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IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED MARCH, 2024. 1
A Cylindrical Halbach Array Magnetic Actuation
System for Longitudinal Robot Actuation across 2D
Workplane
Hongzhe Sun1and Shing Shin Cheng1,2∗
Abstract—Magnetic actuation has been widely investigated for
miniature robot control due to its wireless control capability.
As a permanent magnetic (PM) actuation system, the Halbach
array can provide strong and controllable magnetic fields with
large motion workspace. However, existing cylindrical Halbach
array systems can only generate axial force along its central
axis and require the workspace (i.e., patient anatomy) to be
manipulated inside the system for any useful robot manipulation,
severely limiting their application for robotic surgery. In this
work, we introduce a cylindrical Halbach array actuation system
capable of generating a magnetic field with longitudinal gradients
across a 2-dimensional (2D) workplane instead of only along the
central axis, effectively extending the longitudinal force actuation
coverage from 1D to a 2D plane. This is achieved by optimizing
the magnet sizes and roll angles of the Halbach rows arranged
circumferentially around the system. Co-alignment between the
field and gradient directions is also achieved through proper
configuration of the magnet pitch angles along each Halbach
row, resulting in tip-leading robot motion capability. A series of
model-based simulations were performed during the optimization
process and later verified experimentally. The actuation system
was experimentally demonstrated to stably drive a 2 mm diameter
magnetic robot longitudinally at different locations within the
workplane and at different velocities. This represents a signifi-
cant advancement towards deploying cylindrical Halbach array
systems for robot manipulation in clinical cases.
Index Terms—Medical Robots and Systems, Automation at
Micro-Nano Scales, Magnetic Actuation System, Halbach Array
I. Introduction
AS minimally invasive surgery becomes more widely
adopted in procedures traditionally performed via the
open approach, miniature robots ranging from robotic nee-
dles, catheters, guidewires, capsules, and endoscopic instru-
ments have been explored extensively. These robots can
be tele-operated to perform deep lesion targeting via 1-
degree of freedom (DOF) straight needle insertion, complex
Manuscript received November 13, 2023; Revised January 11, 2024;
Accepted March 7, 2024.
This paper was recommended for publication byEditor Jessica Burgner-Kahrs
upon evaluation of the Associate Editor and Reviewers’ comments. This work
was supported in part by Research Grants Council (RGC) of Hong Kong (GRF
14217822, GRF 14207823), in part by Multi-scale Medical Robotics Center,
InnoHK, Hong Kong, and in part by The Chinese University of Hong Kong
(CUHK) Direct Grant. The content is solely the responsibility of the authors
and does not necessarily represent the official views of the sponsors.
1The authors are with the Department of Mechanical and Automation
Engineering and T Stone Robotics Institute, The Chinese University of Hong
Kong (CUHK), Shatin, N.T., Hong Kong.
2The author is with the Shun Hing Institute of Advanced Engineering and
Multi-Scale Medical Robotics Center, CUHK Shatin, N.T., Hong Kong (e-mail:
sscheng@cuhk.edu.hk).
Digital Object Identifier (DOI): see top of this page.
catheter/endoscope navigation in tortuous anatomical struc-
tures, and tissue manipulation in keyhole procedures. While
most robots are tethered to their typically bulky actuation
and controller system, remote actuation via magnetic field
application has been gaining interest due to its ability to deploy
robots [1], [2] in a fully untethered, wireless manner, enabling
significant downscaling of the robot size and potentially more
complex navigation, manipulation, and other functional capa-
bilities deep in the human body. Electromagnetic coils and
permanent magnets in various configurations are currently the
main magnetic actuation systems (MASs) for these robots.
Electromagnetic systems (EMS) can be divided into dis-
tributed electromagnets [3], [4] and paired coils (e.g.,
Helmholtz and Maxwell coils) [5], which can generate gradient
fields and homogeneous fields, respectively [6]. EMS provides
current-dependent magnetic field with precise force and torque
control, and can be turned off, offering high safety [7].
Permanent magnetic systems (PMS) are typically configured
as mobile systems to manipulate one or more permanent
magnet(s) for robot actuation [8]. There are also stationary
PMS whereby PMs are arranged in pairs or into an array to
superimpose the fixed and unalterable field of each magnet to
provide stronger field [9], [10]. Multiple distributed rotating
magnets have been explored to generate magnetic fields and
forces in any direction [11], but similar to the EMS, the
system’s space utilization rate (workspace-to-system size ratio)
and the magnetic flux density remain relatively small, limiting
its potential for practical clinical application.
Halbach array is a PM configuration that has been inves-
tigated for portable magnetic resonance imaging (MRI) [12],
magnetic particle imaging (MPI) [13], [14], and planar actu-
ation of the magnetic particles [15], [16]. In a linear Halbach
array, PMs are arranged with different rotation angles and
distances to generate a larger and adjustable superimposed
field on one side and offset the field on the other [17].
A circular Halbach array [18] can be configured as dipoles
and quadrupoles to generate homogeneous and gradient field
in the radial direction, respectively. The field can be ad-
justed by combining multiple concentric layers and changing
their rotation angles [19]. Among various configurations of
PMS, the Halbach array affords higher magnet density in
terms of space occupation, thus producing a stronger field
and larger workspace-to-size ratio. Recently a few Halbach-
inspired magnetic actuation systems have been developed to
generate magnetic fields inside a cylindrical space (instead
of within a 2D circular plane) with the ultimate objective of
This article has been accepted for publication in IEEE Robotics and Automation Letters. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/LRA.2024.3382485
© 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
Authorized licensed use limited to: Chinese University of Hong Kong. Downloaded on March 28,2024 at 13:51:26 UTC from IEEE Xplore. Restrictions apply.
