System Dynamic Models and Real-time Simulation of Complex Material Flow Systems

Transcription

1 S. Hoher a P. Schindler b S. Göttlich b V. Schleper c S. Röck a System Dynamic Models and Real-time Simulation of Complex Material Flow Systems Stuttgart, Mai 0 a Institute for Control Engineering of Machine Tools and Manufacturing Units (ISW, University of Stuttgart Seidenstraße 36, 7074 Stuttgart/ Germany {Simon.Hoher, Sascha.Roeck}@isw.uni-stuttgart.de b School of Business Informatics and Mathematics, University of Mannheim, A5, 66 Mannheim/ Germany Goettlich@uni-mannheim.de b Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, Stuttgart Veronika.Schleper@mathematik.uni-stuttgart.de, Abstract In this paper a multi-scale simulation approach based on system dynamics is investigated that is divided into a microscopic and a macroscopic model scale. On the microscopic model scale small amounts of parts are simulated. The motion of each single discrete element is explicitly realized by means of a physically-based simulation. On the macroscopic model scale a simulation of the material flow is realized with a great amount of parts. A two-dimensional hyperbolic partial differential equation (PDE is applied. We explicitly examine the requirements on the virtual commissioning, which are a strongly timedeterministic computation in the range of one millisecond, robust and efficient computing algorithms and system-dynamic features. The simulation concept is validated against a real conveyor belt. Keywords Material flow system Real-time simulation System dynamic models Preprint Series Issue No. 0-4 Stuttgart Research Centre for Simulation Technology (SRC SimTech SimTech Cluster of Excellence Pfaffenwaldring 7a Stuttgart publications@simtech.uni-stuttgart.de

2 System Dynamic Models and Real-time Simulation of Complex Material Flow Systems S. Hoher, P. Schindler, S. Göttlich, V. Schleper 3, S. Röck Institute for Control Engineering of Machine Tools and Manufacturing Units (ISW, University of Stuttgart 3 School of Business Informatics and Mathematics, University of Mannheim Institute of Applied Analysis and Numerical Simulation, University of Stuttgart Abstract In this paper a multi-scale simulation approach based on system dynamics is investigated that is divided into a microscopic and a macroscopic model scale. On the microscopic model scale small amounts of parts are simulated, whereby the motion of each single discrete element is explicitly realized by means of a physically-based simulation. On the macroscopic model scale, based on a two-dimensional hyperbolic partial differential equation (PDE, a simulation of the material flow with a large amount of parts is realized. We explicitly examine the requirements on the virtual commissioning, which are a strongly time-deterministic computation in the range of one millisecond, robust and efficient computing algorithms and system-dynamic features. Both simulation models are validated against a real conveyor belt. Keywords: Material flow system, Real-time simulation, System dynamic models Design INTRODUCTION In order to manufacture products with the same quality, optimum material utilization, long-term profitability and short time cycles the entire material flow through a manufacturing unit needs to be planned and controlled in detail. The required productivity and product flexibility in the process is thereby achieved by means of highly automated machining centers and production lines. The individual cycle times of the machine tools and processing units as well as the material flow must be considered over the complete production process. The design of a new factory layout and the following set-up is mostly very time-consuming and complex. One possibility is to use simulation models of manufacturing units to improve and simplify the design process. So, the time of the control set-up procedure can be reduced and, at the same time, the quality can be increased significantly []. For testing the operability of the control, it is essential to connect the original control to a simulation. The operation of the control is identical with the real scenario and it does not require retooling or individual adjustments. This allows realistic tests of all control functionalities against its virtual counterpart and without any risk to damage machine components. In order to be able to use the simulation for the above mentioned applications, the simulation cycle time is limited to the control cycle time. If the simulation needs more time than the specified control cycle, as shown in Figure, undefined (unreal results will appear and the test of the control fails. Therefore, the computation of the simulation models needs to be done in a strong time-deterministic way. Conventional controls are operated with cycle times between ms (e.g. NC of machine tools up to 00 ms (e.g. PLC of part flow Control??? Simulation 𝑡 ~ ms 𝑡 Data synchronization 𝑡3 Figure : Control cycle vs. simulation cycle remove Virtual Rapid Prototyping