Autonomous robots navigate unknown and known environments. Regions of the environment that are not suitable for navigation may be in the form of stationary obstacles, limitations of the robot, unfavourable terrain/structure of the environment and sudden appearance of unaccounted obstacles. In the context of robotics for known environments such as an automated industrial environment or a warehouse, the environment is known apriori. That is, locations of regions not favourable for path planning, called static constraints, are known. However, there is still a possibility of encountering obstacles that are not part of the known environment, called dynamic constraints. They could be human beings, other robots (either part of them or as a whole), components belonging to the environment (boxes, cables, tools, manufactured products) and anticipated dangers (spills, compromised structures). So, path planning in such an environment consists of the following general two steps. First, a path between the desired source and target representation is generated. Second, segments of the path are evaluated for any encounter with constraints. The two general steps are accomplished differently by different algorithms, each with merits and demerits. The differing success of approaches used, depends on how the environment is represented. In methods that aim to save memory, the map is generated by sampling; so, the map is only as good as the sampling method. Then, the produced path has to be periodically checked for whether a segment of the path is truly constraint-free (static and dynamic). Sometimes, the method may stagnate at a non-optimal path, or may even not be able to complete the process of finding one. Alternatively, in approaches that store a detailed grid based map, changes in terrain and structure are expensive memory costs. The problem thus remains open, with the aim of representing the map with only constraint-free, navigable regions and generating paths as a reaction to, or in anticipation of, encountering new constraints. To solve this open problem, the Constraint Free Discretized Manifold based Path Planner is proposed. The algorithm divides the problem into two parts: the first focuses on maximizing knowledge of the map using manifolds, and the second uses homology and homotopy classes to compute paths. The first step is instrumental in constructing a complete representation of the navigable space as a manifold, free of constraints known apriori. Using topological tools, this representation is shown to have favourable properties, such that any path generated on it is guaranteed to be constraint-free. So, on this constraint-free manifold, no segment of the path has to be explicitly evaluated for a collision with a known constraint. It is shown that alternative spaces associated with the environment also share the same properties under certain conditions. Thus, one can transform the constraint-free path to other equivalent spaces. The second step deals with new knowledge of constraints that render the originally produced path as invalid. Using homology and homotopy, paths on the original manifold can be recognized and avoided by tuning a parameter, thus resulting in an alternative constraint free path. By operating on the discretized constraint free manifold, path classes characterize uniqueness of paths around constraints. This designation provides the ability to avoid a specific path class, should that not be desirable in light of newly encountered constraints. Then, the algorithm can be queried for a new path class free of constraints, without any explicit modification of the original map created and even when there is no physical indication of constraints. Tuning may be performed to produce more than one alternative path to be on the manifold. The proposed algorithm is seen to produce paths on the manifold with an average percentage path length deviation of 29.6%, which is over 70% less than those produced by sampling algorithms. The proposed algorithm also provides an increase in retention of usable samples by a margin of at least 30%, when compared with sampling algorithms. This is while maintaining on-par run times at worst, and better run times in most cases, when evaluated against other algorithms. These general trends hold true even when the proposed algorithm is utilized to generate alternative paths. Any deviation in path length related trend is seen only when a query is made to generate an alternative path that avoids the shortest path previously generated; a feature not present in sampling algorithms.

No description available.

We investigate quotient-fine nearness frames, showing that they are reflective in the category of strong nearness frames, and that, in those with spatial completion, any near subset is contained in a near grill. We construct two categories, each of which is shown to be equivalent to that of quotient-fine nearness frames. We also consider some subcategories of the category of nearness frames, which are co-hereditary and closed under coproducts. We give due attention to relations between these subcategories. We introduce totally strong nearness frames, whose category we show to be closed under completions. We investigate N-homomorphisms and remote points in the context of totally bounded uniform frames, showing the role played by these uniform N-homomorphisms in the transfer of remote points, and their relationship with C -quotient maps. A further study on grills enables us to establish, among other things, that grills are precisely unions of prime filters. We conclude the thesis by showing that the lattice of all nearnesses on a regular frame is a pseudo-frame, by which we mean a poset pretty much like a frame except for the possible absence of the bottom element. / Mathematical Sciences / Ph.D. (Mathematics)

