Modeling and motion control of a pick and place machine with air bearings

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Modeling and motion control of a pick

and place machine with air bearings

ELISABETH AMANN

Master of Science Thesis

Stockholm, Sweden 2012

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Modeling and motion control of a pick and

place machine with air bearings

Elisabeth Amann

Master of Science Thesis MMK 2012:40 MDA 436

KTH Industrial Engineering and Management

Machine Design

SE-100 44 STOCKHOLM

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Master of Science Thesis MMK 2012:40 MDA 436

Modeling and motion control of a pick and place

machine with air bearings

Elisabeth Amann

Approved

2012-06-08

Examiner

Jan Wikander

Supervisor

Mikael Hellgren

Commissioner

Micronic Mydata AB

Contact person

Henrik Linde

Abstract

To minimize friction related heat generation Micronic Mydata is interested in the option

of using air bearings in future surface mounting machines, such as the successors of

today's pick and place machine. The axes of interest is the linear motions in the

horizontal plane whose objective is to place a component in the correct position on a

PCB.

The purpose of this thesis is to investigate the characteristics of air bearings and

evaluate the achievable motion control performance for a new design with such

bearings. For simplicity the smallest axis, the Y-axis which positions the PCB, has been

used for testing. A background study has been done researching possible benefits and

disadvantages of these bearings compared to the linear guide ball bearings used today.

The properties of other parts of the system has also been studied, such as the

permanent magnet linear motor used to actuate the motion, in order to design a good

controller for the system. For the same reason similar motion control applications have

been examined to see what approaches have been used to achieve as good positioning

and trajectory following as possible. The information from the background study was

used to identify system parameters such as friction and motor ripple which were used

when designing a controller. The final version of the controller had a state feedback with

integral action, a dual observer consisting of a state observer combined with a

perturbation observer and feedforward of friction and motor ripple.

The velocity estimation turned out to be problematic and even though it could be proved

that a better velocity estimation would give better tracking performance no sufficient

replacement for the current observer was found. Since the air bearings almost entirely

lack friction disturbances such as erroneous velocity estimates have a larger effect than

on the system with ball bearings which is well damped by friction. It has therefore not

been possible to reach as good tracking performance with the air bearing design but

there is reason to believe that sufficient performance can be achieved with a better

velocity estimate.

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Examensarbete MMK 2012:40 MDA 436

Modellering och rörelsestyrning av en pick and

place maskin med luftlager

Elisabeth Amann

Godkänt

2012-06-08

Examinator

Jan Wikander

Handledare

Mikael Hellgren

Uppdragsgivare

Micronic Mydata AB

Kontaktperson

Henrik Linde

Sammanfattning

För att minimera värmealstring relaterad till friktion är Micronic Mydata intresserade av

att använda luftlager i framtida maskiner, såsom efterföljarna till dagens pick and place

maskin MY100. De axlar som man är intresserad av att byta lager på är de linjära

rörelserna i det horisontella planet som placerar komponenter på förutbestämda platser

på ett kretskort. Syftet med detta arbete är att undersöka egenskaperna hos luftlager

och utvärdera vilken precision som går att uppnå i rörelsestyrningen för en ny design

med sådana lager. För enkelhetens skull har den kortaste axeln använts i

experimenten, Y-axeln, vilken har till uppgift att positionera kretskortet. En

bakgrundsstudie har gjorts i vilken möjliga för- och nackdelar med luftlager har jämförts

med de linjärskenstyrningar som används i maskinen idag. Egenskaperna även hos

andra delar av systemet såsom linjärmotorn som driver rörelsen har också undersökts

för att kunna designa en så bra reglering som möjligt. Av samma anledning har liknande

tillämpningar för rörelsestyrning undersökts för att se vilka metoder som använts för att

uppnå så bra positionering som möjligt. Informationen från bakgrundsstudien användes

för att identifiera systemparametrar som friktion och motorrippel vilka användes för att

designa regleringen. Den slutliga versionen av regulatorn bestod av en

tillståndsåterkoppling med integrerande verkan, en dubbel observatör bestående av en

tillståndsobservatör och en störningsobservatör samt framkoppling av friktion och

motorrippel. Hastighetsuppskattningen visade sig vara den svåraste delen och även om

det kunde bevisas att en bättre uppskattning av hastigheten skulle ge bättre

positionering hittades ingen lämplig ersättare för observatören. Eftersom luftlagren är

nästan helt friktionsfria får störningar såsom felaktig hastighetsmätning en större effekt

än på systemet med kullager som är väl dämpat av friktionen. Det har därför inte varit

möjligt att uppnå lika bra positionering med designen med luftlager men det är troligt att

tillräckligt bra prestanda skulle kunna uppnås om ett bättre sätt att uppskatta

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Acknowledgements

I would like to thank several people who have provided valuable advice and support throughout the project:

Thanks to Mikael Hellgren, my supervisor at KTH, for valuable input.

At Micronic Mydata I would first and foremost like to thank my supervisor Henrik Linde for supporting me in everything from building test rigs to discussing research articles.

I would also like to thank the rest of the servo software group: Joel Rundgren, Robert Axelsson, Adrian Marklund and Dan Zemac for valuable discussions.

Thanks to Markus Halonen for designing the great test rig with air bearings.

Thanks to Johan ¨Ohlund for arranging all the practical details.

Thanks to all the other great people at Micronic Mydata who have helped me throughout the project!

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Acronyms

CAN Controller Area Network.

DSP Digital Signal Processors.

IC Integrated Circuit.

MEMS Microelectromechanical Systems.

PCB Printed Circuit Board. PM Permanent Magnet.

RMS Root Mean Square.

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Glossary

Conveyor A table placed on the Y-wagon which holds the PCB in place while it is within the machine. It also has slots for different picking tools suitable for different components and for discarded components.

MY100 A pick and place machine manufactured by Micronic Mydata which picks electrical components and places them on PCB’s.

Observed velocity Velocity estimated by the state observer.

Pick head The tool which gets components from the storage units and places them on the PCB.

Raw velocity Velocity calculated by differentiating the position measurement, vraw= xk+1−xk

h , where x is the position measurement and h the sampling time..

X-axis The axis of the MY100 machine along which the wagon that holds the pick heads travel.

X-wagon The wagon which moves the pick head between the component storages and the PCB.

Y-axis The axis of the MY100 machine along which the wagon that holds the PCB travels.

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1. Introduction

This chapter will introduce the reader to the background and problem formulation of the thesis as well as the methods used. A brief overview of the report structure will also be given.

Company The research in this master thesis has been done on behalf of Micronic Mydata AB, a company which develops, manufactures and markets high precision pat-tern generators for manufacturing of photomasks and advanced Surface Mount Tech-nology (SMT) equipment for flexible electronics production. The pattern generators are used for manufacturing of screens for e.g. TVs and computers, manufacturing of semiconductor Integrated Circuits (ICs) and for advanced electronic packaging. The sur-face mounting machines are used for placing electronic components on Printed Circuit Boards (PCBs) and stencil free application of solder paste.

