Sensing and Control of Tip-Sample Interaction Force of a Three-Axis Compliant Micro-Manipulator

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					     SENSING AND CONTROL OF TIP-SAMPLE
     INTERACTION FORCE OF A THREE-AXIS
       COMPLIANT MICRO-MANIPULATOR



                                  Thesis




Presented in Partial Fulfillment of the Requirements for the Degree Master of

        Science in the Graduate School of the Ohio State University




                                    By

                             Shiwen Ai, B.Eng.

               Graduate Program in Mechanical Engineering



                         The Ohio State University

                                   2011




                           Thesis Committee:

                        Chia-Hsiang Menq, Advisor

                             Manoj Srinivasan
c Copyright by

  Shiwen Ai

    2011
                                   ABSTRACT



The atomic force microscope is able to measure sample topography and manipulate

nano objects by maintaining a certain tip-sample interaction force that is sensed

through laser deflection measurement. This work propose a systematic way to solve

the drift issue due to ambient temperature change, which is detrimental to AFM

metrology and force spectroscopy. A magnetic actuator is introduced to integrate with

traditional AFM control system to precisely control the force of tip-sample interaction

with magnetic actuator force model continuously updated using real-time calibration

during the tip-sample interaction process. Conventionally AFM experiments need to

wait about one hour after turning the measurement laser after the thermal drift caused

by laser heating reaches thermal steady state. We show with the proposed techniques

how the AFM can maintain its tip-sample interaction force and be immune to thermal

drift. Therefore, there is no need to wait until the thermal balance of cantilever before

AFM experiments can be performed.

   This idea is also extended to a multi-axis probe developed in our group. The

combined techniques together permit precise tip-sample interaction force control in




                                           ii
two-axis and drift-free scanning on samples with unknown geometry and steep fea-

tures such as sidewall and reentrant.




                                        iii
Dedicated to my parents.




           iv
                        ACKNOWLEDGMENTS



    I would like to express my thanks to my advisor Dr. Chia-Hsiang Menq for his

guidance through my whole study at the Ohio State University. I am very grateful

for the support I received from Dr. Menq for this research project. I appreciate that

Dr. Manoj Srinivasan being my examination committee member and spending time

to read this thesis.

   I am thankful to all the colleagues at Precision measurement & control lab for their

stimulating discussion and help on my research, particularly for Dr. Younkoo Jeong’s

help on experimental setup of the AFM system and the manipulator fabrication. I

also like to thank Dr. Denis Pelekhov and Dr. Dan Huber for training me on FIB

machining. I appreciate Barrett Clark’s efforts on proof-reading my thesis.

   On the personal side, I like to thank my parents, my friends at OSU and Columbus

International Friendships for providing endless support and help to create a wonderful

environment for my study.




                                          v
                                                                    VITA


 1988 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        Born in Yuzhou, Henan, China

 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        B.Eng. in Automatic Control, Zhejiang
                                                                                 University

 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        Graduate Fellow, The Ohio State Univer-
                                                                                 sity

 2010-Present . . . . . . . . . . . . . . . . . . . . . . . . . .                Graduate Research Associate, Mechani-
                                                                                 cal and Aerospace Engineering
                                                                                 The Ohio State University



                                                     FIELDS OF STUDY


Major Field: Mechanical Engineering

Specialization: System Dynamics and Control




                                                                            vi
                           TABLE OF CONTENTS


Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .           ii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .           iv

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .              v

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .           vi

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .           ix

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .            x

CHAPTER                                                                                      PAGE

1      Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .             1

       1.1 Background and motivation . . . . . . . . . . . . . . . . .       .   .   .   .   .    1
       1.2 Literature review . . . . . . . . . . . . . . . . . . . . . . .   .   .   .   .   .    5
             1.2.1 AFM based metrology . . . . . . . . . . . . . . .         .   .   .   .   .    5
             1.2.2 AFM based force spectroscopy and manipulation .           .   .   .   .   .    7
             1.2.3 Thermal drift issues in AFM researches . . . . . .        .   .   .   .   .   10
       1.3 Research scope and objective . . . . . . . . . . . . . . . .      .   .   .   .   .   12
       1.4 Thesis overview . . . . . . . . . . . . . . . . . . . . . . .     .   .   .   .   .   14

2      Simultaneous control of tip deflection and tip-sample interaction force
       on single axis with real time drift compensation . . . . . . . . . . . . .                16

       2.1 Conventional AFM control systems . . . . . . . . . . . . . . . . . .                  16
       2.2 Principles of direct force actuation using magnetic actuator . . . .                  18
       2.3 Dual-actuator control of deflection and tip-sample force . . . . . .                   20
              2.3.1 Dynamic estimation of tip-sample interaction force . . . .                   23
              2.3.2 Quasi-static estimation of tip-sample interaction force . . .                25
       2.4 Real-time calibration of the magnetic actuation model and deflec-
           tion drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .            25
       2.5 Experimental validation of real time model calibration, drift com-
           pensation and interaction force control . . . . . . . . . . . . . . . .               28
                                            vii
       2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     36

3      Tip-sample interaction force control on three-axis probing system . . .         37

       3.1 Three-axis compliant manipulator design . . . . . . . . . . . . . .         37
       3.2 Fabrication of multi-axis manipulator . . . . . . . . . . . . . . . .       39
       3.3 Static modeling of the manipulator compliances . . . . . . . . . . .        43
       3.4 Measurement and actuation schemes . . . . . . . . . . . . . . . . .         45
             3.4.1 Two axis deflection laser measurement . . . . . . . . . . .          45
             3.4.2 Magnetic actuation modeling . . . . . . . . . . . . . . . . .       48
       3.5 Real-time calibration of the quadratic model for magnetic actuation         52
       3.6 Two-axis force sensing . . . . . . . . . . . . . . . . . . . . . . . . .    53
       3.7 Experimental evaluation . . . . . . . . . . . . . . . . . . . . . . . .     55
             3.7.1 Evaluation of tip orientation . . . . . . . . . . . . . . . . .     56
             3.7.2 Calibration of the quadratic magnetic force actuation model         58
             3.7.3 2D interaction force control . . . . . . . . . . . . . . . . .      59
             3.7.4 Force controlled two-axis scanning of micro pipette . . . .         60

4      Conclusion and future work . . . . . . . . . . . . . . . . . . . . . . . .      67

       4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   67
       4.2 Recommended future work . . . . . . . . . . . . . . . . . . . . . .         69

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   71

Appendix A: Recursive linear least squares estimator . . . . . . . . . . . . . .       74




                                           viii
                           LIST OF TABLES



TABLE                                                                     PAGE

 1.1    Major nano/micro devices used for force spectroscopy and manipulation   8




                                      ix
                              LIST OF FIGURES



FIGURE                                                                               PAGE

 1.1     Scheme of an overall AFM control system. (Reproduced from G. Schit-
         ter et al.: A Tutorial on the Mechanisms, Dynamics, and Control of
         Atomic Force Microscopes. ) . . . . . . . . . . . . . . . . . . . . . .          3

 2.1     Block diagram of conventional contact mode AFM control system. . .              17

 2.2     Scheme of magnetic force actuation on Z axis. . . . . . . . . . . . . .         19

 2.3     Lumped SHO model for the dual actuated AFM cantilever. . . . . .                21

 2.4     Block diagram of simultaneous deflection and tip-sample interaction
         force control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    22

 2.5     Picture of the customized Agilent 5500 AFM with cooling jacket host-
         ing magnetic solenoids. . . . . . . . . . . . . . . . . . . . . . . . . . .     29

 2.6     Deflection drift due to laser heating that reaches steady state in 1.5
         hours after turning on the measurement laser. . . . . . . . . . . . . .         30

 2.7     Magnetic actuation model fitting results and residual errors using lin-
         ear and 2nd order polynomial models. (Blue: measurement data; Red:
         fitting) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   30

 2.8                                                  ˆ ˆ
         Magnetic model parameters variation (ˆ, g , λ) as the probe location
         changes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    31

 2.9     Comparison of interaction force estimation results when the tip is free
         of contact at different locations. . . . . . . . . . . . . . . . . . . . . .     32

 2.10    Deflection and force profiles during tip-sample interaction process. . .          33

 2.11    Magnetic actuation model parameter estimation during tip-sample in-
         teraction process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    33

                                            x
2.12   When the tip is driven by piezo scanner towards the sample, the tip
       experiences jump due to attraction force before contacting the sample
       surface; while the tip is detached from sample surface, there is also
       jump of the tip due to adhesive force. . . . . . . . . . . . . . . . . . .     35

2.13   Interaction force comparison during the contact periods: with vs.
       without drift compensation. . . . . . . . . . . . . . . . . . . . . . . .      36

3.1    Multi-axis manipulator with neck and body compliant sections and
       micro mirrors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   38

3.2    Design of the neck axis and magnetic moment directions orthogonal
       to each other. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   39

3.3    Micro mirror assembly process. . . . . . . . . . . . . . . . . . . . . .       40

3.4    Overview of the AFM probe attached with two micro mirrors. . . . .             41

3.5    A SEM image of a magnetic particle showing the magnetic moment
       direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   42

3.6    A SEM image of a fabricated multi-axis probe. . . . . . . . . . . . .          43

3.7    Scheme of the new two-axis laser measurement. . . . . . . . . . . . .          46

3.8    Linear dependence of θp / cos θp on the voltage input Vθp to the mag-
       netic torsion actuator. . . . . . . . . . . . . . . . . . . . . . . . . . .    56

3.9    The coupling effects of torsion actuation on deflection measurement.             57

3.10   Magnetic actuation fitting results using quadratic model as the tip
       orientation is fixed (θp = 0◦ ). . . . . . . . . . . . . . . . . . . . . . .    58

3.11   2D Force profiles on sample surfaces with different topography orien-
       tations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   60

3.12   Comparison of micro pipette topography between conventional scan-
       ning and two-axis scanning. . . . . . . . . . . . . . . . . . . . . . . .      61

3.13   Degradation of spatial resolution of two-axis scanning due to the larger
       measurement noise in X piezoelectric scanner. . . . . . . . . . . . . .        62

3.14   (a) Comparison of micro pipette scanning with and without real-time
       drift compensation; (b) Plot of drift against time. . . . . . . . . . . .      63

3.15   Conventional scanning topography and interaction force sensing. . . .          63

3.16   Two-axis scanning topography and interaction force sensing. . . . . .          64


                                          xi
3.17   Two-axis scanning topography and interaction force sensing with real-
       time drift compensation. . . . . . . . . . . . . . . . . . . . . . . . . .    65

3.18   Composite force on the surface normal direction over the scanning line. 65

3.19   Parameter changes of quadratic model for magnetic force actuation
       over the scanning line. . . . . . . . . . . . . . . . . . . . . . . . . . .   66




                                         xii
                                 CHAPTER 1


                             INTRODUCTION




1.1    Background and motivation



With the increasing growth of nanotechnology in academia and industry, there have

been great demands on developing solutions to perform force controlled manipulation

of engineering or biological samples at micro/nano scales. Focused ion beam (FIB)

milling and nano-lithography have provided advanced top-down approaches to pro-

ducing nano devices. On the other hand, a bottom-up approach of nano-assembly will

depend on the reliable interaction between the objects and the end-effector of manip-

ulators. The atomic force microscope (AFM) has became a good candidate to meet

this demand considering its working principles of direct contact with samples and

wide applications in diverse fields as biology, engineering, medicine and chemistry.




