Pythagorean-hodograph Curves Research Articles

Algebraic-Trigonometric Pythagorean-Hodograph space curves

We introduce a new class of Pythagorean-Hodograph (PH) space curves - called Algebraic-Trigonometric Pythagorean-Hodograph (ATPH) space curves - that are defined over a six-dimensional space mixing algebraic and trigonometric polynomials. After providing a general definition for this new class of curves, their quaternion representation is introduced and the fundamental properties are discussed. Then, as previously done with their quintic polynomial counterpart, a constructive approach to solve the first-order Hermite interpolation problem in ℝ3 is provided. Comparisons with the polynomial case are illustrated to point out the greater flexibility of ATPH curves with respect to polynomial PH curves.

Rational frames of minimal twist along space curves under specified boundary conditions

An adapted orthonormal frame (f1(ξ),f2(ξ),f3(ξ)) on a space curve r(ξ), ξ ∈ [ 0, 1 ] comprises the curve tangent $\mathbf {f}_{1}(\xi ) =\mathbf {r}^{\prime }(\xi )/|\mathbf {r}^{\prime }(\xi )|$ and two unit vectors f2(ξ),f3(ξ) that span the normal plane. The variation of this frame is specified by its angular velocity Ω = Ω1f1 + Ω2f2 + Ω3f3, and the twist of the framed curve is the integral of the component Ω1 with respect to arc length. A minimal twist frame (MTF) has the least possible twist value, subject to prescribed initial and final orientations f2(0),f3(0) and f2(1),f3(1) of the normal–plane vectors. Employing the Euler–Rodrigues frame (ERF) — a rational adapted frame defined on spatial Pythagorean–hodograph curves — as an intermediary, an exact expression for an MTF with Ω1 = constant is derived. However, since this involves rather complicated transcendental terms, a construction of rational MTFs is proposed by the imposition of a rational rotation on the ERF normal–plane vectors. For spatial PH quintics, it is shown that rational MTFs compatible with the boundary conditions can be constructed, with only modest deviations of Ω1 about the mean value, by a rational quartic normal–plane rotation of the ERF. If necessary, subdivision methods can be invoked to ensure that the rational MTF is free of inflections, or to more accurately approximate a constant Ω1. The procedure is summarized by an algorithm outline, and illustrated by a representative selection of computed examples.

Time-Optimal Trajectory Planning for Delta Robot Based on Quintic Pythagorean-Hodograph Curves

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.

Obstacle avoidance method of three-dimensional obstacle spherical cap

Focusing on obstacle avoidance in three-dimensional space for unmanned aerial vehicle (UAV), the direct obstacle avoidance method in dynamic space based on three-dimensional velocity obstacle spherical cap is proposed, which quantifies the influence of threatening obstacles through velocity obstacle spherical cap parameters. In addition, the obstacle avoidance schemes of any point on the critical curve during the multi-obstacles avoidance are given. Through prediction, the insertion point for the obstacle avoidance can be obtained and the flight path can be re-planned. Taking the Pythagorean Hodograph (PH) curve trajectory re-planning as an example, the three-dimensional direct obstacle avoidance method in dynamic space is tested. Simulation results show that the proposed method can realize the online obstacle avoidance trajectory re-planning, which increases the flexibility of obstacle avoidance greatly.

Reduced difference polynomials and self-intersection computations

A reduced difference polynomialf(u,v)=(p(u)−p(v))/(u−v) may be associated with any given univariate polynomial p(t), t ∈ [ 0, 1 ] such that the locus f(u,v)=0 identifies the pairs of distinct values u and v satisfying p(u)=p(v). The Bernstein coefficients of f(u, v) on [ 0, 1 ]2 can be determined from those of p(t) through a simple algorithm, and can be restricted to any subdomain of [ 0, 1 ]2 by standard subdivision methods. By constructing the reduced difference polynomials f(u, v) and g(u, v) associated with the coordinate components of a polynomial curve r(t)=(x(t),y(t)), a quadtree decomposition of [ 0, 1 ]2 guided by the variation-diminishing property yields a numerically stable scheme for isolating real solutions of the system f(u,v)=g(u,v)=0, which identify self-intersections of the curve r(t). Through the Kantorovich theorem for guaranteed convergence of Newton–Raphson iterations to a unique solution, the self-intersections can be efficiently computed to machine precision. By generalizing the reduced difference polynomial to encompass products of univariate polynomials, the method can be readily extended to compute the self-intersections of rational curves, and of the rational offsets to Pythagorean–hodograph curves.

