Construction of planar quintic Pythagorean-hodograph curves by control-polygon constraints

In the construction and analysis of a planar Pythagorean–hodograph (PH) quintic curve r(t), t∈[0,1] using the complex representation, it is convenient to invoke a translation/rotation/scaling transformation so r(t) is in canonical form with r(0)=0, r(1)=1 and possesses just two complex degrees of freedom. By choosing two of the five control–polygon legs of a quintic PH curve as these free complex parameters, the remaining three control–polygon legs can be expressed in terms of them and the roots of a quadratic or quartic equation. Consequently, depending on the chosen two control–polygon legs, there exist either two or four distinct quintic PH curves that are consistent with them. A comprehensive analysis of all possible pairs of chosen control polygon legs is developed, and examples are provided to illustrate this control–polygon paradigm for the construction of planar quintic PH curves

Path planning of multiple autonomous vehicles

Safe and simultaneous arrival of constant speed, constant altitude UAVs on target is solved by design of paths of equal lengths. The starting point of the solution is the well-known Dubins path which is composed of circular arcs and line segments, thus requiring only one simple manoeuvre - constant rate turn. An explicit bound can be imposed on the rate during the design and the resulting paths are the minimum time solution of the problem. However, transition between arc and line segment entails discontinuous changes in lateral accelerations (latax), making this approach impractical for real fixed wing UAVs. Therefore, the Dubins solution is replaced with clothoid and also a novel one, based on quintic Pythagorean Hodograph (PH) curves, whose latax demand is continuous. The clothoid solution is direct as in the case of the Dubins path. The PH path is chosen for its rational functional form. The clothoid and the PH paths are designed to have lengths close to the lengths of the Dubins paths to stay close to the minimum time solution. To derive the clothoid and the PH paths that way, the Dubins solution is first interpreted in terms of Differential Geometry of curves using the path length and curvature as the key parameters. The curvature of a Dubins path is a piecewise constant and discontinuous function of its path length, which is a differential geometric expression of the discontinuous latax demand involved in transitions between the arc and the line segment. By contrast, the curvature of the PH path is a fifth order polynomial of its path length. This is not only continuous, also has enough design parameters (polynomial coefficients) to meet the latax (curvature) constraints (bounds) and to make the PH solution close to the minimum time one. The offset curves of the PH path are used to design a safety region along each path. The solution is simplified by dividing path planning into two phases. The first phase produces flyable paths while the second phase produces safe paths. Three types of paths are used: Dubins, clothoid and Pythagorean Hodograph (PH). The paths are produced both in 2D and 3D. In two dimensions, the Dubins path is generated using Euclidean and Differential geometric principles. It is shown that the principles of Differential geometry are convenient to generalize the path with the curvature. Due to the lack of curvature continuity of the Dubins path, paths with curvature continuity are considered. In this respect, initially the solution with the Dubins path is extended to produce clothoid path. Latter the PH path is produced using interpolation technique. Flyable paths in three dimensions are produced with the spatial Dubins and PH paths. In the second phase, the flyable paths are tuned for simultaneous arrival on target. The simultaneous arrival is achieved by producing the paths of equal lengths. Two safety conditions: (i) minimum separation distance and (ii) non-intersection of paths at equal distance are defined to maneuver in free space. In a cluttered space, an additional condition, threat detection and avoidance is defined to produce safe paths. The tuning is achieved by increasing the curvature of the paths and by creating an intermediate way-point. Instead of imposing safety constraints, the flyable paths are tested for meeting the constraints. The path is replanned either by creating a new way-point or by increasing the curvature between the way-points under consideration. The path lengths are made equal to that of a reference path.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Contour parallel milling tool path generation for arbitrary pocket shape using a fast marching method

Contour parallel tool paths are among the most widely used tool paths for planer milling operations. A number of exact as well as approximate methods are available for offsetting a closed boundary in order to generate a contour parallel tool path; however, the applicability of various offsetting methods is restricted because of limitations in dealing with pocket geometry with and without islands, the high computational costs, and numerical errors. Generation of cusps, segmentation of rarefied corners, and self-intersection during the offsetting operations and finding a unique offsetting solution for pocket with islands are among the associated problems in contour tool path generation. Most of methods are inherently incapable of dealing with such problems and use complex computational routines to identify and rectify these problems. Also, these rectifying techniques are heavily dependent on the type of geometry, and hence, the application of these techniques for arbitrary boundary conditions is limited and prone to errors. In this paper, a new mathematical method for generation of contour parallel tool paths is proposed which is inherently capable of dealing with the aforementioned problems. The method is based on a boundary value formulation of the offsetting problem and a fast marching method based solution for tool path generation. This method handles the topological changes during offsetting naturally and deals with the generation of discontinuities in the slopes by including an "entropy condition” in its numerical implementation. The appropriate modifications are carried out to achieve higher accuracy for milling operations. A number of examples are presented, and computational issues are discussed for tool path generatio

Contour parallel milling tool path generation for arbitrary pocket shape using a fast marching method

Contour parallel tool paths are among the most widely used tool paths for planer milling operations. A number of exact as well as approximate methods are available for offsetting a closed boundary in order to generate a contour parallel tool path; however, the applicability of various offsetting methods is restricted because of limitations in dealing with pocket geometry with and without islands, the high computational costs, and numerical errors. Generation of cusps, segmentation of rarefied corners, and self-intersection during the offsetting operations and finding a unique offsetting solution for pocket with islands are among the associated problems in contour tool path generation. Most of methods are inherently incapable of dealing with such problems and use complex computational routines to identify and rectify these problems. Also, these rectifying techniques are heavily dependent on the type of geometry, and hence, the application of these techniques for arbitrary boundary conditions is limited and prone to errors. In this paper, a new mathematical method for generation of contour parallel tool paths is proposed which is inherently capable of dealing with the aforementioned problems. The method is based on a boundary value formulation of the offsetting problem and a fast marching method based solution for tool path generation. This method handles the topological changes during offsetting naturally and deals with the generation of discontinuities in the slopes by including an "entropy condition" in its numerical implementation. The appropriate modifications are carried out to achieve higher accuracy for milling operations. A number of examples are presented, and computational issues are discussed for tool path generation