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\begin{document}

\title{Magnetic miniature actuators with six-DOF multimodal soft-bodied
locomotion}



\author[1]{Changyu Xu}%
\author[1]{Zilin Yang}%
\author[1]{Shaun Wee Kiat Tan}%
\author[1]{Jianhuang Li}%
\author[1]{Guo Zhan Lum}%
\affil[1]{Affiliation not available}%


\vspace{-1em}



  
  \date{May 20, 2024}


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\selectlanguage{english}
\begin{abstract}
Magnetic miniature robots (MMRs) are mobile actuators that can exploit
their size to non-invasively access highly confined, enclosed spaces. By
leveraging on such unique abilities, MMRs have great prospects to
transform robotics, biomedicine and materials science. As having high
dexterity is critical for MMRs to enable their targeted applications,
existing MMRs have developed numerous soft-bodied gaits to locomote in
various environments. However, there exist two critical limitations that
have severely restricted their dexterity: (i) MMRs capable of multimodal
soft-bodied locomotion have only demonstrated five-degrees-of-freedom
(five-DOF) motions because the sixth-DOF rotation about their net
magnetic moment axis is uncontrollable; (ii) six-DOF MMRs have only
realized one mode of soft-bodied, swimming locomotion. Here we propose a
six-DOF MMR that can execute seven modes of soft-bodied locomotion and
perform 3-dimensional pick-and-place operations. By optimizing its
harmonic magnetization profile, our MMR can produce 1.41-63.9 folds
larger sixth-DOF torque than existing MMRs with similar profiles,
without compromising their traditional five-DOF actuation capabilities.
The proposed MMR demonstrated unprecedented dexterity; it could jump
through narrow slots to reach higher grounds; use precise orientation
control to roll, two-anchor crawl and swim across tight openings with
strict shape constraints; perform undulating crawling across three
different planes in convoluted channels.

~

\textbf{Keywords:} Magnetic materials; soft actuators; miniature robots;
locomotion.

Corresponding author(s)
Email:~{\emph{~~\href{https://sites.google.com/site/ntuminiaturesoftroboticslab/members}{gzlum@ntu.edu.sg}~~}}%
\end{abstract}%



\sloppy


\section*{}

{\label{731740}}

\section*{Introduction}

{\label{731740}}

Miniature robots are mobile actuators that are in the length scales of
millimeter or smaller~~\hyperref[csl:1]{(Nelson et al., 2010}; \hyperref[csl:2]{Hines et al., 2017}; \hyperref[csl:3]{Sitti, 2018}; \hyperref[csl:4]{Miskin et al., 2020)}. Because these actuators can
leverage on their size and mobility to non-invasively access highly
confined and enclosed spaces~\hyperref[csl:5]{(Sitti et al., 2015}; \hyperref[csl:6]{Ng et al., 2021)}, they are hypothesized
to have the potential to revolutionize medical treatments such as
minimally invasive surgery and targeted drug
delivery~\selectlanguage{ngerman}\hyperref[csl:7]{(Li et al., 2017}; \hyperref[csl:8]{Palagi et al., 2016}; \hyperref[csl:9]{Jeon et al., 2019}; \hyperref[csl:10]{Hu et al., 2018}; \hyperref[csl:11]{Gultepe et al., 2013}; \hyperref[csl:12]{Tottori et al., 2012}; \hyperref[csl:13]{Felfoul et al., 2016}; \hyperref[csl:14]{Esteban-Fernández et al., 2020}; \hyperref[csl:15]{Alapan et al., 2020}; \hyperref[csl:16]{Wu et al., 2018}; \hyperref[csl:17]{Wang et al., 2021}; \hyperref[csl:18]{Wang et al., 2021)}. These small robots have also shown to be
powerful tools that can enable unprecedented lab-on-chip applications
\hyperref[csl:19]{(Ceylan et al., 2017}; \hyperref[csl:20]{Diller \& Sitti, 2014}; \hyperref[csl:21]{Hu et al., 2021}; \hyperref[csl:22]{Chung et al., 2015)}, and facilitate critical fundamental studies in
material sciences~\hyperref[csl:23]{(Tasoglu et al., 2014}; \hyperref[csl:24]{Zhang et al., 2021)} and biology~\hyperref[csl:25]{(Palagi \& Fischer, 2018}; \hyperref[csl:26]{Kim et al., 2013}; \hyperref[csl:27]{Wani et al., 2017}; \hyperref[csl:28]{Dong et al., 2020}; \hyperref[csl:29]{Gu et al., 2020}; \hyperref[csl:30]{Lum et al., 2016)}.

