This dissertation presents the development of a hexapole 3D magnetic actuator that can actively control a magnetic bead in three-dimensional space. The actuator achieves magnetic bead stabilization, trajectory tracking, accurate force modeling, and dynamic force sensing. An optimal inverse model of the over-actuated hexapole system minimizes input currents to produce a desired magnetic force. Experimental results validate the inverse model and magnetic force model. Hall sensors are also integrated to directly measure magnetic fields and further improve force modeling accuracy. The actuator enables both force control applications and automatic scanning of unknown environments through dynamic force sensing and parameter estimation.

1. Three-Dimensional Motion Control and Dynamic Force
Sensing of a Magnetically Propelled Micro Particle Using a
Hexapole Magnetic Actuator
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy
in the Graduate School of The Ohio State University
By
Fei Long, M.S.
Graduate Program in Mechanical Engineering
The Ohio State University
2016
Dissertation Committee:
Chia-Hsiang Menq, Advisor
Manoj Srinivasan
Rama Yedavalli
Vadim I. Utkin

2. Copyright by
Fei Long
2016

3. ii
Abstract
This dissertation presents the development of a hexapole 3D magnetic actuator that
can be used as a probing system by actively controlling a magnetic bead in three
dimensional space. The magnetic force, which is a noncontact force, is an ideal force for
biological applications due to its biocompatibility and magnetic susceptibility. This
magnetic actuator can achieve magnetic bead stabilization, trajectory tracking, accurate
force modeling and dynamic force sensing. These capabilities will transform the
magnetic actuator from the traditional force applier to a three-dimensional scanning
probe system, which is not achieved in other magnetic actuator systems.
An over-actuated Hexapole magnetic actuator is employed to realize 3D motion
control of the magnetic bead. A lumped parameter magnetic force model is derived to
characterize the nonlinear relationship from the input current to the output magnetic
force. This electromagnetic actuating system achieves significantly greater force
generation capability compared with existing magnetic actuators [1, 2]. These
improvements are accomplished through enhanced design and optimization of the current
allocation of the over-actuated system. A magnetic bead can be stably controlled and
steered and the magnetic force model is experimentally validated.
The fundamental issues in this over-actuated multi-pole actuator are caused by the
following four characteristics of the magnetic force: a) redundancy and coupling, b)
instability, c) nonlinearity, and d) position dependency. An optimal inverse model of the

4. iii
over-actuated hexapole electromagnetic actuating system over the 3-D workspace is
derived to minimize 2-norm of the six input currents when applied to produce the desired
3-D magnetic force on the magnetic bead. This inverse model greatly facilitate the
feedback linearization in the feedback control. Due to the compact form of the optimal
inverse model, it can be implemented in high speed real-time control to achieve stable
magnetic bead trapping and precise motion control.
Another challenge in electromagnetic actuation system is the hysteresis effect. The
existing current-based magnetic force model relies on the assumption that the magnetic
flux generation is proportional to the input current, which is not valid under hysteresis
effect. The hysteresis effect will greatly degrade the magnetic force model accuracy. As
soft magnetic material is used in this magnetic actuator, the hysteresis effect is not only
nonlinear but also rate-dependent. Moreover, the magnetic coupling among six magnetic
poles will make the hysteresis issue beyond the capability of modeling. To solve this
problem, Hall sensors are introduced to directly measure the magnetic field in each pole
and a so called Hall-Sensor-based magnetic force model is proposed. Owning to the 3-D
motion control capability, the Hall-Sensor-based and current-based magnetic force
models can be experimentally calibrated and compared by steering the magnetic bead
wherein the viscous force can serve as the reference force. It is clearly seen that the Hall-
Sensor-based magnetic force model greatly outperforms the current-based magnetic force
model in term of force modeling accuracy. When the magnetic field becomes larger, the
magnetization saturation of the magnetic bead begin to emerge and a more accurate Hall-

5. iv
sensor based model is proposed in which the nonlinear magnetization effect of the
magnetic bead is modeled.
With accurate magnetic force model, a dynamic force sensing estimator can be
developed to achieve real-time dynamic force sensing and parameter estimation
simultaneously. With the measurement information of the magnetic bead and the Hall-
sensor based magnetic force model, the bead-sample interaction force can be dynamically
estimated. Moreover, the drag coefficient can be also estimated, which is an indication of
the environment change such as the wall effect, fluid property change and etc. The
Kalman filter algorithm is used to estimate the state variables since the dynamics subject
to random thermal force and measurement noises. Combined with motion control
capability, this magnetic actuator can achieve force control application and automatic
scanning of an unknown environment.

6. v
To my grandfather

7. vi
Acknowledgments
In the past one fifth of my life, I was very lucky to spend a meaningful time in the
Precision Measurement and Control Lab (PMCL). I would like to sincerely thank my
advisor Dr. Chia-Hsiang Menq, who continuously encourages me, challenges me and
advises me with his enthusiasm. His inquisitiveness and logical thinking inspired me all
the time. I am also very thankful for Dr. Manoj Srinivasan, Dr. Rama Yedavalli, and Dr.
Vadim Utkin for agreeing to be my committee members and giving me valuable advices.
I am very grateful to dear PMCL friends, with who I shared scores of achievements
and frustrations. I want to thank Dr. Peng Cheng for his strong support in all aspects,
especially the image processing work, which might be the most beautiful real-time visual
sensing system in this world. I sincerely thank my colleagues, Zhen Liu and Yanhai Ren,
with who I spent countless days and nights. They are always willing to give their
intelligent suggestions and kind encouragement. I want to thank my seniors Dr. Zhipeng
Zhang and Dr. Yanan Huang, who gave me a lot of constructive opinions in research and
life.
I want to thank Dr. Daisuke Matsuura, who finished the mechanical design part of
this project and put in all his effort to facilitate my work at the beginning of this project.
Most of all, I am so grateful to my beloved parents and grandparents. Especially, I
want to thank my grandparents. I am not able to become who I am if I were not brought
up by them. I cannot imagine how lucky I was to be influenced by their noble characters.
Finally, I am deeply indebted to my grandfather, whose interest in mathematics inspired
me to become a researcher.

8. vii
Vita
Feb, 1987 ....................................... Born – Dandong, Liaoning, China
2005 ............................................... Huainan No.2 Middle School, Huainan, Anhui,China
2009 ............................................... B.S. Mechanical Engineering, Shanghai Jiao Tong
University, Shanghai, China
2009 – 2010 .................................. University Fellowship, The Ohio State University
2010 - Present ............................... Graduate Research Assistant, PMCL, The Ohio State
University
Publications
1. Z. Zhang, F. Long, and C. H. Menq,“Three-dimensional visual servo control of a
magnetically propelled microscopic bead,” IEEE/ASME Trans. Robot. 29, 373–
382 (2013)
2. F. Long, D. Matsuura, and C. H. Menq,“ Actively Controlled Hexapole
Electromagnetic Actuating System Enabling 3-D Force Manipulation in Aqueous
Solutions,” IEEE/ASME Trans. Mechatronics., (accepted)
3. F. Long, P. Cheng, and C. H. Menq, “Optimal Inverse Modeling and Control of
Hexapole Electromagnetic Actuation”, (in process)
4. F. Long, P. Cheng and C. H. Menq, “A Hall Sensors Based 3-D Magnetic Force
Modeling and Experiment of an Actively Controlled Hexapole Electromagnetic
Actuator”, (in process)
5. F.Long, P. Cheng and C. H. Menq, “Accurate 3D Magnetic Force Modeling and
Experiments of Superparamagentic Magnetic Bead Propelled By Hexapole
Electromagnetic Actuator”, (in process)
Fields of Study
Major Field: Mechanical Engineering

9. viii
Table of Contents
Abstract............................................................................................................................... ii
Acknowledgments.............................................................................................................. vi
Vita....................................................................................................................................vii
List of Figures..................................................................................................................xiii
Chapter 1: Introduction ................................................................................................... 1
1.1. Background and Motivation ............................................................................. 1
1.2. Objectives and specific aims.............................................................................. 5
1.3. Dissertation Overview........................................................................................ 8
Chapter 2: design, force modeling and inverse modeling of the hexapole magnetic
actuator............................................................................................................................ 10
2.1 Introduction........................................................................................................... 10
2.2 Design of Hexapole Electromagnetic Actuator................................................... 12
2.2.1 Design, synthesis, and fabrication .................................................................... 12
2.2.2 Finite element analysis of the magnetic field ................................................... 15
2.2.3 Hexapole magnetic field model........................................................................ 17
2.3 Force Model and Inverse Modeling..................................................................... 21
2.3.1 General magnetic force model.......................................................................... 21

10. ix
2.3.2 Current-Based Magnetic Force Model ............................................................. 22
2.3.3 Inverse Model Based on Constant Constraints................................................. 24
2.3.4 Optimal inverse model at the center of the workspace..................................... 25
2.4. Experimental Verification of the Optimal Inverse Model................................ 28
2.5. Calibration and Validation of the Force Model ................................................ 31
2.6. Force Generation Capability............................................................................... 38
2.7. Conclusion............................................................................................................. 41
Chapter 3: The Optimal Inverse Model and Control of Hexapole Electromagnetic
Actuation.......................................................................................................................... 43
3.1 Introduction........................................................................................................... 43
3.2 Hardware Implementation of High Speed Control............................................ 44
3.3 The inverse model.................................................................................................. 46
3.3.1 The position dependent inverse model based on constant constraints ............. 46
3.3.2 The Optimal Inverse Model in the Entire Workspace...................................... 47
3.4 Active feedback control: stabilization, Brownian motion control and tracking
control........................................................................................................................... 53
3.4.1 Stabilization and Brownian motion control...................................................... 53
3.4.2 Trajectory tracking ........................................................................................... 57
3.5 Comparison of High Speed Control in FPGA and Low Speed Control in PC 60

11. x
3.6. Conclusion............................................................................................................. 63
Chapter 4: Hall Sensors Based 3-D Magnetic Force Modeling and Experiment ..... 65
4.1 Introduction........................................................................................................... 65
4.2. The Magnetic force models.................................................................................. 66
4.2.1 The Hall-sensor-based magnetic force model .................................................. 66
4.2.2 The current-based magnetic force model ......................................................... 68
4.3 Hardware Integration and Validation of Hall Sensors based Hexapole
Magnetic Actuator....................................................................................................... 70
4.3.1 Hall sensors integrated with magnetic actuator................................................ 70
4.3.2 Validation of the Hall Sensor measurement..................................................... 71
4.4 Hall Sensor Measurement in Magnetic Bead Control and Calibration Of
Magnetic Force Models............................................................................................... 73
4.4.1 Hall Sensor measurement in bead trapping ...................................................... 73
4.4.2 Force model calibration.................................................................................... 74
4.5 Application of Hall-Sensor-Based Model and application of Hall Sensors ..... 82
4.5.1 Force Prediction................................................................................................ 82
4.5.2 Magnetic Field Feedback Control Using Hall-Sensor inner loop control........ 86
4.6 Accurate Hall sensor based Magnetic Force Modeling for large magnetic force
generation..................................................................................................................... 90

12. xi
4.6.1 The modeling of the magnetic bead magnetization.......................................... 90
4.6.2 The Modeling of The Accurate Hall Sensor Based Magnetic Force................ 93
4.6.3 Calibration of the Magnetic Force Model ........................................................ 95
4.6.4 Application of the Accurate Hall-Sensor-Based Model................................. 101
4.6.4.1 Magnetic Force Prediction Using the Accurate Hall-Sensor-Base Force
Model.................................................................................................................. 101
4.6.4.2 Parameter calibration and magnetization study ...................................... 105
4.6.4.3 Accurate Hall-sensor-based optimal inverse modeling .......................... 106
4.7 Conclusion............................................................................................................ 109
Chapter 5: Dynamic force sensing and parameter estimation ................................. 111
5.1. Introduction........................................................................................................ 111
5.2 Joint State-Parameter Estimator Algorithm.................................................... 113
5.2.1 Drag coefficient estimation based on the magnetic force model.................... 113
5.2.2 Drag coefficient estimation based on the thermal variance measurement ..... 118
5.3 Simulation result the joint state-parameter estimator..................................... 120
5.3.1 The simulation of the drag coefficient estimation in water............................ 120
5.3.2 The simulation of the drag coefficient estimation in Glycerol....................... 121
5.3.3 The simulation of the simultaneous estimation of the drag coefficient and the
external force ........................................................................................................... 123

13. xii
5.4 Experiment results .............................................................................................. 125
5.5 Conclusion............................................................................................................ 129
Chapter 6: Conclusion and future works ................................................................... 131
6.1 Conclusion............................................................................................................ 131
6.2 Future Works....................................................................................................... 134
BIBLIOGRAPHY......................................................................................................... 136

14. xiii
List of Figures
Fig. 2.1. (a) CAD model of motion stage and lower poles. (b) Assemble yoke ring and
three upper poles............................................................................................................... 12
Fig. 2.2. (a) Fabricated prototype, integrated on an inverted microscope. (P2, P4, P5)
forms the upper layer and (P1, P3, P6) forms the lower layer. Each pole is associated with
an actuation coil (for flux generation) and a measurement coil (for flux measurement). P2
→P1, P4→P3 and P6→P5 are +x, +y and +z directions of the actuation coordinate
system. (b) CAD model and meshing of the hexapole actuator ....................................... 12
Fig. 2.3. (a) The top view (measurement coordinate) of the vector plot of the magnetic
flux density distribution (unit: Tesla). (b) The magnetic flux density vectors near the
workspace center (actuation coordinate). (c) The magnetic field vectors near the tip of
pole 1................................................................................................................................. 16
Fig. 2.4. Magnitude of the magnetic flux density, i.e., |B|, associated with the
measurement coordinate system. (a) |B| in the horizontal plane (top view). (b) |B| in the
vertical plane (side view).................................................................................................. 17
Fig. 2.5. (a) Hexapole magnetic actuator integrated on an inverted microscope. (P2, P4,
P5) forms the upper layer and (P1, P3, P6) forms the lower layer. Each pole is associated
with an actuation coil (for magnetic flux generation). P2→P1, P4→P3 and P6→P5 are
+x, +y and +z directions of the actuation coordinate system. (b) CAD model of six tips of
the hexapole magnetic actuator and the associated Measurement Coordinate {O;x ,y ,z }m m m

15. xiv
and Actuation Coordinate {O;x ,y ,z }a a a . (c) FEM analysis in ANSYS about magnetic
charge model of a sharp-tipped magnetic pole. (d) The sketch of the magnetic bead and
six magnetic charges......................................................................................................... 17
Fig. 2.6. Validation of the hexapole magnetic field model: (a) comparison of magnetic
induction vectors, and (b) normalized norms of error vectors. (c) the definition of the
fitting error........................................................................................................................ 21
Fig. 2.7. Illustration of the desired force in the spherical coordinate system. .................. 26
Fig. 2.8. Three orientation-dependent optimal constraints compared with their
corresponding constant constraints................................................................................... 27
Fig. 2.9. Optimal current allocation compared with that obtained using constant
constraints. ........................................................................................................................ 28
Fig. 2.10. Block diagram of the feedback motion control ( dP is the desired position, mP is
the measurement position, e is the error signal, dF is the desired force calculated by the
controller, '
(t) (t )d d a F F is caused by the actuator delay, MTF is the magnetic force, F is
the modeling error, TF is the thermal force, a and m are the delay in actuator and
measurement).................................................................................................................... 28
Fig. 2.11. Six input currents of the hexapole actuator ...................................................... 30
Fig. 2.12. Stabilization of the magnetic particle at the center of the workspace .............. 31
Fig. 2.13. Actuation current applied to coil 1 and voltage readings from the six
measurement coils............................................................................................................. 32
Fig. 2.14. Twelve linear trajectories on the horizontal plane passing the center of the
workspace (in measurement coordinate) .......................................................................... 34

16. xv
Fig. 2.15. Particle motion and viscous force displayed with reference to the actuation
coordinate, wherein black dashed lines are target motion and color solid lines are
measured motion (plotted in time sequences)................................................................... 35
Fig. 2.16. Six actuation currents associated with motion control ..................................... 35
Fig. 2.17. The best-fitted results when using the nominal flux distribution matrix to
calibrate the force gain vector........................................................................................... 37
Fig. 2.18. The best-fitted results when using the measured flux distribution matrix to
calibrate the force gain vector........................................................................................... 37
Fig. 2.19. The best-fitted results when using the modified flux distribution matrix to
calibrate the scaling factor and the force gain vector simultaneously. ............................. 38
Fig. 2.20. Testing the linear range of electromagnetic actuation: P2 (left) and P3 (right). 39
Fig. 2.21. Comparing three force envelopes calculated using nominal force models: 3-D
actuator using current allocation based on constraint constraints (blue), newly developed
actuating system using current allocation based on constraint constraints (red), and newly
developed actuating system using optimal current allocation (grey)................................ 40
Fig. 2.22. Comparing three force envelopes calculated using calibrated force models: 3-D
actuator using current allocation based on constraint constraints (blue), newly developed
actuating system using current allocation based on constraint constraints (red), and newly
developed actuating system using optimal current allocation (grey)................................ 41
Fig. 3.1. (a) Magnetic setup integrate with the inverted microscope (b) High speed
embedded control system using FPGA (c) real-time display of the reference bead and the

17. xvi
control bead (the reference sticks to the cover glass surface to provide the information
such as drift, vibration and etc.)........................................................................................ 44
Fig. 3.2. Inverse model based on constant constraints. (a) The percent force error at
(20,0,0)um, desired force ˆ
dF =1; (b) The percent force error at (20,0,0)um, desired force
ˆ
dF =10; (c) The percent force error at (40,0,0)um, desired force ˆ
dF =1; (d) The percent
force error at (40,0,0)um, desired force ˆ
dF =10................................................................. 48
Fig. 3.3. Optimal inverse model. (a) The percent force error at (20, 0, 0) um; (b) The
percent force error at (40, 0, 0) um ................................................................................... 48
Fig. 3.4. Locations where the optimal inverse model are obtained .................................. 50
Fig. 3.5. Comparison of percentage force error between Taylor expansion and least-
square fitting ..................................................................................................................... 52
Fig. 3.6. (a) Force generation envelops at (0,0,0)um. (b) Force generation envelops at
(20,0,0)um. (c) Force generation envelops (40,0,0)um .................................................... 53
Fig. 3.7. Block diagram of the feedback motion control using proportional controller ( dP
is the desired position, mP is the measurement position, e is the error signal, Kp is the
proportional gain, dF is the desired force calculated by the controller, '
(t) (t )d d a F F is
caused by the actuator delay, a and m are the delay in actuator and measurement, MTF is
the magnetic force, F is the modeling error, TF is the thermal force) ............................. 53
Fig. 3.8. 3D plot in bead stabilization experiment (the result in from The Optimal inverse
model) ............................................................................................................................... 55

18. xvii
Fig. 3.9. Actuation effort using optimal inverse model and the inverse model based on the
constant constraints. (For the inverse model based on constant constraints, I5 and I6 are
negative with much larger absolute value than optimal invers model) ............................ 56
Fig. 3.10. The positioning standard deviation in x,y and z axis using the optimal inverse
model and constant constraints......................................................................................... 56
Fig. 3.11. The positioning performance from Proportional controller, using optimal
inverse model and constant constraints............................................................................. 56
Fig. 3.12. Trajectories in 3D tracking control................................................................... 58
Fig. 3.13. Actuation effort in 3D tracking control. (For the inverse model based on
constant constraints, I5 and I6 are negative with much larger absolute value than optimal
invers model) .................................................................................................................... 59
Fig. 3.14. Tracking error using the optimal inverse model and inverse model based on
constant constraints........................................................................................................... 59
Fig. 3.15. The measurement PSD curves and calibrated PSD curves (The peak at 71.37Hz
is due to structure vibration) ............................................................................................. 61
Fig. 3.16. The positioning result using the optimal trapping stiffness in 200Hz case and
1606Hz case...................................................................................................................... 62
Fig. 3.17. Trapping stiffness vs. standard deviation of Brownian motion, 200Hz control
and 1606Hz control........................................................................................................... 63
Fig. 4.1. Magnetic field around the tip of the magnetic pole and a suppositional Hall
Sensor................................................................................................................................ 67

19. xviii
Fig. 4.2. (a). Hall sensors integrated with Hexaple magnetic actuator. (b) Zoom-in plot of
Hall sensors associated with upper poles. (c) Zoom-in plot of Hall sensors associated with
lower poles........................................................................................................................ 70
Fig. 4.3. (a). The setup for studying the relationship between the surface-mount Hall
Sensor measurement and the Hall Sensor measurement at the tip. P1 and P2 are actuated
by current-driven coils. (b). Positions and dimensions of hall elements. ......................... 71
Fig. 4.4. Input-output plot between Vtip and Vsurface in two cases: 1). P1 is self-actuated, 2)
P2 is actuated, P1 is magnetized by the magnetic coupling between P1 and P2............... 72
Fig. 4.5. input-output plot between current and Hall Sensor voltage. .............................. 74
Fig. 4.6. Motion control result in trajectory tracking........................................................ 75
Fig. 4.7. The input-output of six poles in two tracking experiments, the second
experiment is conducted by reversing the sign of the actuation currents. (To make the
input-output plot orientated with positive slope, the signs of some plots are changed
accordingly) ...................................................................................................................... 76
Fig. 4.8. Control loop for magnetic force analysis; ( dP is the desired position, mP is the
measurement position, pK is the proportional control gain, dF is the desired force
calculated by the controller, a and a are the delay in actuator and measurement, MTF is the
magnetic force, TF is the thermal force, N is measurement noise) ................................... 77
Fig. 4.9. PSD analysis without low-pass-filter ................................................................. 78
Fig. 4.10. PSD analysis with low-pass-filter..................................................................... 78
Fig. 4.11. Hall-sensor-based magnetic force model along circles in xy plane (as Fig.4.6
(a))..................................................................................................................................... 80

20. xix
Fig. 4.12. Hall-sensor-based magnetic force model along circles in yz and xz planes (as in
Fig. 4.6 (b) and Fig.4.6 (c))............................................................................................... 80
Fig. 4.13. Current-based magnetic force model along circles in xy plane (as Fig.4.6 (a))81
Fig. 4.14. Current-based magnetic force model along circles in yz and xz planes (as in
Fig.4.6 (b) and Fig.4.6 (c))................................................................................................ 81
Fig. 4.15. Histogram of modeling error in force calibration (Hall-sensor-based model vs.
Current-based model; the black histogram are from Fig.4.11 and Fig.4.12, the green
histogram are from Fig.4.13 and Fig.4.14) ....................................................................... 82
Fig. 4.16. Bead motion trajectory for force prediction ..................................................... 83
Fig. 4.17. Force prediction using Hall-Sensor-based model (motion along circles in xy
plane)................................................................................................................................. 83
Fig. 4.18. Force prediction using Hall-Sensor-based model (motion along straight line in
xy plane)............................................................................................................................ 84
Fig. 4.19. Force prediction using Current-based model (motion along circles in xy plane)
........................................................................................................................................... 84
Fig. 4.20. Force prediction using Current-based model (motion along straight lines in xy
plane)................................................................................................................................. 84
Fig. 4.21. Histogram of modeling error in force prediction (Hall-sensor-based model vs.
Current-based model; the black histogram are from Fig.4.17, the green histogram are
from Fig.4.19)................................................................................................................... 85

