Order Geometric Continuity Research Articles

η3D-splines for the generation of 3D Cartesian paths with third order geometric continuity

The paper proposes a new planning primitive, named η3D-splines, based on 7th order polynomials, which is suited to generate three-dimensional paths characterized by third-order geometric continuity. The third order continuity represents an important property, since it allows continuous-jerk reference signals for the joints actuators of robotic systems. Differently from other approaches in the literature, η3D-splines are efficiently evaluated by means of closed form expressions as function of the assigned interpolation conditions. This allows an intuitive real-time generation of composite paths: from the knowledge of the geometric characteristics of the curve which is currently executed, and by choosing a novel end-point together with the desired interpolating conditions, a new path can be efficiently generated by simultaneously maintaining the overall third order geometric continuity. Additionally, the η3D-splines can be shaped by acting on a set of six free parameters, so as to emulate other planning primitives, like, for example, linear segments, circular arcs, clothoids, helical curves, and conic spirals. Furthermore, by means of the same parameters, all possible 7th order polynomials, which fulfill the given interpolating conditions, can be generated. The accompanying video shows an anthropomorphic manipulator executing a composite trajectory generated by means of the η3D-splines.

Accurate and Efficient Approximation of Clothoids Using Bézier Curves for Path Planning

An accurate and efficient clothoid approximation approach is presented in this paper using Bézier curves based on the minimization of curvature profile difference. Compared with existing methods, the proposed approach is able to guarantee higher order geometric continuity with smaller approximation error in terms of position, orientation, and curvature. The approximation scheme takes place in three stages. First, a subset of clothoids with specific winding angle constraints referred to as elementary clothoids is approximated using quintic Bézier curves. Then, a basic clothoid defined in the first quadrant is formulated, which is composed of a series of transformed elementary clothoids. An adaptive sampling stra-tegy is applied to ensure that the resulting Bézier segments are computed within a specified accuracy and all the required information can be obtained offline and stored in a lookup table. Finally, a general clothoid with arbitrary parameters can be conveniently approximated based on the lookup table through appropriate geometric transformations. A comparison with the recent circular interpolation and rational Bézier curve based approximation shows that the proposed approach is able to achieve equivalent or greater computational efficiency in most scenarios.

A Geometric Modeling Method of Reconstruction Vertices and Edges with Smooth Surface

In surface modeling, it is important to reconstruct the smooth connecting surface for the polyhedron vertex and edge rendering. The general method by using the spherical and cylindrical surfaces do not possess the geometric continuity. This paper proposed a geometric modeling method of reconstruction vertices and edges with smooth surface for the polyhedron corner rendering with high order geometric continuity. The proposed method is based on the principle of the theory of geometry continuity through a geometry drawing algorithm. The reconstructed surface can be connected neighboring surfaces with G1 geometry continuity. The results show that the reconstructed surfaces can polish and render the polyhedron corner with good geometry continuity property.

${ \mmb{\eta } }^{3}$-Splines for the Smooth Path Generation of Wheeled Mobile Robots

The paper deals with the generation of smooth paths for the inversion-based motion control of wheeled mobile robots. A new path primitive, called eta3-spline, is proposed. It is a seventh order polynomial spline which permits the interpolation of an arbitrary sequence of points with associated arbitrary tangent directions, curvatures, and curvature derivatives, so that an overall G3-path is planned. A G3-path or path with third order geometric continuity has continuous tangent vector, curvature, and curvature derivative along the arc length. Adopting this planning scheme and a dynamic path inversion technique, the robot's command velocities are continuous with continuous accelerations. The new primitive depends on a vector (eta) of six parameters that can be used to finely shape the path. The eta3-spline can generate or approximate, in a unified framework, a variety of curve primitives such as circular arcs, clothoids, spirals, etc. The paper includes theoretical results, path planning examples, and a note on general etak-splines.

Smooth control of a wheeled omnidirectional robot

The paper investigates and proposes a solution for the smooth control of a wheeled omnidirectional mobile robot. The flexibility of this holonomous robot is fully exploited by means of the control strategy proposed. It is based on two main concepts: a Cartesian path planning and a wheel command generation based on the dynamic inversion of the robot kinematic model. The planned path has to exhibit appropriate geometric characteristics in order to generate wheel command signals with continuous accelerations. In particular, the second order geometric continuity of the path, i.e. the continuity of the curvature, has to be guaranteed. Simulation results concerning a perturbed system are reported

Geometrical criteria on the higher order smoothness of composite surfaces

A generalization of a theorem by Pegna and Wolter—called Linkage Curve Theorem—is presented. The new theorem provides a condition for joining two surfaces with high order geometric continuity of arbitrary degree n. It will be shown that the Linkage Curve Theorem can be generalized even for the case when the common boundary curve is only G 1.

Massively parallel approximation of irregular triangular meshes of arbitrary topology with smooth parametric surfaces

This paper describes a parallel implementation of a previously developed mathematical model intended for the approximation of 3D triangular meshes with smooth surfaces yielding first order geometric continuity G 1. This represents a novel application of SIMD architectures to the approximation of irregular meshes. Previous related works have focused on the approximation of rectangular meshes using tensor-product approximants, such as B-splines, that suit the regular structure of most parallel architectures. A parallel implementation of the proposed surface model at three degrees of granularity shows that a coarse grain scheme yields maximum performance when each control triangle is approximated by a single processor avoiding inter-processor communication. The data distribution necessary to attain an independent task-farm topology is studied. The different algorithms have been implemented using a data-parallel model and tested on two Connection Machine 200 parallel computers with 4 and 16 K processors, respectively. The algorithms achieve efficiencies close to a 100% and scale linearly in the number of processors.

The geometric continuity of rational Bézier triangular surfaces

The problems of geometric continuity for rational Bezier surfaces are discussed. Concise conditions of first order and second order geometric continuity for rational triangular Bezier surfaces are given. Meanwhile, a geometric condition for smoothness between adjacent rational Bezier surfaces and the transformation formulae between rational triangular patches and rational rectangular patches are obtained.

Local surface interpolation with shape parameters between adjoining Gregory patches

A surface interpolation method with local control for meshes of cubic curves is described. This mesh is filled with Gregory patches, so that the surface is represented by a union of geometrically continous bicubic quadrilateral and/or quartic triangular Gregory patches. Along any boundarycurve the shape of the two adjoining patches can be locally controlled with shape parameters, suchas tilt, bulge and shear. The procedure is a simple extension of Chiyokura's method [Chiyokura & Kimura '83] that gives extra control to the user while preserving first order geometric continuity between the patches.

Smooth curves under tension

A new type of interpolating spline under tension is presented. The curves are obtained by a variational formulation. Unlike the closely related tau-splines, they possess third order parametric continuity and a smooth torsion plot. The special case of uniform tension parameters yield even spline curves with fourth order geometric continuity.

A locally controllable spline with tension for interactive curve design

We describe a new type of parametrically defined space curve. The parameters which define these curves allow for the convenient control over local shape attributes while maintaining global second order geometric continuity. The coordinate functions are defined by piecewise segments of rational functions, each segment being the ratio of cubic polynomials and a common quadratic polynomial. Each curve segment is a planar curve and where two segments meet the curvature is zero. This simple mathematical representation permits these curves to be efficiently manipulated and displayed.