Eindhoven University of Technology MASTER Modelling, identification and multivariable control of an active vibration isolation system Rademakers, N.G.M.

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Eindhoven University of Technology

MASTER

Modelling, identification and multivariable control of an active vibration isolation system

Rademakers, N.G.M.

Award date:

2005

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Multivariable Control of an Active Vibration Isolation System

Master’s Thesis

N.G.M. Rademakers Report No. DCT 2005.63

Supervisor: Prof. Dr. H. Nijmeijer Coaches: Dr. Ir. I. Lopez

Dr. Ir. R. Hensen, DAF Trucks. N.V.

Ir. S. Kerssemakers, Irmato Group Eindhoven University of Technology Department of Mechanical Engineering Dynamics and Control Group

Eindhoven, June 7, 2005

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This report, Modelling, Identification and Multivariable Control of an Active Vibration Isolation System, covers the master’s thesis study of the author, which has been performed within the Dynamics and Control Group of the faculty of Mechanical Engineering at the Eindhoven Technical University, under the supervision of prof. dr. Henk Nijmeijer. The experimental set-up that is considered in this thesis, the AVIS, has been developed and built at IDE Engineering.

The report is accompanied by a CD-ROM, which contains an electronic version of this report, as well as a presentation and a poster of this research.

Eindhoven, May 2005

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ii

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In various leading-edge technology areas, such as semiconductor and optical instruments industry, the requirement for micro-vibration isolation is increasing. Vibrations of unknown frequency and amplitude may limit the performance of the sensitive equipment that is used in these industries dramatically. One approach to reduce vibrations, is to use a vibration isolation system.

The vibration isolation system that is considered in this thesis, the Active Vibration Isolation System (AVIS), has been developed at IDE Engineering. The AVIS is capable of six degree- of-freedom active vibration control and mainly consists of a payload and a chassis which are interconnected by means of four isolator modules. The current control system of the AVIS is a six-loop Single-Input-Single-Output (SISO) feedback controller, which is iteratively designed using classical loop-shaping techniques. However, FRF measurements show that there are strong couplings between the six axes, which means that the AVIS should be treated as a Multi-Input- Multi-Output (MIMO) system rather than a combination of six SISO systems.

The long-term goal of the vibration isolation project is to develop a six degree-of-freedom multivariable controller for the AVIS. A first step towards this goal and the objective of this study is to develop a six degree-of-freedom multivariable model of the AVIS. Moreover, to identify its physical parameters and design a robust Multi-Input-Multi-Output (MIMO) feedback controller for a selection of two coupled degrees-of-freedom.

For the development of a six degree-of-freedom, multivariable model of the AVIS, the payload and chassis of the AVIS are considered as rigid bodies, whereas the isolator modules are modelled as three orthogonal spring-damper combinations, complemented with two actuators working in horizontal and vertical direction. The equations of motion of the payload are derived using Newton- Euler equations.

A Continuous-Discrete Extended Kalman Filter (CDEKF) is selected for the identification of the physical parameters of the AVIS. The physical parameters of the AVIS that are identified with the CDEKF are the mass inertia and the damping coefficients and stiffness of the springs for each isolator. The CDEKF has been implemented in Matlab resulting in a converging parameter- set. Validation of the model and parameter-set is done by comparison of FRFs using the model and parameter-set with FRF-measurements. This comparison has demonstrated that a accurate mathematical description of the AVIS has been obtained.

Furthermore, multivariable controller design for the AVIS is studied. The so-called µ-theory has been adopted to design a multivariable feedback controller for two axes of the AVIS, i.e. the translation along the z-axis and the rotation around the x-axis. The µ-theory provides powerful tools to design a robust controller for systems in which structured perturbations are present, such as the AVIS. The feedback control problem is translated into a mathematical optimization problem, which is solved yielding an optimal controller. The controller has been implemented and its performance has been evaluated.

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Voor verscheidene vooruitstrevende technologie¨en, zoals de half-geleider en de optische industrie, neemt de vraag naar het onderdrukken van micro-trillingen alsmaar toe. Trillingen met een on- bekende frequentie en amplitude kunnen de prestatie van de gevoelige apparatuur, die gebruikt wordt in deze industrie, drastisch beperken. E´en manier om trilling-isolatie te realiseren, is door gebruik te maken van een trilling-isolatie systeem.

Het trilling-isolatie systeem dat gebruikt wordt in dit onderzoek, de AVIS (Active Vibration Iso- lation System), is ontwikkeld door IDE Engineering. De AVIS is in staat om trilling-isolatie in 6 vrijheidsgraden te realiseren. De AVIS bestaat voornamelijk uit een tafelblad en een chassis die met elkaar verbonden zijn door middel van vier isolator modules. Het huidige regelsysteem van de AVIS bestaat uit zes SISO regelaars, die op een iteratieve wijze ontworpen zijn, gebruik makend van klassieke regeltechnieken. Echter, frequentie responsie metingen laten zien dat er een sterke koppeling bestaat tussen de zes assen van de AVIS. Dit betekent dat de AVIS beschouwd dient te worden als een multivariabel systeem in plaats van een combinatie van zes SISO systemen.

Het lange termijn doel van het trilling-isolatie project is het ontwerpen van een multivariabele regelaar, waarmee trilling-isolatie gerealiseerd kan worden voor alle zes vrijheidsgraden van de AVIS. Het doel van dit onderzoek is ontwerpen van een multivariabel model voor de zes vrijheidsgraden van de AVIS, de identificatie van de parameters van dit model en het ontwerpen van een robuuste multivariabele regelaar voor een combinatie van twee gekoppelde vrijheidsgraden. Deze doelstelling kan gezien worden als een eerste stap om het lange termijn doel te bereiken.

Voor het ontwikkelen van een multivariabel model voor zes vrijheidsgraden van de AVIS, zijn het tafelblad en het chassis beschouwd als starre lichamen. De isolator modules zijn gemodelleerd als drie orthogonale veer/demper combinaties. Bovendien zijn per isolator twee actuatoren aanwezig, die werken in horizontale en verticale richting. De bewegingsvergelijkingen voor het tafelblad zijn afgeleid met behulp van Newton-Euler vergelijkingen.

Een Continue-Discreet Extended Kalman Filter (CDEKF) is gekozen voor de identificatie van de fysische parameters van de AVIS. De geschatte parameters zijn de massa, de massa-traagheid en de demping-constanten en veer-constanten voor iedere isolator module. Het CDEKF is ge¨ımplemen- teerd in Matlab, resulterend in een convergerende parameter verzameling. Het model en de bi- jbehorende parameters zijn gevalideerd door de frequentie responsies van het model te vergelijken met frequentie responsies die gemeten zijn aan het echte systeem.

