京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21540号 / 理博第4447号 / 新制||理||1639(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 玉川 安騎男, 教授 向井 茂, 教授 望月 新一 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21544号 / 理博第4451号 / 新制||理||1639(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 星 裕一郎, 教授 向井 茂, 教授 望月 新一 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20158号 / 理博第4243号 / 新制||理||1610(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 望月 新一, 教授 岡本 久, 教授 玉川 安騎男 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

A configuration space is a space whose points represent the possible states of a given physical system. As such they appear naturally both in theoretical physics and technical applications. For an example of the former, in analytical mechanics, the Lagrangian and Hamiltonian formulations of classical mechanics depend heavily on the use of a physical system’s configuration space for the description of its kinematical and dynamical behavior, and importantly, its evolution in time. As an example of a technical application, consider robotics, where the space of possible configurations of the mechanical linkages that make up a robot is an important tool in motion planning. In this case it is of particular interest to study the singularities of these mechanical linkages, to see if a given configuration is singular or not. This can be done with the help of configuration spaces and their topological properties. Arguably, the simplest configuration space possible arises when the system is just a collection of point-like particles in a plane. Despite its simplicity, the corresponding configuration space has substantial complexity and is of great interest in mathematics, physics and technology: For instance, it arises naturally in the mathematical modelling of robots performing tasks in a warehouse. In this thesis we go through the mathematics necessary to study the behaviour of paths in this space, which corresponds to motions of the particles. We use the theory of groups, algebraic topology, and manifolds to examine the properties of the configuration space of point-like particles in a plane. An important role in the discussion will be played by braids, which are certain collections of curves, interlaced in three-space. They are connected to many different topics in algebra, geometry, and mathematical physics, such as representation theory, the Yang-Baxter equation and knot theory. They are also important in their own right. Here we focus on their relation to configurations of points.

A graph configuration space is generated by the movement of a finite number of robots on a graph. These configuration spaces of points in a graph are topologically interesting objects. By using local, combinatorial properties, we define a new classification of graphs whose configuration spaces are pseudomanifolds with boundary. In algebraic topology, graph configuration spaces are closely related to classical braid groups, which can be described as fundamental groups of configuration spaces of points in the plane. We examine this relationship by finding a presentation for the fundamental group of one graph configuration space. / Master of Science

The three-body bound state problem is considered using configuration-space Faddeev equations within the framework of the total-angular-momentum representation. Different three-body systems are considered, the main concern of the investigation being the i) calculation of binding energies for weakly bounded trimers, ii) handling of systems with a plethora of states, iii) importance of three-body forces in trimers, and iv) the development of a numerical technique for reliably handling three-dimensional integrodifferential equations. In this respect we considered the three-body nuclear problem, the 4He trimer, and the Ozone (16 0 3 3) system. In practice, we solve the three-dimensional equations using the orthogonal collocation method with triquintic Hermite splines. The resulting eigenvalue equation is handled using the explicitly Restarted Arnoldi Method in conjunction with the Chebyshev polynomials to improve convergence. To further facilitate convergence, the grid knots are distributed quadratically, such that there are more grid points in regions where the potential is stronger. The so-called tensor-trick technique is also employed to handle the large matrices involved. The computation of the many and dense states for the Ozone case is best implemented using the global minimization program PANMIN based on the well known MERLIN optimization program. Stable results comparable to those of other methods were obtained for both nucleonic and molecular systems considered. / Physics / D.Phil. (Physics)

The three-body bound state problem is considered using configuration-space Faddeev equations within the framework of the total-angular-momentum representation. Different three-body systems are considered, the main concern of the investigation being the i) calculation of binding energies for weakly bounded trimers, ii) handling of systems with a plethora of states, iii) importance of three-body forces in trimers, and iv) the development of a numerical technique for reliably handling three-dimensional integrodifferential equations. In this respect we considered the three-body nuclear problem, the 4He trimer, and the Ozone (16 0 3 3) system. In practice, we solve the three-dimensional equations using the orthogonal collocation method with triquintic Hermite splines. The resulting eigenvalue equation is handled using the explicitly Restarted Arnoldi Method in conjunction with the Chebyshev polynomials to improve convergence. To further facilitate convergence, the grid knots are distributed quadratically, such that there are more grid points in regions where the potential is stronger. The so-called tensor-trick technique is also employed to handle the large matrices involved. The computation of the many and dense states for the Ozone case is best implemented using the global minimization program PANMIN based on the well known MERLIN optimization program. Stable results comparable to those of other methods were obtained for both nucleonic and molecular systems considered. / Physics / D.Phil. (Physics)