                                         1
   AFM was first invented [1] as a direct metrology tool to characterize three dimen-

sional sample topography by physical contact between a measuring stylus and the

sample surface. Due to this measuring principle, AFM can measure various types of

materials in different environments rather than only being feasible in a vacuum. It

has found applications in semiconductors, metals, polymers, biological membranes,

live cells and film coatings. During its continuous development, it has also become a

versatile force probing tool for force sensing and manipulation. It is used as a force

probe for tribological studies such as abrasion, indentation, friction and lubrication

at nano scales. Moreover, AFM based force spectroscopy has been widely used to

study mechanical properties of a range of biological samples from DNA to live cells.

AFM is also used as a micro/nano manipulator to push, pick up and assemble small

sized entities.


   The key component of the AFM system is a compliant cantilever mounted with a

sharp tip at its end. The whole compliant probe is rigidly attached to a high voltage

piezoelectric actuator. A scheme of the overall control system of AFM is shown in

figure 1.1. In its conventional working principles, the tip makes physical interaction

with the sample; the cantilever deflection due to the tip-sample interaction force is

measured by an optical lever with a laser beam reflected by the cantilever’s back

surface. Another type of probe is functionalized with a conducting material coating,

which serves as electrode to make electrical contact with samples; the conducting

current is measured between the tip and the sample. For both types of probing


                                          2
principles with physical or electrical contacts, the readout signal of the cantilever

deflection or conducting current is feedback to command the piezoelectric actuator

to control the tip sample interaction. This thesis focuses on the type of interaction

with direct physical contact. There are a variety of different tip-sample contact modes

that have been developed for different operations of AFM. Of these, Contact Mode

[2] and Tapping Mode [3] are the two most commonly used.




Figure 1.1: Scheme of an overall AFM control system. (Reproduced from G. Schit-

ter et al.: A Tutorial on the Mechanisms, Dynamics, and Control of Atomic Force

Microscopes. )




                                          3
   A novel multi-axis AFM probe was designed in our group to tackle the accessi-

bility issues associated with a conventional AFM, whose tip can only move up and

down. The compliant cantilever is modified to posses high axial torsional compliance

such that the tip orientation can be actively changed during scanning. The compliant

structure is further developed to have comparable force sensing sensitivity on its X

and Z axes with modified measurement laser path. Great improvement on accessi-

bility and 3D sub-nanometer metrology of samples with unknown geometry are both

demonstrated.


   Besides the accessibility issue, another fundamental issue limiting the use of AFM

imaging is the potential damage to the sample caused by the tip sample interaction

force, especially for the delicate and sensitive biological samples such as live cells.

Moreover, the development of precision tools for manipulation and transport of nano

objects with directed forces is still a heated topic in the precision engineering society.

This has led to the research challenges of precise control of the tip force, which

requires several new technologies to be developed to achieve: 1) direct force actuation

at the tip; 2) reduce the system time variance associated with the measurement and

actuation. The first challenge is fulfilled by introducing an additional force actuator

at the head of the cantilever where the tip is mounted. The variations of the system

mainly come from two resources: the deflection drift due to thermal effect and the

position dependency of the probe with the magnetic actuator in the magnetic field.

A real-time calibration method is needed to be developed to address these issues.


                                            4
This research proposes a new AFM operation mode, in which the tip intermittently

contacts with sample surface with interaction phase and separation phase. Deflection

drift is estimated and compensated in real time during tip-sample interaction and the

tip-sample force is sensed and controlled.




1.2     Literature review



As described in the previous section, the key motivation of this research is to solve the

thermal deflection drift issue that has been existing in AFM imaging and nanoma-

nipulation and to apply these techniques to control the tip-sample force of the newly

designed multi-axis probing system. A survey of the current AFM’s application as

nano/micro-scale metrology and manipulation tool is conducted regarding the achiev-

able position and/or force resolution, the range of samples that can be applied on

and the issues limiting the expansion of its applications, especially the deflection drift

caused by the thermal effect.




1.2.1   AFM based metrology



AFM has proved to be a promising technology for performing three dimensional

metrology of samples in the range of a few microns to a few hundred nanometers

with sufficient force sensitivity and lateral resolution. Several research approaches of


                                             5
AFM have been made to improve AFM imaging qualities. The first category includes

dealing with the issues associated with piezoelectric scanners such as positioning

errors caused by hysteresis and creep and the scanning direction control. While the

second category focuses on extracting more information about the sample surface

by calibrating the geometry of the tip; the third takes efforts on modifying the tip

geometry to improve the accessibility.


   Griffith et al. [6] discussed the methods of eliminating positioning nonlinearity

errors by using position sensors to feedback control the scanner. The piezoelectric

positioning showed much greater repeatability using an inner feedback loop. In their

research, a method is also developed to determine the shape of the tip by scanning

a calibration sample with known geometry and use it to remove the effects of tip

shape from the imaging of other samples. High aspect ratio tips were fabricated by

means of FIB milling to improve the shape of the tip itself. However, even with

high aspect ratios tips, features such as side-walls, reentrants and undercuts are still

not accessible. Martin et al. [7] proposed a scanning method in which the scanning

direction is changed to align along the local surface tangential to better image steep

3D feature. It was demonstrated in their paper to image vertical and near-vertical

surfaces with controlling the scan direction and feedback direction. A specialized

boot-shaped probe was utilized to allow the tip to point horizontally to scan sidewalls.

However, the use of this kind of tip is limited to specific samples. In order to scan

general 3D structures with unknown geometry, Jayanth [4] developed a multi-axis


                                           6
probe to actively change the tip orientation so that this probe can access any 3D

structure with an unknown geometry.




1.2.2   AFM based force spectroscopy and manipulation



The pool of available precision force tools is ever-expanding and includes atomic force

microscopes [11, 16], magnetic tweezers [10], optical tweezers [8], needle-like micro-

manipulators [12] and microgrippers [13, 14]. The first three are the most commonly

used in today’s precision engineering research and have major applications in single-

molecular manipulation, cell mechanical properties mapping and nano particle push-

ing and assembly. Magnetic tweezers use a set of magnetic monopoles to generate

fields to control the motion of a magnetic bead as a force probe. The bead position

is measured by machine vision. For optical tweezers, a micro glass bead is trapped

by a high intensity laser beam and is used to probe samples. The beam position can

be measured by either machine vision or an extra measurement laser beam. Both of

these devices mentioned above use gradient forces, optical and magnetic respectively,

to trap a small bead as a force probe. Due to their low trapping stiffness, they can

have very high force resolutions that are practically limited by thermal force. Their

main drawback is the lateral spatial resolution which is limited by the bead size.

The minimum bead size reported is 2 microns [9]. The detailed working principles of

AFM are discussed in section 1.1. A comparison among Magnetic tweezers, optical

tweezers and atomic force microscopes is shown in table 1.1.

                                          7
        Devices
                          Magnetic tweezers Optical tweezers               AFM
      Properties

  Spatial resolution            ∼ 10nm                   ∼ nm                ˚
                                                                            ∼A

   Force resolution               ∼ fN                 ∼ 0.1pN            ∼ 10pN

  Actuation method           magnetic fields        optical trapping     piezoelectric

   Sensing method            machine vision        backscatter laser    optical lever

     Environment                  liquid                 liquid          liquid/air



Table 1.1: Major nano/micro devices used for force spectroscopy and manipulation




   Compared with other devices, AFM is a better force manipulation tool for its

advantages of flexibility and high throughput required in 3D manipulation. The

applications of using an AFM as a force probe fall into two categories: 1) to push

or pull samples such that the force is determined by the amount of deflection, which

leads to the estimation of mechanical properties of the studied sample. 2) to position,

pick up and assemble small sized objects in order to construct nano-structures with

high complexity.


   The first category is also known as the study of nanotribology, in which customized

AFM tips are usually used to scratch samples to study the nano scale friction and also

indent the samples to characterize the surface mechanical properties. The samples

studied range from engineering materials such as semiconductors to biological samples

                                           8
such as single molecular protein, DNA and live cells. Bhushan et al.[15] used AFM to

investigate tribological and mechanical characterization such as adhesion, friction and

wear of carbon nanotubes. Morii et al.[16] employed AFM to study elastic properties

of DNA molucules. The elastic modulus of DNA is obatined using the force-extension

curves. AFM is also greatly used to understand the mechanical behavior of polymeric

materials. Several contact mechanics models are proposed to calculate the elastic

modulus.


   In the second category, researchers have investigated using AFM as a force con-

trolled manipulation tool for nano object positioning and structure assembly[17].

Akita etal. [18] developed nanotweezers composed of carbon nanotubes that can be

operated in an AFM. Two carbon nanotube arms are attached on a Si AFM tip with

electric lead lines and can be controlled by applying voltage to close the gap between

two arms to pick up nanomaterials. Park et al. [19] demonstrated that an AFM

tip can be modified to serve as an electrode to make electrical contact with carbon

nanotubes to perform cutting and nicking. Voltage pulses from a metal-coated AFM

tip were used to permanently modify the electrical properties of carbon nanotubes

and cut and nick at any point along the tube. Requicha et al. [20] developed a

method of robotic assembly of nanoparticles to construct nanostructures using AFM.