2 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED MARCH, 2024.
navigating robots in a large 3D volume and therefore within the
human body. Two dipole circular Halbach arrays of opposite
field directions were arranged co-axially at a certain distance
apart to generate a gradient along the central axis of the virtual
cylinder formed by the two circular arrays [20]. Another work
uses many straight Halbach rows arranged circumferentially
to form a cylinder, producing a high bidirectional gradient
along the central axis for navigating a robot in soft tissue [21].
The existing cylindrical Halbach arrays are however limited to
producing longitudinal magnetic gradients and thus exerting
longitudinal force on a magnetic robot along only the central
axis of the cylinder. Beyond the central axis, the gradient
directions become more deflective as the field eccentricity
increases, leading to the inability to properly control the robot
motion along any specific direction. Besides, most Halbach
systems cannot enable tip-leading motion in two opposite
directions, important for the manipulation of cylindrical or
tubular robots with distinct polarities at head (or tip) and tail.
In this work, a new cylindrical Halbach array is designed
to produce high longitudinal gradients within a plane instead
of only along the central axis of the cylinder, expanding the
longitudinal robot motion workspace from a single line to a
2D workplane (defined as the plane in which the robot can be
controlled for longitudinal motion), as shown in Fig. 1. From
the clinical perspective, unlike the previous Halbach system
with actuation capability along only the central axis [21], our
proposed actuation system can exert longitudinal force on a
magnetic robot (e.g., a PM-equipped needle or a magnetic
capsule) at any location within the 2D workplane without
manipulating the patient anatomy in which the robot is placed,
thus enabling true clinical implementation in needle-based
procedures, such as deep lesion targeting via a straight in-
sertion trajectory once the predetermined insertion trajectory
is aligned with the workplane of the actuation system. The
co-alignment of the gradient directions and magnetic field
directions is also proposed for the first time in a cylindrical
Halbach array to allow tip-leading motion of magnetic robots
during their bidirectional longitudinal movement, fundamen-
tally enabling motion capabilities such as follow-the-leader
motion in magnetic continuum robots. The main technical
contributions include: (1) design and optimization of a new
cylindrical Halbach array to achieve longitudinal gradients
across an entire workplane and enable bidirectional tip-leading
motion of magnetic robots. (2) Simulation and experimental
validation of the proposed magnetic actuation system.
The remaining part of the paper is organized as follows.
Section II introduces the system design requirements and
description of the new cylindrical Halbach array. Section III
establishes the mathematical model of the actuation system
and optimizes its parameters to minimize the lateral gradients
in the workplane. A series of magnetic field simulations
are provided in Section IV. Experimental verification with a
physical prototype and discussions are presented in Section V.
Finally, concluding remarks are provided in Section VI.
II. System Design
It is envisioned that a cylindrical Halbach array actuation
system will be capable of actuating a magnetic robot inside the
(a)
x
j
y
j
P
j
(b)
θ
i
φ
i
x
y
zRotation axis
R
+y
+y
Pole (m
i
)
+x
d
i
O
x
zy
Halbach row
Row number
nth magnet
in a row
O
Pole
φ
i
Fig. 1. (a) Proposed optimized cylindrical Halbach array with the motion
workplane of a magnetic robot; (b) Coordinate diagram showing the roll angle
𝜃and pitch angle 𝜑of a single magnet 𝑖and the measurement point 𝑃𝑗on the
workplane.
human head while providing high space utilization rate, and
large magnetic field and gradient. In this work, we focus on
expanding the longitudinal motion actuation capability from
along only the central axis to an entire workplane with a
downscaled prototype, A few design requirements have thus
been identified for such a system as follows. (1) The accessible
space within the cylindrical array should be at least one-
third the size of a human head, and can accommodate small
animals (e.g. mice). (2) The overall system should occupy
minimal footprint to maximize its workspace-to-size ratio; (3)
The maximum longitudinal (y-direction) magnetic gradient is
at least 1.0 T/m regardless of the eccentricity from the y-
axis, to ensure sufficient force for the robot to move in a
high-resistance environment (e.g. viscous silicone oil and agar
[15]), which could simulate biological tissue environment. (4)
The maximum magnetic flux density is at least 40 mT to
supply sufficient torque to ensure that the robot always aligns
with the field direction in a fluid environment [22]. Based on
these design requirements, a cylindrical Halbach array-based
magnetic actuation system is developed to have an accessible
space of 100 mm diameter and 152 mm longitudinal length.
It provides a motion workplane of 80 mm (x-direction) by 76
mm (y-direction) for a magnetic robot, as shown in Fig. 1.
This workplane can be easily extended to a 3D workspace of
a cylinder with 80 mm diameter and 76 mm length when the
Halbach array is rotated for a full revolution along its y-axis.
In order to produce longitudinal force in the entire work-
plane within the cylindrical Halbach array with bidirectional
tip leading motion, the desired magnetic fields should have the
following properties: (1) bidirectional longitudinal gradients
(to generate longitudinal forces in two opposite directions);
(2) co-aligned magnetic field directions and gradient directions
(to align the tip orientations with the motion orientations
or in other words enable tip-leading motion at all times);
(3) near-zero lateral gradients (i.e., gradients along the x-
direction) for each y-value (to allow longitudinal gradients
along the y-direction to be equally implemented across the
entire workplane instead of along the central y-axis only).
