Virtual prototyping of mechatronic systems
Annual Reviews in Control 28 (2004) 193–206 www.elsevier.com/locate/arcontrol
Virtual prototyping of mechatronic systems Gianni Ferretti, GianAntonio Magnani, Paolo Rocco* Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza L. da Vinci 32, 20133 Milano, Italy Received 19 December 2003; received in revised form 26 January 2004; accepted 23 February 2004
Abstract Mechatronic systems abound in technological fields such as robotics and machine tools industry. Significant advances in dynamic performance of these systems can be achieved provided that mutual interactions of different domains (such as mechanics, electronics, hydraulics and control) are thoroughly understood. Virtual prototyping entails integration of multi-domain dynamic simulation in the design process, in order to reproduce and analyze the effects of design choices on the overall performance. This paper states the requirements of the modeling language and software tool for simulation of mechatronic systems and describes an experience of use of DYMOLA with Modelica language in the simulation of a complete machining center. The model has been successfully validated against experimental results collected on the real machining center. # 2004 Elsevier Ltd. All rights reserved. Keywords: Mechatronics; Modeling; Object-orientation; Simulation; Virtual prototyping
1. Introduction Every technological system where mechanics, electronics and control concur in a mutually supportive way to the overall performance belongs to the family of the mechatronic systems. Examples are found in robotics, machine tools or machining centers. While the traditional design process of such systems did not entail significant interactions between the mechanical designers and the experts of control and drive electronics, the market demand for ever increasing precision calls now for a synergic design, where the mutual interactions of choices made in the mechanical design (possibly including hydraulics) and in the drive electronics design are thoroughly understood. Dynamic performance of the machine has become the real issue, rather than static dimensioning, and dynamic performance cannot be improved unless the machine is studied with electronics and control operating in closed loop with the mechanics.
* Corresponding author. Tel.: +39 022 399 3685; fax: +39 022 399 3412. E-mail address: [email protected] (G. Ferretti), [email protected] (G. Magnani), [email protected] (P. Rocco). 1367-5788/$ – see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.arcontrol.2004.02.002
Experimental study on physical prototypes is however expensive and time consuming. Virtual prototyping, a new way of defining dynamic simulation (Cellier, 1991; Ferretti, Filippi, Maffezzoni, Magnani, & Rocco, 1999) of the controlled machine, is more compliant with time and budget constraints of R&D industrial centers. If design choices on the mechanical as well as on the electronic side can be tested before assembling the physical prototype, the time to market is shortened and the knowledge of the system is improved, obviously as long as the simulation plant is a reliable replication of reality and the simulation tool is reasonably easy to use for people coming from different areas. The general impression is that today several companies operating in fields where competition and advances in dynamic precision of the machines are vital issues are somewhat ready to introduce these kinds of tools in the design process, but also that there is not enough consciousness on the real potentialities of the new approaches. Uncertainties often arise also on the requirements for the modeling language and software tool to be adopted. Simulation environments such as SIMULINK seem to be extremely powerful to replicate the control system, but difficulties are evident as soon as physical system modeling is attempted, as the unidirectional signal flow implied by the
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procedural description of the blocks does not seem to match the physical description of the system adequately. On the other hand commercial packages (e.g. ADAMS, Altair, SimMechanics) abound for simulation of the mechanics, from static dimensioning to vibration study, but users might experience troubles when attempts are made to simulate closed loops with electronics and control. Meanwhile, the research on modular physical modeling languages and tools, started at the end of the 1970’s, has evolved into several modeling languages and simulation environments: ASCEND (Piela, Epperly, Westerberg, & Westerberg, 1991), OMOLA (Mattsson & Andersson, 1993), gPROMS (Barton & Pantelides, 1994; Oh & Pantelides, 1996), MOSES (Maffezzoni & Girelli, 1998), Modelica (Mattsson, Elmqvist, & Otter, 1998; Tiller, 2001) and DYMOLA (Elmqvist, Bruck, & Otter, 1996), to name just a few. This paper first presents a discussion on the requirements of a software tool for system simulation of mechatronic devices. Multidomain scope, software reuse, reliability and efficiency of the numerical simulation, integration with mechanical CADs are found to be valuable aspects. An experience is then described of use of DYMOLA with Modelica language, which seems to comply with most of the above requirements. The standard libraries of Modelica for simulation of mechatronic systems are first described. Then some customized modules, such as brushless motors and current drivers, are presented, emphasizing the advantages related to acausal physical system modeling. As a case study, the results of a research on the sources of performance limitations in a wood machining center are presented. The performance limiting factors have been investigated through linear models of the single axes of the machining center, suitable to predict the resonant behavior experimentally identified, and through simulation of a detailed model developed in the DYMOLA environment. The overall model of the machine can predict the performance with noticeable accuracy, thus it can be used in the design of new machines, with similar structure and components as the machine studied in this work.
2. Requirements for the modeling language and the simulation environment Modeling and simulation of mechatronic systems are difficult tasks, with several critical points. The choice of the modeling approach and of the simulation environment thus deserves particular attention, especially for large scale industrial projects. The main user requirements for the modeling approach and the simulation environment can be specified as follows. 2.1. Multi domain simulation The simulation environment must deal with equal comfort with mechanical, hydraulic, electrical, control
and possibly other physical domains. As already mentioned, a systematic study of dynamic performance of a mechatronic system should neither privilege aspects relevant only to a single domain, nor ignore the mutual interactions among domains. For instance, software tools where mechanics are simulated in detail, but electronics and control require to the user ad-hoc additions of modules, written in a different modeling language, are not advisable for the user due to longer development times and reduced accuracy 2.2. Modular modeling and software reuse Modular modeling is a major requirement, if not a need, for simulation of complex systems like mechatronic ones. However, real modularization of mechanical, hydraulic ad electrical systems is hampered in most simulation tools, which do not allow, for instance, to establish algebraic relations (constraints) among state variables. Constraints on state variables arise in the rigid connection of the dynamic models of a motor shaft and of a gearbox, separately written, or when more electric currents or liquid flows from different circuits enter a common node or mixing volume. Modular modeling permits to split a large model in a number of smaller and more manageable models, and is a necessary condition to software reuse. The reusability of modeling software from project to project is a key requirement for industrial users. 2.3. Built-in model libraries and customisation Use of built-in libraries of components keeps modeling costs low and development times short. Libraries that cover the needs for simulation of simple mechatronic systems are usually available in most of commercial simulation tools. Mechatronic systems simulation is especially demanding in this respect, since libraries from several physical domains are needed, easy to be integrated one with the others. Nevertheless, situations occur where customised modules must be written, as is shown in the following Sections. In this respect, expansion of library modules, rather than writing customised modules from scratch, obviously facilitates the task of the user. 2.4. Reliability and efficiency of numerical simulation Reliability and efficiency of the numerical solver are key requirements for non-skilled users, generally unwilling to deal with details in numerical integration of equations. A differential algebraic equations (DAE) (Brenan, Campbell, & Petzold, 1989) solver is needed, able to handle events, such as discontinuities and abrupt changes in external variables. The time instant when the event is triggered must be detected with precision, thanks to variable step size of integration. Also simulation of closed loop systems must be dealt with in the most reliable way: solutions where the physical part of the system is simulated with a DAE solver while the procedural control part is simulated with an ODE solver, and synchronisation is kept between the two solvers,
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do not seem to provide the same reliability as a single solver integrating the whole system. 2.5. Integration with mechanical CADs Mechanical designers feel comfortable with software tools that can be easily integrated with the currently used mechanical CADs (for example SolidWorks1). Generally speaking, integration means that the mechanical CAD generates the mass and inertial parameters to be used in the dynamic simulation environment, without any manual transcription, which is both time consuming and error prone. Also integration with finite element analysis (FEA) packages turns out to be useful for studies where flexibility of materials is an issue. 2.6. Interfaces for pre- and post-processing Assembly of modules should be performed through a Graphical User Interface (GUI). Both 2D and 3D interfaces are used, the 2D being more common and easy to use. Each library module should be associated to a graphical symbol as evocative as possible of the physical nature of the system. Visualisation of simulation results can be performed both with 2D plots or 3D animation. The 2D plots are more useful for detailed analysis of dynamic properties, while 3D animation helps understanding the gross motion of the system. Compliance with the above requirements 2.1–2.3 (customisation) entails the following properties of a simulation environment and its modeling language. 2.7. Simulation paradigm Declarative modeling languages should be used to simulate the physical parts of the system, be they mechanical, electrical, hydraulic or pneumatic components. Procedural languages, where the unidirectional flow of information from an input to an output is postulated, are inadequate to simulate systems where instantaneous exchanges of power are present. Acausal models, described by DAE systems, should be used to represent in the most natural and physically consistent way each system component, while the task of defining the computational causality of the assembled model should be charged to the simulation environment. 2.8. Object orientation The modeling language should comply with modern object-oriented paradigms of software engineering. The most relevant aspects of object-orientation to the benefit of physical system modeling are modularization (the system is composed by aggregation of modules), abstraction (the internal description of a module is separated from its interface), information encapsulation (only the interface variables are 1
SolidWorks is a trademark of SolidWorks Corporation.