21. xx
Fig. 4.22. Histogram of modeling error in force prediction (Hall-sensor-based model vs.
Current-based model; the black histogram are from Fig.4.18, the green histogram are
from Fig.4.20)................................................................................................................... 85
Fig. 4.23. Magnetic bead position control integrated with Hall-sensor inner loop
control. dP and mP are the desired and measurement position, e is the positioning error, dF
is the desired force calculated from the motion controller. dV and mV are the desired and
measured Hall-Sensor voltage. ......................................................................................... 86
Fig. 4.24. Normalized Response of different desired voltage dV ....................................... 87
Fig. 4.25. Step response for different voltage inputs. ....................................................... 87
Fig. 4.26. Positioning performance using Hall-Sensor inner loop control. ...................... 89
Fig. 4.27. dV is the desired voltage and mV is the measurement voltage)............................ 89
Fig. 4.28. Step Response of Hall-sensor feedback control. .............................................. 89
Fig. 4.29. Magnetization of the M450 superparamagnetic bead....................................... 91
Fig. 4.30. The plot of the Langevin Function (assume the saturation limit sm =1)............ 92
Fig. 4.31. Motion control result in trajectory tracking...................................................... 95
Fig. 4.32. The input-output of six poles in two tracking experiments, the second
experiment is conducted by reversing the sign of the actuation currents. (To make the
input-output plots orientated with positive slope, the signs of some plots are changed
accordingly) ...................................................................................................................... 97
Fig. 4.33. Calibration result of Hall-Sensor-based force model using Langevin function.
(a) Force calibration along circles in xy plane (as in Fig.4.31) (b) zoom-in plot of force
calibration ......................................................................................................................... 99

22. xxi
Fig. 4.34. Calibration result of Hall-Sensor-based model without using Langevin-
function. (a) Force calibration along circles in xy plane (as in Fig.4.31) (b) zoom-in plot
of force calibration............................................................................................................ 99
Fig. 4.35. Calibration result of Hall-Sensor-based force model using Langevin function.
(a) Force calibration along circles in yz and xy planes (as in Fig.4.31) (b) zoom-in plot of
force calibration .............................................................................................................. 100
Fig. 4.36. Calibration result of Hall-Sensor-based force model without using Langevin
function. (a) Force calibration along circles in yz and xy planes (as in Fig. 4.31) (b)
zoom-in plot of force calibration .................................................................................... 100
Fig. 4.37. Histogram of the force calibration error for Hall-sensor-based model with
Langevin-function and without Langevin-function. (The red histograms are from the
modeling errors of Fig.4.33 and Fig.4.35, the blue histograms are from the modeling
errors of Fig.4.34 and Fig.4.36) ...................................................................................... 101
Fig. 4.38. Bead motion trajectory for force prediction ................................................... 101
Fig. 4.39. Force prediction of Hall-Sensor-based force model using Langevin function.
(a) Force calibration along circles in xy plane (as in Fig.4.38) (b) zoom-in plot of force
calibration ....................................................................................................................... 102
Fig. 4.40. Force prediction of Hall-Sensor-based force model without using Langevin
function. (a) Force calibration along circles in xy plane (as in Fig.4.38) (b) zoom-in plot
of force calibration.......................................................................................................... 103

23. xxii
Fig. 4.41. Force prediction of Hall-Sensor-based force model using Langevin function.
(a) Force calibration along line trajectories in xy plane (as in Fig.4.38) (b) zoom-in plot
of force calibration.......................................................................................................... 103
Fig. 4.42. Force prediction of Hall-Sensor-based force model without using Langevin
function. (a) Force calibration along line trajectories in xy plane (as in Fig.4.38) (b)
zoom-in plot of force calibration .................................................................................... 104
Fig. 4.43. Histogram of the force calibration error for Hall-sensor-based model with
Langevin-function and without Langevin-function. (The red histograms are from the
modeling errors of Fig.4.39, the blue histograms are from the modeling errors of
Fig.4.40).......................................................................................................................... 104
Fig. 4.44. Histogram of the force calibration error for Hall-sensor-based model with
Langevin-function and without Langevin-function. (The red histograms are from the
modeling errors of Fig.4.41, the blue histograms are from the modeling errors of
Fig.4.42).......................................................................................................................... 104
Fig. 4.45. relationship between external magnetic field ||B|| (Gauss) and the magnetic
moment ||m||.................................................................................................................... 106
Fig. 4.46. Plots of || ||B ,|| ||V and || || / || ||B V .................................................................... 107
Fig. 4.47. (a). The plot of   2ˆ 20 || || || ||Lf a V V and  ˆ 20 || ||Lf a V . (b). the plot of  ˆ 20 || ||Lf a V .
(c). Zoom-in plot of   2ˆ 20 || || || ||Lf a V V when || ||V is small. (d). the plot of
  2ˆ 20 || || || ||Lf a V V and the inverse function   2ˆ( 20 || || || || )Linverse f a V V ............................ 109

24. xxiii
Fig. 5.1. Joint state-parameter estimator (using the Hall sensor based force model)
running simultaneously with feedback controller. ( dP is the desired position, mP is the
measurement position, e is the error signal, a and m are the delay in actuator and
measurement, ˆI is the current from the nominal inverse model, dF is the desired force
calculated by the controller, '
(t) (t )d d a F F is caused by the actuator delay, F is the
modeling error, MTF is the magnetic force, TF is the thermal force, EF is the bead-sample
interaction force, B is the magnetic flex density, mV is the Hall sensor measurement
voltage, ˆ
MTF is the modeling magnetic force, ˆ is the estimated drag coefficient, ˆ
EF is the
estimated external force)................................................................................................. 113
Fig. 5. 2. Estimation result in water using the magnetic force model to estimate the drag
coefficient ....................................................................................................................... 121
Fig. 5. 3. Estimation result in water using the thermal variance to estimate the drag
coefficient ....................................................................................................................... 121
Fig. 5. 4. Estimation result in glycerol using the magnetic force model to estimate the
drag coefficient ............................................................................................................... 122
Fig. 5. 5. Estimation result in glycerol using the thermal variance to estimate the drag
coefficient ....................................................................................................................... 123
Fig. 5. 6. Estimation result in glycerol using the thermal variance to estimate the drag
coefficient (no measurement noise)................................................................................ 123
Fig. 5. 7. Estimation result using the magnetic force model when the drag coefficient and
external force are changing simultaneously.................................................................... 124

25. xxiv
Fig. 5.8. Joint state-parameter estimation using Hall sensor based model (a) estimation
result in x direction (b) estimation result in y direction (c) estimation result in z direction
(d) trajectories used to test joint state-parameter estimation algorithm.......................... 125
Fig. 5.9. Joint state-parameter estimator (using current-based magnetic force model)
running simultaneously with feedback controller. (The meaning of all parameters is the
same as that in Fig.5.1, the only difference is that the modeling force ˆ
MTF is from current-
based model instead of Hall-sensor based model).......................................................... 127
Fig. 5.10. Joint state-parameter estimation, Hall sensor based force model vs. Current-
based force model. (a) Joint state-parameter estimation result in x direction. (b) Joint
state-parameter estimation result in y direction. (c) Joint state-parameter estimation result
in z direction. (d) Trajectories used to test joint state-parameter estimation algorithm. 128

26. 1
Chapter 1: Introduction
1.1. Background and Motivation
Probing biological samples and manipulating biological processes have become
important techniques in the study of cell mechanics and mechanobiology [3], since it is
widely discovered that the interaction force between cells/biomolecules plays an
important role in many physiological processes [4-6]. Especially, a dream of using
modern instruments to probe biomolecules in live cells has been shared by many
researchers in the field.
In the scanning probe microscopy (SPM) family [7], Atomic Force Microscopy
(AFM) offers high spatial resolution and enables force probing and scanning [8]. AFM,
however, remains as a 2-D surface tool [9-12] due to its kinematic constraints and the
restriction of mechanical connections employed [13]. It has been applied to study
biomolecules from isolated model systems [14, 15] rather than from their native and fully
active environment.
Optical trapping is another modern technique that is useful for the study of biological
systems under physiological conditions [16-22]. It is well suited for quasi-static force
measurement [23] and dynamic force sensing [24]. Whereas it has been theoretically
derived and experimentally verified that trapping smaller objects can be achieved by
increasing the power of the trapping laser, heating is a main issue that needs to be

27. 2
resolved [25]. One feasible approach is to implement real-time Brownian motion control
to reduce heating [26, 27]. However, due to its underlying working principle, unwanted
trapping of debris may easily mix with the measurement probe. Lack of specificity in
optical trapping is, therefore, an important limitation that must be examined before it is
employed to probe biological samples [19].
Magnetic tweezers use magnets and/or electromagnets to generate magnetic field to
propel microscopic magnetic particles, which serve as measurement probes and exert
force on biological samples. They have several advantages over optical tweezers.
Magnetic field is intangible and safe to most biological materials. It is specific to
magnetic particles and there is no heat generated in the process. It is, however, necessary
to employ feedback control to achieve stable magnetic trapping since the magnetic force
field is inherently unstable [28]. Therefore, without active control, the magnetic
particles/probes are often functionalized and anchored to target bio samples to avoid
instability. Many magnetic actuators are, thus, actually simple force appliers, wherein the
force is adjusted according to a pre-calibrated force function [29]. They were used in a
wide range of applications, ranging from manipulating biological macromolecules [30-
34], probing cell membranes [35-37], to characterizing intracellular properties [38-41].
Among various magnetic tweezers, some could only apply forces in a single
direction, using only one coil-actuated pole [29, 42] or two poles facing each other [30,
39], some were 2-D systems with multiple poles [40, 43], and a few were able to generate
forces in 3-D space [34]. These magnetic tweezers were employed as simple force
appliers, without motion control, wherein magnetic forces were applied and the induced

28. 3
motions of magnetic particles were recorded by appropriate measurement systems. One
development of magnetic tweezers did use motion control to achieve stabilization of a
4.5µm magnetic particle suspended in water [44]. It employed a six-pole magnetic
apparatus to generate an upward magnetic force, whereas the downward force was caused
by the gravity. Swimming robots propelled in liquid using external magnetic fields have
also been successfully demonstrated [45-48]. Moreover, an electromagnetic system was
designed to control intraocular micro-robots for delicate retinal procedures [49-51]. An
error less than 0.5 mm was achieved when controlling the micro-robot in a visual-
servoing framework [51].
Accurate force modeling and inverse modeling of electromagnetic actuation are
essential for effective force manipulation and for enabling stable magnetic trapping via
feedback control. The development of a 2-D quadrupole magnetic tweezers was reported
in [52, 53], wherein four tip-shaped electromagnetic poles were employed to control 2-D
magnetic force to propel a microscopic magnetic particle, serving as a force-sensing
probe. It was applied to characterize the mechanical property of live cells using a
functionalized probe [54]. A lumped-parameter analytical magnetic force model was
derived to characterize the nonlinearity of the magnetic force with respect to the input
currents and its position dependency in the workspace. Its extension to the realization of a
3-D hexapole magnetic actuator was detailed in [1, 2]. Employing a visual particle
tracking system [55], a visual servo control system was developed to enable precise
motion control of the magnetically propelled particle in the 3-D workspace [1], wherein a

29. 4
simple inverse model was derived and used to implement a nonlinear feedback control
law to realize stable magnetic trapping.
These two multi-pole electromagnetic actuating systems had electromagnetic poles
made of thin (approximately 100μm thick) high-permeability nickel-iron magnetic alloy
(permalloy) film. Whereas the design of these systems had advantages, such as easy in
fabrication and assembly, it had several limitations. First, it was necessary to use
specially made sample chambers, therefore, standard culture dishes commonly used for
live cell experiments could not be employed due to space limitations. Second, small
cross-sectional area of the thin film resulted in large magnetic reluctance, and thus
yielded small magnetic flux and saturation flux density.
Other than hardware design, inverse modeling is also important for multi-pole
electromagnetic actuating systems, which are usually over-actuated systems. The issue
raised by redundancy was solved through applying constant constraints in [1, 52]. But the
use of constant constraints resulted in excessive actuation effort, severely limiting the
force generation capability. A micro-robot system [50] implemented a suboptimal inverse
model from pseudo inverse, which cannot be applied in our system due to the nonlinear
nature of the force model. A multi-pole magnetic actuator [56, 57] could steer a ferrofluid
drop by optimal control effort. But the sampling rate was limited to 15Hz due to the
cumbersome calculation. Moreover, 3-D realization has not been achieved.
Besides the position control capabilities, the magnetic force model accuracy and
dynamic force sensing capabilities are also crucial for biological applications since many
biological processes happen in a short time and the interaction forces between biological

30. 5
samples can be as small as pN scale. Most researchers did not directly verify the accuracy
of the magnetic force models. The stability of the model-based feedback control [1, 2, 52]
is far less than enough to verify the accuracy of the magnetic force model since the
feedback control can stabilize the bead even when the modeling error exists. For the force
application, however, the inaccuracy of the magnetic force model will lead to biased
force estimation between the magnetic bead and biological samples. A common issue for
soft magnetic material is the hysteresis problem [58], which is usually rate-dependent
[59, 60] in soft magnetic material. Moreover, the poles in the hexapole magnetic actuator
are magnetically coupled together, which made the hysteresis effect very difficult to
model. Some researchers estimated the magnetic force from Brownian motion
fluctuations through image measurement microscopy [61, 62], which is actually a quasi-
static force estimation. New method/model need to be proposed to solve the complicated
hysteresis effect and achieve dynamic force estimation to capture the transient bead-
sample interaction force.
This research project aims to enable a haxapole magnetic actuator to perform active
scanning tasks, wherein a magnetic bead can be actively controlled and the force can be
accurately sensed/controlled dynamically.
1.2. Objectives and specific aims
The objective of this research project is to develop an over-actuated actively
controlled magnetic actuator that can achieve stabilization, motion control, and accurate

31. 6
force sensing and force control. According the objectives, four specific aims are
identified.
(1). Hardware development, magnetic force modeling and calibration
A hexapole magnetic actuator was employed for 3D motion control and force
application. Six sharp-tipped magnetic poles, each of which is actuated by a coil, are
alignment in three orthogonal directions to control the magnetic bead in 3D space. The
actuator is over-actuated since the magnetic poles can only generate attractive force.
Besides the redundancy, the magnetic force on the magnetic bead is nonlinearly
dependent on the input current and is position dependent. A lumped parameter magnetic
force model is employed to characterize the relationship between the input current and
the resulting magnetic force. Due to the motion capability, the magnetic force model can
be experimentally calibrated and verified.
(2). Optimal inverse model and control: stabilization, Brownian motion control and
trajectory tracking
Inverse model is important for feedback linearization and feedback control. The
magnetic force generation capability also greatly depend on the current allocation. Since
the magnetic force is redundant, infinite many solutions exist for a specific desired force.
Constant constraints are used in [1, 2, 52] to remove the redundancy. However, the force
generation capability is greatly sacrificed under such constraints. Moreover, the
constraints result in excessive large current allocation, which will lead to larger modeling

32. 7
error. An optimal inverse model based on the minimal squared norm criterion will be
developed to achieve improved feedback linearization. Stabilization, Brownian motion
control and trajectory tracking can be achieved and the performance will be compared
with that from constant constraints.
(3). Hall Sensor Based 3-D magnetic force modeling and feedback control
Compared with the motion control capability, accurate force modeling is a more
challenging task. With feedback control, the bead can be steered stably even when
modeling error exists. The force application, however, completely relies on the magnetic
force model. Due to the complicated nature of the hysteresis effect and the magnetic
coupling, hysteresis modeling is a very difficult task for the hexapole magnetic actuator.
The Hall sensors can be introduced to directly measure the magnetic field from each pole.
The feasibility of Hall sensor based magnetic force model is investigated. Moreover, with
direct measurement of the magnetic field, the secondary control loop can be established
to directly control the magnetic field of each pole.
(4). Dynamics force sensing and parameter estimation
Dynamics force sensing is desired to sense the force between the magnetic bead and
the biological samples in real-time. If the magnetic force can be accurately modeled, a
joint state-parameter estimator can be developed to simultaneously estimate the bead-
sample interaction force and the drag coefficient of the liquid environment. This
capability can transform the magnetic actuator into an active scanning probe. Combined

33. 8
with the motion control capability, this magnetic actuator can achieve force control or
automatic scanning. For example, the bead-sample interaction force can map the
topography of an unknown part and the drag coefficient is an indication of local
environment change such as cytoplasm property and etc.
1.3. Dissertation Overview
This dissertation is organized as follow. Chapter 2 presents the design, force modeling
and inverse modeling of the hexapole magnetic actuator. The hardware design and
improved current allocation mechanism greatly increased the force generation capability.
Stabilization of the magnetic bead was achieved and the magnetic force model was
experimentally calibrated. Chapter 3 described the development of the optimal inverse
model in the sense of minimal current norm. This optimal inverse model solved
redundancy, nonlinearity, and position dependency simultaneously. Stabilization,
Brownian motion control and tracking control were achieved and the performance was
compared with the inverse model using constant constraints [1]. Chapter 4 presented the
so called Hall-sensor based magnetic force model. Hall sensors are introduced to directly
measure the magnetic field in each pole and the measurement result is lumped into a
magnetic force model. With Hall sensors, obvious magnetic hysteresis can be observed in
bead stabilization and motion control. By steering the magnetic bead in 3D space, the
Hall-sensor based model can be calibrated and compared with the current-based model in
term of force modeling accuracy. The result showed that the Hall-sensor based model
greatly outperformed the current-based model. With Hall sensor measurement, a

34. 9
secondary control loop can be established to directly control the magnetic field in each
pole. When the magnetic field becomes larger, a more accurate Hall sensor based model
is proposed to consider the nonlinear magnetization effect of the magnetic bead. With the
establishment of Hall-sensor based magnetic force model, the dynamic force sensing and
parameter estimation capability are introduced in Chapter 5. This joint state-parameter
estimator enables the magnetic actuator to dynamically sense the interaction force
between the magnetic bead and biological samples. This capability will transform the
magnetic actuator from a simple force applier to a 3D active probing system. Conclusions
and future work are presented in Chapter 6.

35. 10
Chapter 2: design, force modeling and inverse modeling of the
hexapole magnetic actuator
2.1 Introduction
Magnetic tweezers are widely used in biological engineering since the magnetic force
is a noncontact force with biocompatibility and susceptibility. However, most magnetic
tweezers are force appliers wherein the magnetic bead is anchored to biological samples
to avoid instability. One development of magnetic tweezers did use motion control to
achieve stabilization of a 4.5um magnetic particle in water is shown in [44]. However,
the feedback control is heuristic instead of model-based wherein large Brownian motion
is observed. Actively controlled multi-pole magnetic tweezers are developed by L. Chen
[63], but analytical force model is not proposed while the magnetic field comes from
interpolation and look-up table from FEM analysis. An improved 3-D hexapole magnetic
actuator was developed by Z. Zhang [2], wherein analytical magnetic force model is
proposed. Employing a visual particle tracking system [55], a visual servo control system
was developed to enable 3D motion control of the magnetically propelled particle in the
3-D workspace [1], wherein a simple inverse model was derived and used to implement a
nonlinear feedback control law to realize stable magnetic trapping.
Magnetic actuators in [1, 2] are made of thin permalloy film. However, there are
several limitations. First, specially made sample chambers are need in the experiment

36. 11
since the poles are fixed. Therefore, standard culture dishes commonly used for live cell
experiments could not be employed due to space limitations. Second, the thin-film design
with small cross-sectional area will yield small magnetic flux and the saturation limit of
this material will be relatively low compared to other magnetic material.
For over-actuated multi-pole electromagnetic systems, inverse modeling is also
important, not only for feedback control, but also for force generation capability. The
issue raised by redundancy was solved through applying constant constraint in [1]. But
the use of constant constraints resulted in excessive actuation effort, severely limiting the
force generation capability.
This chapter presents the design, modeling, and calibration of an over-actuated
hexapole electromagnetic actuating system, which is used to stabilize and propel a
microscopic magnetic particle in the liquid environment. The design and synthesis of the
six electromagnetic poles, including geometric design and material selection, result in
high magnetic saturation limit, and thus larger force generation capability. An improved
current allocation scheme is derived and implemented to realize real-time current
allocation to enable the most effective manipulation of the 3-D magnetic force exerting
on the particle at the center of the workspace. Compared to the 3-D magnetic actuator in
[1], the hexapole electromagnetic actuating system achieves significantly greater force
generation capability and much improved controllability of 3-D force in the workspace.