Prototyping nanomanipulation is conducted by positioning a set of nanoparticles in

successive sacrificial layers. The sacrificial layers are removed by washing, and the

nanoparticles are connected together by di-thiols leaving a 3D structure.


                                          9
   Although these applications employ AFM tips as force probes to manipulate nano-

objects, nearly all of them perform the manipulation in an open loop manner or just

focus on the position control and do not directly regulate the tip force. This leaves

a great issue where nanomanipulation applications require very gentle tip-sample

interaction. Therefore, force-controlled nanomanipulation still remains an unsolved

issue to the best of our knowledge.




1.2.3   Thermal drift issues in AFM researches



AFM cantilever deflection is sensitive to thermal effect due to its bimaterial structure

nature. Its tip-sample interaction force is proportional to the amount of deflection

change from the initial laser reading of the cantilever. However, due to the thermal

deformation caused by thermal imbalance, the cantilever will drift towards a certain

direction. This deflection drift will cause imaging artifacts as well as false interaction

force estimation. AFM cantilevers are produced using nanolithography on silicon

(Si) or silicon nitride substrate (Si3 N4 ) and coated with gold or aluminum on the

back surface to enhance optical reflectivity. When the cantilever is experiencing

thermal variation, such as laser heating and/or ambient temperature change, the

two layers with different thermal expansion coefficients will cause deflection drift.

Thundat et al. [21] experimentally studied the relationship between ambient-induced

deflections and environmental temperature and humidity change. It is found that

deflection drift is significant for cantilevers with metal coating layers during thermal

                                           10
imbalance and varies approximately linearly with relative humidity for a thermally

stabilized cantilever. Thermal effects are of more interest to us because it is practically

unavoidable in every AFM setup where the measurement laser is the major heating

source.


   In the imaging applications, thermal drift can be corrected using correlation and

Kalman filter methods. Kindt et al. [22] proposed a method to sense the drift com-

ponent of the cantilever deflection signal by correlating two traces of scanning at

the same area of the sample, but with slightly different control setpoints. The set-

point is automatically adjusted so as to maintain a certain relative deflection change.

Mokaberi et al. [23] developed an algorithm to use a Kalman filter to estimate and

predict the drift in X-Y scanner using an approximate dynamical model. The up-

dated drift estimation is compensated by offsetting the scanner the same amount of

distance. The correlation method needs a priori of the sample topography to estimate

the drift and the Kalman filter cannot be applied to real-time estimate the deflection

drift, although it has real time estimation of the scanner drift. Thus neither of these

methods is sufficient to address the drift issue for force spectroscopy and manipula-

tion experiments where the thermal drift needs to be removed in a real-time manner.

Torun et al. [24] designed a spectroscopy experiment in which the sample is placed

on a microstructure that is thermomechanically matched with the cantilever. Since

the cantilever and sample substrate structure have the same thermal behaviors, the

tip-to-membrane distance is always constant even under thermal influence. However,


                                           11
this method requires the thermomechanical properties of the cantilever to be the same

as the sample stage structure; thus it cannot be applied to general AFM cantilevers.

A general method to solve the thermal drift issue is still not available.


   In a nutshell, AFM has become a powerful 3D imaging tool and force manipulator.

Behind its success, a fundamental unresolved issue is the thermal drift, which is

limiting further applications of AFM. This has led to the research challenges of real-

time estimation and compensation of the thermal drift for the purpose of precise

tip-sample interaction force control.




1.3    Research scope and objective



The objective of this research project is to investigate enabling technologies to achieve

force sensing and control capability of an AFM-probe-based multi-axis compliant ma-

nipulator to deal with uncertain mechanics on the micro/nano scale and to predict

system time variance caused by thermal drift due to laser heating and ambient tem-

perature change. The tip-sample interaction force sensing is realized through esti-

mation using the tip deflection and force model of magnetic actuators; thermal drift

is estimated and compensated in real-time for the purpose of precise force control.

The design and modeling of a three-axis probing system developed in our group is re-

viewed; and the techniques of force sensing and control with real time drift compensa-

tion are integrated with the three-axis probing system, which together can accurately

                                           12
control the-tip sample interaction force magnitude and direction in a 2-dimensional

scanning plane. Two specific aims are identified:


   Aim#1: Real-time calibration of magnetic actuation model and compensation of

thermal drift. The AFM cantilever is sensitive to thermal variations due to its struc-

tured composition of two layers of different types of materials. Major sources of the

thermal variations are the laser heating and ambient temperature change. The ther-

mally induced deflection of the cantilever is detrimental to high precision metrology

and nano-manipulation for long time range operations. Therefore, a real-time drift

compensation method is needed to address these issues generally in AFM applica-

tions such as imaging, biomechanics study and force controlled nanomanipulation.

Specifically, in our research, magnetic actuators are employed to control the periodi-

cal tip-sample interaction and the tip force of the manipulator during contact phase.

The magnetic actuation model is generally dependent on the relative position of the

probe in the magnetic field generated by the actuators. In summary, thermal drift

and position-dependency of the magnetic actuation model are the two factors of the

probing system’s time variance and need to be solved by real-time calibration.


   Aim#2: Two-axis force sensing and control of the active three-axis re-orientable

probe. A traditional AFM cantilever has only one degree of freedom in the defined Z

axis. The system cannot control either the lateral motion or the tip orientation. Our

group has investigated the design, actuation and implementation of an innovative

multi-axis probing system that has one DoF on the axial rotation and two DoF on
                                         13
X and Z translations. A conventional AFM cantilever is modified by FIB milling

such that the structure bears the desired compliances in the designed directions. A

novel measurement scheme is also designed to posses comparable force measurement

sensitivity on X and Z axes of the probe. We propose in this research to extend the

real-time calibration method to integrate with the multi-axis probe and resolve the

couplings between each axis. Finally, two-axis force sensing and control in the X-Z

scanning plane are achieved.




1.4    Thesis overview



This thesis consists of four chapters. Chapter 1 introduces the background infor-

mation of the AFM and a survey of existing research works on AFM applications

and the thermal drift issue. The research scope and aims are defined based on the

studies of prior researches. Chapter 2 proposes the solution to tackle the thermal

drift issue along a single axis. This includes: 1)the employment of an additional

magnetic actuator on top of the conventional AFM to simultaneously control the

deflection and tip-sample force; 2)the real-time calibration of the magnetic actuation

model; and 3)the estimation and compensation of the deflection drift. Chapter 3

reviews the design and kinematic modeling of the three-axis manipulator. The meth-

ods presented in Chapter 2 to real-time compensate the deflection drift and control




                                         14
the tip-sample interaction force are extended to control the tip-sample force in two-

axis. Comparisons of the scanning topographies and force profiles among different

scanning schemes (single-axis, two-axis, with and without drift compensation, etc.)

are shown to demonstrate the significance of performing drift compensation and the

improvement on tip-sample interaction force control. Chapter 4 concludes the thesis

by summarizing the key merits of the research and proposing future work.




                                         15
                                 CHAPTER 2


SIMULTANEOUS CONTROL OF TIP DEFLECTION AND

TIP-SAMPLE INTERACTION FORCE ON SINGLE AXIS

        WITH REAL TIME DRIFT COMPENSATION




2.1    Conventional AFM control systems



In conventional AFM system, when the probe is brought into the proximity of the

sample surface, the force experienced by the tip is sensed through the cantilever de-

flection. Depending on the contact situation, these forces include mechanical contact

force, van der Waals force, capillary force, chemical bonding, electostatic force, etc.

In order to maintain a constant interaction force between tip and sample, there is a

single feedback control loop to adjust the probe height using a piezoelectric actuator

on the Z axis. A complete control block diagram of contact mode AFM is shown

in figure 2.1. The closed loop bandwidth of such a control system is limited by the


                                          16
slowest component, namely the piezoelectric actuator. The tracking performance of

the tip during scanning relies on the sample topography variation, scanning speed

and the control system tuning.




   Figure 2.1: Block diagram of conventional contact mode AFM control system.




   The piezoelectric scanner is typically a tube scanner with three DoF on X, Y and

Z axes. An alternative design has split the Z axis actuation from X-Y scanning with

two independent piezo stacks. This eliminates the coupling effects between different

actuation axes. The X and Y piezo motions are implemented for raster scanning and

slow scanning. Depending on the application requirements, the piezoelectric scanner

motions can be controlled in open loop or closed loop. For the open loop imple-

mentation, the full dynamical range of the piezoelectric actuator can be exploited

for fast scanning. On the other hand, closed loop implementation can eliminate the

effects of positioning error due to the hysteresis of piezoelectric materials. Z motion

of the piezoelectric actuator is employed to regulate the tip-sample interaction force

                                         17
by using the feedback deflection of the tip during scanning. However, for certain

applications such as nanomanipulation and force spectroscopy, the deflection control

loop is turned off and the cantilever is driven by piezoelectric actuator to indent the

samples.


   There are several disadvantages of using the piezoelectric actuator to control the

tip-sample force. First, the piezoelectric actuator is not able to respond to rapid

topography changes, thus resulting in topography tracking error and force error.

Second, since the cantilever deflection is solely dependent on the tip-sample force,

there is no way to design control algorithms such that the deflection and tip-sample

force can be independently controlled. One solution to deal with this issue is to employ

a magnetic actuator to directly control the tip-sample force [25]. The principles and

advantages of magnetic actuation are discussed in the following section.




2.2    Principles of direct force actuation using magnetic actu-

       ator



Figure 2.2 shows the scheme of controlling the tip-sample force using magnetic ac-

tuator. The magnetic moment carried by a micro particle m is aligned along the

Z axis and the magnetic filed B generated by the coils is along the Y axis, i.e.
                T                    T
m= 0 0 m            , B = 0 −B 0         . It is assumed that the bending motion of the


                                            18
cantilever is small, so the relative angle between m and B does not change. The

bending torque on the magnetic particle is given by


                         τmag = m × B = −mB 0 0 ,                             (2.2.1)


which has only one component about the X axis causing the up-and-down tip deflec-

tion. Given that the deflection caused by the application of the bending torque is

small, the magnetic actuation effect is represented by equivalent magnetic force along
                                                              mB
the Z axis according to the cantilever length l: Fmag(eq) =      .
                                                               l




             Figure 2.2: Scheme of magnetic force actuation on Z axis.




