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TECHNICAL PAPERS

# Optimal Feed-Rate Scheduling for High-Speed Contouring

[+] Author and Article Information
J. Dong, J. A. Stori

Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801

J. Manuf. Sci. Eng 129(1), 63-76 (Feb 15, 2004) (14 pages) doi:10.1115/1.2280549 History: Received February 21, 2003; Revised February 15, 2004

## Abstract

The majority of efforts to improve the contouring performance of high-speed CNC systems has focused on advances in feedback control techniques at the single-axis servo level. Regardless of the dynamic characteristics of an individual system, performance will inevitably suffer when that system is called upon to execute a complex trajectory beyond the range of its capabilities. The intent of the present work is to provide a framework for abstracting the capabilities of an individual multiaxis contouring system, and a methodology for using these capabilities to generate a time-optimal feed-rate profile for a particular trajectory on a particular machine. Several constraints are developed to drive the feed-rate optimization algorithm. First, simplified dynamic models of the individual axes are used to generate performance envelopes that couple the velocity versus acceleration capabilities of each axis. Second, bandwidth limitations are introduced to mitigate frequency related problems encountered when traversing sharp geometric features at high velocity. Finally, a dynamic model for the instantaneous following error is used to estimate the contour error as a function of the instantaneous velocity and acceleration state. We present a computationally efficient algorithm for generating a minimum-time feed-rate profile subject to the above constraints, and demonstrate that significant improvements in contouring accuracy can be realized through such an approach. Experimental results are presented on a conventional two-axis $X−Y$ stage executing a complex trajectory.

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## Figures

Figure 1

Parallelogram constraints on an axis

Figure 2

Block diagram for simplified amplifier and servo dynamics

Figure 3

Experimental response of each axis to open-loop command reversals

Figure 4

Dynamic model of a single axis of a CNC system including PD and feed-forward controller

Figure 5

Tangential and curvature-based estimates of the contour error

Figure 6

Complex trajectory for experimental case studies. The trajectory begins and ends at the origin with zero velocity. The sharp reversals and varying directions of motion make this a challenge for accurate tracking at high velocities. The performance in regions A–D will be shown in more detail in Figs.  1113.

Figure 7

Comparison between contour error model predictions and experimentally observed errors on the trajectory of Fig. 6

Figure 8

Optimized velocity scheduling, case I: Actuator constraints only. Top left: Actuator constraints are binding at all points on trajectory. Top right: Accompanying velocity, acceleration, and parametric speed (solid line is tangential, dashed is x-axis, dash-dot is y-axis). Bottom: Optimal trajectory states of each axis viewed in velocity versus acceleration space.

Figure 9

Optimized velocity scheduling, case II: Actuator and bandwidth constraints. Top left: Actuator constraints are binding at all points on trajectory. Top right: Accompanying velocity, acceleration, and parametric speed (solid line is tangential, dashed is x-axis, dash-dot is y-axis). Bottom: Optimal trajectory states of each axis viewed in velocity versus acceleration space.

Figure 10

Optimal velocity scheduling, case III: Actuator, bandwidth, and contour error constraints. Top left: Either actuator (x), bandwidth (o), or contour error (Δ) constraints are active at all points on trajectory. Top right: Accompanying velocity, acceleration, and parametric speed (solid line is tangential, dashed is x-axis, dash-dot is y-axis). Bottom: Optimal trajectory states of each axis viewed in velocity versus acceleration space.

Figure 11

Comparison between the constant feed-rate trajectory and the optimized trajectory subject to actuator, bandwidth, and contour-error constraints at regions A–D on the path

Figure 12

Comparison between the G-code trajectory and the optimized trajectory subject to actuator, bandwidth, and contour-error constraints at regions A–D on the path

Figure 13

Comparison between the optimized trajectory subject to actuator and contour-error constraints and the optimized trajectory subject to actuator, bandwidth, and contour-error constraints at regions A–D on the path

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