37. 12
2.2 Design of Hexapole Electromagnetic Actuator
2.2.1 Design, synthesis, and fabrication
Fig. 2.1. (a) CAD model of motion stage and lower poles. (b) Assemble yoke ring and three upper poles.
Fig. 2.2. (a) Fabricated prototype, integrated on an inverted microscope. (P2, P4, P5) forms the upper layer
and (P1, P3, P6) forms the lower layer. Each pole is associated with an actuation coil (for flux generation)
and a measurement coil (for flux measurement). P2→P1, P4→P3 and P6→P5 are +x, +y and +z directions
of the actuation coordinate system. (b) CAD model and meshing of the hexapole actuator
Since a single electromagnetic pole can only generate attractive force exerting on the
magnetic particle, three pairs of electromagnetic poles are employed and their pole tips
are placed symmetrically on three orthogonal axes to enable generation of 3-D forces. In
order to significantly increase the force generation capability, these sharp-tipped poles are
made of cone-shaped iron steel rods, and are assembled to concentrate the magnetic flux

38. 13
into the workspace. Each electromagnetic pole is actuated through an individual coil. All
the coils and poles are then magnetically connected through a magnetic yoke, which
helps increase the magnetic flux density and the flux gradient in the workspace. The six
pole tips enclose the workspace, wherein the specimen and the magnetic particle are
placed. The hexapole electromagnetic actuator is assembled with two overlaid motion
stages to form a unique experimental apparatus. The apparatus is integrated with an
inverted microscope equipped with a visual particle tracking system [55]. In order to have
the optical path free of blockage, a rigid body rotation is applied to the three pairs of
electromagnetic poles along with their three orthogonal axes such that their tips are on
two parallel horizontal planes, i.e., one upper plane and the other lower plane. Fig.2.1 (a)
shows the design of a manual x-y stage and the lower three electromagnetic poles,
assembled on an x-y-z piezo motion stage. The coarse x-y stage achieves an 8mm×8mm
working range, and can be used to locate a specimen cell. The more detailed design of the
moving stage can be found in [64, 65]. The lower three poles are fixed under the culture
dish, which holds live sample cells so that the bottom cover glass of the culture dish,
whose thickness is approximately 100μm, can be placed in-between upper three and
lower three poles; the gap between them is as small as possible to generate strong
magnetic force. The upper three magnetic poles are assembled on the yoke ring and their
pole tips sunk into medium filled in the dish, as shown in Fig.2.1 (b). They can be easily
disassembled to replace culture dish to perform live cell experiments in practical use. The
arrangement also makes cleaning easy, necessary to avoid cell contamination. The
magnetic particle is to be stabilized and steered within the 3-D workspace, whereas the

39. 14
position of the sample dish with respect to the 3-D workspace is controlled by the 3-axis
piezo stage.
The fabricated prototype is shown in Fig.2.2 (a). The electromagnetic poles and the
magnetic yoke are fabricated with 1018 steel, low-carbon (0.18% carbon) steel with high
saturation limit (over 2T). However, one drawback of using such material is the
hysteresis effect, which will be addressed in the future through hysteresis modeling or
real-time sensing and control. At the present time, superparamagnetism is assumed to
characterize force generation capability of the actuating system. The diameter of the
poles is about 6mm. Three upper poles are about 45mm long, whereas the lower poles are
42mm long and are milled to form a flat platform to support the culture dish. Whereas
sharper tips can be produced using advanced machining processes, the radius of the pole
tips of the prototype is 40μm, which is included in the finite element analysis (Fig.2.2
(b)). The distance from the workspace center to the tips of all six poles is adjustable due
to the flexibility achieved in the new design. Its nominal value of 500µm is used in all
analyses and experiments reported in this chapter. The actuation coils, realized for
experiments and winded around the protrusions of the yoke, have 70 turns. The whole
setup is integrated upon an inverted microscope (Olympus IX 81) with bright field
illumination and a dry 60x objective lens is used for visual measurement. A vision-based
measurement method with sub-nanometer resolution [55] is employed to provide position
feedback, wherein a speed CMOS camera (Mikrotron MC3010) and Image
grabbing/processing card (Matrix Odyssey XCL) are used. A 4.5um bead (Dynabead M-
450 Epoxy) is used in the experiment. The coils are driven by DA converters

40. 15
(Measurement Computing, PCI-DAS6032), followed by six linear power amplifiers
(Micro Dynamics, BTA-28V-6A) working in current mode with 10K bandwidth. A piezo
positioner (PI P-721 PIFOC) is used to drive the lens for image calibration. The whole
setup is put on a Smart Table (Newport) and a vibration isolation table (Herzan TS-150)
to remove vibrations.
2.2.2 Finite element analysis of the magnetic field
Finite Element Method (FEM) is employed to analyze quantitatively the magnetic
field produced by the actuating system and to visualize its spatial distribution,
particularly within the workspace. A CAD model imitating the real setup is built and
meshed (Fig.2.2 (b)) using the ANSYS environment for FEM calculation. When applying
current to the coil, the magnetic field produced by the hexapole actuator can be computed
using FEM analysis.
Fig.2.3 shows the ANSYS analysis result by actuating coil 1 with 1A current. It can be
seen from Fig.2.3 (a) (top view of measurement coordinate) that the magnetic field forms
a closed loop under the guidance of magnetic poles and the yoke, which makes the
magnetic flux density and the flux gradient in the workspace significantly higher. Fig.2.3
(b) shows the magnetic flux density vectors, associated with the actuation coordinate
system, within a 100μm cube centered at the origin of the workspace. Due to the direction
of the input current, which is counter clockwise, these vectors all point to the tip of pole
1. The magnetic field vectors near the tip of the electromagnetic pole 1 are shown in
Fig.2.3 (c). It can be seen that they strongly converge to the tip of the electromagnetic

41. 16
pole. Fig.2.3 (b) and Fig.2.3 (c) validate the assumption that sharp tip behaves like a point
charge. The magnetic flux density is measured in Tesla (104
Gauss) in Fig.2.3.
Fig. 2.3. (a) The top view (measurement coordinate) of the vector plot of the magnetic flux density
distribution (unit: Tesla). (b) The magnetic flux density vectors near the workspace center (actuation
coordinate). (c) The magnetic field vectors near the tip of pole 1.
Fig.2.4 shows the contour plot of the magnitude of the magnetic flux density
(measured in Gauss) when applying 1A current to coil 1. A similar analysis for the thin-
foil design was given in Fig. 6 of [2]. Specifically, the contour plots of two sections, i.e.
the horizontal plane and the vertical plane, are displayed and the gradient of the magnetic
field is clearly visualized. The magnetic flux density in the vicinity of the workspace
center of this hexapole actuator increases by twofold from that of the actuator with the
thin-foil design. Moreover, the saturation limit of the magnetic pole is over 2T, which is
much higher than that of permalloy pole with 0.9T saturation limit in the previous design.
This improvement allows each 70-turn coil to be actuated up to 3 Amperes, the force

42. 17
generation capability is greatly increased. The detailed force generation calculation will
be addressed later in Chapter 2.6.
Fig. 2.4. Magnitude of the magnetic flux density, i.e., |B|, associated with the measurement coordinate
system. (a) |B| in the horizontal plane (top view). (b) |B| in the vertical plane (side view).
2.2.3 Hexapole magnetic field model
Fig. 2.5. (a) Hexapole magnetic actuator integrated on an inverted microscope. (P2, P4, P5) forms the upper
layer and (P1, P3, P6) forms the lower layer. Each pole is associated with an actuation coil (for magnetic
flux generation). P2→P1, P4→P3 and P6→P5 are +x, +y and +z directions of the actuation coordinate
system. (b) CAD model of six tips of the hexapole magnetic actuator and the associated Measurement
Coordinate {O;x ,y ,z }m m m and Actuation Coordinate {O;x ,y ,z }a a a . (c) FEM analysis in ANSYS about
magnetic charge model of a sharp-tipped magnetic pole. (d) The sketch of the magnetic bead and six
magnetic charges.

43. 18
Fig.2.5 shows the hexapole model associated with the measurement coordinate
system and the actuation coordinate system, wherein the three upper poles/charges are
associated with input currents I2, I4 and I5, and lower poles/charges I1, I3 and I6.
It can be seen from Fig.2.5(c) that the magnetic field produced by a sharp tipped pole
looks to the magnetic particle in the workspace of the actuator as though it is generated
by a point source [66],
0
i
iq


 (2.1)
Where qi is the magnetic charge of the ith
pole, Φi is the magnetic flux going through the
corresponding pole, μ0 is the permeability of vacuum. The magnetic field at the location
of the magnetic bead results from six charges/poles (Fig.2.5 (d)), wherein the
contribution of each can be modeled by the following relationship,
2
( , ) ( , )
( , )
i
i i m i i
i i
q
k
r
B p b u p b
p b
, i=1~6, (2.2)
where ( , )i iB p b is the magnetic flux density contributed by the ith
magnetic charge qi.
( , )i iB p b is a function of the bead’s position p = [x,y,z]T
and charge’s bias
=[ , , ]i i i ix y z  b since the optimal location to model the magnetic charge iq is not
necessarily at the tip of the pole. 0mk    =1.0×10-7
N/A2
. ( , )i ir p b is the norm of the
displacement vector pointing from the ith
magnetic charge iq to the location of the bead,

44. 19
and ( , )i iu p b is the unit directional vector, i.e., ( , ) ( , ) ( , )i i i i i iru p b r p b p b (Fig.2.5 (d)).
The resulting magnetic field sensed by the magnetic bead can be obtained by superposing
the magnetic field generated by six magnetic charges/poles, i.e. 𝐁(𝐩, 𝐛) =
∑ 𝑘 𝑚
𝑞 𝑖
𝑟𝑖
2(𝐩,𝐛 𝑖)
𝐮𝑖(𝐩, 𝐛𝑖)6
𝑖=1 , where ( , )B p b is the total magnetic flux density at the location p
of the magnetic bead, 1 2 3 4 5 6=[ , , , , , ]b b b b b b b is defined as the 18-domensional Bias
Vector. The position p can be normalized with respect to , which is the radius of the
workspace. The total magnetic flux density can thus be expressed in the following form,
1
2
33 5 61 2 4
2 3 3 3 3 3 3
41 2 3 4 5 6
ˆ ˆ( , ) 5
6
ˆ ˆ ˆˆ ˆ ˆ
ˆ( , )
ˆ ˆ ˆ ˆ ˆ ˆ
m
q
q
qk
qr r r r r r
q
q
 
 
 
  
   
  
 
 
  
R p b
Q
r r rr r r
B p b (2.3)
Where ˆ ˆ( , ) ( , )i ir p b r p b , ˆ ˆ ˆˆ( ) || ( )||i ir p r p . ˆ ˆ( , )R p b is a 3×6 Charge-Bead
Distribution Matrix depending on the special distribution of magnetic bead, six charges
and bias vector, Q is the Charge Vector.
For the current-actuated magnetic actuator, the magnetic flux in Eq. (2.1) is
determined by the magnetomotive force and the reluctance of the air according to
Hopkinson’s law: the magnetic Flux Vector 1 2 3 4 5 6[ , , , , , ]T
      Φ =
1 2 3 4 5 6[ , , , , , ]T
I aK F F F F F F , wherein the magnetomotive force is proportional to the input
current, i.e., i c iN IF , cN is the turns of the coil, a is the lumped magnetic reluctance

45. 20
from the pole tip to the workspace center in the air, and IK is the 6×6 flux distribution
matrix describing the magnetic coupling among 6 poles since they are connected by a
magnetic yoke (Fig.2.2, Fig.2.3). The magnetic charges vector Q can thus be related to
the input currents, 1 2 3 4 5 6[ ]T
I I I I I II , as described by the following equation,
0
0
c
I
a
N


 

Q Φ K I (2.4)
The spatial distribution of the magnetic field within the workspace of the actuating
system and its dependence on the input currents can be determined using the above
equations once the reluctance and workspace radius, the distance from the magnetic
charge to the workspace center, of the actuating system are known. They are determined
by best fitting the magnetic flux density vector based on Eq. (2.3) with that calculated
using FEM analysis (Fig.2.3 (b)). The sum of the squared norm of error vectors is
selected as the objective function, which is minimized to determine the two optimal
values, i.e., workspace radius 900 m , which is the distance variable defining the
location of iq , and the air reluctance a =6.3×108
A/Wb. The fitted result is shown in
Fig.2.6, wherein vector plots are compared in (a) and normalized norms of error vectors
in (b). It can be seen that norms of error vectors are mostly smaller than 1% of the norm
of the associated flux density. This validates the hexapole magnetic field model, i.e. Eq.
(2.3).

46. 21
Fig. 2.6. Validation of the hexapole magnetic field model: (a) comparison of magnetic induction vectors,
and (b) normalized norms of error vectors. (c) the definition of the fitting error.
2.3 Force Model and Inverse Modeling
2.3.1 General magnetic force model
Without being magnetized to saturation, the magnetic force exerted on a
superparamagnetic microscopic particle placed in the field is (1 2) ( )  F m B , where F
is the gradient force, 0 0 0(3 ) [( ) ( 2 )]V        m B is the effective magnetization of
the magnetic particle [67], μ is the permeability of the particle, and V is the volume of the
particle. An analysis beyond the linear magnetization of the particle can be found in [49].
Assume b 0 for the nominal magnetic force model. By substituting Eq. (2.3) into
the gradient force, the magnetic force can be expressed as
2
0 0 0(3 (2 )) [( ) ( 2 )] (|| || )V         F B = 0 0 0(3 ) [( ) ( 2 )]|| || (|| ||)V         B B ,
where 2 1/2ˆ ˆˆ ˆ|| ||= ( ( ) ( ) )T T
mkB Q R p R p Q from Eq. (2.3). After manipulation, the magnetic
force can be expressed in the following form,

47. 22
 
2
0
4
0 0
3 ( ) 1 ˆ ˆˆ ˆ( ) ( )
2 ( 2 )
T Tm
f
V k 
  

  

Q
F Q R p R p Q (2.5)
where Qf is called the Magnetic Charge Force Gain, the term 1 is due to that the
position is normalized with respect to . From Eq. (2.1), 0/ Q Φ , the force model can
therefore be lumped as a quadratic form of magnetic flux as follow,
2
0
1
ˆ ˆ( , ) ( )T
i Q i
f
F f


p Φ Φ L p Φ, i=x,y,z (2.6)
where 2
0= Qf f  is the Magnetic Flux Force Gain. Define ˆ( )xL p , ˆ( )yL p and ˆ( )zL p as
6×6 Charge-Bead Gradient Matrix and [ (m,n)
ˆ( )xL p , (m,n)
ˆ( )yL p , (m,n)
ˆ( )zL p ] =
 (m,n)
ˆ ˆˆ ˆ( ( ) ( ))T
 R p R p , which means the entries in the mth
row and nth
column of xL , yL
and zL consist the gradient of the entry in the mth
row and nth
column of ˆ ˆˆ ˆ( ( ) ( ))T
R p R p .
Eq. (2.6) is called the General Magnetic Force Model.
2.3.2 Current-Based Magnetic Force Model
Substitute Eq. (2.4) into Eq. (2.6), the current-based magnetic force model can be
modeled as follow,

48. 23
2
2
0
5
0 0 0
3 ( )
ˆ ˆ( , ) ( ) , , ,
2 ( 2 )
I
T Tm c
i I i I
a
g
V k N
F i x y z
 
   
 
  
  
p I I K L p K I , (2.7)
where Ig is the force gain determined by the size and magnetic property of the magnetic
particle and that of the magnetic circuit, the 6×6 flux distribution matrix IK is adopted
from the same magnetic circuit analysis reported in [53],
5 6 1 6 1 6 1 6 1 6 1 6
1 6 5 6 1 6 1 6 1 6 1 6
1 6 1 6 5 6 1 6 1 6 1 6
1 6 1 6 1 6 5 6 1 6 1 6
1 6 1 6 1 6 1 6 5 6 1 6
1 6 1 6 1 6 1 6 1 6 5 6
I
     
      
     
  
     
     
 
     
K (2.8)
An normalized force gain NF can be normalized with the maximum input current, maxI ,
2
2 2 20
max max5
0 0 0
3
2 ( 2 )
c
N I m
a
N
F g I Vk I
 
   
  
    
   
(2.9)
to characterize the force generation capability of a regular hexapole electromagnetic
actuating system. It is used to normalized the magnetic force and form a dimensionless
expression,
ˆ( , )ˆ ˆ ˆ ˆˆ ˆ( , ) ( )T Ti
i I i I
N
F
F
F
 
p I
p I I K L p K I , i=x,y,z (2.10)

49. 24
where max
ˆ II I is the normalized input current. It is evident that this dimensionless force
field characterizes the spatial distribution of the force field produced by the hexapole
electromagnetic actuating system.
2.3.3 Inverse Model Based on Constant Constraints
Whereas the force model described above establishes the relationship between the
input currents and the resulting magnetic force exerting on the magnetic particle, inverse
modeling is necessary for the practical use of the implemented electromagnetic actuating
system. Since the actuator is an over-actuated system, direct inverse of Eq. (2.10) is
impossible. It was shown in [1] that the redundancy could be removed by imposing three
constant constraints. This approach will be briefly summarized and the associated
limitations will be discussed.
Since three pairs of electromagnetic poles are placed symmetrically on the three
orthogonal axes of the actuation coordinate system, in the following, all analysis refer to
this coordinate system. Imposing three constant constraints, i.e., 1 2
ˆ ˆ
xI I c  , 3 4
ˆ ˆ
yI I c  ,
and 5 6
ˆ ˆ
zI I c  , and denoting three effective input currents, i.e., ˆI = ˆ ˆ ˆ[ , , ]T
x y zI I I   =
1 2 3 4 5 6
ˆ ˆ ˆ ˆ ˆ ˆ[ , , ]T
I I I I I I   , an exact linear relationship between the effective input currents
and the dimensionless force at the center, i.e., ˆ ˆ ˆˆ( , )c  F F p 0 I , is derived,
ˆ ˆ2c F A I (2.11)

50. 25
where A is a constant actuation matrix,
[(2 ),( 2 ),( 2 )]x y z x y z x y zdiag c c c c c c c c c        A (2.12)
The inverse model at the center can then be derived from Eq. (2.11) and Eq. (2.12),
1
1 3 5 0
1 1ˆ ˆ ˆ ˆ[ , , ]
2 4
cI I I 
 c A F , (2.13)
and
1
2 4 6 0
1 1ˆ ˆ ˆ ˆ[ , , ]
2 4
cI I I 
 c A F , (2.14)
where 0 [c ,c ,c ]T
x y zc is a constant offset introduced by the three constant constraints. It
is worth noting that whereas the use of three constant constraints leads to exact inverse
relationship at the center, the constant offset also results in unnecessarily large input
currents as evident in Eq. (2.13) and Eq. (2.14). It can also be seen from Eq. (2.11) that
due to the imposed constant constraints the magnetic force increases linearly, instead of
increasing quadratically as in Eq. (2.10), with the input currents, whereby the force
generation capability is severely degraded.
2.3.4 Optimal inverse model at the center of the workspace
It can be seen from Eq. (2.10) that the hexapole actuating system is capable of
generating 3-D force in arbitrary direction at any spatial position in the 3-D workspace,

51. 26
and the resulting force is scalable, i.e., 2ˆ ˆ ˆˆ|| ( , )|| || ||F p I I . When the desired force is
expressed in magnitude and orientation in the spherical coordinate system (Fig.2.7), i.e.,
ˆ ˆ ˆ|| || ( , )d d  F F r , where ˆ [cos cos ,cos sin ,sin ]T
    r is the unit force in the radial
direction, the optimal inverse solution can, therefore, be cast as,
 
1/2
ˆ ˆ ˆˆ ˆ ˆ( , ) ( , ),opt d d opt  I F p F I r p (2.15)
Fig. 2.7. Illustration of the desired force in the spherical coordinate system.
At the center, the optimal current intˆ ( , )u
opt  I associated with the unit force, i.e.,
ˆ ˆ( , )d  F r , can be obtained by finding three optimal constraints, namely ( , )xc   ,
( , )yc   , and ( , )zc   , and minimizing the squared norm of the input current vector, i.e.,
2ˆ|| ( , )|| I . This solution will improve current allocation of the over-actuated system. On
one hand, minimizing the norm of input current vector to generate the desired force will
reduce the heat generation from the coils. On the other hand, it will enhance the force
generation capability of the actuating system.

52. 27
The squared norm of the input current vector is related to the effective input current
vector and three constraints as,
 2 2
2 2 21ˆ ˆ( , ) ( , )
2
x y zc c c       I I (2.16)
where 1ˆ ˆ( , ) (1 2) ( , )    
I A r according to Eq. (2.11). The objective function can then
be cast as
22 2 2 11 1
ˆ( , , ; , ) { } ( , )
2 8
x x x x y zJ c c c c c c   
    A r (2.17)
Minimizing this objective function yields the three optimal constraints, ( , )xc   , ( , )yc  
and ( , )zc   , which are orientation dependent. They are compared with constant
constraints used in [1, 52] and displayed in Fig.2.8.
Fig. 2.8. Three orientation-dependent optimal constraints compared with their corresponding constant
constraints.

53. 28
Let ˆ ˆ( , )c  F r and substitute ( , )xc   , ( , )yc   and ( , )zc   into 0c and A in Eq.
(2.13) and Eq. (2.14), the optimal current allocation, intˆ ( , )u
opt  I , associated with the unit
force can be determined. The result is compared with that obtained using constant
constraints and displayed in Fig.2.9. It can be seen that the absolute value of each of the
six input currents resulted from optimal current allocation is significantly smaller than
that associated with constant constraints.
Fig. 2.9. Optimal current allocation compared with that obtained using constant constraints.
2.4. Experimental Verification of the Optimal Inverse Model
Fig. 2.10. Block diagram of the feedback motion control ( dP is the desired position, mP is the measurement
position, e is the error signal, dF is the desired force calculated by the controller, '
(t) (t )d d a F F is
caused by the actuator delay, MTF is the magnetic force, F is the modeling error, TF is the thermal force,
a and m are the delay in actuator and measurement)

54. 29
A feedback control system was implemented and used to stabilize the magnetic
particle placed in the workspace of the hexapole electromagnetic actuating system. A
block diagram of the control system is shown in Fig.2.10. The total delay D in the
feedback control system is contributed by actuator delay a and measurement delay m ,
and Zero-Order-Hold delay ZOH due to digital control, i.e. D a m ZOH      . Denote
a ZOH  as the effective actuator delay A . The position of the particle was measured
using a 3-D vision-based particle tracking system [55], wherein a CMOS camera was
employed to acquire the image of the particle at 200 frames per second (fps). The image
grabbing board is used to process the visual measurement algorithm and 200 fps is close
to the limit of calculation capability. Current allocation derived from inverse modeling
was implemented to overcome the issue raised by over-actuation and to achieve feedback
linearization. Together with the constant-gain feedback controller, it stabilized the
magnetic particle and suppressed the disturbances introduced by the random thermal
force. The bead dynamics can be described by the Langevin equation [68],
( ) ( )MT Tm t t  P P F F , where m is the mass of the particle,  is the drag coefficient of
the particle in aqueous solution. The inertia term mP is very small and can be neglected
compared to the damping force P . Therefore, the bead dynamics can be described by a
1st
order system.
    ( ), ( )MT d A D Tt t t     P F I F p F (2.18)

55. 30
Both methods for current allocation, i.e. the one based on optimal inverse modeling
and the other using constant constraints, were implemented. The feedback controller was
a constant-gain PI controller to stabilize the magnetic bead at the desired location.
Experiments were conducted to stabilize the particle in water and the results were shown
in Fig.2.11 and Fig.2.12 to compare the performance of the two methods.
It can be clearly seen from Fig.2.11 that when using optimal current allocation each
of the six input currents absolute value is significantly smaller than its counterpart. This
result validates the theoretical analysis developed for optimal inverse modeling.
Fig. 2.11. Six input currents of the hexapole actuator
The Brownian motion displayed in Fig.2.12 also shows improved performance of
stabilization when using optimal current allocation. As shown in the block diagram of the
feedback control system (Fig.2.10), the positioning fluctuation of the particle attributed to
at least two disturbances, i.e., the random thermal force FT and the force modeling error
ΔF. Whereas the two methods are based on the same force model and both yield

56. 31
theoretically exact inverse solutions, the force error ΔF is likely different as the
realization of the two methods operate with different actuation currents (Fig.2.11). The
results shown in Fig.2.12 imply that the modeling error likely becomes greater when
increasing the actuation currents.
Fig. 2.12. Stabilization of the magnetic particle at the center of the workspace
2.5. Calibration and Validation of the Force Model
Whereas the hexapole magnetic force model has a sound theoretical basis, numerical
values of two model parameters, i.e., the force gain and the flux distribution matrix, are
not exactly known without experimental calibration. Any discrepancies introduce errors
in inverse modeling and degrade the effectiveness of current allocation for force
generation and control.
The nominal flux distribution matrix IK given in Eq. (2.8) was obtained through the
magnetic circuit analysis presented in [53]. Due to the magnetic leakage, which is not

57. 32
negligible as can be seen from the results in Fig.2.3 (a), the exact values of IK differ
from the nominal values. An experimental setup along with an experimental procedure
was devised to calibrate the flux distribution matrix by utilizing the electromagnetic
induction, which is widely used to extract information from the magnetic field [69, 70].
In the experiment, each individual actuation coils were excited one by one sequentially
and the six induction voltages were measured simultaneously using the six measurement
coils (Fig.2.2 (a)). The current applied to the ith
actuation coil together with the voltage
readings from the six measurement coils was used to determine the ith
column of the flux
distribution matrix according to the Faraday’s Law, i.e., ( ) (t)mE t N d dt   , where mN
is the number of turns of the measurement coil, (t) is the magnetic flux, and (t)E is the
induction voltage. A typical experimental result is shown in Fig.2.13, wherein a
sinusoidal actuation current applied to coil 1 and six measured induction voltages are
displayed.
Fig. 2.13. Actuation current applied to coil 1 and voltage readings from the six measurement coils.

