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UBC Theses and Dissertations

Optimal trajectory generation and precision tracking control for multi-axis machines Erkorkmaz, Kaan


This thesis presents experimentally verified smooth trajectory generation, feedrate optimization, and high performance control algorithms developed for multi-axis Cartesian machine tools. New spline parameterization and interpolation schemes are introduced that yield smooth contouring motion with minimal feedrate fluctuation along arbitrarily shaped toolpaths. The first approach is based on optimizing the toolpath geometry to yield minimal discrepancy between the spline parameter and arc length increments, resulting in an Optimally Arc Length Parameterized (OAP) quintic spline. This spline exhibits minimal feedrate fluctuation when interpolated at constant parameter increments. The second approach is based on scheduling the spline parameter to yield the desired arc displacement, hence the desired feedrate profile accurately, without having to re-parameterize the spline toolpath. The feedrate correction polynomial and iterative interpolation techniques developed for this purpose are shown to improve the feedrate consistency with reliable convergence properties, at small computational cost, making these methods viable for real-time implementation in the CNC executive. A jerk continuous feedrate optimization technique is introduced for minimizing the cycle time, while preserving the motion smoothness and tracking accuracy for traveling along spline toolpaths. Feedrate modulation is achieved by varying the travel duration of each segment and fitting the resulting C3 continuous minimum jerk displacement profile. This results in continuous velocity, acceleration, and jerk transitions spanning the entire motion along the toolpath, allowing smoother feed motion with shorter cycle time compared to piecewise constant feedrate modulation method used in current CNC systems. The feed drive dynamics of a three axis machining center are identified in detail in order to develop a controller with high tracking accuracy and bandwidth. The linear rigid body dynamic comprising of inertia and viscous friction are determined using a modified least squares scheme, which considers the existence of Coulomb friction in the parameter estimation process. The nonlinear friction model is refined by jogging the axes back and forth at various speeds, and observing the equivalent friction torque through a Kalman filter. The structural dynamics of the ball screw mechanism are identified by conducting frequency response tests, while the axis is already in motion, in order to decouple the interfering effect of stick-slip friction. The torsional vibrations of the lead screw, as well as translational vibrations of the table resulting from the torsional and axial vibrations of the lead screw, are experimentally identified. The modal characteristics of the ball screw are combined with the rigid body dynamics and guideway friction, resulting in a detailed drive model which is used for motion control law design. Two robust, adaptive sliding mode controllers have been designed, one which considers only the rigid body motion, and the second which also considers the torsional vibrations of the ball screw. Notch filtering of the first resonant mode is also investigated as a practical alternative to active vibration control, yielding successful experimental results when used in conjunction with the rigid body based sliding mode controller. Feedforward friction compensation has been added, to improve the contouring performance at circular arc quadrants and sharp corners, where the friction disturbance undergoes a discontinuous change due to axis velocity reversal. The proposed control techniques have been validated in simulations and high speed tracking and contouring experiments. The methods developed in this thesis have been evaluated on a three axis machining center, and are directly applicable to other Cartesian configured multi-axis systems, such as electronic component assembly or photolithography machines. Their extension to non-Cartesian axis configurations would require the kinematic chain and dynamic model of the machine to be considered in the control law design and trajectory generation algorithms.

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