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Modeling and Simulating an IndustrialMeasurement Robot
Laboratory Exercises 1 and 2EL1820, Modeling of Dynamical Systems
August 2012
Christian Larsson, Oscar Flardh, Mikael Johansson and Bjorn JohanssonAutomatic Control/School of Electrical Engineering/KTH
Based on material developed by
Thomas Schon, Krister Edstrom, and Jan-Erik StrombergAutomatic Control, Linkopings Universitet
Goal: The goal is to gain experience and insight in modeling and simulating arealistic industrial system. Models of an industrial measurement robot will bedeveloped using Simulink and Dymola.
NB: You need to have a good understanding of the underlying phenomenawhen you develop a mathematical model of a real system. This means thatmodeling will take a lot of time. It is therefore crucial for you tothoroughlypreparebefore the laboratory exercises! Do the preparation tasks before thelaboratory session.
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Contents
1 Introduction 3
2 System Desription 42.1 Servo Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Toothed Belt Drive . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Lead Screw Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Robot Arm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Current Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Laboratory Exercise 1 Modeling the Servo Motor 73.1 Purpose and Content . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Preparatory Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 Laboratory Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Laboratory Exercise 2 Modeling of the Complete Robot 94.1 Purpose and Content . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Modeling Tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.3 Preparatory Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 94.4 Laboratory Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 10
A Tables 11
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1 Introduction
We will develop a model of an industrial measurement robot and verify thatthe model behaves as expected in simulations. The robot under considerationis shown in Figure 1.
Figure 1: Two pictures of the robot.
The exercise is divided into two four-hour sessions. In the first session, we
will model the servo controlling the robot arm. We will develop a model basedon a block diagram, which can be simulated in Simulink, and we will developan object oriented model, which can be simulated in Dymola. Both Simulinkand Dymola are described in the appendix. In the second session, the modelwill be extended to a more complete model of the complete robot arm. Thisextension is only done in Dymola.
There are preparation tasks to be donebeforeboth sessions. It is very im-portant that you have carefully done the preparation tasks, otherwise there willnot be time to complete the exercise.
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2 System Desription
An intuitive picture of the system is shown in Figure 2: an external voltagesource serves as reference signal for a current regulator, which drives a servomotor. The servo motor is connected to transmissions, a toothed belt drive anda lead screw drive, that change the vertical position of the robot arm.
Figure 2: Explanatory sketch of the robot arm.
An alternative picture of the subsystems and how they interact is shown in
Figure 3. More detailed descriptions of the subsystems follow below.
Figure 3: The main components in the system and their interaction. The inter-action quantities are expressed in SI-units.
2.1 Servo Motor
The servo is a simple DC motor, see Figures 4 and 5.
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Figure 4: The electrical subsystem.
Figure 5: The motor.
The electrical data and mechanical data for the DC motor are given in Fig-ure 9. The motor under consideration is M-586-0585.
In Figure 9, there is a diagram with two curves corresponding to M-586-0585. The upper curve specifies three constraints on the motor: maximumnumber of revolutions, maximum torque, and maximum power (try to find outhow!). The lower curve specifies the relationship between torque and angular
velocity when the current to the motor is kept constant. This current is calledcontinuous stall currentand it can be read to be3.9A.In the section winding specificationsin Figure 9, there are two resistors and
one inductor. The reason for this is that the winding is not an ideal inductor.The winding is both inductive and resistive. In Figure 5, RI corresponds tothe armature resistance, Rr +RI corresponds to the terminal resistance, and IIcorresponds to thearmature inductance.
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In reality, the servo is connected to a tachometer, see Figure 4. The tachome-
ter makes it possible to use the angular velocity of the servo in the feedbackcontroller. However, since we are not going to do controller design, we neglectthe tachometer.
2.2 Toothed Belt Drive
The servo motor is connected to a toothed pulley with an outer diameter of20mm and a thickness of10 mm. The toothed pulley drives another toothedpulley, with an outer diameter of80 mm and a thickness of15 mm, using atoothed belt. The toothed belt has a length of750 mm, and it is elastic, whichmakes it 0.4% longer at full load (200 N). Also assume that the belt is tightened.The pulleys are made of aluminum, with a density of2.7 103 kg/m3. Theformulas for the moment of inertia for the pulleys can be found in, e.g., the
Physics Handbook or a book in mechanics.There is energy loss at both pulleys. The smaller pulley has a friction of
2105 Nms/rad and the larger pulley pulley has a friction of5105 Nms/rad.
2.3 Lead Screw Drive
The larger pulley in the belt drive is connected to a lead screw drive with apitch of1inch per revolution. Data for the lead screw drive (the serial numberis B-8000) is given in Figure 7. The nut in the lead screw drive is connected tothe robot arm through a spring with the spring constant 75 kN/m. The totallength of the screw is 1 m (note that the moment of inertia in Figure 7 is perlength unit). The friction in the screw can be neglected.