Figure 1.1: The MY100 pick and place machine [1]

Problem The thesis work concerns the MY100 pick and place machine which can be seen in Figure 1.1. The motions in the machine are today driven by linear motors on caged ball linear guides. The company experiences that these guides have disadvantages like high noise level, heat generation, use of lubrication that might drip on the PCB and emission of particles. The heat generated from the bearings and the linear motor causes deflection of the frame which must be compensated with a complicated and expensive solution to get the desired accuracy. An alternative would be to replace the ball bearings with air bearings which do not have these disadvantages. Air bearings are non contact bearings which are fed with pressurized air and thereby provide a thin film on which the load can glide. Since there is no contact between the surfaces these bearings does not generate heat and are silent and free from lubricants. Micronic Mydata’s pattern generators already use air bearings, but the motion is slower, has significantly lower acceleration and the machines are operated in cleanrooms. The company is now interested in investigating whether air bearings would be a feasible solution for future pick and place machines.

Purpose The thesis work concerns investigating, testing and evaluating the properties and possibilities of air bearings in the environment and operating conditions of a pick and place machine. The purpose is to investigate if equal performance can be achieved

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with air bearings as with the current design. Questions of interest are:

Can the motion control be as precise with air bearings as with the current design? How will the mechanical properties of the bearings, such as stiffness and damping, affect the motion control?

Delimitations The project is limited to the motion control of the linear motors. The design of a test rig with air bearings has been done by the company and is outside the scope of this project. The mechanical properties of the bearings have only been evaluated with respect to how they can affect the motion control, not how the durability of the machine will be influenced.

Limitations As it was not feasible to redesign an entire machine, a test rig consisting of one of the axes was used. This means that it has been used in a protected environment without any of the disturbances that are normally present in the machine. Also the limited time of the project has not allowed for any long term testing and it is thus not known how the performance varies over time.

Method Two test rigs have been provided by Micronic Mydata, one with the same design as the current pick and place machine and one that has been redesigned for air bearings. The methods which have been used to evaluate the air bearings are:

1. Gathering of knowledge about air bearings and how these are used in precise motion control applications by literature reviews and interviews with Micronic Mydata personnel.

2. Modeling of the current system and the version with air bearings in Simulink. The models have been verified on the test rigs by using the log function of the pick and place machine software.

3. A controller has been designed for the new system in Matlab/Simulink and the theoretical performance of the two systems has been compared.

4. The controller has been implemented in the existing servo code and the perform-ance of the test rig with air bearings has been tested.

5. Based on the test results conclusions has been drawn about the possibility of using air bearings in future pick and place machines.

An alternative to using the machine software would have been to use dSpace which would have allowed for easier integration between the model and the measurements. This was not used since it was not available until late in the project, and since it was desired to see that the control algorithms could be implemented with the computational power available in the MY100.

Report structure The report starts by giving an overview of the system that the tests were performed on and what requirements were posed on the air bearing design in Chapter 2. After that the frame of reference (Chapter 3) consisting of the theory present on air bearings and motion control of linear servo motors with and without air bearings is presented. A description of the mathematical models, simulation models and the test rigs in Chapter 4 is followed by the design of the controller in Chapter 5. The results are accounted for in Chapter 6 and an analysis of the tests with regard to the frame of reference is presented in Chapter 7. Finally in Chapter 8 a discussion of the advantages and disadvantages of the air bearings is held and recommendations are be given on the possibilities of using air bearings in future pick and place machines.

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2. Background

This chapter gives an overview of the system that has been examined and describes the important components. First the MY100 machine is described with focus on the parts that was used for testing in this project. Some of the important factors which have been fixed throughout the project are described, such as the linear motor and the design of the air bearing test rig. To give the reader an understanding of the starting point for the controller design also the structure of the controller used in the current machine is explained. Finally the requirements on the system are listed, as well as methods for testing them.

2.1 System overview

The machine of interest is the MY100 pick and place machine. The machine works with a split axis motion where the axes can be seen in Figure 2.1. The Y-wagon positions the PCB along the Y-axis and the X-wagon carries the pick head which gets components and places them on the PCB along the X-axis. The capacity of the fastest machine with two X-wagons is up to 50 000 components per hour, so the movements are quite fast with accelerations up to 35 m/s2 _{and speeds up to 4.5 m/s.}

Figure 2.1: Directions of movement in the MY100 machine [2]

The tests were performed on the Y-axis movement since it was more convenient in size for building test rigs, but the essential components such as the motor is the same for both X and Y. The following part of this section will therefore give an overview of the Y-axis and how it communicates with the rest of the system. A sketch of the important units can be seen in Figure 2.2.

The central computer runs a software designed by Micronic Mydata called TPSys, which controls all the parts of the machine and decides which movements should be done. This computer communicates with the different nodes via Controller Area Network (CAN). A node can be for example a motor controller card, like in this case the controller for the Y-axis.

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Controller

card Servo drive Motor Y-wagon

Central computer

voltage _current _force

CAN command

encoder position data

Y-axis MY100

Figure 2.2: Overwiew of the system

The controller card, called the CMOT (CAN Motor controller card), is located nearby the Y-axis. The CMOT contains hardware for communicating with the central computer over CAN and for controlling the movement of the Y-wagon. The node receives a command, typically a position that the Y-wagon should move to, and calculates the reference trajectories for the movement and implements a position feedback loop. The desired output current is converted to an analog voltage of ±10V which is sent to the servo amplifier.

The servo amplifier is used to maintain the desired current fed to the linear motor which moves the Y-wagon along the axis. It takes in the analog voltage control signal from the CMOT and uses this as reference in an internal feedback loop based on motor parameters which controls the current. This is done by a 16 bit Digital Signal Processors (DSP). The output on the phases is a PWM signal. To align the phases correctly the amplifier must know the position of the motor relative the magnets which is why the encoder signal is fed back.

The motoris a brushless linear motor where the coils move along a permanent magnet rail. This is an important part of the system and will be explained in more detail later in this chapter.

Figure 2.3: Drawing of the Y-module (source: Micronic Mydata)

The Y-wagonis as mentioned earlier responsible for positioning the PCBs. The PCB is held in place by a conveyor, mounted on top of the wagon, which is responsible for

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keeping the board still and transporting it in and out of the MY100 when the machine is standing in a production line. The motor is mounted underneath the wagon which is mounted on linear guides. The bearings are preloaded by the magnetic force between the motor and the magnet rail. A drawing of the entire Y-module can be seen in Figure 2.3.

Position measurementThe position is measured by a linear incremental encoder with a resolution of 0.5 µm [22]. The redhead is mounted on the Y-wagon and the linear scale on the frame of the Y-module. The TTL(transistor-transistor logic)-pulses are counted by a counter on the CMOT which is sampled with a period of 420 µs. The encoder signal is also read by the servo amplifier as described earlier.

2.2 Motor

The motor used can be seen in Figure 2.4 and is a Permanent Magnet (PM) synchronous brushless linear motor where the coil set is moving along a rail with skewed magnets. All the motor parameters from the data sheet [23] can be seen in Table 2.1.

Figure 2.4: The linear motor used in the MY100, Anorad LCK-S-3-P-NC [23]

Table 2.1: Motor characteristics. All specifications are ±10%, for stand still conditions multiply continuous current and force by 0.9 [23].