In the literature, there exist multiple methods for actuating miniature
robots~\hyperref[csl:2]{(Hines et al., 2017}; \hyperref[csl:31]{Ilami et al., 2021)}. For example, these robots can be actuated via
electric fields~\hyperref[csl:32]{(Chen et al., 2019}; \hyperref[csl:33]{Yang et al., 2015}; \hyperref[csl:34]{Gupta et al., 2018}; \hyperref[csl:35]{Liu et al., 2021)}, pressure~\hyperref[csl:36]{(Paek et al., 2015)},
heat~\hyperref[csl:37]{(Mourran et al., 2017)}, light~\hyperref[csl:8]{(Palagi et al., 2016}; \hyperref[csl:38]{Zeng et al., 2018}; \hyperref[csl:39]{Guo et al., 2021)}, chemicals
\hyperref[csl:40]{(Lee et al., 2010}; \hyperref[csl:41]{Maeda et al., 2007)}, or magnetic fields~\hyperref[csl:42]{(Kummer et al., 2010}; \hyperref[csl:43]{Xu et al., 2021}; \hyperref[csl:44]{Kim et al., 2018}; \hyperref[csl:45]{Xu et al., 2019}; \hyperref[csl:46]{Ceylan et al., 2021}; \hyperref[csl:47]{Salehizadeh \& Diller, 2020}; \hyperref[csl:48]{Kuang et al., 2021}; \hyperref[csl:49]{Kim et al., 2011}; \hyperref[csl:50]{Son et al., 2021)}. Of these
methods, magnetic actuation is especially robust because the actuating
magnetic fields can easily and harmlessly penetrate through most
synthetic and biological materials~\hyperref[csl:44]{(Kim et al., 2018}; \hyperref[csl:51]{Pham et al., 2021}; \hyperref[csl:52]{Sitti \& Wiersma, 2020)}. As a result,
these magnetic miniature robots (MMRs) have great potential to operate
across a diverse range of enclosed environments~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:13]{Felfoul et al., 2016}; \hyperref[csl:53]{Ren et al., 2021)}. In
addition, unlike actuation signals such as pressure, heat and chemicals,
magnetic fields can be specified not only in magnitude but also in their
direction and spatial gradients~\hyperref[csl:42]{(Kummer et al., 2010}; \hyperref[csl:43]{Xu et al., 2021}; \hyperref[csl:54]{Diller et al., 2016)}. By having more
independent control parameters, MMRs have also shown to be the most
functional among all the classes of small-scale actuators
\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:30]{Lum et al., 2016}; \hyperref[csl:55]{Cui et al., 2019}; \hyperref[csl:56]{Alapan et al., 2020}; \hyperref[csl:57]{Zhang et al., 2021}; \hyperref[csl:58]{Zhang et al., 2021}; \hyperref[csl:59]{Xie et al., 2020}; \hyperref[csl:60]{Pauer et al., 2021}; \hyperref[csl:61]{Nguyen et al., 2021}; \hyperref[csl:62]{Miyashita et al., 2017)}.