Based on the working principle of one straight Halbach
row [17] arranged along the y-direction, as seen in Fig. 1(a),
rotating the individual magnets in sequence in a Halbach row
can generate a y-directional gradient on one side of the row. In
This article has been accepted for publication in IEEE Robotics and Automation Letters. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/LRA.2024.3382485
© 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
Authorized licensed use limited to: Chinese University of Hong Kong. Downloaded on March 28,2024 at 13:51:26 UTC from IEEE Xplore. Restrictions apply.
SUN et al.: A CYLINDRICAL HALBACH ARRAY MAGNETIC ACTUATION SYSTEM FOR LONGITUDINAL ROBOT ACTUATION ACROSS 2D WORKPLANE 3
our design, starting with the first magnet having a pitch angle
𝜑= 90°, the individual magnets in the Halbach row rotate by
45°in the clockwise direction (see subfigure in Fig. 1(a)) with
the last magnet having 𝜑= 90°as well (i.e. 360°pitch angle
difference between the first and the last magnets). This generates
two bidirectional gradients with magnetic field directions and
gradient directions aligned, and the maximum magnetic field
regions are located at y-values of the magnets whose 𝜑are
0°and 180°. In this way, the desired properties (1) and (2) of
the generated magnetic fields have been attained. To provide
sufficiently large flux density for the two gradient fields, nine (n
= 9) (instead of five) magnets are used in each Halbach row, as
shown in Fig. 1(a). The distance between adjacent magnets in the
row is set as 19 mm, having considered about the space required
for full rotation of each magnet in the row and the minimal
space occupation of the system in the longitudinal direction
(requirement (2)).
In the effort to achieve property (3), the magnets in each
Halbach circular array are smaller and magnetically weaker as
they get closer to the workplane, as shown in Fig. 1(a). This helps
to smooth out the lateral gradients. Besides, the two magnets
that intersect the workplane are removed entirely to reduce the
lateral gradients near the edge of the workplane. It should be
noted that the overall magnetic flux density is traded off in the
process, as smaller magnets are employed in parts of the array.
In our design, to satisfy requirement (1), each circular array of
the cylindrical setup has a diameter of 120 mm, with 11 magnets
arranged circumferentially around the central axis and on each
side of the workplane as shown in Fig. 1(a). In other words, there
are 22 Halbach rows in our proposed cylindrical Halbach array.
The largest magnets located in the middle (i.e. top and bottom)
of each circular array have a side length of 12 mm to provide
flux density of at least 40mT (requirement (4)). The roll angle
of magnet i, where i= 1, 2, 3,...22, within each circular array is
defined as 𝜃𝑖, as shown in Fig. 1(b). For example, the topmost
magnet in Fig. 1(a) has a roll angle of 𝜃6= 90°. Note that each
magnet used in this work is cubic and the volume of a magnet
is in direct proportion to its magnetic dipole moment m𝑖.
III. System Optimization
The side length of each magnet within the circular array
and the roll angles of the magnets can be optimized through
a mathematical formulation to minimize the lateral gradients.
As illustrated in Fig. 1(b), the magnetic field generated by the
cylindrical Halbach array at the measurement point 𝑃𝑗(𝑥𝑗,𝑦𝑗,
𝑧𝑗) on the workplane can be calculated as:
B(𝑃𝑗)=
𝑁
𝑖=1"𝜇0
4𝜋r𝑖 𝑗 3(3ˆ
r𝑖 𝑗 ˆ
rT
𝑖 𝑗 −I3)m𝑖#(1)
where:
r𝑖 𝑗 =
𝑥𝑗−𝑅cos 𝜃𝑖
𝑦𝑗−𝑑𝑖
−𝑅sin 𝜃𝑖
;m𝑖=
𝑚𝑖sin 𝜑𝑖cos 𝜃𝑖
𝑚𝑖cos 𝜑𝑖
𝑚𝑖sin 𝜑𝑖sin 𝜃𝑖
𝜇0is the air permeability; ris the displacement vector from the
magnet center to the measurement points; ˆ
ris the unit vector
of r;𝑑is the y-value of the magnet position; I3is the three-by-
three identity matrix; 𝑁=198 is the number of magnets. 𝑅is the
layout radius of the magnets, which is set to 60 mm. The scalar
value of the magnetic dipole moment m𝑖can be approximately
calculated as:
𝑚𝑖=
𝐵𝑟𝑉𝑖
𝜇0
;𝑉𝑖=𝑙3
𝑖(2)
where 𝐵𝑟is the remanence of the magnet. 𝑉𝑖and 𝑙𝑖are the
volume and the side length of the cubic magnet 𝑖, respectively.
During the formulation of the optimization problem, the field
variance at locations along the x-direction within the workplane
is computed for a particular yposition and denoted as 𝜎2
𝑦. The
objective function to be minimized, with 𝑙𝑖and 𝜃𝑖used as the
optimization parameters, is defined as the sum of 𝜎2
𝑦for all y
positions within the workplane (i.e., from 𝑦=−38 mm to 𝑦=38
mm with a 1 mm interval), as follows:
𝑓𝑚𝑖𝑛 (𝜃𝑖, 𝑙𝑖)=
38
𝑦=−38
𝜎2
𝑦(3)
where:
𝜎2
𝑦=
1
81
40
𝑥=−40(B(𝑥 , 𝑦)−𝜇𝑦)2;𝜇𝑦=
1
81
40
𝑥=−40 B(𝑥, 𝑦 )
𝜇𝑦denotes the average scalar value of 3D field vector among
the 81 xsample locations evaluated from 𝑥=−40 mm to
𝑥=40 mm with a 1 mm interval at a particular yposition.
