Introduction to Geometry. By H. S. M. Coxeter. Pp. xiv, 443. 1961. (John Wiley)

G01 codes generated by CAM (Computer Aided Manufacturing) system are the most common form of tool path in CNC (Computer Numerical Control) machining. For the piecewise linear path, tangential and curvature discontinuities bring about large fluctuation of feedrate and acceleration, which produces vibration of machine tool. In recent studies, the methods for G 2 (curvature-continuous) tool-path smoothing and jerk-limiting feedrate scheduling were developed. However there still exist the deficiencies when these methods are employed in CNC machining. It is difficult to simultaneously ensure that the tool path is chord-error-constrained and G01-point-interpolated in real time. In addition, heavy computational load hinders realtime processing in CNC system. Recently the scholars experimentally found the potential of G 3 (curvature-smooth) trajectory and jerk-continuous motion in reducing the vibration of machinery. This work proposes a realtime tool-path smoothing algorithm, generating G 3 interpolative tool path composed by mixed linear and quartic Bezier segments. The purpose of the smoothing scheme is the simultaneous considerations of G 3 continuity, confined chord error, G01 points interpolated, and realtime performance. And the tool path generated is optimized in curvature variation energy (CVE) and analytical curvature extrema is available. To reduce the vibration, a high-efficient algorithm of jerk-continuous (JC) feedrate scheduling for G 3 tool path is provided. Finally, a realtime tool-path processing scheme is developed, including G 3 interpolation and motion planning functions. As shown in the simulation, the contour error, curvature of tool path, feedrate fluctuation and machining time are reduced compared with G 2 transition scheme. The experiment on a machine tool is conducted to demonstrate the advantages of the proposed algorithm in vibration reduction and surface quality, compared with G 2 transition scheme.

A realtime curvature-smooth interpolation scheme and motion planning for CNC machining of short line segments

Preface. 1. The Geometry of Curves. Introduction. Arclength Parametrization. Frenet Formulas. Nonunit Speed Curves. Some Implications of Curvature and Torsion. The Geometry of Curves and MAPLE. 2. Surfaces. Introduction. The Geometry of Surfaces. The Linear Algebra of Surfaces. Normal Curvature. Plotting Surfaces in MAPLE. 3. Curvature(s). Introduction. Calculating Curvature. Surfaces of Revolution. A Formula for Gaussian Curvature. Some Effects of Curvature(s). Surfaces of Delaunay. Calculating Curvature with MAPLE. 4. Constant Mean Curvature Surfaces. Introduction. First Notions in Minimal Surfaces. Area Minimization. Constant Mean Curvature. Harmonic Functions. 5. Geodesics, Metrics and Isometries. Introduction. The Geodesic Equations and the Clairaut Relation. A Brief Digression on Completeness. Surfaces not in R3. Isometries and Conformal Maps. Geodesics and MAPLE. 6. Holonomy and the Gauss-Bonnet Theorem. Introduction. The Covariant Derivative Revisited. Parallel Vector Fields and Holonomy. Foucault's Pendulum. The Angle Excess Theorem. The Gauss-Bonnet Theorem. Geodesic Polar Coordinates. 7. Minimal Surfaces and Complex Variables. Complex Variables. Isothermal Coordinates. The Weierstrass-Enneper Representations. BjA-rling's Problem. Minimal Surfaces which are not Area Minimizing. Minimal Surfaces and MAPLE. 8. The Calculus of Variations and Geometry. The Euler-Lagrange Equations. The Basic Examples. The Weierstrass E-Function. Problems with Constraints. Further Applications to Geometry and Mechanics. The Pontryagin Maximum Principle. The Calculus of Variations and MAPLE. 9. A Glimpse at Higher Dimensions. Introduction. Manifolds. The Covariant Derivative. Christoffel Symbols. Curvatures. The Charming Doubleness. List of Examples, Definitions and Remarks. Answers and Hints to Selected Exercises. References. Index.

Differential Geometry and Its Applications

A novel extension of the Bézier model and its applications to surface modeling

This paper presents a path planning algorithm for autonomous vehicles to generate feasible and smooth maneuvers in unstructured environments. Our approach uses a graph-based search algorithm and a discrete kinematic vehicle model to find a primitive path that is non-holonomic, collision-free, and safe. The algorithm then improves the quality and smoothness of the identified path by making local shortcuts. This shortcut for smoothing the path is determined by connecting the internal state of the primitive path to the initial state based on the Pythagorean Hodograph (PH) cubic curve. The formulation of the PH cubic curves in the smoothing path provides arc-length parameterization of the curve in a closed form. We present examples and experimental results for the real time path planning in unstructured road from an implementation on an autonomous vehicle.

Real-time path planning of autonomous vehicles for unstructured road navigation

In this paper, the geometric interpolation of planar data points and boundary tangent directions by a cubic G 2 Pythagorean-hodograph (PH) spline curve is studied. It is shown that such an interpolant exists under some natural assumptions on the data. The construction of the spline is based upon the solution of a tridiagonal system of nonlinear equations. The asymptotic approximation order 4 is confirmed.

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On interpolation by Planar cubic G^2 pythagorean-hodograph spline curves

In this paper the problem of geometric interpolation of planar data by parametric polynomial curves is revisited. The conjecture that a parametric polynomial curve of degree < n can interpolate 2n given points in R 2 is confirmed for n < 5 under certain natural restrictions. This conclusion also implies the optimal asymptotic approximation order. More generally, the optimal order 2n can be achieved as soon as the interpolating curve exists.

/pdf/on-geometric-interpolation-by-planar-parametric-polynomial-1woua4yewp.pdf

On geometric interpolation by planar parametric polynomial curves

http://valjhun.fmf.uni-lj.si/~emil/papers/GCI-last-rev-2.pdf

On geometric interpolation of circle-like curves

http://www.karlin.mff.cuni.cz/~sir/papers/biarc.pdf

C1 Hermite interpolation with spatial Pythagorean-hodograph cubic biarcs

In this paper, geometric Hermite interpolation by planar cubic G 1 splines is studied. Three data points and three tangent directions are interpolated per polynomial segment. Sufficient conditions for the existence of such a G 1 spline are determined that cover most of the cases encountered in practical applications. The existence requirements are based only upon geometric properties of data and can easily be verified in advance. The optimal approximation order 6 is confirmed, too.

Marjeta Krajnc

Papers

On interpolation by Planar cubic G^2 pythagorean-hodograph spline curves

On geometric interpolation by planar parametric polynomial curves

On geometric interpolation of circle-like curves

C1 Hermite interpolation with spatial Pythagorean-hodograph cubic biarcs

Geometric Hermite interpolation by cubic G1 splines