Chapter 1
Wireless Magnetic Actuation at Small
Scales
The control of magnetic nanoobjects is challenging in two aspects: firstly,
physics changes and fluid viscosity and surface effects dominate volumetric effects such as inertia. Secondly, the force generated by a magnetic pulling gradient on a magnetized object scales with the inverse of the third power of the
dimension. The following sections present the relevant aspects of fluid dynamics
followed by the fundamentals of the forces acting on a magnetic object exposed
to an external magnetic field.
1.1
Actuation Strategies at Low Reynolds Numbers
Consider an object of characteristic dimension l which is moving through a
liquid at velocity u. The viscosity of the fluid is expressed by η and the density
by δ. Introduced by Osborne Reynolds, the ratio of the inertial force to the
viscous force is called the Reynolds number and can be expressed as
Re =
u·l·δ
η
(1.1)
From the equation we can understand that for low Re we are in a world that
is either very viscous, very slow or very small. Low Re flow around a body
is referred to as creeping flow or Stokes flow. Further, the flow pattern does
not change appreciably whether it is slow or fast, and the flow is effectively
reversible. Consequently, reciprocal motion results in negligible net movement
[Abbott et al., 2009].
For a better understanding of how different physics are in this world of low
Re, consider the locomotion principles of microorganisms in nature. We distinguish between three different basic methodologies of how microorganisms swim
[Brennen and Winet, 1977; Vogel, 2013], as illustrated in Fig. 1.1. For example,
cilia are active organelles that are held perpendicular to the flow during the
power stroke and parallel to the flow during the recovery stroke. Many cilia are
used simultaneously. Another kind of active organelles are eukaryotic flagella.
They deform to create paddling motions, such as traveling waves or circular
translating movements. Bacterial (prokaryotic) flagella work differently and use
1
1
WIRELESS MAGNETIC ACTUATION AT SMALL SCALES
(a)
er s troke
pow
(b)
(c)
Figure 1.1: Locomotion of microorganisms. (a) Cilia move across the flow
during the power stroke, and fold near the body during the recovery stroke.
(b) Eukaryotic flagella create patterns such as traveling waves. (c) A molecular
motor spins a passive bacterial flagella (reprinted with permission from [Abbott
et al., 2009]).
instead a molecular motor to turn the base of a passive flagellum [Abbott et al.,
2009].
A number of bio-inspired robotic swimming methods have been published
that try to mimic the locomotion principle for an effective way of transportation. However, many of these techniques require mechatronic components that
present challenges in microfabrication and wireless power and control [Behkam
and Sitti, 2006; Kosa et al., 2007]. Almost all these methods use magnetic
fields. As discussed above, this actuation principle is attractive since no other
method offers the ability to transfer such large amounts of power wirelessly over
relatively large distances. A variety of control strategies have been proposed
for navigating wireless microrobots such as gradient magnetic fields, stick-slip
actuation based on rocking magnetic fields, or by exploiting resonating structures. Bio-inspired magnetically driven propulsion techniques of helical shaped
microagents that mimic a bacteria flagellum have been explored. A rotating
magnetic field can be used to rotate a helical propeller [Honda et al., 1996;
Zhang et al., 2009], eliminating the need to replicate a molecular motor in a
microrobot.
1.2
Magnetic Actuation of Microstructures
A controllable external pulling source is not available for microorganisms,
however we can generate such a source and utilize gradients in magnetic fields
to apply forces and torques directly on untethered microrobots; a strategy that
2
1
WIRELESS MAGNETIC ACTUATION AT SMALL SCALES
obviously could not have evolved through natural selection.
Direct-gradient propulsion is a non-contact manipulation method that can be
achieved by the use of either permanent or soft magnetic materials. In the first
case, the magnetization of the object does effectively not depend on the applied
magnetic field, and the object can be modeled as a simple magnetic dipole. The
resulting equations for the torque and the force acting on the object are easily
determined [Abbott et al., 2007].