                                          19
2.3    Dual-actuator control of deflection and tip-sample force



The solenoid generating the magnetic field has first order dynamics characterized by

the coil inductance L and resistance R. Its actuation bandwidth is much higher com-

pared to that of the piezoelectric actuator that typically has the roll-off frequency

of about 1kHz. It means that it is able to regulate the high order dynamic compo-

nents of the interaction force with proper force estimation techniques. Besides, using

both piezoelectric and magnetic actuators enables the control of the deflection and

tip-sample force simultaneously. Given the above mentioned advantages of using a

magnetic actuator, it is added on top of the conventional AFM control loop to form

a more sophisticated control system.


   As far as only the first mode of the cantilever probe is considered, it is lumped as

a simple harmonic oscillator (SHO) model with multiple inputs shown in figure 2.3.

The block diagram of simultaneous control of tip deflection and tip-sample interac-

tion force is shown in figure 2.4. The cantilever deflection and tip-sample interaction

force are independently regulated by PI controllers. The deflection feedback signal is

available from the laser measurement. Interaction force feedback is based on estima-

tion using deflection measurement and magnetic force input. It is noticed that the

two control loops are coupled together in that the magnetic force causes deflection

changes, and the deflection also has effects on the interaction force estimation. The




                                         20
      Figure 2.3: Lumped SHO model for the dual actuated AFM cantilever.




dynamical behaviors of each component of the system are studied to fully understand

the system dynamics.


   The piezoelectric actuator is modeled as a empirical second order system with

driving voltage input Vpzt (s) and displacement output Zpzt (s).




                                                2
                                             ωz Γz
                         Zpzt (s) =                         V (s)
                                                          2 pzt
                                                                                   (2.3.1)
                                      s2 + (ωz /Qz ) s + ωz

Here ωz is the resonant frequency; Qz is the quality factor; and Γz is a scaling constant.




                                             21
Figure 2.4: Block diagram of simultaneous deflection and tip-sample interaction force

control.




   The magnetic actuator which is a home-made solenoid applying magnetic field is

modeled as a first order system with voltage input Vmag (s) and force output Fmag (s).

                                            Γsol
                              Fmag (s) =          Vmag (s)                     (2.3.2)
                                           τs + 1

where τ is the time constant and Γsol is the scaling factor.


   The cantilever is represented by a second order model with three inputs: mag-

netic force Fmag (s), tip sample interaction force Fint (s) and piezoelectric actuator

displacement Zpzt (s). The output is the tip displacement Ztip (s) that is given by

equation




                                           22
                           1                                      cc s + kc
   Ztip (s) =                         (Fmag (s) + Fint (s)) +                   Zpzt (s)   (2.3.3)
                mc   s2   + cc s + kc                         mc s2 + cc s + kc

where mc , cc , kc are the mass, damping coefficient and stiffness of the SHO model,

respectively.


   The interaction between the tip and sample involves complicated physical pro-

cesses and the exact mechanics during this process is still under study. Based on the

above dynamical modeling of the cantilever, an augmented state estimator [26] can

be designed to estimate transient interaction force.




2.3.1   Dynamic estimation of tip-sample interaction force



For a system that is observable, all its state variables with no direct measurement

available can be estimated using a state observer. Once, the dynamical model for

the cantilever is identified, it can be written in state space representation with state

                                     ˙
variables x1 (t) = ztip (t) and x2 = ztip (t) with piezo actuator’s position fixed.




                                      ˙
                                      x = Ax + Bu + Bw                                     (2.3.4)

                                      y = Cx + Du + Dw                                     (2.3.5)




                                                  23
where the system matrices are defined as
                      
   0   1       0
A=
   k
           ,B =  ,C = 1 0 ,D = 0 ,
         c      1
   −   −
     m   m        m
                                T
and the states are x = x1 x2        , input u = Fmag , disturbance w = Fint .


   The interaction force is treated as an unknown disturbance and is assigned with

         ˙
dynamics w = Awd w. Then the disturbance is augmented into the state variables by

rewriting the system and output equations.



                                              
                         x  A BAwd   x 
                          ˙
                         =            + Bu                                (2.3.6)
                                    
                         w˙    O Awd     w
                                               
                                                x
                                       y = C OT  
                                                                              (2.3.7)
                                                 w


   A state observer can be constructed based on the augmented state space repre-

sentation of the system.
                              
                    ˙
                    ˆ               ˆ
                   x  A BAwd   x 
                   =            + Bu + Lp (y − y )
                                                     ˆ                          (2.3.8)
                              
                    w
                    ˆ˙   O Awd     wˆ
                                                   
                                                          ˆ
                                                         x
                                                ˆ
                                                y = C OT  
                                                                              (2.3.9)
                                                          ˆ
                                                          w

By properly choosing the pole placement vector Lp according to the designed observer

                                                                    ˆ
dynamics, the interaction force can be obtained from the estimation w.
                                           24
2.3.2   Quasi-static estimation of tip-sample interaction force



For the proposed implementation in this project, quasi-static modeling of the can-

tilever is used and the control scheme is implemented based on the quasi-static model.

In such a case, the magnetic actuator and piezoelectric actuator are treated as ‘zero’

dynamics components, and the manipulator is operated in the range far below its res-

onance frequency. The interaction force estimation is greatly simplified and is given

by




                            Fint = kc (Ztip − Zpzt ) − Fmag .                   (2.3.10)




2.4     Real-time calibration of the magnetic actuation model

        and deflection drift



The proposed control scheme has two independent actuators to actuate the probe.

This enables a special scanning mode in which the tip and sample contact and sep-

arate periodically; during the separation cycle, the magnetic actuation model and

deflection drift can be estimated, and the estimated results are used for the following

contact period. During the contact period, the interaction force is precisely controlled

for scanning or manipulation applications. In this proposed method for solving the

thermal drift issue and interaction force control, a key element is the magnetic force
                                           25
actuator. By substituting the magnetic field B = vΓ, the quasi-static magnetic force

model is obtained:
                                              mvΓ
                                    Fmag =        ,                            (2.4.1)
                                               l

where v is the actuator voltage input and Γ is the actuation gain. The deflection

caused by magnetic force ∆m,mag is given by

                                               Fmag
                                   ∆m,mag =                                    (2.4.2)
                                                kc

Due to the small misalignment errors of the probe and the uncertainties associated

with determining these property numbers with instruments, the magnetic actuation

is characterized by a calibration model to directly relate the magnetic actuator input

voltage to the caused deflection. The equivalent tip-sample force from the magnetic

actuation can be obtained by calibrating the deflection measurement sensitivity to

the tip displacement and calculating the cantilever lumped stiffness using the thermal

noise method [27]. A second order model for the magnetic actuation is applied with

the form




                               ∆m,mag = v 2 + gv + b,                          (2.4.3)

where   represents the small nonlinearity, g is the actuator linear gain and b is the

bias.




                                         26
   Based on the quasi-static model of the cantilever, the laser measurement of can-

tilever deflection, ∆m , is attributed to three parts:

                                      Fint
                               ∆m =        + ∆m,mag + d,                       (2.4.4)
                                       kc

where Fint is the tip-sample interaction force, kc is the cantilever stiffness on the

measurement direction, ∆m,mag is the deflection caused by magnetic force and d is

the thermal drift. During the separation phase when the tip is free of contact, i.e,

Fint = 0, substituting 2.4.3 into 2.4.4 yields


                                ∆m = v 2 + gv + b + d.                         (2.4.5)


A recursive least squares method based model estimator can be employed to estimate

                                                                g
these model parameters: nonlinearity (ˆ), actuator linear gain (ˆ) and deflection

drift, which is the combination of the model bias and thermal drift       ˆ b ˆ
                                                                          λ =ˆ+d .

This magnetic actuation model needs to be very precise for two reasons: 1) The

interaction force estimation relies on the magnetic force input; 2) The estimated bias

is important to remove the drift effect in the interaction force.


   The force sensing equation is given according to measured deflection, calibrated

magnetic actuation model and estimated deflection drift:


                                                      ˆ
                          Fint = kc ∆m − ˆv 2 + g v + λ
                                                ˆ           .                  (2.4.6)


The model parameters are updated during the calibration in the separation phase for

each tapping cycle.

                                           27
2.5    Experimental validation of real time model calibration,

       drift compensation and interaction force control



A customized commercial AFM (Agilent 5500) is employed to carry out the exper-

iments. The magnetic actuation solenoids are homemade coils hosted in a donut-

shaped cooling jacket which is attached on the frame of the AFM system. The sam-

ple holder and scanner are placed within the central hollow of the cylindrical cooling

jacket so that the probe is within the magnetic fields of all the solenoids. A magnetic

bead is attached at the end of the cantilever using another micro manipulator and

magnetized with an electromagnet (GMW 3470) for magnetic actuation of the probe.

A dSpace real time control module (DS1104) including a processor based controller

and inboard 8-channel I/Os is available in our lab for control algorithm prototyping.

Figure 2.5 shows the configuration for the experiment setup.


   The control algorithm with real time magnetic actuation model calibration and

drift compensation on single axis can be implemented on the real time controller

with 25kHz sampling rates. Figure 2.6 shows the deflection drift due to laser heating.

The deflection signal is recorded immediately after turning on the measurement laser

while the cantilever is free of any contact. The deflection changes rapidly at the

initial stage after the laser measurement is turned on and gradually reaches steady

state (thermal balance) after about 1.5 hours. Typically, AFM imaging tasks requires




                                         28
Figure 2.5: Picture of the customized Agilent 5500 AFM with cooling jacket hosting

magnetic solenoids.




a 30 minute to 1 hour wait after turning on the laser measurement system before the

thermal drift does not severely affect the imaging results.


   The magnetic actuation model calibration is performed by inputing a slow-changing

voltage to the actuator and measuring the corresponding deflection. Figure 2.7 shows

that a second order model fits well with the input-deflection relationship while using

a linear model can result in notable nonlinearity error.