Reconfiguration Hard/Soft Production and Maintenance Redesign add component / function new requirements Figure : Manufacturing systems life cycle (according [] models have to be operations. Consequently, the used to simulation very time-efficient. To consider both, static and dynamic system behavior as realistic as possible the simulation model should be based on physical principles. Thus, time-efficient and numerically robust simulation models and methods are indispensable. The design of today s mass production processes [, 3], includes a high flexibility in material handling, routing, volume and control. During the life cycle of a flexible production facility the factory layout will be adjusted to the requirements of the market (see Figure. This can only be realized by using simulation technologies in the design and redesign processes during the lifecycle of the facility. One of these technologies is the hardware-in-the-loop simulation to set-up the control systems without interrupting or disturbing the production process. STATE OF THE ART Since the fifties of the last century simulation models of manufacturing facilities have been examined and successfully applied in the production process. In this case especially discrete event system models (DES and models according to the queuing theory were focused on, caused by the modeling simplicity for these kinds of problems. A current overview is given by Jahangirian et al. [4]. For the sequence coordination within a production process they are highly significant, but they are quite disadvantageous for the applications described in the introduction. These models should consider the following problems:

3 Set-up of control hardware, runtime errors in the control hardware, runtime errors in the sensor and actor interfaces, influences of the production process dynamics, design and redesign of the facility, control behavior in case of exceptional situations. System-dynamic (SD models have features that meet these requirements. The potentials of system-dynamic models for production facilities have already been presented by Baines and Harrison in 999 [5]. Baines and Harrison show the usability of SD models of production facilities and their competitive advantages in industrial applications. The presented models are related to a global/business/operation level, but they are not dealing in detail with the production level. Figure 3 shows the different modeling level of manufacturing systems and the applications dealt with. Models on the production level have been described as interesting objects of research, in order to ensure the success of the entire production process [6]. Reinhart, Zäh et al. [7] presented in the year 009 a physical model based on the system dynamics for the simulation of the material flow of a production plant on production level. The objective of the model is to simulate the motion of each individual discrete good through the production process based on physical laws. However, questions in regard to the modeling and real-time simulation (according to figure remain unanswered and will be examined in detail in this paper. In the following, we will distinguish between two different model approaches: First the microscopic model, dealing with small amounts of parts and second the macroscopic model, dealing with a great amount of parts. The complete microscopic model consists of collision detection [9], collision response [], physical model of the parts and time integration, as seen in figure 4. Borough [4] gives an integrated summary of these physics engines. Commercial physics engines like the NVIDIA PhysX are also avaiable. Figure 4 gives an overview of a common physics engine. The physical model describes the physical motion of every discrete part including physical properties like mass, inertia and deformation. The interaction of the parts, including the contact damping and friction, is element of the collision response. Due to the complexity of collision detection it is usually divided into two independent Global Level: Dealing with political policy, national strategies, world issues. Business Level: Dealing with business planning, forming strategies, testing, marketing and financial scenarios. Operation Level: Dealing with manufacturing system modeling, production planning, operations management. Production Level: Dealing with material flow, Commissioning. Production Goods Figure 3: Different modeling levels of manufacturing systems Factory 6 Warehouse object transformation Physically-based simulation + Time integration response forces Figure 4: subcomponents. In the Partitioning with Broad Phase it will be checked if a collision is theoretically possible (f. e. tree hierarchies or spatial partitioning and simple bounding volumes tests are done. In the Near Phase the exact collision point is calculated. Then the model is solved by means of numerical time integration. MICROSCOPIC MODEL OF A MATERIAL FLOW SYSTEM Contrary to other approaches (see f. e. Hoher and Röck [3] the deformation of objects remains unconsidered. The material is reduced to simple geometries like cuboids or cylinders. This approach is especially useful for high computational efficiency with large amount of objects in the collision detection phase. In the collision response the interaction between the parts and the (complex environment, e.g. obstacles like deflectors are considered. In addition, the contact between conveyor belts and parts must be taken into account. The following microscopic model addresses the topics speed and accuracy in manufacturing simulation. The model has to been very time-efficient and has to produce repeatable and strongly timedeterministic results. It has to comply with the requirements of virtual rapid prototyping, such as virtual commissioning, in which simulation is combined with control. Individual dynamic of physically-based objects The complete material flow process through a production plant can be described as interacting discrete parts. Since we assume that these parts are rigid we can describe their motion by the Newton Euler equations. We write for each part ( with the mass, the position vector and the external forces. The rotational dynamics results in ( with the inertia tensor, the angular velocity and the external torques. Collisions of the objects Collision detection (partitioning + broad phase Collision response overlapping pairs Collision detection (near phase colliding pairs Architecture for Multi Body Collision D The collision response has to fulfill the energy conservation laws, which means that the total linear and the total angular momentum of the entire system will be maintained. Another task of the collision response is the calculation of the impact damping due to friction and deformation of the material surface. The most intuitive method for dealing with the collision case is to apply opposite forces to both objects. A physical model of the collision response can simply be computed with force-based methods [9, 0]. These forces depend on the material features, like Young s moduli, and the intersection depths. The main problem of the force-based collision response is the required computational effort, caused by small time step sizes. Due

4 , (5c, (5d with and (6. (7 Figure 5: collision of two rigid bodies to further idealizations in the physical collision the impulse-based collision response can be used. This allows for larger step sizes for the above mentioned transport processes. This method is in line with the work of Moore and Wilhems [9], Bourg [] and Baraff []. We insert an adequate friction model in the impulse-based collision response, and choose an adequate integration method for our application area. The approach described below allows solving the transport problem efficiently with only one algebraic equation, without iterations, time-deterministically and in consideration of the contact friction. Pilling up material can also be simulated. Any possible step size can be chosen as long as the spatial coherence is not violated. Impulse-based collision response: Using an algebraic solution The spring stiffness in the collision point is now theoretically set towards infinity. In case of collision there will be a very strong force for a very short moment, the so-called impulse. An infinitely high force in an infinitely short time step changes the velocities of the collision objects immediately. Therefore, the impulse-based collision response provides the same results as the force-based collision response presented in chapter 3. with a theoretically infinitely high stiffness. The impact damping due to friction and deformation in the contact point can also be considered with the impulse-based collision response. In the following the applied variables will be introduced. Every rigid body is defined by its physical mass and its tensor of inertia. is related to the object coordinate system. In the center of mass it has the linear velocity and the angular speed. is given in the object coordinate system. The collision point is defined by the position vector from the center of mass to the collision point and the normal vector of the collision plane. The sign of the normal vector is directed at the center of mass of object. The collision point is additionally defined by the collision plane and the relative velocity. In Figure 5 the collision problem is shown for two rigid bodies. The impulse is to be directed towards the normal of the collision plane and is normalized to the length :. (4 The linear and angular velocities and shortly before the collision transfer into the linear and angular velocities und shortly after the collision:, (5a, (5b is the transformation matrix from the object coordinate system to the inertial coordinate system. The total linear impulse as well as the total angular momentum is maintained according to equations (5. The only unknown in equations (5 is the parameter.it can be determined by means of energy conservation in the center of masses:. ( describes