Soft-magnetic materials result in easier fabrication methods as well as different issues in control. Additionally, soft magnetic materials can reach levels
of magnetization as high as the remanence magnetization of permanent magnets [Kasap, 2006; O’Handley, 1999; Jiles, 1998]. However, with soft magnetic
materials, the magnetization of the body is a nonlinear function of the applied
magnetic field. Hence, the relationship between the applied field and the resulting torque and force is nontrivial [Abbott et al., 2007].
1.2.1
Magnetization Model and Resulting Torque
The control of soft-magnetic beads has been widely studied [Martel et al.,
2007; Ergeneman et al., 2008; Amblard et al., 1996]. Here, a spherical shape
simplifies the control problem since there is no preferred direction of magnetization. Its symmetry results in a uniform magnetisation, and hence, it is not
possible to apply a magnetic torque. A magnetic torque can be only applied
to a sphere if it is made of a permanent magnet and hence, has a permanent
magnetization along a determined direction resulting in a permanent dipole.
A magnetic torque can only be applied to soft-magnetic bodies of nonsymmetric shape. If a randomly shaped soft-magnetic body is exposed to an
external magnetic field, the determination of the resulting magnetization vector
is nontrivial. However, it has been shown that many simple geometries can
be accurately modeled magnetically as ellipsoids [Beleggia et al., 2006]. For
example, in the case of magnetic nanowires as potential drug carriers, the magnetisation of their cylindrical body can be modelled based on its ellipsoidal
equivalent. The exact result for a sphere can be derived from the same model
for the degenerated case where all axes of the ellipsoid are equal. Hence, in the
following the ellipsoidal case is treated specifically.
As illustrated in Fig. 1.2, the body coordinate frame is located at the center
of mass and the x-axis is aligned with the axis of symmetry. The body is
magnetized to a volume magnetization M in units ampere per meter [A/m ] by
an external magnetic field H in [A/m ] at the bodys center of mass. Note that
in this manuscript M will always refer to the volume magnetisation and not to
the mass magnetization in units [Am2 /kg ], if not stated differently. Because of
3
1
WIRELESS MAGNETIC ACTUATION AT SMALL SCALES
Figure 1.2: Axially symmetric bodies in an external magnetic field. The xaxis of the body frame is aligned with the axis of symmetry. The field H, the
magnetization M, and the axis of symmetry are coplanar. θ ∈ [0, 90] is the
angle between H and the axis of symmetry, and φ ∈ [0, 90] is the angle between
M and the axis of symmetry (from [Abbott et al., 2007]).
the symmetry of the body, the field H, the magnetization B, and the axis of
symmetry are coplanar. The applied magnetic field can also be expressed as an
applied magnetic flux density H with units tesla [T]. They are related as B=
µ0 H, where µ0 = 4π × 10−7 Tm/A is the permeability of free space [Abbott
et al., 2007]. At low applied fields, the magnetization grows linearly with the
field until it reaches a saturation magnitude. As the field strength increases, the
constant-magnitude saturated magnetization vector rotates toward the applied
field.
Consider first the linear magnetization region for relatively low applied fields.
As stated in [Abbott et al., 2007], the magnetization is related to the internal
field by the susceptibility of the material χ as
M = χHi
(1.2)
The internal field Hi can be described as linear superposition of the the
applied field H and a demagnetizing field Hd
Hi = H + Hd
(1.3)
The demagnetizing field Hd , is related to the magnetization by a tensor N
of demagnetization factors based on the body geometry as Hd = −N M. The
matrix N is diagonal if the body coordinate frame is defined such that it aligns
with the principle axes of the body: N = diag(nx , ny , ny ), with the demagnetization factors nx , ny , ny along x, y, and z. Combining the earlier assumptions,
we can relate the magnetization to the applied field by an susceptibility tensor
M = χa · H
(1.4)
with a tensor of the form
4
1
WIRELESS MAGNETIC ACTUATION AT SMALL SCALES
χa = diag
χ
χ
χ
,
,
(1 + nx χ 1 + ny χ 1 + nz χ
(1.5)
where M, H and χa are all written with respect to the body frame. Because
of the symmetry of the elliptical body, one needs only to consider two demagnetization factors - the factor along the axis of symmetry na , and the factor in
