Recent papers [Yang DCH, Kong T. Computer Aided Design 1994;26:225–34; Yeh S-S, Hsu P-L. Computer Aided Design 1999;31:349–57] formulate real-time CNC interpolators for variable feedrates along parametric curves. These interpolators employ truncated Taylor series to compute successive reference-point parameter values, but in both papers an erroneous coefficient for the highest (quadratic) term is cited. The derivation of the proper coefficients is a straightforward, although somewhat convoluted, exercise in chain-rule differentiation. Compact recursive formulae are presented here to compute the correct coefficients, up to the cubic term, in cases where the feedrate depends on: (i) elapsed time; (ii) curve arc length; or (iii) local path curvature. The local and cumulative effects of truncation errors on the accuracy of such interpolator schemes are also assessed, and compared with the essentially exact interpolators for Pythagorean-hodograph curves.

Exact Taylor series coefficients for variable-feedrate CNC curve interpolators

A practical approach for generating motion paths with continuous steering for car-like mobile robots is presented here. This paper addresses two key issues in robot motion planning; path continuity and maximum curvature constraint for nonholonomic robots. The advantage of this new method is that it allows robots to account for their constraints in an efficient manner that facilitates real-time planning. B-spline curves are leveraged for their robustness and practical synthesis to model the vehicle's path. Comparative navigational-based analyses are presented to selected appropriate curve and nominate its parameters. Path continuity is achieved by utilizing a single path, to represent the trajectory, with no limitations on path, or orientation. The path parameters are formulated with respect to the robot's constraints. Maximum curvature is satisfied locally, in every segment using a smoothing algorithm, if needed. It is demonstrated that any local modifications of single sections have minimal effect on the entire path. Rigorous simulations are presented, to highlight the benefits of the proposed method, in comparison to existing approaches with regards to continuity, curvature control, path length and resulting acceleration. Experimental results validate that our approach mimics human steering with high accuracy. Accordingly, efficiently formulated continuous paths ultimately contribute towards passenger comfort improvement. Using presented approach, autonomous vehicles generate and follow paths that humans are accustomed to, with minimum disturbances.

Continuous Path Smoothing for Car-Like Robots Using B-Spline Curves

An NC system that machines a curved shape at fixed depth of cut experiences time-varying cutting forces due to the ‘curvature effect’—the material removal rate is higher than nominal in concave regions, and lower in convex regions. A curvature-dependent feedrate function that automatically compensates for this effect is formulated, and it is shown that, for Pythagorean-hodograph (PH) curves, the periodic real time computation of reference points in accordance with this function can be analytically reduced to a sequence of root-finding problems for simple monotone functions. Empirical results from an implementation of this variable-feedrate interpolators on an open-architecture CNC milling machine are presented and compared with results from fixed-feedrate interpolators. The curvature-compensated feedrate scheme has important potential applications in ensuring part accuracy and in optimizing part programs consistent with a prescribed accuracy.

Variable-feedrate CNC interpolators for constant material removal rates along Pythagorean-hodograph curves

The interpolation of first-order Hermite data by spatial Pythagorean-hodograph curves that exhibit closure under arbitrary 3-dimensional rotations is addressed. The hodographs of such curves correspond to certain combinations of four polynomials, given by Dietz et al. [4], that admit compact descriptions in terms of quaternions – an instance of the “PH representation map” proposed by Choi et al. [2]. The lowest-order PH curves that interpolate arbitrary first-order spatial Hermite data are quintics. It is shown that, with PH quintics, the quaternion representation yields a reduction of the Hermite interpolation problem to three “simple” quadratic equations in three quaternion unknowns. This system admits a closed-form solution, expressing all PH quintic interpolants to given spatial Hermite data as a two-parameter family. An integral shape measure is invoked to fix these two free parameters.

Hermite Interpolation by Rotation-Invariant Spatial Pythagorean-Hodograph Curves

This chapter provides an introduction to pythagorean-hodograph (PH) Curves, which offer many advantages, such as rational offsets, superior shape properties, and real-time interpolators. Apart from these practical advantages, the investigation of PH curves is of considerable intellectual appeal for the wealth of mathematical ideas it entails, including the geometry of complex numbers, inaugurated by Wessel, Argand, and Gauss; Hamilton's quaternions and the geometrical algebras of Clifford; medial axis transforms and the Minkowski metric of special relativity; projective geometry and dual representations; the cyclographic map and Laguerre geometry; and connections with classical geometrical optics. The distinguishing feature of a polynomial PH curve is that the components of its hodograph satisfy a Pythagorean condition—that is, the sum of their squares is equal to the square of a polynomial. By virtue of their special algebraic structure, PH curves provide exact solutions to a number of basic geometrical problems in computer-aided design and manufacturing.

Chapter 17 – Pythagorean-Hodograph Curves

https://ykoren.engin.umich.edu/wp-content/uploads/sites/122/2014/05/78.-Adaptive-fuzzy-logic-controller-for-feed-drives-of-a-CNC-machine-tool-Mechatronics-2004.pdf

Adaptive fuzzy logic controller for feed drives of a CNC machine tool

Design of rational cam profiles with pythagorean-hodograph curves

In CAD systems, a surface to be machined is expressed by a series of curves, such as B-spline, Bezier and NURBS curves, which compose the surface and then in CAM systems the curves are divided into a large number of line or arc segments. These divided movement commands, however, cause many problems including their excessive size of NC data that makes almost impossible local adjustment or modification of the surface. To cope with those problems, the necessity of real-time curve or surface interpolators was embossed. This paper presents an efficient real-time tool-path generation method for interpolation of NURBS surfaces in CNC machining. The proposed tool-path generation method is based on an improved iso-scallop strategy and can provide better precision than the existing methods. The proposed method is designed such that tool-path planning is easily managed in real-time. It proposed a new algorithm, for regulation of a scallop height, which can efficiently generate tool-paths and can save machining time compared with the existing method. Through computer simulations, the performance of the proposed method is analyzed and compared with the existing method in terms of feedrate, total machining time and a degree of constraint on the scallop height.

Tool-path generation for NURBS surface machining

Variation of the effective moment of inertia due to load disturbance in a CNC feed drive system may increase control error for each axis of motion and thereby degrading total precision including contouring accuracy. Although some investigations have been made on estimation and control of variation of the effective moment of inertia, most of them are for individual axis control: there has been no research that considered the improvement of contouring accuracy in CNC machine tools under load variation. In this paper, a real-time inertia compensation method is proposed including a newly defined contour error model that represents error caused by load disturbance. Position error due to load variation is calculated based on the estimated disturbance torque by a full-order disturbance observer for each drive axis. The position error is then used to establish the new contour error model considering the effect of load variation. The proposed compensation method is realized by adjusting velocity loop gains in the control system using the least square method such that the effect of variation of the effective moment of inertia on the contour error is minimized. The proposed method is evaluated with a 3-axis open CNC testbed.

Sungchul Jee

Papers

Variable-feedrate CNC interpolators for constant material removal rates along Pythagorean-hodograph curves

Adaptive fuzzy logic controller for feed drives of a CNC machine tool

Design of rational cam profiles with pythagorean-hodograph curves

Tool-path generation for NURBS surface machining

Real-time inertia compensation for multi-axis CNC machine tools