Abstract: The task of planning trajectories plays an important role in transportation, robotics, information systems (sending messages), etc. In robot motion planning, the robot should pass around obstacles from a given starting position to a given target position, touching none of them, i.e. the goal is to find a collision-free path from the starting to the target position. Research on path planning has yielded many fundamentally different approaches to the solution of this problem that can be classified as roadmap methods (visibility graph method, Voronoi diagram) and methods based on cell decomposition. Assuming movements only in a restricted number of directions (eight directional or horizontal/vertical) the task, with respect to its combinatorial nature, can be solved by decomposition methods using heuristic techniques. We present drawbacks of this approach (combinatorial explosion, limited granularity and generating infeasible solutions). Then, using the Voronoi diagrams, we need only polynomial time for finding a solution and, choosing a Euclidean or rectilinear metric, it can be adapted to tasks with general or directional-constrained movements.

Abstract: This paper addresses the problem of deploying a group of robotic agents equipped with omnidirectional vision in a simply connected orthogonal environment to achieve complete visibility. The agents are point masses with discrete-time first- order dynamics with no prior knowledge of the environment. Each agent can sense distances to the environment boundary and to other agents within line-of-sight. Communication is possible only between collocated agents. The agents operate asynchronously. The paper also addresses the problem of complete visibility deployment under the additional constraint that the visibility graph of the final agent locations is connected. We provide distributed algorithms that are guaranteed to solve the above problems if a sufficient number of agents are available. Remarkably, this number is identical to the number assuming complete prior knowledge of the environment. A final contribution of the paper is the characterization of the robustness properties of the algorithms to agent failures in the case of deployment with connectivity constraints.

Abstract: Computing optimal paths for mobile robots is an interesting and important problem. This paper presents a method to compute the shortest path for a differential-drive mobile robot, which is a disc, among piecewise smooth and convex obstacles. To obtain a well-defined notion of shortest, the total amount of wheel rotation is optimized. We use recent characterization of minimum wheel-rotation paths for differential-drive mobile robots with no obstacles. We reduce the search for the shortest path to the search on a finite nonholonomic visibility graph. Edges of the graph are either minimum wheel-rotation trajectories inside the free space or trajectories on the boundary of obstacle region. Vertices of the graph are initial and goal configurations and points on the boundary of obstacle region. We call the search graph a nonholonomic visibility graph because the jump condition of the Pontryagin maximum principle gives a necessary condition which is reminiscent of bitangency in well-known visibility graphs. To the best of our knowledge, this is the first progress on the problem

Abstract: The subject of this dissertation is motion coordination for mobile robotic networks with visibility sensors. Such networks consist of robotic agents equipped with sensors that can measure distances to the environment boundary and to other agents within line of sight. We look at two fundamental coordination problems: (i) deploying over an unknown nonconvex environment to achieve complete visibility, and (ii) gathering all agents initially scattered over the environment at a single location. As a special case of problem (i), we first address the problem of optimally locating a single robotic agent in a nonconvex environment. The agent is modeled as a point mass with continuous first-order dynamics. We propose a nonsmooth gradient algorithm for the problem of maximizing the area of the region visible to the observer in a non-self-intersecting nonconvex polygon. First, we show that the visible area is almost everywhere a locally Lipschitz function of the observer location. Second, we provide a novel version of the LaSalle Invariance Principle for discontinuous vector fields and for Lyapunov functions with a finite number of discontinuities. Finally, we establish the asymptotic convergence properties of the nonsmooth gradient algorithm and we illustrate numerically its performance. Second, we address problem (i) by proposing a novel algorithm to the deploy a group of robotic agents in an unknown nonconvex environment to achieve complete visibility. The agents are point masses with discrete-time first-order dynamics. The agents operate asynchronously and two agents can communicate when mutually visible. We also address this deployment problem under the additional constraint that the visibility graph of the final agent locations is connected. We provide distributed algorithms that are guaranteed to solve the two deployment problems if a sufficient number of agents is available. Remarkably, this number is identical to the upper bound established in the famous art gallery problem, i.e., the number of agents sufficient to achieve complete visibility in a known environment through centralized computation. We additionally provide time complexity bounds for the proposed algorithms. Third, we address problem (ii) by proposing a novel motion coordination algorithm for a group of robotic agents to achieve rendezvous, that is, to move to a common location inside a nonconvex environment. The robots move synchronously in discrete time, they have a range-limited visibility sensor, and no communication ability is required. The algorithm is designed using the notions of robust visibility, connectivity-preserving constraint sets, and proximity graphs. We rigorously establish the correctness of the algorithm and we illustrate through simulations the algorithm's performance in asynchronous setups with sensor measurement and control errors.