Parameter Unit Value

Continuous force (FcT max) N 304

Peak force (Fp) 1s on. N 738

Force constant (Kf) N/Apk 38.3

Back EMF (Ke) Vp / m/s 45.3

Motor constant (Km= √F_Pc

c) N/

√

W 23.3

Max power dissipation (PcT max) W 170 Continuous current (IcT max) Apk(Arms) 7.9(5.6) Peak current (Ip1) 1s on Apk(Arms) 25.0(17.7) Resistance p-p (R25) @ 25◦C Ω 2.6

Inductance (L) p-p mH 20.0

Maximum applied bus voltage (VDC V 325 Electrical time constant (τe) ms 7.7 Electrical cycle length (Ec) mm 60 Maximal coil temperature (tmax) ◦C 125 Magnetic attraction (Fa) N 3416 Thermal resistance (Rth) ◦C/W 0.59

Coil mass kg 4.0

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75%) above the maximum continuous current. The motor has three phases configured in a Y-winding and essentially works like an unrolled rotary PM synchronous motor, described in [18]. A force is generated by driving a current through the windings where the magnetic field from the permanent magnets has the largest magnitude. The force can be written as F = BIL where B is the magnetic flux density in Tesla, I is the current in Ampere and L is the length of the wire in meters. Since the desired current through the different windings is dependent on the relative position of the moving coil set the encoder position of the Y-wagon is fed back to the servo amplifier which produces sine shaped current with a 120◦ offset between the phases.

2.3 Air bearing setup

The air bearing design was done by the company and is outside the scope of the thesis. A simple drawing of the Y-wagon with air bearings is shown in Figure 2.5. The bearings (shaded in the figure) are attached to the wagon with ball joints to take up force in the normal direction of the bearing but allow movement caused by irregularities in the rail. The bearings are also fixed to prevent rotation around the z-axis. The picture shows a cross section of the Y-wagon with bearings and motor, there are two bearings on the left side of the wagon and four on the right side. The bearings run on a flat rail on the left side and a triangular on the right side. The magnetic attraction force between the coil and the permanent magnets of the motor, which is around 3500 N, keeps the wagon fixed vertically and the triangular rail controls the horizontal position. The lift generated by the pressurized air fed to the air bearings is large enough to keep the bearings flying some microns above the rails. The moving coil of the motor is mounted underneath the Y-wagon and the rail of permanent magnets is mounted between the bearing rails.

m y-wagon motor x z y Fz Fx Fn 40◦

Figure 2.5: Principle drawing of the Y-wagon with air bearings

2.4 Control structure

The starting point of the controller design was the structure that is used in the cur-rent MY100 machines, which can be seen in Figure 2.6. A trajectory planner, denoted ”References” in the figure, generates third order trajectories, which means that jerk, acceleration, velocity and position is defined for the entire movement. The box denoted ”System” is the servo drive, motor and Y-wagon from Figure 2.2 and the measured output y is the position of the Y-wagon. The control signal u is the sum of the feed-forward control signal uf f and the controller’s control signal uc. The feedforward block calculates the current needed to get the desired acceleration aref and compensates for static and dynamic friction. The controller compensates for model errors by regulating the position and velocity errors, ep and ev to zero. Since only the position is measured an observer is implemented to estimate the velocity of the wagon.

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References _Controller u _System Observer Feedforward pref vref ep ev _y − vobs − p uc uf f aref 1

Figure 2.6: Schematic of the controller used in the MY100 today

2.5 Requirements and testing

The goal of this project has been to see if the performance of the a machine with air bearings can be at least as good as the MY100 is today. The scope was narrowed down to the motion control of the system. To compare performance concrete requirements were necessary, which is why the following list was compiled by tests on a MY100 and interviews with people involved in developing the machine.

2.5.1 Requirements

1. The test rig with air bearings should have equal or better performance as the one with ball bearings in the following cases

(a) Tracking of the acceleration trajectory (b) Tracking of the velocity trajectory

(c) Tracking of the position trajectory

(d) Overshoot in µm — current performance is around 8 µm (e) Settling time in ms — current performance is around 25 ms

2. Normal operation of the Y-axis should be possible, which means movement along trajectories with the specifications below. This should be done with normal load on the Y-wagon, corresponding to the most common conveyor T4 which weighs approximately 30 kg.

(a) Accelerations up to 10 m/s2 (b) Velocities up to 2 m/s

(c) Nominal smoothing time of 0.015 s → jerk of 2₃ · 103 _m/s3

3. It is also desirable to simulate movement of the X-wagon, which means that the load is decreased to around 10 kg with the following requirements for the trajectories:

(a) Accelerations up to 35 m/s2 _{or, due to limitation in travel distance, as high} as possible below that

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(b) Velocities up to 4.5 m/s or, due to limitation in travel distance, as high as possible below that

(c) Maintain constant velocity of 1.8 m/s (±0.1m/s) after braking from a higher velocity. This is done when the pick head travels over a camera which inspects that the component is aligned correctly before it is placed on the PCB 4. The wagon should remain at the same position at standstill while exposed to

typical disturbance forces present in the MY100, such as vibrations caused by movements from other axes

2.5.2 Test cases

Suitable tests for the requirements in Section 2.5.1 are described below. The list num-bering corresponds to that in the earlier mentioned section.

1. The comparison with the linear bearing rig can be performed by plotting data from typical cases (below) for both rigs and compare them

(a) – (c) Can be compared by largest absolute error and RMS of the deviation (d) – (e) Can be compared by direct measurements in the plot

2. The performance specification can be tested by making sure that the criteria in (1) are fulfilled at maximum jerk, acceleration and velocity

3. Can be tested by running typical cases (below)

4. Induce vibrations in the rig while regulating at a certain position

Typical cases

For testing a list of interesting movements of the MY100 has been compiled. These correspond to the work that is usually done by the machine or are considered hard and is thus a good way of testing the performance of the controller.

• Short point to point movement (20-100 mm). This is a normal movement for the y-axis.

• Long point to point movement (entire axis). This is a normal movement for the x-axis.

• Constant speed (low speeds also: about 20 mm/s). This is hard since there will be fewer encoder ticks per sampling time.

• ”Cat ear” movement: fast-slow-fast. This kind of movement is used by the x-axis when passing the camera. It is important that the speed is kept constant since the picture will be useless if the speed is too high.

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3. Frame of reference

This chapter gives an overview of the relevant theory present about the area. The focus of this project is on the controller and the dynamics of the motor, wagon and frame. Neither the servo drive, nor the software for the central computer were altered and are thus not discussed in further detail. The focus is on the dynamics of the motor and bearings, both the ball and the air bearings, and motion control applications where similar motors and bearings have been used. Different control approaches are examined, starting from state feedback with a Luenberger observer and continuing to more advanced designs such as the perturbation observer.

3.1 Bearings

The main focus of the project is the performance of air bearings and since the bearings are the major difference between the test rigs it is interesting to investigate the properties of both kinds of bearings further. The problem formulation states that the desired advantage is the lack of friction but the question is whether this is only positive. It is also desirable to examine if there are other significant differences that might affect the performance of the machine, such as stiffness and damping of the bearings.