Chan, Kam Fung. / "September 2010." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 61-62). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- Main result --- p.6 / Chapter 1.2 --- Useful definition --- p.7 / Chapter 1.3 --- Outline of proof of Theoreml.l --- p.11 / Chapter 2 --- Proof of Theorem1.2 and 13 --- p.12 / Chapter 2.1 --- The spaces \hV\ and \hW\ --- p.13 / Chapter 2.2 --- The space \hWloc\ --- p.19 / Chapter 2.3 --- The space \Wloc\ --- p.23 / Chapter 3 --- Proof of Theoreml4 --- p.26 / Chapter 3.1 --- Sheaves with category structure --- p.26 / Chapter 3.2 --- W° and hW° --- p.29 / Chapter 3.3 --- Armlets --- p.29 / Chapter 4 --- Proof of Theorem15 --- p.36 / Chapter 4.1 --- Homotopy colimit decompositions --- p.36 / Chapter 4.2 --- Introducing boundaries --- p.50 / Chapter 4.2.1 --- Proof of Theorem4.21 --- p.53 / Chapter 4.2.2 --- Proof of Lemma4.20 --- p.56 / Chapter 4.3 --- Using the Harer-Ivanov stabilization theorem --- p.58 / Bibliography --- p.61

We define an extension of the nth homotopy group which can distinguish a larger class of spaces. (E.g., a converging sequence of disjoint circles and the disjoint union of countably many circles, which have isomorphic fundamental groups, regardless of choice of basepoint.) We do this by introducing a generalization of homotopies, called component-homotopies, and defining the nth extended homotopy group to be the set of component-homotopy classes of maps from compact subsets of (0,1)n into a space, with a concatenation operation. We also introduce a method of tree-adjoinment for "connecting" disconnected metric spaces and show how this method can be used to calculate the extended homotopy groups of an arbitrary metric space.

<p>Let X be a smooth projective variety over C equipped with a C*-action whose fixed points are isolated. Let Y and Z be subvarieties of complementary dimentions in X which intersect properly. In this thesis we present an algorithm for computing the points of intersection between Y and Z based on homotopy continuation and the Bialynicki-Birula decompositions of X into locally closed invariant subsets. As an application we present a new solution to the inverse kinematic problem of a general six-revolute serial-link manipulator.</p>

There are many different notions of stratified spaces. This thesis concerns homotopically stratified spaces. These were defined by Frank Quinn in his paper Homotopically Stratified Sets ([16]). His definition of stratified space is very general and relates strata by “homotopy rather than geometric conditions”. This makes homotopically stratified spaces the ideal class of stratified spaces on which to define and study stratified homotopy theory. In the study of stratified spaces it is useful to examine spaces of popaths (paths which travel from lower strata to higher strata) and holinks (those spaces of popaths which immediately leave a lower stratum for their final stratum destination). It is not immediately clear that for adjacent strata these two path spaces are homotopically equivalent and even less clear that this equivalence can be constructed in a useful way. The first aim of this thesis is to prove such an equivalence exists for homotopically stratified spaces. We will define stratified analogues of the usual definitions of maps, homotopies and homotopy equivalences. Then we will provide an elementary criterion for deciding when a strongly stratified map is a stratified homotopy equivalence. This criterion states that a strongly stratified map is a stratified homotopy equivalence if and only if the induced maps on strata and holink spaces are homotopy equivalences. Using this criterion we will prove that any homotopically stratified space is stratified homotopy equivalent to a homotopically stratified space where neighborhoods of strata are mapping cylinders. Finally we will develop categorical descriptions of the class of homotopically stratified spaces up to stratified homotopy. The first of these categorical descriptions will involve categories with a topology on their object and morphism sets. The second categorical description will involve only categories with discrete object spaces.