   The interaction force estimation results rely on the accuracy of the magnetic ac-

tuation model calibration. Since the model parameters vary as the probe moves into

different locations within the magnetic field, a one-time calibration result is only valid

for the specific condition where the calibration is performed. A recursive calibration


                                          29
Figure 2.6: Deflection drift due to laser heating that reaches steady state in 1.5 hours

after turning on the measurement laser.




                   (a) Linear                          (b) Second order



Figure 2.7: Magnetic actuation model fitting results and residual errors using linear

and 2nd order polynomial models. (Blue: measurement data; Red: fitting)




                                          30
method is used to continuously update the model parameters. The parameters vari-

ation as the probe changes its location in Z axis is shown in figure 2.8.




                                                    ˆ ˆ
Figure 2.8: Magnetic model parameters variation (ˆ, g , λ) as the probe location

changes.




   The interaction force estimation is very sensitive to these variations in parameters.

Therefore, it is important to employ real time calibration in the control algorithm

to capture these changes. This can be verified in figure 2.9. The interaction force

estimation should be zero as the probe is far away from the sample substrate and

the probe has only slow-changing input from the magnetic actuation. The magnetic

actuation model is calibrated once and used continuously for interaction force esti-

mation. As it is shown, the interaction force estimation became erroneous when the

tip deviates from its original position.


                                           31
                      (a) Interaction force estimation at the original lo-
                      cation where the magnetic actuation model is cali-
                      brated




                      (b) Interaction force estimation with the pre-
                      calibrated magnetic actuation model at a different
                      location



Figure 2.9: Comparison of interaction force estimation results when the tip is free of

contact at different locations.




   Figure 2.10 shows the tip-sample interaction process with the proposed periodical

contact-and-separation control scheme. When the separation phase starts, the control

gains for the deflection and interaction PI controllers are set to zero; the magnetic

actuator is commanded to lift off the tip from sample. As a result, the interaction

force decreases and becomes negative due to the adhesive force. The tip sample

adhesion accumulates as the magnetic actuator continue to lift up the tip until the

jump of adhesion occurs. After that, the tip force becomes zero as it is retracted from

the sample surface.




                                              32
  Figure 2.10: Deflection and force profiles during tip-sample interaction process.




Figure 2.11: Magnetic actuation model parameter estimation during tip-sample in-

teraction process.




   The magnetic actuation model real time calibration algorithm starts to work after

the tip is detached from the sample surface. In other words, the deflection change


                                        33
of the cantilever is exclusively caused by magnetic actuation. The estimator is im-

plemented based on a recursive linear least squares (RLS) method (See appendix A

for details), in which every new separation cycle takes the previous step’s estimation

result as the initial guess for the estimator, and the estimation process can converge

quickly to capture the small variations in the model parameters. The parameter

estimation is shown in figure 2.11.


   After the magnetic actuation model identification is finished, i.e., the model es-

timation has converged, the estimated parameters are registered for the following

contact period for deflection compensation and interaction force control. Then the

controllers for the deflection and interaction force control loops are turned on so that

the tip is driven by the control efforts to maintain contact with the sample surface

again and the deflection and interaction forces are regulated according to their set-

point values.


   It can be seen from figure 2.10 that the tip motion in response to the adhesion jump

can be suppressed by magnetic force. The cantilever deflection increases smoothly

due to the magnetic force increase. In conventional AFM nanoindentation, the probe

is actuated solely by the piezoelectric scanner to indent on the sample surface, in

which the force is not controlled; the large adhesive force can cause the jump of tip

motion when the tip is detached from sample surface which is showed in figure 2.12.

This demonstrates another benefit of using a magnetic actuator to control the tip

force.
                                          34
Figure 2.12: When the tip is driven by piezo scanner towards the sample, the tip

experiences jump due to attraction force before contacting the sample surface; while

the tip is detached from sample surface, there is also jump of the tip due to adhesive

force.




   This technology enables doing AFM experiments without concerning the thermal

drift issue. Figure 2.13 shows the comparison of interaction force control with and

without drift compensation. Only the interaction force during the contact periods are

plotted, and it can be seen that without drift compensation, the actual interaction

force deviates significantly from the controlled value.




                                         35
Figure 2.13: Interaction force comparison during the contact periods: with vs. with-

out drift compensation.




2.6    Summary



So far we have developed the algorithms for real-time calibration of the magnetic

actuation model and estimation of the deflection drift, which are two important ele-

ments for precise interaction force control. In the current implementation, a simplified

quasi-static model is used for interaction force estimation.




                                          36
                                 CHAPTER 3


  TIP-SAMPLE INTERACTION FORCE CONTROL ON

                 THREE-AXIS PROBING SYSTEM




3.1    Three-axis compliant manipulator design



Our group has proposed a novel design of a multi-axis AFM probe by introducing

compliant neck and compliant body sections into the conventional AFM cantilever.

The neck is designed to bear high axial torsion compliance and low transverse bending

compliances, and its axis passes through the end point of the tip, as shown in figure

3.2. The body compliant section is designed to allow the cantilever has equal compli-

ance on X and Z axes and relatively low torsional compliance. The measurement laser

path is also modified by introducing two micro reflectors that are rigidly attached to

the cantilever, enabling comparable measurement sensitivity of the cantilever bending

deformation on X & Z axes. This design scheme has three distinguishing features:

1)the orientation and end position of the tip in the scanning (X-Z) plane are actively

                                         37
controlled; 2) the tip’s end point does not have translational deviation when the tip

orientation changes; 3)the tip-sample interaction forces in two axes can be sensed and

controlled. A set of magnetic actuators are employed to apply torques and gradient

forces to the manipulator head to control the tip orientation and the tip-sample in-

teraction force in the scanning plane. Figure 3.1 shows the schematic design of the

multi-axis manipulator.




Figure 3.1: Multi-axis manipulator with neck and body compliant sections and micro

mirrors.




                                         38
Figure 3.2: Design of the neck axis and magnetic moment directions orthogonal to

each other.




3.2    Fabrication of multi-axis manipulator



The multi-axis probe fabrication process begins with attaching the micro mirrors on

a bear cantilever using nanomanipulators (Kleindeik) installed within a dual beam

FIB/SEM chamber (FEI Helios Nanolab). According to the design of the manip-

ulator [5], the floating mirror and the base mirror are both mounted with respect

to the cantilever with angles of 50◦ to reach the optimal measurement sensitivities.

The assembly process involves picking up a piece of micro mirror with desired size,

transporting the micro mirror to the target probe and aligning the angle and sol-

dering the micro mirror on the probe with platinum. An illustration of the process

is shown in figure 3.3. The target AFM probe and the micro mirror (another AFM

cantilever with the tip removed) are first placed on a sample holder according to the

relative location to the nanomanipulator within the FIB/SEM chamber. Secondly,

the manipulator is attached to a piece of micro mirror by platinum deposition and the

mirror is picked up. In the third step, the nanomanipulator brings the mirror to the

                                         39
target location on the probe and the mirror-probe connection is welded by platinum

deposition. Finally, the nanomanipulator is detached from the mirror by FIB milling

and retracted. This process is repeated so that two micro mirrors are attached on

the AFM probe, as figure 3.4 shows, according to the designed location and angle.




                (a) Sample configuration on (b) Pick-up a piece of mirror
                sample holder




               (c) Transport to target location(d) Attch mirror with desired
                                               angle and location



                     Figure 3.3: Micro mirror assembly process.




   Thereafter, a magnetic bead is attached at the end of the cantilever similar to

the configuration of the single axis probe discussed in previous chapter. The special
                                            40
     Figure 3.4: Overview of the AFM probe attached with two micro mirrors.




requirements on the magnetic particle for multi-axis probe are twofold: 1)The particle

needs to have a fairly large diameter in order to bear large magnetic moment which is

in proportion to the particle volume. Thus ohmic heating of the actuation coils can

be avoided by applying relative low input current to change the tip orientation by the

same amount; 2)The magnetic moment is aligned along the direction normal to the

tilted neck whose axis pass through the end point of tip. These two features are meant

to avoid the tip translation deviation when the tip orientation is changed. Plus, in

order to be able to fabricate the tilted neck on top of a conventional AFM cantilever,

large beam thickness is required to realize the desired neck angle. A commercially

available AFM probe (Bruker AFM probes, CONT40) with 7µm thickness is used as

the manipulator basis. A magnetic particle having about 70µm diameter is attached

at the front end of the cantilever using another micromanipulator with micro pipettes.

                                         41
A scanning electron microscope (SEM) image of the magnetized bead is shown in

figure 3.5. The electron traces due to magnetic field in the image can be used to

indicate the magnetic moment direction.




Figure 3.5: A SEM image of a magnetic particle showing the magnetic moment

direction.




   Finally the designed compliant sections are created by FIB milling according to the

design dimensions. The cantilever made of silicon has superior mechanical properties

such as low elastic modulus and high strength, which makes it suitable to be machined

for prototyping. An image of the complete functionalized probe is shown in figure

3.6.




                                          42
            Figure 3.6: A SEM image of a fabricated multi-axis probe.




3.3    Static modeling of the manipulator compliances



Since the dual-beam neck design gives the neck very high bending stiffness on X and

Z axes compared to the body compliant section, a presumption can be made that

the body deformation that is measured by the optical lever can approximate the tip

translations with pre-calibrated measurement sensitivity. The body can be viewed as




                                        43
a simple beam structure whose compliance is given by a 6 × 6 matrix:
                                                          
                 tbx  ctxf x   0      0      0      0    ctxτ z  Fx 
                                                                 
                                                                 
                t   0        ctyf y   0      0      0      0   Fy 
                 by                                              
                                                                 
                                                                 
                t   0          0    ctzf z ctzτ x   0      0   Fz 
                 bz                                              
                 =                                                           (3.3.1)
                                                                 
                θ   0          0    cθxf z cθxτ x   0      0   τx 
                 bx                                              
                                                                 
                                                                 
                  θby   0       0      0      0    cθyτ y   0   τy 
                                                                 
                
                                                                 
                                                                 
                  θbz    cθzf x   0      0      0      0    cθzτ z    τz

where Fi and τi are the applied load (force, torque) components on ith axis; tbi

and θbi are the translations and rotations of the end of the body section; c(t,θ)i(f,τ )i

(i, j = x, y, z) are the intrinsic compliance elements of a simple cantilever beam. On

the left hand side of the equation, the angular deformations are of particular interests

because they induce the change of laser reflection path. Each element in the com-

pliance matrix Cb can be theoretically calculated using the body section’s geometry

lb , wb , tb and elastic modulus E, G:
                                                                              
                              3                                            2
                           4lb                                          6lb
                         Et w3     0        0        0         0            3
                         b b                                          ETb wb 
                                   L
                                                                              
                         0                  0        0         0        0 
                                                                              
                                 Etb wb                                       
                                             3        2                       
                                           4lb      6lb                       
                         0         0                           0        0 
                                          Ewb t3
                                                b   Ewb t3b
                                                                               
                 Cb =  
                                              2
                                                                                  (3.3.2)
                         0                 6lb      12lb                      
                                    0                           0        0 
                                           Ewb t3   Ewb t3
                                                                              
                                               b         b                    
                                                               3lb
                                                                              
                                                                              
                         0         0        0        0                  0 
                        
                                                             Gwb t3
                                                                   b
                                                                               
                                                                               
                         6lb 2
                                                                        12lb 
                                3
                                    0        0        0         0            3
                          Etb wb                                       Etb wb



                                              44
3.4     Measurement and actuation schemes



3.4.1   Two axis deflection laser measurement



In conventional AFM systems, the laser beam is reflected by the cantilever’s top

surface and sensed by a quadrant photodiode (QPD) to establish an optical lever.

