Polyhedral Obstacles Research Articles

Quad-Rotor Avoidance Trajectory Generation for Convex Polyhedron Obstacles

This study investigates a method for generating obstacle avoidance trajectories for arbitrary convex polyhedrons. We propose a formulation that converts discrete conditions into continuous equation forms to avoid convex polyhedron obstacles. The condition that the evaluation point be located outside of the convex polyhedron can be transformed into a constraint in the continuous equation form and incorporated into the optimization calculation to generate avoidance trajectories. Avoidance trajectory generation using the Legendre pseudospectral method is performed for convex polyhedral obstacles of various shapes. The results show that the proposed method successfully generates avoidance trajectories for arbitrary convex polyhedral obstacles.

Dual Collision Detection in Model Predictive Control Including Culling Techniques

An optimization-based approach for collision avoidance of fully dimensional autonomous vehicles in confined environments is considered. We discuss the dual re-formulations of indicator, distance, and signed distance functions which yield an exact re-formulation of the usually nonlinear, noncontinuously differentiable respective original formulations. Embedding the resulting dual collision avoidance constraints to a model predictive control (MPC) problem induces additional decision variables, which greatly increases problem complexity. Therefore, a culling procedure is combined with the dual approach to reduce the problem size by identifying and eliminating decision variables associated with faces of the polyhedral obstacles and the controlled vehicle, respectively. Three culling conditions are proposed, namely, frustum, occlusion, and backface culling. The proposed method is illustrated using a simulative study where a path-following task is performed for a ship autopilot model, and near-field obstacle information is extracted based on a set of automatic identification system (AIS) data from the Kiel canal.

Chance-Constrained Trajectory Planning With Multimodal Environmental Uncertainty

We tackle safe trajectory planning under Gaussian mixture model (GMM) uncertainty. Specifically, we use a GMM to model the multimodal behaviors of obstacles’ uncertain states. Then, we develop a mixed-integer conic approximation to the chance-constrained trajectory planning problem with deterministic linear systems and polyhedral obstacles. When the GMM moments are estimated via finite samples, we develop a tight concentration bound to ensure the chance constraint with a desired confidence. Moreover, to limit the amount of constraint violation, we develop a Conditional Value-at-Risk (CVaR) approach corresponding to the chance constraints and derive a tractable approximation for known and estimated GMM moments. We verify our methods with state-of-the-art trajectory prediction algorithms and autonomous driving datasets.

Polyhedral potential field constructions for obstacle avoidance in a receding horizon formulation

We revisit and enhance previous work pertaining to the use of polyhedral potential fields for motion planning in a receding horizon framework. We adapt the standard vertex-based mixed-integer (MI) representations of piecewise affine functions and propose three equivalent formulations which balance the complexity of the geometric representation with the size of the associated MI representation. In all cases, we arrive at formulations whose polyhedral support comes (either directly or as a refinement of it) from an associated hyperplane arrangement induced by the hyperplanes bounding the polyhedral obstacles.

Unique continuation from a generalized impedance edge-corner for Maxwell’s system and applications to inverse problems

We consider the time-harmonic Maxwell system in a domain with a generalized impedance edge-corner, namely the presence of two generalized impedance planes that intersect at an edge. The impedance parameter can be 0, ∞ or a finite non-identically vanishing function. We establish an accurate relationship between the vanishing order of the solutions to the Maxwell system and the dihedral angle of the edge-corner. In particular, if the angle is irrational, the vanishing order is infinity, i.e. strong unique continuation holds from the edge-corner. The establishment of those new quantitative results involve a highly intricate and subtle algebraic argument. The unique continuation study is strongly motivated by our study of a longstanding inverse electromagnetic scattering problem. As a significant application, we derive several novel unique identifiability results in determining a polyhedral obstacle as well as its surface impedance by a single far-field measurement. We also discuss another potential and interesting application of our result in the inverse scattering theory related to the information encoding.

