Würzburg, Univ., Diss., 2008

In this thesis the cubic spline guidance law and its variants are derived. A detailed analysis is carried out to find the initial conditions for successful interception. The results are applied to three dimensional guidance design and for solving waypoint following problems. The basic cubic spline guidance law is derived for intercepting a stationary target at a desired impact angle in a surface-to-surface engagement scenario. The guidance law is obtained using an inverse method, from a cubic spline curve based trajectory. For overcoming the drawbacks of the basic cubic spline guidance law, it is modified by introducing an additional parameter. This modification has an interesting feature that the guidance command can be obtained using a single cubic spline polynomial even for impact angles greater than π/2, while resulting in substantial improvement in the guidance performance in terms of lateral acceleration demand and length of the trajectory. For imparting robustness to the cubic spline guidance law, in the presence of uncertainties and acceleration saturation, an explicit guidance expression is also derived. A comprehensive capturability study of the proposed guidance law is carried out. The capturability for the cubic spline guidance law is defined in terms of the set of all feasible initial conditions for successful interception. This set is analytically derived and its dependence on various factors, such as initial engagement geometry and interceptor capability, are also established. The basic cubic spline guidance and its variants are also derived for a three dimen- sional scenario. The novelty of the present work lies in the particular representation of the three dimensional cubic spline curve and the adoption of the analytical results available for two dimensional cubic spline guidance law. This enables selection of the boundary condition at launch for given terminal boundary condition and also in avoiding the singularities associated with the inverse method based guidance laws. For establishing the feasibility of the guidance laws in the real world, the rigid body dynamics of the interceptor is presented as a 6 degrees-of-freedom model. Further, using a simplified model, elementary autopilots are also designed. The successful interception of the target in the presence of the rigid body dynamics proves practical applicability of the cubic spline based guidance laws. Finally, the theory developed in the first part of the thesis is applied to solve the waypoint following problem. A smooth path is designed for transition of vehicle velocity from incoming to outgoing direction. The approach developed is similar to Dubins’ path, as it comprises line–cubic spline–line segments. The important feature of this method is that the cubic spline segments are fitted such that the path curvature is bounded by a pre-specified constrained value and the acceleration demand for following the smooth path obtained by this method, gradually increases to the maximum value and then decreases. This property is advantageous from a practical point of view. All the results obtained are verified with the help of numerical simulations which are included in the thesis. The proposed cubic spline guidance law is conceptually simple, does not use linearised kinematic equations, is independent of time-to-go es- timates, and is also computationally inexpensive.

JPS Teknik AB in Färila, located in the middle of Sweden, is a company in manufacturingsaw units to harvester heads. This thesis is about the company's best selling saw unit, JPSR5500. The saw unit is powered by a hydraulic motor, the torque is transmitted from themotor into the saw unit thought a spline coupling. In this spline coupling, some of the unitshave been weared out after about 2000 harvester machine hours. When the splines wear outthe failure induce a total stop, the bolt connecting the hub and the motor shaft breaks, and thesaw unit assembly get loose. The purpose with this study is to find the root cause of thefailure and the goal is to give suggestions to improve the design to avoid failure. The method used to achieve the purpose is based on Ullman's mechanical design processwhere a root cause analysis is a central part. This analysis is built on two main pillars, aliterature study concerning previous research about wear in spline couplings and a knowledgebased study about the product with a customer focus. A sample of previous research in thistopic is about the load distribution in a spline coupling due to different load cases and angularmisalignment, wear mechanisms and the effect of washers in a pretension bolt joint with adynamic working condition. The main conclusions of this study is that the hydraulic motor axis should be extended. Theoperator's manual and assembly instructions should be updated and revised annually, thecompany should also invest in education and training for the users of the product.