Abstract: The reduced visibility graph (RVG) is an important structure for computation of shortest paths for mobile robots. An efficient bit representation is proposed to construct segments that are part of the RVG. Based on the bit representation, a hardware-efficient scheme is presented whose computational complexity is O(k2log(n/k)), where k is the number of objects and n is the total number of vertices. An architecture that accomplishes the construction of the RVG without division or explicit intersection point calculations is proposed. An efficient field-programmable gate array implementation using block random access memory on an XCV3200E device is presented

Abstract: This paper is inspired by a wayfinding study by Holscher et al. (2005), who find serious wayfinding difficulties in a complex multi-level building and identify architectural properties related to the difficulties. The present study re-analyzes the qualitative results in terms of Space Syntax measures and thus ties them down to formal properties of the architectural structure. The analysis carefully models the spatial interconnections between different floors, as stairs are considered to cause many of the usability issues. Axial maps of the separate floors were interconnected via additional staircase axes using the manual connection feature of Depthmap (Turner, 2004). With respect to Visibility Graph Analysis (VGA) staircases were represented by “widgets” – additional space with representative spatial properties. These were connected to the staircases in the floor plan by merging visibility graph nodes. Considering the building as a whole, the poor intelligibility score of 0.15 is remarkable. Analyzed as separate systems the floors' intelligibility ranges from .09 (second floor) to .71 (basement). With respect to usability issues, several techniques revealed valuable results. Along a typical trajectory through the entrance hall the primary isovist changes rapidly. At no point all relevant navigation choices are visible simultaneously. The lack of survey is best demonstrated by the distribution of integration and connectivity: E.g. the entrance hall neither contains the most connective nor the most integrated part of the system. For the analysis of dead ends in the basement we regarded them as blockages. The visual step depth from one side to the other quantifies the complexity of the detours to overcome the dead ends. (17 and 8 turns respectively). A similar technique was applied to measure the amount of turning occurring in the staircases. As all staircases are offset from the main axis one needs to travel along a minimum of 7 axial lines from the entrance hall to the corresponding main intersection in the basement. To validate our findings we constructed two layout re-designs. Eliminating the dead ends in the basement increases integration in a formerly segregated region. Also, the distribution of spatial measures becomes more similar between different floors. The second re-design removes visual clutter near the entrance where a local Integration maximum emerges together with the most connective area. Although the analyses are post-hoc at this stage the re-designs point at the potential of Space Syntax as a predictive tool for wayfinding design.

Abstract: Over the past twenty years, rectangle visibility graphs have generated considerable interest, in part due to their applicability to VLSI chip design. Here we study unit rectangle visibility graphs, with fixed dimension restrictions more closely modeling the constrained dimensions of gates and other circuit components in computer chip applications. A graph $G$ is a unit rectangle visibility graph (URVG) if its vertices can be represented by closed unit squares in the plane with sides parallel to the axes and pairwise disjoint interiors, in such a way that two vertices are adjacent if and only if there is a non-degenerate horizontal or vertical band of visibility joining the two rectangles. Our results include necessary and sufficient conditions for $K_n$, $K_{m,n}$, and trees to be URVGs, as well as a number of general edge bounds.

Abstract: Until present visibility has been assessed as a Boo lean phenomenon in various kinds of Geographical Information Systems (GIS) and Computer Aided Design (CAD) systems: either things can or cannot be seen. Probabilistic Visibility (PV) is presented as an advanced alternative. PV represents the probability that one location of obj ect can be seen from another. Results of PV analysis are stored in Probabilistic Visibility Graphs (PVG) which stores the probability of visual contact for all relevant pairs of locations (e.g. raster cells) . A visibility graph will be calculated based on fi eld experiments and literature visibility decay - as a function of terrain, distance, viewing-angle and vegetation type -. The presentation takes its point of departure from analysis of visibility in forest ed environments. It describes the field experiment, an d how the graph is calculated and visualized. Furthermore, perspectives of the further developmen t and application of probabilistic visibility graph s will be given.