all radial directions perpendicular to the axis of symmetry nr . For relatively
large susceptibility values, as they are typical for soft magnetic materials on the
order of 103 to 106 , and if we assume the demagnetization factors are not too
close to zero, equation (1.5) can be approximated with
χa = diag
1 1 1
, ,
na nr nr
(1.6)
From 1.5 it can be seen that magnetization is insensitive to changes in susceptibility if the susceptibility is relatively high. And in turn, the susceptibility
is dominated by the body geometry, since it is determined by the demagnetization factor, which is a geometry-dependent value. The demagnetization factors
for general ellipsoid bodies are part of the well-understood results of magnetostatics and were first computed in 1945 [Osborn, 1945]. Generally, they are
related by the constraint nx + ny + nz = 1. This can be rewritten for an axially
symmetric body as na + 2nr = 1. To understand the influence of the geometry
on the magnetization vector, we need to look at the magnetization angle φ that
describes the offset between the magnetization vector and the body symmetry
axis. The longer the axis of the body at constant radial dimension, the smaller
the angle will become, since the demagnetization factor na along the body’s
symmetry axis will get smaller and, hence, the magnetization along this axis
stronger. This can be understood from the following expressions
1
na = 2
R −1
R2
na = 2
R −1
R
√
ln
2 R2 − 1
1− √
1
R2 − 1
!
!
√
R + R2 − 1
√
− 1 prolate ellipsoid
R − R2 − 1
√
−1
sin
R2 − 1
R
(1.7)
!!
oblate ellipsoid
(1.8)
where R ≥ 1 is the ratio of the long and short dimensions of the body. We
can compute the magnetization angle φ directly, assuming 1.5, as
−1
φ = tan
na
nr tan θ
(1.9)
Thus, the larger R gets, the smaller na , and hence the smaller the angle φ.
The offset angle between φ and θ determines how much the body is turning in
5
1
WIRELESS MAGNETIC ACTUATION AT SMALL SCALES
order to align its magnetization axis with the vector of the applied field which
is known as the torque. For a uniform magnetization throughout the body, it is
assumed the volume contributes linearly to the torque and force that the body
experiences in an applied external field. Hence, we can express the magnetic
torque acting on the body in an applied external field as
T = υM × B
(1.10)
where υ is the volume of magnetic material in [ m3 ] [Furlani, 2001]. The
torque, in Newton meters [Nm], acts on the magnetic moment of the body such
that M aligns with the direction of the magnetic field. A soft magnetic material
in a magnetic field will become magnetized along its long axis, also called its
easy axis and the torque tends to align the longest axis of the body with the
field.
From (1.9) we can already see that for a soft magnetic sphere, where nx =
ny = nz or na /nr = 1 applies, there will be no torque acting on the body,
since the magnetization vector and the field vector are always aligned which
each other. This assumes the absence of any remanent magnetization. However, when taking hysteresis effects into account, the magnetization along the
field vector will always be dominant and almost no rotational movement can
be translated. This is why, if rotary motion of magnetic beads is required, a
permanent magnetic material is generally used.
If the magnetization vector computed in (1.4) results for a certain applied
field H in a smaller magnitude than the saturation magnetization ms of the
material, then we take M and φ as accurate. However, for |M| > ms , we move
into the saturated magnetization region. We set |M| = ms and compute the
rotation of M by minimizing the magnetic energy with respect to
e=
1
µ0 υ (nr − na ) ms 2 sin2 φ − µ0 υms |H| cos(θ − φ)
2
(1.11)
where e is the energy in units of Joule. This equation is typically applied as
a model for single-magnetic-domain samples, but it is a good approximation of
a multidomain body once saturation has been reached. To minimize e in 1.11,
we get the derivative of e and equalize it to zero
(nr − na )ms sin(2φ) = 2|H| sin(θ − φ)
(1.12)
M will rotate such that φ satisfies (1.11). Within that model the magnetization
vector M changes continually with changes in the applied field H as proven in
[Abbott et al., 2007].
6
1
WIRELESS MAGNETIC ACTUATION AT SMALL SCALES
!
B
!
y
M
!
!
r
x
L
Figure 1.3: Magnetization model and magnetic torque and force acting on a
cylindrical body in an externally applied magnetic field. The cylindrical body
