Electric Motor Controlled Joint Simulator

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Master of Science Thesis in Electrical Engineering

Department of Electrical Engineering, Linköping University, 2016

Electric Motor Controlled

Joint Simulator

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Master of Science Thesis in Electrical Engineering

Electric Motor Controlled Joint Simulator

Sofie Griph

LiTH-ISY-EX--16/4976--SE Supervisor: Erik Hedberg

isy, Linköpings universitet

Johan Nåsell

Atlas Copco

Staffan Molinder

Atlas Copco

Examiner: Johan Löfberg

isy_{, Linköpings universitet}

Division of Automatic Control Department of Electrical Engineering

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Sammanfattning

Åtdragningssystem används inom flera olika branscher, till exempel för monte-ring inom fordonsindustrin och vid hopsättning av hårdvara i datorer. Både fö-retaget som gör verktygen och kunderna måste veta prestandan hos åtdragnings-verktyget för att kunna försäkra sig om att skruvförbandet har dragits åt som önskat. Det kan göras med hjälp av ett testförband.

Höga krav på säkerhet så väl som monteringshastighet ger höga krav på test-utrustningen. Problemet med de existerande testförbanden är att de är svåra att göra repeterbara tester på. De vanligaste testförbanden består av mekaniska eller hydrauliska system. De mekaniska systemen har problem med förslitningar på skruvar, skillnad på mängd smörjmedel med mera, medan de hydrauliska ibland är för långsamma.

Detta examensarbete är i en studie kring om det är möjligt att göra ett test-skruvförband med hjälp av en elmotor. Testförbandet ska skapa reperterbarhet i testerna. Motorn tillsammans med en regulator ska simulera ett skruvförband så att verktyget uppfattar det som ett verkligt förband. All undersökning har ge-nomförts genom systemmodellering och simulering i MATLAB. Fyra olika regler-strukturer har evaluerats: en PID-regulator, en kombinerad regulator som använ-der sig av framkoppling från referens samt störning, en som baseras på den andra regulatorn men som har en extra inre strömåterkoppling och en LQ-regulator.

Slutsatsen är att det är möjligt att göra ett testskruvförband genom att använ-da en elmotor och att LQ-regleringen är den som lämpar sig bäst för det. För att styrka resultatet finns det några aspekter som skulle kunna utredas mer i detalj. En är att lägga till en verktygsmodell som innehåller verktygets dynamik, nu an-vänds verktygets referens. En annan är att implementera lösningen på hårdvara.

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Abstract

Tightening systems are used in several industries, including assembly in the auto-motive industry and installation of computer hardware. Both the company mak-ing the tools and the customers need to know the performance of the tightenmak-ing system to ensure that the screw joints tighten as desired. This can be done using a test joint system.

High demands on safety as well as fast assembly speeds, puts high demands on the test equipment. The problem with the existing test joints is that they are hard to do repeatable tests on. The most common test joints are constructed us-ing mechanic or hydraulic systems. The mechanical systems have problems with wear of screws, changes in lubrication etc., while the hydraulic systems some-times are too slow.

This master’s thesis is a study of whether it would be possible to construct a test joint using an electric motor. The electric motor together with a controller should simulate a screw joint so that the tool would perceive it as a real one. All investigation has been performed by system modeling and simulations in MATLAB. Four different control structures have been evaluated: a PID controller, one combined controller which uses feedforward from reference as well as distur-bance, one which is based on the same structure as the second but with an added inner current loop and the last one is an LQ controller.

The conclusion is that it is possible to make a test joint using an electric motor and that the LQ controller seems to be the best choice. To prove the result, a few more aspects could be investigated more closely. One is to add a dynamic model of the tool, now only the reference to the tool is used. Another is to implement it on hardware.

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Acknowledgments

First of all I would like to thank Atlas Copco for giving me the opportunity to work with this master’s thesis. A special thanks to my supervisors Johan Nåsell and Staffan Molinder for all your support and knowledge about tightening sys-tems. I would also like to thank Maria Södergren, who is the manager of the tightening technique R&D team at Atlas Copco.

A special thanks to my supervisor Erik Hedberg and examiner Johan Löfberg at ISY, Linköping University, for all fruitful discussions and support during my work. I would also like to thank Martin Enqvist for giving me some of your time and the idea of one of my control structures.

Last but not least, I would like to thank my family and friends. A special thanks to my love Michael Andersson who has inspired and supported me through everything, you are the best.

Linköping, May 2016 Sofie Griph

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Notation

Nomenclatures Nomenclature Meaning β Filter parameter c Prevailing torque cn Constant, n = 1, 2, 3 e Control error em Back EMF F Force h(ω) Motor friction i Current Jm Motor inertia Js Screw inertia k Slope

km Motor proportionality constant

km,f Viscous friction constant

ks Spring constant

kstif f Stiffness constant

K Proportionality constant

Lm Motor inductance

m Constant

µ Friction constant

nr Number of revolutions

ps Pitch of the screw threads

r Reference Rm Motor resistance Rn Covariance matrix, n = 1, 2 t Time T Time constant T (t) Torque

T (θ) Torque dependent on angle (screw)

Td Derivative time

Tf inal Final target torque

Tf irst First target torque

Ti Integration time

Tm Motor torque

Tsi Moment of inertia (screw)

u Input

v Motor voltage

v1 System disturbance

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Notation xiii

Nomenclature Meaning

ω Motor angular speed

ωf irst First speed

ωr Reference tool angular speed

ωrundown Rundown speed

ωsecond Second speed

ωt Tool angular speed

y Output

θf inal Angle at final target torque

θf irst Angle at first target torque

θmax Maximum angle

θmin Minimum angle

θr Reference angle

θsnug Snug angle

θt Tool angle

∆θr Angle difference reference

Abbrevations

Abbrevation Meaning

dc _{Direct Current}

pid _{Proportional-Integral-Derivative}

mpc _{Model Predictive Control}

lq Linear Quadratic

ff Feed-Forward

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1

Introduction

The master’s thesis has been carried out at Atlas Copco Industrial Technique AB, Stockholm, and the division of Automatic Control, the department of Electrical Engineering at Linköping University. This chapter will describe the background, the problem formulation and discuss related work in the area.