Thesis (MSc)--University of Stellenbosch, 2000. / ENGLISH ABSTRACT: In this thesis we study the underlying mathematical principles of stationary subdivision, which can be regarded as an iterative recursion scheme for the generation of smooth curves and surfaces in computer graphics. An important tool for our work is Fourier analysis, from which we state some standard results, and give the proof of one non-standard result. Next, since cardinal spline functions have strong links with subdivision, we devote a chapter to this subject, proving also that the cardinal B-splines are refinable, and that the corresponding Euler-Frobenius polynomial has a certain zero structure which has important implications in our eventual applications. The concepts of a stationary subdivision scheme and its convergence are then introduced, with as motivating example the de Rahm-Chaikin algorithm. Standard results on convergence and regularity for the case of positive masks are quoted and graphically illustrated. Next, we introduce the concept of interpolatory stationary subdivision, in which case the limit curve contains all the original control points. We prove a certain set of sufficient conditions on the mask for convergence, at the same time also proving the existence and other salient properties of the associated refinable function. Next, we show how the analysis of a certain Bezout identity leads to the characterisation of a class of symmetric masks which satisfy the abovementioned sufficient conditions. Finally, we show that specific special cases of the Bezout identity yield convergent interpolatory symmetric subdivision schemes which are identical to choosing the corresponding mask coefficients equal to certain point evaluations of, respectively, a fundamental Lagrange interpolation polynomial and a fundamental cardinal spline interpolant. The latter procedure, which is known as the Deslauriers-Dubuc subdivision scheme in the case of a polynomial interpolant, has received attention in recent work, and our approach provides a convergence result for such schemes in a more general framework. Throughout the thesis, numerical illustrations of our results are provided by means of graphs. / AFRIKAANSE OPSOMMING: In hierdie tesis ondersoek ons die onderliggende wiskundige beginsels van stasionêre onderverdeling, wat beskou kan word as 'n iteratiewe rekursiewe skema vir die generering van gladde krommes en oppervlakke in rekenaargrafika. 'n Belangrike stuk gereedskap vir ons werk is Fourieranalise, waaruit ons sekere standaardresuJtate formuleer, en die bewys gee van een nie-standaard resultaat. Daarna, aangesien kardinale latfunksies sterk bande het met onderverdeling, wy ons 'n hoofstuk aan hierdie onderwerp, waarin ons ook bewys dat die kardinale B-Iatfunksies verfynbaar is, en dat die ooreenkomstige Euler-Frobenius polinoom 'n sekere nulpuntstruktuur het wat belangrike implikasies het in ons uiteindelike toepassings. Die konsepte van 'n stasionêre onderverdelingskema en die konvergensie daarvan word dan bekendgestel, met as motiverende voorbeeld die de Rahm-Chaikin algoritme. Standaardresultate oor konvergensie en regulariteit vir die geval van positiewe maskers word aangehaal en grafies geïllustreer. Vervolgens stelons die konsep van interpolerende stasionêre onderverdeling bekend, in welke geval die limietkromme al die oorspronklike kontrolepunte bevat. Ons bewys 'n sekere versameling van voldoende voorwaardes op die masker vir konvergensie, en bewys terselfdertyd die bestaan en ander toepaslike eienskappe van die ge-assosieerde verfynbare funksie. Daarna wys ons hoedat die analise van 'n sekere Bezout identiteit lei tot die karakterisering van 'n klas simmetriese maskers wat die bovermelde voldoende voorwaardes bevredig. Laastens wys ons dat spesifieke spesiale gevalle van die Bezout identiteit konvergente interpolerende simmetriese onderverdelingskemas lewer wat identies is daaraan om die ooreenkomstige maskerkoëffisientegelyk aan sekere puntevaluasies van, onderskeidelik, 'n fundamentele Lagrange interpolasiepolinoom en 'n kardinale latfunksie-interpolant te kies. Laasgenoemde prosedure, wat bekend staan as die Deslauriers-Dubuc onderverdelingskema in die geval van 'n polinoominterpolant, het aandag ontvang in onlangse werk, en ons benadering verskaf 'n konvergensieresultaat vir sulke skemas in 'n meer algemene raamwerk. Deurgaans in die tesis word numeriese illustrasies van ons resultate met behulp van grafieke verskaf.

No description available.

