附录Control of a Non-Orthogonal Reconfigurable Machine ToolReuven KatzJohn YookYoram KorenReceived: January 3, 2003; revised: September 16, 2003AbstractComputerized control systems for machine tools must generate coordinated movements of the separately driven axes of motion in order to trace accurately a predetermined path of the cutting tool relative to the workpiece. However, since the dynamic properties of the individual machine axes are not exactly equal, undesired contour errors are generated. The contour error is defined as the distance between the predetermined and actual path of the cutting tool. The cross-coupling controller (CCC) strategy was introduced to effectively decrease the contour errors in conventional, orthogonal machine tools. This paper, however, deals with a new class of machines that have non-orthogonal axes of motion and called reconfigurable machine tools (RMTs). These machines may be included in large-scale reconfigurable machining systems (RMSs). When the axes of the machine are non-orthogonal, the movement between the axes is tightly coupled and the importance of coordinated movement among the axes becomes even greater. In the case of a non-orthogonal RMT, in addition to the contour error, another machining error called in-depth error is also generated due to the non-orthogonal nature of the machine. The focus of this study is on the conceptual design of a new type of cross-coupling controller for a non-orthogonal machine tool that decreases both the contour and the in-depth machining errors. Various types of cross-coupling controllers, symmetric and non-symmetric, with and without feedforward, are suggested and studied. The stability of the control system is investigated, and simulation is used to compare the different types of controllers. We show that by using cross-coupling controllers the reduction of machining errors are significantly reduced in comparison with the conventional de-coupled controller. Furthermore, it is shown that the non-symmetric cross-coupling feedforward (NS-CC-FF) controller demonstrates the best results and is the leading concept for non-orthogonal machine tools. 2004 ASME Contributed by the Dynamic Systems, Measurement, and Control Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received by the ASME Dynamic Systems and Control Division January 3, 2003; final revision September 16, 2003. Associate Editor: J. Tu. Keywords:machine tool, cross-coupling controller, non-orthogonal, RMT1 IntroductionCurrently manufacturing industries have two primary methods for producing medium and high volume machined parts: dedicated machining systems (DMSs) and flexible manufacturing systems (FMSs) that include CNC machines. The DMS is an ideal solution when the part design is fixed and mass production is required to reduce cost. On the other hand, the FMS is ideal when the required quantities are not so high and many modifications in the part design are foreseen. In contrast to these two extremes, Koren describes an innovative approach of customized manufacturing called reconfigurable manufacturing systems (RMS). The main advantage of this new approach is the customized flexibility in the system to produce a part family with lower investment cost than FMS. A typical RMS includes both conventional CNC machines and a new type of machine called the Reconfigurable Machine Tool . The Engineering Research Center (ERC) for Reconfigurable Machining Systems (RMS) at the University of Michigan with its industrial partners has designed an experimental Reconfigurable Machine Tool (RMT) 。This machine allows ERC researchers to validate many of the new concepts and machine tool design methodologies that have been already developed in the center. There are many types of RMTs. This paper describes an arch-type non-orthogonal multi-axis RMT machine 。The economic justification of RMTs is given in section 2 of this paper. A contouring motion requires that the cutting tool moves along a desired trajectory. Typically, computerized control systems for machine tools generate coordinated movements of the separately driven axes of motion in order to trace a predetermined path of the cutting tool relative to the workpiece. To reduce the contouring error, which is defined as the distance between the predetermined and the actual path, there have been two main control strategies. The first approach is to use feedforward control in order to reduce axial tracking errors .however, they are limited when non-linear cuts are required. The other approach is to use cross-coupling control in which axial-feedback information is shared between the moving axes. The cross-coupling controller is used in addition to the conventional axial servo controller. At each sampling time, the cross-coupling controller calculates the current contour error and generates a command that moves the too