3.1.1 Air bearings

Air bearings are non-contact bearings where the bearing is separated from the guideway or shaft by a thin air film. There are two kinds, aerodynamic and aerostatic bearings, where the air film in the aerodynamic one is generated by the relative motion between the surfaces whereas the aerostatic one needs an external pressure source. In the case of the aerostatic bearing, the air is supplied through a valve and led out towards the guideway either by orifices or by a porous material [6]. The bearings used in this project were aerostatic linear porous media bearings so this is the kind that will be considered in the background study. The parameters of the bearings used in the test rig can be seen in Table 3.1.

Table 3.1: Characteristics of the air bearings used in the test rig: New Way S124002 [21].

Parameter Unit Value

Size mm×mm 40×80

Viable pressure range bar 4-5.5

Fly height µm 5

Flow NLPM 1.8

Ideal load N 623

Stiffness N/µm 58

A picture of the bearings used in the test rig can be seen in Figure 3.1. The pressurized air is provided through the hole on the side of the bearing and the bearing is mounted with a ball joint from the top. It is important to use this kind of mounting where the bearing is fixed only radially so that is can adapt to unevenness in the guide material. Despite this adaptability it is important to have a smooth surface to fly on, no abrupt changes can be tolerated as this would cause the bearing to hit the guide. A slow incline or decline can be tolerated though. The manufacturer recommends guide surfaces like

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granite, hard coated aluminum, ceramics, glass, stainless steel and plated steel with a suggested surface finish of 0.4 Ra [21].

The material used for the air supply is porous carbon. This should give the bearing a uniform pressure distribution, compared to traditional orifice type air bearings which have a pressure gradient across the air gap. According to the manufacturer this soft carbon material will also help protect the bearings at air failure since they can be moved along the support surface without damaging it.

Figure 3.1: The air bearings used in the test rig [21]

Possible advantages of the air bearings were described briefly in the introduction. The manufacturer of the air bearings used in the test rig state the following advantages [3]: • No static friction — since the bearings are non contact there will be no static friction and thus no positioning problems caused by the stick-slip effect. This gives an application with air bearings almost infinite resolution.

• Zero wear — since there is no contact there will be no wear on the bearings. This gives both reduced maintenance and more consistent machine characteristics since the bearings do not change over time.

• Smooth and silent motion — movement of the balls in a rolling element bearing give rise to both velocity ripple (although at nanometer level) and noise, which is eliminated with air bearings which have no contact with the guideway and no moving parts.

• Higher speed and acceleration — since there are no moving parts with inertia in the air bearings, speed and acceleration can be significantly higher.

• No lubrication means no contamination to the rest of the system.

• No unwanted motion. Difference in size between the balls in a rolling element bearing give rise to unwanted and unpredictable motion in all directions. This is avoided with air bearings since there are no internal parts that can give rise to these motions. The possible cause of movement of the air bearings, height variations in the rails, is constant and can be compensated for in software. • Radial positioning — the air cushion of the air bearing adapts to small flatness

deviations in the rail giving a smooth motion. The momentum of the balls on the other hand will amplify the unwanted motion and further damage the rail.

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Given only this information the air bearings sound ideal, but it is important to bear in mind that this is marketing material with the sole purpose to sell air bearings. There are also disadvantages such as that the low flying height require a clean surface in order not to get stuck. The main problem would be fat or fluids since this is not easily pushed away by the bearing. There is also a case of instability called pneumatic hammer which will be described soon. The axial and radial characteristics will now be described further with respect to independent research that has been done in the field.

Axial characteristics: The axial characteristic of interest is the friction, or lack thereof. Air bearings are often said to be frictionless which is not entirely true. From the research of Fujii [16], who investigated the frictional characteristics of air bearings, it can be concluded that there should only be viscous friction from the bearings. This friction was assumed to arise due to Couette flow, which is the laminar flow of a viscous fluid in the space between two parallel plates where one is moving relative to the other. According to the article the viscous friction can be calculated as in Equation (3.1).

Fviscous= µair· S

h · v (3.1)

where µair is the coefficient of viscosity of air, S is the surface area of the bearing, h is the thickness of the air film and v is the velocity of the bearing. For the air bearings of the Y-wagon this would approximately be

Fviscous= 0.0185 · 10−3P a · s ·

0.0032m2

0.000005m· vm/s = 0.0118N s/m · v

which is at most (at 2 m/s) 0.0236 N per bearing or 0.1417 N for all six bearings. The viscosity coefficient of air is assumed to be approximately 1.85 · 10−5 Pa·s at room temperature.

Radial characteristics: The radial characteristics of the bearings will determine how the system reacts to forces in other directions than along the Y-axis. The air film under the bearing has a stiffness that depends on the surface area of the bearing and a damping that depends also on the resistance of the air channels through the bearing, i.e. a porous media bearing has higher damping than an orifice bearing. The stiffness of the air bearings is given in the data sheet to be 58 N/µm [21], where the manufacturer describes that it has been measured experimentally by applying a known force to the bearing and measuring the change in flying height [21].

The damping on the other hand is not as easily defined and since porous media bear-ings is a relatively new phenomenon most literature concerns orifice bearbear-ings. Slocum et al. describes an estimation of the damping of rectangular porous air bearings shown in Equation (3.2) where w is the width of the bearing, L is the length, h is the flying height and µeff is the effective viscosity which is calculated according to Equation (3.3). This is based on the theory of squeeze film damping, a damping which arises when two close similar surfaces move towards or away from each other, and is governed by Reynold’s equation. Research has mostly been done on Microelectromechanical Systems (MEMS) but the results have been applied also to air bearings since the film thickness is similar.

b =0.991 − 0.578w L ·µeffw 3_L h3 , w L ≤ 1 (3.2) µeff = µ 1 + 9.64(λ/h)1.159 (3.3)

The value of λ is stated in the article to be approximately 0.1µm for air and the viscosity is around 1.85 · 10−5 _{Pa·s. This gives a damping coefficient of approximately 477 kNs/m}

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for a 40x80 mm bearing and a flying height of 5µm. The experimental results of the article show however that the squeeze film approximation is not entirely true due to reflow through the porous material and suggests a reduction factor of 4.4 for the width and length of the bearing. This would give a damping coefficient of only 1.3 kNs/m. This is of course only valid for the particular setup in the article and cannot be assumed to hold for the test rig used in this project.

Pneumatic hammer Pneumatic hammer is a state of instability where the bearing is vibrating radially, which is due to that the bearing cannot find a stable flying height. If the air is restricted by a gap that is slightly too small to allow equilibrium it will increase air pressure and push the bearing upward to widen the gap. This makes it easier for the air to escape and thus the pressure decreases and the bearing falls down again. This cycle repeats infinitely since there is no equilibrium point to be found. The manufacturer claims that the porous media bearings should however have a larger resistance to change in air flow due to the millions of tiny passageways on the face and therefore it should be difficult to change the air volume in the gap quickly, giving a naturally stable bearing [5].

3.1.2 Linear ball bearings

In a linear ball bearing the bearing slides against the balls which are then led back through a groove in the bearing. In the bearing used in the machine a ball cage is used to separate the balls and eliminate the friction between them [25].