It is highly desirable to enhance the dexterity of MMRs because these
actuators are expected to navigate across highly unstructured
environments for their targeted applications~\hyperref[csl:1]{(Nelson et al., 2010}; \hyperref[csl:3]{Sitti, 2018}; \hyperref[csl:6]{Ng et al., 2021}; \hyperref[csl:19]{Ceylan et al., 2017)}. To
enhance the agility of these actuators, the bodies of MMRs can be made
soft so that they can produce a series of time-varying deformations to
realize various types of locomotive gaits~\hyperref[csl:6]{(Ng et al., 2021}; \hyperref[csl:10]{Hu et al., 2018}; \hyperref[csl:30]{Lum et al., 2016}; \hyperref[csl:45]{Xu et al., 2019}; \hyperref[csl:53]{Ren et al., 2021}; \hyperref[csl:63]{Diller et al., 2014}; \hyperref[csl:64]{Ren et al., 2019}; \hyperref[csl:65]{Wang et al., 2021}; \hyperref[csl:66]{Zhang \& Diller, 2015}; \hyperref[csl:67]{Zhang et al., 2016}; \hyperref[csl:68]{Huang et al., 2021}; \hyperref[csl:69]{Ng et al., 2022)}. Some
critical advantages of having soft-bodied locomotion are that such gaits
can allow the MMRs to potentially position themselves more precisely and
better adapt to various types of environments~\hyperref[csl:6]{(Ng et al., 2021}; \hyperref[csl:10]{Hu et al., 2018}; \hyperref[csl:53]{Ren et al., 2021}; \hyperref[csl:63]{Diller et al., 2014}; \hyperref[csl:69]{Ng et al., 2022}; \hyperref[csl:70]{Ze et al., 2020}; \hyperref[csl:71]{Ng \& Lum, 2021}; \hyperref[csl:72]{Du et al., 2020}; \hyperref[csl:73]{Wu et al., 2020)}. There
are generally two types of MMRs with soft bodies: robots that are formed
by either (i) ferrofluid droplets~\hyperref[csl:74]{(Fan et al., 2020}; \hyperref[csl:75]{Fan et al., 2020}; \hyperref[csl:76]{Liu et al., 2019}; \hyperref[csl:77]{Ahmed et al., 2021)} or (ii)
elastomeric bodies~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:30]{Lum et al., 2016}; \hyperref[csl:45]{Xu et al., 2019}; \hyperref[csl:53]{Ren et al., 2021}; \hyperref[csl:64]{Ren et al., 2019)}. Of these two types of MMRs,
those that have elastomeric bodies are especially promising. Unlike
their droplet-based counterparts, the elastomer-based MMRs can produce
not only dexterous locomotion but they also possess adequate stiffness
necessary for performing 3-dimensional (3D) pick-and-place
operations~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:20]{Diller \& Sitti, 2014}; \hyperref[csl:45]{Xu et al., 2019)}. In comparison to the droplet-based MMRs,
the elastomer-based ones also possess higher stiffness that can allow
them to transmit forces more effectively to the environment and other
objects~\hyperref[csl:28]{(Dong et al., 2020}; \hyperref[csl:30]{Lum et al., 2016}; \hyperref[csl:64]{Ren et al., 2019)}. By having adequate stiffness, the
elastomer-based MMRs can induce sufficient impact from the substrate to
produce a jumping locomotion too~\hyperref[csl:10]{(Hu et al., 2018)}, and such complex
gaits have not been demonstrated by the droplet-based soft MMRs. Due to
the advantages of elastomer-based soft MMRs, we will be focusing on this
class of actuators, and soft MMRs will only be referred to those that
have elastomeric bodies in the subsequent text.

In recent years, the creation of soft MMRs that can generate multiple
modes of locomotion represents a significant advancement for small-scale
actuators~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:53]{Ren et al., 2021}; \hyperref[csl:72]{Du et al., 2020)}. For instance, the amphibious soft MMRs of
Hu et al. are able to execute seven modes of soft-bodied locomotion,
which can enable them to navigate across terrestrial and aquatic
environments \hyperref[csl:10]{(Hu et al., 2018)}. Likewise, the soft MMRs of Ren et al.
can use three different modes of locomotion to move in fluidic
environments~\hyperref[csl:53]{(Ren et al., 2021)}. Despite these advancements, soft MMRs
with multimodal locomotive gaits are restricted to having
five-degrees-of-freedom (five-DOF) motions because the sixth-DOF
rotation about their net magnetic moment axis is not possible
\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:53]{Ren et al., 2021}; \hyperref[csl:72]{Du et al., 2020)}. As a result, these MMRs can only rotate about two
axes and translate along three axes~\hyperref[csl:42]{(Kummer et al., 2010)}, and their
dexterity is still far from being optimized. While soft MMRs with
six-DOF do exist, such actuators have so far only realized one mode of
swimming locomotion~\hyperref[csl:43]{(Xu et al., 2021)}. This is because existing
six-DOF actuators are currently limited to having a 1D magnetization
profile for their soft components~\hyperref[csl:43]{(Xu et al., 2021)}, so that their
changes in geometry and magnetization profile are easy to account for
during deformation. Although it is easy to implement six-DOF control on
soft MMRs with 1D magnetization profile, such simple profiles have also
severely constrained the dexterity of these actuators. The feasibility
of executing six-DOF control on sophisticated MMRs, which have 2D or 3D
magnetization profiles for their soft components, has not been explored
yet. Due to these aforementioned limitations, the dexterity of all the
existing soft MMRs is still severely restricted, and these actuators
will not be able to overcome difficult barriers that have strict shape
constraints. As a result, existing MMRs would not have sufficient
agility to effectively negotiate across highly unstructured
environments, and this poses a critical limitation for their targeted
applications. In view of these challenges, it is therefore highly
desirable to create dexterous MMRs that can concurrently possess six-DOF
and perform multiple modes of soft-bodied locomotion.