58. 33
The calibrated flux distribution matrix is denoted as ˆ
IK . It was determined after
completing the calibration procedure,
0.6022 0.0124 0.0285 0.1507 0.1668 0.0229
0.0103 0.9322 0.1740 0.0787 0.0680 0.1780
0.0294 0.1655 0.6291 0.0121 0.1458 0.0319ˆ
0.1540 0.0712 0.0112 0.9040 0.0746 0.1501
0.1805 0.0712 0.1521 0.0769 0.90
I
    
    
    

    
   
K
26 0.0095
0.0235 0.1726 0.0331 0.1506 0.0123 0.6122
 
 
 
 
 
 
 
 
     
(2.19)
Whereas the off-diagonal elements of the matrix quantify the degree of couplings
among the six electromagnetic poles, diagonal elements are actuation gains of the six
poles. It is seen that the actuation gains associated with the three lower poles are
significantly smaller than those with the three upper poles. This is due to the fact that
significant amount of the three lower poles’ material was removed to form a flat platform
to support the culture dish. It is worth noting that whereas the calibrated flux distribution
matrix in Eq. (2.19) precisely quantifies the effect of the geometric difference between
the upper and lower poles on the flux distribution in the six electromagnetic poles, it does
not fully quantify the effect in the workspace. The geometric difference also changes the
way that the magnetic field converges to the pole tip and thus the field distribution in the
workspace.
Another experiment was performed to calibrate the force gain of the actuating system
and to improve the result given by Eq. (2.19). In the experiment, the magnetic particle
was steered along a linear trajectory and propelled at a constant speed in viscous liquid.

59. 34
Moreover, the setup is degaussed after each trajectory tracking. When traveling at
constant velocity, pv , the magnetic force exerting on the particle is balanced by the
viscous force, i.e. v p F v , where  is the drag coefficient. The viscous fluid used in
the experiment was glycerol, which was much denser than water. Therefore, it required
greater magnetic force to balance the viscous force. The particle employed was 4.5μm
magnetic spherical bead and the drag coefficient was calibrated to be about 8.5×10-
6
N.s/m.
Fig. 2.14. Twelve linear trajectories on the horizontal plane passing the center of the workspace (in
measurement coordinate)
With reference to the measurement coordinate, twelve linear trajectories, as shown in
Fig.2.14, having distinct directions on the horizontal plane passing the center of the
workspace were planned. In the experiment, the particle was controlled to travel along
each trajectory one by one at a speed of6.5 m s . The viscous force was, therefore, 55pN
along the travel direction. The particle’s position in the workspace was measured using
the vision-based particle tracking system. With reference to the actuation coordinate, the

60. 35
three components of the particle’s position, measured and target, are displayed as time
sequences in the left column of Fig.2.15. It can be seen that the particle followed the
target trajectories accurately. Three components of the calculated viscous force in the
actuation coordinate are displayed in the right column of Fig.2.15 as time sequences
synchronized with the particle’s motion. They were used to serve as measurements of the
magnetic force.
Fig. 2.15. Particle motion and viscous force displayed with reference to the actuation coordinate, wherein
black dashed lines are target motion and color solid lines are measured motion (plotted in time sequences).
Fig. 2.16. Six actuation currents associated with motion control

61. 36
Associated with the real-time motion control in the experiment, six actuation currents
were known and displayed in Fig.2.16. Since the magnetic force is related to the
actuation currents through the hexapole magnetic force model, calibration can be
accomplished through best fitting by minimizing the following objective function,
 
2
1
( , ) (j) (j), (j); , (c)
N
I v I I
j
J c

 g F F p I g K , (2.20)
Where N denotes the number of total time instants, i.e., N=200Hz×72.6sec=14412, vF is
the viscous force. The magnetic force model is cast as ( (j), (j); , (c))I IF p I g K from the
point view of calibration. It differs from Eq. (2.7) in two ways. First, it adopts a force
gain vector with three components instead of a scalar gain. This allows the force model to
have different force gains in different directions. Second, the flux distribution matrix is
denoted as (c)IK to provide options for model calibration.
The results of three options are presented in this paper. First, the nominal flux
distribution matrix is used in Eq. (2.20), i.e., (c)I IK K , and the force gain vector Ig is
determined while minimizing the objective function. The calibrated force gain vector is
[5.37,6.82,6.93]T
I g , measured in pN , and the best-fitted results are shown in Fig.2.17.
Second, the measured flux distribution matrix is used, i.e., ˆ(c)I IK K as Eq. (2.19).
The force gain vector is [9.08,9.64,8.39]T
I g , and the fitted results are shown in

62. 37
Fig.2.18. It can be seen that model calibration using the second option yields better
results.
Fig. 2.17. The best-fitted results when using the nominal flux distribution matrix to calibrate the force gain
vector.
Fig. 2.18. The best-fitted results when using the measured flux distribution matrix to calibrate the force
gain vector
Third, in order to further improve model calibration, a weighting coefficient, c , is
employed, i.e., row 1, 3, and 6 of ˆ
IK is scaled by c , to account for the difference
between the lower poles and the upper poles in the distribution of the magnetic flux
density in the workspace, and the modified flux distribution matrix is used in Eq. (2.20).

63. 38
The improved results are shown in Fig.2.19. The value of c and the force gain vector Ig
are determined simultaneously, i.e., 1.2c  and [7.56,8.55,7.62]T
I g .
Fig. 2.19. The best-fitted results when using the modified flux distribution matrix to calibrate the scaling
factor and the force gain vector simultaneously.
A quantitative measure, i.e., ( , ) (3 )Ie J c N g , is defined to evaluate and compare
the three calibration results. The average error reduces from 12.42pN to 10.04pN, and
further to 8.25pN. It is worth noting that a significant component of the average error is
attributed to the thermal force, which is about 4pN in our calibration experiments. The
actual modeling error is, therefore, significantly smaller than the calculated average error
and it is very small compared to the calibration force range, which is 110pN (±55pN).
2.6. Force Generation Capability
The force generation capability of a hexapole electromagnetic actuating system is
dictated by the design and synthesis of the electromagnetic poles and by the inverse
modeling derived for current allocation. According to Eq. (2.9), a design yielding a
higher force gain and allowing larger maximum input current will lead to greater

64. 39
effective force possibly generated by the actuating system, i.e., 2
maxN IF g I . Compared
with the design of the 3-D actuator in [1], the new design of the hexapole electromagnetic
actuating system yields a fourfold increase in force gain. While the maximum input
current allowed by the 3-D actuator in [1] was 1.5 A, the new design was expected to
have larger maximum input current.
An experiment was conducted to determine the maximum input current allowed by the
newly developed actuating system. In the experiment, an actuation current was applied to
an individual electromagnetic pole and its strength was increased monotonously while a
Gaussmeter was used to measure the magnetic flux density near the pole tip. The test was
applied sequentially to two poles, namely P2 (an upper pole) and P3 (a lower pole). As
shown in Fig.2.20, the flux density increased linearly with respect to the input current and
neither test reached magnetic saturation while the actuation current was increased to 3A.
It is worth noting that the placement of the Gaussmeter sensor tip strongly affects the
absolute value of the readings of the hall-effect sensor. The objective of the test was to
examine the linear range of the electromagnetic actuation.
Fig. 2.20. Testing the linear range of electromagnetic actuation: P2 (left) and P3 (right)

65. 40
The hexapole magnetic force model of the newly developed actuating system and that
of the 3-D actuator in [1] are used to calculate three force envelopes when a 4.5µm
magnetic bead is placed at the center of the workspace. Force generation capabilities are
compared and shown in Fig.2.21, wherein the three force envelopes are calculated using
nominal force models and they are spatially symmetric. It can be seen that the force
generation capability of the newly developed actuating system is significantly increased
through design and synthesis of the actuating system and the realization of optimal
current allocation. Fig.2.22 shows the comparison of three force envelopes, which are
calculated using calibrated force models. Whereas reduction of the force generation
capability of the newly developed system is noticeable due to the removing of significant
amount of the material from the three lower poles, significant improvement of force
generation capability using the optimal inverse model remains evident.
Fig. 2.21. Comparing three force envelopes calculated using nominal force models: 3-D actuator using
current allocation based on constraint constraints (blue), newly developed actuating system using current
allocation based on constraint constraints (red), and newly developed actuating system using optimal
current allocation (grey).

66. 41
Fig. 2.22. Comparing three force envelopes calculated using calibrated force models: 3-D actuator using
current allocation based on constraint constraints (blue), newly developed actuating system using current
allocation based on constraint constraints (red), and newly developed actuating system using optimal
current allocation (grey).
2.7. Conclusion
An actively controlled hexapole electromagnetic actuating system was designed and
implemented for use with live cell experiments. It can be applied to stabilize and propel a
microscopic magnetic particle in aqueous solutions to serve as a measurement probe for
force sensing and mechanical property characterization. A hexapole magnetic force
model was employed to investigate the force generation capability of the actuating
system and to lay the foundation for inverse modeling. Optimal inverse modeling was
derived. It solved the redundancy problem of the over-actuated system and led to the
realization of optimal current allocation to enable the most effective manipulation of the
3-D magnetic force exerting on the particle at the center of the workspace.
Several aspects to improving the performance of the hexapole elesctromagnetic
actuating system are identified. First, optimal inverse modeling needs to be extended to
the entire workspace to enable superior motion control of the particle away from the

67. 42
center and to enhance its applications that require larger workspace. Second, the delay in
the feedback control loop needs to be shortened to improve the control performance;
which can be realized through developing high-speed vision-based particle tracking
techniques. Third, it is necessary to investigate and reduce the effect of hysteresis on
force generation and control through modeling [58] or real-time estimation [24].

68. 43
Chapter 3: The Optimal Inverse Model and Control of
Hexapole Electromagnetic Actuation
3.1 Introduction
Lumped-parameter analytical force models were presented in the previous chapter.
The performance of the hexapole magnetic actuator relies on accurate inverse modeling
of the electromagnetic force, which is essential for effective current allocation and
feedback control. The optimal inverse modeling at the workspace center is developed in
Chapter 2. However, the inverse modeling in the center cannot be directly used in the
workspace since the magnetic force model is position dependent.
A simplified inverse model was derived and employed to realize feedback
linearization in the implementation of active feedback control [1]. Whereas stable
magnetic trapping and motion control were successfully demonstrated, the simplification
and approximation adopted in inverse modeling severely limit the performance of the
developed actuating systems on two fronts. First, the use of constant constraints to derive
the simple inverse model, which is valid at the center of the workspace, results in
excessive current flow in the coil. It significantly degrades the force generation capability
of the actuating system. Second, extending the inverse model at the center to the entire
workspace through linear approximation in the spatial domain leads to significant error

69. 44
when generating magnetic force exerting on the magnetic particle placed away from the
center of the workspace.
This chapter presents the derivation of an optimal inverse model of the over-actuated
hexapole electromagnetic actuating system to minimize a specially weighted norm of the
six input currents when applied to produce the desired 3-D magnetic force in the
workspace. The optimal inverse model is then employed to realize real-time current
allocation to enable feedback linearization in the implementation of active feedback
control. Stable magnetic trapping and precise motion control of the microscopic magnetic
particle in aqueous solutions are demonstrated in experiments.
3.2 Hardware Implementation of High Speed Control
Fig. 3.1. (a) Magnetic setup integrate with the inverted microscope (b) High speed embedded control
system using FPGA (c) real-time display of the reference bead and the control bead (the reference sticks to
the cover glass surface to provide the information such as drift, vibration and etc.)

70. 45
It is widely known that the time delay in the system will greatly degrade the
performance of the feedback control. However, the feedback control implemented in PC
cannot achieve high sampling rate since the computation capability is limited and the
image grabbing card processing speed is under 300 fps. Moreover, the time consistency
of PC is not very good due to operation systems.
To achieve high sampling rate control, FPGA is used to calculate the control effort
and realize input/output functionalities. The FPGA based real-time image processing is
accomplished by P. Cheng [71], which can achieve 10,000 fps real-time image
processing with reduced image size. The complete FPGA system in shown in Fig.3.1. To
increase the image resolution, a superluminescent diode (SLD) illumination system
(QPhotonics) is used to improve the image sensitivity [72]. The visual sensing resolution
is about 1e-3 pixels in x and y direction, and sub nm in z direction. The CMOS camera
used in this system has 8um pixel size. Therefore, the visual sensing resolution is about
0.13 nm in x and y direction when 60x lens is used and 0.2 nm when 40x lens is used.
DA converters (DAC8814 EVM, Texas Instruments) are updated by the main FPGA to
drive linear power amplifiers (Micro Dynamics, BTA-28V-6A). USB modules (DLP-
USB1232H, FTDI) are connected and programmed to accomplish PC-FPGA
communication. In such a system, PC is used as a coordinator such as downloading
control parameters, data logging and etc. The real-time image is displayed on the monitor
by connecting it to a VGA port in the FPGA. A DE2-115 FPGA (Terasic Inc.) is used as
a main FPGA to calculate the control effort and output the actuation effort through DA

71. 46
converters. The real-time image processing is performed in two TR4-230 FPGA (Terasic
Inc.) boards.
The calculation and input output functionalities are all realized in FPGA systems.
1606Hz sampling rate, which is the maximum achievable sampling rate for 512×512
image, is used in real-time control. Higher sampling rate can be easily achieved by
reducing the image size. A reference bead (Fig.3.1 (c)), which is attached to the coverslip
surface, is measured simultaneously to compensate the drift, such as structure thermal
drift.
3.3 The inverse model
3.3.1 The position dependent inverse model based on constant constraints
An inverse model based on constant constraints is proposed in [1] to remove
redundancies. It is briefly summarized here. It is already shown in Eq. (2.11) that the
force at the center can be expressed as ˆ ˆ ˆˆ( , )c  F F p 0 I = ˆ2 A I , where 1 2
ˆ ˆ
xI I c  ,
3 4
ˆ ˆ
yI I c  , 5 6
ˆ ˆ
zI I c  , ˆ ˆ ˆ ˆ[ , , ]T
x y zI I I   I = 1 2 3 4 5 6
ˆ ˆ ˆ ˆ ˆ ˆ[ , , ]T
I I I I I I   , A = diag[
(2 )x y zc c c  , ( 2 )x y zc c c   , ( 2 )x y zc c c   ]. Denote ˆ 2δI
J A , then ˆ
ˆ ˆ
c  δI
F J δI ,
whereas it is not straightforward to obtain the exact inverse solution for the entire
workspace. Nonetheless, by keeping the first-order terms in Taylor expansion of Eq.
(2.10), a linear approximation of the magnetic force around the center can be derived,
ˆ ˆ
ˆ ˆ ˆˆ ˆ( , )   pδI
F p δI J δI J p (3.1)

72. 47
Where ˆpJ = 2∙diag [ 2
(2 )x y zc c c  , 2
( 2 )x y zc c c   , 2
( 2 )x y zc c c   ]. The effective
actuation current required to produce the desired force is derived as follow,
1 1
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ( )d
 
  pδI δI
δI J F p J J p (3.2)
The effective actuation current ˆI along with the three linear constraints[ , , ]x y zc c c can be
employed to calculate the required actuation current,I ,
( , )dInverseI F p (3.3)
It is worth noting that the desired force, dF , and the spatial location, p , are associated
with the actuation coordinate frame. When employing Eq. (3.3) to realize inverse model
calculation, the rotational matrix a
m R between the two coordinate frames needs to be
applied to both vectors.
3.3.2 The Optimal Inverse Model in the Entire Workspace
The optimal inverse model in the center is presented in Chapter 2.3.4. However, the
magnetic force model strongly depends on the position of the magnetic bead as shown in
Eq. (2.10). The linear inverse model based on the 1st order Taylor expansion, i.e. Eq.
(3.2), will result in significant error for both constant constraints and optimal constraints.
For constant constraints, the error will grow when the desired force is large or the

73. 48
position is further away from center (Fig.3.2). For optimal constraints, the error does not
depend on force magnitude due to scalable characteristics. However, the error does grow
significantly when the position is far away from center (Fig.3.3).
Fig. 3.2. Inverse model based on constant constraints. (a) The percent force error at (20,0,0)um, desired
force ˆ
dF =1; (b) The percent force error at (20,0,0)um, desired force ˆ
dF =10; (c) The percent force error at
(40,0,0)um, desired force ˆ
dF =1; (d) The percent force error at (40,0,0)um, desired force ˆ
dF =10
Fig. 3.3. Optimal inverse model. (a) The percent force error at (20, 0, 0) um; (b) The percent force error at
(40, 0, 0) um
Due to the insufficiency of the linear inverse model, the optimal inverse model in the
entire workspace is desired to solve the position dependency of the magnetic force model.
However, due to the asymmetrical magnetic charge configuration at a non-center point,
there is no analytical inverse model as Eq. (2.13) and Eq. (2.14). Therefore, the objective

74. 49
function as Eq. (2.17) is not available. However, minimizing the norm of currents while
still satisfying the force model is essentially an optimization problem under nonlinear
equality constraints, and Lagrange multipliers ( , , )x y z   can therefore be employed to
construct the following objective functions:
 
   
2ˆ ˆ ˆ ˆˆ( , , ) || || ( ) cos cos
ˆ ˆ ˆ ˆˆ ˆ( ) cos sin ( ) sin
T T
x I x I
T T T T
y I y I z I z I
J     
    
   
  
P I I K L p K I
I K L p K I I K L p K I
(3.4)
The force model Eq. (2.10) is used in Eq. (3.4). The position dependent optimal
inverse solution satisfying ˆmin( ( , , ))J   P criterion in Eq. (3.4) is denoted as ˆ ˆ( , , )unit
opt  I P
. Minimizing ˆ( , , )J   P is an optimization problem with nine parameters, including six
currents and three Lagrange multipliers. The local minimum occurs at the values where
the gradient of ˆ( , , )J   P is a zero vector, i.e. ˆ( ( , , ))J   P 0 , which is still a nonlinear
functions. MATLAB optimization toolbox is used to minimize Eq. (3.4), wherein the
optimal inverse model at the center, i.e. intˆ ( , )u
opt  I can serve as a good initial guess for the
numerical calculation of ˆ ˆ( , , )unit
opt  I P at different locations ˆP . In the first Octant, the
locations where ˆ ˆ( , , )unit
opt  I P is numerically calculated are shown in Fig.3.4. The range is
a 45um×45um×45um cube in the first Octant, leading to a range of 90um×90um×90um
in the entire workspace.

75. 50
Fig. 3.4. Locations where the optimal inverse model are obtained
After numerically obtaining the optimal inverse model at different locations, it is
desired that a usable function can be found to describe ˆ ˆ( , , )unit
opt  I P so that real-time
control can be implemented. ˆ ˆ( , , )unit
opt  I P can be fit with respect to position ˆP at different
orientations ( , )  since this inverse model needs to capture the position dependent
information of the inverse model. A frequently used fitting scheme for capturing the
nonlinear trend is the Taylor expansion. Specifically, the 2nd order Taylor expansion
with respect to ˆP is computed for each orientation( , )  , which can be expressed in the
following quadratic form:
1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( , , ) ( , , ) ( ) ( ) = ( , )
2
( , ) ( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( , )
=
( , ) (
1
unit unit T T Taylor
opt i opt i i
T
xx i xy i xz i x i
yx i yy i yz i y i
zx i zy
c c c cx
c c c cy
c cz
     
       
       
  
    
 
 
 
 
 
 
I P I 0 G P P P H P P C+ P P
, ) ( , ) ( , )
( , ) ( , ) ( , ) ( , ) 1
i zz i z i
x i y i z i i
x
y
c c z
c c c c
    
       
   
   
   
   
   
    
(3.5)

76. 51
where i=1, 2,… , 6 is the index of 6 currents, ˆ( )G P and ˆ( )H P are the gradient vector and
hessian matrix, [ , , ,1]T
x y zP is the augmented position vector, ( , )Taylor
i C is the Taylor
expansion matrix, ( , )ic   , which is the last entry of matrix ( , )Taylor
i C , is just the
optimum inverse model at the center, ( , )k ic   , k = x, y, z, is from the gradient vector,
( , )mn ic   , where m,n = x,y,z, is from the Hessian matrix.
To test the performance of the inverse model, Eq. (3.5) is used to solve the actuation
current at different locations within the 45um×45um×45um cube in the 1st Octant,
wherein the desired forces are in different orientations. The real force generated by the
actuation current are calculated and compared with the desired force to evaluate the
performance of the inverse model in Eq. (3.5). Define the percent force modeling error
as| | | |*100dF F , which is shown in Fig.3.5 (a) ,the maximum force modeling error at
each location is selected and plotted in Fig.3.5 (b), where the x axis is ˆ|| ||P , denoted as R
and sorted in ascend order, y axis is the percent force error. It can be clearly seen that the
force modeling error grows with respect to R, which is expected for Taylor expansion.
Although the Eq. (3.5) captured the characteristics of Taylor expansion, the growing
error as in Fig.3.5 (b) is not desired. However, the format of the inverse model in Eq.
(3.5) can be kept due to its simplicity. Alternatively, the least square fitting can be
employed to fit a similar format but reduce the norm of error in the whole workspace:
ˆ ˆ( , , ) ( , )unit T LS
opt i i   I P CP P (3.6)

77. 52
The matrix ( , )LS
i C follows the same pattern as ( , )Taylor
i C in Eq. (3.5), but the
entries of ( , )LS
i C comes from Least-Square fitting. The force modeling error is
calculated again at each point and is plotted in Fig.3.5 (b), together with error from
Taylor expansion approximation. It can be seen that the force model error using Least-
Square fitting is almost always under 5%.
Fig. 3.5. Comparison of percentage force error between Taylor expansion and least-square fitting
The inverse model is thus constructed by following Eq. (3.6). Interpolation is used to
complete the inverse model since the inverse model is only fit at discrete orientations
where optimal solutions are numerically obtained.
The optimal inverse model can not only accurately solve the position dependency
issue in the force model, but also can greatly increase the force generation capability
since it minimized the norm of the actuation current. The comparison of the force
generation capability using the optimal inverse model (Eq. (3.6)) and the inverse model
based on constant constraints (Eq. (3.2)) are plot together in Fig.3.6.