Obstacle Avoidance Model for UAVs with Joint Target based on Multi-Strategies and Follow-up Vector Field

A multi-strategies obstacle avoidance method based on follow-up rotating vector field is proposed for UAV flight planning. The inter-UAVs repulsion with distance factor and the target directional gravity are used as the inter-UAVs control strategy. In the process of obstacle avoidance, the artificial potential field method is used outside the obstacle to guide the UAVs to sail to the target point. In the obstacle avoidance scope, the follow-up rotating vector field method is proposed to avoid convex polyhedral obstacles. Finally, smoothing strategy is conbined to the follow-up rotating vector field to obtain safety and smoothing path. The simulation results show that the proposed algorithm can solve the problem of “dead zone” and “jitter” in the track by combining various strategies, and realize the obstacle avoidance behavior while avoiding collision between machines.

Planning the Shortest Path in Cluttered Environments: A Review and a Planar Convex Hull-Based Approach

AbstractRouting or path-planning is the problem of finding a collision-free and preferably shortest path in an environment usually scattered with polygonal or polyhedral obstacles. The geometric algorithms oftentimes tackle the problem by modeling the environment as a collision-free graph. Search algorithms such as Dijkstra’s can then be applied to find an optimal path on the created graph. Previously developed methods to construct the collision-free graph, without loss of generality, explore the entire workspace of the problem. For the single-source single-destination planning problems, this results in generating some unnecessary information that has little value and could increase the time complexity of the algorithm. In this paper, first a comprehensive review of the previous studies on the path-planning subject is presented. Next, an approach to address the planar problem based on the notion of convex hulls is introduced and its efficiency is tested on sample planar problems. The proposed algorithm focuses only on a portion of the workspace interacting with the straight line connecting the start and goal points. Hence, we are able to reduce the size of the roadmap while generating the exact globally optimal solution. Considering the worst case that all the obstacles in a planar workspace are intersecting, the algorithm yields a time complexity of O(n log(n/f)), with n being the total number of vertices and f being the number of obstacles. The computational complexity of the algorithm outperforms the previous attempts in reducing the size of the graph yet generates the exact solution.

A discretization result for some optimization problems in framework spaces with polyhedral obstacles and the Manhattan metric

In this work we consider the shortest path problem and the single facility Weber location problem in any real space of finite dimension where there exist different types of polyhedral obstacles or forbidden regions. These regions are polyhedral sets and the metric considered in the space is the Manhattan metric. We present a result that reduce these continuous problems into problems in a “add hoc” graph, where the original problems can be solved using elementary techniques of Graph Theory. We show that, fixed the dimension of the space, both the reduction and the resolution can be done in polynomial time.

Recovering an electromagnetic obstacle by a few phaseless backscattering measurements

We consider the electromagnetic scattering from a convex polyhedral PEC or PMC obstacle due to a time-harmonic incident plane wave. It is shown that the modulus of the far-field pattern in the backscattering aperture possesses a certain local maximum behavior. Using the local maximum indicating phenomena, one can determine the exterior unit normal directions, as well as the face areas, of the front faces of the obstacle. Then we propose a recovery scheme of reconstructing the obstacle by phaseless backscattering measurements. This work significantly extends our recent study in Li and Liu (2014 preprint) from two dimensions and acoustic scattering to the more challenging three dimensions and electromagnetic scattering.

Stable determination of sound-hard polyhedral scatterers by a minimal number of scattering measurements

The aim of the paper is to establish optimal stability estimates for the determination of sound-hard polyhedral scatterers in RN, N≥2, by a minimal number of far-field measurements. This work is a significant and highly nontrivial extension of the stability estimates for the determination of sound-soft polyhedral scatterers by far-field measurements, proved by one of the authors, to the much more challenging sound-hard case.The admissible polyhedral scatterers satisfy minimal a priori assumptions of Lipschitz type and may include at the same time solid obstacles and screen-type components. In this case we obtain a stability estimate with N far-field measurements. Important features of such an estimate are that we have an explicit dependence on the parameter h representing the minimal size of the cells forming the boundaries of the admissible polyhedral scatterers, and that the modulus of continuity, provided the error is small enough with respect to h, does not depend on h. If we restrict to N=2,3 and to polyhedral obstacles, that is to polyhedra, then we obtain stability estimates with fewer measurements, namely first with N−1 measurements and then with a single measurement. In this case the dependence on h is not explicit anymore and the modulus of continuity depends on h as well.