Here we propose a six-DOF, soft MMR that can roll, two-anchor and
undulating crawl, jump, swim with two different gaits, and climb a
meniscus. Because the proposed MMR has six-DOF, its orientation can be
precisely controlled when it executes these soft-bodied locomotive
gaits, thereby allowing it to achieve very high dexterity. As the
proposed six-DOF MMR can perform seven modes of soft-bodied locomotion,
its dexterity will also be significantly higher than existing six-DOF
soft MMRs that can only execute one type of swimming
gait~\hyperref[csl:43]{(Xu et al., 2021)}. Our proposed actuator is made up of a central
magnetic beam with two identical buoyant components attached to its free
ends (Fig. 1A(i), and please see supplementary information (SI) section
S1 for the fabrication process and material properties of the MMR). The
buoyant components are used to make the proposed soft MMR more neutrally
buoyant so that it can easily swim against gravity in aquatic
environments. To investigate the feasibility of implementing six-DOF
control for a soft MMR with 2D magnetization profile, a
single-wavelength, harmonic magnetization profile~\((\vec{M})\) is
programmed into our compliant beam component (Fig. 1A(ii)). We have
selected a harmonic magnetization profile because this type of profile
has proven to be effective for enabling multimodal
locomotion~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:53]{Ren et al., 2021)}.To maximize the producible sixth-DOF
torque of our soft MMR, the phase shift angle of its magnetization
profile,~\(\varphi\), has been optimized to -90° (Fig. 1A(ii) and
SI section S3). With the selected~\(\varphi\), our proposed MMR
can theoretically produce 1.41-63.9 folds larger sixth-DOF torque than
existing soft MMRs that have a harmonic magnetization profile, i.e.,
those with~\(\varphi\) of 0°~\hyperref[csl:66]{(Zhang \& Diller, 2015}; \hyperref[csl:67]{Zhang et al., 2016}; \hyperref[csl:68]{Huang et al., 2021}; \hyperref[csl:78]{Zhang \& Diller, 2018)} and
45°~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:53]{Ren et al., 2021}; \hyperref[csl:79]{Demir et al., 2021}; \hyperref[csl:80]{Culha et al., 2020)} (SI section S3). Because the magnitude of our
MMR's achievable net magnetic moment,~\(\vec{m}\), is equal to
all other soft MMRs with a harmonic magnetization profile (SI section
S3), this implies that all of these MMRs will have the same actuation
capabilities for their traditional five-DOF motions~\hyperref[csl:42]{(Kummer et al., 2010}; \hyperref[csl:43]{Xu et al., 2021)}.
The enhancement in the sixth-DOF torque for our proposed MMR has
therefore been achieved without compromising their traditional five-DOF
actuation capabilities. To have a fair comparison for our actuation
analysis in SI section S3, we assume that all of these MMRs have the
same geometries and material properties, and the key difference between
them is that they possess different values of~\(\varphi\) in
their~\(\vec{M}\). We have also constrained the harmonic
magnetization profile of all the MMRs to be single wavelength in our
analysis (SI section S3), because this has been a critical requirement
for enabling multimodal soft-bodied locomotion~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:53]{Ren et al., 2021)}.
Using our optimized harmonic magnetization profile, the proposed MMR can
be actuated to produce its desired soft-bodied locomotion and
concurrently achieve optimal six-DOF motions. Since our proposed soft
MMR has adequate stiffness, it can also execute 3D pick-and-place
operations.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.91\columnwidth]{figures/figure1/figure1}
\caption{\selectlanguage{ngerman}{The proposed six-DOF soft MMR. (A) The configuration of the MMR: (i)
Based on the top view, the proposed MMR is made up of a central magnetic
beam with two identical buoyant components attached to its free ends.
(ii) The front view of the MMR that illustrates the harmonic
magnetization profile of the beam with a phase shift angle
(\(\varphi\)) of -90°. A material reference frame is attached to
the MMR and its axes are denoted by the subscript \{M\}. (B) The two
types of deformed configurations of the soft MMR: (i) The MMR assumes a
`U'-shaped configuration when the magnetic field (\(\vec{B}\)) is
applied along the positive~\(z_{\{M\}}\)-axis. The deformed MMR
will produce a corresponding net magnetic moment (\(\vec{m}\))
that is parallel to~\(\vec{B}\). (ii) The MMR will adopt an
inverted `V'-shaped configuration when the direction
of~\(\vec{B}\) is reversed. In this configuration, the MMR will
possess an effective~\(\vec{m}\) along the negative
~\(z_{\{M\}}\)-axis, parallel to~\(\vec{B}\). The
theoretical deformation of the magnetic beam is represented by the
yellow curves, and the rotation about the MMR's~\(\vec{m}\)
represents the sixth-DOF rotation. (C) The spatial reference frames used
for describing the MMR's orientation: (i) the global, (ii) intermediate
and (iii) local reference frames, and their axes are denoted by the
subscript \{\emph{G}\}, \{\emph{I}\} and \{\emph{L}\}, respectively. For
the intermediate and local reference frames, their respective z-axis is
aligned to the MMR's~\(\vec{m}\). (D) The proposed six-DOF
control strategy of our MMR. We determine the
required~\(\vec{B}\) and~\({\vec{B}}_{\text{grad}} \) that can make the
MMR's desired orientation into a minimum potential energy configuration.
Scale bars: 2 mm.
{\label{159397}}%
}}
\end{center}
\end{figure}\selectlanguage{ngerman}