78. 53
Fig. 3.6. (a) Force generation envelops at (0,0,0)um. (b) Force generation envelops at (20,0,0)um. (c) Force
generation envelops (40,0,0)um
3.4 Active feedback control: stabilization, Brownian motion control and tracking
control
3.4.1 Stabilization and Brownian motion control
With the inverse model available, the nonlinear controller can be designed to achieve
motion control, wherein the nonlinearity between force and the actuation current can be
handled by the optimal inverse model. Therefore, the feedback controller design is easier.
Fig. 3.7. Block diagram of the feedback motion control using proportional controller ( dP is the desired
position, mP is the measurement position, e is the error signal, Kp is the proportional gain, dF is the desired
force calculated by the controller, '
(t) (t )d d a F F is caused by the actuator delay, a and m are the delay
in actuator and measurement, MTF is the magnetic force, F is the modeling error, TF is the thermal force)

79. 54
Looking at the feedback control block diagram in Fig.3.7, the real magnetic force MTF
consists of desired force '
dF and modeling error F . For the optimal inverse model, F is
the error that is not modeled by the nominal model, such as imperfect magnetic charge
approximation, hardware assembly and etc. For the inverse model based on constant
constraints, however, a great part of F also comes from the linear approximation of the
nominal model of Eq. (3.1) and Eq. (3.2), the effect of which is shown in Fig.3.2.
The basic functionality to enable the magnetic actuator working as an active probing
system is the stabilization of the magnetic bead. A stabilization experiment using P
controller is performed. Similar to Eq. (2.18), the bead dynamics using proportional
control is described as follow,
          p d A D Tt t t t t       P K P P F F (3.7)
The magnetic bead is stabilized at different 3D positions as shown in Fig.3.8. There
are three layers in which each layer contains 25 points where the magnetic bead are
stabilized. At each layer, the magnetic bead is first stabilized at the lower left corner in
the xy plane. The magnetic bead is stabilized at each point for 10sec, and then is steered
at 5um/s to the next point, which is 10um away from previous point. The bead is steered
following a zigzag path until it reached the last point at the upper right corner of that
layer. After the transient response vanished at each position, the data corresponded to
7sec was selected and plotted. At each layer, the bead is steered back to the starting point,
i.e. the lower left point, after finishing the data collection of 25 points of that layer.

80. 55
Fig. 3.8. 3D plot in bead stabilization experiment (the result in from The Optimal inverse model)
The optimal inverse model and the inverse model based on constant constraints are
compared to show the current allocation performance of the optimal inverse model. From
Fig.3.9 it can be seem that the current in the optimal inverse model is much smaller than
that based on constant constraints. It is worth mentioning that current I5 and I6 from
constant constraints are negative with much larger absolute value compared with the
optimal inverse model.
Experimentally, the optimal inverse model will not only lead to much smaller
actuation effort but also lead to better Brownian motion control with smaller standard
deviation. The Brownian motion standard deviations of 75 points in 3 layers (as Fig.3.8)
are plotted in Fig.3.10. It can be seen that the positioning standard deviation from the
optimal inverse model is obvious smaller compared to the inverse model based on
constant constraints, which will lead to better special resolution. It is likely that the larger
actuation current in the constant constraints will lead to larger modeling error, which can
lead to greater positioning fluctuation.

81. 56
Fig. 3.9. Actuation effort using optimal inverse model and the inverse model based on the constant
constraints. (For the inverse model based on constant constraints, I5 and I6 are negative with much larger
absolute value than optimal invers model)
Fig. 3.10. The positioning standard deviation in x,y and z axis using the optimal inverse model and constant
constraints
Fig. 3.11. The positioning performance from Proportional controller, using optimal inverse model and
constant constraints.
The magnetic force modeling error in block diagram Fig.3.7 is inevitable in the real
system due to different factors such as hardware assembly, magnetic charge biases and
etc. From Eq. (2.10), with smaller actuation effort, the optimal inverse model will lead to

82. 57
smaller modeling error. According to Eq.(3.7), the response from dP to e and F to e
are  (s) ( )DS
d pe P s s K e 
 
  ,  (s) ( )m Ds s
pe F e s K e 
 
    . Assuming dP and
F are step input, the steady-state error caused by dP and F can be calculated by the
Final Value Theorem: | 0dPe  , | 1F pe K   . Where | dPe and | Fe  are steady-state errors
from dP and F respectively. The bead dyanmics is a type 1 system. Therefore, the
positioning error using P controller in the stabilization experiment is a good indication of
moeling error. The steady-state error | Fe  is inverse proportional to the control gain pK .
Therefore, a larger gain can lead to smaller stedy-state error. The open loop transfer
function is (s) / ( )Ds
OL pG K e s

 , and the Nyquist stability criterion requires that
(2 )p DK   . Therefore, the control gain has to be limited to avoid instability. From
Fig.3.11 it can be seen that the optimal invese model can lead to much smaller
positioning error, which indicate a much smaller modeling error. From the force model
Eq.(2.10), i.e. ˆ ˆˆF ( )T T
i I i I I K L p K I , the modeling error such as magnetic charge biases
and assembly error will be lump into ˆ( )iL p , which indicate that larger actuation current
will lead to larger modeling error.
3.4.2 Trajectory tracking
Tracking and steering capability is necessary to enable the magnetic actuator to
perform 3-D probe sensing tasks, such as cell scanning, binding a receptor with the
magnetic bead and etc. To demonstrate the capability of the inverse model performance

83. 58
at different locations and directions, the magnetic particle are steered along different
trajectories as shown in Fig.3.12. The magnetic bead is steered at 5um/s along circular
trajectories at different locations in xy plane, yz plane and xz plane. In xy plane, the bead
is firstly steered to track the lowest 40um-diameter circle, then is moved to the plane in
the middle, which is 5um above the lowest one. After tracking the circle in the middle,
the bead is moved to the highest plane within which is another circle of the same shape.
After tracking 3 circles in xy planes, the bead is moved from the highest plane to the
middle plane, and is steered along 10um-diameter circles in yz plane and xz plane. First,
the bead is steered along circles in yz plane and xz planes in the center (labeled as 0 in
Fig.3.12 (b) and (c)). It is worth mentioning that the circles in yz plane and xz plane are
plotted separately in Fig.3.12 for clarity, while the yz circular tracking is followed by xz
circular tracking immediately. Then the magnetic bead is steered to position 1, 2, 3 and 4
sequentially, circular tracking in yz plane and xz plane is performance at each point.
Fig. 3.12. Trajectories in 3D tracking control.

84. 59
Fig. 3.13. Actuation effort in 3D tracking control. (For the inverse model based on constant constraints, I5
and I6 are negative with much larger absolute value than optimal invers model)
It can be seen from Fig.3.13 that the optimal inverse model can achieve the 3D
trajectory tracking with much smaller actuation effort, which is as expected. Moreover,
the optimal inverse model can achieve much smaller tracking error, which is shown in
Fig.3.14. It can be seen that by using the optimal inverse model, the standard deviation of
the tracking error reduced to (57.82nm, 43.81nm, 42.01nm) compared with (123.10nnm,
63.08nm, 109.83nm) while using constant constraints.
Fig. 3.14. Tracking error using the optimal inverse model and inverse model based on constant constraints.

85. 60
3.5 Comparison of High Speed Control in FPGA and Low Speed Control in PC
Due to the compact form of the inverse model Eq. (3.5) and Eq. (3.6). The inverse
model can be easily implemented in FPGA and the feedback control performance is
addressed in Chapter 3.4. The advantage of using high speed control is demonstrated in
this sections with a simple controller, i.e. the P controller. It is analyzed in Chapter 3.4.1
that (2 )p DK   from Nyquist criterion, which means that the stability margin is
inversely proportional to the time delay in the feedback control loop. The time delay in
the control loop is directly determined by the sampling rate. Therefore, it is desired that
the sampling rate is high enough to reduce the time delay, thus better suppressing the
Brownian motion and increasing the stability margin.
Most researcher implemented the control algorithm in PC [1, 2, 52, 53, 56, 63].
However, the sampling rate in PC is limited by the computation capability and time
consistency of PC. The magnetic bead stabilization is performed in PC at 200 fps in
Chapter 2, before the FPGA system is developed. The data from the 200Hz control in PC
is compared with the high speed 1606Hz control in this Chapter.
Power Spectrum Density (PSD) curves [52] can be used to determine trapping
stiffness pK , damping ration  and time delay D . From the block diagram Fig.3.7 it can
be seen that the transfer function from thermal force to bead position is
1 ( e )Ds
T ps K 
 
 P F . The PSD of the thermal force is known to be (F ) 4T BPSD k T
[73], where Bk is the Boltzmann constant, T is the absolute temperature. The PSD of the
magnetic bead can be formulated as follow,

86. 61
2
4
(P)
2 exp( 2 )
B
p D
k T
PSD
i f K i

  

 
(3.8)
pK , and D can be known by comparing Eq. (3.8) with PSD of the measurement data.
The calibration result in x direction are shown in Fig.3.15. It is calibrated from the PSD
curves that the time delay D under 200Hz sampling rate is about 12ms, which is 2.4
steps of the sampling interval. Theoretically, 2 steps of the delay come from the
measurement and 0.5 step is caused by Zero-Order-Hold (ZOH) of the digital control.
Therefore, the calibrated 2.4 steps delay is very close to the theoretical prediction. The
calibrated drag coefficient is about 5.51e-8 N.s/m in x direction. There is a peak at PSD
curve at 71.37 Hz, which is due to structure vibration of the setup.
Fig. 3.15. The measurement PSD curves and calibrated PSD curves (The peak at 71.37Hz is due to
structure vibration)

87. 62
For high sampling rate control in FPGA, the parameters can be calibrated with similar
method using PSD curves. From the PSD calibration, the time delay in the high speed
control system is dropped to 2.9ms, which is a great reduction compared with 12ms in PC
based control loop using 200Hz control. As expected, the standard deviation of the
Brownian motion is much smaller and the stability margin is greatly increased. Therefore,
a much larger control gain can be used (Fig.3.17). The minimal Brownian motion
standard deviations under 1606Hz control is 31.68nm, 30.65nm and 27.34nm in x, y and
z directions, while in 200Hz control the standard deviations are 51.97nm, 51.55nm and
36.32nm in three directions (Fig.3.16).
Fig. 3.16. The positioning result using the optimal trapping stiffness in 200Hz case and 1606Hz case
It is worth mentioning that for 1606Hz case the calibrated drag coefficient in x and y
direction is 3.71e-8 N.s/m and 6.43e-8 N.s/m in z direction, which is much smaller than
that in 200Hz case, i.e. 5.51e-8 N.s/m in x and y direction and 1.31e-7 N.s/m in z
direction. Therefore, the improvement of the high speed control is even better if the drag
coefficient is the same in these two cases.

88. 63
The increase of sampling rate will reduce the measurement delay m and Zero-Order-
Hold delay ZOH . The actuator delay a , however, is limited by the actuator dynamics,
which cannot be changed by changing control sampling rate. If the actuator delay can be
further reduced, the increase of the control sampling rate is even more significant.
Fig. 3.17. Trapping stiffness vs. standard deviation of Brownian motion, 200Hz control and 1606Hz control
3.6. Conclusion
An accurate optimal current allocation scheme based on minimal 2-norm of currents of
an over-actuated hexapole electromagnetic actuator is accomplished, which solved the
redundancy, nonlinearity and position-dependency at the same time. Compared with
linearized inverse model by applying constant constraints, the optimized currents
allocation greatly increased the force accuracy and force generation capability, which is
demonstrated theoretically and experimentally. Stabilization and trajectory tracking are
experimentally studied. It can be seen that the optimal inverse model will reduce the
current magnitude, reduce the Brownian motion standard deviation and reduce the

89. 64
tracking error. Due to the compact form of the optimal inverse model, the high speed
control can be implemented in the FPGA system and will greatly improve the feedback
control performance compared with the control implemented in PC.
There are still several aspects can be improved to fully enable the magnetic actuator to
perform 3-D scanning probe tasks. First, the magnetic force modeling accuracy need to
be improved. While the motion capability can be accomplished by feedback control
mechanism, the magnetic force modeling error, such as hysteresis, need to be reduced to
accurately know the interaction force between the bead and samples. Second, a dynamics
joint state-parameter estimator need to be developed to dynamically estimate the bead-
sample interaction force and the parameters such as drag coefficient, which can be time-
varying due to environment change.

90. 65
Chapter 4: Hall Sensors Based 3-D Magnetic Force Modeling
and Experiment
4.1 Introduction
Since the soft magnetic material is used to fabricate the magnetic actuator, the
analytical magnetic force model based on actuation current in Chapter 2 and 3 usually
subjects to the hysteresis effect that will greatly degrade the magnetic force model
accuracy. For high speed actuated multi-pole magnetic actuator in which magnetic poles
are located close to each other, the magnetic field not only has rate-dependent hysteresis
effect [59, 60] that is difficult to model, the magnetic coupling among six pole is an even
bigger issue that is usually beyond the capability of modeling.
In this chapter, the difficulty of characterizing the relationship from input current to
output magnetic field is solved by introducing Hall Sensors. Six high bandwidth Hall
Sensors (Asahi Kasei EQ-730L), corresponding to six magnetic poles, are introduced to
directly measure the magnetic field of each pole and thus the total magnetic field. A so-
called Hall-Sensor-based magnetic force model, is proposed to describe the magnetic
force and completely solve the hysteresis issue. The Hall-sensor based model can be
experimentally calibrated by steering the magnetic bead in the 3-D workspace. By
comparing the viscous force and modeling force, the unknown parameters in the
magnetic force model can be calibrated. The Hall-Sensor-based force model is compared

91. 66
with the current-based force model in Chapter 3. It is clearly seen that the Hall-Sensor-
based magnetic force model greatly outperforms the current-based magnetic force model
in term of force accuracy. Moreover, the calibrated Hall sensor based magnetic force
model can not only fit well with the measurement force, but can also be employed to
predict the magnetic force when different trajectories are traversed. Moreover, with Hall
sensor measurement capabilities, the magnetic field of each pole can be directly
controlled using a secondary control loop.
In most magnetic force models, the magnetic moment of the magnetic bead is
modeled linearly dependent on the external magnetic field. When the magnetic field
becomes large, however, the magnetic moment will be nonlinearly dependent on the
magnetic field. An accurate Hall sensor based model is proposed to characterize the
nonlinear effect of the magnetization and the force model is also calibrated by steering
the magnetic bead in Glycerol with large magnetic force.
4.2. The Magnetic force models
4.2.1 The Hall-sensor-based magnetic force model
From the analysis in Chapter 2.2 and Chapter 2.3, it can be seen that two factors
mainly determine the force modeling quality, 1) the accuracy of magnetic charge model,
i.e. Eq. (2.1), 2) the value of magnetic flux in Eq. (2.6). It is already verified in Chapter
2.2 that the point charge model can accurately model the magnetic field. However, the
uncertainties in the magnetic flux calculation, which is mainly caused by the hysteresis
effect, often result in significant modeling errors. The difficulty of modeling the

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hysteresis in this hexapole magnetic actuator comes from three aspects: 1) the rate-
dependent hysteresis of the soft magnetic material [59, 60], 2) transient magnetization
effect such as accommodation effect [74-76], and 3) the magnetic coupling among six
poles.
Fig. 4.1. Magnetic field around the tip of the magnetic pole and a suppositional Hall Sensor.
To solve the hysteresis issue, suppose a miniature hall sensor is available at the tip of
the magnetic pole (Fig.4.1) to measure the magnetic field. Then the voltage measurement
of the Hall Sensor of the ith
pole, denoted as iHv , is proportional to the corresponding
magnetic flux i ,
i ii H Hd v  (4.1)
where iHd is a constant gain from Hall sensor voltage to the magnetic flux. With the
measurement of each Hall Sensor, the Magnetic Flux Vector can be represented by the
following equation,

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H HΦ D V (4.2)
where 1 2 3 4 5 6
= ( , , , , , )H H H H H H Hdiag d d d d d dD is defined as the Flux-Gain Matrix and
1 2 3 4 5 6
=[ , , , , , ]T
H H H H H H Hv v v v v vV is the Hall Sensor Voltage Vector. Substituting Eq. (4.2)
into Eq. (2.6), the Hall sensor based force model can be written as,
( , , ) ( , )T T
H H H H HfF p b V V D L p b D V (4.3)
HD can be normalized with respect to the first entry 1Hd , define the Normalized Flux-
Gain Matrix ˆ
HD = 1H HdD = 2 3 4 5 6
ˆ ˆ ˆ ˆ ˆ(1, , , , , )H H H H Hdiag d d d d d , the force model can be
lumped as the following form,
1
2
ˆˆ ( , , )
ˆˆ ˆ ˆ( , , ) ( , ) ( , , )
H HH
T T
H H H H H H H H H
f
f d f 
F p b V
F p b V V D L p b D V F p b V (4.4)
where ˆ
Hf is the force gain of the quadratic form about voltage vector HV .
4.2.2 The current-based magnetic force model
From Hopkinson’s Law, the theoretical value of the 6×6 flux distribution matrix IK
has 5/6 as the diagonal entries and -1/6 as the off-diagonal entries as in Eq. (2.8), but it is
shown in Chapter 2 that the real value does not follow the theoretical value due to the

94. 69
magnetic leakage in the magnetic circuit and the geometry difference between upper
poles and lower poles, the following form is proposed to describe IK :
1
2
3
4
5
6
I
k s m m s
k m s s m
s m k m s
m s k s m
m s m s k
s m s m k






     
      
     
  
     
     
 
      
K (4.5)
Where k1~k6 means the portion of magnetic flux directly from the corresponding pole,
which takes the biggest weight among the 6 entries of each row/column vector. m , s and
 describe the coupling between the neighboring poles, the poles that are separated by
one pole and the opposite poles. It is obvious that k1~k6 > m > s > , which can also be
inferred from the measurement IK in Eq. (2.19). Referring to Eq. (2.6) and Eq. (2.4), the
current-based magnetic force model can be written as:
2
( , , ) ( , )T Tc
I I
a
N
f
 
  
 
F p b I I K L p b K I (4.6)
Define 1
ˆ =I I kK K , max
ˆ= II I the force model can be lumped in the following form,
2
2 2
max 1
ˆ
ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ( , , ) ( , ) ( , , )
I
T Tc
I I I I
a
f
N
f I k f
 
   
 
F p b I I K L p b K I F p b I (4.7)

95. 70
Where ˆ
If is the lumped force gain of the quadratic form about normalized current vector
ˆI .
4.3 Hardware Integration and Validation of Hall Sensors based Hexapole Magnetic
Actuator
4.3.1 Hall sensors integrated with magnetic actuator
Fig. 4.2. (a). Hall sensors integrated with Hexaple magnetic actuator. (b) Zoom-in plot of Hall sensors
associated with upper poles. (c) Zoom-in plot of Hall sensors associated with lower poles.
The Hall Sensors utilized in this electromagnetic actuator, Asahi Kasei EQ-730L, are
high bandwidth Hall Sensors (100K Hz) with high sensitivity (130mv/mT typ.). It is
hoped that the magnetic field diverged from the tips of six poles can be directly measured
as in Fig.4.1. However, this cannot be realized since the size of the workspace enclosed

96. 71
by the tips is 500um in radius, while the Hall Sensor is a 4.1mm×3mm×1.15mm cuboid.
Therefore, six Hall Sensors are placed at the surface of each magnetic pole, as in Fig. 4.2.
4.3.2 Validation of the Hall Sensor measurement
Fig. 4.3. (a). The setup for studying the relationship between the surface-mount Hall Sensor measurement
and the Hall Sensor measurement at the tip. P1 and P2 are actuated by current-driven coils. (b). Positions
and dimensions of hall elements.
The Hall Sensors are attached to the surface of six magnetic poles. Therefore, it is not
clear whether the measurement at the surface can represent the magnetic field at the tip.
This section is to investigate the possibility of modeling the magnetic field at the tip with
the measurement at the surface of the magnetic pole.
The setup for evaluation is in Fig.4.3. As shown in the figure, the Hall element within
the IC chip is a 0.3mm diameter plate, which is 0.41mm to the surface. The first Hall
Sensor H1 is attached to the surface of the first pole P1, the second Hall Sensor H2 is
placed near the tip of the P1. The second pole P2 is introduced to study the influence of
the magnetic coupling, which will happen in real situation since the six poles are placed

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near each other. It is worth mentioning that the Hall Sensor element of H2 and the axis of
P2 are in the same plane so that H2 only measures the magnetic flux density diverged
from P1. The locations and dimensions of the Hall Sensor elements of H1 and H2 are
illustrated in Fig.4.3 (b). It is hoped that the measurement of H1 at the surface, denoting
as Vsurface, is proportional to the measurement of H2 at the tip, denoting as Vtip, so that
Vsurface can represent Vtip, which is used in the Hall-Sensor-based model Eq. (4.4).
The following two experiments are conducted:
1) Actuate P1 with sinusoidal input of different frequencies (1000Hz, 2000Hz, and
3000Hz).
2) Actuate P2 with sinusoidal input of different frequencies (1000Hz, 2000Hz and
3000Hz). In this case, P1 is magnetized by P2 instead of self-actuated. But H2 still only
measures the magnetic flux diverged from the tip of P1.
Fig. 4.4. Input-output plot between Vtip and Vsurface in two cases: 1). P1 is self-actuated, 2) P2 is actuated, P1
is magnetized by the magnetic coupling between P1 and P2.
The experiment result of above two cases is shown in Fig.4.4. First, when P1 is
actuated, it can be seen that the ratio between Vsurface and Vtip is almost the same when the