1.1 Background

Atlas Copco is a global industrial company with headquarters in Nacka, Sweden. The company was founded in 1873 and started their business making railway cars. Today, the product focus has changed and they are now manufacturing in-dustrial tools and equipment. They have a broad spectra of products such as air compressors, assembly systems and rock drills. The department Industrial Tech-nique develops handheld electric and pneumatic assembly systems, e.g. nutrun-ners and screwdrivers, which are often called tightening systems. They develop both software and hardware solutions for their products, including automatic control design.

Tightening systems are used in several areas, including the assembly in the automotive industry, installation of wind turbines and installation of computer hardware. Both the company making the tools and the customers need to know the performance of the tightening system to ensure that the joints tighten as de-sired and this can be done using a test joint. A test joint can be designed in different ways and the complexity varies a lot. Most of them are mechanical sys-tems, either breaking systems or fixed. The fixed systems can be constructed by spring washers and by choosing different washers the test joint can represent dif-ferent screw joints. The problem with the mechanical test joints are the difficulty of doing identical tests. Wear of screws, changed amount of lubrication, move-ments between washers etc. affect the outcome. Other test joints are made using

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2 1 Introduction

hydraulics or pneumatics. The hydraulic systems are the most common among the customers of Atlas Copco. These systems are large and sometimes too slow at simulating screw joints. Because of the disadvantages of the existing test joints the idea of this thesis is to investigate if it is possible to use an electric motor. To be able to tell if it is possible or not, modeling, control and simulations of the test joint have to be performed.

1.2 Problem Formulation

The ability to tell how well performing a tightening tool is and to do repeatable tests are desirable both for customers and manufacturers. A fast and exact test system can save a lot of time and by that facilitate research and development of new tools.

The goal of the thesis is to determine whether it is possible or not to design a test joint using an electric motor. The purpose of the test joint, also called joint simulator, is to emulate the behavior of a screw joint during rundown and tightening. The output of the simulator is the simulated screw joint torque and it should be equivalent to what a tool would feel if it would tighten a real joint. The modeling and simulations are based on a simple motor model and the motor data is based on a motor type that is expected to be used for this purpose. All simulation and validation is performed in MATLAB using Simulink or Simscape.

The main challenges of this work are

• Modeling: In order to do simulations a model of the system is needed. This includes the electric motor, a stiffness between the tool and the motor, the tool, the desired screw joint torque function (the expected output of the test system) and the reference angular speed (the input to the tool).

• Control: A controller is necessary to make the test system act as a real screw joint. Different control strategies are evaluated to find the most suitable design.

• Large motor inertia: The motor inertia is larger than the screw inertia and therefore it can be problematical simulating a real screw joint in the transi-tion between standstill and rotatransi-tion.

• Test evaluation: Since there is no data describing the screw joint torque or the reference angular speed, both of them must be modeled and they are also used to evaluate the system. Therefore, it is important that they represent reality well. Another issue is to determine what it would take to make the test joint work if it does not work with the modeled motor. Is it possible to find an electric motor that could behave desirable?

To be able to reach the goal of the thesis the problem is divided into smaller subproblems and a primary and a secondary goal are formulated.

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1.3 Requirements on the Controlled System 3

1.2.1 Subproblems

The resulting subproblems are

• Evaluating the motor performance: The electric motor must be sufficiently fast in order to be used for this purpose. Therefore, the settling time is measured when a step in reference torque is taken.

• Evaluating the test joint using reference angular speed: The test joint is controlled and evaluated without a tool model affecting the reference angular speed (the input and reference to the tool) and the system. Hence, the tool is assumed to give the reference angular speed as output.

• Evaluating the test joint using a tool model: The test joint is controlled and evaluated with a tool model affecting the system. Now, the controller in the tool and the test joint are affecting each other.

1.2.2 Goals

The primary goal concerns the first two subproblems. Thus, evaluate the mo-tor performance and evaluate the test joint using the reference angular speed as output from the tool. The secondary goal is to model the tool and evaluate the test joint using the output from the tool model. The secondary goal will not be treated in this study but can be seen as a future development.

1.3 Requirements on the Controlled System

In order to evaluate the performance of the different control strategies and tell if the result is good enough some requirements are stated on the controlled system. The test joint should satisfy the following criteria

• At rotation the torque output, which corresponds to the torque at the motor shaft, should at most differ 1% of the final target torque from the ideal torque given by the modeled screw joint torque function. This should be valid after snug. (See section 2.2 for the definition of the final target torque and snug.)

• At standstill the screw position should not differ more than 2 degrees from the rotation angle of the screw representing standstill.

These requirements implies high demands on reference tracking and fast re-sponse.

1.4 Limitations and Delimitations

The electric motor used in this thesis has hardware limitations that will affect the performance of the controlled system. The input voltage is limited to the interval [−300, 300] V , which corresponds to a limited control signal.

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4 1 Introduction

Another limitation on this master’s thesis is that the work only stretches over a limited period of time. Because of this, the investigation will only consider a limited amount of control designs and there might also remain some improve-ment opportunities on the tuning of these. Hence, there might exist a better set of control parameters that will give a slightly better result.

Since the purpose of the study is to see if it is possible to design a test joint using an electric motor, the investigation is performed in continuous time even though an eventual future development might be to try out the results of this master’s thesis on hardware. The justification to this is that if it is not possible to make the control work in continuous time it will not be possible in discrete time. The continuous time implementation does also speed up the simulations and facilitate the tuning and evaluation of the control designs.

1.5 Related Research

Electric motors are used in various industrial applications, e.g. in Electro-Hydraulic systems as in André [2014], as actuators in vehicle applications which is eval-uated in Pettersson and Storm [2012] and in tightening systems described in Lagerholm and Molinder [2014]. A lot of research has been made in the field of controlling motor speed or torque. The fundamentals of electric motor drives and how to control them are described in Krishnan [2001]. In order to make a joint simulator some areas need to be covered. An overview of related work in these areas will be presented here.

1.5.1 Modeling

How to model a one-phase direct current (DC) motor is described in Glad and Ljung [2004] and Krishnan [2001]. This type of electric motor will be used in the simulated test joint modeled and evaluated in this study.

In order to model the screw joint torque function, the output of the test joint, the friction in the screw joint has to be modeled. Different friction models are discussed in Olsson et al. [1998] and typical friction modeling for screw joints is described by Swedish Fasteners Network.