The minimization of (3) ensures that the gradients along the
x-direction is minimized while maintaining the y-directional
gradients. Since the cylindrical Halbach array is symmetrical
along the xy-plane and the yz-plane, the parameters of the
magnets from only the first to sixth rows (𝑖= 1, 2, ... , 6) on
the upper right octant of the workplane, instead of all 22 rows,
need to be determined, as illustrated in Fig. 2. Six groups of
magnetic field data when 𝑦=8,18,28,38 mm are shown in
Fig. 3. The data at 𝑦=38 was chosen due to its coincidence
with the maximum field region. The region between 𝑦=−8
mm and 𝑦=8 mm is hardly used for robot manipulation due to
the low flux density (i.e., small torque) and the deflective field
directions on the workplane that make it difficult to provide
tip-leading actuation.
θ
3
θ
4
θ
5
θ
6
R
x
z
O
l
1
l
2
l
3
l
4
l
5
l
6
θ
1
θ
2
α
1
α
2
α
4
α
3
α
5
α
6
ε
ε
ε
ε
ε
Fig. 2. Schematic showing the magnet parameters involved in the optimization
of the objective function to minimize the lateral gradients.
In the effort to search for the optimal side lengths of the
magnets in a circular array, many side length sequences (𝑙1to
𝑙6) have been adopted in the optimization trials to compute (3).
This article has been accepted for publication in IEEE Robotics and Automation Letters. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/LRA.2024.3382485
© 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
Authorized licensed use limited to: Chinese University of Hong Kong. Downloaded on March 28,2024 at 13:51:26 UTC from IEEE Xplore. Restrictions apply.
4 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED MARCH, 2024.
σ2=0.21 σ2=0.23
σ2=0.14 σ2=0.23
Magnets’ side length sequence l
i
(mm)
-40 -30 -20 -10 0 10 20 30 40
x-axis (mm)
-40 -30 -20 -10 0 10 20 30 40
x-axis (mm)
-40 -30 -20 -10 0 10 20 30 40
x-axis (mm)
9, 9, 10, 11, 12, 128, 8, 8, 10, 12, 126, 7, 8, 10, 11, 12
(a)
Halbach rows’ roll angle sequence θ
i
(°)
(b)
-40 -30 -20 -10 0 10 20 30 40
x-axis (mm)
-40 -30 -20 -10 0 10 20 30 40
x-axis (mm)
-40 -30 -20 -10 0 10 20 30 40
x-axis (mm)
21.5, 33.3, 45.0, 58.0, 73.4, 90.0
Flux density (mT)
y = 38 mm y = 28 mm y = 18 mm y = 8 mm
0
10
20
30
40
50
60
70
0
10
20
30
40
50
60
70
0
10
20
30
40
50
60
70
0
10
20
30
40
50
60
70
0
10
20
30
40
50
60
70
0
10
20
30
40
50
60
70
Flux density (mT)
Flux density (mT)
Flux density (mT)
Flux density (mT)
Flux density (mT)
σ2=1.18 σ2=0.97
σ2=0.52 σ2=0.57
σ2=0.21 σ2=0.23
σ2=0.14 σ2=0.23
σ2=4.83 σ2=4.50
σ2=2.62 σ2=0.75
σ2=1.92 σ2=1.56
σ2=0.85 σ2=1.28
σ2=4.49 σ2=3.97
σ2=2.33 σ2=0.76
16.0, 28.9, 41.7, 55.8, 72.3, 90.028.0, 38.5, 48.9, 60.6, 74.7, 90.0
Fig. 3. (a) Magnetic fields along the x-direction in different ywith different
magnets’ side length sequences; (b) Magnetic fields along the x-direction in
different ywith different Halbach rows’ roll angle sequences.
All side lengths are set to integers for ease of manufacture. 12
mm is the largest side length and adopted as 𝑙6. The other magnet
sizes are reduced at different shrinkage rates with the smallest
side length attempted in our trials being 6 mm to provide the
minimum desired gradients. Among all the trials, the magnetic
flux densities and variances for three representative side length
sequences are shown in Fig. 3(a). In these three exemplary data,
the roll angle sequence is set as (21.5, 33.3, 45.0, 58.0, 73.4,
90.0 (°)).
The roll angle sequences used in the optimization trials were
chosen by setting different roll angles for the first Halbach row
(𝜃1). For every 𝜃1attempted, the roll angles of the other rows
(𝜃2to 𝜃5) were computed accordingly as follows (also see Fig.
2):
𝜃𝑖=𝜃𝑖−1+𝜖+𝛼𝑖+𝛼𝑖−1
2𝑖=2,3,4,5 (4)
where the interval angle between each row 𝜖can be expressed
as
𝜖=
1
5 𝜋
2−𝜃1−𝛼1+𝛼6
2−
5
𝑖=2
𝛼𝑖!
and the angle 𝛼𝑖occupied by each row 𝑖is defined as
𝛼𝑖=2tan−1𝑙𝑖
2𝑅−√2𝑙𝑖
In this way, each roll angle sequence is uniquely represented
by 𝜃1. It is also essential to ensure that the magnets do
not physically interfere with each other when adjusting the
roll angles. Magnetic flux densities and variances for three
representative roll angle sequences are shown in Fig. 3(b). In all
these examples, the side length sequence is set as (8, 8, 8, 10,
12, 12 (mm)).