Figure 3.2: The caged ball guide used in the machine today [26]

Axial characteristics The friction of the bearings consists of a static component referred to as seal resistance and a dynamic component that originates from the viscous force of the lubricant. The bearings used today have a seal resistance, which is the force needed to bring them in motion, of 3.5 N each. The static friction force, Ff, of rolling element bearings could be described as in Equation (3.4):

Ff = µ · P + (Fsr· n) (3.4)

where P is the load in N, µ is the coefficient of friction, Fsr is the seal resistance and n is the number of bearings in the system. According to the manufacturer of the bearings µ should be approximately 0.002-0.003. The viscous friction depends on factors as the lubrication of the bearing must be tested for the actual conditions [25]. Assuming a load of 3500 N this would give a static friction force of 21 N to 24.5 N.

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Radial characteristics The stiffness of the ball bearings is essentially determined by the stiffness of the material used in the balls and the preload of the bearings. A greater preload will give a larger contact area due to deformation of the balls and thus an increased stiffness. A similar guide to the one in MY100 is tested in [11] and found to have a stiffness of approximately 42 N/µm without preload. According to the manufacturer of the bearings used in MY100 the effect of a preload gives up to 2.8 times increased rigidity [25], which would mean an approximate stiffness of up to 117 N/µm for the bearings. Given this information it is reasonable to assume that the ball bearings have a stiffness slightly above the one of the air bearings but still in the same range. The damping is harder to predict, in [24] it is described as very small and hard to measure since the damping of other parts of the structure will drown the damping of the bearings in the measurements. The damping is said to be due to friction inside the bearing and the lubrication. An attempt to measure damping in axial ball bearings was done in [14] and the value was about 10 kNs/m for a particular bearing which is significantly smaller than the value predicted for the air bearings (477 kNs/m) but larger than the value achieved after scaling of the bearing dimensions (1.3 kNs/m). In conclusion can be said that there is nothing indicating that this kind of air bearings would have inferior radial properties to the ball bearings.

3.2 Motor

The dynamics of the linear motor is essential when creating a model of the system. Several articles have been written about modeling and control of linear motors. Yao and Xu [27] modeled an iron core brushless DC linear motor from the Anorad LCK series (same series as the motor in the MY100) with the equation seen in (3.5)

m¨xL= Fm− B ˙xL− ff ric( ˙xL) − fdis(t, xL, ˙xL) (3.5)

Where xL is the displacement, m is the translating mass, FM is the force generated by the motor, B is the combined damping and viscous friction on the load, ff ric is a combination of stiction and Coulomb friction and fdis represents external disturbances. A simplified model is mentioned as commonly used, shown in Equation (3.6)

Fm= KFi Ld

dti + Ri + KE˙xL= u (3.6)

Yao and Xu assumed that the simplified model might not describe the dynamics of an iron-core linear motor good enough since these motors also exhibit force ripple and cogging force. The cogging force is due to the magnetic attraction between the iron cores of the translator and the permanent magnets and is only dependent of the relative position between them. The magnetic attraction varies along the axis and the iron core will thus want to align at certain positions. The cogging force is periodic if the magnets are identical and equally pitched. The force ripple is described as a position dependent periodic variation of the force constant Kf. The enhanced model of the motor is described as

Fm = KF(xL)i + fcogging(xL)

KF(xL) = KF 0+ KF x(xL) (3.7)

Bascetta et al. [7] also described a model of the force ripple in a permanent magnet linear synchronous motor. Opposed to the assumption in Equation (3.7) it is stated that

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the aligning force of the motor (Fm in Equation (3.6) and (3.7)) is independent of the relative position of the slider and the magnets. Two main groups of forces are presented in the article; those which are functions of the current and those which are independent of the current. To the former group belongs:

• Variations of the reluctance due to variations of the self inductance in the phase windings give variations in the motor force.

• If the amplitude of the three phases are not exactly equal there will be a periodic disturbance of the overall force.

• Imperfections in the current and back EMF waveforms.

The current independent force is called cogging and is due to the attraction between the iron core of the slider and the permanent magnet rail. The slider will want to align at certain positions, seeking a path of minimum reluctance. Skewing the permanent magnets reduces but does not eliminate this force. The period of the disturbances will be proportional to the period of the permanent magnets, Ec [m], and the frequency of the cogging force is stated to be

fcogging= 2π Ec

˙x (3.8)

3.3 Applications

This section will describe different applications where air bearings are used today and which advantages and disadvantages that have been experienced in previous research. New Way Air Bearings which manufacture the bearings used in the test rig mention markets where they have delivered solutions on their website [4].

• Computed tomography where rotating x-ray assemblies scan patients use air bear-ings due to that a higher rotating speed that can be obtained and that a low noise level is desired in medical applications.

• The Flat Panel Display industry use air bearings due to their cleanliness and increased precision in motion control.

• In metrology and precision machine tools the air bearings are used due to the precision gained from the lack of friction.

There are also examples in research where the implementation of air bearings has been investigated. The most common objective is to get rid of the stick-slip effect that regular bearings have due to friction and thus increase the precision, many focus on achieving nanometer precision. In [13] an air bearing application for high precision machine tools is implemented. It is stated in the article that the air bearings have the advantages of high stiffness and damping perpendicular to the direction of movement and no frictional effects like stick-slip. However the lack of friction in the feed direction leads to the absence of damping which according to the authors is a setback for the traditional cascaded control strategy commonly used. It is also said that modern linear roller guideways show no damping either and that the reasoning in the article is valid for those as well. Simulations are shown where the PID controller gives permanent oscillations when regulating at a desired position. A state space approach is suggested since it allows for almost arbitrary pole placement.

In [15] Slocum et al. use a test rig with an open face linear motor (like the one used in the MY100) and porous media air bearings was built. The objective was that the

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iron core motor could cause excessive loading and premature failure of rolling element bearings. Experiments were done to measure damping with the conclusion that it was better than for typical rolling element bearings but not as good as for hydrostatic bear-ings. Tests were implemented for a grinding application with satisfactory results, the process was said to be ”very smooth and quiet” and the machine was stiff enough to grind a part chatter free and remove stock better than a similar production machine.

The focus of [9] is to compensate motor ripple and experiments are performed on two test rigs: one with air bearings and one with rolling element guideways. The motor used was also here an open face linear motor. It is stated that the test setup with rolling element bearings is more rigid than the one with air bearings and thus allows for a higher controller bandwidth and higher accelerations without excitation of resonance frequencies.

3.4 Control strategies

This section describes different approaches to designing a controller for the system. First the nomenclature that is used throughout the thesis is defined. A traditional state feedback controller with an observer is explained and the feedback is later extended with integral action. Finally some common strategies for dealing with the nonlinearities of iron core linear motors are investigated.

3.4.1 Nomenclature

The system definitions will be described in the state space as the controller preferably can be designed as state feedback [13]. In continuous time the system description will be the one in Equation (3.9) and the Laplace variable s is used for derivation.

