Influence-based motion planning algorithms for games

In games, motion planning has to do with the motion of non-player characters (NPCs) from one place to another in the game world. In today’s video games there are two major approaches for motion planning, namely, path-finding and influence fields. Path-finding algorithms deal with the problem of finding a path in a weighted search graph, whose nodes represent locations of a game world, and in which the connections among nodes (edges) have an associated cost/weight. In video games, the most employed pathfinders are A* and its variants, namely, Dijkstra’s algorithm and best-first search. As further will be addressed in detail, the former pathfinders cannot simulate or mimic the natural movement of humans, which is usually without discontinuities, i.e., smooth, even when there are sudden changes in direction. Additionally, there is another problem with the former pathfinders, namely, their lack of adaptivity when changes to the environment occur. Therefore, such pathfinders are not adaptive, i.e., they cannot handle with search graph modifications during path search as a consequence of an event that happened in the game (e.g., when a bridge connecting two graph nodes is destroyed by a missile). On the other hand, influence fields are a motion planning technique that does not suffer from the two problems above, i.e., they can provide smooth human-like movement and are adaptive. As seen further ahead, we will resort to a differentiable real function to represent the influence field associated with a game map as a summation of functions equally differentiable, each associated to a repeller or an attractor. The differentiability ensures that there are no abrupt changes in the influence field, consequently, the movement of any NPC will be smooth, regardless if the NPC walks in the game world in the growing sense of the function or not. Thus, it is enough to have a spline curve that interpolates the path nodes to mimic the smooth human-like movement. Moreover, given the nature of the differentiable real functions that represent an influence field, the removal or addition of a repeller/attractor (as the result of the destruction or the construction of a bridge) does not alter the differentiability of the global function associated with the map of a game. That is to say that, an influence field is adaptive, in that it adapts to changes in the virtual world during the gameplay. In spite of being able to solve the two problems of pathfinders, an influence field may still have local extrema, which, if reached, will prevent an NPC from fleeing from that location. The local extremum problem never occurs in pathfinders because the goal node is the sole global minimum of the cost function. Therefore, by conjugating the cost function with the influence function, the NPC will never be detained at any local extremum of the influence function, because the minimization of the cost function ensures that it will always walk in the direction of the goal node. That is, the conjugation between pathfinders and influence fields results in movement planning algorithms which, simultaneously, solve the problems of pathfinders and influence fields. As will be demonstrated throughout this thesis, it is possible to combine influence fields and A*, Dijkstra’s, and best-first search algorithms, so that we get hybrid algorithms that are adaptive. Besides, these algorithms can generate smooth paths that resemble the ones traveled by human beings, though path smoothness is not the main focus of this thesis. Nevertheless, it is not always possible to perform this conjugation between influence fields and pathfinders; an example of such a pathfinder is the fringe search algorithm, as well as the new pathfinder which is proposed in this thesis, designated as best neighbor first search.Em jogos de vídeo, o planeamento de movimento tem que ver com o movimento de NPCs (“Non-Player Characters”, do inglês) de um lugar para outro do mundo virtual de um jogo. Existem duas abordagens principais para o planeamento de movimento, nomeadamente descoberta de caminhos e campos de influência. Os algoritmos de descoberta de caminhos lidam com o problema de encontrar um caminho num grafo de pesquisa pesado, cujos nós representam localizações de um mapa de um jogo, e cujas ligações (arestas) entre nós têm um custo/peso associado. Os algoritmos de descoberta de caminhos mais utilizados em jogos são o A* e as suas variantes, nomeadamente, o algoritmo de Dijkstra e o algoritmo de pesquisa do melhor primeiro (“best-first search”, do inglês). Como se verá mais adiante, os algoritmos de descoberta de caminhos referidos não permitem simular ou imitar o movimento natural dos seres humanos, que geralmente não possui descontinuidades, i.e., o movimento é suave mesmo quando há mudanças repentinas de direcção. A juntar a este problema, existe um outro que afeta os algoritmos de descoberta de caminhos acima referidos, que tem que ver com a falta de adaptatividade destes algoritmos face a alterações ao mapa de um jogo. Ou seja, estes algoritmos não são adaptativos, pelo que não permitem lidar com alterações ao grafo durante a pesquisa de um caminho em resultado de algum evento ocorrido no jogo (e.g., uma ponte que ligava dois nós de um grafo foi destruída por um míssil). Por outro lado, os campos de influência são uma técnica de planeamento de movimento que não padece dos dois problemas acima referidos, i.e., os campos possibilitam um movimento suave semelhante ao realizado pelo ser humano e são adaptativos. Como se verá mais adiante, iremos recorrer a uma função real diferenciável para representar o campo de influência associado a um mapa de um jogo como um somatório de funções igualmente diferenciáveis, em que cada função está associada a um repulsor ou a um atractor. A diferenciabilidade garante que não existem alterações abruptas ao campo de influência; consequentemente, o movimento de qualquer NPC será suave, independentemente de o NPC caminhar no mapa de um jogo no sentido crescente ou no sentido decrescente da função. Assim, basta ter uma curva spline que interpola os nós do caminho de forma a simular o movimento suave de um ser humano. Além disso, dada a natureza das funções reais diferenciáveis que representam um campo de influência, a remoção ou adição de um repulsor/atractor (como resultado da destruição ou construção de uma ponte) não altera a diferenciabilidade da função global associada ao mapa de um jogo. Ou seja, um campo de influência é adaptativo, na medida em que se adapta a alterações que ocorram num mundo virtual durante uma sessão de jogo. Apesar de ser capaz de resolver os dois problemas dos algoritmos de descoberta de caminhos, um campo de influência ainda pode ter extremos locais, que, se alcançados, impedirão um NPC de fugir desse local. O problema do extremo local nunca ocorre nos algoritmos de descoberta de caminhos porque o nó de destino é o único mínimo global da função de custo. Portanto, ao conjugar a função de custo com a função de influência, o NPC nunca será retido num qualquer extremo local da função de influência, porque a minimização da função de custo garante que ele caminhe sempre na direção do nó de destino. Ou seja, a conjugação entre algoritmos de descoberta de caminhos e campos de influência tem como resultado algoritmos de planeamento de movimento que resolvem em simultâneo os problemas dos algoritmos de descoberta de caminhos e de campos de influência. Como será demonstrado ao longo desta tese, é possível combinar campos de influência e o algoritmo A*, o algoritmo de Dijkstra, e o algoritmo da pesquisa pelo melhor primeiro, de modo a obter algoritmos híbridos que são adaptativos. Além disso, esses algoritmos podem gerar caminhos suaves que se assemelham aos que são efetuados por seres humanos, embora a suavidade de caminhos não seja o foco principal desta tese. No entanto, nem sempre é possível realizar essa conjugação entre os campos de influência e os algoritmos de descoberta de caminhos; um exemplo é o algoritmo de pesquisa na franja (“fringe search”, do inglês), bem como o novo algoritmo de pesquisa proposto nesta tese, que se designa por algoritmo de pesquisa pelo melhor vizinho primeiro (“best neighbor first search”, do inglês)