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accessible to the other modules), reuse of the modules through parameterisation. Structuring the object-oriented models (modules) in libraries facilitates the extension of existing models and the simulation of large and complex systems. 2.9. Handling of equations and constraints Assembling of modules described by DAE equations generally leads to higher index2 systems and algebraic loops. Symbolic manipulation of the equations is then required to yield a set of equations amenable to numerical integration. Some simulation tools use Pantelides’ (1988) algorithm (or similar) to reduce the index of the system. Others use constraint relaxation techniques, generating an ODE system where, however, constraints are not identically satisfied during transients. The efficiency of this pre-processing software layer might be crucial for the efficiency of the overall simulation environment.
3. Modelica standard libraries for mechatronics None of the commercial packages on the market fully complies with all the above user requirements, to our knowledge. On the other hand, the choice of the package to adopt may depend on specific issues of the problem and may require a trade off among the degrees of fulfillment of different requirements. In the present work, an experience is reported of use of DYMOLA (with Modelica language) for simulation of machining centers. This package fully complies with the requirements of multi-domain simulation and ease of customization, and features the acausal declarative modeling paradigm as well as the object-oriented approach. The efficiency of the numerical integration is guaranteed by a sophisticated software layer where the assembled model is first translated, i.e. equations are symbolically manipulated, in order to reduce the index and produce efficient simulation code. A state of the art numerical solver of DAE’s is then used for numerical integration. A 2D GUI for model assembling is available, as well as a simple 3D animation module for post processing of simulation results. Concerning interfaces with mechanical CAD packages, a translator from SolidWorks to Modelica is also available. It converts bodies and joints of SolidWorks models to Modelica models, also filling in the templates of the relevant geometrical and inertia parameters. Additional mechanical components (external forces, torques, frictions, etc.) can be easily added using a Modelica editor. In DYMOLA all models are written in the Modelica language, the outcome of the efforts of the Modelica Association, a non-profit, non-governmental organization 2 The index of a DAE system is the number of times all or part of the equations of the system must be differentiated, in order to obtain an ODE system. When the index is higher than one, explicit or hidden algebraic constraints on the state variables of the system are present.
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born in 1996. The aim of the association is to develop and promote the Modelica modeling language for modeling, simulation and programming of physical and technical systems and processes. The association maintains specifications of the modeling language and develops free of use Modelica standard libraries. DYMOLA is one of the software tools (together with MathModelica (Fritszon, Gunnarsson, & Jirstrand, 2002) and OpenModelica (Fritszon, Aronsson, et al. 2002)) that fully supports the Modelica language and offers the software environment to actually perform the simulation. There are several Modelica libraries relevant for a mechatronic project: the Modelica.Mechanics.Rotational package (shown, as an example, in Fig. 1) offers elements to simulate one dimensional rotational mechanical systems, useful to reproduce transmission chains, gearboxes, shafts with inertia, spring/damper elements, frictional and backlash elements. Remarkably, rigid connection of two inertia elements, an operation which is essential for the modular simulation of transmission chains but generates a higher (three) index DAE system, can be made without any concern. The simulation environment is in charge of resolving the constraint of equality of the angular positions and velocities of the two inertia elements. A similar library (Modelica.Mechanics.Translational) is available to simulate one dimensional translational mechanical systems. Three-dimensional mechanical systems, such as robots, machine tools or vehicles, can be modelled with the ModelicaAdditions.Multibody library. The components of this library can be combined with the above 1D-mechanical libraries through flanges in the Joint modules (the joints introduce the degrees of freedom in the mechanical systems). The Body elements require specification of
geometrical as well as inertial (mass, center of mass and inertia tensor) parameters, that can be easily introduced if an interface with a solid modeller is available. Differently from other simulation environments, the electrical and control domains can be simulated with equal ease as the mechanical domain. The Modelica.Electrical.Analog library allows simulation of simple electrical circuits. The declarative paradigm is fully respected in this library, too. All components have electrical cuts where exchanges of electrical power (through voltage and current variables) take place. This implies that each physical element (be it a resistor or a capacitor, for example) has just one library counterpart, regardless that the element is actually voltage or current driven. Complete reusability of the models is thus guaranteed. The Modelica.Blocks library contains input/output blocks to build up simple block diagrams for simulation of controllers (both analogue and digital ones). Based on our experience, the library, although somewhat limited, allows to simulate most of the functions of commercial controllers, obviously through aggregation of the library elements. An example is given in next section for the simulation of a current driver for a brushless motor. Other major libraries suitable for mechatronic systems are the hydraulic (HyLib) and pneumatic (PneuLib) libraries. They contain the most important basic components, useful to model hydraulic and pneumatic circuits. Coupling elements exist to connect hydraulic and pneumatic components with control blocks of the blocks library and elements of the mechanics library. The libraries support modeling of one-dimensional rotational and translation hydraulic and pneumatic systems, and consist of large varieties of pumps, boosters, motors, cylinders, valves, restrictions, tanks, reservoirs, sensors, lines and volumes.
4. An example of a customized module
Fig. 1. Modelica standard library for rotational mechanics.
