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Spline Interpolation and Contour Error Pre-Compensation for 5-Axis Machining

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Spline Interpolation and Contour Error Pre-Compensation for 5-Axis Machining SPLINE INTERPOLATION AND CONTOUR ERROR PRE-COMPENSATION FOR 5-AXIS MACHINING by Alexander Yuen B.A.Sc., The University of British Columbia, 2011 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The Faculty of Graduate Studies (Mechanical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) July 2013 ⃝c Alexander Yuen 2013 Abstract This thesis presents experimentally verified smooth spline interpolation and contour error pre- compensation algorithms developed for 5-axis machine tools. In the smooth spline interpola- tion algorithm, the cutter location data from the computer-aided manufacturing system is first fitted independently to decouple the relative changes in position and orientation. The tool tip positions are fitted to a quintic B-spline, achieving geometric jerk continuity. Next, the tooltip orientations are fitted to another quintic B-spline in spherical coordinates, achieving geometric jerk continuity and feasible orientations at all points of the spline. The nonlinear relationship between spline parameters and displacement are approximated with 9th order and seventh order feed correction splines for position and orientation, respectively. The 9th order feed correction spline is fit adaptively to minimize fitting error while preserving C3 continuity. The seventh order feed correction spline is optimized to minimize jerk while preserving C3 continuity as well. In the contour error pre-compensation algorithm, the position commands generated in the smooth spline interpolation algorithm, are first fitted to piecewise quintic splines while respect- ing velocity, acceleration and jerk continuity at the spline joints. The transfer function of each servo drive is kept linear by compensating the disturbance effect of friction with a feed-forward block. Using the analytically represented 5-axis, splined tool path, splined tracking errors and kinematic model of the five-axis machine tool, contouring errors are predicted ahead of axis control loops. The contouring errors are decoupled into three linear and two rotary drives, and the position commands are modified before they are sent to servo drives for execution. The methods developed in this thesis have been evaluated on a 5-axis machining center with a tilting-table configuration, and are directly applicable to other 5-axis kinematic configurations ii Abstract such as spindle-tilting or hybrid configurations. The experiments show improvements in fit- ting accuracy, reduction in vibrations, reduction in tracking errors, and significant reductions in contour error for five-axis tool paths. iii Preface Chapter 3. A version of this material has been published in the International Journal of Ma- chine Tools and Manufacture [Yuen A, Zhang K, Altintas Y. Smooth trajectory generation for five-axis machine tools. International Journal of Machine Tools and Manufacture doi: 10.1016/j.ijmachtools.2013.04.002]. I was the lead investigator, responsible for the major development of the algorithms, experiment design, data collection and analysis, as well as manuscript composition. Zhang K was involved in the early development of the algorithms and contributed to the manuscript edits. Altintas Y was the supervisory author on this project and was involved throughout the project in concept formation and manuscript composition Chapter 4. A version of this material has been accepted by the International Journal of Machine Tools and Manufacture [Zhang K, Yuen A, Altintas Y. Pre-Compensation of Contour Errors in Five-Axis CNC Machine Tools. International Journal of Machine Tools and Manufacture]. Zhang K was the lead investigator, responsible for the major development of the algorithms, analysis, and manuscript composition. In this work, I was involved in the early development of the algorithms, experiment design, data collection, and analysis. Furthermore, I contributed a major portion to the manuscript revision. Altintas Y was the supervisory author on this project and was involved throughout the project in concept formation and manuscript composition Chapter 5. The results found in this chapter were previously published in the above two men- tioned works. Modifications include a more detailed analysis of the results. All figures and tables found in this thesis are used with permission from applicable sources. iv Table of Contents Abstract .......................................... ii Preface ........................................... iv Table of Contents ..................................... v List of Tables ........................................ viii List of Figures ....................................... ix Nomenclature ....................................... xii Acknowledgments ..................................... xix 1 Introduction ...................................... 1 2 Literature Review ................................... 7 2.1 Overview . 7 2.2 Three-Axis Spline Interpolation Techniques . 8 2.3 Five-Axis Spline Interpolation Techniques . 12 2.4 Five-Axis Spline Contour Error Pre-compensation . 15 2.5 Conclusions . 19 3 Spline Parametrization and Interpolation for 5-Axis Machines .......... 20 3.1 Overview . 20 v Table of Contents 3.2 B-Spline Background . 22 3.3 Decoupled B-spline Polynomial Representation . 28 3.4 C3 Tool Tip Position Spline Generation . 31 3.4.1 Parametric Position Spline Fitting . 31 3.4.2 Feed Correction Polynomial . 32 3.5 C3 Tool Orientation Spline Generation . 40 3.5.1 Parametric Orientation Spline . 40 3.5.2 Orientation Spline Reparametrization . 43 3.5.2.1 Windowing for a High Number of Data . 47 3.6 Conclusions . 50 4 Contour Error Pre-compensation .......................... 51 4.1 Overview . 51 4.2 Definition of Position and Orientation Contour Errors . 52 4.3 Tracking Error Prediction . 54 4.3.1 Fitting a Quintic Time Spline to Axis Position Commands . 57 4.3.2 Analytical Solution of Tracking Error . 63 4.4 Prediction of Contour Errors . 66 4.5 Compensation of Contour Errors . 69 4.6 Conclusions . 74 5 Simulations and Experiments ............................ 76 5.1 Overview . 76 5.2 Experimental Test Bed . 76 5.3 Simulations and Experiments for 5-Axis Spline Parametrization and Interpola- tion ........................................ 78 vi Table of Contents 5.3.1 Controller Design . 78 5.3.2 Simulation and Experimental Results . 79 5.4 Simulations and Experiments for 5-Axis Contour Error Pre-compensation . 87 5.4.1 Controller Design . 87 5.4.2 Simulation and Experimental Results . 89 5.5 Conclusions . 97 6 Conclusions ...................................... 98 6.1 Conclusions . 98 6.2 Future Research Directions . 99 Bibliography ........................................ 101 vii List of Tables 5.1 Plant parameters for Fadal VMC 2216 . 77 5.2 PID control parameters for 5-Axis spline parameterization and interpolation tests 79 5.3 Comparison of max./min. jerk values for test tool path at constant feed segment 86 5.4 Percentage tracking error . 86 5.5 PID control parameters for 5-Axis contour error pre-compensation tests . 88 viii List of Figures 1.1 Geometric deviations in position and orientation for a typical 5-axis machine tool 2 1.2 Overview of the work in this thesis, starting from the CAD/CAM system to the generated reference commands. The dashed boxes highlight the main contri- butions . 5 1.3 Overview of the main contributions in this thesis. Figure a) shows the basic overview of the spline interpolation algorithm developed. Figure b) shows the basic overview of the contour error pre-compensation algorithm developed. 6 2.1 Basic methodology of the spline fitting process . 8 2.2 Discrepancy between geometric parameter (u) and actual arc length (s) .... 9 2.3 Basic methodology of the 5-axis spline fitting process . 13 2.4 Approaches to reduce geometric deviations in a multi-axis machine . 16 3.1 Use of geometrically smooth splines to interpolate CAM data, and the resulting improvement in feed profiling . 21 3.2 Overview of proposed trajectory generation scheme. 23 3.3 B-spline curve representation. 26 3.4 Visualization of basis functions and the first derivative of basis functions over the parametric interval [0,1] . 27 3.5 Decoupled representation for a 5-axis toolpath . 29 3.6 Sequence of functions for reparametrization . 30 3.7 Different methods of interpolating along a given spline toolpath . 33 ix List of Figures 3.8 Adaption results of the feed correction polynomial fitting algorithm . 39 3.9 Mappings between spherical coordinates and (θ; ϕ) plane. 41 th 3.10 7 order Bezier´ spline and its control coefficients fitted to (sk; w¯k) data. 44 3.11 Comparison of w(s) with initially guessed control points and converged control points . 47 3.12 Comparison of reference jerks with w(s) using initially guessed control points and w(s) using converged control points . 48 3.13 Windowing scheme for generation of Orientation Reparameterization Spline . 49 4.1 Tool position and orientation contour errors . 52 4.2 Five-axis machine tool configuration. 55 4.3 Offsets of the rotary table that are included in the kinematic calculations for 5-axis rotary table kinematic configuration . 55 4.4 Tracking error prediction model. 57 4.5 Axis position fitting. 61 4.6 Block diagram for the PID controller and x-axis drive. 63 4.7 Tool tip position and orientation contour errors. 66 4.8 Orthogonality of On and Oact to O’n . 69 4.9 Contouring errors pre-compensation strategy. 70 5.1 Experimental setup . 77 5.2 Control Law for 5-Axis Spline Parameterization and Inteprolation Experiments 78 5.3 Simulation of Dual Spline and Decoupled Methods
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