the absorbed energy i.e. the impact damping during the collision. For no energy is absorbed by the contact area which means the collision is elastic. Inserting the equations (5 in equation ( with leads to For (. (9 the whole kinetic energy of both objects is absorbed in the contact surface. Therefore, so much kinetic energy is converted that the collision objects adhere to each other and thereafter have the same velocity. This means the collision is totally inelastic (plastic. Inserting the equations (5 in equation ( with ( Merging of equation (9 and (0 leads to ( results in The collision behavior can be determined as follows: { s s s s s s s. (0 ( (0 where is the coefficient of restitution. can be compared to an infinite bouncing ball. With a totally inelastic collision ( the collision objects stick together. The most frequent case is the inelastic collision. The contact areas absorb kinetic energy but the bodies still separate again from each other. The parameter is usually determined experimentally or through empirical knowledge. The friction can be calculated as ( ( The friction velocity is then added to the collision velocity. Consequently it is possible to calculate the collision response for the unilateral collision of two rigid bodies analytically without any iteration. With the assumption of infinitesimal collision time, Poisson s hypothesis and an approximated Coulomb friction model can be calculated very time-efficiently. In theory, one can choose any step size with this method without calculating an instable solution. But it has to ensured that a collision between two steps is not overseen by the collision detection system. Based on the assumption of a spatial coherence the objects move only slightly between two step sizes. If further physical effects as gravity are taken into account or parts are stacking, it could happen that the

5 objects get jammed. This can be prevented if penetrating objects are detected in each time step and returned to the object surface. Numerical solution method In order to identify the position and additional physical effects, the equations of motion ( and ( have to be solved numerically. The impulse-based collision response abruptly changes the velocities at certain times. This leads to a discontinuity in the equations of motion. Multi-step methods cannot handle these discontinuities. The colliding objects would seize in the point of collision. This effect also occurs with explicit one-step (Runge Kutta methods with an order higher than, when collision detection is done in the intermediate steps. We apply the semi-implicit Euler method for solving the equation of motion: ( Figure 6: Velocity profile 𝒗 of the moving in -direction secondly. The major advantage of this procedure is the application of well-investigated numerical schemes for onedimensional conservation laws [5] which allows for fast simulation times. VALIDATION (3 Only one function evaluation is necessary and in our problem there is a sufficient area of stability. MACROSCOPIC MODEL OF A MATERIAL FLOW SYSTEM Due to inefficient simulation times for an increasing amount of parts, we started to think about a different model. Certainly, the new approach should capture the right dynamical behavior of the material flow and provide suitable simulation times as well. This can be achieved using a so-called macroscopic model avoiding the individual tracking of parts through the system using averaged quantities as density (parts per length and flux (parts per time. As a first approximation, we propose a two-dimensional hyperbolic partial differential equation (PDE which determines the motion of parts on a conveyor belt in a rather simple way. That means the main ingredients are conservation of mass and an appropriate velocity field. Some similar approaches can be found in [4]. Following the above mentioned ideas, we set up an equation for the evolution of the part density at position and time. For simplicity the velocity field is given by a fixed and smooth vector field describing the moving conveyor belt, see figure 6. Then, mathematically, the flow of material depends obviously on the density. The corresponding PDE which is in fact a conservation law can be stated as ( ] (4a (4b (4c where is given as a user-defined constant (maximum possible number of parts, is the initial distribution of parts and denotes the Heaviside-function which is either or 0. That means, in the first case, if parts do not collide and are transported with velocity. Otherwise, if, the parts are immediately redirected so that the density does not become higher than. Moreover, the flux in (4 can be rewritten as where and velocity field. ] ] ], (5 denote the first and second component of the Numerically, an efficient way to solve the problem (4 is the usage of dimensional splitting. More details can be found in [5]. Here, the two-dimensional equation is split into a sequence of one dimensional scalar equation. More precisely, solving (4 for just one time-step means