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.     

Contributing vertices-based Minkowski sum computation of convex polyhedra

Minkowski sum is an important operation. It is used in many domains such as: computer-aided design, robotics, spatial planning, mathematical morphology, and image processing. We propose a novel algorithm, named the Contributing Vertices-based Minkowski Sum (CVMS) algorithm for the computation of the Minkowski sum of convex polyhedra. The CVMS algorithm allows to easily obtain all the facets of the Minkowski sum polyhedron only by examining the contributing vertices—a concept we introduce in this work, for each input facet. We exploit the concept of contributing vertices to propose the Enhanced and Simplified Slope Diagram-based Minkowski Sum (ESSDMS) algorithm, a slope diagram-based Minkowski sum algorithm sharing some common points with the approach proposed by Wu et al. [Wu Y, Shah J, Davidson J. Improvements to algorithms for computing the Minkowski sum of 3-polytopes. Comput Aided Des. 2003; 35(13): 1181-92]. The ESSDMS algorithm does not embed input polyhedra on the unit sphere and does not need to perform stereographic projections. Moreover, the use of contributing vertices brings up more simplifications and improves the overall performance. The implementations for the mentioned algorithms are straightforward, use exact number types, produce exact results, and are based on CGAL, the Computational Geometry Algorithms Library. More examples and results of the CVMS algorithm for several convex polyhedra can be found at http://liris.cnrs.fr/hichem.barki/mksum/CVMS-convex.     

Improvements to algorithms for computing the Minkowski sum of 3-polytopes

A Minkowski sum is a geometric operation that is equivalent either to the vector additions of all points in two operands or to the sweeping of one operand around the profile of the other without changing the relative orientation. Applications of Minkowski sums are found in computer graphics, robotics, spatial planning, and CAD. This paper presents two algorithms for computing Minkowski sum of convex polyhedron in three space (3-polytopes). Both algorithms are improvements on current ones found in the literature. One is based on convex hulls and the other on slope diagrams. The original convex hull based Minkowski algorithm is costly, while the original slope diagram based algorithms require the operation of stereographic projection from 3D to 2D for merging the slope diagrams of the two operands. Implementation of stereographic projection is complicated which increases the computation time and reduces the accuracy of the geometric information that is needed for constructing the resultant solid. This paper reports on improvements that have been made to these two algorithms and their implementation. These improvements include using vector operations to find the interrelations between points, arcs and regions on a unit sphere for the slope diagram algorithm, and addition of a pre-sorting procedure before constructing convex hull for convex hull based Minkowski sum algorithm. With these improvements, the computation time and complexity for both algorithms have been reduced significantly, and the computational accuracy of the slope diagram algorithm has been improved. This paper also compares these two algorithms to each other and to their original counterparts. The potential for extending these algorithms to higher dimensions is briefly discussed.     

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.     

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.     

Shortest Path Geometric Rounding

Exact implementations of algorithms of computational geometry are subject to exponential growth in running time and space. In particular, coordinate bit-complexity can grow exponentially when algorithms are cascaded : the output of one algorithm becomes the input to the next. Cascading is a significant problem in practice. We propose a geometric rounding technique: shortest path rounding . Shortest path rounding trades accuracy for space and time and eliminates the exponential cost introduced by cascading. It can be applied to all algorithms which operate on planar polygonal regions, for example, set operations, transformations, convex hull, triangulation, and Minkowski sum. Unlike other geometric rounding techniques, shortest path rounding can round vertices to arbitrary lattices, even in polar coordinates, as long as the rounding cells are connected. (Other rounding techniques can only round to the integer grid.) On the integer grid, shortest path rounding introduces less combinatorial change and geometric error than the other rounding methods. Three algorithms are given for shortest path rounding, one of which we have used in industrial application software since 1992. In combination with recent advances in exact floating point evaluation of numerical primitives, shortest path geometric rounding yields a practical solution to numerical issues in computational geometry. Geometric algorithms can be implemented exactly on floating point input coordinates; the exact output coordinates can be rounded to accurate floating point approximations; and the cost of each arithmetic operation is only a little more than if it were implemented as a single hardware floating point operation. Received February 7, 1997; revised September 9, 1998.     

Voxelized Minkowski sum computation on the GPU with robust culling

We present a new approach for computing the voxelized Minkowski sum (excluding any enclosed voids) of two polyhedral objects using programmable Graphics Processing Units (GPUs). We first cull out surface primitives that will not contribute to the final boundary of the Minkowski sum, analyzing and adaptively bounding the rounding errors of the culling algorithm to solve the floating point error problem. The remaining surface primitives are then rendered to depth textures along six orthogonal directions to generate an initial solid voxelization of the Minkowski sum. Finally we employ fast flood fill to find all the outside voxels. We generate both solid and surface voxelizations of Minkowski sums without enclosed voids and support high volumetric resolution of 1024³ with low video memory cost. The whole algorithm runs on the GPU and is at least one order of magnitude faster than existing boundary representation (B-rep) based algorithms. It avoids the large number of 3D Boolean operations needed in most existing algorithms and is easy to implement. The voxelized Minkowski sums can be used in a variety of applications including motion planning and penetration depth computation.     