The angular deformations of the cantilever’s body where the laser beam is reflected

is picked up by the laser spot location change on the QPD. The readout signals from

QPD are usually converted to tip displacement through calibration of measurement

sensitivity. The deflection measurement on Z axis is sensitive to the up-and-down tip

motion; however, the lateral deflection measurement is insensitive to the tip lateral

motion because a very small lateral motion of the tip can result in axial twist of the

cantilever body. From the point of view of measuring tip motions, this measurement

scheme is insufficient to measure two-axis tip motions since the lateral measurement

range is very limited due to its low measurement sensitivity. Usually the lateral laser

measurement range is about 1◦ which is correspondent to about 200nm of tip motion.


   A novel laser measurement scheme is design to measure the two-axis tip motion.

Micro-mirrors are introduced to modify the laser path on the cantilever for the pur-

pose that the QPD readouts have comparable measurement sensitivities in both X

and Z directions. Details about the kinematic laser measurement design is in the

following paragraph.


                                          45
             Figure 3.7: Scheme of the new two-axis laser measurement.




   As figure 3.7 shows, two pieces of mirrors are rigidly attached to the manipulator.

The base mirror is fixed at the rear end of the cantilever and remains stationary; the

floating mirror, which is attached before the body compliant section, has rotations θf

due to the body angular deformation θb . The mirror rotations θf are described in the

coordinate frame attached to the initial position of the mirror with the Z axis pointing

to the normal direction of the mirror plane. There is a kinematic transformation from

the global coordinate to the coordinate associated with the floating mirror, which is

given by                                                 
                        θf x  1   0     0  θbx 
                                              
                                              
                        θ  = 0 cosθ   −sinθm  θby                          (3.4.1)
                         fy         m         
                                              
                                              
                         θf z    0 sinθm cosθm     θbz

where θm is the mounting angle of the floating mirror.




                                          46
     The laser beam is reflected twice on the two micro mirrors and falls at the location
                         T
xbeam = xbeam zbeam          relative to the initial laser spot on the photo detector. The

laser spot location is related to the floating mirror motion through kinematic relations:
                                                           
                                                      θ
                                                           f x
                            xbeam   0          R0 0  
                                                           
                                  =                   θ                    (3.4.2)
                                                       fy
                             zbeam      R0 cosθ0 0 0     
                                                           θf z

where R0 is the effective length from laser source to QPD and θ0 is the incident angle

on the floating mirror. It is noticed that neither measurements on X or Z directions

is sensitive to the floating mirror motion θf z because this in-plane rotation of the

mirror does not cause any angle change. Given that the output voltage readings of

the QPD are proportional to the X and Z positions of the laser spot center on the

detector,

                                    ∆m = kph xbeam ,                               (3.4.3)

where kph is the photo-detector conversion constant ([V /m]). The overall measure-

ment kinematics relating the body angular deformation and sensor outputs is given

by




                                                                 
                                          1     0       0  θbx 
                                              
        ∆mx   0      kph R0 cos θ0      0 
                                                                 
                                                                  
              =                                                                  (3.4.4)
                                                      m − sin θm  θby 
                                            0 cos θ            
                                         
          ∆mz    kph R0      0             0 
                                                                  
                                                                  
                                               0 sin θm cos θm      θbz



                                            47
3.4.2   Magnetic actuation modeling



In a conventional AFM, the tip orientation and tip-sample interaction direction are

both fixed, resulting in disadvantages relating to the accessibility of the probe. In

contrast, the three-axis compliant manipulator is actuated by three magnetic actua-

tors, including one torsion actuator to twist the tip along the neck axis and two force

actuators to apply magnetic force in arbitrary directions within the X-Z scanning
                                                                                   T
plane; it is capable to control the tip-orientation θp and 2D force Fmagx Fmagz        .


   The torsion actuator, namely the pair of solenoids placed along the X direction,

provides a magnetic field to change the tip orientation θp . It enables large twist of

the neck due to its high torsional compliance 1/kθ . The relation between the applied

actuation voltage Vθp and the tip orientation θp is obtained by balancing torques

according to the actuation gain and compliant probe properties:

                                   θp     mΓθ
                                        =     Vθp .                             (3.4.5)
                                 cos θp    kθ

               mΓθ
The constant          is obtained by calibration using an external visual sensing sys-
                kθ
tem and used for real-time orientation control.


   Besides the outcome of torsion actuation to twist the tip along the neck axis, this

torsional actuation will also cause coupling deformations at the laser measurement

point due to the body compliance. The coupling effects can be modeled by the

equation

                                          48
                                                         
                           θbx  cθxτ x   0      0  τx 
                                                      
                                                      
                           θ  =  0     cθyτ y   0   τy                      (3.4.6)
                            by                        
                                                      
                                                      
                            θbz      0      0    cθzτ z   τz
The torque component τx is due to the small misalignment angle δα of the solenoids

along the X axis and is given by


                                         τx = τ sin δα.                           (3.4.7)


The main actuation torque is along the neck axis; because the neck is tilted by θn

from the horizontal cantilever body, this torque is decomposed into τy and τz by the

equations

                                    τy = τ cos δα cos θn                          (3.4.8)

                                    τz = τ cos δα sin θn                          (3.4.9)

τ is given by the equation

                                     τ = Vθp mΓθ cos θp                          (3.4.10)

By substitution, it is found that the resulting coupling deflections are linearly depen-

dent on the tip orientation θp , which is given by


            ∆mx,τ = kph R0 cos θ0 (cos θm cos θn + sinθm sin θn ) cos δα · θp    (3.4.11)

                                             ∆mz,τ = kph R0 cos θn sin δα · θp   (3.4.12)



   For the pair of parallel wires and the solenoid along the Y axis, they are employed

to provided 2D magnetic force which can be used to directly control the tip-sample
                                              49
interaction force. These two are named X- and Z- actuators in the sense that they

apply force in the X and Z direction, respectively, at the untwisted condition of the

probe. The magnetic forces generated by the two actuators are cast by a quadratic

model in equation 2.7b to capture the nonlinearity         and coupling c errors and linear

gains g and bias b .
                                                               
                               2
    Fmagx      11   12  Vx  g   g  Vx  c            b 
           =              +  11 12    +  1  Vx Vz +  1               (3.4.13)
                                                     
      Fmagz       21   22    Vz2   g21 g22   Vz    c2            b2

where Vx and Vz are the input voltages to the magnetic force actuators. As the tip

orientation changes, the magnetic forces are expressed in the reference coordinates

by a rotational transformation of the forces commutated by the quadratic model.
                                                      
                       Fmagx  cos θp − sin θp  Fmagx 
                              =                                              (3.4.14)
                                                     
                         Fmagz     sin θp cos θp    Fmagz


   Because the force application point is at the center of the magnetic particle, it is

transformed to the equivalent loads at the measurement location by



                                                
                           Feqx   1    0
                                         
                                         
                           F   0       0
                            eqy          
                                                
                                         
                           F   0          F
                                          1   magx 
                            eqz  
                                 =                                           (3.4.15)
                                                
                           τ   0 d  F
                            eqx         y   magz
                                         
                                         
                            τeqy   dz dx 
                                         
                                         
                                         
                             τeqz     −dy 0

                                             50
where d is a translation vector from the center of magnetic particle to the measure-

ment point.


     Given the body intrinsic compliance from equation 3.3.1, the relation between the

voltage inputs to the two magnetic force actuators and the laser readouts is readily

obtained:
                                                                                
                 ∆mx,f magx   0      kph R0 cos θ0 cos θm −kph R0 cos θ0 sin θm 
                             =                                                  
                                                                                
                   ∆mz,f magz    kph R0          0                    0
                                            
                                      1  0
                                            
                                            
                                   0   0
                                            
  0     0 cθxf z cθxτ x   0      0                                          
                                            
                                    0       cos θ − sin θ
                                          1                p  Fmagx (Vx , Vz )
 
                                                 p
  0     0   0      0    cθyτ y   0                                          
                                                                            
                                    0 d  sin θ    cos θp      Fmagz (Vx , Vz )
                                        y       p
  cθzf x 0   0      0    cθzτ z      
                                     
                                             
                                             
                                      dz dx 
                                            
                                            
                                            
                                      −dy 0

                                                                               (3.4.16)

                                          T
where Fmagx (Vx , Vz ) Fmagz (Vx , Vz )       is the quadratic magnetic force model. In

this modeling, the magnetic force is described by a quadratic model as function of

the two input voltages. The force is transformed to the measurement point in X

and Z axes and through the body compliance and measurement kinematics causes

laser measurement readouts. This entire relation includes parameters that depend

on the orientation change θp . For the purpose of real-time drift estimation and tip-

sample interaction force control, this overall relationship between the actuator inputs
                                              51
and resulting laser readouts is described by a new quadratic model with varying

parameters depending on tip orientation θp .
                                                                       
      ∆mx,mag      11   (θp )   12 (θp ) Vx2  g11 (θp ) g12 (θp ) Vx 
               =                          +                        
                                                                   
        ∆mz,mag       21   (θp )   22 (θp ) Vz2       g21 (θp ) g22 (θp )  Vz
                                                                           
                                                    c1 (θp )         b1 (θp )
                                                   +
                                                    
                                                              Vx Vz + 
                                                                      
                                                                                 
                                                                                    (3.4.17)
                                                     c2 (θp )            b2 (θp )



3.5    Real-time calibration of the quadratic model for mag-

       netic actuation



Similar to the ideas presented in Chapter 2, the magnetic actuation model parameters

are continuously updated after each tip-sample separation phase using recursive model

estimation.


