基于启发式搜索分离向量的凸多面体碰撞检测

碰撞检测是计算机模拟物理过程的基础，在计算机图形学、CAD／CAM、虚拟现实和机器人等领域有着广泛的应用．该文给出了一个新的用于凸多面体碰撞检测的算法——HP-jump．HP-jump建立了一个有效的碰撞检测模型用于报告物体的碰撞，同时提供了一个快速的启发式的策略用于搜索两个凸多面体的分离向量．该算法是利用凸多面体的层次表示来搜索支撑顶点对，用平衡二叉树来记录球面凸多边形的顶点，同时还利用了时间、空间相关性，这些都加速了算法的执行．该文的最后给出了HP-jump与GJK，I-COLLIDE算法的比较．     

Set-theoretic approach to uncertain dynamical systems using convex polytopes

A set-theoretic approach to uncertain dynamical systems by using convex polytopes is treated. In some points, this approach is superior to the one using ellipsoids, though it also has some disadvantages. First, some fundamental operations are discussed, such as addition, intersection, and the linear transformation of convex polytopes. The results are applied to the analysis of reachable sets for discrete-time linear dynamical systems and also to the state estimation.     

Robust stability of systems of linear differential equations with periodic impulsive influence

We give sufficient robust stability conditions for matrix polytopes of linear systems with impulsive influence. The methods of our study are based on logarithmic matrix measure theory and linear operator theory in Banach spaces. Our results reduce the robust stability problem to the feasibility problem for a system of linear matrix inequalities in the class of positive definite matrices.     

Robust control of uncertain multi-inventory systems via linear matrix inequality

We consider a continuous time linear multi-inventory system with unknown demands bounded within ellipsoids and controls bounded within ellipsoids or polytopes. We address the problem of ε-stabilising the inventory since this implies some reduction of the inventory costs. The main results are certain conditions under which ε-stabilisability is possible through a saturated linear state feedback control. All the results are based on a linear matrix inequalities approach and on some recent techniques for the modelling and analysis of polytopic systems with saturations. Numerical simulations are provided.     

Approximate set-valued observers for nonlinear systems

A set-valued observer (SVO) produces a set of possible states based on output measurements and a priori models of exogenous disturbances and noises. Previous work considered linear time-varying systems and unknown-but-bounded exogenous signals. In this case, the sets of possible state vectors take the form of polytopes whose centers are optimal state estimates. These polytopic sets can be computed by solving several small linear programs. An SVO can be constructed conceptually for nonlinear systems; however, the set of possible state vectors no longer takes the form of polytopes, which in turn inhibits their explicit computation. This paper considers an “extended SVO”. As in the extended Kalman filter, the state equations are linearized about the state estimate, and a linear SVO is designed along the linearization trajectory. Under appropriate observability assumptions, it is shown that the extended SVO provides an exponentially convergent state estimate in the case of sufficiently small initial condition uncertainty and provides a nondivergent state estimate in the case of sufficiently small exogenous signals     

Fuzzy Linear Programming with Interactive Uncertain Parameters

In this paper, we treat fuzzy linear programming problems with uncertain parameters whose ranges are specified as fuzzy polytopes. The problem is formulated based on fractile optimization model using a necessity measure. It is shown that the problem can be reduced to a semi-infinite linear programming problem and that a solution algorithm based on a relaxation procedure can be applied. A simple numerical example is given to illustrate the solution procedure. This revised version was published online in June 2006 with corrections to the Cover Date.     

Fast interference detection between geometric models

We present efficient algorithms for interference detection between geometric models described by linear or curved boundaries and undergoing rigid motion. The set of models include surfaces described by rational spline patches or piecewise algebraic functions. In contrast to previous approaches, we first describe an efficient algorithm for interference detection between convex polytopes using coherence and local features. Then an extension using hierarchical representation to concave polytopes is presented. We apply these algorithms along with properties of input models, local and global algebraic methods for solving polynomial equations, and the geometric formulation of the problem to devise efficient algorithms for convex and nonconvex curved objects. Finally, a scheduling scheme to reduce the frequency of interference detection in large environments is described. These algorithms have been successfully implemented and we discuss their performance in various environments.     