The most common actuation systems adopted in robotics, machine tools industry and a number of mechatronic systems are by far servomechanisms with permanent magnet brushless motors, connected to the loads by transmission chains (or gearboxes). In a brushless motor the electromechanical commutation typical of brushed dc motors is replaced by an electronic commutation of the currents in the three phases of the stator windings. This should in principle guarantee that the electromagnetic torque delivered on the motor shaft is independent of the rotor position. However, some constructive imperfections in the motor or in the drive, where electronic commutation is implemented, produce an undulation (ripple) (Ferretti, Magnani, & Rocco, 1998; Jahns & Soong, 1996) on the actual torque. While this problem could be considered minor in the static dimensioning of the actuation system, it is of utmost importance for its dynamic performance. Torque ripple might in fact excite the resonances of the mechanical system, usually associated to
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the elastic couplings between motors and loads (Ferretti, Magnani, & Rocco, 1999). Dynamic models and simulation are valuable tools to study these phenomena, and in particular to separate the effects of the single sources of disturbances on the performance of the system. Mechanics, electronics and control are different domains involved in this truly mechatronic problem. The model of the system in DYMOLA is obtained by the feedback connection of two sub-models, one representing the brushless motor, the other the driver (Fig. 2). The two models are connected through three electrical connectors (for the phases of the motor) as well as through a control connector (the measure of the rotor position). The brushless motor model is shown in Fig. 3. The three phases are Y connected in the block emf3, that generalizes the emf model in the Modelica.Electrical.Analog.Basic library. The equations of the emf3 model are the following ones: 3 X
ii ¼ 0
i¼1
Vi ¼ Vn þ e i ei ¼ ki v
2 2 ki ¼ kt sin p# ði 1Þ p 3 3 #˙ ¼ v
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Fig. 2. The brushless motor and the driver.
t m ¼ ðk1 i1 þ k2 i2 þ k3 i3 Þ; where ii, Vi, ei and ki (i = 1, 2, 3) are the phase currents, the phase input voltages, the back emf’s and the back emf profiles for each phase, respectively; Vn is the voltage of the central node of the Y connection; # and v are the angular position and velocity of the rotor; kt is the torque constant of the motor, p the number of pole pairs and tm is the electromagnetic torque generated by the motor. The code of the emf3 model is reported in the Appendix A. Thanks to declarative coding it is straightforward to write code from equations and read out equations from code. Remarkably, the algebraic constraint on the currents (which are state variables) can be specified in the most natural way
Fig. 3. Modelica model of the brushless motor.
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(the sum of the currents equal zero). Assigning, in the simulation code, back emf profiles ki different from the nominal sinusoidal ones, ripple due to higher order harmonics can be reproduced. In addition to the emf3 module, the model of the brushless motor includes inductance and resistance of stator windings and the mechanical part, consisting of rotor inertia and viscous friction. Just for comparison, Fig. 4 shows the SIMULINK model of the electrical part of the motor, obtained by resolving the algebraic constraint and expressing the whole model in terms of two out of three currents. This operation is time consuming and error prone and produces a model description rather difficult to read. A full digital version of the current controller has been implemented (Fig. 5). A vector control scheme (Texas, 1997) has been adopted, where the phase currents are first transformed into direct and quadrature currents through Park’s transformation. Two digital loops are closed on these currents, the quadrature reference being the output of the outer position/velocity controller, the direct reference being zero usually. The voltage commands output of the two anti-windup PI controllers are then back transformed to voltages on the three phases through inverse Park’s transformation. The PWM amplifier has not been simulated since, operating with a frequency of 10 or 20 kHz and with a modulation of the pulse width of 1 ms, it requires an integration step size smaller than 1 ms, which might be acceptable for the simulation of the electronics of a motor drive, but is far too small in the joint simulation of the mechanics and the electronics. This is particularly true if the
model of the motor is instantiated several times, for the simulation of a complete machine.
5. A case study 5.1. Preliminaries The model of the brushless motor described in the previous section has been included in a library of elements used to simulate a complete three axes machining center. This simulation is part of a research aimed at understanding the sources of performance limitations in mechanical machining centers and at deriving mathematical models that can support its mechatronic design. The structure of the machining center studied in this work is sketched in Fig. 6. This kind of structure is common to several machining centers used for wood and metal, characterized by three linear axes (X, Y, Z). Motion transmission for X axis is realized through two reduction stages with belts and pulleys, followed by a pinion-rack stage, that allows to convert rotational motion to linear motion. In the Y axis, only one stage with belts and pulleys is present, besides the pinion-rack pair. In the following, axis Z will not be considered, as it does not present critical elements. 5.2. Experimental results Experimental tests have been performed on the machine, in order to calculate frequency responses. A new software package, shipped by the manufacturer of the CN, has been
Fig. 4. SIMULINK model of the brushless motor.
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Fig. 5. Model of the digital driver.
Fig. 6. Three linear axes machining center.
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used, that allows to estimate a closed loop frequency response with the frequency sweep method. The sweep signal (a sinusoidal signal whose frequency grows with time) is summed to the control variable, as shown in Fig. 7. The closed loop frequency response is obtained from the ratio of the Fourier transform of the output of the regulator (TCMD) and of the sweep itself. The other axes, not involved in the identification experiment, are kept in assigned positions. In Figs. 8 and 9 the frequency responses computed by the software tool for the X axis and for the Y axis are reported, respectively. The frequency response shows that both the axes are characterized by two antiresonance (AR)–resonance (R) pairs (namely lightly damped complex zeros followed by complex poles). In the case of the X axis the first resonance is found at about 50 Hz, while the second one is beyond 200 Hz: a clear frequency separation thus emerges. This is not true for the Y axis, where the two resonances are located at 20 Hz and 60–70 Hz, respectively. In the following, lumped parameters models of the two axes are presented, that give an interpretation of these results.
Fig. 7. Test for the computation of the closed loop frequency response.
5.3. Modeling and analysis The goal of the analysis is to determine models of the X axis and of the Y axis that are able to predict with acceptable accuracy the dominant AR–R pairs. For the X axis the base of the machine is assumed to be rigid and the inertias of the motor, of the pulleys of the belts and of the load, including the pinion and the cart of the X axis, also loaded with the other axes, are explicitly accounted for. Consistent with this, the stiffness constants considered are those associated to the transmission belts (first and second stage), to the bearings, to the teeth and to the bending and torsion of the shaft of the pinion.
Fig. 8. Closed loop frequency response for X axis.
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Fig. 9. Closed loop frequency response for Y axis.