   For force controlled two-axis scanning, the local sample surface slope is estimated

to actively control the tip orientation along the surface normal. At the same time, the

deflection drifts in the two-axis measurements are compensated, and the tip-sample

interaction force is precisely controlled. The AFM is operated based on tapping-

mode-like interaction-and-separation scheme. During the interaction phase, piezo-

electric scanner moves the tip along the surface tangential in the raster scanning

direction while the deflection and interaction force are simultaneously regulated. Tip

orientation is held constant during the interaction phase to avoid the coupling effects
                                              52
on deflections introduced by changing tip orientation. At the beginning of the sepa-

ration phase, the magnetic actuators start lifting up the tip until the adhesion force

between the tip and sample is overcome. Then the tip orientation is changed accord-

ing to the previous interaction phase’s estimated surface slope; the magnetic force

actuation model parameters are then estimated. The effects of the torsional actuation

coupling on deflections are constants for each step of orientation change, and they

are combined together with the model bias and thermal drift (λ = b + d + ∆m,τ ).

Finally, the magnetic actuation model for calibration is given by




                                                                            
                                                    2
  ∆mx,mag Vx , Vy , Vθp  ˆ11 (θp ) ˆ12 (θp ) Vx  g11 (θp ) g12 (θp ) Vx 
                                                           ˆ         ˆ
                         =                      +                       
                                                                        
    ∆mz,mag Vx , Vy , Vθp     ˆ21 (θp ) ˆ22 (θp ) Vz2      ˆ         ˆ
                                                           g21 (θp ) g22 (θp )    Vz
                                                                                   
                                                       c1 (θp )
                                                         ˆ                   ˆ
                                                                           λ1 (θp )
                                                      +
                                                       
                                                                 Vx Vz + 
                                                                          
                                                                                      
                                                                                      
                                                        c2 (θp )
                                                         ˆ                   ˆ
                                                                             λ2 (θp )

                                                                                  (3.5.1)



3.6    Two-axis force sensing



The models of magnetic torsion and force actuation have already been developed in

the above sections. The laser readings of the two-axis deflections are attributed to

the tip-sample interaction, magnetic actuation and thermal drift. After removing the

contribution from magnetic actuation and thermal drift,the two-axis interaction force

                                           53
sensing is given by the equation
                                                                                              
∆mx − ∆mx,mag Vx , Vy , Vθp   0      kph R0 cos θ0 cos θm −kph R0 cos θ0 sinθm 
                             =                                                 
                                                                               
  ∆mz − ∆mz,mag Vx , Vy , Vθp    kph R0          0                    0
                                                                      
                                                                         1    0 
                                                                                 
                                                                                 
                                                                      0     0 
                                                                                 
                                  0     0 cθxf z cθxτ x   0         0 
                                                                                       
                                                                                  
                                                                      0     1  Fintx 
                                                                               
                                  0     0   0      0    cθyτ y      0 
                                                                                      .
                                                                                       
                                                                      0    d1y  Fintz
                                                                               
                                  cθzf x 0   0      0    cθzτ z         
                                                                        
                                                                                  
                                                                                  
                                                                         d1z d1x 
                                                                                 
                                                                                 
                                                                                 
                                                                         −d1y 0

                                                                                            (3.6.1)

where d1 is the vector from the tip end point to the measurement location. This

equation is simplified one step further into
                                                                                                 
                                                                 ∆mx − ∆mx,mag (Vx , Vy , Vθ )
                                                                                                =
                                                                                                
                                                                   ∆mz − ∆mz,mag (Vx , Vy , Vθ )
                                                                                               
kph cos θm cθyτ y d1z − kph sin θm (cθzf x − cθzτ z d1y )      kph cos θm cθyτ y d1x        Fintx 
                                                                                                  .
                                                                                                  
                            0                              kph cos θ0 (cθxf z + cθxτ x d1y )   Fintz

                                                                                            (3.6.2)

Here a 2 × 2 matrix is obtained and is named as the effective compliance for the

tip-sample interaction force. The off-diagonal term is zero because the deviation

dx = 0 due to the symmetrical geometry about the center plane. In the real world,

there will be residual error in the deviation dx due to machining error and mechanical

                                                54
misalignment in the range of less than one micrometer. The significance of the off-

diagonal term is very small compared to the diagonal terms; thus the off-diagonal

term can be neglected, leaving the effective compliance for interaction force a diagonal

matrix. The numerical values for these two elements in the compliance matrix are

obtained using thermal noise from two-axis deflection measurement. Therefore, the

two-axis tip-sample interaction force sensing equation is written as
                                                               
                Fintx  kxx 0  ∆mx − ∆mx,mag (Vx , Vy , Vθ )
                       =                                                  (3.6.3)
                                                              
                  Fintz     0 kzz    ∆mz − ∆mz,mag (Vx , Vy , Vθ )

where kxx and kzz are the two effective stiffness obtain from thermal noise method

calibration.




3.7    Experimental evaluation



From single axis to multi-axis implementation, the system becomes more complicated

in that more magnetic actuators are required to control the tip force, and the actu-

ation on different axes are coupled together. Because the complexity of the control

algorithm increased, the real-time controller sampling rate needs to be decreased to

5kHz given the computing power. Detailed experimental results are discussed in the

following paragraphs.




                                          55
3.7.1   Evaluation of tip orientation



The tip orientation actuation is first modeled to obtain the relations between the input

voltage to the torsion actuator and the tip orientation change, which is estimated

using a standalone CCD camera that provides the top view of the manipulator. From

equation 3.4.5, it is evident that the applied actuation voltage Vθp is proportional to

θp /cosθp and is given by
                                                  θp
                                   Vθp = KV θ         .                         (3.7.1)
                                                cosθp

So the proportionality constant KV θ is calibrated by recording the actuation voltage

                                     ˆ
Vθp at the estimated tip orientation θp . The calibrated model, shown in figure 3.8, is

used later for real time tip orientation control.




Figure 3.8: Linear dependence of θp / cos θp on the voltage input Vθp to the magnetic

torsion actuator.


                                           56
   During the changing of tip orientation, the actuation torque also results in deflec-

tion changes on the X and Z axes due to coupling effects. It is developed from equation

3.4.11 that the deflection changes in both measurement directions are proportional

to the tip orientation change θp and are given by


                                 ∆mx,τ = Kτ,x θp                               (3.7.2)

                                 ∆mz,τ = Kτ,z θp                               (3.7.3)


Therefore, the model is fit into the experimental results to calibrate the two propor-

tionality constants, which is shown in figure 3.9.




  Figure 3.9: The coupling effects of torsion actuation on deflection measurement.




                                         57
3.7.2   Calibration of the quadratic magnetic force actuation model



From equation 3.5.1, a quadratic model is used to relate the input voltages of the two

magnetic force actuators to the resulting deflection changes. The model parameters

are time varying due to the actuator’s position dependence and the changes of the

tip orientation. An off-line calibration is performed by feeding slow changing inputs

to the pair of magnetic force actuators and measuring the corresponding deflections

as the tip orientation is fixed. Figure 3.10 shows that the magnetic force actuation

can be well captured by the proposed quadratic model.




             (a) Calibration on X                      (b) Calibration on Z



Figure 3.10: Magnetic actuation fitting results using quadratic model as the tip

orientation is fixed (θp = 0◦ ).




                                         58
3.7.3   2D interaction force control



To demonstrate the capabilities of the multi-axis probing system with drift compen-

sation and interaction force control, a micro pipette is scanned with the smallest

scanning section possessing a diameter of several hundred nanometers. During the

interaction phase, the cantilever deflection and tip-sample force is maintained con-

stant; the tip orientation is held constant within a contact cycle. At the beginning of

the next separation cycle, the tip orientation is corrected using the previous scanning

topography. Immediately after the tip is detached from the sample surface, the mag-

netic force actuation model is calibrated using the recursive estimator. The deflection

changes caused by the orientation correction are included in the magnetic actuation

model’s bias constants and can be compensated as deflection drifts. Therefore, the

following cycle’s interaction force in controlled based on the magnetic model calibra-

tion. The scanning direction is also controlled during the interaction phase where the

X and Z piezoelectric scanners work together to move the tip along the tangential

direction of the sample surface.


   For the purpose of two-axis interaction force control, the effects on deflection

change caused by magnetic actuation (orientation change and force) are removed

from the total laser reading, leaving the deflection changes caused by the tip-sample

interaction force. The tip-sample interaction process during the scanning of the micro




                                          59
pipette is shown in figure 3.11. It demonstrates the 2D force sensing ability by

comparing the force profiles when the topography has different orientations θT .




                  (a) θT = 45◦                         (b) θT = 90◦



Figure 3.11: 2D Force profiles on sample surfaces with different topography orienta-

tions.




3.7.4    Force controlled two-axis scanning of micro pipette



Figure 3.12 shows that compared with conventional scanning, the two-axis scanning

technology enables the tip to access the undercuts up to ±105◦ on the micro pipette.

The tip orientation change is limited by ±28◦ due to the ohmic heating generated

by the torsion actuation coils. However, the deflection and interaction force control

direction are alway on the surface normal direction. The topography of two-axis

                                        60
Figure 3.12: Comparison of micro pipette topography between conventional scanning

and two-axis scanning.




scanning is fit by a circle and the topography along the arc of pipette section is

plotted. When the sample topography orientation θT goes beyond the limitation

of tip orientation θp range, the tip-sample interaction point has small shift from

the end point of tip. This contact point shift causes the mean error of the micro

pipette topography, shown as the red trend in figure 3.13a. When the tip scans

on sidewalls, the Z piezoelectric actuator controls the scanning direction and the X

actuator regulates the deflection. Thus as the tip scans close to the side walls, the

topography relies more on the measurement of X piezo actuator, as given by equation


                            σr = | sin θp |σX + | cos θp |σZ                 (3.7.4)




                                          61
 (a) Topography resolution along the arc of micro(b) Caculation of standard deviation σr using
 pipette and the trend of mean error.            500 points in every section and comparison with
                                                 mathematical prediction.