1.5.2 Control

To control an electric motor drive cascade control is often used. The most com-mon control strategy is to use a Proportional-Integral (PI) controller in the in-ner and outer loop, as for instance in Krishnan [2001] and Pettersson and Storm [2012]. The inner loop is current control and the outer loop can be speed, torque or position control. It is also common to use more than two controllers in cascade. In Chen et al. [2008] force feedback is used in order to simulate the aerodynamic force acting on the rudder of an aircraft. The control design consists of three PID (Proportional-Integral-Derivative) controllers: one controlling the armature current, one the speed of the DC motor and one the force. A feedforward is also

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1.6 Outline 5

added to restrain the coupling action between the Electric Load Simulator (ELS) and the rudder.

In Shabani et al. [2015] fuzzy control is mentioned as another possible control strategy. This design use control laws of the type "If angular speed is low the current should be high". The basics of fuzzy control are discussed in Enqvist et al. [2014]. The control can be compared to what an operator would do in the same situation. Therefore an extensive knowledge about the system is needed in order to make a well performing control. Hence it will not be evaluated in this thesis.

Because of undesirable effects such as cost, volume and reliability issues sen-sorless control of electric motors has been studied a lot. A method that uses feedforward, an adaptive observer and fuzzy logic to control motor speed is eval-uated in Thanh et al. [2014]. This method is compared with a design that uses a traditional PI controller instead of a fuzzy logic controller, described in Yachi-angkam et al. [2004]. In this thesis, sensors measuring the motor output are used and therefore these types of control designs will not be considered.

Another possible approach is to make a controller that utilizes the model of the system. Model Predictive Control (MPC) is a general model based control strategy that is suitable to use if there are constraints on the system, e.g. limited control signals or safety limits on the output, see Enqvist et al. [2014]. The control design uses the fact that the system model enables prediction of the controlled quantities as a function of possible control signals, which is mentioned in Glad and Ljung [2003]. This results in an optimization problem. Thus, the MPC con-troller will perform well if the model describes the system well. Linear Quadratic (LQ) control is also a model based controller. LQ minimizes a quadratic criteria in order to choose an optimal control signal, see Glad and Ljung [2003]. MPC is actually a general and approximated method to solve LQ problems with con-straints. A disadvantage with the MPC is that it requires high computational power. This can be problematical if the system has fast dynamics or limited com-putational capability, see Enqvist et al. [2014]. Both the LQ controller and MPC are tested on a Electro-Hydraulic system, which has some similarities with the test joint system, in André [2014]. In simulations, both of them perform well but the MPC controller is not successfully implemented on the real system.

1.6 Outline

The thesis includes the following chapters:

Chapter 1, Introduction: An introduction to the work and the background of the master’s thesis.

Chapter 2, System Description: A description of the system.

Chapter 3, General Theory on Modeling: A presentation of related theory on modeling.

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6 1 Introduction

Chapter 4, Modeling: Modeling of the system and a presentation of the re-sulting state-space model.

Chapter 5, Control: A discussion of the control problem and a presentation of the evaluated control strategies.

Chapter 6, Simulation: A comparison between Simscape and Simulink and a presentation of the simulation models.

Chapter 7, Results: A presentation of the results of the master’s thesis. Chapter 8, Conclusions: The conclusions of this work and suggestions for further development.

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2

System Description

This chapter describes the involved systems and signals needed in order to un-derstand the issues of this thesis. A lot of the information is provided by Atlas Copco and is based on the knowledge of making tightening tools and test joints. The fundamentals of screw joints are described in the handbook Swedish Fasten-ers Network.

2.1 Joint

The test joint simulated in this study is a screw joint, which is a type of threaded fastener. The screw joint consists of a screw, joint members to be clamped to-gether and sometimes a nut. The screw will clamp the parts toto-gether with the nut or directly to a threaded hole in one of the joint members.

The clamping force is the traction acting on the screw strain. It can also be defined as the axial force holding two or more parts together. The force can be described as a linear function of both the rotation angle of the screw and the pitch of the thread, explained in Section 4. The value of the clamping force can be seen as a measurement of how hard the joint is tightened. Unfortunately, there is no practical way to measure the force and therefore it is measured indirectly via the tightening torque. The torque can be measured directly and within the elastic range of the screw elongation there is a direct relation between the torque and the clamping force. More information about how the force and torque are related to each other can be found in Section 4. Of all the torque applied to a joint only 10% is transferred into tightening torque. The rest is consumed by friction in the threads or under the screw head. An illustration of a screw joint and the clamping force is shown in Figure 2.1.

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8 2 System Description

(a)Screw joint (b)Bolted joint

Figure 2.1:A screw joint and a bolted joint The clamping force is illustrated with arrows in (a). How a nut can be used to tighten the joint members is shown in (b). Source: Yuri Raysper

2.2 Tightening Process

There are several tightening processes that can be used by a tightening tool. The choice of tightening process depends on the desired quality, speed and ergonomic behavior. For example, it is extremely important to ensure that a safety belt is fas-tened sufficiently hard and correct. On the other hand, it is not equally important to ensure the safety of a screw connecting the rear-view mirror to the car roof, but it might be a higher demand on assembly speed. This is why different tightening strategies are used for different applications.

The tightening strategy in focus in this thesis is the two-stage model, illus-trated in Figure 2.2. In the beginning of the tightening process the tool ramps up to a reference angular speed called rundown speed. At a point called snug level, the point at which the other components of the joint and the screw head be-come tight, the tool will decrease the speed to a constant level called first speed and hold it until the first target torque is reached. After snug, the parts of the joint become more tightened and this part is called tightening. The torque will increase until it reaches the first target. When the tool register first target torque, the speed will drop to zero. After the speed has become zero the torque decreases fast against zero too, this results in a higher torque output than the first target torque. The tool speed will be zero until the next stage should begin. This stage follows nearly the same procedure as the first one. The differences between the second stage and the first are that the reference angular speed (now called sec-ond speed) will be lower, the torque will be higher and after final target torque is reached the speed will drop almost directly towards zero. Thus, there will not be another constant speed between second speed and zero.

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2.3 Tightening Tool 9 t ωr(t) T (t) rundown speed first speed second speed rundown tightening snug first target final target

first target torque final target torque

Figure 2.2:Two-stage tightening process. The red line divides the tightening process into the two parts: rundown and tightening. The dotted gray lines mark first and final target and the gray, thin lines mark the zero crossings of the angular speed, ωr. Note that the torque T starts to decrease after the tool

speed has become zero.

joints set after tightening, which means that after a short time the clamping force is less than it was when the tightening stopped. The pause make sure that the joint has settled before the final target torque is reached and the total tightening process is over. The time it takes for a joint to set is less than 30 milliseconds.