\section*{Actuation Principles}

{\label{279374}}

To describe the physics of our proposed soft MMR, we first attach a
material reference frame to this actuator (Fig. 1A(ii)). The axes and
vectors expressed in this reference frame will be denoted by a subscript
\{\emph{M}\}. When the MMR is undeformed, its~\(\vec{m}\) will be
a null vector (SI section S2B). However, when a magnetic
field,~\(\vec{B}\), is applied along the
positive~\(z_{\{M\}}\)-axis, the MMR will deform and assume a
`U'-shaped configuration (Fig. 1B(i) and please see SI section S2A for
the physics that describes this deformation). In this deformed
configuration, the MMR will possess an effective~\(\vec{m}\) that
is parallel to the applied~\(\vec{B}\). Conversely, the MMR will
assume an inverted `V'-shaped configuration if the direction
of~\(\vec{B}\) is reversed (Fig. 1B(ii)). In the inverted
`V'-shaped configuration, the~\(\vec{m}\) of the soft MMR will be
along the negative~\(z_{\{M\}}\)-axis, parallel to the
applied~\(\vec{B}\) (Fig. 1B(ii)). By adjusting the magnitude
of~\(\vec{B}\), we can also actively control the curvatures of
the MMR in these deformed configurations. Specifically, the curvatures
will become sharper and gentler when the magnitude
of~\(\vec{B}\ \) is increased and decreased, respectively (SI
section S2A). In general, our theoretical model of the MMR's deformation
characteristics in SI section S2A agrees with the experimental data
(e.g., Fig. 1B), and this enables us to control the geometries of the
MMR well. Having precise control over the curvatures of the MMR will be
a critical requirement for the actuator to enable its desired modes of
soft-bodied locomotion.

To analyze the proposed MMR's achievable motions in its deformed
configurations, a spatial reference frame is attached to its body (Fig.
1C(iii)). This reference frame represents the local reference frame of
the actuator, and the axes and vectors expressed in this frame are
denoted by a subscript \{\emph{L}\}. Because it will be easier to
analyze the proposed MMR if its sixth-DOF axis is parallel to one of the
principal axes in the local reference frame~\hyperref[csl:43]{(Xu et al., 2021}; \hyperref[csl:54]{Diller et al., 2016)}, here we
assign~\(z_{\{L\}}\)-axis to be aligned with the
MMR's~\(\vec{m}\) in its `U'-shaped configuration (Fig.
1C(iii)). Based on our local reference frame assignment, the MMR's
rotation about the~\(z_{\{L\}}\)-axis is therefore its sixth-DOF
motion~\hyperref[csl:43]{(Xu et al., 2021}; \hyperref[csl:54]{Diller et al., 2016)}. This frame assignment also indicates that the
MMR's~\(\vec{m}\) will be along the
negative~\(z_{\{L\}}\)-axis when the actuator deforms to its
inverted `V'-shaped configuration. To fully describe the orientation of
the proposed MMR, we also introduce an intermediate reference frame and
a global reference frame (Fig. 1C(i)-(ii)), and the axes and vectors
expressed in these frames are denoted by the subscripts \{\emph{I}\} and
\{\emph{G}\}, respectively. The transformation between the global and
intermediate reference frames can be described mathematically as:

\[\begin{equation}
\begin{matrix}{\vec{v}}_{\left\{G\right\}}=\mathbf{R}_{x}\left(\alpha\right)\mathbf{R}_{y}\left(\beta\right){\vec{v}}_{\left\{I\right\}},\left(1\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

where \(\mathbf{R}_{x}\) and \(\mathbf{R}_{y}\) represent the
standard~\emph{x} -axis and \emph{y} -axis rotational matrices,
respectively (SI section S2B). The variables, \(\alpha\) and
\(\beta\), represent the corresponding desired angular
displacements about these axes (Fig. 1C(i)-(ii)), while the vector,
\(\vec{v}\), represents an arbitrary vector. It is noteworthy
that the \(z_{\{I\}}\)-axis of the MMR defines the desired
orientation of its \(\vec{m}\), implying that the direction of
the sixth-DOF axis can be fully defined based on the values of
\(\alpha\) and \(\beta\). In a similar way, the
transformation between the intermediate and local reference frames can
be described mathematically as:

\[\begin{equation}
\begin{matrix}{\vec{v}}_{\left\{I\right\}}=\mathbf{R}_{z}\left(\theta\right){\vec{v}}_{\left\{L\right\}},\left(2\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

where \(\mathbf{R}_{z}\) represents the standard \emph{z} -axis
rotational matrix (SI section S2B) and \(\theta\) represents
the desired sixth-DOF angular displacement of the MMR (Fig.
1C(ii)-(iii)).

To enable precise orientation control for the proposed MMR, we
control~\(\vec{B}\) and its independent spatial
gradients,~\({\vec{B}}_{\text{grad}}\), in such a way that the desired
orientation of the MMR becomes a minimum potential energy configuration
(SI section S2B-C). Under the influence of these actuating magnetic
signals, the deformed MMR will constantly experience three axes of
restoring torques until it self-aligns to the desired orientation (Fig.
1D and SI section S2B). The presence of the restoring torques will also
help the proposed MMR to reject external disturbances, enabling this
actuator to maintain its desired orientation at all times (SI section
S2B). Of the three axes of restoring torques, two of them are induced by
the phenomenon in which the MMR's~\(\vec{m}\) will tend to align
with the applied~\(\vec{B}\)~\hyperref[csl:23]{(Tasoglu et al., 2014}; \hyperref[csl:30]{Lum et al., 2016}; \hyperref[csl:42]{Kummer et al., 2010)}. By exploiting
this phenomenon, we can therefore control the MMR's two axes of angular
displacements,~\(\alpha\) and~\(\beta\), via
specifying the direction of~\(\vec{B}\) (SI section S2B).
However, controlling the direction of~\(\vec{B}\) will not allow
us to exert the third restoring torque on the MMR, i.e., the restoring
torque about its sixth-DOF axis~\hyperref[csl:42]{(Kummer et al., 2010}; \hyperref[csl:43]{Xu et al., 2021}; \hyperref[csl:54]{Diller et al., 2016}; \hyperref[csl:81]{Diller et al., 2014)}. The sixth-DOF
restoring torque can instead be induced on the MMR via
controlling~\({\vec{B}}_{\text{grad}}\), and this torque will enable us to
precisely control~\(\theta\) (SI section S2B-C). By
controlling~\({\vec{B}}_{\text{grad}}\), we can also apply desired magnetic
forces on the MMR and allow it to translate along three axes so as to
achieve another three-DOF (SI section S2B). A notable advantage of our
control scheme is that we can fully decouple all the magnetic torques
and forces applied to the MMR, effectively allowing the soft actuator to
achieve six-DOF motions (SI section S2B-C). Together with controlling
the time-varying deformations of the MMR, such six-DOF control will be
critical for empowering our proposed actuator to execute multiple modes
of soft-bodied locomotion with precise orientation control. Our six-DOF
control strategy can be implemented successfully because we have
modelled how the critical robotic parameters of our MMR will vary as the
actuator changes its geometry and~\(\vec{M}\) during deformation
(SI sections S2B-C and S3). Examples of such robotic parameters will
include but not limited to the MMR's~\(\vec{m}\)~and producible
normalized sixth-DOF torque (SI sections S2B-C and S3).