In order to assign the values of the stiffness constants, analytical relations have been derived, that express these values in terms of geometrical and physical parameters of the elements of the transmission, as well as of a numerical coefficient to be adapted on an experimental basis, so as to maximize the adherence of the results obtained from the model to those gathered on the machining center. These relations can then be used in future projects for the best a priori estimation of similar transmission structures employed in newly designed machines. A linear model composed of masses, springs and dampers has been derived and analyzed in the Matlab environment, together with the model of the control system of the X axis, in order to study the closed loop behavior. The control system parameters have been obtained from the configuration data and from documentation and, whenever necessary, with ad hoc measurements on the machine. The damping coefficients, quite hard to be obtained on an analytical basis, have been determined so as to obtain the best approximation of the experimental results. The Bode diagrams of the closed loop frequency response derived in this way are reported in Fig. 10. As it can be seen, comparing with the diagrams in Fig. 8, the model correctly
predicts the frequencies of the two pairs AR–R. To this purpose, the stiffness constants computed for the bearing and for the teeth of the pinion and the rack and for the second stage of the belt transmission had to be halved from their nominal values. In general, a stiffness constant less than the computed one is admissible, as it is not easy to account for all the compliant elements in the transmission chain. As far as the Y axis is concerned, the closed loop frequency response of Fig. 9 cannot be interpreted with a model that attributes the compliance to the transmission chain only, as the one used for the X axis. It is in fact necessary to consider, besides the compliance of the transmission between the two dominant inertias of the motor and the load, the compliance at the base of the rotated ‘‘L shaped’’ structure on which the cart of the X axis slides. This structures is constrained by two rails that block all the degrees of freedom, except the translation in the X direction. A compliance of this constraint to moments in the X direction induces a compliance of the support of the motion actuator for the Y axis. A linear model can thus be derived, whose parameters are computed with the same methodology used for X axis. The
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Fig. 10. Closed loop frequency response of the X axis from the linear model.
closed loop frequency response with Bode diagrams shown in Fig. 11 is obtained. Comparing with the diagrams in Fig. 9, it can be noted that the model predicts with remarkable accuracy the experimental results.
5.4. Derivation of the complete model in DYMOLA Fig. 12 shows the top level view of the simulator of the machining center developed in DYMOLA. The model
Fig. 11. Closed loop frequency response of the Y axis from the linear model.
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Fig. 12. Complete DYMOLA model of the machining center.
Fig. 13. Model of the reductor.
is composed of three parts: the simulation of the CN and the servodrives, entirely realized with the DYMOLA blocks, the simulation of the transmission chain for each axis, where the model of the brushless motor has been used, and the simulation of the kinematic chain, realized with the elements of the ModelicaAdditions.MultiBody library.
The models of the reductor and of the belt transmission are shown, as examples, in Figs. 13 and 14, respectively. Both of them contain a customized module: in the reductor an element which allows simulation of either a rigid transmission or an elastic (with backlash) one has been introduced for debugging purposes. In the belt transmission, an element describing a simplified model
Fig. 14. Model of the belt transmission.
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Fig. 15. Model of the CN.
of a belt is present. The first element is realized by composition of Modelica library elements, the second one by direct writing of the balance equations on the belt. Fig. 15 shows the internal structure of the CN module, which is made up by an interpolator (responsible for trajectory generation) and as many closed loop controllers (drives) as the axes in the kinematic chain. In turn, each drive has the structure shown in Fig. 16, where a digital position–velocity controller is closed around the digital current controller described in the previous section. The multi-domain nature of DYMOLA and the physics driven assembly of the model turned out to be essential elements to realize a reliable simulation environment, easy to use for a non specialist of dynamic modeling. In order to validate the model, comparisons have been made with experimental results related to the single axes X and Y, using the responses to the sweep signal. The comparisons with these responses are shown in Figs. 17 and 18. As it can be seen, the model is able to predict with noticeable accuracy this type of responses, confirming the validity of the modeling assumptions.
Fig. 16. Model of the drive for each axis.
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Fig. 17. Responses to the frequency sweep for X axis.
Fig. 18. Responses to the frequency sweep for Y axis.
6. Conclusions
Acknowledgements
The choice of modeling language and simulation environment for virtual prototyping of a machine is a crucial step for the success of the operation. Guidelines have been given in this paper for a balanced choice. DYMOLA with Modelica language has been found to comply with most of the requirements. An experience of its use in the simulation of a complete machining center has been illustrated. Some choices in the derivation of the model have been discussed. The results obtained in terms of experimental validation confirm the reliability of the model, that can be proposed as a useful tool for prototyping newly designed machines.
The authors wish to thank Luciano Bonometti and Matteo Vicini for working together during parts of this project and CMS SpA for financial support.
Appendix A Modelica code of emf3 module: 0 = P0.i + P1.i + P2.i; 0 = vn + e0 P0.v; 0 = vn + e1 P1.v;
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0 = vn + e2 P2.v; k = 2*kt/3; k0 = k*sin(p*flange_b.phi); k1 = k*sin(p*flange_b.phi 2/3*PI); k2 = k*sin(p*flange_b.phi 4/3*PI); e0 = k0*w; e1 = k1*w; e2 = k2*w; w = der(flange_b.phi); flange_b.tau = (k0*P0.i + k1*P1.i + k2*P2.i).
References Barton, P. I., & Pantelides, C. C. (1994). Modeling of combined discrete/ continuous processes. AiChE Journal, 6, 966–979. Brenan, K. E., Campbell, S. L., & Petzold, L. R. (1989). Numerical solution of initial-value problems in differential algebraic equations. Cellier, F. (1991). Continuous system modeling. Springer Verlag. Elmqvist, H., Bruck, D., & Otter, M. (1996). DYMOLA—User’s manual. Research Park Ideon, Lund: Dynasim AB. Ferretti, G., Magnani, G., & Rocco, P. (1998). Modeling, identification and compensation of pulsating torque in permanent magnet ac motors. IEEE Transactions on Industrial Electronics, 46(6), 912–920. Ferretti, G., Magnani, G., & Rocco, P. (1999). Force oscillations in contact motion of industrial robots: An experimental investigation. IEEE/ASME Transactions on Mechatronics, 4(1), 86–91. Ferretti, G., Filippi, S., Maffezzoni, C., Magnani, G., & Rocco, P. (1999). A modular approach to dynamic virtual reality modeling of robotic systems. IEEE Robotics and Automation Magazine, 6, 13–23. Fritszon, P., Gunnarsson, J., Jirstrand, M. (2002). MathModelica: An extensible modeling and simulation environment with integrated graphics and literate programming, Proceedings of the 2nd International Modelica Conference, Oberpfaffenhofen, Germany. Fritszon, P., Aronsson, P., Bunus, P., Engleson, V., Saldalmi, L., Johansson, H., Karstro¨ m, A. (2002). The open source Modelica project, Proceedings of the 2nd International Modelica Conference, Oberpfaffenhofen, Germany. Jahns, T. M., & Soong, W. L. (1996). Pulsating torque minimization techniques for permanent magnet ac motor drives—A review. IEEE Transactions on Industrial Electronics, 43(2), 321–330. Maffezzoni, C., & Girelli, R. (1998). MOSES: Modular modeling of physical systems in an object-oriented database. Mathematical Modeling of Systems, 4(2), 121–147. Mattsson, S. E., & Andersson, M. (1993). Omola—An object-oriented modeling language. In M. Jamshidi & C. J. Herget (Eds.), Recent Advances in Computer Aided Control Systems, Studies in Automation and Control (pp. 291–310). Elsevier Science.