2.3 Tightening Tool

The joint simulator should act as a test system for a tightening tool. The tool that is modeled is an electric angle nutrunner called Tensor ETV STR61-50-10. It is a 50 Nm tool meaning that the final target torque is 50 Nm. The nutrunner is a high speed tool designed with productivity and low weight in mind. Figure 2.3 shows a Tensor ETV ST61-50-10 that looks basically the same as the one used in this thesis.

The tightening system also includes a control unit called controller and a ca-ble connecting the tool with the controller, see Figure 2.4. In this study all three parts is referred as the tool.

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10 2 System Description

Figure 2.3:Electric angle nutrunner, ETV ST61-50-10.

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2.4 System Overview 11

2.4 System Overview

An overview of the simulated system without the controller is shown in Fig-ure 2.5. The electric motor is controlled by the input voltage v and generates a rotation with angular speed ω. The angular speed ωr is the input (reference)

to the tightening tool and the tool is also generating a rotation but with angular speed ωt. A stiffness is connecting the tool with the electric motor. The torque

acting on the tool will be the same as the torque at the motor shaft Tm.

The stiffness together with the electric motor is called the test system. The idea of the joint simulator is that the tool will cause a rotation through the angular speed ωt and the test joint will simulate the screw joint torque by controlling

the output Tm of the test system. The inputs to the test system are the voltage

controlling the electric motor and the angular speed generated by the tool. The measured signals are the current flowing in the electric circuit and the angle, angular speed and torque at the motor shaft.

Figure 2.6 depicts the included parts of the test joint.

Electric

Motor Stiffness Tool

Tm, ω Tm, ωt

ωr

v

Test System

Figure 2.5:Overview of the simulated system without the controller.

Test Joint

Test System Electric Motor Stiffness

Controller

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3

General Theory on Modeling

The modeling performed in the master’s thesis is based on general theory in the research area modeling and simulation, discussed in Glad and Ljung [2004]. This chapter presents theory on modeling and the notation utilized in the report.

3.1 Modeling

Real systems can often be described by mathematical models defining relations between the quantities included in the system. One way of modeling systems is to divide the system into smaller subsystems with known behavior. Knowledge like natural laws and literature in the area can be the foundation of the mathematical equations describing the subsystems. This type of modeling is called physical modeling. Another way of constructing a system model is to observe the system and conduct experiments on it, this is called system identification. System iden-tification is commonly used as a complement to a physical model. The natural laws used in the physical models are actually all mathematical models with their origin in observations of reality. By using both types of modeling it is more cer-tain that the resulting mathematical model really describes the real system. More information about modeling can be found in Glad and Ljung [2004].

3.2 Differential Equations

The relation between the input u and the output y of a system can in most cases, both linear and nonlinear, be described exactly or approximately by a linear dif-ferential equation

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14 3 General Theory on Modeling dn dtny(t) + a1 dn−1 dtn−1y(t) + · · · + an−1 d dty(t) + any(t) = = b0 dm dtmu(t) + · · · + bm−1 d dtu(t) + bmu(t) (3.1)

where a1, . . . , anand b0, . . . , bmare constant coefficients. A simplified notation

for derivatives is ˙y(t) = d dty(t), y(t) =¨ d2 dt2y(t), · · · (3.2)

3.3 Transfer Functions

The differential equation describing the system can also be written as a transfer function. In continuous time the derivation operator p is representing a deriva-tive according to pu(t) = ˙u(t).

Using the operator an arbitrary differential equation can be rewritten as y(t) = G(p)u(t), G(p) = b0p m_{+ · · · + b} m pn_{+ a} 1pn−1+ · · · + an (3.3)

3.4 State-Space Representation

A dynamic system is in general characterized by a number of variables that change their value with time. The output y(t) of the system is therefore not only depen-dent on the input u(t) at the current time t but also at time instants before it. Mathematically, a dynamic system is distinguished by the presence of derivatives of variables with respect to time. The information about the system at the current time t is called the state of the system at time t and the number of states differ dependent on the system model. The states make it possible to predict the effect on the output of a certain choice of input, see Glad and Ljung [2006], and this is very useful in automatic control. A time continuous system can be described as equations of states by representing the system equations as first order differential equations. The result is a state-space model

˙x(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t)

(3.4a) (3.4b) where x represents the state vector.

An advantage with the state-space representation is that it is easy to handle multivariable systems with several inputs and outputs. Another is that physical modeling generally results in a state-space model.

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4

Modeling

To be able to simulate the total system and design model based control strategies, models for all subsystems are needed. The models included in the test system are all physical models. In order to perform simulations and evaluate the system the screw joint torque and the reference angular speed are both necessary. These are modeled as mathematical functions based on physics and information about the tightening process.

4.1 Electric Motor

If the test joint simulated in this thesis would be implemented on hardware it would probably be some sort of direct current three-phase motor. Since the focus of this thesis is to investigate if it is possible to construct a test joint using an electric motor, a simplified motor model is used. The model chosen is a one-phase DC motor, illustrated in Figure 4.1.

v i Rm Lm em ω, θ, Tm Jm Figure 4.1:DC motor

The external voltage v, acting as input, results in a current i and an angular speed ω. The armature circuit consists of a resistance Rmand an inductance Lm.

Some of the electrical energy fed into the system will get lost in the resistor and

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16 4 Modeling Se s I:Lm R:Rm GY km s I:Jm R:h(ω) Se v i em i T ω Tm ω

Figure 4.2:Bond graph of the DC motor

stored in the coil. If there is no energy loss, the rest of the electrical energy will be transformed into mechanical energy.

The current i generates a torque proportional to the current. At the same time a back EMF proportional to the angular speed is generated by the rotation. This represents the energy conversion and can be seen in Equation 4.1.

T = kmi

em= kmω

(4.1a) (4.1b) The rotating parts of the electric motor has an inertia Jm and there is also

friction. These two components will decrease the amount of the mechanical en-ergy representing the outputs. The outputs are the angular speed ω, angle θ and torque Tmof the load. The outputs belong to the outgoing shaft of the motor.

The bond graph in Figure 4.2 shows an overview of the equations, 4.2-4.3, de-scribing the electric motor. The arrows mark the energy flow in the system. Both the electrical and the mechanical part can be described by a s-point, meaning that the flow is constant but the efforts can differ, see Glad and Ljung [2004]. The ef-fort and flow are only illustrated at the input and output of the system as well as at the gyrator, GY. The gyrator represents the conversion between electrical and mechanical energy corresponding to Equation 4.1. The function h(ω) describes the friction.

