Mapping rational rotation-minimizing frames from polynomial curves on to rational curves

Given a polynomial space curve r(ξ) that has a rational rotation–minimizing frame (an RRMF curve), a methodology is developed to construct families of rational space curves r˜(ξ) with the same rotation–minimizing frame as r(ξ) at corresponding points. The construction employs the dual form of a rational space curve, interpreted as the edge of regression of the envelope of a family of osculating planes, having normals in the direction u(ξ)=r′(ξ)×r″(ξ) and distances from the origin specified in terms of a rational function f(ξ) as f(ξ)/‖u(ξ)‖. An explicit characterization of the rational curves r˜(ξ) generated by a given RRMF curve r(ξ) in this manner is developed, and the problem of matching initial and final points and frames is shown to impose only linear conditions on the coefficients of f(ξ), obviating the non–linear equations (and existence questions) that arise in addressing this problem with the RRMF curve r(ξ). Criteria for identifying low–degree instances of the curves r˜(ξ) are identified, by a cancellation of factors common to their numerators and denominators, and the methodology is illustrated by a number of computed examples

Optimization of Corner Blending Curves

The blending or filleting of sharp corners is a common requirement in geometric design applications — motivated by aesthetic, ergonomic, kinematic, or mechanical stress considerations. Corner blending curves are usually required to exhibit a specified order of geometric continuity with the segments they connect, and to satisfy specific constraints on their curvature profiles and the extremum deviation from the original corner. The free parameters of polynomial corner curves of degree ≤6 and continuity up to G3 are exploited to solve a convex optimization problem, that minimizes a weighted sum of dimensionless measures of the mid-point curvature, maximum deviation, and the uniformity of parametric speed. It is found that large mid-point curvature weights result in undesirable bimodal curvature profiles, but emphasizing the parametric speed uniformity typically yields good corner shapes (since the curvature is strongly dependent upon parametric speed variation). A constrained optimization problem, wherein a particular value of the corner curve deviation is specified, is also addressed. Finally, the shape of Pythagorean-hodograph corner curves is compared with that of the optimized “ordinary” polynomial corner 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

Helical polynomial curves and double Pythagorean hodographs II. Enumeration of low-degree curves

AbstractA “double” Pythagorean-hodograph (DPH) curve r(t) is characterized by the property that |r′(t)| and |r′(t)×r″(t)| are both polynomials in the curve parameter t. Such curves possess rational Frenet frames and curvature/torsion functions, and encompass all helical polynomial curves as special cases. As noted by Beltran and Monterde, the Hopf map representation of spatial PH curves appears better suited to the analysis of DPH curves than the quaternion form. A categorization of all DPH curve types up to degree 7 is developed using the Hopf map form, together with algorithms for their construction, and a selection of computed examples of (both helical and non-helical) DPH curves is included, to highlight their attractive features. For helical curves, a separate constructive approach proposed by Monterde, based upon the inverse stereographic projection of rational line/circle descriptions in the complex plane, is used to classify all types up to degree 7. Criteria to distinguish between the helical and non-helical DPH curves, in the context of the general construction procedures, are also discussed

Distributed cooperative trajectory generation for multiple autonomous vehicles using Pythagorean Hodograph Bézier curves

This dissertation presents a framework for multi-vehicle trajectory generation that enables efficient computation of sets of feasible, collision-free trajectories for teams of autonomous vehicles executing cooperative missions with common objectives. Existing methods for multi-vehicle trajectory generation generally rely on discretization in time or space and, therefore, ensuring safe separation between the paths comes at the expense of an increase in computational complexity. On the contrary, the proposed framework is based on a three-dimensional geometric-dynamic approach that uses continuous Bézier curves with Pythagorean hodographs, a class of polynomial functions with attractive mathematical properties and a collection of highly efficient computational procedures associated with them. The use of these curves is critical to generate cooperative trajectories that are guaranteed to satisfy minimum separation distances, a key feature from a safety standpoint. By the differential flatness property of the dynamic system, the dynamic constraints can be expressed in terms of the trajectories and, therefore, in terms of Bézier polynomials. This allows the proposed framework to efficiently evaluate and, hence, observe the dynamic constraints of the vehicles, and satisfy mission-specific assignments such as simultaneous arrival at predefined locations. The dissertation also addresses the problem of distributing the computation of the trajectories over the vehicles, in order to prevent a single point of failure, inherently present in a centralized approach. The formulated cooperative trajectory-generation framework results in a semi-infinite programming problem, that falls under the class of nonsmooth optimization problems. The proposed distributed algorithm combines the bundle method, a widely used solver for nonsmooth optimization problems, with a distributed nonlinear programming method. In the latter, a distributed formulation is obtained by introducing local estimates of the vector of optimization variables and leveraging on a particular structure, imposed on the local minimizer of an equivalent centralized optimization problem