˙x = Ax + Bu

y = Cx + Du (3.9)

The discrete time description will be the one in Equation (3.10) which is the definition given in [29] to define a time invariant system with periodic sampling time h. Differen-tiation is done with the Z-transform variable z.

x(kh + h) = Φx(kh) + Γu(kh)

y(kh) = Cx(kh) + Du(kh) (3.10)

The conversion between the systems is done as in Equation (3.11), for a more detailed explanation see [29, pp. 32-37]. Φ = eAh Γ = Z h 0 eAsdsB (3.11)

Feedback gains will be denoted L and observer gains K.

3.4.2 Pole placement

The basic state feedback approach is described in [29, ch. 4]. The state feedback is calculated to place the poles in the desired position by setting the input to

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which gives the state space system

˙x = (Φ − ΓL)x y = Cx

The poles of the closed loop system are determined by the eigenvalues of the matrix Φ − ΓL and can thus be chosen arbitrarily by setting the values of the matrix L, given that the system is reachable. The reachability of the system can be determined by examining the controllability matrix (3.12). If the matrix has full rank n it is possible to find a control sequence such that any an arbitrary state can be reached from any initial state in finite time [29, p. 94].

Wc= Γ ΦΓ · · · Φn−1Γ

(3.12)

3.4.3 State feedback with integral action

An improved method of state feedback, with integral action, is described in [12]. The motive for introducing the integral term is that the traditional state feedback controller would introduce a steady state error which could be eliminated with integral control. The state vector is therefore extended with the integrated position error xN(t). This gives the continuous time system shown in Equation (3.13), where r(t) is the position reference. ˙x(t) ˙xN(t) = A 0 −C 0 x(t) xN(t) + B 0 u(t) + 0 1 r(t) y = C 0 x xN (3.13)

The control law is described as in Equation(3.14)

u(t) = Lx(t) + LexN = − L −Le x xN (3.14)

Given this augmented system pole discretization and pole placement can be done as described earlier. It is concluded in the article that this kind of controller is superior to the traditional state feedback controller in terms of primarily eliminated steady state error but also diminished overshoot.

3.4.4 Observers

Since the velocity is not measured an observer can be used to estimate it. The estimated state vector ˆx can be calculated with a full order observer as in equation (3.15).

ˆ

x(k + 1|k) = Φˆx(k|k − 1) + Γu(k) + K (y(k) − C ˆx(k|k − 1)) (3.15)

If the estimation error is denoted ˜x = x − ˆx this error is described by Equation (3.16).

˜

x(k + 1) = (Φ − KC) ˜x(k) (3.16)

which means that the error will be stable if the matrix Φ − KC has all eigenvalues, or poles, within the unit circle. The observer poles can be placed in the same way as the

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controller poles described earlier, as long as the observability matrix in Equation (3.17) has full rank [29, p. 99].

Wo =      C CΦ .. . CΦn−1      (3.17)

The observer is typically chosen to be faster than the controller. It can be shown that if both the observer and the state feedback are stable the combination of them will also be stable and they can therefore be designed independently given that the system can be considered linear and time invariant [17, p. 200].

3.4.5 Further development

Starting in this basic technique of state feedback with observed states, many researchers try to find a way to compensate for the nonlinearities of the linear motor. One of the commonly used strategies is identification and feedforward of the motor ripple [9, 7] where the hardest part is the identification. Since cogging is position dependent the frequency will be well within the bandwidth of the controller at slow speeds. This is utilized in [7] where the cogging is identified by what is referred to as ”closed loop position dependent identification”. Essentially the motor is run at a slow and constant speed along the entire axis and the variations in the control signal is measured. Braembussche [9] use a test rig with air bearings and assume that since there is zero friction motor ripple will be the only force acting on the motor. The measurement is done by holding the motor at a known position and measuring the force with a force cell while gradually increasing the input voltage to the motor from 0 to 10 V. This is done at several positions to get the force variations along the axis.

Another approach to taking care of the nonlinearities is to handle model uncertainties with the dual observer described in [10]. This consists not only of the state observer described earlier but also of a perturbation observer. A disturbance w is added to the system so that the state space description now looks like

x(k + 1) =Φx(k) + Γu(k) + Γ_Dw(k) (3.18)

y(k) =Cx(k) + Du(k) (3.19)

This adds a term to the observer described in Equation (3.15) which now can be written as in Equation (3.20).

ˆ

x(k + 1|k) = Φˆx(k|k − 1) + Γu(k) + K (y(k) − C ˆx(k|k − 1)) + ΓDw(k)ˆ (3.20) the disturbance is estimated as

ˆ

w(k) = Qweq(k)

where weq is calculated from Equation (3.18) as in Equation (3.21) where Γ+_D is the Moore-Penrose pseudo inverse of ΓD and Q is a low pass filter.

weq(k) = Γ+_D(ˆx(k + 1) − Φˆx(k) − Γu(k)) (3.21) The purpose of the disturbance observer is to make the state observer more robust and to compensate for the disturbances in the controller. It is concluded that the perturbation observer eliminated bias effects in the estimated velocity which arise due to model uncertainties and thus greatly enhance the control system performance. The trajectory tracking performance was significantly improved compared to using only the state observer.

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3.5 Summary

To summarize the frame of reference the air bearings are said to have many advantages, such as higher precision, silent movement and no wear. The radial properties are in-teresting to research further to see if there is any risk that the bearings will oscillate, but if the theory holds there is reason to believe that the damping and stiffness of the air bearings are sufficient. The motor properties have been investigated and it seems that the most problematic part is the motor ripple which is why this will be examined further. Some control strategies have been proposed and will be tested on the Y-axis later on.

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4. System identification

This chapter concerns the identification of different system parameters. The first section deals with the bearing dynamics to get an idea of if and how these will affect the performance of the positioning. The second section deals with the dynamics of the entire Y-wagon to create a model which can be used to design and test a controller. Since much is similar in the air bearing design and the MY100 as it is today the starting point has been to model and perform tests on a test rig consisting of a Y-wagon with the current machine design. A mathematical model of the system was created and implemented in Simulink. This could then be verified at the test rig by closed loop parameter identification and open loop step responses. This aided the creation of a model for the air bearing system and thus the controller design.

4.1 Test rigs

Two test rigs have been used, both are based on the Y-axis of the machine: one Y-wagon with frame identical to the one used in the MY100 machine and one modified version where air bearings are used. The difference was only in the guide rails and the wagon, the motor and the frame were identical in the two rigs. They were driven by the same software and hardware which control the Y-axis in the MY100, described in Section 2.1, but as the Y-module was used outside the machine the CAN commands were sent from a standalone computer. As the same setup was used for both test rigs and the focus has been to compare the performance between the rigs not so much effort has been put into investigating the performance of the particular hardware.

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To simulate real conditions weights were added on top of the wagons to resemble the most common conveyor, T4 which weighs approximately 30 kg. Since the bolt that the weights were mounted on with the air bearing design was too short to fit more plates the air bearing test rig was only loaded with 20 kg. The ball bearing test rig was however loaded with 30 kg. This should not be an issue as the mass was changed in the software to fit the actual Y-wagon in both cases. The test rigs were propped up on steel profiles since the cable chain travels underneath the frame of the Y-axis. The frame was fixed between a wall and some weights to make it as stable as possible. A picture of the test rigs can be seen in Figure 4.1. It turned out that the bearings were dimensioned for a lower force than they were actually exposed to so the motor had to be mounted at a flying height above the recommended to decrease the attraction force between the coils and the magnets. The pressure was also increased to 6 bar to give increased lift power. A close-up of the wagon with air bearings can be seen in Figure 4.2.