Hurwitz testing sets for parallel polytopes of polynomials

In considering robustness of linear systems with uncertain paramenters, one is lead to consider simultaneous stability of families of polynomials. Efficient Hurwitz stability tests for polytopes of polynomials have earlier been developed using evaluations on the imaginary axis. This paper gives a stability criterion for parallel polytopes in terms of Hurwitz stability of a number of corners and edges. The ‘testing set’ of edges and corners depends entirely on the edge directions of the polytope, hence the results are particularly applicable in simultaneous analysis of several polytopes with equal edge directions.It follows as a consequence, that Kharitonov's four polynomial test for independent coefficient uncertainties is replaced by a test of 2q polynomials, when the stability region is a sector Ω = { e^iv | > 0, rπ/q < | v | ≤ π } and r/q is a rational number.     

New stability conditions via reflection coefficients of polynomials

The geometry of stable discrete polynomials using their coefficients and reflection coefficients is investigated. Starting from so-called barycentric simplex some necessary stability conditions in terms of unions of polytopes are obtained by splitting the unit hypercube of reflection coefficients. Sufficient stability conditions in terms of linear covers of reflection vectors of a family of stable polynomials improve the Cohn stability criterion.     

Linear adaptive control for nonstationary uncertain systems under bounded noise

A discrete-time nonstationary linear control system is considered to be given by the algebraic difference equation in the state space. The control system is subject to a bounded additive noise. Uncertain parameters of the system take their values on the given polytopes which evolve in time. The objective is to generate a linear feedback, which provides the minimization of a given performance criterion in adaptive way. In general, the control problem is reduced to the convex programming one of an insignificant computational complexity. Therewith, the control problem can be solved analytically in the case of interval set-valued parameter estimates.     

参数空间中多变量系统的鲁棒－稳定性

为了保证具有参数不确定性的多变量系统的－稳定性，它们的参数最大允许摄动范围将受到复平面中区域　以及标称系统的限制。利用系统临界　－稳定时的特性和线性算子范数的特性，得到了这个范围半径的解析表达式。由于这个半径是以欧氏空间的一般范数表示的，所以对于参数摄动范围是菱形。矩形、椭圆、对称多边形等情况，均可以利用它求出系统参数的最大允许摄动范围。     

A generalization of strong Kharitonov theorems to polytopes ofpolynomials

The root clustering property of a certain class of polytopes of polynomials is examined. It is shown that the polytopes of polynomials have only zeros within the left sector if and only if a subset of their edge polynomials has only zeros within the left sector. Certain conditions are derived under which the root clustering property of the polytopes of polynomials can be inferred from a subset of their vertices     

参数空间中多变量系统的鲁棒D-稳定性

为了保证具有参数不确定性的多变量系统的D-稳定性,它们的参数最大允许摄动范围将受到复平面中区域D以及标称系统的限制.利用系统临界D-稳定时的特性和线性算子范数的特性,得到了这个范围半径的解析表达式.由于这个半径是以欧氏空间的一般范数表示的,所以对于参数摄动范围是菱形、矩形、椭圆、对称多边形等情况,均可以利用它求出系统参数的最大允许摄动范围.     

基于中心对称多胞体的故障可分离性评价

针对包含幅值有界而分布形式未知的故障及输入干扰项的线性离散系统, 提出了一种新的系统故障可分离性的量化评价方法. 故障可分离性是故障可诊断性中的重要部分, 针对现有方法中基于方向相似度的故障可分离性评价方法存在的不足加以补充, 提出了利用中心对称多胞体对故障可分离性进行分析, 将中心对称多胞体集合转化为多面体的表示形式, 以达到对故障可分离性量化评价的目的, 同时给出了具体评价原理和评价指标. 最后, 通过数值仿真算例, 验证了该方法的有效性和优越性.     

