Localization is the process of determining the rigid body translations and rotations that must be performed on the set of points measured on a manufactured surface to move those points into closest correspondence with the ideal design surface. In unconstrained localization all points have equal effect on the determination of the rigid body transformation, while constrained localization allows a subset of the points to have stronger influence on the transformation. The measured points are physical points in space obtained by direct measurement of a manufactured marine propeller blade. The ideal design surface is the surface description of the propeller blade provided by the blade designer. Given that the measured blade is manufactured from the design surface description, the localization determines an Euclidean motion that brings the measured points of the manufactured surface as close as possible to the design surface. An additional option is to determine an offset distance, such that the Euclidean motion brings the measured points as close as possible to the offset of the design surface. For this optimization problem the offset distance is a seventh parameter that must be determined in addition to the six parameters of the Euclidean motion.
After localization the offset of the design surface that was determined can be used to extract the gross geometric features of the manufactured blade. These features have important hydrodynamic function and include the camber surface, section thickness function, pitch, rake, skew, chord length, maximum thickness, maximum camber, and the leading edge curve. The approximation of the camber surface, which is the basis of most of the remaining features, is an intricate problem relying on an extension of the concept of a Brooks ribbon. It requires the solution of a system of non-linear differential equations and a complicated error evaluation scheme.
Keywords: accuracy control, CAD/CAM, features, inspection, localization, offsets.
Robustness and accuracy are two of the most fundamental outstanding problems in computational geometry and geometric modeling, despite significant advances of these fields in the last three decades. Since curves and surfaces are typically represented by parametric piecewise polynomial equations, the governing equations for geometric processing and shape interrogation in general reduce to solving systems of nonlinear polynomial equations or irrational equations involving nonlinear polynomials and square roots of polynomials. The square root arises for example from the normalization of the normal vector and from the analytical expressions of curvatures. This thesis addresses the development of a new robust and accurate computational method to compute all real roots of such systems within a finite box. A key component of our method is the reduction of the problem involving irrational equations into solution of systems of nonlinear polynomial equations of higher dimensionality through the introduction of auxiliary variables. A fundamental component of our method for solving general non-linear polynomial equation systems is rounded interval arithmetic in the context of Bernstein subdivision. Rounded interval arithmetic leads to numerical robustness and provides results with numerical certainty and verifiability. Several computational geometry applications of the new nonlinear solver are studied to evaluate the method in a realistic context. These applications include continuous decomposition of parametric polynomial surface patches into a set of trimmed patches each with a specified range of curvature (Gaussian, mean, maximum principal, minimum principal and root mean square curvature). Such surface decomposition has applications in sculptured surface fairing for design, tessellation for analysis, and manufacture. Another application includes the computation of self-intersections of offset curves or of intersections of two offsets of two curves. Such algorithm can be applied to design specification, feature recognition through construction of skeletons of geometric models, and manufacture. Finally, the nonlinear solver is used in computing all umbilics of parametric polynomial surface patches for application in surface recognition and tessellation.
Navigation currently limits the existing and potential capabilities of small autonomous underwater vehicles (AUVs). Methods of internal AUV navigation (e.g. dead reckoning and inertial) have temporally unbounded accumulation of position error. On the other hand, beacon based systems require the costly and time consuming placement, calibration, and maintenance of an array of underwater beacons. In this work, we investigate an alternative approach, that is, geophysical map based navigation wherein we match the measurement of simple geophysical properties by vehicle-mounted sensors to a priori maps (computer models) stored on board the vehicle. The objective being an algorithm applicable to small, power-limited AUVs and which performs in real time to a required resolution with bounded position error (or degree of accuracy). To accomplish this, we address issues of representation and storage and derive methods for the interrogation of these computer maps.
Interval B-splines (IBS) are introduced for the non-linear representation of two-dimensional geophysical parameters. A method for the creation of IBS surfaces from measured data is developed such that the topography and uncertainty in the data are represented in a manner that provides for high accuracy and resolution, data reduction, and efficient interrogation with a guaranteed bounded approximation error. Interrogation is divided into macro- and micro-navigation. Macro-navigation involves two processes for global position estimation. A standard filter mechanism predicts the position of the vehicle and is based on a bounded interval error model to incorporate non-Gaussian errors in vehicle translation due to the underwater environment. It limits the solution space (i.e. possible vehicle positions) and, therefore, avoids extraneous solutions. When multiple solutions are found, a multiple hypotheses tracking mechanism is applied until a unique vehicle location is determined. Micro-navigation involves the solution of a system of nonlinear polynomial equations with interval coefficients and provides fine-scale, quantitative position determination. This system is formulated as the intersection of contours established on each map that are derived from the simultaneous measurement of associated geophysical parameters by vehicle mounted sensors. Since these contour intersections represent locations where such measurements could occur, they also represent the complete set of possible vehicle locations. Finally, a simulation model for the underwater environment is developed to study the behavior and illustrate the algorithm.MIT Ocean Engineering Design Laboratory
Copyright © 1997, Massachusetts Institute of Technology
URL: http://deslab.mit.edu/DesignLab/abstracts93.html
Revised: July 23, 1997