Figure 3.13: Degradation of spatial resolution of two-axis scanning due to the larger

measurement noise in X piezoelectric scanner.




where σX and σZ are the resolutions of X and Z piezo positioner measurement signals.

The X piezo actuator has measurement resolution about 2.5nm and Z piezo actuator

has measurement resolution about 0.7nm. When we plot the topography of micro

pipette along the arc length, it is seen the degradation of the spatial resolution when

scanning close to the sidewalls, which is shown in figure 3.13b.


   The 3D topography of the micro pipette is also showed to demonstrate the drift

effects along the slow scanning direction. The scanning starts from the smaller end of

the pipette and gradually scans six sections with 6 minutes wait between each section.

Each section is scanned twice, with and without real-time drift compensation. It is



                                               62
                    (a)                                      (b)



Figure 3.14: (a) Comparison of micro pipette scanning with and without real-time

drift compensation; (b) Plot of drift against time.




clearly seen that the drift effects on the topography become larger and larger as the

scanning proceeds in the Y direction.




               (a) Topography                             (b) Force



   Figure 3.15: Conventional scanning topography and interaction force sensing.

                                          63
               (a) Topography                               (b) Force



     Figure 3.16: Two-axis scanning topography and interaction force sensing.




   In order to further demonstrate the advantages of using the multi-axis manipula-

tor and real time drift compensation, three cases of scanning are performed on micro

pipette to compare the force sensing during interaction: 1)single-axis scanning with-

out drift compensation (figure 3.15); 2)two-axis scanning without drift compensation

(figure 3.16); 3)two-axis scanning with drift compensation (figure 3.17). For all the

three cases, the tip and the sample have periodical interaction phases and separation

phases for the purpose of drift estimation. All have the same scanning speed at 1

line/sec with 100Hz tapping cycle. The scanning range is controlled to be within

±28◦ in this situation so that the tip is always pointing normal to the sample surface.

For each scanning scenario, the scanning is repeated four times on the same section,

with six minutes wait between each scanning. The different colored lines show the

sequence of scanning: blue→green→red→black.

                                          64
              (a) Topography                              (b) Force



Figure 3.17: Two-axis scanning topography and interaction force sensing with real-

time drift compensation.




Figure 3.18: Composite force on the surface normal direction over the scanning line.




   It is evident that thermal drift affects both the extracted topography and tip-

sample interaction force. With real-time compensation of the drift, the topography

is consistent among every scan and the interaction force is well controlled along

the surface normal direction. So far, the capabilities of the multi-axis manipulator

with tip-sample interaction control have been demonstrated. Figure 3.18 also shows

                                        65
Figure 3.19: Parameter changes of quadratic model for magnetic force actuation over

the scanning line.




the composite force magnitude over the pipette. During this scanning, the quadratic

model for magnetic force is recursively calibrated. The model parameters vary within

the scanning range due to the changes in tip orientation θp . The magnetic force

quadratic model parameters during one scanning line are shown in figure 3.19.




                                        66
                                 CHAPTER 4


              CONCLUSION AND FUTURE WORK




4.1    Conclusions



The main objective of this research is to precisely control the tip-sample interaction

force of an AFM probe based micro-manipulator in the presence of thermal drift

and model variations of actuation system. The quasi-static input-output model of

the manipulator is derived. Bases on this model, the tip-sample interaction force is

estimated.


   A magnetic force actuator is used to control the interaction and separation be-

tween the tip and sample. The tip-sample force is sensed and regulated during the

contact of the tip and sample. One important concern is the accuracy of the inter-

action force sensing (estimation). Due to the uncertainties from deflection drift and

magnetic actuation model parameters variation, it is required to conduct real-time


                                         67
calibration of the magnetic force actuator model during each separation cycle because

the interaction force estimation relies on the magnetic force input. The deflection

drift due to laser heating and ambient temperature change is also estimated dur-

ing the tip-sample separation and used for compensation during the contact. These

two techniques are essential to achieve precise control of the tip-sample force of the

manipulator.


   In order to address the accessibility issue of the conventional AFM probe, a three-

axis compliant manipulator was designed in our group that enables active control

of orientation of the manipulator tip and the scanning direction. Control of the

tip orientation is realized through an open loop calibration model of the magnetic

torsion actuator; two-axis interaction force control is achieved by means of a modified

laser measurement detection system and two-axis magnetic force actuation. The

techniques of real-time model calibration and drift compensation are integrated with

the tip orientation control, two-axis scanning and two-axis deflection measurement

of the three-axis probing system. Together, they enable 2D tip-sample interaction

force control with real-time thermal drift compensation. Finally, it is demonstrated

that, on samples with steep features such as sidewalls and undercuts, the tip-sample

interaction direction is controlled always along the sample surface normal, and the

force magnitude is also regulated.


   The significance of this research can be visualized by comparing the new system

with a conventional scanning probe system. In the conventional probing system,
                                         68
imaging and force manipulation are very sensitive to the thermal drift, resulting in

imaging artifacts and uncontrolled forces that can be detrimental to delicate samples;

however, this new system provides the ability to scan and control the tip-sample

force even when the thermal drift caused by laser heating is overwhelming during

the thermal imbalance when the laser measurement system is just turned on. In

other words, the issue that microscopy applications need to wait about an hour until

the thermal balance, will no longer exist with the new technologies developed in

this thesis. Even the deflection drift due to ambient temperature change during the

experiment is corrected by real time estimation. In a nutshell, this versatile multi-axis

micromanipulator has proven to be a successful candidate to meet the challenges on

precision tools for the advancement of nanotechnology.




4.2    Recommended future work



The following are a list of issues that have emerged from this project and can be

addressed in the future:

(1)Due to the time limit of this project, the current force sensing is based on the

quasi-static model of the probe. However, the tip-sample interaction process has a

rather complex nature, so dynamical force estimation needs to be implemented in

order to deal with the transient components of the interaction mechanics.

(2)The tip orientation control is essentially an open-loop control scheme relying on


                                           69
the pre-calibration of the model of tip orientation change. A real-time estimation

procedure of the tip orientation can be developed so that the tip orientation can be

controlled in a closed-loop manner.

(3)Applications such as to study mechanical properties of biological samples with

controlled force magnitude and direction can be performed in the future.




                                        70
                          BIBLIOGRAPHY



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                                       73
                                     APPENDIX A

   RECURSIVE LINEAR LEAST SQUARES ESTIMATOR




Suppose a sequence of t observations are available for a n-parameter linear model is

described by


                 y (k) = φ (k) β, k = 1 · · · t, φ (k) ∈ R1×n , β ∈ Rn×1             (A.0.1)


its estimation is given by

                                t                          −1    t
                                              T
                   ˆ
                   β (t) =           φ (k) φ (k)                      φ (k)T y (k)   (A.0.2)
                               k=1                              k=1


Define P (t) as
                                                  t                    −1
                                                            T
                               P (t) =                φ (k) φ (k)                    (A.0.3)
                                               k=1

Then P (t) is related to its previous step P (t − 1) by

                                         t
                             −1
                       P (t)        =         φ (k)T φ (k)
                                        k=1
                                        t−1
                                    =         φ (k)T φ (k) + φ (t)T φ (t)
                                        k=1


                                    = P (t − 1)−1 + φ (t)T φ (t)                     (A.0.4)

                                                      74
The estimation equation A.0.2 can also be written as
                                        t
                     ˆ
                     β (t) = P (t)           φ (k)T y (k)
                                       k=1
                                       t−1
                            = P (t)          φ (k)T y (k) + φ (t)T y (t)                   (A.0.5)
                                       k=1

Substituting the following equation into A.0.5
                  t−1
                        φ (k)T y (k) = P (t − 1)−1 β (t − 1)
                                                   ˆ
                  k=1


                                      = P (t)−1 − φ (t)T φ (t) β (t − 1)
                                                               ˆ                           (A.0.6)


Then


       β (t) = P (t) P (t − 1)−1 β (t − 1) − φ (t)T φ (t) β (t − 1) + φ (t)T y (t)
       ˆ                         ˆ                        ˆ

            = β (t − 1) − P (t) φ (t)T φ (t) β (t − 1) + P (t) φ (t)T y (t)
              ˆ                              ˆ                                             (A.0.7)


The equation A.0.7 can be written as

                               ˆ       ˆ
                               β (t) = β (t − 1) + K (t) e (t)                             (A.0.8)


where K (t) = P (t) φ (t)T and e (t) = y (t) − φ (t) β (t − 1). Using the Matrix
                                                     ˆ

Inversion Lemma given by

                                                                      −1
                (A + BCD)−1 = A−1 − A−1 B C −1 + DA−1 B                    DA−1            (A.0.9)


on equation A.0.4 and let A = P (t) , B = φ (t)T , C = I 1×1 , D = φ (t) It is obtained

that

                                                                           −1
    P (t) = P (t − 1) − P (t − 1) I 1×1 + φ (t) P (t − 1) φ (t)T                φ (t) P (t − 1)

                                                                                          (A.0.10)
                                                75
and K (t) and be simplified as

                                                                   −1
         K (t) = P (t − 1) φ (t)T I 1×1 + φ (t) P (t − 1) φ (t)T           (A.0.11)



   In summary, the set of equations for recursive linear least squares estimation is

given


                                       ˆ       ˆ
                                       β (t) = β (t − 1) + K (t) e (t)     (A.0.12)

                                                              ˆ
                                        e (t) = y (t) − φ (t) β (t − 1)    (A.0.13)
                                                                     −1
           K (t) = P (t − 1) φ (t)T I 1×1 + φ (t) P (t − 1) φ (t)T         (A.0.14)

                                P (t) = [I n×n − K (t) φ (t)] P (t − 1)    (A.0.15)


where equation A.0.12 is the update of parameter estimation and the equation A.0.13

is the evaluation of prediction compared to observation.




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