The electrical part of the motor is described by Lm

di

dt = v − Rmi − kmω (4.2)

where Lmis the motor inductance, i the armature current, v the external

volt-age, Rm the motor resistance, km represents the back emf and torque constant

(here assumed to be the same because of no energy loss) and ω is the angular speed at the outgoing shaft of the electric motor.

The mechanical part is described by Jm dω dt = kmi − h(ω) − Tm h(ω) = km,fω (4.3a) (4.3b)

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4.2 Spring Stiffness 17

where Jm is the motor inertia. The meaning of ω, km and i are the same as

above, km,f is the viscous friction constant and Tmis the motor torque. The

fric-tion is modeled as a viscous fricfric-tion because it is a common way of modeling the friction in a DC motor. See for example Krishnan [2001].

4.2 Spring Stiffness

Between the electric motor and the tool there is a spring stiffness added in the simulation model. This is done to facilitate for the controller and because there will always be some stiffness between the tool and joint in reality. The stiffness is modeled as a torsion spring with the stiffness constant kstif f according to

Fig-ure 4.3. motor axis kstif f Tool Tm, ωt, θt Tm, ω, θ

Figure 4.3:Spring stiffness between motor and tool

The energy flow between the tool and motor using a spring stiffness can be described by the bond graph in Figure 4.4. Equation 4.4 describes the torque as a function of rotation angle. The angular speed wtrepresents the tool output, ω

the angular speed generated by the electric motor and the integral corresponds to the rotation angle of the spring.

Tm= kstif f t Z 0 (ω − ωt)(s) ds = kstif f · (θ − θt) = kstif f · ∆θ (4.4) Se p C:kstif f Sf Tm ω Tm ωt

Figure 4.4:Bond graph of the spring stiffness between the electric motor and the tightening tool. Only the effort and flows of interest are marked.

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18 4 Modeling

Figure 4.5:A joint can be modeled as a system of connected springs in order to derive the clamping force acting on the screw strain. Source: Swedish Fasteners Network

4.3 Screw Joint Torque

The screw joint torque function simulated by the test system consists of two parts. The first one corresponds to the screw inertia, Tsi, and the other one is the torque

needed to overcome the friction and to tighten the joint, T (θ). The total torque encountered by the tool is given by

Tm= Tsi + T (θ) (4.5)

where Tsi = Jsdωdt, Jsis the screw inertia, ω the angular speed of the screw and

T (θ) is derived below.

A screw joint can be modeled as a system of connected springs, see Figure 4.5 and Swedish Fasteners Network. According to this approach the clamping force, the traction acting on the screw strain, can be derived as

F(θ) = psksθ (4.6)

where psis the pitch of the screw, ksis the spring constant and θ is the rotation

angle of the screw. To be able to tighten a screw joint as desired a certain amount of torque is needed. The tightening torque needs to overcome the friction and the rest represents the clamping force. A simplified model of the tightening torque is

T (F) = µF (4.7)

where F is the clamping force and µ is a simplified friction constant corre-sponding to the friction under the screw head as well as the friction acting on the threads. By combining Equation 4.6 and 4.7, the tightening torque could be derived as a function of the rotation angle of the screw

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4.3 Screw Joint Torque 19 or T (θ) = kθ k = µpsks (4.9a) (4.9b) The constants µ, psand ksare dependent on which screw joint that is going to

be simulated. The model described above corresponds to the tightening part of the tightening process. A more advanced model and the basis of the model above is described by Swedish Fasteners Network.

Another simplification made in the model is that k will be considered a con-stant with the unit Nm/rad. Meaning that it will not be necessary to know all the parameters in Equation 4.9b in order to perform simulations of the screw joint. The value of k will be set according to Equation 4.10. The angles θminand

θmax corresponds to the minimum angle representing the tightening part of the

tightening process respectively the maximum angle representing the same thing. Normally the minimum angle is ten degrees and the maximum angle 720 degrees, which represents that the screw rotates between approximately 0.02 and 2 revolu-tions during tightening. How far a screw rotates depends on how stiff the joint is. A weak joint will be able to rotate longer than a stiff joint. The slope k represents the stiffness of the joint. The torque Tf inal is the final target torque.

Tf inal

θmax

≤_{k ≤} Tf inal θmin

(4.10)

During the rundown part of the tightening process the torque will be low and constant. Therefore T (θ) will be described according to Equation 4.11. The number nr stands for the number of revolutions corresponding to the rundown

and θsnugis the screw angle at snug given in radians. The constant c is called the

prevailing torque and is often 0-10% of the final target torque. The constant m is equal to c − k · θsnug. T (θ) =        c if 0 ≤ θ ≤ θsnug kθ + m if θ > θsnug θsnug = 2πnr (4.11a) (4.11b)

Final target torque will be reached at a certain screw angle θf inal. But the tool

will not stop immediately and therefore the function described in Equation 4.11 needs to include angles over θf inal. This explains why it covers angles over the

expected maximum.

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20 4 Modeling θ T (θ) θf inal θsnug c Tf inal rundown tightening

Figure 4.6:The second part of the screw joint torque function given in Equa-tion 4.12a. Tf inalis the value of T (θ) corresponding to the final target torque.

The tightening process consists of a rundown part followed by a tightening part.

summarized in Equation 4.12. Figure 4.6 depicts the parts describing T (θ). Tm= Js dω dt + T (θ) T (θ) =        c if 0 ≤ θ ≤ θsnug kθ + m if θ > θsnug θsnug = 2πnr m = c − k · θsnug (4.12a) (4.12b) (4.12c) (4.12d)

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4.4 Reference Angular Speed 21

4.4 Reference Angular Speed

The angular speed used as reference to the tightening tool is modeled in order to simulate the impact from the tool on the joint. This signal specifies the type of tightening process that is used. As mentioned in the system description, the process in question is the two-stage process (see Figure 2.2 or 4.7).

4.4.1 Known and Unknown Data

Some data needed for modeling the reference angular speed have been provided by Atlas Copco, based on knowledge about the tightening process in relation to the type of tool. The tool is a 50 Nm tool and the following data are considered to be known:

• Angular speeds: rundown speed ωrundown, first speed ωf irst and second

speed ωsecond.

• Times: time to ramp up to rundown speed ∆t1, time to go from rundown

speed to first speed ∆t3, time to go from first speed to zero ∆t5, time during

stationary screw position ∆t6, time to ramp up to second speed ∆t7and time

to go from second speed to zero ∆t9.

