组合机床设计的模块化建模方法.pdf

Proceedings of IMECE 2004 2004 ASME International Mechanical Engineering Congress and RD see /terms/Terms_Use.cfm RMTs unique is that even though there is a single machine tool, there exist several configurations, which separate models have to be developed for. Developing dynamic models for all possible configurations could be a cumbersome and time- consuming task if ad hoc methods are utilized. Moreover, without a systematic methodology modeling would require a lot of expertise and would be prone to errors, which would degrade the efficiency of using models in the design. In this paper we present a methodology that could help make the RMT modeling task less time demanding, less error- prone and less challenging. The key idea of this methodology is to take advantage of the modular structure of the RMTs and adopt modular modeling concepts into the RMT modeling methodology. First, the physical components of an RMT are modeled in a modular way using the bond graph modeling tool 7. The bond graph model is encapsulated in a schematic representation with defined connection ports. Then, the schematic component models are assembled by following the topology of a given configuration to obtain the model of the configuration. The configuration model can be easily integrated with the modules of non-energetic components such as interpolators and controllers, which can be conveniently represented with block diagrams; however this is beyond the scope of this paper. The paper is organized as follows: The next section introduces the nomenclature used in the models. The section after that briefly reviews the background work. Then, the proposed RMT modeling methodology is presented. Examples of the modeling methodology are given, followed by a discussion of the approach. The paper concludes with summary and conclusions. NOMENCLATURE C Pr ? : position vector of point P with respect to the inertial frame I expressed in frame C C Pr ? ? : time derivative of C Pr ? with respect to the inertial frame I C B ? : angular velocity vector of frame B with respect to the inertial frame I expressed in frame C 21 ,BB A : coordinate transformation matrix from frame to frame 1 B 2 B BACKGROUND The RMT concept was introduced by Koren and Kota 2, and since their introduction, the design of RMTs has been an active research area. Methodologies and tools for designing RMTs 4 as well as evaluating structural stiffnesses 5 and tool tip errors 6 of design alternatives have been developed. However, the problem of developing a system level modeling methodology for RTMs has not been addressed yet. Traditionally, machine tool models depict the machine tool as a group of servomotor and feed drive assemblies that are modeled as first or second order systems 8,9. Chen and Tlusty, however, showed that the structural dynamics of the feed drive could affect the system performance once high- speed machine tools are considered 10. Many researchers identified the necessity to use higher order models for high- speed machine tools to cope with structural dynamics in order to be able to design the control system successfully 11-13. These publications clearly indicate that modeling a machine tool is not a trivial task and care must be taken when deciding on the complexity of the model, but they do not provide a systematic way of modeling and, therefore, remain application specific approaches. There have been research efforts to help the design and control of machine tool feed drives by automatically providing simulation models. Wilson and Stein developed a software program called Model-Building Assistant to automatically synthesize a minimum order model of the machine tool drive system for a given frequency range of interest (FROI) 14. The complexity of the model, which includes a flywheel, a torsional shaft, a ballscrew, a ballnut, a DC motor, a torsional coupling, a belt-drive and a gear-pair as components, is automatically increased until the eigenvalues of the system fall beyond the specified FROI. This work was a proof of concept for a model deduction algorithm and can not be applied to any real machine tool system. However, such algorithm can be used to determine the appropriate model complexity after the development of the system model. Gautier et al. have developed a software package called SICOMAT (SImulation and COntrol analysis of MAchine Tools) which helps with the modeling, simulation, modal analysis and controller tuning of one or two decoupled or two coupled machine tool axes 15. Their models describe the dynamics of the mechanical system by a number of masses and springs. This work makes the modeling of a machine tool process more systematic, and is therefore a valuable tool to the modeling engineer; however, it lacks the generality, modularity and flexibility that the RMT design methodology demands. THE RMT MODELING METHODOLOGY Figure 1 shows the envisioned RMT modeling environment. It is desired to automate the task of RMT modeling, where the model of a given RMT configuration is automatically assembled from a library of modular component models. This way, all the candidate designs, which are generated either manually or automatically 4, can be modeled quickly and the models can be used to evaluate the candidates in terms of their servo axis dynamic performance and help with their design. As Figure 1 also implies, the modular component model library is a key part for the automated RMT modeling environment. Therefore, the first step of the proposed methodology is to develop modular models for the components that are used to generate the RMT configurations. This paper puts the emphasis on mechanical parts and discusses their modeling in a modular way, because the energy interaction between the mechanical components makes their modular 2 Copyright 2004 by ASME Downloaded 13 Apr 2011 to 15. Redistribution subject to ASME