Recovering a polyhedral obstacle by a few backscattering measurements

We propose an inverse scattering scheme of recovering a polyhedral obstacle in Rn, n=2,3, by only a few high-frequency acoustic backscattering measurements. The obstacle could be sound-soft or sound-hard. It is shown that the modulus of the far-field pattern in the backscattering aperture possesses a certain local maximum behavior, from which one can determine the exterior normal directions of the front sides/faces. Then by using the phaseless backscattering data corresponding to a few incident plane waves with suitably chosen incident directions, one can determine the exterior unit normal vector of each side/face of the obstacle. After the determination of the exterior unit normals, the recovery is reduced to a finite-dimensional problem of determining a location point of the obstacle and the distance of each side/face away from the location point. For the latter reconstruction, we need to make use of the far-field data with phases. Numerical experiments are also presented to illustrate the effectiveness of the proposed scheme.

FRÉCHET DISTANCE PROBLEMS IN WEIGHTED REGIONS

We discuss two versions of the Fréchet distance problem in weighted planar subdivisions. In the first one, the distance between two points is the weighted length of the line segment joining the points. In the second one, the distance between two points is the length of the shortest path between the points. In both cases, we give algorithms for finding a (1 + ∊)-factor approximation of the Fréchet distance between two polygonal curves. We also consider the Fréchet distance between two polygonal curves among polyhedral obstacles in [Formula: see text] (1/∞ weighted region problem) and present a (1 + ∊)-factor approximation algorithm.

Recovery of polyhedral scatterers by a single electromagnetic far-field measurement

We prove that a polyhedral obstacle in R3 consisting of finitely many polyhedra with mixed perfect electric conductor and perfect magnetic conductor boundary conditions can be uniquely determined by a single electric or magnetic far-field measurement, namely, the far-field pattern corresponding to a single incident wave. A unique novelty of our new technique for proving the uniqueness is to realize that the existence of an “unbounded” perfect plane implies certain symmetries of the underlying scatterer.

The No Response Test for the reconstruction of polyhedral objects in electromagnetics

We develop a No Response Test for the reconstruction of a polyhedral obstacle from two or few time-harmonic electromagnetic incident waves in electromagnetics. The basic idea of the test is to probe some region in space with waves which are small on some test domain and, thus, do not generate a response when the scatterer is inside of this test domain. We will prove that the No Response Test checks analytic continuability of a time-harmonic field from the far field pattern into the domain B e ≔ R 3 ∖ B ¯ for a non-vibrating test domain B . We show that two incident waves, defined by one incident direction and two appropriately chosen directions of polarization, are enough to recover the convex hull of polyhedrals. Based on this uniqueness result, we build up the No Response Test and we prove convergence in the sense that it fully reconstructs a convex polyhedral scatterer D or the convex hull of an arbitrary polyhedral scatterer. Further, we will describe the algorithmic realization of the No Response Test and show the feasibility of the method by reconstruction of convex polyhedral objects in three dimensions. This is the first formulation of the No Response Test for electromagnetics.