Geometric characteristics of planar quintic Pythagorean-hodograph curves

This study examines necessary and sufficient conditions for a planar quintic Bézier curve to be a Pythagorean-hodograph (PH) curve. Quintic PH curves can be categorized into two classes according to the representation of their derivatives. While the first class has been studied by Farouki (1994) to be a family of regular curves already, a more succinct proof by introducing auxiliary control points is provided in this paper. Geometric characteristics of the second class of quintic PH curves are also studied. The key technique to simplify the discussion is to represent a planar Bézier curve with a complex polynomial in Bernstein form. Benefiting from such complex expression, algebraic characteristics of quintic PH curves can be described by nonlinear complex systems with respect to control points. By treating these systems with geometric methods, conditions for a quintic planar curve to be a PH curve can be described in terms of geometric constraints on its control polygon. Furthermore, we provide methods for the construction of the second class of quintic PH curves. In particular, parameter values of cusps can be explicitly determined in advance for irregular curves.

Interpolation with spatial rational Pythagorean-hodograph curves of class 4

The paper presents a construction of rational Pythagorean-hodograph curves of class 4 and reveals their properties. In particular, it is shown that each such curve depends on twelve free parameters and has a piecewise rational arc-length function. Geometric interpolation of two data points and two tangent directions is considered in detail and a closed form solution that depends on two shape parameters is given. Regions of shape parameters are derived that imply the interpolant to be regular and admissible. Further, it is shown that one of the shape parameters can be fixed by additionally prescribing also the length of the interpolant. Theoretical results and the construction of G1 Hermite interpolation splines are illustrated with numerical examples.

Helical polynomial curves interpolating G1 data with prescribed axes and pitch angles

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.

Sparse Pythagorean hodograph curves

We introduce cubic-like sparse Pythagorean hodograph curves which are possibly of high degrees though they can be handled as cubic curves. They offer more flexibility than the classical Pythagorean hodograph cubic curves with which they share many properties. We give an elegant geometric characterization of their control polygons. This characterization leads to many interesting computational algorithms for curve design. We show how to extend such algorithms to quintic-like sparse PH curves.

Rectifying control polygon for planar Pythagorean hodograph curves

A Bézier control polygon is not appropriate to control a Pythagorean hodograph curve since it has redundant degrees of freedom. So we propose an alternative, which is the rectifying control polygon. A rectifying control polygon of a PH curve has the same degrees of freedom as the PH curve. It interpolates the end points of the PH curve, but not the end tangents. Most of all, it has the same arc length as the PH curve. In this paper, we present the method to compute the rectifying control polygon from the Bézier control polygon of the PH curve. We also present the procedure to compute the PH curves from a given rectifying control polygon. For the development of these algorithms, we employ the Gauss–Legendre quadrature method and the Bernstein–Vandermonde linear system.

Path planning with Pythagorean-hodograph curves for unmanned or autonomous vehicles

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 (a) maintenance of minimum safe separations within vehicle swarms; (b) construction of paths of different shape but identical arc length, ensuring simultaneous arrival of vehicles travelling at a constant speed; (c) determination of the curvature extrema of PH paths, and their modification to satisfy a given curvature bound; and (d) construction of curvature-continuous paths of bounded curvature through fields of polygonal obstacles.

A two-stage path planning approach for multiple car-like robots based on PH curves and a modified harmony search algorithm

ABSTRACTIn this article, collision-avoidance path planning for multiple car-like robots with variable motion is formulated as a two-stage objective optimization problem minimizing both the total length of all paths and the task’s completion time. Accordingly, a new approach based on Pythagorean Hodograph (PH) curves and Modified Harmony Search algorithm is proposed to solve the two-stage path-planning problem subject to kinematic constraints such as velocity, acceleration, and minimum turning radius. First, a method of path planning based on PH curves for a single robot is proposed. Second, a mathematical model of the two-stage path-planning problem for multiple car-like robots with variable motion subject to kinematic constraints is constructed that the first-stage minimizes the total length of all paths and the second-stage minimizes the task’s completion time. Finally, a modified harmony search algorithm is applied to solve the two-stage optimization problem. A set of experiments demonstrate the effectiveness of the proposed approach.

Evaluation of Clothoids in Manufacturing Context

Polynomial based curves are the de facto standard methods used in numerical control systems for curvature continuous corner blending, i.e. smoothing path transitions with different tangent vectors. Disadvantages of polynomial based curves are mostly the non-linearity of the curvature profile over the arc length and the approximation of the arc length. A curve which does not suffer from these disadvantages is Clothoids. In this paper, an algorithmic concept to use Clothoid as blending curves is addressed. The algorithms consider the corner-based and the intermediate point-based blending. Based on theoretical foundations, these two blending types are analyzed. Further, Clothoids and Pythagorean hodograph curve are compared to outline the benefit of applying Clothoids as a blending curve.