The magnitude of lateral gradients in each parameter combi-
nation (i.e. each subfigure) in Fig. 3is reflected in the flatness
12 14 16 18 20 22 24 26 28 30 32
10
1
10
2
10
3
10
4
Objective function
θ
1
(°)
(f
min
mT
2
)
12.05 15.14
30.03
90.54
160.10
8, 8, 8, 10, 12, 12
10, 10, 10, 11, 12, 12
9, 9, 10, 11, 12, 12
7, 7, 8, 10, 11, 12
6, 7, 8, 10, 11, 12
Side length sequences
Fig. 4. The objective function graph where the curves represent different side
length sequences of the magnets and the lowest point indicates the optimal 𝜃1
of each sequence.
of the curves. The flatter the curves, the smaller the lateral
gradients. The graphs of the two-dimensional objective function
𝑓𝑚𝑖𝑛 for a range of 𝜃1is also plotted for a few selected side
length sequences (𝑙1to 𝑙6) in Fig. 4. The optimal values of
the parameters are selected at the lowest point of the function.
While the objective function values for the first two side length
sequences in Fig. 4are the minimum, their maximum flux
densities are 38.2 mT and 39.4 mT, respectively, which do not
satisfy requirement (4). Therefore, it can be confirmed that the
third side length sequence of (8, 8, 8, 10, 12, 12 (mm)) with 𝜃1
= 21.5°, which provides a maximum flux density of 46.0 mT,
should be chosen as the optimal parameter combination.
IV. Simulation
The optimized roll angle sequence and the side length
sequence of the cylindrical Halbach array are listed in Table
I, together with the pitch angles in a Halbach row. The magnetic
flux density of this optimized system on the workplane is
calculated by (1) using MATLAB (MathWorks Inc., USA),
as shown in Fig. 5(a). However, since (1) approximates each
magnet as dipole moment point without volume, the flux density
computed is sufficiently accurate only when the distance from
the measured point is more than 1.5 times the side length of the
magnet [23]. Therefore, to further verify the optimization result,
the parameters were also implemented to obtain a flux density
simulation on a finite element analysis software (COMSOL
Multiphysics, COMSOL Inc., Sweden) with the results shown
in Fig. 5(b).
TABLE I
The Parameter Values of The Cylindrical Halbach Array
Row number 1 2 3 4 5 6
Roll angles (°)
22.0
33.7
45.3
58.2
73.5
90.0
Side length (mm)
8 8 8 10 12 12
nth in a row 1
2
3
4
5
6
7
8
9
Pitch angles (°)
90
45
0
315
270
225
180
135
90
As illustrated in Figs. 5(a)-5(b), it can be seen that the contours
(black lines with equal flux density) in the workplane are mostly
parallel to the x-axis and the flux density gets smaller when
This article has been accepted for publication in IEEE Robotics and Automation Letters. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/LRA.2024.3382485
© 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
Authorized licensed use limited to: Chinese University of Hong Kong. Downloaded on March 28,2024 at 13:51:26 UTC from IEEE Xplore. Restrictions apply.
SUN et al.: A CYLINDRICAL HALBACH ARRAY MAGNETIC ACTUATION SYSTEM FOR LONGITUDINAL ROBOT ACTUATION ACROSS 2D WORKPLANE 5
Magnetic flux densi ty (mT)
-40 -30 -20 -10 0 10 20 30 40
-50
-40
-30
-20
-10
0
10
20
30
40
50
5
10
15
20
25
30
35
40
45
-80 -60 -40 -20 0 20 40 60 80
0
5
10
15
20
25
30
35
40
45
50
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Magnetic flux densi ty (mT)
x-axis (mm) x-axis (m m)
y-axis (mm)
y-axis (m m)
Magnetic flux densi ty (mT)
z-axis (mm)
y-axis (mm)
y-axis (mm)
Flux densi ty (mT) G radient (T /m)
Workplane
2.5
-2.5
(b)
(d) (e) (f)
(a)
Magnetic gradient (T/m)
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
-40 -30 -20 -10 0 10 20 30 40
-50
-40
-30
-20
-10
0
10
20
30
40
50
y-axis (m m)
x-axis (mm)
Workplane
Workplane
Workplane
Workplane
(c)
Magnetic force (mN)
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
-40 -30 -20 -10 0 10 20 30 40
-50
-40
-30
-20
-10
0
10
20
30
40
50
y-axis (m m)
x-axis (m m)
Workplane
2
2.2
2.4
2.6
Fig. 5. (a) Magnetic field on the workplane computed by mathematical model (1) and (b) FEA software; (c) Magnetic field on yz-plane; (d) Magnetic gradients
on the workplace; (e) Magnetic flux density and the gradients along y-axis when x=0; (f) Force map of the magnetic robot described in V-B on the workplane.
approaching y= 0. This means that the longitudinal gradients,
and thus longitudinal forces, have been achieved along the y-
direction with near-zero lateral gradients along the x-axis. It
can also be observed that the flux lines (i.e. red arrows that
indicate the field direction) are perpendicular to all the contours
in the workplane, indicating that the field directions and the
gradient directions are co-aligned. As seen in Fig. 5(c), the field
directions on the y-axis are horizontal, and the tangent directions
of the contours are perpendicular to the y-axis. This suggests
that the z-directional fields and gradients are zero at the xy-
workplane. Fig. 5(d) shows the computed magnetic gradients
on the workplane, where the red arrows denote the gradient
directions. It can be seen that the largest gradient is 1.9 T/m,
satisfying requirement (3). The gradients are larger at locations
closer to the x-axis, suggesting the force acting on a magnetic
robot is larger when it is placed closer to the x-axis in the
workplane. Referring to both Figs. 5(a) and 5(d), it can also be
observed that the field direction in Fig. 5(a) are always aligned
with the gradient directions in Fig. 5(d) in the workplane with
two directions achieved on both sides of y= 0, indicating a tip-
leading motion can always be achieved during the bidirectional
motion.