Mattsson, S. E., Elmqvist, H., & Otter, M. (1998). Physical system modeling with Modelica. Control Engineering Practice, 6, 501–510. Oh, M., & Pantelides, C. C. (1996). A modeling and simulation language for combined lumped and distributed parameter systems. Computers and Chemical Engineering, 20, 611–633. Pantelides, C. C. (1988). The consistent initialization of differential-algebraic systems. Siam Journal of Scientific and Statistical Computing, 9, 213–231. Piela, P. C., Epperly, T. G., Westerberg, K. M., & Westerberg, A. W. (1991). ASCEND: An object oriented computer environment for modeling and analysis: The modeling language. Computers and Chemical Engineering, 15(1), 53–72. Texas. (1997). Digital signal processing solution for permanent magnet synchronous motor (Application Note, Literature Number: BPRA044). Tiller, M. (2001). Introduction to physical modeling with Modelica. Kluwer. Gianni Ferretti received the laurea degree in electronic engineering from Politecnico di Milano, Italy. From 1990 to 1998 he was with the Department of Electronics and Information of the Politecnico di Milano, as assistant professor. His principal research activities are: modeling, simulation and control of thermohydraulic processes, modeling, identification and simulation of mechanical systems, hybrid force/position control of industrial robots, compensation of torque disturbances in drives. He is currently associate professor at the Politecnico di Milano, where he teaches automatic controls. GianAntonio Magnani received the laurea degree in electronic engineering from Politecnico di Milano, Italy, in 1978. From 1978 to mid 1984 he was with the Automatica Research Center of the Italian Electricity Board (ENEL), where he participated in research on modeling, simulation and control of electric power plants. From 1986 to 1992 he was with Fiar Spa, Robotics and Artificial Intelligence Division and with Tecnospazio, a Fiar and Comau (Fiat Group) joint company, where he was in charge of research and development units and led several projects for industrial and space robotics applications. He is currently with Politecnico di Milano as a professor of control system technologies and engineering. He also teaches industrial robotics, and is in charge of research projects awarded by industrial companies and research agencies. His current research interests concern robot position and force control, servomechanisms and mechatronics. He is a senior member of IEEE. Paolo Rocco received the laurea degree in electronic engineering and the doctorate degree in computer science and automation, both from Politecnico di Milano, Italy, in 1991 and 1995, respectively. During 1995 he was a visiting scholar at the School of Mechanical Engineering of Georgia Tech, Atlanta, GA, under a NATO fellowship. He is currently associate professor in automatic control at Politecnico di Milano. His research interests include the force/position control of both industrial and flexible robots, the identification and suppression of torque disturbances in brushless motors and the modeling, simulation and control of mechatronic systems.
Virtual prototyping of mechatronic systems Gianni Ferretti, GianAntonio Magnani, Paolo Rocco* Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza L. da Vinci 32, 20133 Milano, Italy Received 19 December 2003; received in revised form 26 January 2004; accepted 23 February 2004
Abstract Mechatronic systems abound in technological fields such as robotics and machine tools industry. Significant advances in dynamic performance of these systems can be achieved provided that mutual interactions of different domains (such as mechanics, electronics, hydraulics and control) are thoroughly understood. Virtual prototyping entails integration of multi-domain dynamic simulation in the design process, in order to reproduce and analyze the effects of design choices on the overall performance. This paper states the requirements of the modeling language and software tool for simulation of mechatronic systems and describes an experience of use of DYMOLA with Modelica language in the simulation of a complete machining center. The model has been successfully validated against experimental results collected on the real machining center. # 2004 Elsevier Ltd. All rights reserved. Keywords: Mechatronics; Modeling; Object-orientation; Simulation; Virtual prototyping
1. Introduction Every technological system where mechanics, electronics and control concur in a mutually supportive way to the overall performance belongs to the family of the mechatronic systems. Examples are found in robotics, machine tools or machining centers. While the traditional design process of such systems did not entail significant interactions between the mechanical designers and the experts of control and drive electronics, the market demand for ever increasing precision calls now for a synergic design, where the mutual interactions of choices made in the mechanical design (possibly including hydraulics) and in the drive electronics design are thoroughly understood. Dynamic performance of the machine has become the real issue, rather than static dimensioning, and dynamic performance cannot be improved unless the machine is studied with electronics and control operating in closed loop with the mechanics.
* Corresponding author. Tel.: +39 022 399 3685; fax: +39 022 399 3412. E-mail address: [email protected] (G. Ferretti), [email protected] (G. Magnani), [email protected] (P. Rocco). 1367-5788/$ – see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.arcontrol.2004.02.002
Experimental study on physical prototypes is however expensive and time consuming. Virtual prototyping, a new way of defining dynamic simulation (Cellier, 1991; Ferretti, Filippi, Maffezzoni, Magnani, & Rocco, 1999) of the controlled machine, is more compliant with time and budget constraints of R&D industrial centers. If design choices on the mechanical as well as on the electronic side can be tested before assembling the physical prototype, the time to market is shortened and the knowledge of the system is improved, obviously as long as the simulation plant is a reliable replication of reality and the simulation tool is reasonably easy to use for people coming from different areas. The general impression is that today several companies operating in fields where competition and advances in dynamic precision of the machines are vital issues are somewhat ready to introduce these kinds of tools in the design process, but also that there is not enough consciousness on the real potentialities of the new approaches. Uncertainties often arise also on the requirements for the modeling language and software tool to be adopted. Simulation environments such as SIMULINK seem to be extremely powerful to replicate the control system, but difficulties are evident as soon as physical system modeling is attempted, as the unidirectional signal flow implied by the
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procedural description of the blocks does not seem to match the physical description of the system adequately. On the other hand commercial packages (e.g. ADAMS, Altair, SimMechanics) abound for simulation of the mechanics, from static dimensioning to vibration study, but users might experience troubles when attempts are made to simulate closed loops with electronics and control. Meanwhile, the research on modular physical modeling languages and tools, started at the end of the 1970’s, has evolved into several modeling languages and simulation environments: ASCEND (Piela, Epperly, Westerberg, & Westerberg, 1991), OMOLA (Mattsson & Andersson, 1993), gPROMS (Barton & Pantelides, 1994; Oh & Pantelides, 1996), MOSES (Maffezzoni & Girelli, 1998), Modelica (Mattsson, Elmqvist, & Otter, 1998; Tiller, 2001) and DYMOLA (Elmqvist, Bruck, & Otter, 1996), to name just a few. This paper first presents a discussion on the requirements of a software tool for system simulation of mechatronic devices. Multidomain scope, software reuse, reliability and efficiency of the numerical simulation, integration with mechanical CADs are found to be valuable aspects. An experience is then described of use of DYMOLA with Modelica language, which seems to comply with most of the above requirements. The standard libraries of Modelica for simulation of mechatronic systems are first described. Then some customized modules, such as brushless motors and current drivers, are presented, emphasizing the advantages related to acausal physical system modeling. As a case study, the results of a research on the sources of performance limitations in a wood machining center are presented. The performance limiting factors have been investigated through linear models of the single axes of the machining center, suitable to predict the resonant behavior experimentally identified, and through simulation of a detailed model developed in the DYMOLA environment. The overall model of the machine can predict the performance with noticeable accuracy, thus it can be used in the design of new machines, with similar structure and components as the machine studied in this work.