Unobscured reflective optical systems can be more transmissive and of higher diffraction quality than classical systems. Unobscured systems are generated by decentering symmetric systems, tilting elements to correct coma or astigmatism along a real ray, or by cross-tilting elements to control astigmatism. Such a system of relatively high quality may be further corrected with a general spline surface. For spline surfaces, optical aberration coefficients are undefined. This study developed real ray analysis and design techniques for general optical systems. A decentered symmetric system with a field correcting spline surface was designed. The optical design program ACCOS V was used for most design and analysis tasks. Design and analysis of general systems are considered first. Basic system quantities of image location, scaling, and irradiation are defined with real rays. Spline surfaces are discussed with special emphasis on features important in optical design. Real ray analytical techniques of composite spot diagrams across the image, footprints on spline surfaces, wavefront aperture maps, and spline surface maps are described. The use of these tools in general system design procedures is discussed. Standard telescope objectives of f/8.5 were considered as base designs for systems with spline surfaces. A spline surface was added to the decentered Schmidt-Cassegrain. Optimization yielded diffraction-limited performance across a 0.85 degree square field. The spline system was compared to the Galileo spacecraft narrow angle lens and a three-mirror decentered design. It had a far wider field than the Galileo but at a lower quality. Diffraction quality was better than that of the three-mirror system. Simple tolerances were considered for the spline system. The allowable effect of a thermal gradient was estimated by bending the reference axis. Decentration and figure tolerances for the spline were commensurate with classical surfaces. Techniques presented were shown to be useful for design and analysis of general systems. Spline surfaces were found to be useful in optimization of such systems. This work was supported by the Director's Discretionary Fund, Jet Propulsion Laboratory, California Institute of Technology.

The purpose of this investigation is to examine synthesis for reconstruction of electrostatic lenses having an axial potential distribution four times continuously differentiable. The solution of the electrode and pole piece reconstruction is given. Spline functions are used to approximate a continuous function to fit a curve. The present method of synthesis is based on cubic spline functions, which have only two simultaneous continuous derivatives, and all the other higher derivatives are ignored. The fifth-order or quintic spline is introduced simply because it has four simultaneous continuous derivatives. So the reconstruction program would have three terms appearing in the series expansion of the off-axis potential distribution, with regard to two terms when using cubic functions.

Creation of accurate and coherent surface models is vital to the effective planning and construction of flood control and hurricane protection projects. Typically, topographic surface models are synthesized from Delaunay triangulations or interpolated raster grids. Although these techniques are adequate in most general situations, they do not effectively address the specific case where topographic data is available only as cross-section and profile centerline data, such as the elevation sampling produced by traditional hydrographic surveys. The hydraulic spline algorithm was developed to generate irregular two-dimensional channel grids from hydrographic cross-sections at any desired resolution. Hydraulic spline output grids can be easily merged with datasets of higher resolution, such as LIDAR data, to build a complete model of channel geometry and overbank topography. In testing, the hydraulic spline algorithm faithfully reproduces elevations of known input cross-section points where they exist, while generating a smooth transition between known cross-sections. The algorithm performs particularly well compared to traditional techniques with respect to aesthetics and accuracy when input data is sparse. These qualities make the hydraulic spline an ideal choice for practical applications where available data may be limited due to historic or budgetary reasons.

No description available.

Splines are piecewise polynomial functions of a given order of smoothness r. Given complex delta the set of splines of degree less than or equal to d forms a vector space and is denoted by Sr d(delta). For a simplicial complex delta, Strang conjectured a lower bound on the dimension of spline space Srd(delta) and it is known that the equality holds for sufficiently large d. It is called the dimension formula. In this dissertation, we approach the study of splines from the viewpoint of algebraic geometry. This dissertation follows the works of Lau and Stiller. They introduced the conformality conditions which lead to the machinery of sheaves and cohomology which provided a powerful type of generalization of linear algebra. First, we try to analyze effects in the dimensions of spline spaces when we remove or add certain faces in the given complex. We define the cofactor spaces and cofactor maps from the given complexes and use them to interpret the changes in the dimensions of spline spaces. Second, given polyhedral complex delta, we break it into two smaller complexes delta1 and delta2 which are usually easier to handle. We will find conditions for delta1 and delta2 which guarantee that the dimension formula holds for the original complex delta. Next, we use the previous splitting method on certain types of triangulations. We explain how to break the given triangulation and show what kind of simple complexes we end up with. Finally, we study the "2r+1" conjecture on a certain triangulation. The "2r+1" conjecture is that the dimension formula holds on any triangulation for d >/= 2r + 1. We know that the conjecture is sharp because the dimension formula fails on a certain triangulation for d = 2r, but we do not know if it holds on the same triangulation when d = 2r + 1. It is related to a Toeplitz matrix.