A comprehensive characterization of the set of polynomial curves with rational rotation-minimizing frames

A rotation-minimizing frame $({\bf f}_1,{\bf f}_2,{\bf f}_3)$ on a space curve ${\bf r}(\xi)$ defines an orthonormal basis for $\mathbb{R}^3$ in which ${\bf f}_1={\bf r}'/|{\bf r}'|$ is the curve tangent, and the normal-plane vectors ${\bf f}_2$, ${\bf f}_3$ exhibit no instantaneous rotation about ${\bf f}_1$. Polynomial curves that admit rational rotation-minimizing frames (or RRMF curves) form a subset of the Pythagorean-hodograph (PH) curves, specified by integrating the form ${\bf r}'(\xi)={\cal A}(\xi)\,{\bf i}\,{\cal A}^*(\xi)$ for some quaternion polynomial ${\cal A}(\xi)$. By introducing the notion of rotation indicatrix and of core of the quaternion polynomial ${\cal A}(\xi)$, a comprehensive characterization of the complete space of RRMF curves is developed, that subsumes all previously known special cases. This novel characterization helps clarify the structure of the complete space of RRMF curves, distinguishes the spatial RRMF curves from trivial (planar) cases, and paves the way toward new construction algorithms.

Medial axis transforms yielding rational envelopes

Minkowski Pythagorean hodograph (MPH) curves provide a means for representing domains with rational boundaries via the medial axis transform. Based on the observation that MPH curves are not the only curves that yield rational envelopes, we define and study rational envelope (RE) curves that generalise MPH curves while maintaining the rationality of their associated envelopes.To demonstrate the utility of RE curves, we design a simple interpolation algorithm using RE curves, which is in turn used to produce rational surface blends between canal surfaces. Additionally, we initiate the study of rational envelope surfaces as a surface analogy to RE curves.

Efficient high-speed cornering motions based on continuously-variable feedrates. I. Real-time interpolator algorithms

The problem of high-speed traversal of sharp toolpath corners, within a prescribed geometrical tolerance 𝜖, is addressed. Each sharp corner is replaced by a quintic Pythagorean–hodograph (PH) curve that meets the incoming/outgoing path segments with G 2 continuity, and deviates from the exact corner by no more than the prescribed tolerance 𝜖. The deviation and extremum curvature admit closed-form expressions in terms of the corner angle 𝜃 and side-length L, allowing precise control over these quantities. The PH curves also permit a smooth modulation of feedrate around the corner by analytic reduction of the interpolation integral. To demonstrate this, real-time interpolator algorithms are developed for three model feedrate functions. Specifying the feedrate as a quintic polynomial in the curve parameter accommodates precise acceleration continuity, but has no obvious geometrical interpretation. An inverse linear dependence on curvature offers a purely geometrical specification, but incurs slight initial and final tangential acceleration discontinuities. As an alternative, a hybrid form that incorporates the main advantages of these two approaches is proposed. In each case, the ratio $f=V_{\min }/V_{0}$ of the minimum and nominal feedrates is a free parameter, and the improved cornering time is analyzed. This paper develops the basic cornering algorithms—their implementation and performance analysis are described in detail in a companion paper.

The Arc Length Formula for the Offset of Convex Pythagorean Hodograph Curves

The Arc Length Formula for the Offset of Convex Pythagorean Hodograph Curves

Algorithm 952

The implementation of a library of basic functions for the construction and analysis of planar quintic Pythagorean-hodograph (PH) curves is presented using the complex representation. The special algebraic structure of PH curves permits exact algorithms for the computation of key properties, such as arc length, elastic bending energy, and offset (parallel) curves. Single planar PH quintic segments are constructed as interpolants to first-order Hermite data (end points and derivatives), and this construction is then extended to open or closed C 2 PH quintic spline curves interpolating a sequence of points in the plane. The nonlinear nature of PH curves incurs a multiplicity of formal solutions to such interpolation problems, and a key aspect of the algorithms is to efficiently single out the unique “good” interpolant among them.

On control polygons of Pythagorean hodograph septic curves

For a given septic Bézier curve with a distinct ordered sequence of control points, how to determine whether it is a PH curve via exact symbolic computation in theory. This problem motivated the study of a necessary and sufficient condition for a planar septic Bézier curve to possess a Pythagorean hodograph (PH). Based on the definition of a PH curve and the complex representation of a planar curve, we develop geometric conditions in terms of the leg-lengths and angles of a control polygon that must be separated to guarantee the PH property. The relation between the compatibility of solutions with respect to the complex coefficients of PH equations and geometric constraints is analyzed. Moreover, PH septic curves with inflections are extended to construct S-shaped transition curves.

UAV Flyable Trajectory Generation and Its Tracking Control

For the complex combat environment, considering the flyable performance constrains and the safety constrains, an online trajectory planning algorithm of UAVs(Unmanned Aerial Vehicles) based on Pythagorean Hodograph (PH) curve is put forward, which is easy to traced. When the unexpected obstacles are detected, based on the velocity obstacles avoidance method, the collision avoidance trajectory is planned online. The position of the interrupt point and the direction of the UAV turn are calculated. To fully use the PH trajectories performance that the curvature of the curve is continuous and precisely known, the location and crossing heading error equations were established within the Serret-Frenet coordinates. The corresponding asymptotically stable convergence backstepping trajectory tracking control law was designed based on the error formulations. The autonomous collision avoidance trajectory simulation result and its tracking control result show that the proposed trajectory generation method and the corresponding adaptive tracking control method are effective and can improve the tracking accuracy.