SENSING AND CONTROL OF TIP-SAMPLE
INTERACTION FORCE OF A THREE-AXIS
Presented in Partial Fulﬁllment of the Requirements for the Degree Master of
Science in the Graduate School of the Ohio State University
Shiwen Ai, B.Eng.
Graduate Program in Mechanical Engineering
The Ohio State University
Chia-Hsiang Menq, Advisor
c Copyright by
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
two-axis and drift-free scanning on samples with unknown geometry and steep fea-
tures such as sidewall and reentrant.
Dedicated to my parents.
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.
1988 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Born in Yuzhou, Henan, China
2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.Eng. in Automatic Control, Zhejiang
2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graduate Fellow, The Ohio State Univer-
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
TABLE OF CONTENTS
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
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
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
LIST OF TABLES
1.1 Major nano/micro devices used for force spectroscopy and manipulation 8
LIST OF FIGURES
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
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
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
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.
AFM was ﬁrst invented  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
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
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
 and Tapping Mode  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
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
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.
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
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.  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.  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  developed a multi-axis
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 , optical tweezers , needle-like micro-
manipulators  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 . 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.
Magnetic tweezers Optical tweezers AFM
Spatial resolution ∼ 10nm ∼ nm ˚
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
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
such as single molecular protein, DNA and live cells. Bhushan et al. used AFM to
investigate tribological and mechanical characterization such as adhesion, friction and
wear of carbon nanotubes. Morii et al. 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
In the second category, researchers have investigated using AFM as a force con-
trolled manipulation tool for nano object positioning and structure assembly.
Akita etal.  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.  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.  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.
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.  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
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
In the imaging applications, thermal drift can be corrected using correlation and
Kalman ﬁlter methods. Kindt et al.  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.  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.  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,
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
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
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
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.
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
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
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
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 . The principles and
advantages of magnetic actuation are discussed in the following section.
2.2 Principles of direct force actuation using magnetic actu-
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.
m= 0 0 m , B = 0 −B 0 . It is assumed that the bending motion of the
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
the Z axis according to the cantilever length l: Fmag(eq) = .
Figure 2.2: Scheme of magnetic force actuation on Z axis.
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
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).
Zpzt (s) = V (s)
s2 + (ωz /Qz ) s + ωz
Here ωz is the resonant frequency; Qz is the quality factor; and Γz is a scaling constant.
Figure 2.4: Block diagram of simultaneous deﬂection and tip-sample interaction force
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).
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
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,
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  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)
where the system matrices are deﬁned as
0 1 0
,B = ,C = 1 0 ,D = 0 ,
m m m
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
y = C OT
A state observer can be constructed based on the augmented state space repre-
sentation of the system.
x A BAwd x
= + Bu + Lp (y − y )
ˆ˙ O Awd wˆ
y = C OT
By properly choosing the pole placement vector Lp according to the designed observer
dynamics, the interaction force can be obtained from the estimation w.
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
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
actuator. By substituting the magnetic ﬁeld B = vΓ, the quasi-static magnetic force
model is obtained:
Fmag = , (2.4.1)
where v is the actuator voltage input and Γ is the actuation gain. The deﬂection
caused by magnetic force ∆m,mag is given by
∆m,mag = (2.4.2)
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 . A second order model for the magnetic actuation is applied with
∆m,mag = v 2 + gv + b, (2.4.3)
where represents the small nonlinearity, g is the actuator linear gain and b is the
Based on the quasi-static model of the cantilever, the laser measurement of can-
tilever deﬂection, ∆m , is attributed to three parts:
∆m = + ∆m,mag + d, (2.4.4)
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
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:
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.
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
Figure 2.5: Picture of the customized Agilent 5500 AFM with cooling jacket hosting
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
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)
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.
Figure 2.8: Magnetic model parameters variation (ˆ, g , λ) as the probe location
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.
(a) Interaction force estimation at the original lo-
cation where the magnetic actuation model is cali-
(b) Interaction force estimation with the pre-
calibrated magnetic actuation model at a diﬀerent
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.
Figure 2.10: Deﬂection and force proﬁles during tip-sample interaction process.
Figure 2.11: Magnetic actuation model parameter estimation during tip-sample in-
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
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-
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
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
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.
Figure 2.13: Interaction force comparison during the contact periods: with vs. with-
out drift compensation.
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.
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
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
Figure 3.1: Multi-axis manipulator with neck and body compliant sections and micro
Figure 3.2: Design of the neck axis and magnetic moment directions orthogonal to
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 , 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
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
(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
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.
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
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
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
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
t 0 0 ctzf z ctzτ x 0 0 Fz
θ 0 0 cθxf z cθxτ x 0 0 τx
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:
Et w3 0 0 0 0 3
b b ETb wb
0 0 0 0 0
0 0 0 0
b Ewb t3b
0 6lb 12lb
0 0 0
Ewb t3 Ewb t3
0 0 0 0 0
0 0 0 0 3
Etb wb Etb wb
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
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
θf x 1 0 0 θbx
θ = 0 cosθ −sinθm θby (3.4.1)
θf z 0 sinθm cosθm θbz
where θm is the mounting angle of the ﬂoating mirror.