Since there are near-zero lateral gradients along the x-
direction, the flux densities and the gradients of the field along
the y-direction in the workplane can be represented by the values
when x= 0, as shown in Fig. 5(e). The discontinuity of the
gradient curve is caused by the continuous but non-differentiable
field, which reduces to zero linearly as it approaches y=0. Based
on the curve shape of the flux density shown in Fig. 5(e), the
relatively complicated (1) can then be replaced by a simplified
sinusoidal function as follows.
B(𝑃𝑗)=46 sin 𝜋𝑦 𝑗
76 [010]𝑇(5)
The units of y𝑗and 𝐵are mm and mT, respectively. Such
simplification of the field would allow easier robot control since
only the y-position of the robot is needed to calculate the flux
density and magnetic gradient, and therefore the torque and
force acting on the robot. It should be noted that (5) only applies
to the workplane (z= 0). Besides, only the area covering ±38
mm around the y-axis in Fig. 5(e) is adopted as the workplane
in which the robot would be actuated, because it contains high
two-longitudinal gradients and two stable magnetic field regions
with the maximum field with no gradient located at y=±38
mm.
The force exerted on a magnetic target with 1 mm diameter
when it is located at 𝑃𝑗in the magnetic field is illustrated in
Fig. 5(f) and calculated in (6):
𝒇=∇B(𝑃𝑗) · m(6)
where ∇is the gradient operator. mrepresents the magnetic
dipole moment of the robot, which can be calculated by (2).
There is also a torque exerting on the robot to make its polar
direction coincide with the magnetic field direction, which is
calculated as follow:
𝝉=m×B(𝑃𝑗)(7)
However, for the applications in the fluid environment at low
movement and rotational speeds, the magnetic torque on the
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content may change prior to final publication. Citation information: DOI 10.1109/LRA.2024.3382485
© 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
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6 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED MARCH, 2024.
robot is far greater than the resistance from the environment, so
we can assume that the dipole direction of the robot is always
consistent with the field direction.
V. Experimental Results and Discussions
The cylindrical Halbach array prototype was fabricated based
on the optimization results. The PMs (N35 magnet, 𝐵𝑟=1.18
T, Dongguan Jiada Magnet Co., Ltd., China) were distributed
among nine rings with 22 PMs in each ring, forming 22 straight
Halbach rows of magnets. The rings were made in 3D printed
nylon (WeNext Technology Co., Ltd., China). The polarities
of the PMs were first marked with arrows before they were
embedded in the small slots within each ring using ceramic rods
(Soochow Sigma Special Ceramics Co., Ltd., China) and pieces
of carbon fiber (Datong Beilang Technology Co., Ltd., China).
Nine rings were assembled together by two steel shafts and
limiting pins to complete the cylindrical Halbach array assembly
(see Fig. 6(a)). Three experiments were then performed to
evaluate the magnetic field property and actuation capability
of the cylindrical Halbach array actuation system.
Cylindrical Halbach array
3-axis Tesla meter
Camera
Silicone oil
Petri dish
Robot
3-axis displacement
platform
3-axis probe
(a) (b)
Fig. 6. (a) Cylindrical Halbach array actuation system with a Tesla meter
attached to a 3-axis displacement platform; (b) robot platform that can be placed
inside the cylindrical Halbach array actuation system and be translated by the
3-axis displacement platform.
A. Magnetic Field Measurement
First, the magnetic field produced by the Halbach array on
the workplane was measured by a 3-axis Tesla meter (Tunkia
Co., Ltd., China). A 3-axis displacement platform (Shenzhen
Hengyu Laser Instruments Co., Ltd., China) was commanded
to move the probe of the Tesla meter across a plane of size 80
mm ×80 mm (containing the workplane of size 80 mm ×76
mm) and stop to measure the magnetic field every 5 mm interval
along the xand ydirections. The collected data were a matrix
of 17 ×17, with each element in the matrix representing each
measured magnetic flux density and direction. The measured
magnetic field was processed on MATLAB by the contourf
and the quiver functions to show the flux density and the field
directions, respectively in Fig. 7(a). The error map shown in
Fig. 7(b) is obtained by subtracting the measured values from
the theoretical values from Fig. 5(a). It can be seen that the
measured magnetic fields match closely the theoretical data,
with an average error of 1.1 mT and largest absolute error of 2.5
mT. The slight discrepancies may be caused by magnetization
errors, magnet geometry fabrication error, and system assembly
error.
-1
-0.5
0
0.5
1
1.5
2
Magnetic flux density (mT)
y-axis (mm)
-40
-30
-20
-10
0
10
20
30
40
Flux density difference (mT)
y-axis (mm)
-40 -30 -20 -10 0 10 20 30 40
5
10
15
20
25
30
35
40
-40 -30 -20 -10 0 10 20 30 40
x-axis (mm)
(a)
x-axis (mm)
(b)
-40
-30
-20
-10
0
10
20
30
40
Fig. 7. (a) Magnetic field on the workplane measured by 3-axis Tesla meter; (b)
Difference between the measured values and the theoretical values.