Figure 4.2: Close-up of the Y-wagon with air bearings

To ensure that the test rig was reliable comparisons were made between the ball bearing test rig and a MY100 machine. Movements with different lengths were made and the reference tracking of the two Y-wagons were compared. An example of this can be seen in Figure 4.3 which is zoomed in at where the position reference stabilizes at the desired position.

Figure 4.3: Comparison of the MY100 machine and the ball bearing test rig. The x-axis shows time in seconds and the y-axis shows position in meters

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It can be seen that both the machine and the test rig has a small overshoot. The main difference is that the overshoot of the test rig has a second bump which is not present in the machine. The initial idea was that this was caused by movements in the test rig due to the resultant force when the wagon braked but this could not be verified. The movement of the test rig was measured with a dial indicator to be maximum 0.1 mm when the Y-wagon moved with full stroke length, which can be considered sufficient for the testing. The likely cause of the behavior will be explained in detail later.

4.2 Bearing dynamics

As described in Chapter 3, the air bearings are probably not as rigid as the ball bearings so it is interesting to investigate how lateral and vertical forces will affect the wagon, as well as torque due to offsets in the center of mass. These forces might give rise to dislocations and oscillations in the wagon and disturbances to the encoder.

4.2.1 Model

A model of the damping of a porous media air bearing has been presented in Equation (3.2) (p. 11), and the stiffness of the bearings is 58kN/m as seen in Table 3.1 (p. 9). The bearing was modeled as a mass-spring-damper system. To get the dynamics of the wagon it is desired to calculate the equivalent stiffness and damping for all six bearings. The forces acting on the Y-wagon can be a vertical force Fz, caused by gravity, magnetic attraction or disturbances, or a horizontal force Fx, caused by vibrations in the machine. The forces can be divided in two components where the one parallel to the bearing will be absorbed by the structure holding the bearing in place and the normal force will be absorbed by the air film of the bearing. A sketch of the geometry can be seen in Figure 4.4. m y-wagon motor x z y Fz Fx Fn 40◦ 1

Figure 4.4: For a horizontal disturbance forceFxor a vertical force (gravity, magnetic attraction

or disturbances) Fz acting on the Y-wagon it is the component Fn that will be absorbed by the

air film of the bearing

The component Fn will be the one causing displacement of the bearing. Thus the stiffness and damping can be scaled according to Equation (4.1) which shows stiffness, although the same equations hold for damping.

kz = k · cos(40◦)

kx = k · sin(40◦) (4.1)

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m

kz,tot ddz,totz,tot

kx,tot dx,tot dx,tot 2kx 2dx 2dx 2kx 2dx 2dx rb rF Fmotor ϕ L W 2 Figure 4.5: Equivalent mass-spring-damper model of the Y-wagon

The dampings and stiffnesses were estimated as shown in Equation (4.2), once again shown only for k but valid for d also.

kz,tot = 2k + 4kz

kx,tot= 4kx (4.2)

The model in Figure 4.5 is represented by the equations in (4.3)

m¨z = F − dz,tot˙z − kz,totz

m¨x = F − dx,tot˙x − kx,totx (4.3)

Any offset of the motor force from the center of mass will give rise to a torque which will rotate the x and z axis. A sketch of the Y-wagon seen from above is shown in Figure 4.6. The force equation is shown in Equation (4.4), where J is the moment of inertia, calculated as in Equation(4.5). J ¨ϕ = FmotorrF − 4dxrb2ϕ − 4k˙ xrb2ϕ (4.4) J = m L 2_{+ B}2 12 (4.5) 4.2.2 Simulation

The step responses for x, z and ϕ have been simulated in Matlab. The simulations were done both with a damping corresponding to the theoretical damping in Equation (3.2) (p. 11) and for a damping where the dimensions of the bearings were divided by a factor of 4.4 corresponding to the discussion in Section 3.1.1.

Translation Simulations in x and z were done for a unit step, i.e. 1N. The transfer function from force to displacement in z is shown in Equation (4.6) and was derived from the Laplace transform of Equation (4.3). The corresponding transfer function has been used also for the displacement in x but with dx,tot and kx,tot.

Z(s) = 1

ms2_{+ d}

z,tots + kz,tot

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m

kz,tot ddz,totz,tot

kx,tot dx,tot dx,tot 2kx 2dx 2dx 2kx 2dx 2dx rb rF Fmotor ϕ L W 2

Figure 4.6: Sketch of the Y-wagon with with applied motor force seen from above

The results for the simulations can be found in Appendix A, the z-direction can be seen in Figure A.1 and the x-direction can be seen in Figure A.2. The theoretical damping would result in no oscillations whatsoever whereas the scaled damping would give significant oscillations. An illustration of the behavior can be seen in Figure 4.7.

0 0.01 0.02 0.03 0.04 0.05 0 0.5 1 1.5 2 2.5 3 3.5x 10 −9 Step response in z Time (seconds) Displacement [m]

(a) Actual bearing dimensions

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 1 2 3 4 5 6 7x 10 −9 Step response in z Time (seconds) Displacement [m]

(b) Scaled bearing dimensions Figure 4.7: Simulations of a unit step (1N) in thez-direction for different dampings

Rotation Rotation around the z and x axis will give rise to an offset of the encoder which could possibly cause faulty readings. The effect will be larger around the z direction which is shown in Figure 4.6 due to that the distance from the encoder to the center of mass is larger here. Simulations have been done in Matlab with the transfer function from force to angle in Equation (4.7).

Φ(s) = rF Js2_{+ 4d}

xrb2s + 4kxrb2

F (s) (4.7)

The force was chosen to the maximum used force during movement along the y-axis, i.e. at 10 m/s2_{. Like the translational simulations both the theoretical damping and the} scaled one were simulated and the plots can be found in Appendix A, Figure A.3. The simulations showed that the offset angle would be in the order of tenths of milli-degrees. The offset of the encoder in the y-direction was then calculated as in Equation 4.8 where renc is the distance from the center of mass to the encoder.

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This gave that the maximum offset of the motor force from the center of mass can be 10 cm in order for ∆y to be less than 2 µm which is the mechanical resolution of the encoder.

4.2.3 Testing

To measure the stiffness and damping of the air bearings tests were performed by apply-ing a force in the z-direction and measurapply-ing the step response in position of the Y-wagon. The force was applied by pressing down on the center of the wagon with a dynamometer to get a known force and then rapidly removing the pressure to get the step response. Measurements were done with capacitive sensors with a measuring range of 0-200 µm and the measured point was right above the flat rail bearings. To prevent motion in the Y-direction the wagon was secured by clamps. The test setup can be seen in Figure 4.8. The step responses of a step of approximately 100 N and 200 N can be seen in Figure

Figure 4.8: The test setup for measuring step responses in thez-direction

4.9. The capacitive sensors had analog output which is why there is some noise in the measurements. In the simulation the force was halved as it is assumed to spread evenly between the two sides of the wagon and the stiffness and damping 2k and 2d were used to represent the two bearings. No scaling factor was used for the bearing dimensions and it seems like the stiffness and damping corresponds well to the theoretical values. It must however be kept in mind that this is the step response for the entire wagon so the dynamics might differ from the actual bearings since the coupling is not entirely stiff. Another error source is the frame which might sag somewhat, but precautions were made by measuring as close to the supporting steel profiles as possible where very little movement of the frame could be detected.

