15 research outputs found

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

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    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

    Skinning of circles using Biarcs

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

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    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

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    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
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