Als laatste is het ontwerpen van een multivariabele regelaar voor de AVIS bestudeerd. De zoge- naamde µ-theorie is gekozen voor de ontwikkeling van een dergelijke regelaar voor twee vrijheidsgraden van de AVIS. Deze vrijheidsgraden zijn de translatie langs de z-as en de rotatie om de x-as. De µ-theorie verschaft de middelen om een robuuste regelaar te ontwerpen voor systemen zoals de AVIS, waarin verstoringen aanwezig zijn die in een bepaalde structuur gegoten kunnen worden. Het regel-probleem is vertaald in een wiskundig optimalisatie probleem en is vervolgens opgelost, resulterend in een optimale regelaar. De regelaar is ge¨ımplementeerd en de prestatie van deze regelaar is beoordeeld.

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vi

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Preface i

Summary iii

Samenvatting v

1 Introduction 1

1.1 Background . . . 1

1.2 Micro-vibration isolation . . . 2

1.3 Motivations and Objective . . . 3

1.4 Research Approach . . . 4

1.5 Outline . . . 4

2 Active Vibration Isolation System 7 2.1 Experimental Set-up . . . 7

2.1.1 Isolation Table . . . 7

2.1.2 Controller unit . . . 8

2.2 Model of the AVIS . . . 9

2.2.1 Kinematics . . . 9

2.2.2 Dynamics . . . 10

2.2.3 Nonlinear Model of the AVIS . . . 13

2.2.4 Linearized Model of the AVIS . . . 16

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viii CONTENTS

2.3 Discussion . . . 16

3 Identification of the AVIS 17 3.1 Continuous-Discrete Extended Kalman Filter . . . 17

3.1.1 Extended Dynamic Model . . . 18

3.1.2 Propagation . . . 18

3.1.3 Measurement update . . . 18

3.2 Identification Procedure . . . 19

3.3 Setting of the Extended Kalman Filter . . . 20

3.4 Identification Results . . . 21

4 Robust Control Design 27 4.1 General control problem formulation . . . 28

4.1.1 Control Goals . . . 28

4.1.2 H_∞/µ-framework . . . 28

4.2 Robust Control Design based on H_∞Theory . . . 29

4.2.1 Nominal Stability . . . 29

4.2.2 Nominal Performance . . . 30

4.2.3 Robust Stability . . . 30

4.2.4 Robust Performance . . . 32

4.2.5 Computation of the H_∞ Controller . . . 33

4.3 Robust Control Design based on µ Theory . . . 34

4.3.1 µ Analysis . . . 34

4.3.2 µ Synthesis . . . 35

5 Robust Control of the AVIS 37 5.1 2-DOF Control Problem . . . 37

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5.2 Nominal Model . . . 38

5.2.1 Equations of Motion . . . 38

5.2.2 Identification of the 2-DOF model . . . 40

5.3 Robustness and Performance Specifications . . . 40

5.3.1 Robustness Specification . . . 41

5.3.2 Performance Specification . . . 43

5.4 Controller Synthesis and Performance Evaluation . . . 46

6 Conclusions and Recommendations 53 6.1 Conclusions . . . 53

6.2 Recommendations . . . 54

Bibliography 57 A Technical information of the AVIS 59 A.1 Technical specification of the AVIS . . . 59

A.2 Linear Force Actuator . . . 59

A.3 Geophone . . . 60

A.4 Actuator and Sensor Matrix . . . 61

B Linearized Model of the AVIS 65 B.1 Inertia Matrix . . . 65

B.2 Entries of the Damping Matrix . . . 66

B.3 Entries of the Stiffness Matrix . . . 67

B.4 Actuator Matrix . . . 68

C FRFs of the six degree-of-freedom model 69

D SISO Control 77

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x CONTENTS

E Identification of the 2-DOF AVIS model 81

Acknowledgements 83

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Introduction

1.1 Background

In various leading-edge technology areas, the requirement for micro-vibration isolation is increasing. Vibrations of unknown frequency and amplitude may limit the performance of sensitive equipment dramatically. This is particularly true for experiments or processes where the typical amplitudes of the ambient vibration and the dimensions of the investigated or manufactured ob- jects fall in the same range, such as in the field of high-resolution measurement and high-precision manufacturing processes.

A good example of high-resolution measurement equipment is the scanning electron microscope (SEM) as depicted in Figure 1.1(a). Electron microscopes are designed to resolve features of materials down to a few nanometers in size. An electron gun emits a beam of high energy electrons.

This beam travels downward through a series of magnetic lenses designed to focus the electrons to a very fine spot. As the electron beam hits each spot on the sample, secondary electrons are knocked loose from its surface. A detector counts these electrons and sends the signals to an amplifier. The final image is built up from the number of electrons emitted from each spot on the sample. Micro-vibrations can generate internal relative motion along a beam path that either blurs an optical image or causes an electron beam to deviate from its intended path. Therefore, it is essential to isolate electron microscopes from unwanted vibrations.

Another example is found in the semiconductor industry. Lithography systems transfer circuit patterns onto silicon wafers to make every type of integrated circuit (IC) that is used in modern electronic equipment. The circuit pattern is projected onto the wafer through a carefully constructed lens, see Figure 1.1(b). Over the last two decades, the semiconductor industry has been dealing with progressively smaller line widths. One of the future bottlenecks in decreasing the line width, and thus in the miniaturization of ICs, is that the lens of the wafer stepper suffers from vibrations, which affect the focussing of the image and the overlay [9].

These two examples underline that, as technology continues to advance, the need for vibration isolation becomes increasingly necessary.

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2 1.2. MICRO-VIBRATION ISOLATION

e l e c t r o n g u n i l l u m i n a t i n g l e n s s y s t e m

s c a n c o i l s f i n a l l e n s

s p e c i m e n s e c o n d a r y e l e c t r o n s

t o p u m p s

T V s c r e e n

d e t e c t o r

(a) Schematic Representation of a SEM, [1]

a r c l a m p m a s k ( r e t i c l e )

l e n s w a f e r

w a f e r

s t a g e f r a m e

b a s e

(b) Schematic Representation of a wafer stepper, [6]

Figure 1.1: Sensitive equipment

1.2 Micro-vibration isolation

Historically, vibration isolation has been handled with passive systems, which are cost effective and work well to attenuate high frequency base or floor motion. Passive vibration isolation systems generally consist of a payload mounted on elastic springs and damping units, as schematically depicted in Figure 1.2(a). The transmissibility, which is the transfer function between the displacement of the ground and the displacement of the payload, of a passive vibration isolation system is given by:

T_p(s) = q qground

= 2ξω_ns + ω_n²

s²+ 2ξωns + ω_n², (1.1)

where ω_n=pk/m is the natural frequency of the passive system and the amount of damping is defined by the damping ratio ξ = b/(2mω_n). The combination of a mass and the spring is known as a mechanical low-pass system. The mechanical response of the spring-mass system decreases significantly for frequencies above the eigenfrequency, and the damper reduces the vibration amplitude especially within the resonance range, see Figure 1.2(b).

