Equipment industry is the most fundamental industry of any modern society. Modern equipment is characterized by its high automation, efficiency, reliability, flexibility, high precision, energy conservation and environmental protection. This research is motivated by plethora of industry demand of high performance machines.

The performance of a machine is determined by its mechanics and the motion control system. There are three types of motion in general. One is synchronization motion where the reference motion command is determined by feedback measurements. The second type of motion is point-to-point motion where the reference motion command contains only a sequence of target positions. The third type of motion is called path following motion where the tool path is composed of geometric curves. Although the types of reference command are different, all motion control systems consist of two main building blocks: reference motion command generation and servo control system. This dissertation focus on the first building block.

The traditional path following motion control system only accepts lines, circular arcs and other conic tool paths. The feedrate and system accuracy is low because of nonsmoothness of the tool path. If the system can handle free form paths, especially parametric curves, then the accuracy and the speed of the motion control system can be improved.

Although path interpolation is a classical problem in computer numerical controlled system (CNC), a clear formula in the literature is still missing. Given the geometric tool paths and the feedrate along the tool paths, the task of path interpolation is to generate reference motion command satisfying all geometric, kinematic and dynamic requirements. In this dissertation, path interpolation problem for generic parametric tool path is formulated clearly in a mathematical way. By examining various parametrization, the problem becomes an initial value problem for the ordinary differential equation system. From this formula, traditional interpolation algorithms, like digital differential analyzer (DDA) and digital increment algorithm, can be easily derived. Numerical algorithms such as the truncated Taylor expansion algorithm, the Runge-Kutta algorithm and the predictor-corrector algorithm can be utilized. By using criteria such as reliability, accuracy and complexity, all those algorithms are analyzed. The comparison comes from three perspectives - theoretical analysis on numerical computation, simulation and experiments. The relations between interpolation accuracy, interpolation cycling time, feedrate and the curvature of tool path are also investigated. Although the truncated Taylor expansion algorithms and the predictor-corrector algorithms are widely discussed in the literature, the Runge-Kutta algorithms are recommended by theoretical analysis and experiments. The Runge-Kutta algorithm of order 4 is highly recommended if the computation resource is available.

To generate the reference motion command, the feedrate must be designed properly. For example, the integration of the feedrate should be equal to the total arc length of the tool path, and the feedrate should be continuous. The accuracy requirements impose some constraints on the upper bound of the feedrate. For example, the chord error tolerance in the path following motion control system limits the feedrate in path interpolation algorithm. Besides geometric errors, the kinematic limitations of mechatronic system also need to be considered. For example, the velocity and acceleration of each servo axis are bound from above, which gives an upper bound on the feedrate. The dynamic performance of the mechatronic system also has limitations. For example, there are interpolation and integral conditions on closed loop system. To guarantee the tracking accuracy, the reference motion command should contain no signals with harmonic components at high frequencies. Considering all those constraints, feedrate scheduling problem is formulated as a nonlinear constrained feasibility or an optimization problem. From signal processing theory, the feedrate can be reshaped by using FIR lowpass filters. FIR lowpass window filters, such as the rectangular window filter, cascaded window filter, Gaussian window filter and squared sine window filter, are analyzed and compared from both theoretical analysis and experimental results. The results coincide with the industrial practice, but frequency domain requirements should be taken care of.

In the path following motion control system, the geometric error, which measures the quality of machining product, is contouring error. However, the dynamic mismatch of the coordinated servo axes will degrade the contouring accuracy. For the mechatronic system, the main objectives are to robustly stabilize the dynamic system, to reject the disturbances (noises) and to minimize the tracking error. The internal stability requirement imposes interpolation and integral conditions on tracking error dynamics. The trade-off has to be made which gives an upper bound on tracking accuracy. However, the tracking error is the upper bound of the contouring error. Based on the principle of uncertainty, more preference can be given to contouring error dynamics for an optimized tracking error dynamics. But the contouring error is hard to be computed in real time for generic toll paths. In this dissertation, a projection operator is defined to approximate the contouring error, and a novel contouring error compensation structure is proposed. Based on the small gain theorem and the properties of the projection operator, a sufficient condition for internally stabilizing the feedback system is obtained. In this structure, more preference is given to contouring error dynamics by using multiple loop structure. Fixed order contouring error compensators are designed, and the feasibility of the contouring error compensators are verified by experiments. The experiments show that the contouring error can be reduced by more than 20%.

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