98. 73
actuation frequency increased from 1000Hz to 3000Hz, which is obvious from the input-
output plot between Vtip and Vsurface. There is time delay between the measurement of Vtip
and Vsurface since they are non-collocated. However, the time delay is negligible since it is
μs scale (Fig.4.4) while the magnetic actuator system bandwidth is ms scale. Second,
when P2 is the actuation pole, P1 is magnetized through magnetic coupling. The
measurement result in Fig.4.4 shows that the ratio between Vsurface and Vtip almost has the
same value. Therefore, the surface-mount Hall Sensor can represent the magnetic field at
the tip of the magnetic pole.
Due to the proportionality between Vsurface and Vtip, the Hall-Sensor-based model as
Eq. (4.4) can be directly used, wherein the voltage vector VH stands for the voltage
measurement of six surface-mount Hall Sensors.
4.4 Hall Sensor Measurement in Magnetic Bead Control and Calibration Of
Magnetic Force Models
4.4.1 Hall Sensor measurement in bead trapping
With the hall-sensor measurement capability, the relationship between the input
current and the magnetic flux density of each pole can be studied since the hall sensor
voltage is proportional to the magnetic field in the corresponding pole. Therefore, the
current-voltage plot represents the relationship between the input current and resulting
magnetic field.
The bead trapping control is performed in water with 1606Hz sampling rate, using the
current-based optimal inverse model as described in Chapter 3.3.2. A proportional
controller is used in the bead trapping control and the positioning result will be shown

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later in Chapter 4.5.2. Obvious positioning error is observed. Since the bead dynamics is
a type 1 first order system, the positioning error indicates that there is modeling error,
which can be clearly seen from Hall-Sensor measurement in Fig.4.5. It can be seen that
most poles have remanent magnetization since most input-output curves are biased. This
will result in modeling error since current-based actuation relies on the assumption that
the output magnetic flux of each pole is linearly dependent on the input current. The
steady-state error in bead trapping is most likely a result of the remanent magnetization.
Fig. 4.5. Input-output plot between current and Hall Sensor voltage.
4.4.2 Force model calibration
The positioning result in Chapter 4.4.1 is an obvious evidence that the hysteresis will
result in modeling error. Compared with the current-based model, the Hall-sensor based
model is a promising solution for modeling error caused by hysteresis since this model is
based on the direct measurement of the magnetic flux.
The Hall-Sensor-based model and current-based model are formulated in Eq. (4.4) and
Eq. (4.7). There are some parameters to be determined in both models. In Hall-Sensor-

100. 75
based model Eq. (4.4), ˆ
Hf , ˆ
HD andb are unknown parameters to be calibrated, while ˆ
If ,
ˆ
IK andb are to be calibrated in current-based model in Eq. (4.7).
Fig. 4.6. Motion control result in trajectory tracking
To fully explore the magnetic force in the 3-D space, the magnetic bead is steered in
water along different trajectories in the 3-D space as shown in Fig.4.6. The inverse model
and control scheme is based on the nominal current-based magnetic force model, which is
demonstrated in Chapter 3.3.2. The magnetic bead is steered at 10μm/s along circular
trajectories at different locations in xy plane, yz plane and xz plane. In xy plane, the bead
is firstly steered to track the lowest 50μm-diameter circle, then is moved to the plane in
the middle, which is 5um above the lowest one. After tracking the circle in the middle,
the bead is moved to the highest plane within which is another circle of the same shape.
After tracking 3 circles in xy planes, the bead is moved from the highest plane to the
middle plane, and is steered along 10um-diameter circles in yz plane and xz plane. First,
the bead is steered along circles in yz plane and xz planes in the center (labeled as 0 in

101. 76
Fig.4.6 (b) and (c)). It is worth mentioning that the circles in yz plane and xz plane are
plotted separately in Fig.4.6 for clarity, while the yz circular tracking is followed by xz
circular tracking immediately. Then the magnetic bead is steered to position 1, 2, 3 and 4
sequentially, circular tracking in yz plane and xz plane is performed at each point.
The motion control is performed twice, the second time is conducted by reversing the
sign of six actuation currents while the target trajectory is the same. According to Eq.
(4.7), reversing the sign of the current vector does not theoretically change the magnetic
force since the force is quadratic form about current vector. However, due to the
hysteresis effect, the hysteresis loop of each single pole is different after reversing the
signs of the current, which can be seen in Fig.4.7. Moreover, it can be seen that the
hysteresis loop is very complicated to model.
Fig. 4.7. The input-output of six poles in two tracking experiments, the second experiment is conducted by
reversing the sign of the actuation currents. (To make the input-output plot orientated with positive slope,
the signs of some plots are changed accordingly)

102. 77
Fig. 4.8. Control loop for magnetic force analysis; ( dP is the desired position, mP is the measurement
position, pK is the proportional control gain, dF is the desired force calculated by the controller, a and a
are the delay in actuator and measurement, MTF is the magnetic force, TF is the thermal force, N is
measurement noise)
First, the influence of the thermal force and measurement noise are analyzed. The
signal processing scheme is studied for magnetic force analysis. The control loop for
magnetic force analysis is shown in Fig.4.8. To simplify the analysis, proportional control
is employed and modeling error is not included, i.e. (t- )= (t)d A MTF F . From the Langevin
equation [68], the bead dynamics can be modeled by the following 1st
order equations,
 
1
m MT TF F N
s
  P (4.8)
The bead dynamics can be rewritten as =m MT Ts F F s N    P . It is hoped that the
magnetic force information can be obtained from position measurement. The drag
coefficient is calculated by comparing the measurement Power Spectrum Density (PSD)
and the theoretical PSD, as addressed in Chapter 3.5. To attenuate the effect of the
thermal force and the measurement noise, a low-pass-filter is applied on both sides of Eq.
(4.8),

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1 1 1 1
1 1 1 1
m MT Ts F F s N
s s s s
 
   
        
   
P (4.9)
According to Eq. (4.9), it is hoped that ( 1)MTF s  ( 1)TF s  + ( 1)s N s   so
that the left side of Eq. (4.9) can represent the magnetic force.
Fig. 4.9. PSD analysis without low-pass-filter. var(.) operator means the variance of the corresponding
variable.
Fig. 4.10. PSD analysis with low-pass-filter. var(.) operator means the variance of the corresponding
variable.
A simple case is simulated to demonstrate the effect of low-pass-filter: particle tracking
speed is 5um/s along a circle with 50um diameter, control sampling rate is 1606Hz,
measurement noise is 1.5nm in standard deviation, low-pass-filter time constant =0.05,

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the damping coefficient of water =3e-8 N.s/m, and proportional gain is pK =5e-6 N/m.
The PSD analysis and cumulative variance is plotted in Fig.4.9 and Fig.4.10, showing the
effect without and with low-pass-filter. It can be clearly seen that the effect of thermal
force and measurement noise are negligible after applying LPF and the magnetic force
can be measured accurately.
The calibration is accomplished by minimizing the following objective functions, i.e.
Eq. (4.10) and Eq. (4.11), for Hall-Sensor-based model and current-based model
respectively. The measurement position and force are filtered by the same low-pass-filter.
 
2
1
ˆ ˆˆ ˆ ˆ( , , ) (j) (j), , (j);
N
H H H MT H H H H
j
J f f

 D b F F p b V D (4.10)
 
2
1
ˆ ˆˆ ˆ ˆ ˆ( , , ) (j) (j), , (j);
N
I I I MT I I I
j
J f f

 K b F F p b I K (4.11)
Where HJ and IJ are objective functions of Hall-Sensor-based model and current-based
model, N is the number of sampling instants, MTF is the measurement magnetic force as
in Eq. (4.9). The Hall-Sensor-based magnetic force model at each sampling instant is
expressed as ˆ ˆ ˆ( (j), , (j); )H H H Hf F p b V D , where the force gain ˆ
Hf , magnetic charge bias
vectorb and the normalized Hall-Sensor-flux-gain matrix ˆ
HD are constants and position
(j)p and voltage (j)HV are from real-time measurement. The current-based model
ˆ ˆ ˆ ˆ( (j), , (j); )I I If F p b I K is defined similarly.

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Both the Hall-Sensor-based magnetic force model and current-based model are
calibrated and their performances are compared. The calculation is in actuation
coordinate. The measurement force and modeling force are plotted in Fig.4.11 to
Fig.4.14.
Fig. 4.11. Hall-sensor-based magnetic force model along circles in xy plane (as Fig.4.6 (a))
Fig. 4.12. Hall-sensor-based magnetic force model along circles in yz and xz planes (as in Fig. 4.6 (b) and
Fig.4.6 (c))
From Fig.4.11, Fig.4.12 it can be seen that the Hall-Sensor-based magnetic force
model fits very well with the measured viscous force. The value of the objective function
Eq. (4.10) is that HJ = 2434.42, the error at each point is about √J/(3*N) = 0.087pN.

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Fig. 4.13. Current-based magnetic force model along circles in xy plane (as Fig.4.6 (a))
Fig. 4.14. Current-based magnetic force model along circles in yz and xz planes (as in Fig.4.6 (b) and
Fig.4.6 (c))
From Fig.4.13, Fig.4.14 it can be seen that the current-based results in obvious
modeling error. The value of the objective function Eq. (4.11) is 10279.1, the error at
each point is about √J/(3*N) = 0.179pN, while the error in hall sensor based model is
about 0.087pN.

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Fig. 4.15. Histogram of modeling error in force calibration (Hall-sensor-based model vs. Current-based
model; the black histogram are from Fig.4.11 and Fig.4.12, the green histogram are from Fig.4.13 and
Fig.4.14)
The Histogram of modeling errors in Fig.4.11 to Fig.4.14 are plotted in Fig.4.15. It
can be seen that the error in Hall-Sensor based model is not only smaller, but also follows
a reasonable Gaussian distribution, which means that it captured the physics of the
magnetic force model.
4.5 Application of Hall-Sensor-Based Model and application of Hall Sensors
4.5.1 Force Prediction
It is already shown in Chapter 4.4.2 that the Hall-Sensor-based force model can fit
well with the measured viscous force. But it is hoped the calibrated force model can not
only fit well with the measurement force but can also predict the magnetic force in the 3-
D space. Therefore, bead tracking control is conducted along circles and straight lines at
different directions as in Fig.4.16. For the circular trajectory of Fig.4.16, the bead is
steered at 10μm/s to follow circles at different height, similar to the trajectory in Fig.4.6
of force calibration. But the circle diameter is 25um instead of 50um. The bead is first
steered along the lowest circle, then the circle in the middle, followed by the highest

108. 83
circular trajectory tracking. After finishing the circular tracking, the bead is steered from
the highest plane to the middle plane, wherein the bead is steered to follow several linear
motion as Fig.4.16 (b). Starting from the center of the middle plane (labeled as 0 in
Fig.4.16 (b)), the bead is steered from 0 to 1, where the bead stays for 2sec. Then the
bead is steered to 2, 3, 4 and 0 sequentially. The modeling force is calculated to compare
with the measured force, which is processed with the same LPF as in Chapter 4.4.2.
Fig. 4.16. Bead motion trajectory for force prediction
Fig. 4.17. Force prediction using Hall-Sensor-based model (motion along circles in xy plane)
The magnetic force prediction and the viscous force are plot together in Fig.4.17 to
Fig.4.20. From Fig.4.17 and Fig.4.18, it can be seen that the Hall-Sensor-based magnetic
force model can accurately predict the force experienced by the magnetic bead, while the

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current-based force model has obvious error, which can be clearly seen in Fig.4.19 and
Fig.4.20.
Fig. 4.18. Force prediction using Hall-Sensor-based model (motion along straight line in xy plane)
Fig. 4.19. Force prediction using Current-based model (motion along circles in xy plane)
Fig. 4.20. Force prediction using Current-based model (motion along straight lines in xy plane)

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Fig. 4.21. Histogram of modeling error in force prediction (Hall-sensor-based model vs. Current-based
model; the black histogram are from Fig.4.17, the green histogram are from Fig.4.19)
Fig. 4.22. Histogram of modeling error in force prediction (Hall-sensor-based model vs. Current-based
model; the black histogram are from Fig.4.18, the green histogram are from Fig.4.20)
The Histogram of force prediction errors, both in Hall sensor based model and
current-based model, are plotted in Fig.4.21 and Fig.4.22. It can be seen that the error in
Hall sensor based model is not only smaller but also resembles the normal distribution.
But the error in the current-based model is larger and there is no obvious statistics
pattern.

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4.5.2 Magnetic Field Feedback Control Using Hall-Sensor inner loop control
The positioning performance Chapter 4.4.1 and the force calibration result in Chapter
4.4.2 indicate severe drawbacks of the hysteresis problem when the current-based model
is employed. However, with the Hall-sensor measurement capability, the Hall Sensor
voltage can be directly kept at desired value through so called Hall-sensor inner loop
control as in Fig.4.23.
Fig. 4.23. Magnetic bead position control integrated with Hall-sensor inner loop control. dP and mP are
the desired and measurement position, e is the positioning error, dF is the desired force calculated from the
motion controller. dV and mV are the desired and measured Hall-Sensor voltage.
First, the Hall-sensor inner-loop control performance is experimentally verified and
compared with current actuation without Hall-Sensor inner-loop control. Fig.4.24 is the
result of Hall-Sensor inner loop control for different step input dV , i.e. dV =0.1V, 0.2V
and 0.5V. The measurement voltage mV is normalized by dividing mV with dV . The
normalized mV converge to 1 for different dV , which means the Hall-Sensor inner loop
control can work as desired. Moreover, from the zoom-in plot in Fig.4.24, the raise time
is about 0.32ms. The transient response has slight difference since the magnetization of
the pole is a complicated nonlinear process due to hysteresis effect.

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Fig. 4.24. Normalized Response of different desired voltage dV .
Fig. 4.25. Step response for different voltage inputs.
If Hall-Sensor inner loop control is not used, current/voltage input is used to drive the
coil for magnetic flux generation. Most researchers assume there is a linear relationship
between the input current/voltage and the output flux. A linear amplifier working in
current-mode is used to drive the coil, in which 1V input to the amplifier generates 0.3A
output. Two experiments are performed. First, decrease the voltage stepwisely,
specifically 0.1V step and 0.5V step and then increase voltages back to 0V again. Second,

113. 88
input different step voltages and compared the output magnetic flux. The Hall-Sensor
measurement represents the magnetic flux in these two experiments.
From Fig.4.25 (a) and (b), it can be seen that the magnetic flux does not change in a
fixed increment when the input voltage changes in a fixed step. Moreover, the magnetic
flux does not return to the original status when the input voltage returned back to 0V. It is
worth mentioning that the starting point in Fig.25 (a) and (b) is not zero due to remnant
magnetization. From Fig.4.25 (c), normalized step responses are plotted for 0.1V, 0.5V
and 1V inputs. The responses are normalized by dividing the Hall-sensor measurement
mV by the input step size, i.e. 0.1V, 0.5V and 1V. From the result it can be seen that the
normalized output do not coincide with each other, which is an indication of nonlinear
effect of the magnetic pole.
The above experiment reveals that the Hall-Sensor inner loop control is a promising
method to solve the hysteresis in position control. A bead trapping experiment is
performed and the result will be compared with current-based stabilization in Chapter
4.4.1. The nominal optimal inverse model developed in Chapter 3.3.2 is used to calculate
the desired current and thus the desired Hall-sensor voltage dV . The whole setup is
controlled by an FPGA system as shown in Fig.3.1. The Hall-sensor voltage is measured
by an AD converter running at 200K Hz and the feedback control is running at 100K Hz.
Six coils are driven by DA converter followed by an amplifier. The DA converter is
running at 640K Hz while the bandwidth of the amplifier is 10K Hz.

114. 89
Fig. 4.26. Positioning performance using Hall-Sensor inner loop control.
Fig. 4.27. dV is the desired voltage and mV is the measurement voltage)
Fig. 4.28. Step Response of Hall-sensor feedback control.
With the Hall-Sensor inner loop control introduced, it can be seen that the stabilization
steady state error in Fig.4.26 is very small compare with current-based feedback control,
since the remanent magnetization is removed. From the plot of dV and mV in Fig.4.27, it
can be seen that Vd followed Vm accurately. To further demonstrate the Hall-sensor

115. 90
feedback control performance, 3 step responses (denote at step 1, step 2 and step 3) at
different times are selected and the normalized response are plotted together in Fig.4.28.
It can be seen that the normalized step response for different step sizes agree with each
other. The feedback control output is a little different from the test in Fig.4.24 due to two
reasons: 1) different step size result in slight different transient response for nonlinear
systems with hysteresis, 2) the magnetic field in 6 poles are controlled simultaneously
while the magnetic coupling among 6 poles create some disturbances.
4.6 Accurate Hall sensor based Magnetic Force Modeling for large magnetic force
generation
The Hall-sensor-based magnetic force model presented in previous sections is capable
of solving complex hysteresis problem and achieve sub-pN accuracy. But the magnetic
field is relatively small and therefore the magnetization of the magnetic bead is modeled
as a linear function of the external magnetic field. When the external magnetic field is
large, however, the magnetization of the magnetic bead will be nonlinearly dependent on
the external magnetic field [63, 77-79]. An accurate magnetic force model is proposed in
this section wherein the Langevin-function [63] is used to model the nonlinear
relationship between the magnetization of the superparamagnetic magnetic particle and
the external magnetic field.
4.6.1 The modeling of the magnetic bead magnetization
The magnetic bead (Dynabead M450 Epoxy) is aggregated from nanoparticles of
2 3Fe O  , which is experimentally verified to be superparamagnetic [77, 78]. This

116. 91
property will greatly ease the modeling of the magnetic bead. But the magnetization
curve of this particle is only linear up to about 50 Gauss from different measurement
schemes [77-79]. Therefore, the bead magnetization cannot be modeled as a linear
function of the external magnetic field when the magnetic force is large. A Langevin
function is used to model the magnetization of this superparamagnetic particle since it is
a continuous function with relative simple form and therefore can greatly ease the
experimental calibration.
Fig. 4.29. Magnetization of the M450 superparamagnetic bead
The magnetization of the bead is illustrated in Fig.4.29, where B is the external
magnetic field, m is the magnetic moment of the magnetic bead, θ and ϕ are the direction
of the external magnetic field and the magnetic moment of the bead. For a general
ellipsoid bead, the directions of the external magnetic field and the magnetic moment
direction of the bead are not necessarily equal [49]. But since the M450 bead is a
superparamagnetic spherical bead with evenly distributed magnetic nanoparticle in the
bead, it can be assumed that the M450 bead is homogeneous in each directions.
Therefore, it is valid to assume that θ = ϕ.

117. 92
The nonlinear magnetization of the superparamagentic object is usually modeled by
Langevin Function [63, 79]. More specifically, two parameters are employed to describe
the saturation limit and the shape of the Langevin function,
   
ˆ
1 ˆm , , coths s
m
a m a
a
 
   
 
m B B B
B
(4.12)
Where ms is the saturation limit of the magnetic bead, a controls the shape of the
Langevin Function, ˆ = / || ||B B B is the unit directional vector of the external magnetic
field, which is also the direction of the magnetic moment of the magnetic bead,
   ˆ =coth 1m a aB B is defined as the Normalized Magnetization of the magnetic
moment. When a increases, the Langevin Function increases faster (Fig.4.30). The
external magnetic field B will be modeled in the following section as a function of Hall-
sensors measurement.
Fig. 4.30. The plot of the Langevin Function (assume the saturation limit sm =1)

118. 93
Most researchers model the magnetization of the magnetic particle linearly dependent
on the external magnetic field [1, 2, 34, 52, 53]. And the hall-sensor-based model in
Chapter 4.4 has already shown that the linear model is accurate enough if the magnetic
field is relatively small, which means the magnetization is just a linear portion of the
Langevin function around ||B||=0,
 m , , m ,
3
s s
a
a  m B B B 0 (4.13)
Most researchers model the magnetization of the magnetic particle as
0 0 0(3 ) [( ) ( 2 )]V        m B [67], which means m 3sa ≈
0 0 0(3 ) [( ) ( 2 )]V        when || ||B 0 . From [77], the magnetic saturation of one
bead is about ms =1.50e-12 Am2
. Therefore, the nominal value of a is about 227.8125.
4.6.2 The Modeling of The Accurate Hall Sensor Based Magnetic Force
From Eq. (2.1) and Eq. (4.1), the magnetic Charge Vector Q can be modeled as
01 H H Q D V . As presented in Chapter 4.2.1, 1
ˆ
H H HdD D . Therefore, the magnetic
flux density ( , )B p b can be modeled as follow, from Eq. (2.3)
1
2
0 ˆ
ˆ ˆˆ ˆ( , ) ( , )
H
H
m H
H H
b
k d


B
B p b R p b D V (4.14)

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Where ˆ ˆ( , )R p b is the same as in Eq. (2.3), ˆ
HB is the Hall-Sensor-Based Normalized
Flux Density, Hb is the Hall-Sensor-Based Flux Density Coefficient. The force model
can be expressed by the gradient force, i.e.  = 1 2 ( ) F m B =
     ˆ ˆ2 || || || ||sm m m    B B , where  ˆm =   ˆ || || || ||m  B B . Therefore,
 
ˆ
ˆ
2
sm m
m
 
      
F B B
B
(4.15)
From Eq. (4.14), || ||B =  
1/2
ˆ ˆT
H H Hb B B =  
1/2
ˆ ˆ ˆ ˆˆ ˆ( , ) ( , )T T T
H H H H Hb V D R p b R p b D V . The
gradient   B can be formulated as follow,
 
 
 1/2
2
ˆ1 ˆ ˆ ˆ ˆ( , ) ( , )
ˆ ˆ2
1 ˆ ˆ( , ), ( , ), ( , )
2
T T T i
H H H H H
T
i
H H
T TH
H H x y z H H
r
b
r
b

   

   
B V D R p b R p b D V
B B
V D L p b L p b L p b D V
B
(4.16)
The term ˆi ir r  exists because the position is normalized with respect to the
workspace radius . xL , yL and zL have the same meaning as that in Eq. (2.6). The
partial derivative of ˆm with respect to || ||B is ˆm  B =
22
( ) 1 ( )a csch a a  B B .
By utilizing Eq. (4.15) and Eq. (4.16), the force model can be formulated as follow,