• Torques: first target torque Tf irstand final target torque Tf inal.

The unknown data are the following:

• Times: time during rundown speed ∆t2, time during first speed ∆t4 and

time during second speed ∆t8.

The values of the known data are summarized in Table 4.1.

Data Value Unit

ωrundown 650 rpm ωf irst 250 rpm ωsecond 50 rpm ∆_t₁ 200 ms ∆_t₃ 10 ms ∆_t₅ 10 ms ∆_t₆ 50 ms ∆_t₇ 90 ms ∆_t₉ 5 ms Tf irst 16 Nm Tf inal 50 Nm

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22 4 Modeling

It is important that the modeled reference angular speed function is in agree-ment with the modeled screw joint torque function. For example, snug must appear at the same time for both the torque affecting the tool and the reference angular speed. In order to get that behavior the tool speed should drop after snug and the screw angle at snug should be θsnug. The snug angle is also considered to

be known, see Section 4.3, but the angles representing first target θf irstand final

target θf inalmust be derived.

4.4.2 Calculations of Unknown Times

To be able to calculate the values of the unknown times all three screw angles need to be known. The two unknown angles can be derived from the screw joint torque function given in Equation 4.12.

At the time representing first target the angular speed is constant, the screw joint torque is equal to first target torque and the angle is θf irst, which is larger

than θsnug. This information together with Equation 4.12 gives that the screw

joint torque is

Tm= Js

dω

dt + T (θ) = kθf irst+ m = Tf irst (4.13) and the angle at first target can then be derived as

θf irst=

Tf irst−m

k (4.14)

The angle at final target can be derived in the same way and results in θf inal =

Tf inal−m

k (4.15)

Knowing all three angles all unknown times can be given by calculating the areas given in Figure 4.7. The different areas represent different angles, see Fig-ure 4.8. The first area corresponds to the snug angle, the second area represents the difference in angle between first target and snug and the third area is the dif-ference in angle between final target and first target. The exact calculations can bee seen in Appendix A.1.

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4.4 Reference Angular Speed 23 t ωr(t) rundown speed first speed second speed rundown tightening snug first target final target

Figure 4.7:Reference angular speed function representing a two-stage tight-ening process. The light gray area corresponds to θsnug, the gray area

rep-resents the difference between θf irstand θsnugand the dark gray area is the

difference between θf inal and θf irst. The angles can be seen in Figure 4.8.

θ T (θ) θf inal θf irst θsnug c Tf inal rundown tightening

Figure 4.8:The second part of the screw joint torque function given in Equa-tion 4.12a. Here the angles depicted by the different areas in Figure 4.7 are shown on the θ axis and colored by the same color as in Figure 4.7.

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24 4 Modeling

4.5 Tightening Tool

The tightening tool and the simulated test joint affect each other with the torque Tm if the angular speed at the motor shaft ω and the tool output ωt differ, see

Equation 4.4. As the tool is more powerful than the screw, it is considered to be able to keep the angular speed and in that way ωtdesignates the torque that the

motor must generate in order to simulate a screw joint. Thus, the angular speed ωtcan be used to describe the impact of the tool on the test system.

4.5.1 Simulation Model

A simulation model of the tool is needed regardless of the goal (the primary or the secondary) which is relevant to achieve. On the other hand, the complexity varies depending on the goal. If the goal is to evaluate the test joint using only the reference angular speed as the impact from the tool, it is not necessary to have a model describing the tool dynamics besides the reference angular speed function modeled above. But if the goal is to evaluate the system including a tool model, mathematical equations describing the tool dynamics are needed. Though, be-cause of the choice of focus of this study these dynamic equations will not be stated in this report. The tool will therefore, in the simulations, be implemented as the reference angular speed function resulting in ωt= ωr.

4.5.2 Considered as a System Disturbance

The tool angular speed ωt is not a quantity that the controller can control and

therefore it could be considered as a known or an unknown system disturbance. Because the test joint should be able to handle different tightening processes, even though the two-stage model is the one in focus, a general model describing the tool impact is hard to construct. A solution to this can be to assume that the tool behaves stochastically and can be modeled as low-pass filtered white Gaus-sian noise. Another possibility is to see the measurement of the motor torque as a measurement of the disturbance, according to the equation describing the spring stiffness, and use it in the control design in order to decrease the impact of the disturbance. How these different methods can be used either separately or together will be further explained and shown in the next section and in Chapter 5.

4.6 State-Space Model

Some model based controllers requires a state-space model of the test system. The general state-space representation given in Equation 3.4 is extended in order to represent system disturbances v1and measurement noise v2.

˙x(t) = Ax(t) + Bu(t) + N v1(t)

y(t) = Cx(t) + Du(t) + v2(t)

(4.16a) (4.16b)

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4.6 State-Space Model 25

The equations describing the test system are Lm di dt = v − Rmi − kmω Jm dω dt = kmi − km,fω − Tm Tm= kstif f · ∆θ

In the system model the influence from the tool can be modeled as low-pass filtered white Gaussian noise

ωt(t) =

β2

(p + β)2v1(t) (4.18)

where β is a positive constant making the disturbance model stable and the noise is distributed as v1 ∼ N(0, σ ). The value σ corresponds to the standard

deviation of the modeled reference angular speed ωr. Depending on the

charac-teristics of the system disturbance a second order low-pass filter is needed to get a satisfying performance. A constant disturbance can be handled by the controlled system if the open loop system contains one integration. But to handle a ramp disturbance, which is the case during some parts of the tightening process, two integrations are necessary, see Glad and Ljung [2006]. The disturbance can be included in the system model by selecting the states

x =                 x1 x2 x3 x4 x5                 =                 i ∆θ ω ωt αt                 (4.19)

where αtis the angular acceleration of the tool. The two last states are added

to represent the system disturbance ωt. If the inputs are chosen to

u = u1 u2 ! = v θ ! (4.20) the resulting state-space model is

˙x =                  −_R_m_/L_m ₀ −_k_m_/L_m ₀ ₀ 0 0 1 −₁ ₀ km/Jm −kstif f/Jm −km,f/Jm 0 0 0 0 0 0 1 0 0 0 −_β2 −_2β                  x +                 1/Lm 0 0 0 0 0 0 0 0 0                 u +                  0 0 0 0 β2                  v1 y =             1 0 0 0 0 0 0 1 0 0 0 kstif f 0 0 0 0 0 0 0 0             x +                 1/Lm 0 0 0 0 0 0 0 0 1                 u

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26 4 Modeling

The choice to use the motor angle θ as an input is made in order to use all measurable signals in the control designs.

