Quintic Pythagorean-hodograph Curves Research Articles

New algebraic and geometric characterizations of planar quintic Pythagorean-hodograph curves

The aim of this work is to provide new characterizations of planar quintic Pythagorean-hodograph curves. The first two are algebraic and consist of two and three equations, respectively, in terms of the edges of the Bézier control polygon as complex numbers. These equations are symmetric with respect to the edge indices and cover curves with generic as well as degenerate control polygons. The last two characterizations are geometric and rely both on just two auxiliary points outside the control polygon. One requires two (possibly degenerate) quadrilaterals to be similar, and the other highlights two families of three similar triangles. All characterizations are a step forward with respect to the state of the art, and they can be linked to the well-established counterparts for planar cubic Pythagorean-hodograph curves. The key ingredient for proving the aforementioned results is a novel general expression for the hodograph of the curve.

Partition of the space of planar quintic Pythagorean-hodograph curves

The quintics are the lowest–order planar Pythagorean–hodograph (PH) curves suitable for free–form design, since they can exhibit inflections. A quintic PH curve r(t) may be constructed from a complex quadratic pre–image polynomial w(t) by integration of r′(t)=w2(t), and it thus incorporates (modulo translations) six real parameters — the real and imaginary parts of the coefficients of w(t). Within this 6–dimensional space of planar PH quintics, a 5–dimensional hypersurface separates the inflectional and non–inflectional curves. Points of the hypersurface identify exceptional curves that possess a tangent–continuous point of infinite curvature, corresponding to the fact that the parabolic locus specified by the quadratic pre–image polynomial w(t) passes through the origin of the complex plane. Correspondingly, extreme curvatures and tight loops on r(t) are incurred by a close proximity of w(t) to the origin of the complex plane. These observations provide useful insight into the disparate shapes of the four distinct PH quintic solutions to the first–order Hermite interpolation problem.

Construction of planar quintic Pythagorean-hodograph curves by control-polygon constraints

In the construction and analysis of a planar Pythagorean–hodograph (PH) quintic curve r(t), t∈[0,1] using the complex representation, it is convenient to invoke a translation/rotation/scaling transformation so r(t) is in canonical form with r(0)=0, r(1)=1 and possesses just two complex degrees of freedom. By choosing two of the five control–polygon legs of a quintic PH curve as these free complex parameters, the remaining three control–polygon legs can be expressed in terms of them and the roots of a quadratic or quartic equation. Consequently, depending on the chosen two control–polygon legs, there exist either two or four distinct quintic PH curves that are consistent with them. A comprehensive analysis of all possible pairs of chosen control polygon legs is developed, and examples are provided to illustrate this control–polygon paradigm for the construction of planar quintic PH curves.

Singular cases of planar and spatial C1 Hermite interpolation problems based on quintic Pythagorean-hodograph curves

A well–known feature of the Pythagorean–hodograph (PH) curves is the multiplicity of solutions arising from their construction through the interpolation of Hermite data. In general, there are four distinct planar quintic PH curves that match first–order Hermite data, and a two–parameter family of spatial quintic PH curves compatible with such data. Under certain special circumstances, however, the number of distinct solutions is reduced. The present study characterizes these singular cases, and analyzes the properties of the resulting quintic PH curves. Specifically, in the planar case it is shown that there may be only three (but not less) distinct Hermite interpolants, of which one is a “double” solution. In the spatial case, a constant difference between the two free parameters reduces the dimension of the solution set from two to one, resulting in a family of quintic PH space curves of different shape but identical arc lengths. The values of the free parameters that result in formal specialization of the (quaternion) spatial problem to the (complex) planar problem are also identified, demonstrating that the planar PH quintics, including their degenerate cases, are subsumed as a proper subset of the spatial PH quintics.

Quintic Pythagorean-Hodograph Curves Based Trajectory Planning for Delta Robot with a Prescribed Geometrical Constraint

A new trajectory planning approach on the basis of the quintic Pythagorean–Hodograph (PH) curve is presented and applied to Delta robot for implementing pick-and-place operation (PPO). To satisfy a prescribed geometrical constraint, which indicates the distance between the transition segment curve and right angle of PPO trajectory is no greater than a prescribed value, the quintic PH curve is used to produce a connection segment path for collision avoidance. The relationship between the PH curve and constraint is analyzed, based on which PH curve is calculated simply. Afterwards, the trajectory is planned in different phases with different motion laws, i.e. polynomial motion laws and PH curve parameter-dependent motion laws, to obtain a smooth performance both in Cartesian and joint space. The relationship between the PH curve and constraint is also used to improve the efficiency of calculation, and the trajectory symmetry is used to reduce calculation time by direct symmetric transformation. Thus, real-time performance is improved. The results of simulations and experiments indicate that the approach in this paper can provide smooth motion and meet the real-time requirement under the prescribed geometrical constraint.