license or copyright; see /terms/Terms_Use.cfm modeling more intriguing. Modular modeling of components that only exchange signals, e.g. interpolators and controllers, presents a relatively simpler problem and are not discussed here. To promote modularity and to be able to deal with the energy interactions between the components and their environment rather easily, bond graphs are utilized as the modeling language. Bond graphs provide a power-based graphical representation of a physical system. Moreover, bond graphs describe different energy domains in a unified way, which is a relevant advantage for RMT modeling, since their servo axes may include components from different energy domains, such as mechanical, electrical or hydraulic. Bond graphs are only one level in the hierarchy of model representations used in this work. Underneath the bond graph level the mathematical equations represent the physical phenomena captured by the bond graph and this mathematical representation is the lowest level in the hierarchy. In the highest level bond graphs are encapsulated in a schematic representation, which not only allows for a compact representation, but also shows the connection ports where the model can interact with its environment. Figure 2 illustrates this hierarchy of model representations. In this paper all the models are shown in the schematic level, because the goal of this paper is not to discuss their derivation, but rather to show what can be done once those models are obtained. A detailed description of the models used in this paper can be found in 16. Figure 1: The envisioned RMT modeling environment In order to be able to cope with any spatial motion that the mechanical components may go through in different configurations, models that capture the three-dimensional dynamics are used. Moreover, the initial assumption is made that in the mechanical domain all components can be adequately represented as rigid bodies. Figure 3 shows the schematic representation of a generic rigid body with N connection ports, which is one of the main model modules in the library. The ports correspond to points of interest on the rigid body, ,1 i P i = N, where the physical interactions with the environment occur. Bonds (lines with half arrows) are used to indicate that a port is a power port, i.e. the body can exchange energy with its environment through those ports, whereas active bonds (lines with full arrows) indicate signal ports, i.e. only information is transferred through these ports. The model library also contains three-dimensional joint models that can be used to describe the relative motions between the component models. These joint models are also developed in a modular way with ports, where they can be connected to other model modules. The library offers two ways to express the constraints: (1) stiff springs and dampers can be used to implement more realistic constraints or to approximate ideal constraints, or (2) Lagrange multipliers can be introduced to express the constraints ideally. For a discussion of joint models the reader is also referred to 16. Once the model library is populated with some basic modular rigid body and joint models, the modeling procedure can be carried out as follows: The RMT components are broken down into subcomponents and each subcomponent is associated with a model in the library. If none of the model modules in the library can describe the subcomponent adequately, a new model has to be developed for that subcomponent and added to the library. Then, the models are assembled by following the topology of the components and using the necessary joint models. Once a component model is obtained, it can be stored in the library for reuse. Finally, the component models are assembled by following the topology of a given configuration to obtain the model of that configuration. The process is illustrated in Figure 4 as a flowchart and demonstrated in the following section through examples. 1Sf Ck 1 Rb 01 Im Schematic representation Bond graph representation Mathematical representation Figure 2: The hierarchy of model representations Rigid Body-N AI B , ? ? rPB 1 ? ? rPB 2 ? B B ? ? rBB ? ? ? rPB N Figure 3: The schematic representation of a rigid body with N connection points 3 Copyright 2004 by ASME Downloaded 13 Apr 2011 to 15. Redistribution subject to ASME license or copyright; see /terms/Terms_Use.cfm Figure 5: A fictitious slide Provide RMT configuration Break the components down into subcomponents Associate subcomponents with model modules Assemble the component models and add to the database Component models exist in the library? Assemble the configuration model No Yes Subcomponent models exist in the library? Develop subcomponent models and add to the database Yes No Figure 4: The RMT modeling procedure Motor ? ? rPB B2 2 AI B , 2 ? B B 2 2 AI B , V ? B B ? ? rBB ? ? rPB 1 Figure 6: The schematic motor model EXAMPLES The following two examples give an overview of the proposed modeling methodology. The first example shows the modeling of a slide and the second example employs that slide model to develop a model for a RMT. The purpose of these examples is to give a general idea about how the modularity of the components can be exploited in the modeling procedure, rather than to explain the details of how each (sub)component can be identified and modeled. Therefore, the details of the model modules, such as their level of complexity, are not discussed. Modeling a Slide A slide is a basic component of most machine tools, including RMTs. Different RMT configurations can be obtained by adding/removing slides to/from the configuration or by rearranging the existing slides in the configuration. Therefore, it is useful to demonstrate the modeling procedure of a slide. Consider the slide shown in Figure 5. It is assumed that the components are identified as shown in the figure. For