120. 95
   
2
2
ˆ ( )
ˆ
ˆ
1 ˆ ˆˆcoth ( , )
4
H
L
T Ts H
i H H i H H
i
f
f
m b a
a csch a
a
 
    
   F
F B B V D L p b D V
B
,i = x,y,z (4.17)
Where ˆ
HF = [ ˆ ( )H xF , ˆ ( )H yF , ˆ ( )H zF ]is the Normalized Hall-Sensor-Based Force, ˆf
is the Magnetic Force Gain and ˆf =  1
2 2 5 2
04s m Hm k d a  , ˆ
Lf is defined as the Langevin
Force-Gain. The total force gain is the multiplication of ˆf and ˆ
Lf .
4.6.3 Calibration of the Magnetic Force Model
The Accurate Magnetic Force Model is described by Eq. (4.17). There are several
unknown parameters to be calibrated. First, ˆf is a lumped force gain to be calibrated.
Second, The Langevin force-gain ˆ
Lf is a function of || ||a B . From Eq. (4.14), || ||a B =
ˆ|| ||H Hab B , Hab can be lumped as another unknown parameter ˆa , i.e. ˆa = Hab . Third, ˆ
HD
andb need to be calibrated as well.
Fig. 4.31. Motion control result in trajectory tracking

121. 96
The magnetic force model Eq. (4.17) is position dependent. The inverse model and
motion control capability is already achieved in Chapter 3.4. The magnetic bead can be
steered in Glycerol to obtain larger magnetic force. The bead motion trajectory is shown
in Fig.4.31, in xy plane, the bead is firstly steered to track the lowest 50um-diameter
circle at 2um/s, then is moved to the plane in the middle, which is 5um above the lowest
one. After tracking the circle in the middle, the bead is moved to the highest plane within
which is another circle of the same shape. After tracking 3 circles in xy planes at 2um/s,
the bead is moved from the highest plane to the middle plane, and is steered along 10um-
diameter circles in yz plane and xz plane at 1.5um/s instead of 2um/s, since the drag
coefficient in z direction is larger than that in xy plane. First, the bead is steered along
circles in yz plane and xz plane in the center (labeled as 0 in Fig.4.31 (b) and (c)). It is
worth mentioning that the circles in yz plane and xz plane are plotted separately in
Fig.4.31 for clarity, while the yz circular tracking is followed by xz circular tracking
immediately. Then the magnetic bead is steered to position 1, 2, 3 and 4 sequentially,
circular tracking in yz plane and xz plane is performance at each location.
The motion control is performed twice, the second time is conducted by reversing the
sign of six actuation currents while the target trajectory is the same. This is to explore
different part of the hysteresis loop. According to Eq. (4.17), reversing the sign of the
current vector does not theoretically change the magnetic force since the force is a
quadratic form about current vector. However, due to the hysteresis effect, the hysteresis
loop of each single pole is different for all six poles (Fig.4.32) after reversing the signs of

122. 97
the current. Moreover, it can be seen that the hysteresis loop is very complicated to
model.
Fig. 4.32. The input-output of six poles in two tracking experiments, the second experiment is conducted by
reversing the sign of the actuation currents. (To make the input-output plots orientated with positive slope,
the signs of some plots are changed accordingly)
The calibration is accomplished by minimizing the following objective functions,
 
2
1
ˆ ˆ ˆˆ ˆ ˆˆ ˆ( , , , ) (j) ( ) (j), , (j);
N
H H MT L H H H
j
J f a f f a

   D b F F p b V D (4.18)
Where HJ is the objective functions of the Hall-Sensor-based model, N is the number of
sampling instants, MTF is the magnetic force measured with the same method as in
Chapter 4.4.2. The Hall-Sensor-based magnetic force model at each sampling instant is
expressed as    ˆ ˆ ˆ ˆˆ (j), , (j);L H H Hf f a F p b V D , where ˆf , ˆa , the magnetic charge bias

123. 98
vectorb and the normalized Hall-Sensor-flux-gain matrix ˆ
HD are constants and position
(j)p and voltage (j)HV are from real-time measurement.
The Hall-Sensor-based magnetic force model using Langevin-function, i.e. Eq. (4.17),
and Hall-Sensor-based model in Chapter 4.2.1, i.e. Eq. (4.4), are calibrated and their
performances are compared. The measurement force and the modeling force are plot in
Fig.4.33 to Fig.4.36. From Fig.4.33 and Fig.4.35 it can be seen that the Hall-Sensor-based
magnetic force model using Langevin-function fits very well with the measured viscous
force, in all directions. The value of the objective function Eq. (4.18) is that HJ =
4.3063e+05, the error at each point is about (3*N)J =1.5774pN. From Fig.4.34 and
Fig.4.36 it can be seen that the Hall-Sensor-based magnetic force model without using
the Langevin-function followed the trend of the measured viscous force, but had error at
some orientations, especially when the force is large, which is an indication that the
magnetic moment of the magnetic bead cannot be model as an linear function. The value
of the objective function Eq. (4.10) is that HJ = 1.3859e+06, the error at each point is
about (3*N)J =2.8299pN. From the zoom-in plot in Fig.4.34 (b) and Fig.4.36 (b), the
error can be easily seen, while the Hall-sensor-based modeling using Langevin-function
can accurately model the measured force in Fig.4.33 (b) and Fig.4.35 (b).

124. 99
Fig. 4.33. Calibration result of Hall-Sensor-based force model using Langevin function. (a) Force
calibration along circles in xy plane (as in Fig.4.31) (b) zoom-in plot of force calibration
Fig. 4.34. Calibration result of Hall-Sensor-based model without using Langevin-function. (a) Force
calibration along circles in xy plane (as in Fig.4.31) (b) zoom-in plot of force calibration

125. 100
Fig. 4.35. Calibration result of Hall-Sensor-based force model using Langevin function. (a) Force
calibration along circles in yz and xy planes (as in Fig.4.31) (b) zoom-in plot of force calibration
Fig. 4.36. Calibration result of Hall-Sensor-based force model without using Langevin function. (a) Force
calibration along circles in yz and xy planes (as in Fig. 4.31) (b) zoom-in plot of force calibration
To better visualize the force model performance, the histogram of the force modeling
error is plot in Fig.4.37. It can be seen that the maximal force error in the generalized

126. 101
Hall-sensor based model is smaller than the Hall-sensor based model. Moreover, the
modeling error in the Accurate Magnetic Force model is very close to the Gaussian
distribution.
Fig. 4.37. Histogram of the force calibration error for Hall-sensor-based model with Langevin-function and
without Langevin-function. (The red histograms are from the modeling errors of Fig.4.33 and Fig.4.35, the
blue histograms are from the modeling errors of Fig.4.34 and Fig.4.36)
4.6.4 Application of the Accurate Hall-Sensor-Based Model
4.6.4.1 Magnetic Force Prediction Using the Accurate Hall-Sensor-Base Force Model
Fig. 4.38. Bead motion trajectory for force prediction
It is already shown in Chapter 4.6.3 that the Hall-Sensor-based force model with
Langevin-function can fit well with the measured viscous force. But it is hoped the
calibrated force model can not only fit well with the target viscous force but can also

127. 102
predict the magnetic force in the 3-D space. Therefore, bead tracking control is conducted
along circles and straight lines at different directions as in Fig.4.38. The bead is steered at
2um/s to follow circles at different height, similar to the trajectory in Fig.4.31. But the
circle diameter is 25um instead of 50um. The bead is first steered along the lowest circle,
then the circle in the middle, followed by the highest circular trajectory tracking. After
finishing the circular tracking, the bead is steered from the highest plane to the middle
plane, wherein the bead is steered to follow several linear motion as Fig.4.38 (b). Starting
from the center of the middle plane (labeled as 0 in Fig.4.38 (b)), the bead is steered from
0 to 1, where the bead stays for 2sec. Then the bead is steered to 2, 3, 4 and 0
sequentially. The motion speed keeps at 2um/s. The modeling force is calculated to
compare with the viscous force. The modeling force is calculated to compare with the
viscous force.
Fig. 4.39. Force prediction of Hall-Sensor-based force model using Langevin function. (a) Force calibration
along circles in xy plane (as in Fig.4.38) (b) zoom-in plot of force calibration

128. 103
Fig. 4.40. Force prediction of Hall-Sensor-based force model without using Langevin function. (a) Force
calibration along circles in xy plane (as in Fig.4.38) (b) zoom-in plot of force calibration
Fig. 4.41. Force prediction of Hall-Sensor-based force model using Langevin function. (a) Force calibration
along line trajectories in xy plane (as in Fig.4.38) (b) zoom-in plot of force calibration
The magnetic force prediction and the viscous force are plot in Fig.4.39 to Fig.4.42.
From Fig.4.39 and Fig.4.41 it can be seen that the Hall-Sensor-based magnetic force
model with Langevin-function can accurately predict the force experienced by the
magnetic bead, while the Hall-Sensor-based force model has obvious error, which can be
clearly seen in Fig.4.40 and Fig.4.42.

129. 104
Fig. 4.42. Force prediction of Hall-Sensor-based force model without using Langevin function. (a) Force
calibration along line trajectories in xy plane (as in Fig.4.38) (b) zoom-in plot of force calibration
Fig. 4.43. Histogram of the force calibration error for Hall-sensor-based model with Langevin-function and
without Langevin-function. (The red histograms are from the modeling errors of Fig.4.39, the blue
histograms are from the modeling errors of Fig.4.40)
Fig. 4.44. Histogram of the force calibration error for Hall-sensor-based model with Langevin-function and
without Langevin-function. (The red histograms are from the modeling errors of Fig.4.41, the blue
histograms are from the modeling errors of Fig.4.42)

130. 105
Again, the histogram of the force prediction error can be plot in Fig.4.43 and
Fig.4.44. It can be seen that the maximal force error of the Hall-Sensor-based model
without Langevin-function is larger than that of the Hall-sensor based model with
Langevin-function.
4.6.4.2 Parameter calibration and magnetization study
The calibrated parameters are as follow: ˆf = 24.939pN (2.4939e-11N), ˆa = 3.232.
Since ˆ Ha ab , Hb is therefore 1.412e-2. The Magnetic Force Gain 2ˆ (4 )s Hf m b a , with
the values of sm , a and Hb , the calculated Magnetic Force Gain ˆf = 28.664pN, which is
close to the value from calibration, i.e. 24.939pN. The discrepancy may come from the
nominal value of magnetic bead saturation sm , the value of which is actually a statistical
average. From Chapter 4.6.1, m 3sa  0 0 0(3 ) [( ) ( 2 )]V        , with
2ˆ (4 )s Hf m b a and ˆ Ha ab , sm is recalculated as 1.40e-12 instead of nominal value
1.50e-12. Based on this value, a is recalculated as 244.234 instead of 227.8125 as in
Chapter 4.6.1. Therefore, the recalculated Hb is 1.323e-2. Since 1
2
0( )H m Hb k d  , the
Hall-sensor coefficient iHd can be calculated as 5.99e-8.
From the calibrated values and recalculated values, the magnitude of the external
magnetic field and magnetic bead moment can be calculated according to Eq. (4.14) and
Eq. (4.12). The magnetic bead moment with and without Langevin-function model are
both plotted in Fig.4.45, using experiment data. It can be seen that the external magnetic
field can reach about 260 Gauss (1 Gauss = 1e-4 T) in this experiment and the linear

131. 106
model of the magnetization will lead to large modeling error. The linear range of
magnetization is up to 50 Gauss. This agrees with the information provided by the vendor
and other measurement result [77, 78].
Fig. 4.45. relationship between external magnetic field ||B|| (Gauss) and the magnetic moment ||m||.
4.6.4.3 Accurate Hall-sensor-based optimal inverse modeling
The magnetic force model Eq. (4.17) is a quadratic form of Hall-Sensor voltage
vector HV , with is similar to Eq. (4.4), the only difference is that the Langevin Force-Gain
 ˆ || ||Lf a B is also a nonlinear function of the magnetic field B, and thus is a nonlinear
function of HV according to Eq. (4.14). The idea of the optimal inverse model in Chapter
3 can be applied with slight modification. In Eq. (4.17),  ˆ || ||Lf a B is a gain that does not
change the direction of the force. Therefore, the optimal inverse model can be first
developed for the Normalized Hall-Sensor-Based Force ˆ
HF = [ ˆ ˆ( , )T T
H H x H HV D L p b D V ,
ˆ ˆ( , )T T
H H y H HV D L p b D V , ˆ ˆ( , )T T
H H z H HV D L p b D V ]. As shown in Fig.2.7, the desired force dF
can be expressed in spherical coordinate as [cos cos ,cos sin ,sin ]T
d d     F F . Using
similar method as describes in Chapter 3.3.2, the following objective function using

132. 107
Lagrange multipliers can be used to find the optimal inverse model for unit desired force,
i.e. [cos cos ,cos sin ,sin ]T
     ,
 
   
2
ˆ ˆ( , , ) ( , ) cos cos
ˆ ˆ ˆ ˆ( , ) cos sin ( , ) sin
T T
H x H H x H H
T T T T
y H H y H H z H H z H H
J     
    
  
   
p V V D L p b D V
V D L p b D V V D L p b D V
(4.19)
For a particular desired unit force in a specific orientation, the unit inverse model
( , , )unit
H  V p can be obtained by minimizing Eq. (4.19). The next step is to scale the
voltage vector unit
HV to make the force model magnitude equals the magnitude of desired
force, i.e.|| ||dF .
For a particular position and dF orientation, the magnetic field norm || ||B and Hall-
Sensor voltage norm || ||V are linearly dependent, wherein V is a scaled vector of
( , , )unit
H  V p . However, the inverse model is position dependent and orientation
dependent, therefore the ratio between|| ||B and || ||V also depend on the position and
orientation. The plot of || ||B and || ||V in the experiment is shown in Fig.4.46.
Fig. 4.46. Plots of || ||B ,|| ||V and || || / || ||B V

133. 108
Assume that at a particular position p and orientation( , )  , the ratio between || ||V
and || ||B (in Gauss) is 20, then the Langevin Force Gain can be written as  ˆ 20 || ||Lf a V .
The force model can be expressed as,
 
2
2
|| ||ˆ ˆ ˆ ˆ20 || || ( ) ( , )
|| ||
unit T T unit
L H H H Hunit
H
f f a 
V
F V V D L p b D V
V
(4.20)
To satisfy the desired force magnitude, the lumped gain
  2 2ˆ ˆ 20 || || || || / || ||unit
L Hf f a V V V must equal dF . Therefore,
  2 2ˆ ˆ20 || || || || || || /unit
L d Hf a f  V V F V and || ||V need to be solved to finalize the inverse
model. As shown in Fig.4.47, the Langevin Force Gain  ˆ 20 || ||Lf a V is monotonically
decreasing due to the magnetic saturation of the magnetic particle, while
  2ˆ 20 || || || ||Lf a V V is a monotonically increasing function. From Fig.4.47 (c),
  2ˆ 20 || || || ||Lf a V V generally satisfies quadratic relationship when || ||V is small. When
|| ||V is growing, however,   2ˆ 20 || || || ||Lf a V V deviate from quadratic relationship and
resembles a linear relationship when || ||V is large, which is physically expected. Since
when magnetic bead magnetization is saturated, the magnetic force will only grow
linearly with respect to || ||B according to the force model  1 2 ( )  F m B . From
  2ˆ 20 || || || ||Lf a V V = 2 ˆ|| || /unit
d H fF V , || ||V can be easily solved from the inverse
function, which is shown in Fig.4.47 (d). With the magnitude || ||V known, the inverse
model can be expressed as|| || / || ||unit unit
H HV V V .

134. 109
Fig. 4.47. (a). The plot of   2ˆ 20 || || || ||Lf a V V and  ˆ 20 || ||Lf a V . (b). the plot of  ˆ 20 || ||Lf a V . (c).
Zoom-in plot of   2ˆ 20 || || || ||Lf a V V when || ||V is small. (d). the plot of   2ˆ 20 || || || ||Lf a V V and the
inverse function   2ˆ( 20 || || || || )Linverse f a V V
4.7 Conclusion
Hall sensors are introduced to solve the complicated hysteresis issue, which is very
difficult to model due to the complicated hysteresis and magnetic coupling among six
poles. Six Hall Sensors are integrated with the hexapole electromagnetic actuator and the
validity of using surface-mount Hall Sensors is experimentally verified. The magnetic
hysteresis issue can be clearly observed during the bead motion control.
A Hall-Sensor-based magnetic force model is proposed to improve the accuracy of
the magnetic force model, wherein the Hall sensor measurement can accurately present
the magnetic field information. The magnetic force model is experimentally calibrated by
steering the magnetic bead in the 3-D space. The current-based magnetic force model,
which is conventionally used and subject to hysteresis issue is also calibrated and
compared with Hall-Sensor-based magnetic force model. The result clearly shows that
the Hall-sensor-based model greatly outperforms the current-based force model and can

135. 110
achieve sub-pN resolution. With the Hall Sensor measurement capability, the magnetic
force can be accurately predicted, using the calibrated magnetic force model. The Hall
sensors can also be used in an inner control loop to directly control the magnetic field of
each pole, which can remove the modeling error caused by the hysteresis issue.
When the external magnetic field is large, the magnetization of the magnetic bead
cannot be modeled as a linear function of the magnetic field. The nonlinear effect in the
magnetic particle magnetization is modeled with the Langevin-function. The modeling
result is experimentally calibrated and compared with the model in which the particle
magnetization is modeled as the linear function of the magnetic field. The result showed
that the nonlinear magnetization effect exist and the improved model can model the
magnetic force more accurately.
With the introduction of hall sensors, the dynamic force sensing estimator can be
developed to dynamically estimate the interaction force between the magnetic bead and
other objects such as biological samples. This capability will separate this multi-pole
magnetic actuator from other magnetic force appliers. Moreover, parameter estimation
need to be accomplished since parameters such as drag coefficient may change greatly
due to local environment change.

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Chapter 5: Dynamic force sensing and parameter estimation
5.1. Introduction
3D motion control and accurate magnetic force modeling are achieved through
optimal inverse modeling and Hall sensor measurement. However, these capabilities are
not enough to enable the magnetic actuator to perform active scanning tasks unless the
bead-sample interaction force can be sensed. Moreover, it is hoped that the force can be
sensed dynamically to map the transient force of many biological processes. This
capability will complete the magnetic actuator as an active scanning probe and can even
achieve automatic scanning by combining the motion control capability and dynamic
force sensing capability.
In conventional magnetic tweezers, the magnetic beads are often functionalized and
anchored to target bio samples to avoid instability. Under these circumstances, the
magnetic force sensing is mostly quasi-static, wherein the magnetic force is usually
calculated according to a pre-calibrated force function [29] or derived from Brownian
motion fluctuations through image measurement microscopy [61, 62]. Moreover,
automatic scanning is impossible since active control is not achieved.
There are actively controlled magnetic actuators that can stabilize the bead. In some
magnetic actuators [44, 63], the analytical magnetic force model is not proposed, which
makes the dynamic force sensing very difficult. For the magnetic actuator presented in
this project, Hall sensor based model can achieve sub-pN magnetic force accuracy

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(Chapter 4). Moreover, the measurement resolution can reach sub-nm with high
bandwidth implementation in FPGA [71]. The Hall sensor measurement and the motion
measurement can be utilized by a Recursive Least Square Estimator to estimate the bead-
sample force in real-time. Moreover, the bead dynamics, which is describe by the
Langevin equation [68], can be a time-varying process since the drag coefficient can
change significantly as a result of local environment change, such as wall-effect,
temperature change, liquid property change and etc. Therefore, a joint state-parameter
estimation algorithm is proposed to simultaneously estimate the bead-sample interaction
force and the drag coefficient of the magnetic bead. In this estimation algorithm, a
disturbance observer is employed to estimate the external force dynamically and a 1st
order autoregressive (AR) model is utilized to estimate the drag coefficient. Specifically,
Kalman filter algorithm is implemented for the joint state-parameter estimator since the
system subject to random white noise such as thermal force and measurement noise and
the discrete dynamics model is time-varying.
Preliminary studies are performed theoretically and experimentally. From the force
sensing result it can be seen that the proposed joint state-parameter estimator using
Kalman filter algorithm is valid for dynamic force sensing and the Hall sensor based
magnetic force model will lead to improved force sensing result compared to current-
based model.

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5.2 Joint State-Parameter Estimator Algorithm
5.2.1 Drag coefficient estimation based on the magnetic force model
Fig. 5.1. Joint state-parameter estimator (using the Hall sensor based force model) running simultaneously
with feedback controller. ( dP is the desired position, mP is the measurement position, e is the error signal,
a and m are the delay in actuator and measurement, ˆI is the current from the nominal inverse model, dF is
the desired force calculated by the controller, '
(t) (t )d d a F F is caused by the actuator delay, F is the
modeling error, MTF is the magnetic force, TF is the thermal force, EF is the bead-sample interaction force,
B is the magnetic flex density, mV is the Hall sensor measurement voltage, ˆ
MTF is the modeling magnetic
force, ˆ is the estimated drag coefficient, ˆ
EF is the estimated external force)
According to the principle of separation of estimation and control, the observer
design is independent of the feedback controller design. Therefore, the feedback control
and the force sensing estimator can run simultaneously as shown in Fig.5.1. The bead’s
motion dynamics is needed for the estimator design. Since the motion in x, y and z
directions are all governed by the Langevin equation, only the model in x direction is
presented here while the model in y and z direction can be derived similarly. As
mentioned in Chapter 2.4, the inertia term in the Langevin equation can be neglected.

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Therefore, the Langevin function from the block diagram of Fig.5.1 can be described by
the following function:
( ) (t) ( ) ( )MT E Tx t F F t F t    (5.1)
The visual measurement has delay due to image transfer and calculation (1 sampling
for exposure, 1 sampling for image transfer, and 1 sampling for calculation). As a result,
the measurement position ( ) ( )m mx t x t   . The actuator also has delay due to
magnetization dynamics, i.e. ˆ(t) (t )MT MT aF F   . The dynamics based on measurement
position is as follow after considering time delays,
ˆ( ) (t ) ( ) ( )m MT D E m T mx t F F t F t         (5.2)
According to the sources of the force, ( )mx t can be decomposed into three parts, i.e.
( )MTx t , ( )Ex t and ( )Tx t due to the magnetic force, the external force and the thermal force
respectively. The decomposition is described as follow,
 
 
 
'
'
ˆ( ) (t ), 5.3.1
( ) ( ), 5.3.2
( ) ( ), 5.3.3
MT MT d
E E
T T
x t F
x t F t
x t F t
 


  


 
Where '
( ) ( )E E mF t F t   and '
( ) ( )E T mF t F t   . In discrete domain, the increment from
sampling instance 1kt  to kt is consist of three different parts:

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ˆ[ ] [ 1] [ ] [ ] [ ] (t 1 ) [ ] [ ]s
m m MT E T MT E Tx k x k x k x k x k F x k x k


          (5.4)
Where [ ] [ 1]m mx k x k  is the increment within one sampling interval, s is the digital
control sampling interval, /D s   is the number of delay in digital control. [ ]MTx k ,
[ ]Ex k and [ ]Tx k are the motion of the magnetic bead within the time interval 1[t ,t ]k k ,
caused by the magnetic force, the external force and the thermal force. One assumption is
that the force dynamics is much lower compared to the sampling rate and can be
considered constant within 1[t ,t ]k k . That is the reason [ ]MTx k can be described by
ˆ( ) (t 1 )s MTF    in Eq. (5.4). Moreover, if [ ]Ex k is known, the external force can be
similarly describes as '
[ ] [ ]E E sF k x k  .
Since the exact dynamics of [ ]Ex k is very difficult to know, an autoregressive (AR)
model is used to describe the disturbance dynamics. Specifically, the 2nd
order AR model
is used to describe [ ]Ex k ,
[ ] (1 ) [ 1] [ 2] [k]E E E Ex k x k x k w       (5.5)
where  is a weighting factor close to 1 but smaller than 1, [k]Ew is the process noise
representing the discrepancy between the actual dynamics of [k]Ex and the 2nd
order AR
model. The z-transform from [k]Ew to [k]Ex is therefore 2
[z] (( 1)( ))E Ex w z z z    ,
which is an integrator combined with a 1st
order low-pass-filter. Another unknown in Eq.