Mathematical Optimization Approach for Facility Layout on Several Rows

The facility layout problem is concerned with finding an arrangement of non-overlapping indivisible departments within a facility so as to minimize the total expected flow cost. In this paper we consider the special case of multi-row layout in which all the departments are to be placed in three or more rows, and our focus is on, for the first time, solutions for large instances. We first propose a new mixed integer linear programming formulation that uses continuous variables to represent the departments’ location in both x and y coordinates, where x represents the position of a department within a row and y represents the row assigned to the department. We prove that this formulation always achieves an optimal solution with integer values of y, but it is limited to solving instances with up to 13 departments. This limitation motivates the application of a two-stage optimization algorithm that combines two mathematical optimization models by taking the output of the first-stage model as the input of the second-stage model. This algorithm is, to the best of our knowledge, the first one in the literature reporting solutions for instances with up to 100 departments.publishersversionpublishe

Cumulative hydropathic topology of a voltage-gated sodium channel at atomic resolution

Voltage-gated sodium channels (NavChs) are biological pores that control the ow of sodium ions through the cell membrane. In humans, mutations in genes encoding NavChs can disrupt physiological cellular activity thus leading to a wide spectrum of diseases. Here, we present a topological connection between the functional architecture of a NavAb bacterial channel and accumulation of atomic hydropathicity around its pore. This connection is established via a scaling analysis methodology that elucidates how intrachannel hydropathic density variations translate into hydropathic dipole field configurations along the pore. Our findings suggest the existence of a non random cumulative hydropathic topology that is organized parallel to the membrane surface so that pore's stability, as well as, gating behavior are guaranteed. Given the biophysical significance of the hydropathic effect, our study seeks to provide a computational framework for studying cumulative hydropathic topological properties of NavChs and pore-forming proteins in general. This article is protected by copyright. All rights reserved