\section*{Experimental Results}

{\label{468390}}

We first evaluated the sixth-DOF torque,~\(T_{z,\left\{L\right\}}\), producible
by our proposed soft MMR. Because the producible~\(T_{z,\left\{L\right\}}\) of
the MMR would vary when this actuator underwent different amounts of
deformation (SI section S3), here we constrained the MMR to its
undeformed configuration for easier characterization purposes. As the
buoyant components of the proposed MMR were passive and they could not
produce~\(T_{z,\left\{L\right\}}\) for the MMR, these components were removed in
this experiment. The undeformed magnetic beam component of the MMR was
fixed at the free end of another fixed-free beam in this experiment (Fig.
2A). By observing how the fixed-free beam would deform as we varied the
applied actuating magnetic signals to the proposed MMR, we could deduce
the producible~\(T_{z,\left\{L\right\}}\) of our actuator. This could be
achieved by using a camera to record the angular deflection of the
fixed-free beam's free end,~\(\theta_{\text{tip}}\), and subsequently
applying the Euler-Bernoulli equation to deduce the
generated~\(T_{z,\left\{L\right\}}\) (Fig. 2A and SI section S4B). Based on this
methodology, we established the relationship between the MMR's
generated~\(T_{z,\left\{L\right\}}\) and the applied magnetic signals (Fig. 2B).
Each data point in the graph of Fig. 2B had five trials, and the
gradient of the best fit line represented the MMR's
producible~\(T_{z,\left\{L\right\}}\) after it was normalized according to the
strength of the actuating signals. The normalized sixth-DOF torque of
our MMR was experimentally evaluated to be
1.16\(\times\)10\textsuperscript{-7}~N m\textsuperscript{2}
T\textsuperscript{-1}, and there was a 13.7 \% deviation from our
simulation predictions in SI section S3 (predicted value:
1.02\(\times\)10\textsuperscript{-7} N m\textsuperscript{2}
T\textsuperscript{-1}). In general, the experimental data agreed with
the prediction from our simulations, and the deviation could be caused
by minor fabrication errors and parallax measurement errors from the
camera. Another potential source of experimental error was that the MMR
may not remain perfectly in its undeformed configuration during the
experiments, and this might cause the evaluated normalized sixth-DOF
torque to be larger than expected (SI section S3). As our theoretical
model could also predict the MMR's deformation characteristics well
(Fig. 1B and SI section S2A), the data in Fig. 2B supported our
hypothesis that the proposed MMR could indeed produce 1.41-63.9 folds
larger~\(T_{z,\left\{L\right\}}\) than existing soft MMRs with
single-wavelength, harmonic magnetization profiles \hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:53]{Ren et al., 2021}; \hyperref[csl:63]{Diller et al., 2014}; \hyperref[csl:66]{Zhang \& Diller, 2015}; \hyperref[csl:67]{Zhang et al., 2016}; \hyperref[csl:78]{Zhang \& Diller, 2018}; \hyperref[csl:79]{Demir et al., 2021}; \hyperref[csl:80]{Culha et al., 2020)}
(SI section S3). After investigating the producible sixth-DOF torque of
our proposed MMR, we proceeded to evaluate the locomotion of this
actuator.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.91\columnwidth]{figures/figure-2/figure-2}
\caption{{An experiment that investigated the sixth-DOF torque producible by the
proposed MMR. (A) The magnetic beam component of the MMR was fixed at
the free end of a larger fixed-free beam. As the sixth-DOF torque of the
magnetic beam (\(T_{z,\left\{L\right\}}\)) was generated, the fixed-free beam's
angular deflection at the free end,~\(\theta_{\text{tip}}\), would be
recorded by a camera. In this experiment, the magnetic beam was
constrained to be in its undeformed configuration. (B) The experimental
plot of the deduced~\(T_{z,\left\{L\right\}}\) against the actuating magnetic
signals (\(\frac{\partial B_{x,\left\{L\right\}}}{\partial {y\left\{L\right\}}}\)). Five trials were conducted for each data
point and the error bars represented the corresponding standard
deviation of the data points. The gradient of the best fit line was
computed to be~ 1.16\(\times\)10\textsuperscript{-7}~N
m\textsuperscript{2} T\textsuperscript{-1}, and it represented the
normalized sixth-DOF torque of the proposed MMR (in its undeformed
configuration). Scale bar: 2 mm. ~
{\label{280827}}%
}}
\end{center}
\end{figure}