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(5.4) is the drag coefficient , the value of which can not only indicate the environment
change such as wall-effect or temperature change but also is needed for external force
estimation since '
[ ] [ ]E E sF k x k  . The change of  is often slower, therefore, 1st
order
AR is used to model . Specifically, 1  is chosen as the state variable,
1 1
[k] [k 1] [k]w
 
   (5.6)
where [k]w is the process noise of 1  . Therefore, the state-space presentation of [k]Ex
and 1 [k] is a 3rd
order dynamics model as follow,
[ ] [ 1] [k]
[ ] 1 0 [ 1] [ ]
[ 1] 1 0 0 [ 2] 0
(1 )[ ] 0 0 1 (1 )[ 1] [ ]
E E E
E E
k k
x k x k w k
x k x k
k k w k
 
 

        
                
             
X Φ X W
(5.7)
The observation model is derived from Eq. (5.4) and is formulated as follow,
[k] [ ]
[ ]
[ ]
ˆ[ ] [ 1] 1 0 (t 1 ) [ 1] [ ]
1 [ ]m
E
m m s MT E T
O k
k
x k
x k x k F x k x k
k


 
            
  H
X
(5.8)
where [ ]kX is the state variable, Φis the state-space transition matrix, [ ]kW is the process
noise vector, [k]mO , which is the measurement increment between 1[t ,t ]k k , is the

142. 117
observation variable, [ ]kH is the observation matrix with respect to [ ]kX . Kalman filter
algorithm is selected to estimate the state variable since 1) the system subject to random
process noise [k]W and thermal noise [ ]Tx k , 2) the matrix [ ]kH is time varying therefore
linear time invariant (LTI) analysis method such as pole placement is not suitable in this
situation. However, the convergence of the Kalman Filter does require that [ ]kH satisfies
the persistent condition [80], which is naturally satisfied in this system since the magnetic
bead is constantly perturbed by the random thermal force. Therefore, the modeling force
ˆ
MTF is always changing randomly, which will naturally satisfy the persistent excitation
condition.
Similar to the Current Estimator in LTI system [81], the Kalman filter algorithm is a
two-step process, i.e. time update and measurement update. In the time update step, the
Kalman filter predicts the value of the state variables as well as their uncertainty
covariance matrix. When the new measurement information is available, the Kalman
filter gain will be updated and the state variables are corrected based on the new
measurement data. The time update and measurement update algorithm between time
interval 1[t ,t ]k k are as follow, wherein Eq. (5.7) and Eq. (5.8) are used.
Time Update:
| 1 1
| 1 1
ˆ ˆ (5.9.1)
[k] (5.9.2)
k k k
T
k k k
 
 
 

 
X ΦX
P ΦP Φ Q

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Measurement Update:
 
 
| 1
| 1
1
| 1
| 1
| 1
ˆ[k] [k] (5.10.1)
[k] [k] (5.10.2)
[k] (5.10.3)
ˆ ˆ (5.10.4)
[k] (5.10.5)
k m k k
T
k k k k
T
k k k k
k k k k k
k k k k
y O
S R
S
y






  

 



 

 
H X
H P H
K P H
X X K
P I K H P
In Eq. (5.9), 1
ˆ
kX , which is available at the beginning of 1[t ,t ]k k , is the state variable
estimation from previous step, | 1
ˆ
k kX is the state variable prediction using Eq. (5.7), [k]Q
is the process noise covariance, i.e. [k] ( [k])covQ W , | 1k kP characterizes the uncertainty
of | 1
ˆ
k kX using the uncertainty information of 1
ˆ
kX ,i.e. 1kP , and covariance [k]Q . In Eq.
(5.10), ky is called the innovation or measurement residual, kS is the innovation
covariance, kK is the Kalman filter gain, ˆ
kX is the measurement update of the state
variable, kP characterize the uncertainty of ˆ
kX . This Kalman filter algorithm can run
recursively when the new measurement information comes in.
5.2.2 Drag coefficient estimation based on the thermal variance measurement
In Eq. (5.8), the state relates to drag coefficient gamma, i.e.1  , is multiplied by
ˆ
s MTF , in which s is the sampling interval and ˆ
MTF is the magnetic force. Therefore, the
accuracy of the magnetic force is crucial for the drag coefficient estimation. Alternatively,

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the drag coefficient can also be determined according to the innovation information (Eq.
(5.10.1)) since the innovation information is the motion caused by the thermal force (Eq.
(5.8)). A recursive form for estimating the variance of [ ]Tx k in Eq. (5.8) in as follow [21],
 
2
2 2
| 1
ˆ[k] [k 1] 1 [k] [k]T T m k kO   
      H X (5.11)
Where [k]T is the thermal variance. The thermal variance is inverse proportional to
the drag coefficient of the environment,
2
2 /B s Tk T   (5.12)
The thermal force is determined by the drag coefficient, which will determine the
Brownian motion fluctuation. This is the physical reason that the drag coefficient can be
determined from the thermal variance. As long as the drag coefficient is known, the net
force and thus the external force can be known from the bead dynamics (Eq. (5.1)). The
external force can be modeled similarly as that in Chapter. 5.2.1.
However, it is worth mentioning that the measurement information in Eq. (5.8) also
contains the measurement noise. Therefore, the estimation of 2
T in Eq. (5.11) is likely
larger than the actual value of the thermal variance due to measurement noise, which will
lead to smaller estimation value of the drag coefficient according to Eq. (5.12). In dense
liquid where the thermal variance is very small, the additional measurement noise will
result in great estimation bias of the drag coefficient.

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5.3 Simulation result the joint state-parameter estimator
A simulation is performed to test the estimation algorithms in Chapter 5.2. It is
assumed in the simulation program that the magnetic force in Eq. (5.8) is accurately
known and a three-step delay exists in the control loop. The standard deviation of the
measurement noises in x,y and z directions are assumed to be 0.6nm, 0.6nm and 1.2nm.
The thermal force used in the simulation program is a white Gaussian noise, the power of
which is described by the theoretical PSD of the random thermal force [73].
5.3.1 The simulation of the drag coefficient estimation in water
To test the performance of the estimators in Chapter 5.2.1 and 5.2.2, a magnetic bead
stabilization simulation is performed wherein the bead-sample interaction force is set to
zero. The drag coefficient in the simulation changed step-wisely every 10 seconds.
Physically, the change of the drag coefficient will change the random thermal force and
thus the Brownian motion. As described in Chapter 5.2.1 or Chapter 5.2.2, the drag
coefficient can be estimated by using the magnetic force model or the thermal variance.
The performance of these two methods are shown in Fig.5.2 and Fig.5.3. It can be seen
that the drag coefficient estimation can converge to the nominal value using both
methods. The drag coefficient estimation from the thermal variance can converge to the
nominal value since the measurement noise is negligible compared with the thermal
variance. The estimated external force fluctuates around zero in both methods, which is
as expected since the bead-sample interaction force in the simulation is zero. The
parameter estimation using the thermal variance fluctuate more rapidly since this

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estimation is directly from the measurement innovation while the parameter estimation
using the magnetic force model varies slower since the parameter is lumped as a state
variable.
Fig. 5. 2. Estimation result in water using the magnetic force model to estimate the drag coefficient
Fig. 5. 3. Estimation result in water using the thermal variance to estimate the drag coefficient
5.3.2 The simulation of the drag coefficient estimation in Glycerol
It can be seen that in Chapter 5.3.1 that the drag coefficient estimation in water can
converge to the nominal value using both methods. However, it is mentioned in Chapter
5.2.2 that the thermal variance estimation will be greatly biased if the measurement noise
is comparable to the thermal motion. In dense liquid, such as glycerol, the motion caused

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by the thermal force is comparable to the measurement noise. It is therefore expected that
the estimated drag coefficient is smaller than the nominal value since the estimated
thermal variance is larger than the nominal value due to the measurement noise. Similar
simulation as that in Chapter 5.3.1 is performed. It is assumed that the bead is stabilized
in glycerol, the drag coefficient of which is two orders larger than that in water. The drag
coefficient estimation using the magnetic force can converge to the nominal value as
shown in Fig.5.4. When using the thermal variance, however, the estimated drag
coefficient value is much smaller than the nominal value due to the measurement noise,
which can be seen in Fig.5.5. To further confirm the effect of the measurement noise,
another simulation is performed wherein the measurement noise is set to zero and it can
be seen that the drag coefficient can converge to the nominal value in Fig.5.6.
In real situations, the measurement noise can already achieve sub-nm resolution,
which is still not enough for the thermal variance estimation if the magnetic bead is
stabilized in a dense liquid such as glycerol.
Fig. 5. 4. Estimation result in glycerol using the magnetic force model to estimate the drag coefficient

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Fig. 5. 5. Estimation result in glycerol using the thermal variance to estimate the drag coefficient
Fig. 5. 6. Estimation result in glycerol using the thermal variance to estimate the drag coefficient (no
measurement noise)
5.3.3 The simulation of the simultaneous estimation of the drag coefficient and the
external force
From the simulation result in Chapter 5.3.1 and 5.3.2 it can be seen that the estimator
in Chapter 5.2.1 is more robust to measurement noise in different liquids. This estimator
is selected in the following simulation. The purpose of this simulation is to test if the joint
state-parameter estimation algorithm can converge to the nominal value when the drag
coefficient and the bead-sample interaction force change simultaneously. In this

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simulation, the drag coefficient changes in a similar way as that in Chapter 5.3.1 and
5.3.2 and the external force applied on the bead has a step change of 1pN at 10 second. It
can be seen from Fig.5.7 that both the drag coefficient and the external force can
converge to the nominal value, which is due to that the persistent excitation condition is
satisfied. Since this observation matrix Eq. (5.8) is a time-varying matrix, theoretical
estimation bandwidth is difficult to obtain. However, using the method of evaluating 1st
order linear system, the estimation bandwidth of the external force is about 4.42Hz and
the drag coefficient estimation bandwidth is about 1Hz. The estimation bandwidth of the
drag coefficient is enough since the drag coefficient is usually slow varying. The
estimation bandwidth of the external force can be made even higher by implementing
adaptive Kalman filter [82, 83] wherein the process noise [ ]kW in Eq. (5.7) can change
adaptively. However, the process noise [ ]kW is set as a constant here.
Fig. 5. 7. Estimation result using the magnetic force model when the drag coefficient and external force are
changing simultaneously.

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5.4 Experiment results
The estimator mentioned in Chapter 5.2.1 is used here since it is more robust to
measurement noise in different liquids. The motion data of the magnetic bead in water
environment, which is shown in Fig.4.16 of Chapter 4.5.1, is used to test the joint state-
parameter estimator algorithm. The algorithm is not implemented in FPGA yet, but the
position data, Hall sensor measurement are all from experiment and the algorithm update
follow the same way in real-time implementation.
Fig. 5.8. Joint state-parameter estimation using Hall sensor based model (a) estimation result in x direction
(b) estimation result in y direction (c) estimation result in z direction (d) trajectories used to test joint state-
parameter estimation algorithm.

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The Hall sensor based magnetic force model is employed to calculate ˆ
MTF in Eq. (5.8).
The Kalman filter estimation result is shown in Fig.5.2 (a), (b) and (c), which are the
result in x, y and z direction respectively, and the bead’s motion trajectory as in Fig.4.16
is plot again in Fig.5.2 (d). The plot in Fig.5.2 (a), (b) and (c) contains the result from
different trajectories as in Fig.5.2 (d) and are concatenated into one curve. The drag
coefficient [k] , disturbance motion due to external force and [k]EF are plotted. The
nominal drag coefficient which is from PSD calibration (using the method in Chapter 3.5)
can serve as a reference to compare the accuracy of the estimation result. Physically,
bead-sample interaction force is zero when the bead is moving freely. Therefore, the
estimation result of [k]EF represents the modeling error. From Fig.5.2 (a) it can be seen
that the drag coefficient [k] in x direction has a little discrepancy from the nominal
value, which is due to the slight modeling error in x direction that can be seen from Fig.
4.17. From Fig.5.2 (b) and (c) it can be seen that the drag coefficient [k] in y and z
directions fluctuate around the nominal value, which is a natural result of the accurate
magnetic force modeling using the Hall sensor based model. The estimated external
forces [k]EF in all directions are almost always under 0.2pN, which means that the joint
state-parameter estimator is a promising method to achieve sub-pN dynamic force
sensing.
From the estimation result in Fig.5.2 it can be seen that the Hall-sensor based model
can be employed in the joint state-parameter estimator (Fig.5.1) to achieve high accuracy
dynamic force sensing. Moreover, it can also predict the drag coefficient very well, which

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will make the magnetic bead a sensor to detect the local environment change. For
comparison, the current-based model is used in the joint state-parameter estimator, and
the estimator structure is shown in Fig.5.3. The estimator is updated in the same way as
Eq. (5.9) and Eq. (5.10), and the only difference is that the modeling force ˆ
MTF in the
[k]H matrix is form the current-based magnetic force model.
Fig. 5.9. Joint state-parameter estimator (using current-based magnetic force model) running
simultaneously with feedback controller. (The meaning of all parameters is the same as that in Fig.5.1, the
only difference is that the modeling force ˆ
MTF is from current-based model instead of Hall-sensor based
model).

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Fig. 5.10. Joint state-parameter estimation, Hall sensor based force model vs. Current-based force model. (a)
Joint state-parameter estimation result in x direction. (b) Joint state-parameter estimation result in y
direction. (c) Joint state-parameter estimation result in z direction. (d) Trajectories used to test joint state-
parameter estimation algorithm.
The estimation result using current-based magnetic force model is shown in Fig.5.4.
For comparison, the estimation result using Hall sensor based model (Fig.5.2) is plotted
together. From the estimation result in Fig.5.4, it can be seen that the joint state-
parameter estimator using current-based magnetic force model has much degraded
performance. There is a large discrepancy between the estimated drag coefficient [k] and
the nominal value, especially in y and z direction. In the x direction, even though the
value of the estimation result [k] fluctuate around the nominal value due to the fact that
the current-based magnetic force fitting result in x direction is not as worse as y and z

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directions (Fig.4.19 and Fig.4.20), but the estimation error in x direction at certain
location has a large jump and the estimated external force [k]EF can be very large.
Moreover, the estimation result of [k]EF in y and z directions is not as large as x direction
simply because the estimated drag coefficient [k] is very small compared to the nominal
value since the external force is linearly dependent on [k] , i.e. '
[ ] [ ]E E sF k x k  .
It is also worth mentioning that the estimation result using the current-based model
will not be as consistent as estimation using Hall-sensor based model since the hysteresis
history can be very different in different experiments.
5.5 Conclusion
A joint state-parameter estimator is developed to dynamically estimate the bead-
sample interaction force and the drag coefficient, wherein the measurement position and
modeling force are employed to recursively estimate the desired values. The preliminary
experimental study validated the real-time estimation algorithm. Since the bead-sample
interaction force estimation is greatly determined by the magnetic force model accuracy,
the estimator using Hall sensor based model greatly outperforms that using current-based
force model. It can be clearly seen that the estimator using the Hall-sensor based model
can accurately estimate the bead-sample interaction force as well as the drag coefficient,
which can indicate the local environment change such as wall effect. The estimator based
on the current-based model, however, result in large discrepancy for drag coefficient
estimation.

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In the future, the dynamic force sensing algorithm can be implemented in FPGA to
monitor the bead-sample interaction force and drag coefficient in real-time. Moreover, by
combing the motion control capability, the force control can be developed in the future,
which will make the automatic scanning possible.

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Chapter 6: Conclusion and future works
6.1 Conclusion
In this research project, an over-actuated hexapole electromagnetic actuator is
developed, wherein a micro magnetic bead can be stably controlled by actively changing
six input currents. Stabilization, Brownian motion control, trajectory tracking, accurate
magnetic force modeling and dynamic force sensing are accomplished, which will
transform the magnetic actuator into an active scanning probe.
The magnetic actuator design and force modeling are analyzed first. Six sharp-tipped
poles, i.e. three pairs, are aligned in three orthogonal directions to generate 3D magnetic
forces. Six magnetic charge are employed to describe six magnetic poles and the
magnetic charge model are verified in FEM analysis. This magnetic charge model is then
employed to derive the magnetic force model, which can be lumped into a quadratic form
about the current. This force model describes the redundancy, position dependency and
nonlinearity. Moreover, the magnetic actuator can generate larger magnetic force
compared to most existing magnetic actuators since the poles are made of soft magnetic
material, which will lead to larger saturation limit.
The inverse model is very important for facilitating feedback linearization and real-
time feedback control. The inverse model needs to solve redundancy, nonlinearity and
position-dependency. Due to over-actuation, the inverse modeling greatly determines the
force generation capability since the inverse solution is not unique. Even though the

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redundancy/nonlinearity/position-dependency can be solved by linearizing the quadratic
form after imposing constant constraints, the constant constraints and linearization will
greatly limit the force generation capability and result in force generation error. A so-
called optimal inverse model is proposed such that the norm of the actuation current is
minimized while satisfying the desired force at different orientations at different locations.
For each orientation of the desired force, a compact form is proposed to describe six
optimal actuation currents and capture the position dependency. Experiment is conducted
to compare the performance of linear inverse model using constant constraints and the
optimal inverse model. It can be seen that the optimal inverse model will lead to smaller
current consumption and better position control in term of smaller positioning error,
smaller Brownian motion and smaller tracking error. It is also widely known that the time
delay in the feedback control loop can be harmful and reduce the stability margin.
However, the feedback control implemented in PC has limited sampling rate due to
limited computation capability and time consistency. Since the inverse model has a very
compact form for implementation, FPGA board can be used to compute the control
algorithm and coordinate input/output signals. 1606Hz sampling rate can be achieved
with 512×512 image size using FPGA while 200Hz sampling rate is used in PC. The
experimental result showed that high speed control in FPGA can greatly reduce the time
delay in the feedback control, which will greatly increase the gain margin and reduce the
Brownian motion fluctuation.
The soft magnetic material can greatly improve the force generation capability since
the saturation limit is very high. However, the drawback is that the hysteresis problem is

158. 133
obvious in such material. The hysteresis is very difficult to model since the hysteresis is
rate-dependent. Moreover, the magnetic coupling among six poles will make the
hysteresis very complicated. By introducing surface mount Hall sensors, the magnetic
field can be directly measured and it is verified that the measurement from this surface
mount Hall sensor can represent the magnetic field diverge from the tip of the magnetic
pole. A so-called Hall sensor based magnetic force model, which is based on the Hall
sensor voltage, is proposed to solve the complicated hysteresis issue. It can be clearly
seen that the hysteresis issue exist both in stabilization and tracking. Moreover, the
hysteresis is very difficult to model. Owning to the motion control capability, the Hall
sensor based magnetic force model is experimentally calibrated by steering the magnetic
bead in the 3D space. From the calibration result if can be seen that the Hall sensor based
model can accurately model/predict the magnetic force at sub-pN scale, while the
current-based model results in significant errors due to the hysteresis problem. In most
magnetic force models, the magnetic moment of the superparamagnetic bead is modeled
as a linear function of the external magnetic field. When the magnetic field is large,
however, the nonlinear magnetic saturation effect begins to emerge. An accurate Hall
sensor based model is proposed to model the bead magnetization using the Langevin
function. This model is also experimentally verified by steering the magnetic bead in 3D
space using large magnetic force.
Force sensing capability is very important to enable the magnetic actuator performing
probe scanning tasks. Specifically, dynamic force sensing capability is desired to sense
the transient bead-sample interaction force. Since the accuracy of the bead-sample

159. 134
interaction force totally depends on the magnetic force model accuracy, the accurate
force modeling is a very important achievement for force sensing capability. A joint
state-parameter estimation algorithm, which employed the measurement position and
magnetic force model, was developed to simultaneously estimate the bead-sample
interaction force and drag coefficient. The value of the drag coefficient can indicate the
local environment change, such as wall effect, cytoplasm properties and etc. A
disturbance observer is employed to estimate the motion caused by the external force. A
1st
order autoregressive (AR) model is used to estimate the drag coefficient. A Recursive
Least Square estimator is used to update the state variables. Specifically, the Kalman
filter algorithm is used since this system subjects to random noises such as thermal force
and measurement noise, and the discrete model is a time-varying process. A preliminary
experimental study is conducted to show the validity of the estimation algorithm. It can
be clearly seen that the estimator using the Hall sensor based model can accurately
estimate the state variables with very small error while the estimator using the current-
based model results in significant errors.
6.2 Future Works
There are several tasks can be worked on to complete this magnetic actuator as a
scanning probe microscopy and improve the performance of this magnetic actuator.
First, the joint-state parameter estimator algorithm need to be implemented in FPGA
to estimate and monitor the state variables in real-time. The algorithm is validated in
Chapter 5 and the state variables can converge to the nominal values. By implementing

160. 135
the estimator in FPGA, not only the real-time information of the state variables can be
known but also the force control can be accomplished by combining the motion control
capability.
Second, biological experiment can be done to study the cell properties. With motion
control and dynamic force sensing capability, the automatic scanning can be performed to
study the unknown environment, for example, intracellular topography scanning. Since
the joint state-parameter estimator can sense the drag coefficient in real time, the
magnetic bead can also be a sensor to sense the local environment change such as
cytoplasm properties.
Third, the optimal inverse model based on calibrated force model can be developed. It
can be seen from the calibrated force model that there are biases between the equivalent
magnetic charge location and the tips of the magnetic pole. The inverse model in the
project is based on the nominal force model, which can already serve for the feedback
linearization purpose. However, if high bandwidth motion control is desired, such as fast
steering, the modeling error will be the drawback of the high speed tracking.

161. 136
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