Because of the low-pass characteristic, passive isolation systems are designed with very low eigen- frequencies. However, it is still difficult to set the natural frequency of the passive system less than 1.0 Hz in a practical application. Also, amplification of disturbance at the natural frequency is often observed. This problem can, in principle, be solved by increasing the structural damping.

However, a major disadvantage of passive damping is that the disturbance rejection properties deteriorate, since the zero of (1.1) shifts to the left. Moreover, the passive type is, in principle, powerless against a direct disturbance acting on the payload, such as an exciting force generated in a mounted operating machine, airflow from the air conditioning and the sound pressure. From these points of view, significant benefit in vibration isolation can be obtained by active control of the response near the natural frequency of the passive system.

Active vibration isolation can be achieved by feedback, feedforward or a combination of both.

In general, the feedback is based on the absolute velocity of the payload. When a proportional controller gain, ^K_m^p, is used, the active control force is proportional to the absolute velocity of the

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k qqg r o u n d

bs k y h o o k P a s s i v e V i b r a t i o n I s o l a t i o n

A c t i v e V i b r a t i o n I s o l a t i o n

(a) Mass-spring-damper system

10⁻¹ 10⁰ 10¹ 10²

-50 -40 -30 -20 -10 0 10 20

Passive Active

Magnitude[dB]

Frequency [Hz]

(b) Transmissibility

Figure 1.2: Passive and Active Vibration Isolation

payload and the transmissibility becomes:

T_a(s) = q qground

= 2ξω_ns + ω²_n

s²+ (2ξωn+ Kp) s + ω²_n (1.2) This technique is called skyhook damping. The name skyhook damping results from the fact that the force achieved by the controller could conceptually be achieved with a passive damper connecting the payload to a fixed point in space, see Figure 1.2(a).

In Figure 1.2(b), the transmissibility of a passive and an active system is shown. Comparison of the passive and active transmissibility reveals the complete absence of resonance for the active situation. With this the advantage of an active vibration isolation system compared to a passive isolation system is illustrated.

1.3 Motivations and Objective

Much work has been done during the past years toward the development of active isolation systems and many designs are considered such as a Steward Platform using soft actuators, [17], a six degree- of-freedom micro-vibration system using giant magnetostrictive actuators, [13], a micro-gravity Vibration Isolation System for the International Space Station, [21] and many more.

The vibration isolation system that is considered in this thesis, has been developed at IDE En- gineering, the Active Vibration Isolation System (AVIS). The AVIS is a six degree-of-freedom, passive-active system that uses air springs in combination with linear motors and geophones. The current control system of the AVIS is a six-loop Single-Input-Single-Output (SISO) feedback controller, which is iteratively designed using classical loop-shaping techniques, as described in [24].

However, FRF measurements show that there is strong, dynamic coupling between the six axes.

From this it appears that the AVIS should be treated as a Multi-Input-Multi-Output (MIMO) system rather than, a combination of six SISO systems. Consequently, the long term goal of this project is to design a six degree-of-freedom, multivariable controller for the AVIS. However, since the procedure to design a six degree-of-freedom controller is similar to the design of a two degree-of-freedom controller. Moreover, it is easier to study the principles involved for the two degree-of-freedom case and therefore the objective of this study is defined as follows:

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4 1.4. RESEARCH APPROACH

Develop a six degree-of-freedom, multivariable model of the AVIS, and identify its physical parameters. Select a combination of two coupled degrees of freedom and design a robust Multi-Input-Multi-Output (MIMO) feedback controller for these degrees of freedom.

For multivariable control, classical manual loopshaping techniques are not directly applicable, since the concepts of gainmargin and phasemargin for SISO systems do not generalize easily for MIMO- system. Therefore other controller synthesis techniques have to be considered. The traditional multivariable controller, is the Linear Quadratic Gaussian (LQG) regulator. A major disadvantage of this controller is that robustness is not guaranteed.

An alternative way to design MIMO controllers is by H_∞-optimization and µ-synthesis. These methods amount to translating the control design problem to a mathematical optimization problem, from which a (robust) controller results. The H_∞ robust control design method is based on singular values of a closed-loop system and the assumption of unstructured model uncertainty.

However, it turns out that the vibration isolation problem does not readily fit the standard H_∞ problem setup since the involved model uncertainty is structured rather than unstructured. This causes any H∞ controller design to be potentially conservative and thus limits the obtainable performance of the closed loop system. Therefore, the structured singular value or µ-theory is applied, which handles these problems in a non-conservative way.

1.4 Research Approach

To achieve the objective, the following research approach is followed.

• For the development of a full six degree-of-freedom dynamic model, first a kinematic model of the AVIS is formulated. Newton’s second law is applied for the derivation of the translational dynamics, whereas Euler equations are used to describe the rotational dynamics of the AVIS.

• A continuous-discrete extended Kalman filter is designed for the identification of the set of unknown physical parameters of the resulting dynamic model. In order to achieve a faster convergence of the parameters, the filter is used iteratively.

• Subsequently, a combination of two axes of the AVIS is selected which are subjected to MIMO Control. The other degrees-of-freedom are controlled using classical loop-shaping techniques. This makes it possible to neglect the interaction of the SISO controlled axes on one hand and the MIMO controlled axes on the other hand, around the resonance frequency.

Consequently, the MIMO controlled axes can be considered as a 2 x 2 isolated subsystem for which an independent MIMO controller design is justified.

• For the MIMO control problem, the control goals are defined and translated into a mathematical optimization problem. The mathematical optimization problem is solved using the D-K iteration procedure [3], resulting in a robust multivariable controller for the AVIS.

• Finally, the controller is implemented and its performance is evaluated.

