Global Task Coordinate Frame Based Precision Contouring Motion Control

The actual tool path of a numerically controlled (NC) machine tool during contouring tasks is a result of the coordinated motion of all axes, and quality of the products produced is directly re-lated to the contouring error (i.e., the deviation of the actual contour or path to the desired contour). To main the contouring error within certain threshold in order for the produced products to meet the quality requirement, it is not necessary for the axis position tracking errors of the machine tool to be at the same small level for all axes at all time-a more stringent control performance requirement which may not be attainable in reality due to various factors such as the limited bandwidth of certain individual axis servo controllers. As such, how to achieve the required contouring tracking accuracy (i.e., maintaining the contouring error within the required tolerance band of the desired contour), especially during high-speed and large-curvature contouring tasks, has always been an important problem to address in industrial applications. To solve this practically significant control problem, the dissertation presents a global task coordinate frame (GTCF) based precision contouring control strategy. Specifically, the dissertation first develops a novel method that accurately calculates the contouring error while being computationally simple enough to be used in real-time feedback con-trols. The dissertation then constructs a GTCF in which curvilinear coordinates are orthogonal to each other along the desired contour and transforms the usual axes system dynamics into the pro-posed GTCF. High-performance contouring controllers are then synthesized directly based on the strongly coupled and highly nonlinear system dynamics in the task space. The resulting controllers have strong disturbance rejection capability and maintain certain guaranteed robust performance in spite of various parametric uncertainties and uncertain nonlinearities in the system dynamics. Compared with the locally defined task-coordinate frame (LTCF) used in all past contouring con-trol research, the proposed GTCF has the advantages of i) being able to calculate the contouring error accurately regardless how large the actual position tracking errors are, ii) enabling the capabil-ity of re-planning the feedrate or velocity along the desired contour on-line for better coordination of axis motions when some unexpected events happen (e.g., the appearance of sudden large position tracking error of a particular axis), and iii) being computationally efficient for real-time feedback controls. Compared to the existing contouring control algorithms, the proposed adaptive robust control (ARC) schemes havebetter disturbance rejection capability and achieve higher contour-ing accuracy. Overall, the proposed GTCF based ARC contouring controllers are well suited for high-speed/high-accuracy contouring control of biaxial systems with large-curvatures, and provide a solid theoretical framework for the precision contouring control of systems of higher dimensions.Through careful examination of the contouring error definition and the use of differential ge-ometry tools, the dissertation first develops a novel method which can accurately calculate the contouring error whether the actual position tracking error is small or not, while being computa-tionally efficient for real-time feedback controls. Departing from existing methods of estimating the contouring error based on actual position tracking errors with various approximation assump-tions, the proposed contouring error calculation is exact to the first-order approximation of the actual contouring error. In addition, as opposed to the existing methods which heavily rely on the desired motion to be performed on the desired contour, the proposed one depends only on the shape of the desired contour and well reflects the fact that the contouring error is, by definition, a geo-metric quantity and has nothing to do with the desired motion on the contour. With the proposed precise calculation of contouring error, the dissertation then presents an orthogonal GTCF in two dimensional space, in which one curvilinear coordinate represents the contouring error while the other corresponds to the curve length on the desired contour. A systematic way to construct the pro-posed orthogonal GTCF using any geometric description of the desired contour is also given. With the proposed GTCF, the contouring control problem is simplified into the global regulation of the curvilinear coordinate representing the contouring error and the tracking control of the other curvi-linear coordinate representing the motion on the desired contour. To solve this contouring control problem, the axial system dynamics in the Cartesian coordinates are transformed into the proposed GTCF, and ARC contouring controllers are subsequently synthesized directly based on the strongly coupled and highly nonlinear system dynamics in the task space. The resulting ARC controllers achieve excellent contouring performance in spite of various parametric uncertainties and uncertain nonlinearities. The dissertation also explicitly examines the effect of several common nonlineari-ties in multi-axis mechanical systems such as cogging force and dead-zone, and proposes effective corresponding compensation schemes. Finally, the dissertation provides a theoretical framework on the global task coordinate frame for higher dimensional-systems and the construction of effective ARC contouring controllers under both parametric uncertainties and uncertain nonlinearities.All the proposed contouring controllers have been tested on a linear motor driven industrial biaxial gantry with axis position sensing resolution of 0.5μm by linear optical encoders. Experi-mental results show that, when the gantry is commanded to track an ellipse with long-axis of 0.2m and short-axis of 0.1m at an angular speed ofω=7rad/s and a velocity of vmax=1.4m/s (i.e., a high-speed tracking of a contour of normal curvature), the RMS value