A characterization of helical polynomial curves of any degree

Computer Aided Geometric Design

2017

A helical curve, or curve of constant slope, offers a natural flight path for an aerial vehicle with a limited climb rate to achieve an increase in altitude between prescribed initial and final states. Every polynomial helical curve is a spatial Pythagorean-hodograph (PH) curve, and the distinctive features of the PH curves have attracted growing interest in their use for Unmanned Aerial Vehicle (UAV) path planning. This study describes an exact algorithm for constructing helical PH paths, corresponding to a constant climb rate at a given speed, between initial and final positions and motion directions. The algorithm bypasses the more sophisticated algebraic representations of spatial PH curves, and instead employs a simple "lifting" scheme to generate helical PH paths from planar PH curves constructed using the complex representation. In this context, a novel scheme to construct planar quintic PH curves that interpolate given end points and tangents, with exactly prescribed arc lengths, plays a key role. It is also shown that these helical paths admit simple closed-form rotation-minimizing adapted frames. The algorithm is simple, efficient, and robust, and can accommodate helical axes of arbitrary orientation through simple rotation transformations. Its implementation is illustrated by several computed examples.

Section: Constant Climb Rate Paths For Uavsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Helical polynomial curves interpolating G 1 data with prescribed axes and pitch angles

Computer Aided Geometric Design

2017

“…All PH cubics and all helical PH quintics are DPH curves, but non-helical DPH curves of degree ≥ 7 exist. Complete details may be found in [3,13,28].…”

Section: Rational Rmof On Dph Curvesmentioning

confidence: 99%

Rotation-minimizing osculating frames

Computer Aided Geometric Design

Giannelli

Sampoli

et al. 2014

An orthonormal frame (f 1 , f 2 , f 3 ) is rotation-minimizing with respect to f i if its angular velocity ω satisfies ω · f i ≡ 0 -or, equivalently, the derivatives of f j and f k are both parallel to f i . The Frenet frame (t, p, b) along a space curve is rotation-minimizing with respect to the principal normal p, and in recent years adapted frames that are rotation-minimizing with respect to the tangent t have attracted much interest. This study is concerned with rotation-minimizing osculating frames (f , g, b) incorporating the binormal b, and osculating-plane vectors f , g that have no rotation about b. These frame vectors may be defined through a rotation of t, p by an angle equal to minus the integral of curvature with respect to arc length. In aeronautical terms, the rotation-minimizing osculating frame (RMOF) specifies yaw-free rigid-body motion along a curved path. For polynomial space curves possessing rational Frenet frames, the existence of rational RMOFs is investigated, and it is found that they must be of degree 7 at least. The RMOF is also employed to construct a novel type of ruled surface, with the property that its tangent planes coincide with the osculating planes of a given space curve, and its rulings exhibit the least possible rate of rotation consistent with this constraint.

“…It is possible to identify certain special curves with rational Frenet-Serret frames [51], which correspond to the double PH (DPH) curves [16,17,47] -a subset of the spatial PH curves described in Section 3 below, characterized by the property that | r ′ (ξ) | and | r ′ (ξ) × r ′′ (ξ) | are both polynomials in ξ. Every DPH curve of degree ≤ 5 is a helical curve, whose tangent maintains a constant inclination ψ with respect to a fixed direction (the axis of the helix) -equivalently, the ratio of curvature to torsion has the constant value tan ψ.…”

Section: Many Ways To Frame a Space Curvementioning

confidence: 99%

Rational rotation-minimizing frames—Recent advances and open problems

Applied Mathematics and Computation

2016

Recent developments in the basic theory, algorithms, and applications for curves with rational rotation-minimizing frames (RRMF curves) are reviewed, and placed in the context of the current state-of-the-art by highlighting the many significant open problems that remain. The simplest non-trivial RRMF curves are the quintics, characterized by a scalar condition on the angular velocity of the Euler-Rodrigues frame (ERF). Two different classes of RRMF quintics can be identified. The first class of curves may be characterized by quadratic constraints on the quaternion coefficients of the generating polynomials; by the root structure of those polynomials; or by a certain polynomial divisibility condition. The second class has a strictly algebraic characterization, less well-suited to geometrical construction algorithms. The degree 7 RRMF curves offer more shape freedoms than the quintics, but only one of the four possible classes of these curves has been satisfactorily described. Generalizations of the adapted rotation-minimizing frames, for which the angular velocity has no component along the tangent, to directed and osculating frames (with analogous properties relative to the polar and binormal vectors) are also discussed. Finally, a selection of applications for rotation-minimizing frames are briefly reviewedincluding construction of swept surfaces, rigid-body motion planning, 5-axis CNC machining, and camera orientation control.