B. Magnetic robot actuation
In the second set of experiments, a magnetic robot was placed
inside a specimen container (PMMA petri dish containing
viscous silicone oil (50 Pas, Chenghong Silicone Oil Co., Ltd.,
China)), as shown in Fig. 6(b). The petri dish was attached to the
3-axis displacement platform. The robot consists of a cylindrical
magnet (N52 magnet with 1 mm diameter and 2 mm length,
Dongguan Jiada Magnet Co., Ltd., China) and a 3D printed
resin shell with a spike (2 mm diameter and 2.5 mm length,
WeNext Technology Co., Ltd., China), as shown in Fig. 6(b).
The petri dish and a top-view camera (Dechuangxin Image
Technology Co., Ltd., China) were fixed together, as shown in
Fig. 6(b). In these experiments, they would only be translated by
the displacement platform along the y-direction to continuously
position the robot at a certain distance relative to the x-axis.
It should be clarified that in an actual surgical procedure,
the Halbach array actuation system would be translated along
the y-direction instead of the petri dish (human anatomy),
which should remain stationary. The translation mechanism was
applied to the petri dish instead of the Halbach array in these
experiments due to the simpler engineering implementation
involved while maintaining the application context and practical
significance. To qualitatively evaluate via observation whether
the robot moved along a straight line parallel to the y-axis, a
horizontal marker line was placed near the robot, as shown in
Fig. 8. The vertical black marker line represents the x-axis of the
actuation system, and the xy coordinates of the workplane were
fixed at the center of the line. The distance between the robot
and the x-axis marker line is equal to its y-value at a particular
instant, which can be used to calculate the gradient at the robot’s
position and the force exerted on the robot through (5) and (6).
During the experiments, the robot was first placed at different
x-values on the workplane, including x= 4 mm, x=−12 mm,
and x= 25 mm, before the displacement platform was moved to
control the y-value of the robot. The movement of the robot was
recorded by the camera throughout the experiments. Fig. 8shows
the magnetic robot’s ability to move straight along y-direction
at different xvalues, confirming the capability of the actuation
system to exert pure longitudinal force on the robot not only
along the central axis but at any x-value in the workplane. It can
also be seen that the spike side of the robot always aligned with
the motion direction, achieving the desired tip-leading motion.
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content may change prior to final publication. Citation information: DOI 10.1109/LRA.2024.3382485
© 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
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SUN et al.: A CYLINDRICAL HALBACH ARRAY MAGNETIC ACTUATION SYSTEM FOR LONGITUDINAL ROBOT ACTUATION ACROSS 2D WORKPLANE 7
(a)
(b)
(c)
Time = 0 s Time = 31 s Time = 60 s
Time = 0 s Time = 29 s Time = 59 s
Time = 0 s Time = 41 s Time = 82 s
x
y
Marker line of x-axis
Robot
Marker line for
straight movement
Moving track
Moving
direction
Fig. 8. Snapshots of the magnetic robot moving along y-direction at different
xvalues (mm) and with different gradients (determined by y-value (mm) of the
robot): (a) x= 4; gradient = 1.8 T/m (y=−8.3), (b) x=−12; gradient = 1.5 T/m
(y=−16.0), (c) x= 25; gradient = 1.2 T/m (y= 21.4). The direction of the robot
is determined by the sign of the y-value.
C. Robot actuation at different motion velocities
In the third set of experiments, the robot was again placed at
the same three x-values as the previous experiments. At x= 4
mm, x=−12 mm and x= 25 mm, the x-axis marker line was
maintained at three different distances from the robot in an open
loop manner (manual control), namely y=−8.3 mm, y=−16.0
mm and y= 21.4 mm, respectively, throughout the longitudinal
movement of the robot using the displacement platform. This
allows the robot to experience constant magnetic gradients of 1.8
T/m, 1.5 T/m, and 1.2 T/m, leading to mean velocities of 0.59
mm/s, 0.51 mm/s, and 0.30 mm/s, as illustrated in Fig. 9(a).
The other factor influencing the robot speed is the viscosity
of the silicone oil. The robot was therefore tested in oils with
three different viscosities of 12.5 Pas, 50 Pas, 100 Pas under
the magnetic gradient set at 1.5 T/m (i.e. y-value of the robot
was −16.0 mm), as shown in Fig. 9(b). The robot moved at
mean velocities of 1.52 mm/s, 0.51 mm/s, and 0.14 mm/s,
respectively, with the highest oil viscosity resulting in the lowest
robot velocity.
1.1 1 .3 1.5 1.7 1.9
0.2
0.3
0.4
0.5
0.6
Robot’s velocity (mm/s)
Viscosity = 50 Pas
12.5 100
50
Viscosity (Pas)
0
0.4
0.8
1.2
1.6
2.0
Robot’s velocity (mm/s)
Gradient = 1.5 T/m
Gradient (T/m)
(a) (b)
0.7
Fig. 9. (a) The robot velocities under different gradients in the silicone oil of 50
Pas; (b) the robot velocities in silicone oil with different viscosities under the