A parameter-dependent Lyapunov function for a polytope of matrices

A new sufficient condition for a polytope of matrices to be Hurwitz-stable is presented. The stability is a consequence of the existence of a parameter-dependent quadratic Lyapunov function, which is assured by a certain linear constraint for generating extreme matrices of the polytope. The condition can be regarded as a duality of the known extreme point result on quadratic stability of matrix polytopes, where a fixed quadratic Lyapunov function plays the role. The obtained results are applied to a polytope of second-degree polynomials for illustration     

Single-lifting Macaulay-type formulae of generalized unmixed sparse resultants

Resultants are defined in the sparse (or toric) context in order to exploit the structure of the polynomials as expressed by their Newton polytopes. Since determinantal formulae are not always possible, the most efficient general method for computing resultants is rational formulae. This is made possible by Macaulay’s famous determinantal formula in the dense homogeneous case, extended by D’Andrea to the sparse case. However, the latter requires a lifting of the Newton polytopes, defined recursively on the dimension. Our main contribution is a single-lifting function of the Newton polytopes, which avoids recursion, and yields a simpler method for computing Macaulay-type formulae of sparse resultants. We focus on the case of generalized unmixed systems, where all Newton polytopes are scaled copies of each other, and sketch how our approach may extend to mixed systems of up to four polynomials, as well as those whose Newton polytopes have a sufficiently different face structure. In the mixed subdivision used to construct the matrices, our algorithm defines significantly fewer cells than D’Andrea’s, though the matrix formulae are same. We discuss asymptotic complexity bounds and illustrate our results by fully studying a bivariate example.     

Fast and robust retrieval of Minkowski sums of rotating convex polyhedra in 3-space

We present a novel method for fast retrieval of exact Minkowski sums of pairs of convex polytopes in R³, where one of the polytopes frequently rotates. The algorithm is based on pre-computing a so-called criticality map, which records the changes in the underlying graph structure of the Minkowski sum of the two polytopes, while one of them rotates. We give tight combinatorial bounds on the complexity of the criticality map when the rotating polytope rotates about one, two, or three axes. The criticality map can be rather large already for rotations about one axis, even for summand polytopes with a moderate number of vertices each. We therefore focus on the restricted case of rotations about a single, though arbitrary, axis.Our work targets applications that require exact collision detection such as motion planning with narrow corridors and assembly maintenance where high accuracy is required. Our implementation handles all degeneracies and produces exact results. It efficiently handles the algebra of exact rotations about an arbitrary axis in R³, and it well balances between preprocessing time and space on the one hand, and query time on the other.We use Cgal arrangements and in particular the support for spherical Gaussian maps to efficiently compute the exact Minkowski sum of two polytopes. We conducted several experiments (i) to verify the correctness of the algorithm and its implementation, and (ii) to compare its efficiency with an alternative (static) exact method. The results are reported.     

Delaunay partitions in Rn applied to non-convex programs and vertex/facet enumeration problems

Using a pair of theorems linking Delaunay partitions and linear programming, we develop a method to generate all simplices in a Delaunay partition of a set of points, and suggest an application to a piecewise linear non-convex optimization problem. The same method is shown to enumerate all facets of a polytope given as the convex hull of a finite set of points. The dual problem of enumerating all vertices of a polytope P defined as the intersection of a finite number of half-spaces is also addressed and solved by sequentially enumerating vertices of expanding polytopes defined within P. None of our algorithms are affected by degeneracy. Examples and computational results are given.     

Mathematical modeling of the interaction of non-oriented convex polytopes

An ??-function for two non-oriented convex polytopes is set up. The ??-function can be used to construct a mathematical model of packing optimization problem for non-oriented polytopes. An example of an ??-function for two non-oriented parallelepipeds is given.     

State estimation for Markovian Jump Linear Systems with bounded disturbances

In this paper, we investigate the state estimation problem for a class of Markovian Jump Linear Systems (MJLSs) in the presence of bounded polyhedral disturbances. A set-membership estimation algorithm is first proposed to find the smallest consistent set of all possible states, which is shown to be expressed by a union of multiple polytopes. The posterior probabilities of the system jumping modes are then estimated by introducing the Lebesgue measure, based on which the optimal point estimate is further provided. Moreover, a similarity relationship for polytopes is defined and an approximate method is presented to calculate the Minkowski sum of polytopes, which can help reduce the computational complexity of the overall estimation algorithm.