The laser beam is reﬂected twice on the two micro mirrors and falls at the location
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:
xbeam 0 R0 0
= θ (3.4.2)
zbeam R0 cosθ0 0 0
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
∆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
1 0 0 θbx
∆mx 0 kph R0 cos θ0 0
m − sin θm θby
0 cos θ
∆mz kph R0 0 0
0 sin θm cos θm θbz
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
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:
= Vθp . (3.4.5)
cos θp kθ
The constant is obtained by calibration using an external visual sensing sys-
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
θbx cθxτ x 0 0 τx
θ = 0 cθyτ y 0 τy (3.4.6)
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
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
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 .
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
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
F 0 F
τ 0 d F
eqx y magz
τeqy dz dx
τeqz −dy 0
where d is a translation vector from the center of magnetic particle to the measure-
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
∆mx,f magx 0 kph R0 cos θ0 cos θm −kph R0 cos θ0 sin θm
∆mz,f magz kph R0 0 0
0 0 cθxf z cθxτ x 0 0
0 cos θ − sin θ
1 p Fmagx (Vx , Vz )
0 0 0 0 cθyτ y 0
0 d sin θ cos θp Fmagz (Vx , Vz )
cθzf x 0 0 0 cθzτ z
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
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 +
c2 (θp ) b2 (θp )
3.5 Real-time calibration of the quadratic model for mag-
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
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
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
∆mx,mag Vx , Vy , Vθp ˆ11 (θp ) ˆ12 (θp ) Vx g11 (θp ) g12 (θp ) Vx
∆mz,mag Vx , Vy , Vθp ˆ21 (θp ) ˆ22 (θp ) Vz2 ˆ ˆ
g21 (θp ) g22 (θp ) Vz
c1 (θp )
λ1 (θp )
Vx Vz +
c2 (θp )
λ2 (θp )
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
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
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
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
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
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 )
Fintz 0 kzz ∆mz − ∆mz,mag (Vx , Vy , Vθ )
where kxx and kzz are the two eﬀective stiﬀness obtain from thermal noise method
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
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
Vθp = KV θ . (3.7.1)
So the proportionality constant KV θ is calibrated by recording the actuation voltage
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
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.
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◦ ).
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
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-
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
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)
(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
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
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.
(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.
(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
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.
CONCLUSION AND FUTURE WORK
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
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
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
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,
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
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.
 G. Binnig, C.F. Quate and C. Gerber. Atomic force microscope. Phys. Rev.
Lett. 56 (1986), 930–933.
 T. Junno, S. Anand, K. Deppert, L. Montelius, L. Samuelson. Contact mode
atomic force microscopy imaging of nanometer-sized particles. Applied Physics
Letters. 66 (1995), 1063–l066.
 A. Putman, O. Kees, G. Grooth, F. Hulst, J. Greve. Tapping mode atomic
force microscopy in liquid. Applied Physics Letters. 64 (1994), 2454–2456.
 G.R. Jayanth, C.H. Menq. Modeling and design of a magnetically actuated two-
axis compliant micromanipulator for nanomanipulation. IEEE/ASME Trans-
action on Mechatronics 15 (2010), 360–370.
 G.R. Jayanth, C.H. Menq. A re-orientable micromanipulator with two axis
force sensitivity for nanometrology and manipulation. Manuscript submitted
In press (2010).
 J.E. Griﬃth, D.A. Griﬀ, M.J. Vasile, P.E. Russell, E.A. Fitzgerald. Scanning
probe metrology. Journal of Vacuum Science and Technology 10 (1992), 674–
 Y. Martin, H.K. Wickramasinghe. Method for imaging sidewalls by atomic
force microscopy. Applied Physics Letters 64 (1994), 2498–2500.
 M. Michihata, Y. Takaya, T. Hayashi. Development of the nano-probe system
based on the laser-trapping techniqu. CIRP Annals - Manufacturing Technol-
ogy 57 (2008), 493-496.
 Y. Huang, Z. Zhang, C.H. Menq. Minimum-variance Brownian motion control
of an optically trapped probe. Applied Optics 48 (2009), 5871–5880.
 Z. Zhang, k. Huang, C.H. Menq. Design, Implementation and Force Modelling
of Quadrupole Magnetic Tweezers. IEEE/ASME Transactions on Mechatronics
15 (2010), 704–713.