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−0.050 0 0.05 0.1 2 4 6 8 10 12x 10 −7 Step response in z Time (seconds) Displacement [m]

(a) 200 N step response, the red line is the simulated step and the blue is the measured one

−0.05−1 0 0.05 0.1 0 1 2 3 4 5 6x 10 −7 Step response in z Time (seconds) Displacement [m]

(b) 100 N step response, the red line is the simulated step and the blue is the measured one Figure 4.9: Step responses in the z-direction, i.e. when a force is applied on top of the Y-wagon

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4.3 Dynamics of Y-wagon with ball bearings

Starting from the equations in Section 3.2, the motor and Y-wagon can be described by the following equation:

m¨x = Kf · i − Ff ric( ˙x) − Fcog(x) − Fdis (4.9) where the linear displacement is denoted x, Ff ricis combined static and dynamic friction, Fcog is the cogging force and Fdis is a collective term for other disturbances. The servo amplifier has been regarded as an ideal current source and the electrical properties of the motor will thus not be modeled. The motor force constant Kf was regarded as constant even though the motor at maximum acceleration is operated above the region where the relation between current and force is linear. The disturbances were considered unknown so the interesting forces to investigate has been friction and cogging.

4.3.1 Friction

The friction force Ff ric( ˙x) has been modeled as Coulomb friction plus viscous friction which gives the model in Equation (4.10).

Ff ric( ˙x) = (

Fstatic, if ˙x = 0 Fkinetic+ D · ˙x, if ˙x 6= 0

(4.10)

The challenge was to identify the parameters Fstatic, Fkinetic and D. The following methods have been used for doing this:

Static friction

The static friction was tested in two ways:

1. By measuring the force that is required to move the wagon with a dynamometer. The power supply was completely turned off so that no current ran through the motor.

2. By applying a slowly increasing current to the motor and see what value was required to move the wagon.

Due to the cogging force there were variations along the axis but Fstatic was determined to be approximately 40 N.

Kinetic and viscous friction

The combination of kinetic and viscous friction will give rise to a force during movement. Since the viscous friction is proportional to the speed the two parameters were measured by running the wagon at various constant speeds and plot the mean current versus the speed. The tests were done in both directions and for all speeds divisible by 100 ranging from 100 mm/s to 2000 mm/s, giving a total of 40 measurements.The mean current was plotted agains the current and a first order polynomial was fitted to the data points, both for positive and negative speeds. The polynomials for the positive and negative speeds is shown in Equation (4.11). The difference is very small and might depend on measurement errors. The alternative that the frame is slanting was ruled out by measuring with a spirit-level.

Ff ric( ˙x) = (

Ff ric= 10.91 ˙x + 27.92, if ˙x > 0 Ff ric= 9.87 ˙x − 26.34, if ˙x < 0

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The friction was estimated as the mean value of these two, i.e.

Ff ric= 10.34 ˙x ± 27.13

A plot of the estimated friction along with the measured data is shown in Figure 4.10. In addition to the measurements between 0.1 m/s and 2 m/s measurements were made with an interval of 0.01 m/s between 0.01 mm/s and 0.09 mm/s which captures the Stribeck effect. −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −60 −40 −20 0 20 40 60 Velocity [m/s] Force [N] Friction force

Figure 4.10: Measured (asterisks) and estimated (line) friction force

4.3.2 Cogging

The closed loop cogging identification from Section 3.4.5 was used to investigate the cog-ging force. The same data as in the friction identification was used but the mean current at the constant speed phase of the movement was subtracted from the current meas-urements and the residual current times the motor force constant was plotted against position which can be seen as the blue lines in Figure 4.11.

0.2 0.4 0.6 0.8 1 1.2 −50 −40 −30 −20 −10 0 10 20 30 40 50 Position [m] Force [N]

Force variations at constant speed

Figure 4.11: Measured cogging force, the blue line is the measured force variation and the red line is the modeled one

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A theoretical cogging frequency was stated in [7] to be 2π

Ec · ˙x which is shown as the red sine curve in the figure. The figure shows an apparent correlation between the measured force variation and the theoretical cogging force. The equation which generated the sine curve was also used as the inverse estimation of the cogging force and is shown in Equation (4.12).

Fcog= 10 sin( 2π Ec

· x + 0.5) (4.12)

The Y-wagon was modeled in Simulink to be able to compare the model with the test rig. The model can be seen in Appendix B, Figure B.1. The friction model described in Equation (4.10) as well as the cogging force described in Equation (4.12) has been implemented in the model. There is also a limitation in how much voltage the servo amplifier can supply. The model has been compared to the test rig by making a step in the current. For low currents the cogging is clearly visible which can be seen in the comparison between model and test rig shown in Figure 4.12.

0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 time [s] velocity [m/s]

(a) Simulink model

0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 time [s] velocity [m/s] (b) Test rig

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4.4 Dynamics of Y-wagon with air bearings

The model of the Y-wagon with air bearings starts from Equation (4.9) just like the model of the Y-wagon with ball bearings. To as far as possible eliminate errors from the models the same parameter identification tests were done as in the previous section.

4.4.1 Friction

The friction measurements can be seen in Figure 4.13. The values are far above the theoretical zero static friction, which should depend mainly on the cable chain. It will be called friction throughout the report for simplicity but it is implied that this really means the collected forces from the cable chain.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −20 −15 −10 −5 0 5 10 15 Velocity [m/s] Force [N] Friction force

Figure 4.13: Measured (asterisks) and estimated (line) friction force with air bearings

It is interesting that both the values and the shape of the curves are different in the positive and the negative direction. This likely depends on that the negative direction corresponds to pulling the cable chain out of its’ holder, which also includes lifting it. This should require more force than pushing the chain back and allowing it to fall down into the holder again. As opposed to the friction of the ball bearing test rig there is no stick-slip effect but the force steadily decreases as the velocity is lowered. The friction coefficients can be seen in Equation (4.13).

Ff ric( ˙x) = (

Ff ric = 2.56 ˙x + 7.39, if ˙x > 0 Ff ric = 4.54 ˙x − 9.74, if ˙x < 0

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4.4.2 Cogging

The measured cogging force can be seen in Figure 4.14. Here the controller was slower than the one used for measuring cogging with the ball bearings so the signal is less noisy and it is possible to see not only the cogging frequency in Equation (4.12) but also the second harmonic of this. The resulting cogging force can be seen in Equation (4.14)

Fcog = 5 sin( 2π Ec · x + 1.8) + 6 sin(2π Ec · x + 2.0) (4.14) 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 −50 −40 −30 −20 −10 0 10 20 30 40 50 Position [m] Force [N]

Force variations at constant speed

Figure 4.14: Measured cogging force with air bearings, the blue line is the measured force variation and the red line is the modeled one