1.5 Outline

This thesis is organized as follows. In Chapter 2, the Active Vibration Isolation System that is considered in this thesis will be discussed. The description of the AVIS will be provided, after

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which the modelling will be treated. Moreover, the equations of motion of the full six degree-of- freedom model will be presented. The subject of Chapter 3 is the identification of the physical parameters of the AVIS. First, the continuous-discrete extended Kalman filter, which is used for the identification, will be derived. Subsequently, the identification procedure will be described, the setting of the filter will be discussed and the identification results will be presented. Robust control theory is the subject of Chapter 4. The general control problem will be formulated and the control goals will be translated into a mathematical optimization problem using singular values (H_∞) and structured singular values (µ). In Chapter 5, the µ-theory will be applied to two degrees of freedom of the AVIS. Weighting filters will be designed to shape closed-loop transfer functions, such that the performance and robustness criteria are satisfied. The resulting robust µ-controller will be implemented and its performance evaluated. Finally, some conclusions will be drawn and recommendations will be given in Chapter 6.

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6 1.5. OUTLINE

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Active Vibration Isolation System

In this chapter, the active vibration isolation system that is considered in this thesis, the AVIS, is introduced. In Section 2.1, a general description of the elements of the experimental set-up will be provided. Subsequently, a six degree-of-freedom model of the AVIS will be derived in Section 2.2.

First, the kinematics of the AVIS will be formulated and the Newton-Euler approach will be used to derive the nonlinear, dynamic model of the AVIS. Because of the high complexity, the nonlinear model will be linearized around its operation point. A discussion of the results will be given in Section 2.3.

2.1 Experimental Set-up

The AVIS, which is depicted in Figure 2.1, has been designed and built by IDE Engineering and is capable of six degree-of-freedom vibration isolation. This section summarizes the basic system architecture of the AVIS, which mainly consists of an isolation table and a digital controller unit.

The technical specification of the AVIS is given in Appendix A.1.

2.1.1 Isolation Table

The isolation table consist of a tabletop (payload) and a chassis which are interconnected by means of four isolator modules. The top view of the isolation table is schematically represented in Figure 2.2. The isolator module, which is depicted in Figure 2.3, contains a pneumatic air mount.

The mass of the payload in combination with the four air mounts provides passive isolation in six degrees of freedom. The vertical natural frequency is primarily a function of the ratio of the isolator piston area to the isolator air volume, whereas the horizontal natural frequency of the isolator is based solely upon the stiffness characteristics of the all-metal rolling diaphragm, [10].

The resonance frequency of the six logical axes of the passive system lies between 1.5 Hz and 5 Hz.

Furthermore, isolators number 1, 2 and 3 contain a mechano-pneumatic leveling system, which can be considered as a mechanical controller for coarse position control of the payload.

The passive system is complemented with electrodynamic force actuators and geophones to achieve active vibration isolation. Each isolator incorporates two linear motors, which are placed in horizontal and vertical direction. Current amplifiers are used to supply the actuators. The working

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8 2.1. EXPERIMENTAL SET-UP

Figure 2.1: Active Vibration Isolation System

principle of the linear motors is explained in Appendix A.2. The isolators numbered 1, 2 and 3 contain two geophones to obtain velocity information in horizontal and vertical direction. Geophones are inertial sensors based on the production of a voltage in a coil when a magnet is passed through it. This means that the geophone behaves as a velocity sensor above the suspension frequency and as a jerk sensor below the suspension frequency. In order to obtain velocity information for frequencies below the suspension frequency as well, the geophone signals are prefiltered. The geophone-signals are amplified using voltage amplifiers. A detailed description of the geophone compensation is given in Appendix A.3.

Since the actuators are located in the isolators, an actuator matrix is constructed to relate the actuator output of the individual actuators to the actuator output in terms of logical axis of the system. This is done by determination of the polarity of the actuators and using kinematic relations. In the same way, a sensor matrix is constructed. The actuator-matrix and sensor-matrix are given in Appendix A.4.

2.1.2 Controller unit

For implementation of the controllers and communication with the AVIS, a Quanser Q8 board is used in combination with xPC Target in Matlab, [12]. xPC Target provides a high-performance, host-target prototyping environment that enables to connect Simulink models to physical systems and execute them in real-time on PC-compatible hardware. The Quanser Q8 Hardware-in-the- Loop control board is a real-time measurement system and control board, supported by xPC Target, which is used as the interface between the isolation table and the control software. The Q8 has 8 single-ended analog inputs with 14-bit resolution. All 8 channels can be sampled simul- taneously at 100 kHz. The Q8 is equipped with eight analog outputs, with software programmable voltage ranges and simultaneous update capability with an 8 µs settling time.

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x y

z A y 1

A z 1

A z 2

A x 2

A z 3

A y 3

A ^{z 4} A x 4

S ^{z 1}

S z 3

S z 2

S y 3

S x 2

S ^{y 1}1

23 4

Figure 2.2: Schematic representation of the

AVIS Figure 2.3: Isolator Module

2.2 Model of the AVIS

This section presents general approaches to compute a six degree-of-freedom kinematic and dynamic model of the AVIS. Since we are dealing with a vibration isolation isolation system, ground vibrations, which are transmitted through the chassis to the payload, are taken into account as non-manipulated inputs for this model. The kinematic model formulation is based on a rigid body model of the chassis and payload of the AVIS. The Newton-Euler approach is used for the derivation of the dynamics and has been implemented and applied in MATLAB, resulting in an explicit, nonlinear model of the AVIS. The nonlinear model is linearized around its operating point.

2.2.1 Kinematics

The first step in the modelling of the AVIS is the formulation of the kinematics. The kinematic model describes the position and orientation of the payload and the chassis of the AVIS as a function of the time. The kinematic problem is to find a transformation matrix that relates a body-attached coordinate frame to the reference coordinate frame.

For the modelling of the AVIS, there are five primary axes systems considered, which are depicted in Figure 2.4.

• The first axes system is the earth initial axes system, −→e^e. This inertial frame is required for the application of Newton’s laws.

• The second axes system is the payload-carried inertial axes system, −→e^p0. This axis system is obtained if the Earth inertial axes frame is translated to the center of mass of the payload with a vector

−

→rCM=

x y z −→e^e. (2.1)

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10 2.2. MODEL OF THE AVIS

• The payload axes system, −→e^p, is also a payload-carried axes system. The axis system is obtained through successive rotations of the payload-carried inertial frame with Tait-Bryant angles (ψ, θ, φ) about the rotation axes (3,2,1).