On the complexity of umbra and penumbra

Computing shadow boundaries is a difficult problem in the case of non-point light sources. A point is in the umbra if it does not see any part of any light source; it is in full light if it sees entirely all the light sources; otherwise, it is in the penumbra. While the common boundary of the penumbra and the full light is well understood, less is known about the boundary of the umbra. In this paper we prove various bounds on the complexity of the umbra and the penumbra cast on a fixed plane by a segment or convex polygonal light source in the presence of convex polygonal or polyhedral obstacles in R 3 . In particular, we show that a single segment light source may cast on a plane, in the presence of two disjoint triangles, four connected components of umbra and that two fat convex and disjoint obstacles of total complexity n can give rise to as many as Ω ( n ) connected components of umbra. In a scene consisting of a segment light source and k disjoint convex polyhedra of total complexity n, we prove an Ω ( n k 2 + k 4 ) lower bound on the maximum number of connected components of the umbra and a O ( n k 3 ) upper bound on its complexity; if the obstacles may intersect, we only prove an upper bound of O ( n 2 k 2 ) . We also prove that, in the presence of k convex polyhedra of total complexity n, some of which are light sources, the umbra cast on a plane may have in the worst case Ω ( n 2 k 3 + n k 5 ) connected components and has complexity O ( n 3 k 3 ) (the polyhedra are supposed pairwise disjoint for lower bounds and possibly intersecting for the upper bounds). These are the first bounds on the size of the umbra in terms of both k and n. These results prove that the umbra, which is bounded by arcs of conics, is intrinsically much more intricate than the boundary between full light and penumbra which is bounded by line segments and whose worst-case complexity is, as we show, in Ω ( n k + k 4 ) and O ( n k α ( k ) + k 4 ) ; moreover, if there are only O ( 1 ) light sources of total complexity m, then the complexity is in Ω ( n α ( k ) + k m + k 2 ) and O ( n α ( k ) + k m α ( k ) + k 2 ) .

Uniqueness in determining polyhedral sound-hard obstacles with a single incoming wave

We consider the inverse acoustic scattering problem of determining a sound-hard obstacle by far-field measurements. It is proved that a polyhedral scatterer in , consisting of finitely many solid polyhedra, is uniquely determined by a single incoming plane wave.

Stable determination of sound-soft polyhedral scatterers by a single measurement

We prove optimal stability estimates for the determination of a finite number of sound-soft polyhedral scatterers in R 3 by a single far-field measurement. The admissible multiple polyhedral scatterers satisfy minimal a priori assumptions of Lipschitz type and may include at the same time obstacles, screens and even more complicated scatterers. We characterize any multiple polyhedral scatterer by a size parameter h which is related to the minimal size of the cells of its boundary. In a first step we show that, provided the error e on the far-field measurement is small enough with respect to h, then the corresponding error, in the Hausdorff distance, on the multiple polyhedral scatterer can be controlled by an explicit function of e which approaches zero, as e → 0 + , in an essentially optimal, although logarithmic, way. Then, we show how to improve this stability estimate, provided we restrict our attention to multiple polyhedral obstacles and e is even smaller with respect to h. In this case we obtain an explicit estimate essentially of Holder type.

Locating an obnoxious plane

Let S be a set of n points in three-dimensional Euclidean space. We consider the problem of positioning a plane π intersecting the convex hull of S such that min{ d( π, p); p ∈ S} is maximized. In a geometric setting, the problem asks for the widest empty slab through n points in space, where a slab is the open region of R 3 that is bounded by two parallel planes that intersect the convex hull of S. We give a characterization of the planes which are locally optimal and we show that the problem can be solved in O( n 3) time and O( n 2) space. We also consider several variants of the problem which include constraining the obnoxious plane to contain a given line or point and computing the widest empty slab for polyhedral obstacles. Finally, we show how to adapt our method for computing a largest empty annulus in the plane, improving the known time bound O( n 3 log n) [J.M. Díaz-Báñez, F. Hurtado, H. Meijer, D. Rappaport, T. Sellarès, The largest empty annulus problem, International Journal of Computational Geometry and Applications 13 (4) (2003) 317–325].