1.5 T/m gradient.
D. Discussion
This work introduces a cylindrical Halbach array-based PMS
that allows bidirectional 1-DOF longitudinal motion across an
entire workplane intersecting the axis of the cylinder. This for the
first time enables robot navigation forward and backward at any
location on a 2D plane of a specimen placed inside the Halbach
cylinder without requiring the specimen to be manipulated
(rotated or translated sideways), thus fundamentally enabling
practical implementation of the Halbach-based systems for
untethered robot actuation in a surgical procedure during which
the human anatomy should ideally be fixed. When considering
deep brain lesion targeting, such as tumor biopsy, deep brain
stimulation, thermal therapy, as a potential clinical application,
a straightforward calculation using (6) and Fig. 5(f) shows that
our system can exert actuation force of up to 10.8 mN on a
2 mm diameter magnetic target similar to the diameter of a
14 G needle, thus enabling its penetration through agar gel
of 0.6% [24] and porcine brain [21], which feature human
brain-like stiffness. Besides, our magnetic actuation approach
also allows a continuum, tubular-shaped robot (e.g. needle-like
robot) to be pulled by the magnetic force from its distal end
instead of being pushed from its proximal end as in the case
of conventional actuation mechanisms, significantly reducing
the risk of robot buckling especially when inserted deep in the
brain. The alignment of magnetic field directions and gradient
directions, which enables the tip-leading motion, also allows
the robot to have control of its orientation during its motion and
prevent any undesired lateral motion. This in turn allows design
customization of the robot tip to reduce motion resistance from
the environment or enable effective diagnostic or therapeutic
capabilities.
TABLE II
Comparison of This Work with Existing MASs
P
MS
or
EMS
Workspace
(WS)
System size①
(SS)
Ratio of
WS to SS
②
Max.
field
(mT)
③
Max.
gradient
(T/m)
④
Number
of
magnets
This work
PMS
∅
80 mm×
76
mm cylinder
∅
138 mm×172
mm cylinder
1:6.73
45
1.9 198
Ryan et al.
[11]PMS
∅5 mm sphere
300×300×300
mm3 cube* 1:412529
30
0.83
8
Son et al.
[21]PMS
∅
6 mm×
2
0
mm cylinder*
∅
100 mm×80 mm
cylinder* 1:888.89
145
7 100
CardioMag
[7]EMS
200×300×500
mm3 cuboid
2
2
0
0
×
18
0
0
×
18
0
0
mm3 cuboid*
1:237.60
55
0.245
8
OctoMag
[3]EMS
∅
25 mm
sphere
435×435×250
mm3 cuboid*
1:5782.29
15
0.2*
8
Note: *, estimated parameters that the references do not provide; ➀, overall size
of the system; ➁, volume ratio of the workspace to the system size; ➂➃, these
parameters are evaluated at the center of the workspace.
Table II shows the comparison in terms of the physical
and performance characteristics between this work and several
existing stationary MASs. It should be reminded that the
workspaces of our system and the system by Son et al. [21],
both of which are based on the Halbach working principle,
are generated by rotating the cylindrical Halbach array by
360°via a simple rotational mechanism, so that the longitudinal
gradient can be generated at any position in the 3D cylindrical
workspace. Having orders of magnitudes larger ratio of WS to
SS, our system supports a centimeter-scale workspace suitable
for small animal experiments, which is multiple times that of
most typical stationary MASs, while occupying a similar or
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content may change prior to final publication. Citation information: DOI 10.1109/LRA.2024.3382485
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8 IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED MARCH, 2024.
smaller system size. This is an important and favorable condition
for clinical translation. Our maximum flux density is moderate
among the state-of-the-art works, but is sufficiently large to
actuate millimeter-scale robots [3], [5]. Unlike many MASs that
provide large field used only for generating torque for robot
steering, our system provides relatively high gradient and thus
large force to translate millimeter-scale magnetic robots for
tissue penetration. While [7] reported an EMS with large flux
density and workspace, it features an extremely bulky system
and requires large energy and cost for powering the EM coils.
Similar to [21], our system involves the assembly of many PMs.
Since they are fixed in a customized arrangement and do not
individually move during the actuation, the large number of
magnets do not cause any control complexity.
While comparing favorably with many existing MASs in
terms of WS-SS ratio and providing high gradients, our proposed
system features a few limitations. First, when the cylindrical
array is increased three times to accommodate the human head,
the gradients and thus the actuation forces will be reduced
by similar multiples. This will be addressed by adding more
magnets or increasing the magnet remanence in the larger system
through proper design optimization. Second, the magnetic field
cannot be turned off as in the EM-based system to stop the robot
motion. Therefore, if the robot must stay stationary, it needs to be
specifically placed in the gradientless regions, namely 𝑦=±38
mm. Third, the current system provides only longitudinal
actuation. The radial and tangential DOFs could be introduced
by adding dipole and quadrupole cylindrical Halbach arrays to
form a concentrically nested system, potentially providing full
3D actuation capability.
VI. Conclusion
A new cylindrical Halbach array actuation system has been
developed to provide a millimeter-scale magnetic robot with
bidirectional longitudinal movement across a 2D workplane, an
advancement from the previous state-of-the-art work that allows
only motion along the central axis. The magnet sizes and the
roll angles of the Halbach rows were optimized to minimize
the lateral gradients and thus achieve longitudinal gradients
uniformly spread across the 2D workplane. The field and the
gradient directions are always aligned in the workplane, leading
to the tip-leading motion of the robot. Besides, the speed of the
robot can be regulated by actuating it at different gradients or
within environment of different viscosities. In our future work,
we will expand the accessible space of the actuation system
to a human-scale size. We will also increase its DOFs and
develop control strategies to enable execution of more complex
trajectories.
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This article has been accepted for publication in IEEE Robotics and Automation Letters. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/LRA.2024.3382485
© 2024 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.
Authorized licensed use limited to: Chinese University of Hong Kong. Downloaded on March 28,2024 at 13:51:26 UTC from IEEE Xplore. Restrictions apply.