• The fourth axes system is the chassis-carried inertial axes system, −→e^ch0. This axis system is obtained if the earth inertial frame is translated to the chassis of the AVIS with a vector

−

→r_ch =

x_ch y_ch z_ch −→e^e (2.2)

• The last axes system is the chassis coordinate frame, −→e^ch. This is also a chassis-carried axes system. The axis system is obtained through successive rotations of the chassis-carried inertial frame with Tait-Bryant angles (ψ_ch, θ_ch, φ_ch) about the rotation axes (3,2,1).

The rotation from the body-fixed coordinate frames, −→e^p or −→e^ch, to the earth fixed coordinate frame, −→e^ecan be described by the direction cosine matrix A^ie(α):

−

→eⁱ= A^ie(α)−→e^e. (2.3)

where i denotes the body-fixed coordinate frame (p or ch) and α is a vector with the respective rotation angles. The direction cosine matrix is given by:

A^ie=





Cα₃Cα₂ Sα₁Sα₂Cα₃− Cα₁Sα₃ Cα₁Sα₂Cα₃+ Sα₁Sα₃

Sα₃Cα₂ Sα₁Sα₂Sα₃+ Cα₁Cα₃ Cα₁Sα₂Sα₃− Sα₁Cα₃

−Sα2 Sα1Cα2 Cα1Cα2



. (2.4)

The angular velocity of the body-attached frame −→eⁱwith respect to frame −→e^eis described by the angular velocity,

ieω^e =





− ˙α₃sin α₂+ ˙α₁

˙

α₂cos α₁+ ˙α₃sin α₁cos α₂

− ˙α₂sin α₁+ ˙α₃cos α₁cos α₂



 (2.5)

= w^e^T(α) ˙α, (2.6)

where w^e^T(α) is a row of axial-vectors of the rotation tensor,

R = −→eⁱ^T−→e^e, (2.7)

with respect to frame −→e^e.

2.2.2 Dynamics

There are several approaches to derive dynamic models for multibody systems. One suitable method for the derivation of the dynamic model of the AVIS, is the Newton-Euler approach.

Newton’s second law is applied to relate the translational dynamics of the payload of the AVIS to the resultant of all external forces acting on it, while Euler equations are used to describe the rotational dynamics of the payload.

Newton-Euler equations

In this section, the Newton-Euler equations will be discussed briefly. A detailed derivation of the Newton-Euler equations can be found in [20].

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e

1

r

ee3

r ee 2

r03p

er

pe 3

r pe 2

r

pe1

r

01p

er

02p

er

rrC M

f y

q

qy f

rr

c h

03c h

er e3c h

r

e

2c h

r

e 1c h

r

01c h

er ^y ^{c h} ^q ^{c h}

f c h

y c h

q c h

f c h

er

2c h0

Figure 2.4: AVIS coordinate systems

Newton’s second law for a rigid body is:

X−→ F = d

dtm ˙−→r_CM, (2.8)

where P −→

F is the resultant of all external forces applied to the body with mass m. ˙−→rCM is the velocity vector of the center of mass (CM) of the body relative to the earth inertial frame. Because the mass of the payload is constant, (2.8) can be written as:

X−→

F = m ¨−→r_CM, (2.9)

where ¨−→rCM is the linear acceleration vector of the body relative to the earth inertial frame.

−¨

→rCM =

¨

x y¨ z¨ −→e^e (2.10)

Euler equations are applied for the rotational dynamics of the payload. The rotational dynamics can be described as:

X−→

M_CM=−→˙

H_CM, (2.11)

whereP −→

M_CM is the resultant of the external moments applied to a body relative to its center of mass. −→

H_CM is the angular momentum vector relative to the center of mass of the payload. The angular velocity vector can be described by:

−

→H_CM = J_CM−→ω , (2.12)

where J_CM is the inertia tensor of the rigid body and −→ω is the angular velocity vector. From (2.11) and (2.12), the rotational equations of motion derived in the earth inertial axes are:

X−→

M_CM = d dt

A^pe^TJ_CM^p A^{pe pe}ω^eT

−

→e^e, (2.13)

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with:

J_CM^p =





J1 0 0 0 J2 0 0 0 J3



. (2.14)

and A^pe and^peω^e are the direction cosine matrix and angular velocity vector respectively.

Forces and Moments

The forces and moments at the center of mass of the payload consist of forces due to the air mounts and forces due to the actuators. The air mounts are modelled as three springs and three dampers which are aligned along the axes of the earth inertial axis frame as depicted in Figure 2.5.

The force generated by the springs is given by:

−

→F_k,i = −

k_i,1 k_i,2 k_i,3 −→e⁰· (−→r_CM+ −→p_i− −→c_i− −→r_ch) , (2.15) where i denotes the isolator number, k_i,1 is the stiffness of the spring in −→eê₁ direction, k_i,2 the stiffness in −→eê₂ direction and k_i,3the stiffness in −→eê₃direction. −→r_CM is the position vector of the center of mass (CM) of the payload relative to the earth inertial frame and −→r_ch is the position vector of the chassis of the AVIS relative to the earth inertial frame. Moreover, −→p_iis a body-fixed vector of the point where the force is applied on the payload relative to the body-fixed frame and

−

→c_iis a body-fixed vector of the point where the force is applied on the frame of the AVIS relative to the chassis-carried inertial frame. These vectors are depicted in Figure 2.6.

The force generated by the dampers is given by:

−

→Fb,i= −

bi,1 bi,2 bi,3

−→e^e· ˙−→rCM+ ˙−→p_i− ˙−→ci−−→r˙ch

, (2.16)

where i denotes the isolator number, bi,1 is the damping coefficient in −→eê₁ direction, bi,2 the damping coefficient in −→eê₂ direction and b_i,3the damping coefficient in −→eê₃direction. ˙−→r_CM is the linear velocity vector of the center of mass of the payload relative to the earth inertial frame and

−˙

→rch is the linear velocity vector of the chassis of the AVIS relative to the earth inertial frame.

Moreover, ˙−→p_i is the velocity vector of the point where the force is applied on the payload relative to the body-fixed frame and ˙−→ci is the velocity vector of the point where the force is applied on the base of the AVIS relative to the chassis-carried inertial frame.

The AVIS is equipped with 8 linear motors. The force generated by these actuators is given by:

F_a= τ u, (2.17)

where τ is the actuator constant and u the current, which is used as the control input. The actuators for isolator 1 and isolator 3 provide forces in −→eê₂and −→eê₃ direction, while the actuators for isolator 2 and isolator 4 provide forces in −→eê₁ and −→eê₃direction.