is approximated as an ellipsoid on which the shape dependent demagnetization
factors can be computed.
Due to the dominant magnetic shape anisotropy, it is in general more appropriate to use bodies with a preferred magnetization axis, the easy axis. For example, this applies for a cylindrical nanowire. We can apply the same constraints
to any shape by equating them to ellipsoids. Improving these approximations
has been the object of numerous studies, especially for cylinders [Joseph, 1966;
Chen et al., 1991]. From [Beleggia et al., 2006] we can extract the demagnetization factor along a cylinder axis based on the model of an equivalent ellipsoid.
For a cylinder of length l and radius r with aspect ratio τ = l/2r the following
expression for the demagnetization factor along the cylinders axis applies:
na cyl (τ ) = 1 +
1
4
− F − 2)
3πτ
τ
(1.13)
with F (x) ≡2 F1 [− 12 , 12 , 2, x], which is defined by the Gaussian hypergeometric function 2 F1 (a, b, c, z).
We consider a Ni NW as a soft-magnetic cylindrical object with length l =
1 µm and radius r = 50 nm as shown in Fig. 1.3. Computing (1.13) with these
values, where a piecewise polynomial approximation of 2 F1 is created, we get
na = 0.0412, and therefore n=
r 0.4794, and the cylinder long axis is the easy axis.
For precise and fast alignment performance, we try to maximize the magnetic
torque T in (1.10) and look for the external field needed to fully magnetize the
body for a maximum M. For a soft magnetic body this means reaching the
saturation magnetization ms . The resulting magnetization dependent on the
externally applied magnetic field is given by combining (1.3) and (1.4) as
M=
χ
H
1 + χN
which can be approximated as M =
1
NH
(1.14)
for high χ, as for nickel (χ =
7
1
WIRELESS MAGNETIC ACTUATION AT SMALL SCALES
!
!
!
M
M
!
!
!
!
B
FF
(a) Magnetic pulling along the long axis
!
!
F
!
B
(b) Magnetic pulling along the long axis
Figure 1.4: A cylindrical, magnetised body aligned with the external field is
pulled along and perpendicular to its axis through a fluid. a) A force along the
magnetization axis is applied, depicted as an increase of the norm of the flux
density vectors towards increasing y at points in the x − y-plane. b) A magnetic
pulling force acting perpendicular to the magnetization axis is depicted. The
norm of the flux density vector increases with an increasing x-coordinate.
µr − 1 = 599).
If we apply a field parallel to the body’s long axis, which we set to be coplanar
with the x-axis of the coordinate frame, and we set |M| = ms for the desired
reach of the saturation magnetization, we get |Hsat | = na · ms for the field
required to fully magnetize the body. This, however, involves the assumption
that the body is fully, uniformly magnetized along its long axis.
1.2.2
Pulling Through Fluids by Applied Magnetic Gradients
As discussed, when a magnetic body is subjected to an externally applied
magnetic field, it will experience a torque aligning its long axis with the direction
of the applied field. Let us set the field vector coplanar with the x-axis of the
coordinate frame. Now, if a magnetic gradient is applied coplanar with the
x-axis, the object will experience a force and is pulled along its long axis (see
Fig. 1.4(a)). More generally, the magnetic force that body experiences can be
expressed as
F = υ (M · ∇) B.
(1.15)
Further, pulling the object perpendicular to its easy axis, as depicted in
Fig.1.4(b), at the same magnetic field magnitude, hence, same magnetic moment, and same magnetic field gradient, the magnetic force will be the same,
but the resulting velocity will be smaller. This arises from the higher drag that
8
1
WIRELESS MAGNETIC ACTUATION AT SMALL SCALES
2r
y
II
F drag, ||
u
u
F ext
2l
θ
F drag,
x
u
(a)
(b)
Figure 1.5: Drag on a slender body. (a) The drag force perpendicular is approximately twice the drag force parallel to the slender axis. (b) An external
force is applied along y at an angle θ to the slender body axis, which results in
a velocity with components both in x and y.
this configuration experiences. When pulling a magnetic object through Newtonian fluid at low Re, the object nearly instantaneously reaches its terminal
velocity u where the viscous drag force, which is linearly related to velocity
through a drag coefficient, exactly balances the applied magnetic force F.
Assuming no electric current is flowing through the body region, Maxwell’s
equations require that ∇ × B = 0 and, after rearranging (1.15), the force can
be expressed as:

F=υ
∂B
∂x
∂B
∂y
∂B
∂z
T



·M = 


∂Bx
∂x
∂By
∂x
∂Bz
∂x
∂Bx
∂y
∂By
∂y
∂Bz
∂y
∂Bx
∂z
∂By
∂z
∂Bz
∂z
T



Mx




 ·  My 


Mz
(1.16)
An object moving with a velocity u encounters a drag force Fdrag = −D · u.
If an external force is applied to the object, it reaches its steady-state motion
almost instantaneously and the velocity can be estimated from
Fext + Fdrag = 0
(1.17)
For a spherical body the drag coefficient can be described as
Dsphere = 6πrηu
(1.18)
For slender bodies resistive force coefficients (RFC) can be used [Cox, 1970],
9
1
WIRELESS MAGNETIC ACTUATION AT SMALL SCALES
(a) Example of a Helmholtz coil configura- (b) A single
tion
magnetic core
electromagnet
with
soft-
Figure 1.6: Magnetic field generation
and the drag of motion perpendicular to the slender axis of the object is approximately twice the drag of a motion along the slender axis (see Fig. 1.5a)
Dcylk =
4πηl
ln (2l/r) − 0.807
Dcyl⊥ =
8πηl
ln (2l/r) + 0.193
(1.19)
Thus, if an external force is applied to the slender body at an angle θ, the
resulting velocity is not parallel to the applied force due to drag anisotropy (see
Fig. 1.5b).
1.3
Generating Magnetic Fields
Controlled magnetic fields can be generated by stationary current-controlled
electromagnets [Zhang et al., 2009], as shown in Fig. 1.6 b), by electromagnets that are position and current controlled [Yesin et al., 2006], by positioncontrolled permanent magnets, or even by a commercial magnetic resonance
imaging (MRI) systems [Mathieu et al., 2006]. In any case, the rapid decay of
magnetic field strength with distance from its source creates a major challenge
for magnetic control.
Permanent magnets generate strong fields on a per volume basis, however
obstacles, such as the interaction between adjacent magnets and the requirement
of a certain shielding mechanism in order to turn off the magnetic field must
be considered. In order to control the field strength the magnets need to be
externally actuated. In comparison, air-core solenoids have rather weak fields
compared to permanent magnets, but field contributions superpose linearly, so
current controls the field strength and the field can be switched off instantly.
A common approach to the generation of homogenous fields is the use of
Helmholtz coil configurations. By positioning three pairs orthogonally to each
10
1
WIRELESS MAGNETIC ACTUATION AT SMALL SCALES
other a system with three degrees of freedom (3-DOF) is achieved. The magnetic propulsion methods discussed above use Helmholtz coils, Maxwell coils,
or various combinations thereof, that entirely surround the workspace to generate the desired field strength and orientation. Fig. 1.6 b) shows a typical
Helmholtz configuration. In such a 3 DOF system, the direction of the magnetic field and the magnetic gradient are dependent on each other, meaning that
a non-spherical object cannot be freely navigated. This system is consequently
non-holonomic, i.e. constraints exist that prevent the object from moving instantaneously in some directions.
1.3.1
Magnetic Fields and Gradients Generated by Solenoids
Consider a single current loop l at a distance r/2 from the origin of the
axis. We can express the magnetic field Bxl along its axis (off-axis field is not
considered) generated by a current is flowing through the conductor. From the
Biot-Savart-rule we get
Bxl (x, is ) = µ0
is r2
2r2 +
r
2
2
− x )3/2
(1.20)
where x is the distance along the axis of the coil from its center, and r is
the radius of the current loop. For generating higher fields, we can extend the
current loop to a solenoid s consisting of n wire turns. A total current I = n · is
then flows through the coil and the generated field can be expressed as
is r2
Bxs (x, is ) = µ0 n
2(
r2
+
r
2
−x
2 3/2
(1.21)
To generate a homogenous field over a larger volume of space, another
solenoid is added at a distance r from the first solenoid plane, at the position
x = −r/2, as shown in Fig. 1.7.
We can linearly superpose the two fields generated by the same current
flowing through both coils as
=
µ0 n
2