Abstract: This study proposes an evacuation efficiency evaluation model considering both the spatial configuration and the physical/Euclidean distance of the layout. One of the critical issues in the design of high-rise buildings is the evacuation planning, and a tool for evaluating the evacuation efficiency is highly needed. The conventional evacuation efficiency evaluation tools such as SIMULEX focus on the evacuation time, and thus are inappropriate in specifically pointing out the areas with evacuation difficulty within the layout being analyzed. This study focuses on the configurational properties of space because they are easily connected with the evacuation route based on plan layout. The concept of Visibility Graph Analysis (VGA) is adopted as the starting point for measuring the configurational properties. In addition, the physical evacuation distance is considered as another basic factor for the evaluation of evacuation efficiency in order to describe actual physical setting of the building. Evacuation Cost can be inferred as the sum of the traveling distance (Distance Cost) and the change of visual information within the evacuation process (Visibility Cost). Distance Cost from a point to exit is proportionate to the evacuation distance. The Visibility Cost is the degree of effort required to visually survey for exit, and it is related to visual point depth. Essentially, the proposed model calculates the Evacuation Cost from a certain point to exit by substituting the summation of Distance Cost and Visibility Cost for the visual depth of the visibility graph. With reference to this, three hypotheses to examine the relation between Distance Cost and Visibility Cost are initially proposed. Moreover, two additional hypotheses which adopt the concept of Angular Analysis are proposed. In this paper, this new method is applied to an actual high-rise building, I-Park in Seoul, and its results are compared with those of SIMULEX, the evacuation simulation program. The high correlation and stability of the results suggest that the model proposed in this study can replace SIMULEX. The proposed model offers a clear visualization of the evacuation efficiency within a building plan, which can play a major role in the design development process where decisions must be made between alternatives. It can also be used to work out evacuation planning of buildings that are already built.

Abstract: In this paper, we present an efficient method for finding collision-free trajectory for multiple unmanned aerial vehicles (UAVs) with kinematic constraints and for their rendezvous to form a formation. First, we construct a visibility graph that supports a minimum turning radius constraint when constructing the graph, so that additional smoothing process is not necessary. Second, we modify the standard A* to consider velocity conditions for rendezvous and collision avoidance with obstacles or other UAVs. Permitting velocity decrease only when it is required that the robot slow down the speed, unnecessary node expansions are avoided. This multi-vehicle problem is solved in a decoupled manner. In order to show the effectiveness of this approach, we present simulation results of rendezvous and independent flight for multiple UAVs.

Abstract: As new technological achievements take place in the robotic hardware field, an increased level of intelligence is required as well. The most fundamental intelligent task for a mobile robot is the ability to plan a valid path from its initial to terminal configurations while avoiding all obstacles located on its way. The robot motion planning problem came into existence in early 70’s and evolved to a vast and active research discipline as it is today. Numerous solution methods have been developed for robot motion planning since then, many of them being variations of a few general approaches: Roadmap, Cell Decomposition, Potential Fields, mathematical programming, and heuristic methods. Most classes of motion planning problems can be solved using these approaches, which are broadly surveyed in (Latombe, 1991), (Hwang & Ahuja, 1992), and (Choset et al., 2005). This chapter introduces two new offline path planning models which are founded on the Roadmap and Potential Fields classic motion planning approaches. These approaches have their unique characteristics and strategies for solving motion planning problems. In fact, each one has its own advantage that excels others in certain aspects. For instance, the Visibility Graph yields the shortest path; but its computational time exceeds other methods. Or, while the Voronoi Diagram plans the safest path and is easy to calculate in 2D, it often produces overly lengthy paths, and yields poor results in higher space dimensions. On the other hand, Potential Fields are easy to compute and are suitable for high dimensional problems, but they suffer from the local minima problem, and the oscillating paths generated near narrow passages of configuration space reduce their efficiency. A brief review on these underlying methods is given in this section. In order to benefit from the strong aspects of these classic path planning methods and compensate their drawbacks, a policy of combining these basic approaches into single architectures is adopted. In devising the new planners it is intended to aggregate the superiorities of these methods and work out efficient and reliable composite algorithms for robot motion planning.