−

→Fa,i=







0 F_a,i,2 F_a,i,3 −→e^e for i = 1, 3

F_a,i,1 0 F_a,i,3 −→e^e for i = 2, 4

(2.18)

Combination of (2.15), (2.16) and (2.18) results in the sum of forces for isolator i :

−

→Fi=−→ Fk,i+−→

Fc,i+−→

Fa,i. (2.19)

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k 1 k 3 k 2

b 3

b 1 b 2

Figure 2.5: Model of the Isolator

This means that the sum of forces on the payload is:

X−→ FCM=

4

X

i=1

−

→Fi. (2.20)

Since there are no external moments applied on the AVIS, the sum of moments on the payload is given by:

X−→ MCM =

4

X

i=1

−→pi×−→ Fi

. (2.21)

2.2.3 Nonlinear Model of the AVIS

The Newton-Euler approach has been implemented in MATLAB to obtain the AVIS dynamics.

This results in a set of equations which can be written in the form:

M q, η ¨q + C q, ˙q, η + K q, η = Tu q, η u + Tν ν, η

(2.22) where q∈ R⁶ is the vector of generalized coordinates. u∈ R⁸ and ν ∈ R¹² are the input vectors with manipulated inputs and non-manipulated inputs respectively. η ∈ R⁴² is a vector with physical parameters of the AVIS. M (q, η) ∈ R^6×6is the mass-matrix, which is in general dependent on the generalized coordinates, C( ˙q, q, ν, η) ∈ R⁶ is a vector which contains the centripetal and Coriolis terms and the damping forces and K q, η ∈ R⁶ is a vector which contains the spring forces. Moreover, Tu(q, η) ∈ R^6×8 and Tν(ν, η) ∈ R⁶ represent a matrix and a vector which relate the manipulated inputs (actuator inputs) and non-manipulated inputs (chassis vibrations) respectively to the external applied forces and torques on the system.

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e

1c h

r

e

2c h

r

e

3c h

r

p

e

1

r

p

e

2

r

p

e

3

r

rv

C M

e

1

r

e

2

r

e

3

r

rv

c h

2

F r

3

F r

1

F r

4

F r

1

pr

2

pr

3

pr

4

pr

2

cr

3

cr

₄

cr

1

cr

Figure 2.6: Model of the AVIS

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The vector of generalized coordinates of the payload, q, is given by:

q =

x y z φ θ ψ ^T

, (2.23)

where x, y and z are the positions of the center of mass of the payload relative to the earth inertial frame in −→eê₁, −→eê₂ and −→eê₃ direction respectively. φ, θ and ψ are the roll, pitch and yaw angle of the payload.

The vector of manipulated inputs, u, consists of the actuator inputs and is given by:

u =

u_1,2 u_1,3 u_2,1 u_2,3 u_3,2 u_3,3 u_4,1 u_4,3 T

, (2.24)

where the first index indicates the isolator number and the second index the direction of the actuator force with respect to the earth-inertia coordinate frame.

The vector of non-manipulated inputs, ν, is given by:

ν =

q

ch

˙q_ch

, (2.25)

with,

qch =

xch ych zch φch θch ψch ^T

(2.26)

˙qch =

˙

xch y˙ch z˙ch φ˙ch θ˙ch ψ˙ch

^T

, (2.27)

(2.28) where xch, ychand zchare the positions of the chassis-fixed coordinate frame relative to the earth inertial frame in −→eê₁, −→eê₂ and −→eê₃ direction respectively. φch, θch and ψch are the roll, pitch and yaw angle chassis. It should be noted that the vector of non-manipulated inputs will not be used throughout this thesis, since the motion of the chassis cannot be measured in the current experimental set-up. Moreover, the vector of non-manipulated inputs is not directly needed for multivariable controller design. However, for other control strategies, such as the Computed Torque Control technique, a model including chassis vibrations is needed. Therefore, the vector of non-manipulated inputs is included in the model for completeness.

The vector of physical parameters of the AVIS, η, is given by

η =





 ηm

ηb

ηk

ητ

ηp







(2.29)

with:

ηm =

m J1 J2 J3 ^T

(2.30) ηb =

b1,1 b1,2 b1,3 b2,1 b2,2 b2,3 b3,1 b3,2 b3,3 b4,1 b4,2 b4,3 ^T

(2.31) η_k =

k1,1 k1,2 k1,3 k2,1 k2,2 k2,3 k3,1 k3,2 k3,3 k4,1 k4,2 k4,3

T

(2.32) ητ =

τ₁ τ₂ τ₃ τ₄ τ₅ τ₆ τ₇ τ₈ T

(2.33) ηp =

px py pz cx cy cz ^T

. (2.34)

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16 2.3. DISCUSSION

2.2.4 Linearized Model of the AVIS

The rigid-body dynamic model of the AVIS, (2.22), reveals a high complexity of the nonlinear and coupled dynamics. Later on in the thesis, it appears that for the design of a feedback controller using µ-synthesis, a linear model of the AVIS is needed. Therefore, the full non-linear model is linearized around its operating point:

q = 0, q = 0.˙ (2.35)

The linearization is justified, since the rotation angles of the AVIS are very small in a practical application.

Moreover, in (2.22) the forces generated by the motion of the chassis T_ν(ν, η), are taken into account. However, since the motion of the chassis cannot be measured in the current experimental set-up, these forces are neglected for the linearized model.

This results in the following linear model of the AVIS:

M (η˜

m)¨q + ˜C(η

b, η

p) ˙q + ˜K(η

k, η

p)q = ˜T (η

τ, η

p)u, (2.36)

where ˜M (η

m) ∈ R^6×6 is the linearized inertia matrix, ˜C(η

b, η

p) ∈ R^6×6 is the linearized damping matrix , ˜K(η_k, η_p) ∈ R^6×6is the linearized stiffness matrix and ˜T ∈ R^6×8is the linearized actuator matrix . The entries of these matrices are listed in Appendix B.

2.3 Discussion

In this chapter the characteristic of the AVIS, which is the experimental set-up considered in this thesis, has been presented.

The AVIS is a vibration isolation system, consisting of a vibration isolation table and a controller unit, which is capable of six degree-of-freedom vibration isolation. High frequency vibrations are attenuated by a passive system, consisting of a payload mounted on four air mounts. Linear motors in combination with geophones are used for active vibration isolation near the resonance frequency of the passive system.

The kinematic and rigid-body dynamic models of the isolation table have been presented here.

The air mounts are modelled as a combination of 3 orthogonal spring-damper combinations. The equations of motion of the payload of the isolation table are derived using Newton-Euler equations.