4.7 Model Parameters

As mentioned in the introduction, the motor data is based on an electric motor which is expected to be used for this type of joint simulator. All parameters can be found in Appendix A.2.

The only unknown parameter in the spring stiffness model, see Equation 4.4, is the stiffness constant kstif f. It will be set to a value which should give a

behav-ior of the stiffness which is close enough to reality. The value is 750 Nm/rad and was provided by the Atlas Copco. It was based on knowledge about the tighten-ing process.

In the screw joint torque function, see Equation 4.12, three parameters are needed to be set in order to obtain the whole model. The first one is the screw inertia Js, the second is the prevailing torque c and the third is the slope during

tightening k. The screw inertia will be set to 1µkgm2, which is typical for screws tightened by a 50 Nm tool. The prevailing torque is set to 10% of final target torque, corresponding to 5 Nm, and the slope will have the largest value possible according to Equation 4.10.

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5

Control

This chapter contains a discussion of the control problem and a description of the control strategies investigated and evaluated in this study. The chapter ends with a presentation of the evaluation procedure performed on the controllers.

5.1 Control Problem

Beside the requirements and limitations on the control design stated in the in-troduction there are several things that are problematical with simulating a real screw joint. This section present a discussion of these difficulties and proposed solutions to them.

5.1.1 Choice of Reference

The torque at the motor shaft is the most important variable to control during the tightening process, except for when the first target torque is reached and the angular speed drops to zero. Then it is more important to control the screw position. A screw standing still cannot have any motion. On the other hand, the screw joint torque can vary until the torque overcome a threshold meaning that the screw should start to rotate again. This complicates the choice of control strategy and reference signal.

One idea might be to control the torque during the rotating parts of the tight-ening process and control the angle at standstill. But that would result in a con-trol design using different concon-trollers for different parts of the tightening process and it requires well chosen conditions to change between the controllers.

Another approach is to control the angular speed effecting the tool. By using the angular speed zero as reference and the screw joint torque function, given in Equation 4.12, as an over limit of the torque the controller would make the

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28 5 Control

test joint imitate a real behavior of a screw. The result is an easier control design which should handle the different tasks mentioned above. But the limitations on the control signal and the torque gives a structure containing two controllers in cascade, which complicates the ability to troubleshoot the implementation and is unnecessary complex.

To get a more basic control structure, another possibility would be to con-trol the motor angle. Concon-trolling the motor angle is according to the model of the spring stiffness, in Equation 4.4, equivalent to controlling the torque. If the screw joint torque function is used together with the spring stiffness model, a reference could be generated. The reference will then contain the model of the screw joint torque instead of using the model as a torque limitation as above. A modification to this design is to control the difference between motor angle and tool angle instead of motor angle. These approaches are the ones chosen in this study because of their simplicity. Equation 5.1 and 5.2 show how the reference can be generated. θr= Tm kstif f + θt (5.1) or ∆θr = Tm kstif f (5.2) where Tmis the screw joint torque

Tm= Js

dωt

dt + T (θt)

The angle and the angular speed representing the rotation of the screw is the angular speed generated by the tool. These cannot be measured directly and therefore they will be estimated by an observer, see Section 5.2.

5.1.2 Motor Inertia

As mentioned in the introduction, the motor inertia is larger than the screw in-ertia and therefore it can be problematical to simulate the screw joint correctly at the transition between standstill and rotation. A solution to this can be very complicated and therefore this phenomenon will at first be neglected to see how large the effect is. If the difference in inertia turns out to be significant and have a too large impact on the simulation, a more sophisticated solution might be con-sidered.

5.2 Observer

Normally the states are not observed in a system. In a lot of cases the states are needed in the control design either to generate a closed loop or, as in this

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5.2 Observer 29

master’s thesis, to generate a reference signal. The problem is how to estimate or reconstruct the state vector using the measured input signal u and outputs y. One way of solving this problem is to use an observer based on the system model given i Section 4.6. The general equation describing the observer looks as follows, Glad and Ljung [2003],

˙ˆx(t) = A ˆx(t) + Bu(t) + K (y(t) − C ˆx(t) − Du(t))

= (A − K C) ˆx(t) + (B − K D) u(t) + K y(t) (5.4) where ˆx represent the estimated states. The quantity y(t) − C ˆx(t) − Du(t) can be seen as a measurement of how good the estimate ˆx is. If x(t) would be equal to

ˆ

x(t) and if there was no measurement noise, the quantity would be equal to zero. K is called the observer gain and is here chosen to be calculated by a Kalman filter. The Kalman filter uses the state-space representation of the system (4.16) where the system disturbance and the measurement noise is assumed to be white noise with the covariance matrices R1respectively R2. The solution to the Kalman filter

is given by

K =P CT + N R12

R−₂1 (5.5)

where R12 is the cross-covariance between the system disturbance and the

measurement noise and P is the symmetric, positive semi-definite solution to the associated Riccati equation

AP + P AT −_{P C}T _{+ N R}₁₂_R−1 2 P CT + N R12 T + N R1NT = 0 (5.6)

The covariance matrices R1 and R2 acts as weights in the solution of the

Kalman filter and is used to tune the Kalman gain. The ratio between them de-termine whether the filter should rely on the model or the measurements. A high value on R1and a low value on R2means that the measurements are more

trusted than the model. This is because a large R1corresponds to a large system

disturbance and a low R2corresponds to small measurement noise.

The observer implemented in this thesis uses a Kalman gain given by the MAT-LAB function lqe. The covariance matrix R1 was determined by calculating the

standard deviation of the modeled reference angular speed ωr and use the value

to represent the white Gaussian noise v1 ∼ N(0, σ ), where σ is the standard

de-viation. It is known that the system disturbance is very large in this application and therefore R2was set to a lower value than R1and then tuned until satisfying

estimates of the states were obtained. The resulting matrices were R1= Cov(v1) R2= 10

−_x

· I (5.7)

where x is 25 or 20 dependent on which system test that is made, see Sec-tion 5.4 and Appendix A.3. The matrix I is the identity matrix with as many rows and columns as the number of measurements. R1is one-dimensional.

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30 5 Control

5.3 Control Strategies

The control strategies chosen to be investigated in the thesis are described and motivated in this section.