2. Requirements for the modeling language and the simulation environment Modeling and simulation of mechatronic systems are difficult tasks, with several critical points. The choice of the modeling approach and of the simulation environment thus deserves particular attention, especially for large scale industrial projects. The main user requirements for the modeling approach and the simulation environment can be specified as follows. 2.1. Multi domain simulation The simulation environment must deal with equal comfort with mechanical, hydraulic, electrical, control
and possibly other physical domains. As already mentioned, a systematic study of dynamic performance of a mechatronic system should neither privilege aspects relevant only to a single domain, nor ignore the mutual interactions among domains. For instance, software tools where mechanics are simulated in detail, but electronics and control require to the user ad-hoc additions of modules, written in a different modeling language, are not advisable for the user due to longer development times and reduced accuracy 2.2. Modular modeling and software reuse Modular modeling is a major requirement, if not a need, for simulation of complex systems like mechatronic ones. However, real modularization of mechanical, hydraulic ad electrical systems is hampered in most simulation tools, which do not allow, for instance, to establish algebraic relations (constraints) among state variables. Constraints on state variables arise in the rigid connection of the dynamic models of a motor shaft and of a gearbox, separately written, or when more electric currents or liquid flows from different circuits enter a common node or mixing volume. Modular modeling permits to split a large model in a number of smaller and more manageable models, and is a necessary condition to software reuse. The reusability of modeling software from project to project is a key requirement for industrial users. 2.3. Built-in model libraries and customisation Use of built-in libraries of components keeps modeling costs low and development times short. Libraries that cover the needs for simulation of simple mechatronic systems are usually available in most of commercial simulation tools. Mechatronic systems simulation is especially demanding in this respect, since libraries from several physical domains are needed, easy to be integrated one with the others. Nevertheless, situations occur where customised modules must be written, as is shown in the following Sections. In this respect, expansion of library modules, rather than writing customised modules from scratch, obviously facilitates the task of the user. 2.4. Reliability and efficiency of numerical simulation Reliability and efficiency of the numerical solver are key requirements for non-skilled users, generally unwilling to deal with details in numerical integration of equations. A differential algebraic equations (DAE) (Brenan, Campbell, & Petzold, 1989) solver is needed, able to handle events, such as discontinuities and abrupt changes in external variables. The time instant when the event is triggered must be detected with precision, thanks to variable step size of integration. Also simulation of closed loop systems must be dealt with in the most reliable way: solutions where the physical part of the system is simulated with a DAE solver while the procedural control part is simulated with an ODE solver, and synchronisation is kept between the two solvers,
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do not seem to provide the same reliability as a single solver integrating the whole system. 2.5. Integration with mechanical CADs Mechanical designers feel comfortable with software tools that can be easily integrated with the currently used mechanical CADs (for example SolidWorks1). Generally speaking, integration means that the mechanical CAD generates the mass and inertial parameters to be used in the dynamic simulation environment, without any manual transcription, which is both time consuming and error prone. Also integration with finite element analysis (FEA) packages turns out to be useful for studies where flexibility of materials is an issue. 2.6. Interfaces for pre- and post-processing Assembly of modules should be performed through a Graphical User Interface (GUI). Both 2D and 3D interfaces are used, the 2D being more common and easy to use. Each library module should be associated to a graphical symbol as evocative as possible of the physical nature of the system. Visualisation of simulation results can be performed both with 2D plots or 3D animation. The 2D plots are more useful for detailed analysis of dynamic properties, while 3D animation helps understanding the gross motion of the system. Compliance with the above requirements 2.1–2.3 (customisation) entails the following properties of a simulation environment and its modeling language. 2.7. Simulation paradigm Declarative modeling languages should be used to simulate the physical parts of the system, be they mechanical, electrical, hydraulic or pneumatic components. Procedural languages, where the unidirectional flow of information from an input to an output is postulated, are inadequate to simulate systems where instantaneous exchanges of power are present. Acausal models, described by DAE systems, should be used to represent in the most natural and physically consistent way each system component, while the task of defining the computational causality of the assembled model should be charged to the simulation environment. 2.8. Object orientation The modeling language should comply with modern object-oriented paradigms of software engineering. The most relevant aspects of object-orientation to the benefit of physical system modeling are modularization (the system is composed by aggregation of modules), abstraction (the internal description of a module is separated from its interface), information encapsulation (only the interface variables are 1
SolidWorks is a trademark of SolidWorks Corporation.
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accessible to the other modules), reuse of the modules through parameterisation. Structuring the object-oriented models (modules) in libraries facilitates the extension of existing models and the simulation of large and complex systems. 2.9. Handling of equations and constraints Assembling of modules described by DAE equations generally leads to higher index2 systems and algebraic loops. Symbolic manipulation of the equations is then required to yield a set of equations amenable to numerical integration. Some simulation tools use Pantelides’ (1988) algorithm (or similar) to reduce the index of the system. Others use constraint relaxation techniques, generating an ODE system where, however, constraints are not identically satisfied during transients. The efficiency of this pre-processing software layer might be crucial for the efficiency of the overall simulation environment.
3. Modelica standard libraries for mechatronics None of the commercial packages on the market fully complies with all the above user requirements, to our knowledge. On the other hand, the choice of the package to adopt may depend on specific issues of the problem and may require a trade off among the degrees of fulfillment of different requirements. In the present work, an experience is reported of use of DYMOLA (with Modelica language) for simulation of machining centers. This package fully complies with the requirements of multi-domain simulation and ease of customization, and features the acausal declarative modeling paradigm as well as the object-oriented approach. The efficiency of the numerical integration is guaranteed by a sophisticated software layer where the assembled model is first translated, i.e. equations are symbolically manipulated, in order to reduce the index and produce efficient simulation code. A state of the art numerical solver of DAE’s is then used for numerical integration. A 2D GUI for model assembling is available, as well as a simple 3D animation module for post processing of simulation results. Concerning interfaces with mechanical CAD packages, a translator from SolidWorks to Modelica is also available. It converts bodies and joints of SolidWorks models to Modelica models, also filling in the templates of the relevant geometrical and inertia parameters. Additional mechanical components (external forces, torques, frictions, etc.) can be easily added using a Modelica editor. In DYMOLA all models are written in the Modelica language, the outcome of the efforts of the Modelica Association, a non-profit, non-governmental organization 2 The index of a DAE system is the number of times all or part of the equations of the system must be differentiated, in order to obtain an ODE system. When the index is higher than one, explicit or hidden algebraic constraints on the state variables of the system are present.
