A unified Pythagorean hodograph approach to the medial axis transform and offset approximation

AbstractAlgorithms based on Pythagorean hodographs (PH) in the Euclidean plane and in Minkowski space share common goals, the main one being rationality of offsets of planar domains. However, only separate interpolation techniques based on these curves can be found in the literature. It was recently revealed that rational PH curves in the Euclidean plane and in Minkowski space are very closely related. In this paper, we continue the discussion of the interplay between spatial MPH curves and their associated planar PH curves from the point of view of Hermite interpolation. On the basis of this approach we design a new, simple interpolation algorithm. The main advantage of the unifying method presented lies in the fact that it uses, after only some simple additional computations, an arbitrary algorithm for interpolation using planar PH curves also for interpolation using spatial MPH curves. We present the functionality of our method for G1 Hermite data; however, one could also obtain higher order algorithms

Interpolation of G1 Hermite data by C1 cubic-like sparse Pythagorean hodograph splines

open3siProvided that they are in appropriate configurations (tight data), given planar G1 Hermite data generate a unique cubic Pythagorean hodograph (PH) spline curve interpolant. On a given associated knot-vector, the corresponding spline function cannot be C1, save for exceptional cases. By contrast, we show that replacing cubic spaces by cubic-like sparse spaces makes it possible to produce infinitely many C1 PH spline functions interpolating any given tight G1 Hermite data. Such cubic-like sparse spaces involve the constants and monomials of consecutive degrees, and they have long been used for design purposes. Only lately they were investigated in view of producing PH curves and associated G1 PH spline interpolants with some flexibility. The present work strongly relies on these recent results.embargoed_20220415Ait-Haddou R.; Beccari C.V.; Mazure M.-L.Ait-Haddou R.; Beccari C.V.; Mazure M.-L

Smooth path planning with Pythagorean-hodoghraph spline curves geometric design and motion control

This thesis addresses two significative problems regarding autonomous systems, namely path and trajectory planning. Path planning deals with finding a suitable path from a start to a goal position by exploiting a given representation of the environment. Trajectory planning schemes govern the motion along the path by generating appropriate reference (path) points. We propose a two-step approach for the construction of planar smooth collision-free navigation paths. Obstacle avoidance techniques that rely on classical data structures are initially considered for the identification of piecewise linear paths that do not intersect with the obstacles of a given scenario. In the second step of the scheme we rely on spline interpolation algorithms with tension parameters to provide a smooth planar control strategy. In particular, we consider Pythagorean\u2013hodograph (PH) curves, since they provide an exact computation of fundamental geometric quantities. The vertices of the previously produced piecewise linear paths are interpolated by using a G1 or G2 interpolation scheme with tension based on PH splines. In both cases, a strategy based on the asymptotic analysis of the interpolation scheme is developed in order to get an automatic selection of the tension parameters. To completely describe the motion along the path we present a configurable trajectory planning strategy for the offline definition of time-dependent C2 piece-wise quintic feedrates. When PH spline curves are considered, the corresponding accurate and efficient CNC interpolator algorithms can be exploited

Smooth path planning with Pythagorean-hodoghraph spline curves geometric design and motion control

This thesis addresses two significative problems regarding autonomous systems, namely path and trajectory planning. Path planning deals with finding a suitable path from a start to a goal position by exploiting a given representation of the environment. Trajectory planning schemes govern the motion along the path by generating appropriate reference (path) points. We propose a two-step approach for the construction of planar smooth collision-free navigation paths. Obstacle avoidance techniques that rely on classical data structures are initially considered for the identification of piecewise linear paths that do not intersect with the obstacles of a given scenario. In the second step of the scheme we rely on spline interpolation algorithms with tension parameters to provide a smooth planar control strategy. In particular, we consider Pythagorean–hodograph (PH) curves, since they provide an exact computation of fundamental geometric quantities. The vertices of the previously produced piecewise linear paths are interpolated by using a G1 or G2 interpolation scheme with tension based on PH splines. In both cases, a strategy based on the asymptotic analysis of the interpolation scheme is developed in order to get an automatic selection of the tension parameters. To completely describe the motion along the path we present a configurable trajectory planning strategy for the offline definition of time-dependent C2 piece-wise quintic feedrates. When PH spline curves are considered, the corresponding accurate and efficient CNC interpolator algorithms can be exploited

Silhouette-Informed Trajectory Generation Through a Wire Maze for Small UAS

Current rapidly-exploring random tree (RRT) algorithms rely on proximity query packages that often include collision checkers, tolerance verification, and distance computation algorithms for the generation of safe paths. In this paper, we broaden the information available to the path-planning algorithm by incorporating silhouette information of nearby obstacles in conflict. A silhouette-informed tree (SIT) is generated through the flight-safe region of a wire maze for a single unmanned aerial system (UAS). The silhouette is used to extract local geometric information of nearby obstacles and provide path alternatives around these obstacles. Thus, focusing the search for the generation of new tree branches near these obstacles, and decreasing the number of samples required to explore the narrow corridors within the wire maze. The SIT is then processed to extract a path that connects the initial location of the UAS with the goal, reduce the number of line segments in this path if possible, and smooth the resulting path using Pythagorean Hodograph Bezier curves. To ensure that the smoothed path remains in the flight-safe region of the configuration space, a tolerance verification algorithm for Bezier curves and convex polytopes in three dimensions is proposed. Lastly, temporal specifications are imposed on the smoothed path in the shape of an arbitrary speed profile

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version