Abstract: Small Uninhabited Aerial Vehicles that are to be operated at low altitude in obstacle prone environments require efficient mission planning capabilities. A fully autonomous miniature helicopter is being employed as a testbed to explore the development of new subsystems and algorithms for autonomous intelligent functions. An overview of the concepts related to mission planning and management for this research platform is presented. The underlying problem of mission planning for a helicopter differs from high-altitude systems by the operation close to the ground; among known or unknown obstacles. For the global path planning in a three-dimensional space, previously known information about the environment is used. The path planning is based on the visibility graph concept, which returns length-optimal paths in a plane. To account for the whole thee-dimensional space, this algorithm is run in several fixed-height layers, and then connected to the start and finish points, generating near-optimal paths in a very short time. Subsequently, the path planner is completed by a real-time collision avoidance planner. Upon detection of unknown, static or moving obstacles, the global path can be adapted in little time. The arithmetic techniques of this approach are further explained through examples. These mission planning components are integrated into the unmanned helicopter’s ground control station software and thus can be used to plan and monitor missions in obstructed environments. Experience from simulations and flight tests has shown, that the combination of those techniques results in a powerful, computationally efficient mission planning method that could be applied to unmanned vehicles with payload limitations.

Abstract: Visibility is an important topic in computer graphics, motion planning, and computational geometry. To deal with the increasing complexity of the scenes considered, some research has been performed in visibility processing in order to accelerate the visibility determination. Two of the most studied such structures are visibility graph and visibility complex. Visibility graph is a fundamental geometric structure which is used in many applications, including illumination and rendering, motion planning, pattern recognition, and sensor networks. While the concept of visibility graph is widely studied for 2D scenes, there is no acceptable equivalence of visibility graph for 3D space. Similarly, 3D visibility complex, proposed as an extension of visibility complex to 3D, is very complicated and cannot be used for visibility computations. In this paper, we propose a new model for defining the visibility relations in 3D. The main idea is to replace the role of lines and segments in 2D with planes and planar polygons in 3D. Moreover, we define two new structures, namely 3D visibility graph and partial visibility complex, which we believe is the natural way to extend the earlier models. We show how to compute these structures in acceptable times. We also use partial visibility complex to compute the view around a point in 3D in 0((|V(q)|+n2) log n) time, where |V(q)| is the size of the view.

Abstract: Coordination of multiple UAVs is an essential technology for various applications in robotics, automation, and artificial intelligence. In general, it includes 1) waypoints assignment and 2) trajectory generation. In this paper, we propose a new method for this problem. First, we modify the concept of the standard visibility graph to greatly improve the optimality of the generated trajectories and reduce the computational complexity. Second, we propose an efficient stochastic approach using simulated annealing that assigns waypoints to each UAV from the constructed visibility graph. Third, we describe a method to detect collision between two UAVs. Finally, we suggest an efficient method of controlling the velocity of UAVs using A* algorithm in order to avoid inter-UAV collision. We present simulation results from various environments that verify the effectiveness of our approach.

Abstract: This paper studies the question: What is the maximum integer kb,n such that every kb,n-colorable graph has a b-bend n-dimensional orthogonal box drawing? We give an exact answer for the orthogonal line drawing in all dimensions and for the 3-dimensional rectangle visibility representation. We present an upper and lower bound for the 3-dimensional orthogonal drawing by rectangles and general boxes. Particularly, we improve the best known upper bound for the 3-dimensional orthogonal box drawing from 183 to 42 and the lower bound from 3 to 22.

Abstract: Motion graphs have gained popularity in recent years as a means for re-using motion capture data by connecting previously unrelated segments of recorded motion. The techniques for controlling character movement via motion graphs have largely focused on path planning which is difficult because of the dense connections found in the graph. In addition, current use of learning controller directly on motion graph are not meaningful due to intractably large learning space generated by the dense graph. We introduce a novel alternative which allows high level control of character behavior using a dual representation we called a state-annotated motion graph. This special motion graph is generated from labelled data and then bound to a finite state machine with similar labels. At run-time, character behavior is simply controlled by switching states. We show that it is possible to generate rich, controllable motion without the need for deep planning. And learning from the state-level is capable of producing new high-level behaviors with a week of unsupervised training. For our results, we demonstrate that simple state-switching controllers can be coded intuitively to create various effects applied to an interactive fighting testbed.