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born in 1996. The aim of the association is to develop and promote the Modelica modeling language for modeling, simulation and programming of physical and technical systems and processes. The association maintains specifications of the modeling language and develops free of use Modelica standard libraries. DYMOLA is one of the software tools (together with MathModelica (Fritszon, Gunnarsson, & Jirstrand, 2002) and OpenModelica (Fritszon, Aronsson, et al. 2002)) that fully supports the Modelica language and offers the software environment to actually perform the simulation. There are several Modelica libraries relevant for a mechatronic project: the Modelica.Mechanics.Rotational package (shown, as an example, in Fig. 1) offers elements to simulate one dimensional rotational mechanical systems, useful to reproduce transmission chains, gearboxes, shafts with inertia, spring/damper elements, frictional and backlash elements. Remarkably, rigid connection of two inertia elements, an operation which is essential for the modular simulation of transmission chains but generates a higher (three) index DAE system, can be made without any concern. The simulation environment is in charge of resolving the constraint of equality of the angular positions and velocities of the two inertia elements. A similar library (Modelica.Mechanics.Translational) is available to simulate one dimensional translational mechanical systems. Three-dimensional mechanical systems, such as robots, machine tools or vehicles, can be modelled with the ModelicaAdditions.Multibody library. The components of this library can be combined with the above 1D-mechanical libraries through flanges in the Joint modules (the joints introduce the degrees of freedom in the mechanical systems). The Body elements require specification of
geometrical as well as inertial (mass, center of mass and inertia tensor) parameters, that can be easily introduced if an interface with a solid modeller is available. Differently from other simulation environments, the electrical and control domains can be simulated with equal ease as the mechanical domain. The Modelica.Electrical.Analog library allows simulation of simple electrical circuits. The declarative paradigm is fully respected in this library, too. All components have electrical cuts where exchanges of electrical power (through voltage and current variables) take place. This implies that each physical element (be it a resistor or a capacitor, for example) has just one library counterpart, regardless that the element is actually voltage or current driven. Complete reusability of the models is thus guaranteed. The Modelica.Blocks library contains input/output blocks to build up simple block diagrams for simulation of controllers (both analogue and digital ones). Based on our experience, the library, although somewhat limited, allows to simulate most of the functions of commercial controllers, obviously through aggregation of the library elements. An example is given in next section for the simulation of a current driver for a brushless motor. Other major libraries suitable for mechatronic systems are the hydraulic (HyLib) and pneumatic (PneuLib) libraries. They contain the most important basic components, useful to model hydraulic and pneumatic circuits. Coupling elements exist to connect hydraulic and pneumatic components with control blocks of the blocks library and elements of the mechanics library. The libraries support modeling of one-dimensional rotational and translation hydraulic and pneumatic systems, and consist of large varieties of pumps, boosters, motors, cylinders, valves, restrictions, tanks, reservoirs, sensors, lines and volumes.
4. An example of a customized module
Fig. 1. Modelica standard library for rotational mechanics.
The most common actuation systems adopted in robotics, machine tools industry and a number of mechatronic systems are by far servomechanisms with permanent magnet brushless motors, connected to the loads by transmission chains (or gearboxes). In a brushless motor the electromechanical commutation typical of brushed dc motors is replaced by an electronic commutation of the currents in the three phases of the stator windings. This should in principle guarantee that the electromagnetic torque delivered on the motor shaft is independent of the rotor position. However, some constructive imperfections in the motor or in the drive, where electronic commutation is implemented, produce an undulation (ripple) (Ferretti, Magnani, & Rocco, 1998; Jahns & Soong, 1996) on the actual torque. While this problem could be considered minor in the static dimensioning of the actuation system, it is of utmost importance for its dynamic performance. Torque ripple might in fact excite the resonances of the mechanical system, usually associated to
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the elastic couplings between motors and loads (Ferretti, Magnani, & Rocco, 1999). Dynamic models and simulation are valuable tools to study these phenomena, and in particular to separate the effects of the single sources of disturbances on the performance of the system. Mechanics, electronics and control are different domains involved in this truly mechatronic problem. The model of the system in DYMOLA is obtained by the feedback connection of two sub-models, one representing the brushless motor, the other the driver (Fig. 2). The two models are connected through three electrical connectors (for the phases of the motor) as well as through a control connector (the measure of the rotor position). The brushless motor model is shown in Fig. 3. The three phases are Y connected in the block emf3, that generalizes the emf model in the Modelica.Electrical.Analog.Basic library. The equations of the emf3 model are the following ones: 3 X
ii ¼ 0
i¼1
Vi ¼ Vn þ e i ei ¼ ki v
2 2 ki ¼ kt sin p# ði 1Þ p 3 3 #˙ ¼ v
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Fig. 2. The brushless motor and the driver.
t m ¼ ðk1 i1 þ k2 i2 þ k3 i3 Þ; where ii, Vi, ei and ki (i = 1, 2, 3) are the phase currents, the phase input voltages, the back emf’s and the back emf profiles for each phase, respectively; Vn is the voltage of the central node of the Y connection; # and v are the angular position and velocity of the rotor; kt is the torque constant of the motor, p the number of pole pairs and tm is the electromagnetic torque generated by the motor. The code of the emf3 model is reported in the Appendix A. Thanks to declarative coding it is straightforward to write code from equations and read out equations from code. Remarkably, the algebraic constraint on the currents (which are state variables) can be specified in the most natural way
Fig. 3. Modelica model of the brushless motor.
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(the sum of the currents equal zero). Assigning, in the simulation code, back emf profiles ki different from the nominal sinusoidal ones, ripple due to higher order harmonics can be reproduced. In addition to the emf3 module, the model of the brushless motor includes inductance and resistance of stator windings and the mechanical part, consisting of rotor inertia and viscous friction. Just for comparison, Fig. 4 shows the SIMULINK model of the electrical part of the motor, obtained by resolving the algebraic constraint and expressing the whole model in terms of two out of three currents. This operation is time consuming and error prone and produces a model description rather difficult to read. A full digital version of the current controller has been implemented (Fig. 5). A vector control scheme (Texas, 1997) has been adopted, where the phase currents are first transformed into direct and quadrature currents through Park’s transformation. Two digital loops are closed on these currents, the quadrature reference being the output of the outer position/velocity controller, the direct reference being zero usually. The voltage commands output of the two anti-windup PI controllers are then back transformed to voltages on the three phases through inverse Park’s transformation. The PWM amplifier has not been simulated since, operating with a frequency of 10 or 20 kHz and with a modulation of the pulse width of 1 ms, it requires an integration step size smaller than 1 ms, which might be acceptable for the simulation of the electronics of a motor drive, but is far too small in the joint simulation of the mechanics and the electronics. This is particularly true if the
model of the motor is instantiated several times, for the simulation of a complete machine.
5. A case study 5.1. Preliminaries The model of the brushless motor described in the previous section has been included in a library of elements used to simulate a complete three axes machining center. This simulation is part of a research aimed at understanding the sources of performance limitations in mechanical machining centers and at deriving mathematical models that can support its mechatronic design. The structure of the machining center studied in this work is sketched in Fig. 6. This kind of structure is common to several machining centers used for wood and metal, characterized by three linear axes (X, Y, Z). Motion transmission for X axis is realized through two reduction stages with belts and pulleys, followed by a pinion-rack stage, that allows to convert rotational motion to linear motion. In the Y axis, only one stage with belts and pulleys is present, besides the pinion-rack pair. In the following, axis Z will not be considered, as it does not present critical elements. 5.2. Experimental results Experimental tests have been performed on the machine, in order to calculate frequency responses. A new software package, shipped by the manufacturer of the CN, has been
Fig. 4. SIMULINK model of the brushless motor.