Bx (x, is ) = Bx1 (x, is ) + Bx2 (x, is )
!
is r2
is r 2
3/2 +
3/2
r2 + ( 2r − x)2
r2 + ( 2r + x)2
(1.22)
At the center point of the two coils the axial component of the field in
x-direction is
11
1
WIRELESS MAGNETIC ACTUATION AT SMALL SCALES
x
r
x = r/2
is
x=0
r/2
r
x = -r/2
is
Figure 1.7: Electromagnetic coils in Helmholtz configuration as field-generation
hardware for microrobot propulsion. Uniform magnetic fields in the center of
the workspace are generated by a current flowing in the same direction.
Bx (0, is ) =
3/2
4
µ0 nis
.
5
r
(1.23)
Note that this is an accurate expression for on-axis field computation and
does not cover the off-axis field. The off-axis field calculation is non-trivial
and in most cases there is no closed-form analytical solution. It is reasonable,
though, to assume a homogenous field for a certain small workspace around
the center point at x = 0. For Helmholtz coils this can usually be stated for
a volume around the center point as large as 5% of the distance between the
coil planes. Another assumption is that the thickness of the wire is considered
infinitesimally small and, hence, the coil is considered a plane with no expansion
along the x-axis. For long solenoids with many wire turns or with thick wire,
the expression in (1.22) must be integrated over the length a of the coil from
x = r/2tox = r/2 + a.
In order to obtain 3-DOF control, we add two more pairs of coils in an
orthogonal arrangement as previously mentioned and shown in Fig. 1.6 a). We
can now build up the field vector B(P) = [Bx (P), By (P), Bz (P)]T at a point
P = [x, y, z]T of the workspace, consisting of the three axial contributions of
each coil pair.
To generate a gradient an additional coil pair for each axis can be added.
The current in the gradient pair is of the same magnitude, but opposite direction
(Fig. 1.8).
12
1
WIRELESS MAGNETIC ACTUATION AT SMALL SCALES
x
r
x = r/2
is
x=0
r/2
r
x = -r/2
is
Figure 1.8: Maxwell coil configuration. Gradient fields can be generated by
running the current in opposite directions.
The magnitude of the field gradient at a certain point along the axis can
0
be obtained by the derivative of Bx (x, is ) = Bx1 (x, is ) − Bx2 (x, is ), where the
contributions of the two coils are subtracted according to the inverse current
flow. The system consists of 12 solenoids but only 6 current inputs.
An alternative is to work with a single coil pair in each direction for both
the generation of field and gradient, resulting in a system of 6 solenoids but
controlling each current independently. In other words, a gradient can also be
generated with a classic Helmholtz configuration by applying a higher current
in one coil of the pair than in the other, with the result of a constant gradient
and homogenous field in the center of the two coils. Depending on the design
constraints other variations, such as Maxwell coils, can be employed through a
similar derivation.
1.3.2
General Control System
Within a given set of air-core electromagnets, each electromagnet creates a
magnetic field throughout the workspace that can be computed for any point P
of the workspace and any given electromagnet e. We can express the magnetic
field due to this electromagnet by the vector Be (P), whose magnitude varies
linearly with the current through the electromagnet and can be described as a
unit-current vector in units T/A multiplied by a scalar current value in units
A.
Be (P) = B̃e (P)ie
(1.24)
13
1
WIRELESS MAGNETIC ACTUATION AT SMALL SCALES
With soft-magnetic cores, the field induced is related to the current running
through an individual electromagnet as well as the cores of the other electromagnets in the system.
If a material that resembles an ideal soft-magnetic material with negligible
hysteresis is used, and the cores are kept within their linear magnetization regions, we can assume that for n electromagnets the field contributions of the n
individual currents superimpose linearly. This can be denoted in matrix form
by the 3 × n unit-field contribution matrix B(P):

B(P) =
h

i1
i
.. 