to solve the equation in -direction firstly and Test rig and simulation environment Both simulation models are useful to simulate material flow problems in D. The collision response of the microscopic model described in section 3.3 can also handle 3D problems. Using convexity-based 3D collision detection in the microscopic model or additional parameters in the macroscopic model, complex geometries can be calculated, too. We validated the model described above based on measured part trajectories on a real conveyor test rig. The test rig is a linear conveyer with a length of more than five meters, a maximum speed of m s which can transport approximately 0000 objects. In the middle of the conveyer a deflector made of aluminum is mounted as an obstacle. As material we use small hollow and cylindrical spacers, made of stainless steel with a mass of and an outer diameter of mm. A high-speed camera records the scene and sends the pictures to Matlab for calculating the object trajectories via image processing. MatFlow is a simulation code for the above mentioned application which is written in Matlab. MatFlow serves primarily for the microscopic and macroscopic model validation. All of the implemented algorithms of the microscopic model are non-iterative and strongly time-deterministic. If the code is run in a real-time environment it is strongly real-time capable. The microscopic model described above has been realized as ½ D. We used cylindrical and cuboid shapes of the moving objects for simplifying collision detection but results are valid for any other object types as well. Experimentation setup We have thoroughly validated our collision calculations in various problem situations, starting with the individual objects on the transport system up to scenarios with several hundred objects. In experiment the simulation results are validated with a transport process with 3 objects on a conveyor belt, in experiment with 0 objects. The conveyed goods are positioned as shown in Figure.a in front of the deflector. The microscopic model is using the exact part positions. With the macroscopic model the parts are distributed in a square with length m and a density of m. The crucial point is the choice of so that it is time invariant and consistent with the test rig. In Figure 6, a schematic view of the velocity field is given. The whole domain representd the conveyor belt divided into two domains: Domain is obstacle-free and parts are transported with the conveyor belt velocity in direction whereas domain prescribes an obstacle that cannot be passed, i.e. for. In summary, this leads to the following vector field :, s (6 (7

6 Figure : Numerical results of our microscopic model where redirected parts move with velocity obstacle. s towards the accumulation. Apart from this effect, the simulated and measured position trajectories are almost identical. The transport speed of the conveyor belt is m s. The experimental setup is shown as a block diagram in Figure 7. In experiment the trajectories of the microscopic model are almost exact. However, there is a time delay in case of the last object. The simulated object glides and rolls faster along the separator than the measured one. This fact is probably caused by the lacking stickslick effect in the friction model. Numerical results The empirically determined parameters for the microscopic and the macroscopic model can be seen in table. In Figure the measured and simulated trajectories are illustrated. The trajectories of the microscopic model indicate the position of an object versus the simulation time. There is a high correlation between the measured and the simulated trajectories. In experiment the times of hitting the obstacle and exiting the obstacle of the first and the last object nearly correspond except for a few milliseconds. Because of the method of the microscopic model the objects penetrate into each other during the Experimentation setup Test rig TestFlow High-speed camera Transient trajectories Statistical data m s The results prove the capability of system dynamic models to be used within a real-time control cycle. The calculated trajectories have a high degree of convergence with the measured trajectories. Time steps of 0-6 s are required with the microscopic model in case of collision with an explicit integration and a force-based approach. Our impulse-based approach in the microscopic model allows for a stable simulation with time step sizes of only 0-3 s. This means that the impulse-based method is nearly 000 times faster than a comparable simulation with a force-based approach. The computation effort for a force-based and an impulse-based collision response is nearly the same per calculation step. First time measurements on a CoreDuo desktop PC with no special chipset have shown that the collision processing system consisting of collision detection, response and time integration needs ms for about 00 objects. For a small number of objects the simulation Experiment (3 objects Simulation setup MatFlow Micro m s Macro (Real-time capable simulation Transient trajectories Statistical data Figure 7: Block diagram of the experimentation and simulation setup M er Ob M er M er ce Experiment (0 objects 𝑒 𝑒 M er Co veyer 𝑒 M er Ob c e 𝜇 M er M er 9 𝜇 M er Co veyer 7 7 𝜇 𝜌 m 𝜌 m 6 69 Δ𝑡 s Table : Parameters of the physically based model

