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 having no intersection with the obstacles of a given scenario. Variations of the shortest piecewise linear path with angle-based criteria are proposed and discussed. In the second part of the scheme we rely on spline interpolation algorithms with tension parameters to provide a smooth planar control strategy. In particular, we consider the class of curves with Pythagorean structures, because they provide an exact computation of fundamental geometric quantities. A selection of test cases demonstrates the quality of the new motion planning scheme.
•A two-step approach to design planar smooth collision-free paths is presented.•The construction of piecewise linear paths with angle-based criteria is investigated.•The smooth path is based on PH spline interpolation schemes with tension parameters.•A selection of test cases demonstrates the quality of the new motion planning scheme.
A minimal twist frame(f1(ξ),f2(ξ),f3(ξ)) on a polynomial space curve r(ξ), ξ∈0,1 is an orthonormal frame, where f1(ξ) is the tangent and the normal-plane vectors f2(ξ),f3(ξ) have the least variation ...between given initial and final instances f2(0),f3(0) and f2(1),f3(1). Namely, if ω=ω1f1+ω2f2+ω3f3 is the frame angular velocity, the component ω1 does not change sign, and its arc length integral has the smallest value consistent with the boundary conditions. We consider construction of curves with rational minimal twist frames, based on the Pythagorean-hodograph curves of degree 7 that have rational rotation-minimizing Euler–Rodrigues frames(e1(ξ),e2(ξ),e3(ξ)) — i.e., the normal-plane vectors e2(ξ),e3(ξ) have no rotation about the tangent e1(ξ). A set of equations that govern the construction of such curves with prescribed initial/final points and tangents, and total arc length, is derived. For the resulting curves f2(ξ),f3(ξ) are then obtained from e2(ξ),e3(ξ) by a monotone rational normal-plane rotation, subject to the boundary conditions. A selection of computed examples is included to illustrate the construction, and it is shown that the desirable feature of a uniform rotation rate (i.e., ω1=constant) can be accurately approximated.
Optimization of Corner Blending Curves Farouki, Rida T.; Pelosi, Francesca; Sampoli, Maria Lucia
Computer aided design,
December 2019, 2019-12-00, 20191201, Letnik:
117
Journal Article
Recenzirano
Odprti dostop
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.
•Methods for shape optimization of corner blending curves are developed.•An objective function, based on curvature and deviation, is formulated.•Curvature unimodality correlates with uniformity of the parametric speed.•Optimized corner curves of degrees up to 6 and continuity G3 are computed.•The optimized corners are compared Pythagorean-hodograph (PH) corners.
We classify planar polynomial Pythagorean-hodograph curves of any degree with respect to Euclidean similarities. We also analyze possible global shapes of the Pythagorean-hodograph curves of degree ...four and five and describe all possible configurations of their singular points. This description results in a classification of Pythagorean-hodograph quartics and quintics with respect to the homeomorphisms of the plane. We also describe all the Pythagorean-hodograph curves of any degree up to five which exhibit central or axial symmetry.
•Following the recent approach for PH curves, Minkowski Pythagorean B-splines curves are introduced and studied.•Using the Clifford algebra model is shown how to construct an MPH B-spline curve of ...arbitrary degree.•Selected interpolation problems based on symbolic computations with these curves are discussed and illustrated on examples.
Following and extending the recent results of Albrecht et al. (2017) for planar Pythagorean hodograph (PH) B-spline curves to the Minkowski 3-space, we introduce a class of Minkowski Pythagorean hodograph (MPH) B-spline curves. The distinguished property of these curves is that the Minkowski norm of their hodograph is a B-spline function. We focus mainly on the clamped case and using Clifford algebra representation we present formulas for their construction. The closed case is also mentioned. Then we solve two practical problems – construction of MPH B-spline curves with control polygon close to a given control polygon, and construction of MPH B-spline curves going through given points. We emphasize symbolic solutions wherever it is possible. The results and approaches are illustrated on several examples.
