Rounding Spatial G-Code Tool Paths Using Pythagorean Hodograph Curves

2007 ◽  
Vol 7 (3) ◽  
pp. 186-191 ◽  
Author(s):  
Zbyněk Šír ◽  
Elmar Wings ◽  
Bert Jüttler
Keyword(s):  
Hermite Interpolation ◽  
Second Order ◽  
Peak Acceleration ◽  
Curve Segment ◽  
Industrial Environment ◽  
Tool Paths ◽  
Pythagorean Hodograph ◽  
G Code ◽  
Linear Elements ◽  
Ph Curve

We describe and analyze a new algorithm for rounding standard G-code tool paths. The joints of circular/linear elements are replaced by small segments of Pythagorean hodograph (PH) curves so that the final curve is globally C2 continuous. The PH segments are produced via a second order Hermite interpolation. We discuss some implementation details and investigate the error introduced by replacing a part of G-code by a PH curve segment. We also report results of tests within an industrial environment that demonstrate an increase in path velocity while decreasing peak acceleration.

Loci Conversion and Corner Smoothing with PH Curves in CNC System

2009 ◽  
Vol 419-420 ◽  
pp. 161-164
Author(s):  
Qi Kui Wang ◽  
You Dong Chen ◽  
Wei Li ◽  
Tian Miao Wang ◽  
Hong Xing Wei
Keyword(s):  
Tangent Vector ◽  
Machining Process ◽  
Free Form ◽  
Surface Interpolation ◽  
Pythagorean Hodograph ◽  
Linear Elements ◽  
Smoothing Process ◽  
Ph Curve ◽  
Interpolation Functions ◽  
Tangent Vectors

Free-form surface interpolation functions give more advantages in machining than the traditional line and circle functions. A method is developed to convert lines and circles into Pythagorean-hodograph (PH) curves. In order to get smooth machining process the PH curve is used to replace the joints of the circular/linear elements by the connection situation. The slope of line is used to get the tangent vector in the line conversion. When converting a circle to a PH curve, points of the divided circle are introduced to compute the vectors. The methods of computing tangent vectors are proposed according to the slope of the line and the quadrant of the circle. The transformation errors from lines and circles to PH curves are computed. In the corner smoothing process the tangent vectors are computed by the connection between lines and circles. Replacement errors at the joints are computed for the use of PH curve. The results demonstrate the feasibility of the conversion from line and circle to PH curve. The PH curves at the joints of the circular and linear elements show continuous trajectory.

Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics

2005 ◽  
Vol 22 (4) ◽  
pp. 325-352 ◽  
Author(s):  
Francesca Pelosi ◽  
Rida T. Farouki ◽  
Carla Manni ◽  
Alessandra Sestini
Keyword(s):  
Hermite Interpolation ◽  
Pythagorean Hodograph ◽  
Geometric Hermite Interpolation

Path Planning of Multiple UAV’s in an Environment of Restricted Regions

2005 ◽  
Author(s):  
Madhavan Shanmugavel ◽  
Antonios Tsourdos ◽  
Rafal Zbikowski ◽  
Brian White
Keyword(s):  
Path Planning ◽  
Path Length ◽  
Mathematical Property ◽  
Multiple Uavs ◽  
Safety Margins ◽  
Pythagorean Hodograph ◽  
Offset Curves ◽  
Safety Constraints ◽  
Continuous Curvature ◽  
Ph Curve

This paper describes a novel idea of path planning for multiple UAVs (Unmanned Aerial Vehicles). The path planning ensures safe and simultaneous arrival of the UAVs to the target while meeting curvature and safety constraints. Pythagorean Hodograph (PH) curve is used for path planning. The PH curve provides continuous curvature of the paths. The offset curves of the PH paths define safety margins around and along each flight path. The simultaneous arrival is satisfied by generation of paths of equal lengths. This paper highlights the mathematical property — changing path-shape and path-length by manipulating the curvature and utilises this to achieve the following constraints: (i) Generation of paths of equal length, (ii) Achieving maximum bound on curvature, and, (iii) Meeting the safety constraints by offset paths.