5.3.1 Proportional-Integral-Derivative (PID) Control

One of the simplest and most used control designs is the PID controller. The controller uses the control error, which is the difference between the reference and the output, to decide the control signal acting as the input to the system. The PID controller can be described by the following expression

u(t) = K          e(t) + 1 Ti t Z 0 e(τ) dτ + Td de(t) dt          (5.8) where u(t) is the input voltage to the motor, the control error e(t) is the dif-ference between the redif-ference and the measured motor angle (e(t) = θr(t) − θ(t)),

the parameter K is called the gain and Ti and Td are called the integration time

respectively the derivative time. The first term in the equation corresponds to the proportional part, the second the integral part and the third the derivative part of the name Proportional-Integral-Derivative (PID). By changing the values of the gain, integration time and derivative time it is possible to change the impact of these three parts. A little bit simplified one could say that the proportional part effects how fast the response is, the integral part eliminates the steady-state error and the derivative part reduces the oscillations.

Figure 5.1 shows a block diagram of the implementation of the PID controller. The block marked F represents the controller and G is the test system. The ob-server in Section 5.2 is used together with the relationship in Equation 5.1 to generate the reference θr, this is represented in the figure by the Reference

Gen-erator block. F u G Observer Ref. Gen. θr e y ˆ x θ +

-Figure 5.1:Block diagram of the implementation of the PID controller. The control parameters will initially be set by performing a step response experiment and then use the Ziegler-Nichols tuning method, see Enqvist et al. [2014]. Thereafter they will be tuned by hand until a satisfying compromise between reference tracking and speed is reached.

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5.3 Control Strategies 31

The justification to use this control strategy is the simple structure and imple-mentation. In industry, there are also several examples of when a PID controller is sufficient to control various industrial systems. Another advantage with the PID controller is that it does not require a deep understanding of how the system works, thus no model of the system is needed to get a well performing control.

5.3.2 Feedforward Controller Based on a Simplified System

Model

Here, two similar control designs will be described. The last one will be an ex-tension of the first. To understand these control designs better a discussion and explanation of the concept feedforward from reference will be made. Then the control structures will be discussed.

Feedforward from Reference

Good reference tracking is important when simulating a screw joint and one way of improving it is to use feedforward from reference as an extension to the ordi-nary PID controller given in Equation 5.8. The idea is to define a reference model Gmto which the closed loop system Gryshould be equal. The reference model is

formulated as

yr(t) = Gm(p)r(t) (5.9)

where yris the desired response to the reference signal r. The reference model

Gm could be chosen as Gm = 1 or to the sometimes more desirable and realistic

expression

Gm(p) =

1

(1 + pT )n (5.10)

where T is the wanted time constant of the closed system Gryand n is a

posi-tive constant.

Figure 5.2 illustrates the control strategy using feed-forward from reference. The blocks Ff and Gm defines the feed-forward, F the PID controller and G the

test system. The transfer function of the closed loop system is given by Gry =

G(FGm+ Ff)

1 + FG = Gm+

FfG − Gm

1 + FG (5.11)

The first term in the equation is, as mentioned above, the desired transfer function and the second term is the error in the reference tracking. An ideal feed-forward is obtained by choosing Ff according to

Ff =

Gm

G (5.12)

To be able to realize the transfer function of the closed loop system, the feed-forward Ff needs to be stable, causal and proper. Proper means that the transfer

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32 5 Control

proper, Gmshould be selected such that it has equal or higher relative degree as

the system G. Relative degree is the difference between the number of poles and zeros (the number of zeros subtracted from the number of poles), Enqvist et al. [2014]. The positive number n in Equation 5.10 is therefore chosen such that Ff

fulfills the requirements stated above.

Gm F G Ff u r + e y +

-Figure 5.2:Feedforward from reference.

Feedforward Controller (FF)

One of the biggest challenges in this control problem is that the system distur-bance has a large impact on the system and participates in generating the refer-ence signal. If it was possible to measure the disturbance it would be easier to control the system. In contrast to other control problems the goal will in this case not be to eliminate the disturbance completely but to reach the goal θ = θr,

where θris given in Equation 5.1 above.

It is possible to measure the motor torque and motor angle and by using the spring stiffness model

Tm= kstif f · (θ − θt)

the two measurements could be seen as a measurement of the tool angle (the disturbance). One way of controlling the test system could be to use this fact together with a simplified model of the system. If the inductance in the electric circuit is neglected the following model of the test system is obtained

¨ θ = −1 Jm k2 m Rm + km,f ! ˙ θ + km Jm· Rm v − 1 Jm Tm= −c1θ + c˙ 2u − c3Tm Tm= kstif f· (θ − θt) (5.14a) (5.14b) where the angular speed at the motor shaft ω is equal to ˙θ. If the control signal is chosen according to

u = 1 c2

¨_θ_r _{+ c}₁_θ˙_r_{+ c}₃_T_m

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5.3 Control Strategies 33

The resulting control structure will contain a feedforward from reference and a sort of feedforward from disturbance, see Figure 5.3. The first two parts of the first term in Equation 5.15 corresponds to the feedforward from reference

Ffθr = Gm G θr = 1 c2 ¨_θ_r_{+ c}₁_θ˙_r (5.16) To get a proper Ff the reference model Gmis modeled as

Gm(p) =

1

(1 + pT )2 (5.17)

The time constant T is used as a tuning parameter in order to achieve good reference tracking and a stable system. The transfer function G describes the dynamics of the test system from the input signal (motor voltage) to the output equal to the motor angle

G(p) = c2 p(p + c1)

(5.18) The last part of the first term in Equation 5.15 is the feedforward from distur-bance and the second term in the same equation is the control signal given by the controller F.

This control structure is based on a very simplified model of the electric mo-tor, but through this model it is possible to obtain a quite accessible design. The implementation is evaluated on the more advanced model given in Section 4.6 and the controller F is an ordinary PID controller implemented and tuned ac-cording to Section 5.3.1. The reference model Gm, represented by the block Gm

in Figure 5.2, was set to 1 after a lot of tuning of the control structure. This is not according to the general theory of feedforward but it gave the best reference tracking. Note that the reference model was still used to calculate the feedfor-ward Ff. F G Ff c_c3₂ ˜ u u Observer Ref. Gen. θr e y ˆ x θ + + + -Tm

Electric Motor Controlled Joint Simulator

Master of Science Thesis in Electrical Engineering

Department of Electrical Engineering, Linköping University, 2016