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The Pythagorean-hodograph (PH) curves offer distinct advantages in planning curvilinear paths for unmanned or autonomous air, ground, or underwater vehicles. Although several authors have discussed their use in these contexts, prior studies contain misconceptions about the properties of PH curves or invoke heuristic approximate constructions when exact methods are available. To address these issues, the present study provides a basic introduction to the key properties of PH curves, and describes some exact constructions of particular interest in path planning. These include (1) maintenance of minimum safe separations within vehicle swarms; (2) construction of paths of different shape but identical arc length, ensuring simultaneous arrival of vehicles travelling at a constant speed; (3) determination of the curvature extrema of PH paths, and their modification to satisfy a given curvature bound; and (4) construction of curvature-continuous paths of bounded curvature through fields of polygonal obstacles.

We present a configurable trajectory planning strategy on planar paths for offline definition of time-dependent C 2 piecewise quintic feedrates. The more conservative formulation ensures chord tolerance, as well as prescribed bounds on velocity, acceleration and jerk Cartesian components. Since the less restrictive formulations of our strategy can usually still ensure all the desired bounds while simultaneously producing faster motions, the configurability feature is useful not only when reduced motion control is desired but also when full kinematic control has to be guaranteed. Our approach can be applied to any planar path with a piecewise sufficiently smooth parametric representation. When Pythagoreanhodograph spline curves are considered, the corresponding accurate and efficient CNC interpolator algorithms can be exploited.

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