Design of Pythagorean Hodograph Curve Interpolator Based on NiosII Embedded Processor and FPGA

2011 ◽  
Vol 383-390 ◽  
pp. 6868-6872
Author(s):  
Jing Jie Guo ◽  
Wei Tang
Keyword(s):  
High Speed ◽  
Precision Machining ◽  
Interpolation Algorithm ◽  
Interpolation Scheme ◽  
Embedded Processor ◽  
Computational Burden ◽  
Curve Interpolation ◽  
Pythagorean Hodograph ◽  
Fpga Chip ◽  
Ph Curve

In this paper, a novel architecture of Pythagorean Hodograph (PH) curve interpolator based on Nios Ⅱ embedded processor and FPGA is proposed. The whole interpolator including NiosⅡ processor is built in a single FPGA chip. The interpolator uses a two-stage interpolation scheme to reduce the computational burden of PH curve interpolator. The Nios Ⅱ embedded processor implements 1st-stage interpolation, the FPGA receives the command from the Nios Ⅱ processor and implements 2nd-stage interpolation simultaneously. Therefore, the interpolator can implement the real-time PH curve interpolation algorithm steadily to meet the needs of high-speed and high-precision machining.

First order hermite interpolation with spherical Pythagorean-hodograph curves

2007 ◽  
Vol 23 (1-2) ◽  
pp. 73-86 ◽  
Author(s):  
Gwang-Il Kim ◽  
Jae-Hoon Kong ◽  
Sunhong Lee
Keyword(s):  
Hermite Interpolation ◽  
First Order ◽  
Pythagorean Hodograph

scholarly journals KANTBP 4M Program for Solving the Scattering Problem for a System of Ordinary Second-Order Differential Equations

2020 ◽  
Vol 226 ◽  
pp. 02008
Author(s):  
Galmandakh Chuluunbaatar ◽  
Alexander A. Gusev ◽  
Ochbadrakh Chuluunbaatar ◽  
Sergue I. Vinitsky ◽  
Luong Le Hai
Keyword(s):  
Differential Equations ◽  
Hermite Interpolation ◽  
Scattering Problem ◽  
Second Order ◽  
The Finite Element Method ◽  
Multichannel Scattering ◽  
Reaction Region ◽  
Interpolation Polynomials ◽  
Complex Valued ◽  
Hermite Interpolation Polynomials

We report an upgrade of the program KANTBP 4M implemented in the computer algebra system MAPLE for solving, with a given accuracy, the multichannel scattering problem, which is reduced to a boundary-value problem for a system of ordinary differential equations of the second order with continuous or piecewise continuous real or complex-valued coeffcients. The solution over a finite interval is subject to mixed homogeneous boundary conditions: Dirichlet and/or Neumann, and/or of the third kind. The discretization of the boundary problem is implemented by means of the finite element method with the Lagrange or Hermite interpolation polynomials. The effciency of the proposed algorithm is demonstrated by solving a multichannel scattering problem with coupling of channels in both the reaction region and the asymptotic one.

Superconvergence Recovery for the Gradient of the Trilinear Finite Element

2011 ◽  
Vol 268-270 ◽  
pp. 1021-1024
Author(s):  
Jing Hong Liu ◽  
De Cheng Yin
Keyword(s):  
Finite Element ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Second Order ◽  
Three Dimensions ◽  
Elliptic Boundary Value Problem ◽  
Elliptic Boundary Value ◽  
Interpolation Postprocessing ◽  
Linear Elements

For a second-order elliptic boundary value problem in three dimensions, we use an interpolation postprocessing technique to obtain recovered gradients of tri- linear elements over regular meshes. Further, superconvergence of these gradients are proved.

C˜2 Continuous Cubic Hermite Interpolation Splines with Second-Order Elliptic Variation

2021 ◽  
Vol 7 (6) ◽  
pp. 6317-6331
Author(s):  
Jie Li ◽  
Yaoyao Tu ◽  
Shilong Fei
Keyword(s):  
Shape Control ◽  
Spline Function ◽  
Hermite Interpolation ◽  
Second Order ◽  
Basis Functions ◽  
Interpolation Spline ◽  
Selection Scheme ◽  
Spline Curve ◽  
Spline Curves ◽  
Two Parameters

In order to solve the deficiency of Hermite interpolation spline with second-order elliptic variation in shape control and continuity, c-2 continuous cubic Hermite interpolation spline with second-order elliptic variation was designed. A set of cubic Hermite basis functions with two parameters was constructed. According to this set of basis functions, the three-order Hermite interpolation spline curves were defined in segments 02, and the parameter selection scheme was discussed. The corresponding cubic Hermite interpolation spline function was studied, and the method to determine the residual term and the best interpolation function was given. The results of an example show that when the interpolation conditions remain unchanged, the cubic Hermite interpolation spline curves not only reach 02 continuity, but also can use the parameters to control the shape of the curves locally or globally. By determining the best values of the parameters, the cubic Hermite interpolation spline function can get a better interpolation effect, and the smoothness of the interpolation spline curve is the best.

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