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Fig. 5. Model of the digital driver.
Fig. 6. Three linear axes machining center.
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used, that allows to estimate a closed loop frequency response with the frequency sweep method. The sweep signal (a sinusoidal signal whose frequency grows with time) is summed to the control variable, as shown in Fig. 7. The closed loop frequency response is obtained from the ratio of the Fourier transform of the output of the regulator (TCMD) and of the sweep itself. The other axes, not involved in the identification experiment, are kept in assigned positions. In Figs. 8 and 9 the frequency responses computed by the software tool for the X axis and for the Y axis are reported, respectively. The frequency response shows that both the axes are characterized by two antiresonance (AR)–resonance (R) pairs (namely lightly damped complex zeros followed by complex poles). In the case of the X axis the first resonance is found at about 50 Hz, while the second one is beyond 200 Hz: a clear frequency separation thus emerges. This is not true for the Y axis, where the two resonances are located at 20 Hz and 60–70 Hz, respectively. In the following, lumped parameters models of the two axes are presented, that give an interpretation of these results.
Fig. 7. Test for the computation of the closed loop frequency response.
5.3. Modeling and analysis The goal of the analysis is to determine models of the X axis and of the Y axis that are able to predict with acceptable accuracy the dominant AR–R pairs. For the X axis the base of the machine is assumed to be rigid and the inertias of the motor, of the pulleys of the belts and of the load, including the pinion and the cart of the X axis, also loaded with the other axes, are explicitly accounted for. Consistent with this, the stiffness constants considered are those associated to the transmission belts (first and second stage), to the bearings, to the teeth and to the bending and torsion of the shaft of the pinion.
Fig. 8. Closed loop frequency response for X axis.
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Fig. 9. Closed loop frequency response for Y axis.
In order to assign the values of the stiffness constants, analytical relations have been derived, that express these values in terms of geometrical and physical parameters of the elements of the transmission, as well as of a numerical coefficient to be adapted on an experimental basis, so as to maximize the adherence of the results obtained from the model to those gathered on the machining center. These relations can then be used in future projects for the best a priori estimation of similar transmission structures employed in newly designed machines. A linear model composed of masses, springs and dampers has been derived and analyzed in the Matlab environment, together with the model of the control system of the X axis, in order to study the closed loop behavior. The control system parameters have been obtained from the configuration data and from documentation and, whenever necessary, with ad hoc measurements on the machine. The damping coefficients, quite hard to be obtained on an analytical basis, have been determined so as to obtain the best approximation of the experimental results. The Bode diagrams of the closed loop frequency response derived in this way are reported in Fig. 10. As it can be seen, comparing with the diagrams in Fig. 8, the model correctly
predicts the frequencies of the two pairs AR–R. To this purpose, the stiffness constants computed for the bearing and for the teeth of the pinion and the rack and for the second stage of the belt transmission had to be halved from their nominal values. In general, a stiffness constant less than the computed one is admissible, as it is not easy to account for all the compliant elements in the transmission chain. As far as the Y axis is concerned, the closed loop frequency response of Fig. 9 cannot be interpreted with a model that attributes the compliance to the transmission chain only, as the one used for the X axis. It is in fact necessary to consider, besides the compliance of the transmission between the two dominant inertias of the motor and the load, the compliance at the base of the rotated ‘‘L shaped’’ structure on which the cart of the X axis slides. This structures is constrained by two rails that block all the degrees of freedom, except the translation in the X direction. A compliance of this constraint to moments in the X direction induces a compliance of the support of the motion actuator for the Y axis. A linear model can thus be derived, whose parameters are computed with the same methodology used for X axis. The
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Fig. 10. Closed loop frequency response of the X axis from the linear model.
closed loop frequency response with Bode diagrams shown in Fig. 11 is obtained. Comparing with the diagrams in Fig. 9, it can be noted that the model predicts with remarkable accuracy the experimental results.
5.4. Derivation of the complete model in DYMOLA Fig. 12 shows the top level view of the simulator of the machining center developed in DYMOLA. The model
Fig. 11. Closed loop frequency response of the Y axis from the linear model.
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Fig. 12. Complete DYMOLA model of the machining center.
Fig. 13. Model of the reductor.
is composed of three parts: the simulation of the CN and the servodrives, entirely realized with the DYMOLA blocks, the simulation of the transmission chain for each axis, where the model of the brushless motor has been used, and the simulation of the kinematic chain, realized with the elements of the ModelicaAdditions.MultiBody library.
The models of the reductor and of the belt transmission are shown, as examples, in Figs. 13 and 14, respectively. Both of them contain a customized module: in the reductor an element which allows simulation of either a rigid transmission or an elastic (with backlash) one has been introduced for debugging purposes. In the belt transmission, an element describing a simplified model
Fig. 14. Model of the belt transmission.
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Fig. 15. Model of the CN.
of a belt is present. The first element is realized by composition of Modelica library elements, the second one by direct writing of the balance equations on the belt. Fig. 15 shows the internal structure of the CN module, which is made up by an interpolator (responsible for trajectory generation) and as many closed loop controllers (drives) as the axes in the kinematic chain. In turn, each drive has the structure shown in Fig. 16, where a digital position–velocity controller is closed around the digital current controller described in the previous section. The multi-domain nature of DYMOLA and the physics driven assembly of the model turned out to be essential elements to realize a reliable simulation environment, easy to use for a non specialist of dynamic modeling. In order to validate the model, comparisons have been made with experimental results related to the single axes X and Y, using the responses to the sweep signal. The comparisons with these responses are shown in Figs. 17 and 18. As it can be seen, the model is able to predict with noticeable accuracy this type of responses, confirming the validity of the modeling assumptions.
Fig. 16. Model of the drive for each axis.
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Fig. 17. Responses to the frequency sweep for X axis.
Fig. 18. Responses to the frequency sweep for Y axis.
6. Conclusions
Acknowledgements
The choice of modeling language and simulation environment for virtual prototyping of a machine is a crucial step for the success of the operation. Guidelines have been given in this paper for a balanced choice. DYMOLA with Modelica language has been found to comply with most of the requirements. An experience of its use in the simulation of a complete machining center has been illustrated. Some choices in the derivation of the model have been discussed. The results obtained in terms of experimental validation confirm the reliability of the model, that can be proposed as a useful tool for prototyping newly designed machines.
The authors wish to thank Luciano Bonometti and Matteo Vicini for working together during parts of this project and CMS SpA for financial support.
Appendix A Modelica code of emf3 module: 0 = P0.i + P1.i + P2.i; 0 = vn + e0 P0.v; 0 = vn + e1 P1.v;
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0 = vn + e2 P2.v; k = 2*kt/3; k0 = k*sin(p*flange_b.phi); k1 = k*sin(p*flange_b.phi 2/3*PI); k2 = k*sin(p*flange_b.phi 4/3*PI); e0 = k0*w; e1 = k1*w; e2 = k2*w; w = der(flange_b.phi); flange_b.tau = (k0*P0.i + k1*P1.i + k2*P2.i).
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