B̃n (P) 
 .  = B(P)I
in
B̃1 (P) · · ·
(1.25)
It is likewise possible to express the derivative of the field in a given direction
in a specific frame, for example the x direction, as the contributions from each
of the currents:

∂B(P)
=
∂x
∂ B̃1 (P)
∂x
···
∂ B̃n (P)
∂x

i1
 . 
 .  = Bx (P)I
 . 
in
(1.26)
The controllers presented here are based on setting a target flux density,
orientation, and unit force. In this case, these equations can be combined as:

"
B
Funit
#

B(P)

 M̂T Bx (P)
=
 M̂T B (P)
y

M̂T Bz (P)







i1
.. 

.  = A(B̂, P)I
in
(1.27)
By introducing the normalised magnetic moment M̂, the dependancy on
volume and material properties of the particular object has been here removed
resulting. This results in unit force equations of a body in a desired magnetic
field, which can then be readily scaled to the desired body.
The n electromagnet currents are mapped to a desired field and gradient
through a 6 × n actuation matrix A(B̂, P) which depends on the orientation of
the magnetic field and the set point. For a desired target vector, the choice of
currents that best solves for the target can be found using the pseudo-inverse:
"
I = A(B̂, P)
†
B
Funit
#
(1.28)
14
1
WIRELESS MAGNETIC ACTUATION AT SMALL SCALES
This derivation can be similarly extended for controllers that require torque
and/or force control as opposed to the field and unit force control and is discussed in full in [Kummer et al., 2010]. If there are multiple solutions to achieve
the desired field and gradient, the pseudoinverse finds the solution that minimizes the two-norm of the current vector, which is desirable for the minimization
of both power consumption and heat generation. The pseudoinverse of A is of
rank 5 corresponding to the no-torque generation about the magnetization axis,
which is never possible. More details of this can be found in [Kummer et al.,
2010].
In order now to use (1.28), a unit-current field map must be constructed
for each of the electromagnets. This can be done by analytical models, finiteelement-method (FEM) data or system measurements.
15
References
[Abbott et al., 2007] Jake J Abbott, OlgaÇ Ergeneman, Michael P Kummer,
Ann M Hirt, and Bradley J Nelson. Modeling Magnetic Torque and Force
for Controlled Manipulation of Soft-Magnetic Bodies. IEEE Transactions on
Robotics, 23(6):1247–1252, 2007.
[Abbott et al., 2009] J J Abbott, K. E Peyer, M C Lagomarsino, L Zhang,
L Dong, I K Kaliakatsos, and B. J Nelson. How Should Microrobots Swim?
The International Journal of Robotics Research, 28(11-12):1434–1447, October 2009.
[Amblard et al., 1996] F Amblard, B Yurke, A Pargellis, and S Leibler. A magnetic manipulator for studying local rheology and micromechanical properties
of biological systems. Review of Scientific Instruments, 67(3):818–827, March
1996.
[Behkam and Sitti, 2006] B Behkam and M Sitti.
Design methodology for
biomimetic propulsion of miniature swimming robots. Journal of Dynamic
Systems Measurement and Control-Transactions of the Asme, 128(1):36–43,
March 2006.
[Beleggia et al., 2006] M Beleggia, M De Graef, and Y T Millev. The equivalent
ellipsoid of a magnetized body. J. Phys. D.: Appl. Phys., 39(5):891–899,
February 2006.
[Brennen and Winet, 1977] C Brennen and H Winet.
Fluid-Mechanics of