2.4 Robot Arm
The lead screw drive changes the vertical position of the robot arm. The massof the robot arm is 5.5kg and the friction is small,25Ns/m.
2.5 Current Regulator
A reference signal is given to the current regulator through the port (t1, t2).The current regulator supplies electrical current through the port(t3, t4). Theoperational amplifier Q2 in Figure 6 is a power amplifier, which can give upto 10V. The operational amplifierQ3amplifies the voltage across the currentmeter shuntR9, and it gives an output which is proportional to the current inthe motor. By using this signal for feedback, the current can be regulated.
For simplicity, we will use a current source instead of the current regulatorin the laboratory exercises. However, note that if you are using an ideal currentsource, then there cannot be any inductors in the circuit.
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Figure 6: Circuit diagram for the current regulator.
3 Laboratory Exercise 1 Modeling the Servo Motor
3.1 Purpose and Content
We will model the servo motor that drives the robot arm. The models will bedeveloped in Simulink and Dymola to illustrate the differences between objectoriented models and block diagram models.
3.2 Preparatory Exercises
1. Derive a state space model of the servo motor. Assume that the motor isdriven by a voltage source instead of a current source.
2. Use the motor specifications in the Appendix to find numerical values ofthe model parameters. Note that the mechanical damping of the motorcan be found by looking at the slope of an appropriate curve in Figure 9.
3. Translate your state space model to a block diagram, based on integratorelements, which can be implemented in Simulink.
4. If you have not used Simulink before you should briefly read through theGetting started guide (http://www.mathworks.com/help/pdf_doc/simulink/sl_gs.pdf). Especially chapter 2 and 3 might be useful.
5. Try to figure out what the object oriented model should look like. Lookat the available standard components in Modelica, which you can find athttp://www.modelica.org/ModelicaLibrariesOverview .
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6. If you have not used Dymola before it you should briefly read through
the Getting started guide (http://www.3ds.com/fileadmin/PRODUCTS/
CATIA/DYMOLA/PDF/Getting-Started.pdf ).
3.3 Laboratory Exercises
1. Implement your models in Simulink and Modelica.
2. The time constants of the motor are given in Figure 9. They can be com-puted using the parameters from the preparatory exercises and they canalso be computed using simulation. Do they match.
3. Compare the results from the two simulation programs (Simulink andDymola). Do they match? Why or why not?
4. Discuss the differences between object oriented and block diagram basedmodeling. What are the benefits and what are the drawbacks? Which onedo you prefer in this case? Why?
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4 Laboratory Exercise 2 Modeling of the Complete
Robot
4.1 Purpose and Content
This time, we will model the complete robot in Dymola.
4.2 Modeling Tips
We start with giving some general tips on how to create and test models ofcomplex systems in a structured way.
First, divide the model into smaller modules and try to develop an intu-ition on how these smaller modules work.
Make a list of the properties that are most interesting from a modelingpoint of view. Start with a very simple model, which incorporates theessential properties. For example, if you model a gear that mesh withanother smaller gear, the most important property is that they scale thetorque and angular velocity. Then, for an improved model, you can alsoconsider the friction between the meshing teeth and the elasticity of thematerials.
Test the smaller modules first, before you put them together. This willmake debugging easier.
When the simple model is extended with additional dynamics, it is ad-
visable to add the slowest dynamics first.If one of the gears in the example above is connected to a long rod ofaluminum, then the rod has probably slower dynamics than the gears(if you turn the rod by applying torque to the end that is not connectedto the gear, it will take longer time for the motion to propagate throughthe rod, compared to the time it takes for the second gear to rotate whenthe first gear is rotated). In this case, it is more important to model thedynamics of the rod than it is to model the dynamics of the meshing ofthe gears.
4.3 Preparatory Exercises
1. Modify your servo motor model from the first session to be driven by acurrent source instead of a voltage source.
2. Derive a bond graph for the components in Figure 3. Comment on yourassumptions and derive numerical values for the parameters needed.
3. Reflect on how a complete robot model can be structured in Modelica. Asa first choice, use the components in Modelicas standard library. We will
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use the libraries Electrical (Basic and Sources), Mechanical (Rotational
and Translational), and Blocks (Sources). With some effort, you shouldbe able to model the system using only these standard components.
4.4 Laboratory Exercises
1. Show your bond graph to the teaching assistant, and motivate the as-sumptions you made. For example, we assumed that the voltages in theregulator are within the bounds given by the supply voltages.
2. The actual system will oscillate quite a bit. Can you see this in the simu-lations? What is the cause of the oscillations?
3. Is the system sensitive to variations in the friction coefficients? Which
system properties change when the friction coefficients are varied?4. Is it possible to reach the maximum velocity of500 mm/s? Keep your
answer to question 1 in mind. If it is not possible to reach the maximumvelocity, suggest some measure to solve this problem.
5. Why is the current regulator needed?
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APPENDIX
A Tables
Figure 7: Mechanical properties of the lead screw drive.
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Figure 8: The servo motor.
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Figure 9: Properties of the servo motor.
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