Kinodynamic motion planning is an important problem in robotics. It consists of planning the dynamic motion of a robotic system taking into account its kinematic and dynamic constraints. For this class of problems, high dimensionality is a major difficulty and finding an exact time optimal robot motion trajectory is proven to be NP-hard. Probabilistic approximate techniques have therefore been proposed in the literature to solve particular problem instances. These methods include Randomized Potential Field Planners (RPP), Probabilistic Roadmaps (PRM) and Rapidly Exploring Random Trees (RRT). When physical obstacles and differential constraints are added to the problem, applying RPPs or PRMs encounter difficulties. In order to handle these difficulties, RRTs have been proposed. In this study, we consider a multiple camera industrial inspection problem where the concurrent motion of these cameras needs to be planned. The cameras are required to capture maximum number of defect locations while globally avoiding collisions with each other and with obstacles. Our approach is to consider a solution to the kinodynamic planning problem of multiple camera inspection by making use of the RRT algorithm. We explore and resolve issues arising when RRTs are applied to this specific problem class. Along these lines, we consider the cases of a single camera without obstacles and then with obstacles. Then, we attempt to extend the study to the case of multiple camera where we also need to avoid collisions between cameras. We present simulation results to show the performance of our RRT based approach to different instrument configurations and compare with existing deterministic approaches.

Drastic changes in aircraft operational requirements and the emergence of new enabling technologies often occur symbiotically with advances in technology inducing new requirements and vice versa. These changes sometimes lead to the design of vehicle concepts for which no prior art exists. They lead to revolutionary concepts. In such cases the basic form of the vehicle geometry can no longer be determined through an ex ante survey of prior art as depicted by aircraft concepts in the historical domain. Ideally, baseline geometries for revolutionary concepts would be the result of exhaustive configuration space exploration and optimization. Numerous component layouts and their implications for the minimum external dimensions of the resultant vehicle would be evaluated. The dimensions of the minimum enclosing envelope for the best component layout(s) (as per the design need) would then be used as a basis for the selection of a baseline geometry. Unfortunately layout design spaces are inherently large and the key contributing analysis i.e. collision detection, can be very expensive as well. Even when an appropriate baseline geometry has been identified, another hurdle i.e. vehicle scaling has to be overcome. Through the design of a notional Cessna C-172R powered by a liquid hydrogen Proton Exchange Membrane (PEM) fuel cell, it has been demonstrated that the various forms of vehicle scaling i.e. photographic and historical-data-based scaling can result in highly sub-optimal results even for very small O(10-3) scale factors. There is therefore a need for higher fidelity vehicle scaling laws especially since emergent technologies tend to be volumetrically and/or gravimetrically constrained when compared to incumbents. The Configuration-space Exploration and Scaling Methodology (CESM) is postulated herein as a solution to the above-mentioned challenges. This bottom-up methodology entails the representation of component or sub-system geometries as matrices of points in 3D space. These typically large matrices are reduced using minimal convex sets or convex hulls. This reduction leads to significant gains in collision detection speed at minimal approximation expense. (The Gilbert-Johnson-Keerthi algorithm is used for collision detection purposes in this methodology.) Once the components are laid out, their collective convex hull (from here on out referred to as the super-hull) is used to approximate the inner mold line of the minimum enclosing envelope of the vehicle concept. A sectional slicing algorithm is used to extract the sectional dimensions of this envelope. An offset is added to these dimensions in order to come up with the sectional fuselage dimensions. Once the lift and control surfaces are added, vehicle level objective functions can be evaluated and compared to other designs. For each design, changes in the super-hull dimensions in response to perturbations in requirements can be tracked and regressed to create custom geometric scaling laws. The regressions are based on dimensionally consistent parameter groups in order to come up with dimensionally consistent and thus physically meaningful laws. CESM enables the designer to maintain design freedom by portably carrying multiple designs deeper into the design process. Also since CESM is a bottom-up approach, all proposed baseline concepts are implicitly volumetrically feasible. Furthermore the scaling laws developed from custom data for each concept are subject to less design noise than say, regression based approaches. Through these laws, key physics-based characteristics of vehicle subsystems such as energy density can be mapped onto key system level metrics such as fuselage volume or take-off gross weight. These laws can then substitute some historical-data based analyses thereby improving the fidelity of the analyses and reducing design time.

Robots moving about in complex environments must be capable of determining and performing difficult motion sequences to accomplish tasks. As the tasks become more complicated, robots with greater dexterity are required. An increase in the number of degrees of freedom and a desire for autonomy in uncertain environments with real-time requirements leaves much room for improvement in the current popular robot motion planning algorithms. In this thesis, state of the art robot motion planning techniques are surveyed. A solution to the general movers problem in the context of motion planning for robots is presented. The proposed robot motion planner solves the general movers problem using a sample-based tree planner combined with an incremental simulator. The robot motion planner is demonstrated both in simulation and the real world. Experiments are conducted and the results analyzed. Based on the results, methods for tuning the robot motion planner to improve the performance are proposed.