Abstract: Given an n-node plane graph G, the visibility representation of G is concerned with drawing each node of G using a horizontal line segment such that the line segments associated with any two adjacent nodes of G are vertically visible to each other. Finding most compact visibility representations of plane graphs is not only of theoretical importance but also of practical interest, and has received much attention in the community of algorithmic graph theory. In this paper, we give a linear-time algorithm to find a visibility representation of G no wider than ⌊4n/3⌋ - 2. Our result improves upon the previously known upper bound 4n/3 + 2⌈ √n⌉, providing a positive answer to a conjecture suggested in the literature about whether an upper bound 4n/3 + O(1) on the required width can be achieved for an arbitrary plane graph. In fact, our visibility representation achieves optimality in the upper bound of width because the bound differs from the previously known lower bound ⌊4n/3⌋ - 3 only by one unit.

Abstract: Space syntax studies of pedestrian behaviour in building and urban environments have shown that there is a consistent correspondence between the configuration of space and the patterns of usage found within it. In particular, it has been shown that the topological relationships within a spatial system correlate to observed aggregate pedestrian movement. However, there is no proposed mechanism supporting the theory at the level of the individual. Although links between space syntax and individual movement decisions have been suggested through way-finding studies of building environments, virtual reality experiments, and agent-based models, none have proposed a formal link to the axial line analyses used within space syntax. Here we extend work on agent-based models to build a bridge between the line-based topological analyses of space syntax and visually directed agents, through the analysis of what we call ’through vision’. The decision rules for visually directed agents form a Markov transition matrix. We recap the mathematics of Markov chains in order to show that the steady state movement corresponds to an eigenvector of the transition matrix. As the agent transition matrix is extremely complex, we demonstrate that a good approximation of the eigenvector is achieved through the summation of the lines of vision through any one location within an environment. This set of lines forms a superset of the all-line axial map comprising the edges of the visibility graph, or lines of through movement. We show that the lines may be reduced in number (or bundled together) by an algorithmic process and connected into paths, thus making a direct connection between a moving individual with vision and the space syntactic topological analysis of space.

Abstract: Determining whether two segments s and t in a planar polygonal scene weakly see each other is a classical problem in computational geometry. In this problem we seek for a segment connecting two points of s and t without intersecting edges of the scene. In planar polygonal scenes, this problem is 3SUM-hard and its time complexity is Ω(n2) where n is the complexity of the scene. This problem can be defined in the same manner when s and t are any kind of objects in the plane. In this paper we consider this problem when s and t can be points, segments or convex polygons. We preprocess the scene so that for any given pair of query objects we can solve the problem efficiently. In our presented method, we preprocess the scene in O(n2+(Ɛ) time to build data structures of O(n2) total size by which the queries can be answered in O(n1+Ɛ) time. Our method is based on the extended visibility graph [1] and a range searching data structure presented by Chazelle et al. [2].

Abstract: We introduce a new type of diagram called the VV(c)-diagram (the visibility-Voronoi diagram for clearance c), which is a hybrid between the visibility graph and the Voronoi diagram of polygons in the plane. It evolves from the visibility graph to the Voronoi diagram as the parameter c grows from 0 to ∞. This diagram can be used for planning natural-looking paths for a robot translating amidst polygonal obstacles in the plane. A natural-looking path is short, smooth, and keeps--where possible--an amount of clearance c from the obstacles. The VV(c)-diagram contains such paths. We also propose an algorithm that is capable of preprocessing a scene of configuration-space polygonal obstacles and constructs a data structure called the VV-complex. The VV-complex can be used to efficiently plan motion paths for any start and goal configuration and any clearance value c, without having to explicitly construct the VV(c)-diagram for that c-value. The preprocessing time is O(n2 logn), where n is the total number of obstacle vertices, and the data structure can be queried directly for any c-value by merely performing a Dijkstra search. We have implemented a CGAL-based software package for computing the VV(c)-diagram in an exact manner for a given clearance value and used it to plan natural-looking paths in various applications.

Abstract: In this paper, a method for matching symbols in line-drawings is presented. Facing both segmentation and recognition of symbols is a difficult challenge. Starting from the results of a vectorization proce- dure, a visibility graph is built to enhance the main geometric constraints which were specified during the construction of the initial document. The cliques detection, which correspond to a perceptual grouping of primi- tives, is used in the system to detect regions of particular interest. Both opened and perceptually closed curves are identified from aggregation of cliques. Finally, the recognition stage uses an attributed edit distance technique to match approximated curves within the host attributed re- lation graph and the ones from a collection of symbols.