The vibrations of the chassis of the AVIS are also taken into account and are considered as non- manipulated inputs. This has resulted in a highly complex nonlinear dynamic model of the AVIS.

The nonlinear rigid-body dynamic model has been linearized around its operating point and the forces generated by the motion of the chassis have been omitted, since they are not measurable in the current experimental set-up. The obtained linearized model still describes the coupling between the six degrees-of-freedom and is used later on in the thesis for the identification of the physical parameters of the AVIS and the design of a multivariable feedback controller.

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Identification of the AVIS

In Chapter 2, a kinematic and dynamic model of the AVIS is derived. However, a full mathematical description of the AVIS includes, besides the kinematic and dynamic model, a set of physical parameters. Most of these parameters have to be identified or estimated since they cannot be measured or known a priori. Therefore, the identification of the physical parameters of the AVIS is presented in this chapter.

There exist different estimation techniques for the identification of parameters, including nonlinear optimization techniques, least-squares estimation techniques, and Kalman filtering. Nonlinear optimization techniques work well for identification problems with a small number of optimization parameters. Since the number of parameters is relatively large, nonlinear optimization techniques are not suitable for the identification of the physical parameters of the AVIS. A major disadvantage of least-squares estimation techniques, is that they can only be used for the identification of parameters which appear linearly in the dynamic model. Since the position and the acceleration of the payload cannot be measured directly and thus have to be estimated, it can be concluded that the property of linearity in the parameters is not satisfied and therefore the least-squares estimation technique is rejected.

In this thesis the Continuous-Discrete Extended Kalman Filter (CDEKF) will be selected for the identification of the parameters of the AVIS. The extended Kalman filter enables to reconstruct the states of nonlinear system minimizing the variance of the difference between the actual state and the estimated state. The label ‘Continuous’ refers to the fact that the filter deals with continuous- time system dynamics and filter propagation, whereas the label ‘Discrete’ refers to the fact that the measurements are taken at discrete times and the states are updated at these discrete times.

This chapter is organized as follows: In Section 3.1, a brief description of the CDEKF equations will be given. In Section 3.2 the identification procedure will be discussed, followed by the setting of the CDEKF in Section 3.3. The CDEKF will be implemented in Matlab and the identification results will be given in Section 3.4. A discussion of the results will be given in Section 3.5.

3.1 Continuous-Discrete Extended Kalman Filter

The extended Kalman filter is a well established technique for estimating the parameters and states of nonlinear systems. This section summarizes the continuous-discrete extended Kalman

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18 3.1. CONTINUOUS-DISCRETE EXTENDED KALMAN FILTER

filter equations. The CDEKF is elaborately described in [8].

3.1.1 Extended Dynamic Model

Consider the linearized dynamic model of the AVIS given by (2.36), with states x_q = q and x_q_˙= ˙q.

If the physical parameters of the AVIS which have to be identified are taken as additional states, x_η= η_e, the extended dynamic model can be written as:

˙

x(t) = f (x(t), u(t)) + W (t); W (t) ∼ N (0, Q(t)) (3.1) yk = h_k(x(tk)) + Vk; Vk∼ N (0, Rk), (3.2) where

x(t) =



 x_q(t) x_q_˙(t) x_η(t)



, (3.3)

is the system state, u(t) is the system input, f (x(t), u(t)) is a vector of nonlinear functions with zero rows corresponding to the extra states related to the physical parameters and W (t) is a zero-mean white noise process with spectral density matrix Q(t). The noise W (t) is included to indicate the model contains modelling uncertainties. In (3.2), y

k is the discrete measurement model, which is a function of the state, x(t_k), where the instant of time t at sampling k is denoted as t_k. The vector h_k(x(t_k)) contains the values of the measurements. V_k is a discrete-time, zero-mean Gaussian white noise process with covariance matrix Rk, which is included to indicate that the measurements contain measurement noise. The noises W (t) and Vk are assumed to be uncorrelated and thus E[W (t)V_k^T] = 0 for all k and t.

3.1.2 Propagation

For the initial conditions x(0) ∼ N (ˆx₀, P₀), where ˆx₀ and P₀are the initial conditions of the discrete estimated state ˆx_kand error covariance P_k, the continuous time propagation of the estimated state is given by

˙ˆx(t) = f(ˆx(t)), (3.4)

where ˆx(t) is the continuous time state estimate. In between the measurements, the evolution of the error covariance matrix, P (t), is governed by a Riccati equation,

P (t) = F (ˆ˙ x(t))P (t) + P (t)F^T(ˆx(t)) + Q(t), (3.5) where the Jacobian matrix F (ˆx(t)) is defined as

F (ˆx(t)) = ∂f (x(t))

∂x(t)

_x(t)=ˆ_x(t)

. (3.6)

3.1.3 Measurement update

When measurements are available, the state estimates are updated according to ˆ

x_k(+) = ˆx_k(−) + Kk[z_k− h_k(ˆx_k(−))], (3.7)

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where ˆx_k(−) and ˆx_k(+) are the discrete state estimates just before and after update. The discrete state estimates are used to re-initiate the continuous-time state propagation equation (3.4). The Kalman gain matrix K_k given by

Kk = Pk(−)H_k^T(ˆx_k(−))Hk(ˆx_k(−))Pk(−)H_k^T(ˆx_k(−)) + Rk⁻¹

, (3.8)

where Hk(ˆx_k) is a Jacobian matrix which is defined as H_k(ˆx_k) = ∂h_k(x(t_k))

∂x(tk) _x(t

k)=ˆx_k

. (3.9)

The error covariance matrix is updated according to

Pk(+) = [I − KkHk(ˆxk(−))]Pk(−), (3.10) where P_k(−) is the discrete error covariance prior to the update and P_k(+) the discrete error covariance after the update.

3.2 Identification Procedure

Consider again the linearized model of the AVIS, (2.36), and the vector of physical parameters, (2.29). Before the CDEKF is implemented in Matlab, some attention should be given to the parameters which are identified and the way the filter is used.

Firstly, the vector of payload fixed and chassis fixed vectors, (2.34), is considered. It should be noted that the chassis fixed vectors can be chosen arbitrarily. In order to reduce the complexity of the linearized model of the AVIS, (2.36), the elements of the chassis-fixed vectors are chosen equal to the elements of the payload-fixed vectors. Since the elements of the payload-fixed vectors are easy measurable, these elements are not included in the CDEKF. Furthermore, the actuator constants, (2.33), can be removed from set of parameters which are identified by the CDEKF, because these parameters are accurately known. The values of the parameters of the body fixed vectors and the actuator constants are given in Table 3.1. The remaining set of parameters, which have to be identified by the CDEKF, is now given by:

ηe= [m J1 J2 J3

b1,2 b1,3 b2,1 b2,2 b2,3 b3,1 b3,2 b3,3 b4,1 b4,2 b4,3

k1,2 k1,3 k2,1 k2,2 k2,3 k3,1 k3,2 k3,3 k4,1 k4,2 k4,3]^T.