Approximating shortest path for the skew lines problem in time doubly logarithmic in 1/epsilon

We consider two three-dimensional situations when a polytime algorithm for approximating a shortest path can be constructed. The main part of the paper treats a well-known problem of constructing a shortest path touching lines in R 3 : given a list of straight lines L = (L 1,…,L n) in R 3 and two points s and t , find a shortest path that, starting from s , touches the lines L i in the given order and ends at t . We remark that such a shortest path is unique. We show that it can be length–position ε-approximated (i.e. both its length and its position can be found approximately) in time (Rn/ d ̃ α ̃ ) 16+ O(n 2 log log 1/ε) , where d ̃ is the minimal distance between consecutive lines of L, α ̃ is the minimum of sines of angles between consecutive lines, and R is the radius of a ball where the initial approximation can be placed (such a radius can be easily computed from the initial data). As computational model we take real RAM extended by square and cubic roots extraction. This problem of constructing a shortest path touching lines is known for quite some time to be a challenging problem. The existing methods for approximating shortest paths based on adding Steiner points which form a grid and subsequently applying Dijkstra's algorithm for finding a shortest path in the grid, provide a complexity bound which depends polynomially on 1/ ε, while our algorithm for the problem under consideration has complexity linear in log log 1/ε . Our algorithm is motivated by the observation that the shortest path in question is a geodesic in a certain length space of non-positive curvature (in the sense of A.D. Alexandrov), and it relies on the (elementary) theory of CAT(0)-spaces. In the second part of the paper we analyze very simple grid approximations. We assume that a parameter a>0 describing separability of obstacles is given and the part of a grid with mesh size a outside the obstacles is built (for semi-algebraic obstacles all these pre-calculations are polytime). We show that there is an algorithm of time complexity O((1/ a) 6) which, given a-separated obstacles in a unit cube, finds a path (between given vertices s and t of the grid) whose length is bounded from above by (84π ∗+96a) , where π ∗ is the length of a shortest path. On the other hand, as we show by an example, one cannot approximate the length of a shortest path better than 7π ∗ if one uses only grid polygons (constructed only from grid edges). For semi-algebraic obstacles our computational model is bitwise. For a general type of obstacles the model is bitwise modulo constructing the part of the grid admissible for our paths. Observe that the existing methods for approximating shortest paths are not directly applicable for semi-algebraic obstacles since they usually place the Steiner points forming a grid on the edges of polyhedral obstacles.

Pseudo Approximation Algorithms with Applications to Optimal Motion Planning

We introduce a technique for computing approximate solutions to optimization problems. If $X$ is the set of feasible solutions, the standard goal of approximation algorithms is to compute $x\in X$ that is an $\varepsilon$-approximate solution in the following sense: $$d(x) \leq (1+\varepsilon)\, d(x^*),$$ where $x^* \in X$ is an optimal solution, $d\colon X\rightarrow {\Bbb R}_{\geq 0}$ is the optimization function to be minimized, and $\varepsilon>0$ is an input parameter. Our approach is first to devise algorithms that compute pseudo $\varepsilon$-approximate solutions satisfying the bound $$d(x) \leq d(x_R^*) + \varepsilon R,$$ where $R>0$ is a new input parameter. Here $x^*_R$ denotes an optimal solution in the space $X_R$ of $R$-constrained feasible solutions. The parameter $R$ provides a stratification of $X$ in the sense that (1) $X_R \subseteq X_{R'}$ for $R < R'$ and (2) $X_R = X$ for $R$ sufficiently large. We first describe a highly efficient scheme for converting a pseudo $\varepsilon$-approximation algorithm into a true $\varepsilon$-approximation algorithm. This scheme is useful because pseudo approximation algorithms seem to be easier to construct than $\varepsilon$-approximation algorithms. Another benefit is that our algorithm is automatically precision-sensitive. We apply our technique to two problems in robotics: (A) Euclidean Shortest Path (3ESP), namely the shortest path for a point robot amidst polyhedral obstacles in three dimensions, and (B) $d_1$-optimal motion for a rod moving amidst planar obstacles (1ORM). Previously, no polynomial time $\varepsilon$-approximation algorithm for (B) was known. For (A), our new solution is simpler than previous solutions and has an exponentially smaller complexity in terms of the input precision.