In this paper, a time-optimal trajectory planning method based on quintic Pythagorean-Hodograph (PH) curves is proposed to realize the smooth and stable high-speed operation of the Delta parallel ...robot. The trajectory is determined by applying the quintic PH curves to the transition segments in the pick-and-place operation trajectory and the 3-4-5 polynomial motion law to the trajectory. The quintic PH curves are optimized to reduce the cycle time of the pick-and-place operation. In addition, a comparison between different trajectory planning methods has been implemented so as to observe the performance of the obtained results. The MATLAB simulation results reveal that compared with the trajectory planning based on vertical and horizontal motion superposition, the trajectory planning based on quintic PH curves is completed with a shorter motion cycle time and more stable motion performance, with the velocities, accelerations, and jerks in joint space bounded and continuous. Experiments carried out on the prototype also confirm that the trajectory planning based on quintic PH curves has a shorter cycle time, which is of great importance to high-speed operations of Delta parallel robots.
Cubic biarcs are the natural counterpart, in the context of the PH spatial Hermite interpolation, of piecewise quadratics for Hermite interpolation of standard parametric curves. Recently, it has ...been shown that, spatial Hermite interpolation can be always treated with PH cubic biarcs, providing an interesting alternative to the use of higher degrees, see Bastl et al. (2014). Beside the parameter value at the joint of the two arcs, which is typically free in any scheme of this kind, two free angular parameters remain. They should be automatically fixed by some suitable criterion in order to ensure the generation of interpolants with a reasonable shape. In this paper, an alternative to the constant choice of free parameters presented in Bastl et al. (2014) is proposed. It consists of an extension of the so called CC selection strategy introduced in Farouki et al. (2008) to the biarc setting. Such strategy is fully data-dependent, does not require any special configuration of the coordinate system and it guarantees the PH cubic reproduction when such kind of the interpolant exists. Moreover, the obtained scheme possesses other two important features, i.e. it is third order accurate and gives the possibility to control the torsion sign of the produced interpolant, in case with the introduction of an additional tension parameter relaxing the smoothness from C1 to G1. The numerical results, based also on the straightforward spline extension of the scheme, confirm the developed theoretical analysis.
•Construction of rational frame of a Pythagorean hodograph curve by a rational rotation of the Euler–Rodrigues frame.•Hermite interpolation method for rational adapted frame along a prescribed ...spatial Pythagorean hodograph curve.•Rational spline frame interpolating orientation data along a spatial Pythagorean hodograph curve.
The problem of constructing a rational adapted frame (f1(ξ),f2(ξ),f3(ξ)) that interpolates a discrete set of orientations at specified nodes along a given spatial Pythagorean-hodograph (PH) curve r(ξ) is addressed. PH curves are the only polynomial space curves that admit rational adapted frames, and the Euler–Rodrigues frame (ERF) is a fundamental instance of such frames. The ERF can be transformed into other rational adapted frame by applying a rationally-parametrized rotation to the normal-plane vectors. When orientation and angular velocity data at curve end points are given, a Hermite frame interpolant can be constructed using a complex quadratic polynomial that parametrizes the normal-plane rotation, by an extension of the method recently introduced to construct a rational minimal twist frame (MTF). To construct a rational adapted spline frame, a representation that resolves potential ambiguities in the orientation data is introduced. Based on this representation, a C1 rational adapted spline frame is constructed through local Hermite interpolation on each segment, using angular velocities estimated from a cubic spline that interpolates the frame phase angle relative to the ERF. To construct a C2 rational adapted spline frame, which ensures continuity of the angular acceleration, a complex-valued cubic spline is used to directly interpolate the complex exponentials of the phase angles at the nodal points.
•G1 interpolation scheme for motion data with cubic PH biarcs is presented.•The length of the center trajectory is prescribed in advance.•The solution is given in a closed form and depends on four ...shape parameters.•The twist of the Euler–Rodrigues frame is minimized.•The spline construction is provided.
In this paper the G1 interpolation scheme for motion data, i.e., interpolation of data points and rotations at the points, with cubic PH biarcs is presented. The rotational part of the motion is determined by the Euler–Rodrigues frame which matches the given boundary positions. In addition, the length of the biarc is prescribed. It is shown that the interpolant exists for any data and any chosen length greater than the difference between the interpolation points. The interpolant is given in a closed form and depends on some free shape parameters, which are determined so that the curve is of a nice shape and the twist of the Euler–Rodrigues frame is minimized. The spline construction is provided and numerical examples that confirm the derived theoretical results are included.
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.
•An algorithm to construct helical interpolants to G1 Hermite data, with prescribed axes and pitch angles, is presented.•The problem admits a closed-form solution, requiring little more than the solution of a quadratic equation.•The solutions are applicable to path planning for unmanned aerial vehicles with limited climb rates.•The quaternion form and rotation-minimizing frames on the helical interpolants are derived.
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