results provide stringent test scenarios for symbolic-algebra simplification methods.

Solution trajectories of the harmonic-elimination problem

The paper considers the problem of finding the notch angles which are required for implementing programmed-harmonic-elimination switching in pulse-width-modulated (PWM) invertors. For a desired fundamental output voltage, the proposed methodology employs the Krawczyk–Moore test within a linear-programming-based search to determine, with mathematical certainty, a set of N-dimensional boxes enclosing all solutions of the harmonic-elimination problem. Each of these boxes contains a distinct and unique solution, and is a safe starting region for a Newton iterative method applied to solve the transcendental harmonic-elimination equations. Once all solutions for a given fundamental amplitude are determined, the solution trajectories for all feasible amplitudes are traced using a simple continuation method which is essentially a numerical adaptation of the implicit-function theorem. The proposed technique is used for optimising line-to-neutral quarter-wave symmetric PWM waveforms employed in three-phase applications. Switching-angle spreads from 0° to 60° and 0° to 90° are considered. Solution trajectories are computed for up to eight switching angles.

Accelerated Solution of Multivariate Polynomial Systems of Equations

We propose new Las Vegas randomized algorithms for the solution of a square nondegenerate system of equations, with well-separated roots. The algorithms use $\Oc (\delta\, \csttn D^{2} \log(D) \log(b))$ arithmetic operations (in addition to the operations required to compute the normal form of the boundary monomials modulo the ideal) to approximate all real roots of the system as well as all roots lying in a fixed n-dimensional box or disc. Here D is an upper bound on the number of all complex roots of the system (e.g., Bezout or Bernshtein bound), $\delta$ is the number of real roots or the roots lying in the box or disc, and $\epsilon=2^{-b}$ is the required upper bound on the output errors. For computing the normal form modulo the ideal, the efficient practical algorithms of [B. Mourrain and P. Trebuchet, in Proceedings of the International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2000, pp. 231--238] or [J. C. Faugere, J. Pure Appl. Algebra, 139 (1999), pp. 61--88] can be applied. We also yield the bound $\Oc( \csttn D^{2} \log(D) )$ on the complexity of counting the numbers of all roots in a fixed box (disc) and all real roots. For a large class of inputs and typically in practical computations, the factor $\delta$ is much smaller than $D, \delta=o(D)$. This improves by the order of magnitude the known complexity estimates of the order of at least 3n D4 + D3 log(b) or D4, which so far are the record estimates even for the approximation of a single root of a system and for each of the cited counting problems, respectively. Our progress relies on proposing several novel techniques. In particular, we exploit the structure of matrices associated to a given polynomial system and relate it to the associated linear operators, dual space of linear forms, and normal forms of polynomials in the quotient algebra; furthermore, our techniques support the new nontrivial extension of the matrix sign and quadratic inverse power iterations to the case of multivariate polynomial systems, where we emulate the recursive splitting of a univariate polynomial into factors of smaller degree.

An n-tet graph approach for non-guillotine packings of n-dimensional boxes into an n-container

In this paper we propose a simple recursive uniform algorithm for the problem of packing n-dimensional boxes into an n-container. We are particularly concerned about the special case n=3 where the boxes can be packed in a given subset of their six possible positionings. Our method studies symmetries in the packings by the use of an ordered set of three directed graphs with the same edges (a 3- tet or triad) and induced smaller structures of the same kind named minors. With the method, degeneracy and symmetry issues, which curtail the implicit enumeration to practically acceptable running times, become transparent. In order to illustrate the performance of the algorithm, computational results from solving randomly generated 3-D examples are presented and compared with the ones of a layers and knapsack approach. The present study has real world applications for the problems of pallet and container loading.

Robust Solutions of Uncertain Quadratic and Conic-Quadratic Problems

We consider a conic-quadratic (and in particular a quadratically constrained) optimization problem with uncertain data, known only to reside in some uncertainty set ${\cal U}$. The robust counterpart of such a problem leads usually to an NP-hard semidefinite problem; this is the case, for example, when ${\cal U}$ is given as the intersection of ellipsoids or as an n-dimensional box. For these cases we build a single, explicit semidefinite program, which approximates the NP-hard robust counterpart, and we derive an estimate on the quality of the approximation, which is essentially independent of the dimensions of the underlying conic-quadratic problem.

An interval method for global nonlinear analysis

In this paper, the problem of finding the set of all real solutions to a system of n nonlinear equations contained in a given n-dimensional box [the global nonlinear analysis (GNA) problem] is considered. A new iterative interval method for solving the GNA problem is suggested. It is based on the following techniques: (1) transformation of the original system into an augmented system of n'=n+m equations of n' variables by introducing m auxiliary variables, the augmented system being of the so-called semiseparable form; (2) enclosure of the nonlinear augmented system at each iteration by a specific linear interval system of size n'/spl times/n'; (3) elimination of the auxiliary variables; and (4) solution of the resulting reduced size n/spl times/n linear system, using the so-called constraint propagation approach. The method suggested shows a significant improvement over previous techniques for the numerical examples solved.

Learning unions of high-dimensional boxes over the reals

Beimel and Kushilevitz (1997) presented an algorithm that exactly learns (using membership queries and equivalence queries) several classes of unions of boxes in high dimension over finite discrete domains. The running time of the algorithm is polynomial in the logarithm of the size of the domain and other parameters of the target function (in particular, the dimension). We go one step further and present a PAC+MQ algorithm whose running time is independent of the size of the domain. Thus, we can learn such classes of boxes over infinite domains. Specifically, we learn unions of t disjoint n-dimensional boxes over the reals in time polynomial in n and t, and unions of O(log n) (possibly intersecting) n-dimensional boxes over the reals in time polynomial in n.

A New Method for Global Solution of Systems of Non-Linear Equations

In this paper the problem of finding the set of all real solutions to a system of n non-linear equations contained in a given n-dimensional box (the global solution problem) is considered. A new method for solving the global solution problem is suggested. It is based on a transformation of the original system into a larger system of separable form. The global solution of the latter system is then found in a most efficient manner by a new interval method which exploits the separabily property. Numerical examples illustrating the efficiency of the method suggested are provided.