Repairing triangular meshes for reverse engineering applications

When an object is digitized and represented in a triangular model, erroneous facets may exist and affect the accuracy of the downstream data analysis algorithms. We here propose an approach to detect and eliminate erroneous facets that might exist in a triangular model. Five types of erroneous facets are identified in this study: degenerate, non-manifold vertices, self-intersection, incomplete connection and inconsistent plane normal. Of these erroneous facets, the first two types must be processed first since they are correlated to the other three types of errors. An individual algorithm is proposed for each type of the errors, and an integrated procedure is then proposed to detect and eliminate all errors automatically. Finally, several examples are presented to demonstrate the feasibility of the proposed method.     

Effective lattice point counting in rational convex polytopes

This paper discusses algorithms and software for the enumeration of all lattice points inside a rational convex polytope: we describe LattE, a computer package for lattice point enumeration which contains the first implementation of A. Barvinok’s algorithm (Math. Oper. Res. 19 (1994) 769).We report on computational experiments with multiway contingency tables, knapsack type problems, rational polygons, and flow polytopes. We prove that these kinds of symbolic–algebraic ideas surpass the traditional branch-and-bound enumeration and in some instances LattE is the only software capable of counting. Using LattE, we have also computed new formulas of Ehrhart (quasi-)polynomials for interesting families of polytopes (hypersimplices, truncated cubes, etc).We end with a survey of other “algebraic–analytic” algorithms, including a “homogeneous” variation of Barvinok’s algorithm which is very fast when the number of facet-defining inequalities is much smaller compared to the number of vertices.     

A new global accessibility algorithm for a polyhedral model with convex polygonal facets

Global accessibility information of a CAD model has been utilized widely in various manufacturing applications. This information needs fast-computing to improve the efficiency of manufacturability analysis. It needs compact representation to increase the effective utilization in its process-planning task. We propose a new geometric algorithm to explicitly find the global accessibility cones (GAC) of a polyhedral model. The proposed algorithm has three main steps. The first is concave region extraction, collecting facets that are not on the convex hull of the entire model. Second, inaccessibility of convex polygonal facets in these concave regions is analyzed in order to find their inaccessibility cones (IAC). The method is done in 2D instead of 3D. Finally, to compute GACs of those facets, the complement of the IACs union is determined for an exact solution, while the slicing-method is proposed to find a near-exact solution. In this paper, geometric examples are demonstrated and a comparison of the computational complexity with existing algorithms is provided.     

Projective Visual Hulls

This article presents a novel method for computing the visual hull of a solid bounded by a smooth surface and observed by a finite set of cameras. The visual hull is the intersection of the visual cones formed by back-projecting the silhouettes found in the corresponding images. We characterize its surface as a generalized polyhedron whose faces are visual cone patches; edges are intersection curves between two viewing cones; and vertices are frontier points where the intersection of two cones is singular, or intersection points where triples of cones meet. We use the mathematical framework of oriented projective differential geometry to develop an image-based algorithm for computing the visual hull. This algorithm works in a weakly calibrated setting–-that is, it only requires projective camera matrices or, equivalently, fundamental matrices for each pair of cameras. The promise of the proposed algorithm is demonstrated with experiments on several challenging data sets and a comparison to another state-of-the-art method.     

Formula manipulation in ALGOL 68 and application to Routh's algorithm

This paper presents an implementation of Routh's algorithm and the Schur criterion, in order to determine the location of the zeros of a complex polynomial in one variable, over the ring of multivariate polynomials with integral coefficients. Both algorithms are applied to the characteristic polynomials of multistep methods. Moreover, procedures are given for the arithmetic operations +,?,*,÷ with arbitrarily long integral numbers as operands.     

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．     

A GJK-based approach to contact force feasibility and distribution for multi-contact robots

This paper presents a new approach to two fundamental problems concerning the equilibrium of a multi-contact robot: the contact force feasibility (CFF) and the contact force distribution (CFD). The CFF is to determine if there exist feasible contact forces to maintain a robot in equilibrium without breaking its contacts with the environment, while the CFD is to compute the minimum contact forces if a feasible solution exists. A general measure of overall contact force magnitude is defined, which includes the traditional measure (i.e., the sum of normal force components) and a more complex measure (i.e., the maximum of normal force components) as special cases. We first reduce the two problems into verifying the existence of nonnegative solutions and determining the nonnegative minimum one-norm solution to a system of linear equations, respectively. To obtain the explicit formulation of the linear system, it is required to compute the Minkowski sum of point sets, which usually is computationally expensive. Then, based on the GJK distance algorithm, we develop an iterative algorithm, which enables us to solve the linear system without calculating the Minkowski sum and compute the CFF and CFD in real time.     