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.

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.

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.

Identification and “reverse engineering” of Pythagorean-hodograph curves

Methods are developed to identify whether or not a given polynomial curve, specified by Bézier control points, is a Pythagorean-hodograph (PH) curve — and, if so, to reconstruct the internal algebraic structure that allows one to exploit the advantageous properties of PH curves. Two approaches to identification of PH curves are proposed. The first is based on the satisfaction of a system of algebraic constraints by the control-polygon legs, and the second uses the fact that numerical quadrature rules that are exact for polynomials of a certain maximum degree generate arc length estimates for PH curves exhibiting a sharp saturation as the number of sample points is increased. These methods are equally applicable to planar and spatial PH curves, and are fully elaborated for cubic and quintic PH curves. The reverse engineering problem involves computing the complex or quaternion coefficients of the pre-image polynomials generating planar or spatial Pythagorean hodographs, respectively, from prescribed Bézier control points. In the planar case, a simple closed-form solution is possible, but for spatial PH curves the reverse engineering problem is much more involved.

Interpolation by $G^2$ Quintic Pythagorean-Hodograph Curves

Interpolation by $G^2$ Quintic Pythagorean-Hodograph Curves

Interpolation by G 2 Quintic Pythagorean-Hodograph Curves

Interpolation by G 2 Quintic Pythagorean-Hodograph Curves

Rotation-minimizing Euler-Rodrigues rigid-body motion interpolants

A characterization for spatial Pythagorean-hodograph (PH) curves of degree 7 with rotation-minimizing Euler–Rodrigues frames (ERFs) is determined, in terms of one real and two complex constraints on the curve coefficients. These curves can interpolate initial/final positions pi and pf and orientational frames (ti,ui,vi) and (tf,uf,vf) so as to define a rational rotation-minimizing rigid body motion. Two residual free parameters, that determine the magnitudes of the end derivatives, are available for optimizing shape properties of the interpolant. This improves upon existing algorithms for quintic PH curves with rational rotation-minimizing frames (RRMF quintics), which offer no residual freedoms. Moreover, the degree 7 PH curves with rotation-minimizing ERFs are capable of interpolating motion data for which the RRMF quintics do not admit real solutions. Although these interpolants are of higher degree than the RRMF quintics, their rotation-minimizing frames are actually of lower degree (6 versus 8), since they coincide with the ERF. This novel construction of rational rotation-minimizing motions may prove useful in applications such as computer animation, geometric sweep operations, and robot trajectory planning.

Motion design with Euler–Rodrigues frames of quintic Pythagorean-hodograph curves

The paper presents an interpolation scheme for G1 Hermite motion data, i.e., interpolation of data points and rotations at the points, with spatial quintic Pythagorean-hodograph curves so that the Euler–Rodrigues frame of the curve coincides with the rotations at the points. The interpolant is expressed in a closed form with three free parameters, which are computed based on minimizing the rotations of the normal plane vectors around the tangent and on controlling the length of the curve. The proposed choice of parameters is supported with the asymptotic analysis. The approximation error is of order four and the Euler–Rodrigues frame differs from the ideal rotation minimizing frame with the order three. The scheme is used for rigid body motions and swept surface construction.

Topological criterion for selection of quintic Pythagorean-hodograph Hermite interpolants

A topological approach to identifying the “good” interpolant among the four distinct solutions to the first-order Hermite interpolation problem for planar quintic Pythagorean-hodograph curves is presented. An existence theorem is proved, together with a complete analysis of uniqueness/non-uniqueness properties. A simple formula for finding the “good” solution, without appealing to curve fairness or energy integrals, is also presented.

A geometric product formulation for spatial Pythagorean hodograph curves with applications to Hermite interpolation

A novel formulation for spatial Pythagorean hodograph (PH) curves, based on the geometric product of vectors from Clifford algebra, is proposed. Compared to the established quaternion representation, in which a hodograph is generated by a continuous sequence of scalings/rotations of a fixed unit vector n ˆ , the new representation corresponds to a sequence of scalings/reflections of n ˆ . The two representations are shown to be equivalent for cubic and quintic PH curves, when freedom in choosing n ˆ is retained for the vector formulation. The latter also subsumes the original (sufficient) characterization of spatial Pythagorean hodographs, proposed by Farouki and Sakkalis, as a particular choice for n ˆ . In the context of the spatial PH quintic Hermite interpolation problem, variation of the unit vector n ˆ offers a geometrically more-intuitive means to explore the two-parameter space of solutions than the two free angular variables that arise in the quaternion formulation. This space is seen to have a decomposition into a product of two one-parameter spaces, in which one parameter determines the arc length and the other can be used to vary the curve shape at fixed arc length.