(3.11)

Because of the complexity of the CDEKF equations, the filter cannot be used for real-time identification and is used off-line. This is not a problem, since the AVIS is considered as a time-invariant system. Consequently, the model parameters, which are identified with the filter, are time-invariant as well.

Another point of attention is the use of a geophone as a velocity sensor. According to the prob- abilistic Kalman Filter, the discrete measurement model, (3.2), contains measurement noise, V_k, which is a discrete, zero-mean Gaussian white noise process with covariance matrix R_k. Because of the use of geophones in the experimental set-up, the measurement noise can not be considered as a zero-mean Gaussian white noise process. The reason for this is that the geophone voltage is not proportional to the velocity of the payload for low frequencies. Because of the off-line use of the filter, this problem can easily be solved by means of pre-filtering of the identification data with a 1 Hz, sixth order high-pass filter.

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20 3.3. SETTING OF THE EXTENDED KALMAN FILTER

Table 3.1: Body fixed vectors and actuator constant Parameter Value Unit Parameter Value Unit

px 0.475 m τ1 2 N/V

p_y 0.375 m τ₂ 2 N/V

pz 0.175 m τ3 2 N/V

c_x 0.475 m τ₄ 2 N/V

cy 0.375 m τ5 2 N/V

cz 0.175 m τ6 2 N/V

τ2 2 N/V

Furthermore, the CDEKF is used iteratively in order to achieve a faster convergence of the parameters. This means that several data-sets are used for the identification. For the first data-set, initial estimations for the parameters and initial uncertainty of these parameters are defined. Sub- sequently, this data-set is filtered by the CDEKF and the resulting estimations for the parameters are used as initial estimations for the subsequent data-set. However, the uncertainty of the initial estimation is reset to the initial value after each data-set.

3.3 Setting of the Extended Kalman Filter

Whereas the identification procedure is discussed in the previous section, the setting of the CDEKF is presented in this section. The setting of the CDEKF contains specifications of the initial estimation of the states, x₀, and the error covariance matrix, P0, as well as the tuning of the spectral density matrix, Q and measurement covariance matrix R.

Since a completely automated identification is considered, no preliminary manual experiment has to be done. Consequently, the initialization of the augmented states, x_η, is based on an estimation of dimensions and characteristics of the system. However, since the CDEKF is used off-line, the information of the input and output signal is know beforehand and can be used for the initialization of the states, x_q_˙. The initial estimations of the states are listed in Table 3.2.

The initial estimation of the error covariance matrix, P₀, indicates the uncertainty of the initial estimation of the states, x₀. The non-diagonal elements of the error covariance matrix, which are the covariances between the states, are selected to be zero, because it is assumed that the states are mutually independent. The diagonal of the error covariance matrix can be divided into three parts: the variances of the generalized coordinates, Pq, the variances of the derivatives of the generalized coordinates, Pq˙ and the variances of the augmented states, Pη. The initial variance of the generalized coordinates, Pq, depends on the position-range of the payload in operating conditions, since the initial position of the payload is not know a priori. In operating conditions, the maximal amplitude of the position of the payload is in general 5 [µm] for the translations and 5 [µrad] for the rotational degrees of freedom. This means that the standard deviation of the initial estimation of these signals is approximately 2.5 [µm] for the translations and 2.5 [µrad]

for the rotational degrees of freedom. Consequently, the uncertainty of the initial values of the generalized coordinates of the AVIS is set to:

P0_q = (2.5 · 10⁻⁶)². (3.12)

Since the filter is used off-line, the initial estimation of the states, x_q_˙, is accurately known. There-

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fore the uncertainty of the initial estimation of these states is selected very small:

P0q˙ = (1 · 10⁻⁸)². (3.13)

The initial values of the physical parameters are very inaccurate. Because of this a 25% initial uncertainty is supposed.

P₀_η = (0.25 · η₀)² (3.14)

The spectral density matrix, Q, defines the level of model uncertainty. This matrix is selected to be a diagonal matrix as well, because the model equations are mutually independent. As there are no uncertainties in the state equations ^dq_dt = ˙q, the respective noise parameters are set to zero.

Q_q = 0 (3.15)

The uncertainty of the second six equations, is predominantly caused by modelling errors such as neglecting the influence of vibrations of the chassis, omitting higher order dynamics and because of the linearization. The spectral density matrix entries for x_q_˙ are set to

Qq˙= 10⁻⁸ (3.16)

As we are only interested in average parameter values, zero parameter noise is selected.

Qη= 0 (3.17)

The measurement covariance matrix, R, defines the level of measurement noise. The measurement covariance matrix is a square matrix, with a size equal to the number of outputs of the system.

Just like the spectral density matrix, Q, the measurement covariance matrix is a diagonal matrix, because the outputs are assumed to be mutually independent. For the AVIS, six geophones, with a sensitivity of 31.25 [Vs/m], are used for the measurement of the velocity of the payload. The geophones produce a continuous signal, which is sampled with a frequency of 1 [kHz] with a 14-bit resolution over a range of ± 10 [V]. This causes a measurement noise with a variance of 4 · 10⁻¹⁶, which is negligible small. However, this value results in instability of the CDEKF algorithm and therefore the measurement covariance matrix is set to,

R = 10⁻⁸· I⁶, (3.18)

in order to achieve good convergence properties of the parameters.

With this the initial estimations of the states of the CDEKF and their uncertainties, as well as the measurement covariance matrix and the state spectral density matrix are selected and the CDEKF can be used for the estimation of the states and parameters of the AVIS.

3.4 Identification Results

The CDEKF, as described in Section 3.1, has been implemented in Matlab and used for the identification of the parameters of the AVIS. For the identification 10 data-sets of 100 seconds with a sample frequency of 1 [kHz] are taken. The input for the translational degrees of freedom, is a band limited white noise with a covariance of 0.25 [V], whereas the input for the rotational degrees of freedom, is a band limited white noise with a covariance of 0.1 [V]. The CDEKF is used iteratively and off-line as described in Section 3.2.

The parameter estimations as a function of the time are represented in Figure 3.1, Figure 3.2 and Figure 3.3. From these figures, it can be concluded that the parameters have converged after 10