A New Mathematical Programming Framework for Facility Layout Design

We present a new framework for efficiently finding competitive solutions for the facility layout problem. This framework is based on the combination of two new mathematical programming models. The first model is a relaxation of the layout problem and is intended to find good starting points for the iterative algorithm used to solve the second model. The second model is an exact formulation of the facility layout problem as a non-convex mathematical program with equilibrium constraints (MPEC). Aspect ratio constraints, which are frequently used in facility layout methods to restrict the occurrence of overly long and narrow departments in the computed layouts, are easily incorporated into this new framework. Finally, we present computational results showing that both models, and hence the complete framework, can be solved efficiently using widely available optimization software. This important feature of the new framework implies that it can be used to find competitive layouts with relatively little computational effort. This is advantageous for a user who wishes to consider several competitive layouts rather than simply using the mathematically optimal layout

Computing complete Lyapunov functions for discrete-time dynamical systems

A complete Lyapunov function characterizes the behaviour of a general discrete-time dynamical system. In particular, it divides the state space into the chain-recurrent set where the complete Lyapunov function is constant along trajectories and the part where the flow is gradient-like and the complete Lyapunov function is strictly decreasing along solutions. Moreover, the level sets of a complete Lyapunov function provide information about attractors, repellers, and basins of attraction. We propose two novel classes of methods to compute complete Lyapunov functions for a general discrete-time dynamical system given by an iteration. The first class of methods computes a complete Lyapunov function by approximating the solution of an ill-posed equation for its discrete orbital derivative using meshfree collocation. The second class of methods computes a complete Lyapunov function as solution of a minimization problem in a reproducing kernel Hilbert space. We apply both classes of methods to several examples

PeF: Poisson's Equation Based Large-Scale Fixed-Outline Floorplanning

Floorplanning is the first stage of VLSI physical design. An effective floorplanning engine definitely has positive impact on chip design speed, quality and performance. In this paper, we present a novel mathematical model to characterize non-overlapping of modules, and propose a flat fixed-outline floorplanning algorithm based on the VLSI global placement approach using Poisson's equation. The algorithm consists of global floorplanning and legalization phases. In global floorplanning, we redefine the potential energy of each module based on the novel mathematical model for characterizing non-overlapping of modules and an analytical solution of Poisson's equation. In this scheme, the widths of soft modules appear as variables in the energy function and can be optimized. Moreover, we design a fast approximate computation scheme for partial derivatives of the potential energy. In legalization, based on the defined horizontal and vertical constraint graphs, we eliminate overlaps between modules remained after global floorplanning, by modifying relative positions of modules. Experiments on the MCNC, GSRC, HB+ and ami49\_x benchmarks show that, our algorithm improves the average wirelength by at least 2\% and 5\% on small and large scale benchmarks with certain whitespace, respectively, compared to state-of-the-art floorplanners

Categoriality and continuity in prosodic prominence

Prosody has been characterised as a "half-tamed savage" being shaped by both discrete, categorical aspects as well as gradient, continuous phenomena. This book is concerned with the relation of the "wild" and the "tamed" sides of prosodic prominence. It reviews problems that arise from a strict separation of categorical and continuous representations in models of phonetics and phonology, and it explores the potential role of descriptions aimed at reconciling the two domains. In doing so, the book offers an introduction to dynamical systems, a framework that has been studied extensively in the last decades to model speech production and perception. The reported acoustic and articulatory data presented in this book show that categorical and continuous modulations used to enhance prosodic prominence are deeply intertwined and even exhibit a kind of symbiosis. A multi-dimensional dynamical model of prosodic prominence is sketched, based on the empirical data, combining tonal and articulatory aspects of prosodic focus marking. The model demonstrates how categorical and continuous aspects can be inte- grated in a joint theoretical treatment that overcomes a strict separation of phonetics and phonology

Complete lyapunov functions: determination of the chain-recurrent set using the gradient

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