 K.C. Neuman, A. Nagy. Single-molecule force spectroscopy: optical tweezers,
magnetic tweezers and atomic force microscopy. Nature Methods 5 (2008), 491–
 D.A. LaVan, D.R. Overby, J. Karavitis, D.E. Ingber. Electromagnetic needles
with submicron pole tip radii for nanomanipulation of biomelecules and living
cells. App. Phys. Lett. 85 (2004), 2968.
 Y. Ansel, F. Schmitz, S. Kunz, H.P. Gruber, G. Popovic. Development of
tools for handling and assembling microcomponents. J. Micromechanics and
Microengineering 12 (2002) 430–437.
 K. Molhave, O. Hansen. Electro-thermally actuated microgippers with in-
tegrated force-feedback. J. Micromechanics and Microengineering 15 (2005)
 B. Bhushan. Nanotribology of carbon nanotubes. J. Phys.: Condens. Matter
20 (2008), 36.
 T. Morii, R. Mizuno, H. Haruta, T. Okada. An AFM study of the elasticity of
molecules. Proceedings, 7th International Symposium on Atomically Controlled
Surfaces, Interfaces and Nanostructures 464-465 (2004), 456–458
 M. Sitti. Survey of nanomanipulation systems. Proceedings, 1st IEEE Confer-
ence on Nanotechnology (2001), 75–80.
 S. Akita, Y. Nakayama, S. Mizooka, Y. Takano, T. Okawa, Y. Miyatake, S.
Yamanaka, M. Tsuji, T. Nosaka. Nanotweezers consisting of carbon nanotubes
operating in an atomic force microscope. App. Phys. Lett. 79 (2001), 1691.
 J.Y. Park, Y. Yaish, M. Brink, S. Rosenblatt, P.L. McEuen. Electrical cutting
and nicking of carbon nanotubes using an atomic force microscope. App. Phys.
Lett. 80 (2002), 4446.
 A.A.G. Requicha, S. Meltzer, R. Resch, D. Lewis, B.E. Koel, M.E. Thompson.
Layered nanoassembly of three-dimensional structures. Proceedings, IEEE In-
ternational Conference on Robotics and Automation 4 (2001), 3408–3411.
 T. Thundat, R.J. Warmack, G.Y. Chen, D.P. Allison. Thermal and ambient-
induced deﬂections of scanning force microscope cantilevers. App. Phys. Lett.
64 (1994), 2894–2896.
 J.H. Kindt, J.B. Thompson, M.B. Viani, P.K. Hansma. Atomic force micro-
scope detector drift compensation by correlation of similar traces acquired at
diﬀerent setpoints. Review of Scientiﬁc Instruments 73 (2002), 2305–2307.
 B. Mokaberi, A.A.G. Requicha. Drift compensation for automatic nanomanip-
ulation with scanning probe microscopes. IEEE Transaction on Automation
Science and Engineering 3 (2006), 199–207.
 H. Torun, O. Finkler, F.L. Degertekin. Athermalization in atomic force mi-
crospcope based force spectroscopy using matched microstructure coupling.
Rev. Sci. Instrum. 80 (2009), 076103.
 Y. Jeong, G.R. Jayanth, S.M. Jhiang, C.H. Menq. Direct tip-sample interaction
force control for the dynamic mode atomic force microscopy. Applied Physics
Letters 88 (2006), 204102.
 G.F. Franklin, J.D. Powell, M.L. Workman. Digital control of dynamic systems.
Addison-Wesley 9780201820546 (1998).
 D.A. Walters, J.P. Cleveland, N.H. Thompson, P.K. Hansma, M.A. Wendman,
G. Gurley, V. Elings. Short cantilever for atomic force microscopy. Review of
Scientiﬁc Instruments 67 (1996), 3583.
RECURSIVE LINEAR LEAST SQUARES ESTIMATOR
Suppose a sequence of t observations are available for a n-parameter linear model is
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
β (t) = φ (k) φ (k) φ (k)T y (k) (A.0.2)
Deﬁne P (t) as
P (t) = φ (k) φ (k) (A.0.3)
Then P (t) is related to its previous step P (t − 1) by
P (t) = φ (k)T φ (k)
= φ (k)T φ (k) + φ (t)T φ (t)
= P (t − 1)−1 + φ (t)T φ (t) (A.0.4)
The estimation equation A.0.2 can also be written as
β (t) = P (t) φ (k)T y (k)
= P (t) φ (k)T y (k) + φ (t)T y (t) (A.0.5)
Substituting the following equation into A.0.5
φ (k)T y (k) = P (t − 1)−1 β (t − 1)
= P (t)−1 − φ (t)T φ (t) β (t − 1)
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
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
Inversion Lemma given by
(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
P (t) = P (t − 1) − P (t − 1) I 1×1 + φ (t) P (t − 1) φ (t)T φ (t) P (t − 1)
and K (t) and be simpliﬁed as
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
β (t) = β (t − 1) + K (t) e (t) (A.0.12)
e (t) = y (t) − φ (t) β (t − 1) (A.0.13)
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.