of the contouring error is 2.70μm and the maximum contouring error is 8.85μm. When the gantry is commanded to track an ellipse with long-axis of 0.2m and short-axis of only 0.02m at an angular speed ofω=7rad/s and a velocity of vmax=1.4m/s (i.e., a high-speed tracking of a contour of large curvature), the RMS value of the contouring error is 2.20μm and the maximum contouring error is 6.73μm. Compara-tive experimental results also confirm that the proposed GTCF is much better suited for precision contouring controls than the traditional LTCF, especially under high-speed tracking of contours of large curvatures-a contouring tracking accuracy improvement of 61.1% is obtained. Experimental results also verify that the proposed controllers indeed have excellent contouring accuracy and good performance robustness to both parametric uncertainties and uncertain nonlinearities in implemen-tation. Other experimental results also validate the effectiveness of the proposed cogging force compensation (a contouring tracking accuracy improvement of 30% is achieved) and dead-zone compensation. All these results reveal that the proposed GTCF based contouring control strategy is a good solution to the high-speed/high-accuracy large-curvature contouring control problem faced in industrial applications. The method is applicable to the coordinated motion control of other multi-axis mechanical systems as well.The dissertation consists of the following seven chapters:Chapter 1 details the research background and the history of contouring motion control of multi-axis mechanical systems. The chapter also introduces the difficulties faced in the high-speed/high-accuracy and large-curvature contouring tasks, and some typical nonlinearities existed in multi-axis mechanical systems such as cogging force and dead-zone. The chapter concludes with a brief summary of the contributions and significance of the dissertation research.Chapter 2 presents an orthogonal GTCF for biaxial systems in which calculation of the con-touring error is exact to the first-order approximation of the actual contouring error regardless how large the position tracking errors are, A systematic way to construct curvilinear coordinates of the proposed GTCF using any description of the geometry of the desired contour in two-dimensional space is also given. The proposed GTCF based ARC algorithm, along with traditional LTCF based ARC ones, are implemented and comparative experimental results are presented. The results val-idate the effectiveness and the excellent contouring performance of the proposed GTCF approach for free-form contouring control with large-curvatures and arbitrary position tracking errors.Chapter 3 details a GTCF based integrated direct/indirect adaptive robust contouring controller (DIARC) for an industrial biaxial gantry that achieves not only excellent contouring performance but also accurate parameter estimations.Theoretically, a prescribed contouring tracking transient performance and steady-state contouring accuracy is achieved under both parametric uncertainties and uncertain nonlinearities. In addition, asymptotic output tracking is also achieved when only parametric uncertainties exist. Comparative experimental results verify that the proposed GTCF based DIARC algorithm not only achieves the best contouring performance but also has accurate parameter estimations of the physical parameters.Chapter 4 explicitly takes into account the effect of cogging forces in the design of GTCF based ARC controllers to further improve the achievable contouring performance. Comparative experimental results confirm the effectiveness of the proposed cogging force compensation in im-proving the achievable contouring accuracy in practice.Chapter 5 focuses on the contouring motion control of multi-axis mechanical systems having unknown dead-zone nonlinearities. An integrated DIARC with a dead-zone inverse is proposed. The proposed DIARC algorithm achieves a guaranteed robust transient performance and steady-state tracking accuracy even when the entire system may be subjected to other uncertain nonlin-earities and time-varying disturbances. In addition, through on-line monitoring of the persistent excitation (PE) condition and conducting parameter adaptation only when PE condition is actually satisfied, estimates of various system parameters including the unknown dead-zone nonlinearity asymptotically converge to their true values and asymptotic output tracking is achieved even in the presence of unknown dead-zone nonlinearity. Comparative experimental results validate the effectiveness of the proposed dead-zone compensation scheme.Chapter 6 studies the precision contouring control of mechanical systems of any degrees-of-freedom (DOF). A global task coordinate frame in n-dimensional space is presented. Specifically, the first group of curvilinear coordinates of the proposed task space represent the distances from the actual tool position to the set of hyper-surfaces that define the desired contour. This group of curvilinear coordinates are used to guarantee that the actual tool position would move along the desired contour. The other coordinates are for motion tracking. An ARC contouring controller is subsequently synthesized to address the strongly coupled and highly nonlinear system dynamics in the task space, as well as various parametric uncertainties and uncertain nonlinearities. Theoret-ically, the proposed ARC controller achieves a guaranteed transient performance and steady-state tracking accuracy. In addition, an improved steady-state tracking performance-asymptotic output tracking-is achieved when only parametric uncertainties exist.Chapter 7 summarizes main research work that has been done and draws conclusions. Major innovations of the research are highlighted and some possible future research directions are given.Details of the experimental system used in the dissertation research are given in Appendix A, in which physical characteristics of each individual axis and the overall dynamic model of the biaxial linear-motor-driven gantry are presented.