7 y [m] Experiment 3: a3 t = 0 s b3 t = 0.5 s c3 t = s d3 t = s e3 t = 4 s f3 t = 6 s x [m] cycle can be selected approximately like the control cycle of modern machine controls. Compared to Figure, we observe in Figure 9 that the transport and the queuing behavior at the obstacle of the macroscopic model are quite realistic although external forces are prevented so far. Further analysis has to be done in terms of a real-time capable simulation. Note that the macroscopic model (4 is just a phenomenological approach and not a rigorous derivation of the underlying ODE-model described in Section 3. This will be the subject of future considerations. CONCLUSION We have described our impulse-based approach to a microscopic system dynamic simulation and our hyperbolic transportation PDE approach to a macroscopic system dynamic simulation of complex material flow systems. With little adaption of the state of the art models, a high correlation with real conveyor belts has been achieved and the simulation has been run time-deterministically. We have validated the models against a real test rig and have observed quite realistic results. We believe real-time simulation with system dynamic models is ultimately possible and reasonable in terms of a flexible and reconfigurable manufacturing design process. FUTURE WORK Our models open up new opportunities in system dynamic modeling of complex material flow systems. Our future considerations are to design an integrated multi-scale simulation model, which allows for continuous switching between the microscopic and macroscopic model scale. We believe that with a multi-scale approach we are able to simulate large part numbers (in real-time, and additionally can indicate the sensors and actors of manufacturing units. ACKNOWLEDGMENTS The authors would like to thank the German Research Foundation (DFG for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 30/ at the University of Stuttgart. We would also like to thank our reviewers for their concise and helpful comments. REFERENCES Reinhart, G., Wünsch, G. (007, Economic application of virtual commissioning to mechatronic production systems. Production Engineering, Vol., pp , Springer. ElMaraghy, H. A. (005, Flexible and reconfigurable manufacturing systems paradigms. International Journal of Flexible Manufacturing Systems, Vol. 7, No. 4, pp. 6-76, Springer. Koren, Y., Heisel, U., Jovane, F., Moriwaki, T., Pritschow, G., Ulsoy, G. and Van Brussel, H. (999, Reconfigurable Manufacturing Figure 9: Numerical results of our macroscopic model Systems. In: Annals of the CIRP, Vol. 4, No., pp , Elsevier. Jahangirian, M., Eldabi, T., Naseer, A., Stergioulas, L. K., Young, T. (00, Simulation in manufacturing and business: A review. European Journal of Operational Research, 03 (, pp. - 3, Elsevier. Baines, T. S., Harrison, D. K. (999, An opportunity for system dynamics in manufacturing system modeling. Production Planning and Control 0, pp , Taylor & Francis. Lambiase, A., Lambiase, F., Palumbo, F. (007, Effectiveness of Digital Factory for simple repetitive task simulation in mediumsmall enterprices. In: rd International Conference on Changeable, Agile, Reconfigurable and Virtual Production (CARV 007, Toronto. Reinhart, G., Lacour, F.-F. (009, Physically based Virtual Commissioning of Material Flow Intensive Manufacturing Plants. In: Zaeh, M. F.; ElMaraghy, H. A.: 3rd International Conference on Changeable, Agile, Reconfigurable and Virtual Production (CARV 009. pp , Utz, Munich. Cohen, J. D., Lin, M. C., Manocha, D., Ponamgi, M. (995, I-COLLIDE: An interactive and exact collision detection system for large-scale environments. In: Proc. of ACM Interactive 3D Graphics Conference, pp. 9-96, New York. Moore, M., Wilhelms, J. (9, Collision detection and response for computer animation. Computer Graphics, Vol., No. 4, pp. 9-9, ACM. Sekler, P., Verl, A. (009, Real-Time Computation of the System Behaviour of Lightweight Machines. st International Conference on Advances in System Simulation (SIMUL 09, pp , IEEE. Bourg, D. M. (00, Physics for Game Developers, O Reilly & Associates. D. Baraff (993, Issues in computing contact forces for nonpenetrating rigid bodies. Algorithmica, Vol. 0, No. -4, pp. 9-35, Springer. Hoher, S., Röck, S. (0. A Contribution to the Real-time Simulation of Coupled Finite Element Models of Machine Tools - a Numerical Comparison, Simulation Modelling Practice and Theory, Vol. 9, pp , Elsevier. Hughes, R.L. (00, A continuum theory for the flow of pedestrians. Transportation Research Part B 36 (6, pp Leveque, R.J. (00, Finite Volume Methods for Hyperbolic Problems. Cambrigde University Press ρ [/dm ]