the purposes of this example, all the subcomponents except the motor can be modeled as rigid bodies with various number of connection points. The motor dynamics can be broken down into two domains: the three-dimensional rigid body dynamics of the housing and the electromechanical dynamics that drive the relative rotational motion between the rotor and the stator. A model has been developed for the motor that captures the dynamics in both domains and its schematic representation is given in Figure 6. Now that all the subcomponent models are included in the library, the slide model can be assembled as shown in Figure 7. Note that some model modules are reused, e.g. both the Ballscrew and the Adapter are represented by Rigid Body-2. Since the slide model will typically be used in a complete machine tool model, its model as shown in Figure 7 can be encapsulated in an even higher level representation and included in the library. Figure 8 shows the schematic representation of the slide model, which encapsulates the model in Figure 7. For validation purposes, this slide model was compared with a hand driven simple model of the same slide. A simulation was run, where a step voltage input was applied to the motor and the resulting saddle speed was recorded. Figure 9 compares the results of both simple and modular models. Practically, the modular model gives the same response as the simple model. Figure 10 shows the time history of the difference between the two models. This difference occurs because in the modular model the constraints are satisfied only within a numerical tolerance, whereas in the simple model the ideal constraints are eliminated from the equations of motion. Modeling the Arch-type RMT The Arch-type RMT, which was developed by the NSF Engineering Research Center for Reconfigurable Manufacturing Systems at the University of Michigan, is the worlds first full scale RMT. It is a three-axis machine tool that is designed around a part family with five different surface inclinations ranging from -15 to 45 at 15 increments and has the flexibility of doing machining operations such as milling 4 Copyright 2004 by ASME Downloaded 13 Apr 2011 to 15. Redistribution subject to ASME license or copyright; see /terms/Terms_Use.cfm and drilling at any of those angles. The reconfigurability of the Arch-type RMT comes from the spindle unit, which can be configured at the five angles mentioned above by moving it along the curved guideway of the arch module and fixing it at any of the five locations on the arch module that are defined by mechanical stops. Figure 11 shows a CAD model of the Arch- type RMT along with its main components. Rigid Body-3 AI B , ? rPB 1 ? rPB 2 ? rPB 3 ? B B ? rBB Rigid Body-2 AI B , ? rPB 1 ? rPB 2 ? B B ? rBB Motor ? rPB B2 2 AI B ,2 ? B B 2 2 AI B , V ? B B ? rBB ? rPB 1 Fixed Joint AI B ,1 ? rPB B1 1 ? rPB B2 2 AI B ,2 ? B B 1 1 ? B B 2 2 Screw Joint AI B ,1 ? rPB B2 2 AI B ,2 ? B B 1 1 ? B B 2 2 ? rBB 1 1 Fixed Joint AI B ,1 ? rPB B1 1 ? rPB B2 2 AI B ,2 ? B B 1 1 ? B B 2 2 Rigid Body-3 AI B , ? rPB 1 ? rPB 2 ? rPB 3 ? B B ? rBB 1 1 1 Rotational Constraint AI B ,1 AI B ,2 ? B B 1 1 ? B B 2 2 Rigid Body-2 AI B , ? rPB 1 ? rPB 2 ? B B ? rBB 1 BallscrewAdapter RailSaddle Figure 7: The assembled slide model SlideSlide AI B , ? ? rPB 1 ? ? rPB 2 ? B B V Figure 8: The schematic slide model 012345 time ?s? 0 0.005 0.01 0.015 0.02 0.025 saddle speed ?m?s? modular simple Figure 9: Comparison of saddle speeds to a step input 012345 time ?s? 0 1?10?6 2?10?6 3?10?6 4?10?6 5?10?6 saddle speed difference ?m?s? Figure 10: The difference in saddle speeds of the two models For the purposes of this example the base module is assumed to be identical to the ground and it has no effect on the dynamics of the machine tool. The worktable, the column and the spindle are essentially slides and their models are based on the slide model given above. The arch is modeled as a rigid- body with a connection port for each mechanical stop. Finally, the model of the Arch-type RMT is assembled by following the topology of the actual machine as shown in Figure 12. Note that the figure shows the model for one of the configurations only. The models for the other configurations can be obtained by changing the connection port of the arch model. 5 Copyright 2004 by ASME Downloaded 13 Apr 2011 to 15. Redistribution subject to ASME license or copyright; see /terms/Terms_Use.cfm Now that the model is assembled, the equations of motion can be derived from the graphical model automatically, and simulations can be performed. Although the mathematical model is ready, we cannot provide any simulation results in this paper due to the current lack of good estimates of system parameters. Simulations can be carried out easily once the parameter values are available. DISCUSSION In this paper, modular and hierarchical modeling concepts are identified as the key characteristics of the RMT modeling methodology. The modular structure of RMTs makes this modeling approach beneficial, because the models contain all the key characteristics of reconfigurability 17: Figure 11: The Arch-type RMT 1. Modularity: The (sub)components are modeled in a modular way 2. Integrability: The models can be integrated with other modules through their connection ports 3. Customization: The level of detail included in the model modules can be customized for individual components 4. Convertibility: Models can be easily converted from one configuration to another 5. Diagnosability: Model verification can be carried out easily on model modules The approach presented in this paper allows for the separation of the modeling task into two steps: (1) Developing component models, and (2) assembling the configuration model. While the first step still requires a significant modeling expertise, the second step is much more systematic, and can even be automated, whi