Most spacecraft are designed to be maneuvered to achieve pointing goals. This is generally accomplished by designing a three-axis control system. This work explores new maneuver strategies when only two control inputs are available: (i) sequential single-axis maneuvers and (ii) three-dimensional (3D) coupled maneuvers. The sequential single-axis maneuver strategies are established for torque, time, and fuel minimization applications. The resulting control laws are more complicated than the equivalent results for three-axis control because of the highly nonlinear control switch-times. Classical control approaches lead to optimal, but discontinuous control profiles. This problem is overcome by introducing a torque-rate penalty for the torque minimization case. Alternative approaches are also considered for achieving smooth continuous control profiles by introducing a cubic polynomial multiplicative control switch smoother for the time and fuel minimization cases. Numerical and analytical results are presented to compare optimal maneuver strategies for both nominal and failed actuator cases. The 3D maneuver strategy introduces a homotopy algorithm to achieve optimal nonlinear maneuvers minimizing the torque. Two cases are considered: (i) one of the three-axis control actuators fails and (ii) two control actuators fail among four control actuators. The solution strategy first solves the case when all three actuators are available. Then, the failed actuator case is recovered by introducing a homotopy embedding parameter, ε, into the nonlinear dynamics equation. By sweeping ε, a sequence of neighboring optimal control problems is solved that starts with the original maneuver problem and arrives at the solution for the under-actuated case. As ε approaches 1, the designated actuator no longer provides control inputs to the spacecraft, effectively modeling the failed actuator condition. This problem is complex for two reasons: (i) the governing equations are nonlinear and (ii) ε fundamentally alters the spacecraft’s controllability. Davidenko’s method is introduced for developing an ordinary differential equation for the costate variable as a function of ε. For each value of ε, the costate initial conditions are iteratively adjusted so that the terminal boundary conditions for the 3D maneuver are achieved. Optimal control applications are presented for both rest-to-rest and motion-to-rest cases that demonstrate the effectiveness of the proposed algorithm.

A fibrewise coHopf space X over a base B is a sectioned space for which the diagonal map X —> X x _BX may be compressed into X V_BX up to fibrewise pointed homotopy. Such spaces have been investigated by I. M. James in the case where X is a sphere bundle over a sphere. The purpose of this thesis is to demonstrate some of the properties of fibrewise coHopf spaces over more general bases. Particular attention is given to sphere bundles and fibrations with spherical fibre. The fibrewise reduced suspension of a sectioned fibrewise space with closed sec- tion is fibrewise coHopf with associative comultiplication (up to fibrewise pointed homotopy) and a fibrewise inversion. Examples of fibrewise coHopf spaces not of this form are exhibited, and sufficient conditions are given to ensure that a fibrewise coHopf space has the primitive fibrewise pointed homotopy type of a fibrewise re- duced suspension, in terms of the dimension and connectivity of the space, its base and the fibres. It is shown that these conditions may be relaxed if the fibrewise coHopf structure on the space is assumed to be homotopy-associative. An example of a non-associative fibrewise coHopf sphere bundle is given. It is shown that, if q > 1 is odd, a sectioned orientable q-sphere bundle over a finite connected complex is fibrewise coHopf if and only if its fibrewise localisation at the prime 2 is fibrewise coHopf. Moreover, the fibrewise rationalisation of an odd-dimensional sphere bundle over a finite polyhedron whose fibrewise unreduced suspension is fibrewise coHopf is shown to be a trivial fibration. As an application, it is shown that new fibrewise coHopf spherical fibrations may be constructed by mixing. The Thorn space is used to determine the cohomology ring of the total space of a fibrewise coHopf sphere bundle in terms of that of its base, and a generalised Hopf invariant is constructed which vanishes on fibrewise coHopf sphere bundles.

Thesis (Ph. D.)--University of Washington, 2007. / Vita. Includes bibliographical references (leaves 48-49).