一种光线跟踪的包容性检测算法

提出了一维投影判别法和基于右手定则的空间多边形的包容性检测算法,该算法将空间多边形和线面交点投影至一维坐标轴并进行包容性的必要性判定,以少量逻辑比较即可排除大多数无关面片,然后利用基于右手定则的包容性检测算法,进行充分性判定。理论计算和模型中的应用表明,此算法用时显著减少。     

LMI stability conditions for uncertain rational nonlinear systems

This paper presents LMI conditions for local, regional, and global robust asymptotic stability of rational uncertain nonlinear systems. The uncertainties are modeled as real time varying parameters with magnitude and rate of variation bounded by given polytopes and the system vector field is a rational function of the states and uncertain parameters. Sufficient LMI conditions for asymptotic stability of the origin are given through a rational Lyapunov function of the states and uncertain parameters. The case where the time derivative of the Lyapunov function is negative semidefinite is also considered and connections with the well known LaSalle's invariance conditions are established. In regional stability problems, an algorithm to maximize the estimate of the region of attraction is proposed. The algorithm consists of maximizing the estimate for a given target region of initial states. The size and shape of the target region are recursively modified in the directions where the estimate can be enlarged. The target region can be taken as a polytope (convex set) or union of polytopes (non‐convex set). The estimates of the region of attraction are robust with respect to the uncertain parameters and their rate of change. The case of global and orthant stability problems are also considered. Connections with some results found in sum of squares based methods and other related methods found in the literature are established. The LMIs in this paper are obtained by using the Finsler's Lemma and the notion of annihilators. The LMIs are characterized by affine functions of the state and uncertain parameters, and they are tested at the vertices of a polytopic region. It is also shown that, with some additional conservatism, the use of the vertices can be avoided by modifying the LMIs with the S‐Procedure. Several numerical examples found in the literature are used to compare the results and illustrate the advantages of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.     

Composite Geometric Concepts and Polynomial Predictability

We study the predictability of geometric concepts, in particular those defined by boolean combinations of simple geometric objects. First, we give a negative results, showing that the problem of predicting the class of convex polytopes encoded by listing their vertices is prediction complete for P. Thus, an efficient solution to this prediction problem implies the existence of efficient solutions to all prediction problems whose associated evaluation problems are in P. Assuming the existence of a one-way function that is hard on iterates, there are such prediction problems which do not admit efficient solutions. Thus we show under minimal cryptographic assumptions that the class of convex polytopes encoded by listing their vertices is not predictable. As a side effect, we show that determining membership in the convex hull of a given set of points is complete for P with respect to log space reductions. Next, we establish the predictability of the class consisting of unions of a fixed number of flats by reducing its prediction problem to that of the class of flats, which has previously been shown to be predictable. Finally, we give an Occam algorithm for predicting fixed finite unions of boxes. Both constructive results for flats and boxes hold if the dimension is variable.     

An implicitization challenge for binary factor analysis

We use tropical geometry to compute the multidegree and Newton polytope of the hypersurface of a statistical model with two hidden and four observed binary random variables, solving an open question stated by Drton, Sturmfels and Sullivant in (Drton et al., 2009, Ch. VI, Problem 7.7). The model is obtained from the undirected graphical model of the complete bipartite graph _K2,4$K_{2,4}$ by marginalizing two of the six binary random variables. We present algorithms for computing the Newton polytope of its defining equation by parallel walks along the polytope and its normal fan. In this way we compute vertices of the polytope. Finally, we also compute and certify its facets by studying tangent cones of the polytope at the symmetry classes of vertices. The Newton polytope has 17 214 912 vertices in 44 938 symmetry classes and 70 646 facets in 246 symmetry classes.     

Controlled linear perturbation

We present an algorithmic solution to the robustness problem in computational geometry, called controlled linear perturbation, and demonstrate it on Minkowski sums of polyhedra. The robustness problem is how to implement real RAM algorithms accurately and efficiently using computer arithmetic. Approximate computation in floating point arithmetic is efficient but can assign incorrect signs to geometric predicates, which can cause combinatorial errors in the algorithm output. We make approximate computation accurate by performing small input perturbations, which we compute using differential calculus. This strategy supports fast, accurate Minkowski sum computation. The only prior robust implementation uses a less efficient algorithm, requires exact algebraic computation, and is far slower based on our extensive testing.     

Algorithms for Automatic Alignment of Arrays

Aggregate data objects (such as arrays) are distributed across the processor memories when compiling a data-parallel language for a distributed-memory machine. The mapping determines the amount of communication needed to bring operands of parallel operations into alignment with each other. A common approach is to break the mapping into two stages: analignmentthat maps all the objects to an abstract template, followed by adistributionthat maps the template to the processors. This paper describes algorithms for solving the various facets of the alignment problem: axis and stride alignment, static and mobile offset alignment, and replication labeling. We show that optimal axis and stride alignment is NP-complete for general program graphs and give a heuristic method that can explore the space of possible solutions in a number of ways. We show that some of these strategies can give better solutions than a simple greedy approach proposed earlier. We also show how local graph contractions can reduce the size of the problem significantly without changing the best solution. This allows more complex and effective heuristics to be used. We show how to model the static offset alignment problem using linear programming, and we show that loop-dependent mobile offset alignment is sometimes necessary for optimum performance. We describe an algorithm with for determining mobile alignments for objects withindoloops. We also identify situations in which replicated alignment is either required by the program itself or can be used to improve performance. We describe an algorithm based on network flow that replicates objects so as to minimize the total amount of broadcast communication in replication.