Propulsion by Cilia and Flagella. Annual Review of Fluid Mechanics, 9:339–
398, 1977.
[Chen et al., 1991] D X Chen, J A Brug, and R B Goldfarb. Demagnetizing
Factors for Cylinders. IEEE Transactions on Magnetics, 27(4):3601–3619,
July 1991.
[Cox, 1970] R G Cox. The motion of long slender bodies in a viscous fluid. Part
1. General theory. Journal of Fluid Mechanics, 44(part 3):790–810, January
1970.
[Ergeneman et al., 2008] OlgaÇ Ergeneman, Gorkem Dogangil, Michael P
Kummer, Jake J Abbott, Mohammad K Nazeeruddin, and Bradley J Nelson.
16
REFERENCES
A Magnetically Controlled Wireless Optical Oxygen Sensor for Intraocular
Measurements. IEEE Sensors Journal, 8(1-2):29–37, January 2008.
[Furlani, 2001] Edward P Furlani. Permanent Magnet and Electromechanical
Devices. Materials, Analysis, and Applications. Academic Press, August 2001.
[Honda et al., 1996] T Honda, K I Arai, and K Ishiyama. Micro swimming
mechanisms propelled by external magnetic fields. IEEE Transactions on
Magnetics, 32(5):5085–5087, September 1996.
[Jiles, 1998] David Jiles. Introduction to magnetism and magnetic materials,
2nd ed. CRC PressI Llc, 1998.
[Joseph, 1966] R I Joseph. Ballistic Demagnetizing Factor in Uniformly Magnetized Cylinders. Journal of Applied Physics, 37(13):4635–4639, 1966.
[Kasap, 2006] Safa O Kasap. Principles of electronic materials and devices.
McGraw-Hill Science/Engineering/Math, January 2006.
[Kosa et al., 2007] G Kosa, M Shoham, and M Zaaroor. Propulsion Method
for Swimming Microrobots. IEEE Transactions on Robotics, 23(1):137–150,
2007.
[Kummer et al., 2010] Michael P Kummer, Jake J Abbott, Bradley E Kratochvil, Ruedi Borer, Ali Sengul, and Bradley J Nelson. OctoMag : An Electromagnetic System for 5-DOF Wireless Micromanipulation. IEEE Transactions on Robotics, 26(6):1006–1017, October 2010.
[Martel et al., 2007] Sylvain Martel, Jean-Baptiste Mathieu, Ouajdi Felfoul,
Arnaud Chanu, Eric Aboussouan, Samer Tamaz, and Pierre Pouponneau.
Automatic navigation of an untethered device in the artery of a living animal
using a conventional clinical magnetic resonance imaging system. Applied
Physics Letters, 90(11):114105, 2007.
[Mathieu et al., 2006] Jean-Baptiste Mathieu, Gilles Beaudoin, and Sylvain
Martel. Method of Propulsion of a Ferromagnetic Core in the Cardiovascular System Through Magnetic Gradients Generated by an MRI System.
IEEE Transactions on Biomedical Engineering, 53(2):292–299, 2006.
[O’Handley, 1999] Robert C O’Handley. Modern Magnetic Materials. Principles
and Applications. Wiley-Interscience, November 1999.
[Osborn, 1945] J A Osborn. Demagnetizing factors of the general ellipsoid.
Physical Review, 67(11-12):351–357, 1945.
17
REFERENCES
[Vogel, 2013] Steven Vogel. Comparative Biomechanics. Life’s Physical World
(Second Edition). Princeton University Press, May 2013.
[Yesin et al., 2006] K B Yesin, K Vollmers, and B. J Nelson. Modeling and
control of untethered bio-microrobots in a fluidic environment using electromagnetic fields. Int. J. Robot. Res, 25:527–536, 2006.
[Zhang et al., 2009] Li Zhang, Jake J Abbott, Lixin Dong, Kathrin E Peyer,
Bradley E Kratochvil, Haixin Zhang, Christos Bergeles, and Bradley J Nelson.
Characterizing the Swimming Properties of Artificial Bacterial Flagella. Nano
Letters, 9(10):3663–3667, October 2009.
18