Rolling could be considered as one of the fastest modes of terrestrial
locomotion~\hyperref[csl:6]{(Ng et al., 2021}; \hyperref[csl:10]{Hu et al., 2018)}. Our proposed MMR could execute this
locomotion by first applying a~\(\vec{B}\) of 13 mT along its
positive~\(z_{\{L\}}\)-axis so that it could assume a `U'-shaped
configuration (Fig. 3A). It is advantageous to adopt such a
configuration since this would be a favorable shape for the MMR to
roll~\hyperref[csl:10]{(Hu et al., 2018)}. Subsequently, the MMR would be able to start
rolling when we continuously rotate the magnetic field in
the~\(y_{\{L\}}z_{\{L\}}\) or~\(z_{\{L\}}x_{\{L\}}\) planes (\(\left|\vec{B}\right|\)
= 13 mT). Because our proposed MMR had six-DOF, this actuator could
choose to roll along its length (rotating about
the~\(x_{\{L\}}\)-axis) or along its width (rotating about
the~\(y_{\{L\}}\)-axis) (Fig. 3A and SI Video S1). The rolling
direction of the MMR could also be steered via controlling its angular
displacement about the~\(z_{\{L\}}\)-axis with~\({\vec{B}}_{\text{grad}}\)
(Fig. 3B and SI Video S1). In comparison, rolling along the length of
the MMR would be faster than rolling along its width since the length of
our actuator was longer than its width (Fig. 1A and SI section S4C).
This is assuming that the rotating frequencies of~\(\vec{B}\)
were identical for both rolling scenarios. Despite being slower, rolling
along the width of the MMR had its unique merits too. Because the
curvatures of the actuator could be controlled to be much gentler for
this mode of rolling (Fig. S11), the MMR could potentially squeeze
through smaller openings if it was rolling along its width. By having
two modes of rolling, the proposed MMR was therefore very robust. On
obstacle-free terrains, it could choose to roll quickly along its length
( 2.46 mm s\textsuperscript{-1} with a 0.25
Hz~rotating~\(\vec{B}\)). Alternatively, it could choose to roll
along its width if it had to negotiate across barriers with narrow
openings in the environment ( 1.85 mm s\textsuperscript{-1} with a 0.25
Hz rotating~\(\vec{B}\)). While existing rolling MMRs were also
steerable, they had only demonstrated rolling along their
lengths~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:29]{Gu et al., 2020}; \hyperref[csl:53]{Ren et al., 2021}; \hyperref[csl:82]{Yang et al., 2020)}. As a result, our proposed MMR would be much
more dexterous than existing rolling MMRs ~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:29]{Gu et al., 2020}; \hyperref[csl:53]{Ren et al., 2021}; \hyperref[csl:82]{Yang et al., 2020)} because it
would have greater versatility to roll across environments that have
difficult barriers. To demonstrate the remaining DOF of our proposed
MMR, we commanded it to translate along its three principal axes in the
local reference frame (Fig. 3C and SI Video S1). Before executing these
translations, we first applied a~\(\vec{B}\) along
the~\(z_{\{L\}}\)-axis of the MMR so that it could deform into a
`U'-shaped configuration to possess an
effective~\(\vec{m}\).~Corresponding~\({\vec{B}}_{\text{grad}}\) was
then applied to exert the desired magnetic forces on the MMR to
translate it along the~\(x_{\{L\}}\)-,~\(y_{\{L\}}\)-,
and~\(z_{\{L\}}\)-axes (Fig. 3C and SI Video S1). During the
translations along the~\(x_{\{L\}}\)-
and~\(y_{\{L\}}\)-axes, an upward magnetic force was also applied
to the MMR so that it could reduce the normal force and thus friction
induced by the substrate (Fig. 3C(i)-(ii)). A notable observation in the
experiments shown in Fig. 3A-C and SI Video S1 was that the rotations
and translations of the proposed MMR could be fully decoupled, implying
that this actuator was able to attain six-DOF control. It would be
advantageous to be able to apply magnetic forces on the proposed MMR
because these forces could potentially enhance the efficiency of the
actuator's locomotion~\hyperref[csl:10]{(Hu et al., 2018)}. For instance, we could apply
upward forces on the MMR so that it could reduce its friction with the
ground, jump higher or swim against gravity easier. However, translating
the MMR solely by magnetic forces might be detrimental because it would
be challenging to position the actuator precisely with such control
strategies~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:83]{Zhang et al., 2015)}.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.91\columnwidth]{figures/figure3/figure3}
\caption{\selectlanguage{ngerman}{The rolling, translations, and two-anchor crawling locomotion of the
proposed MMR. Except for (B), the remaining sub-figures were created by
superpositioning the snapshots of the MMR at different timestamps in SI
Videos S1-2. (A) The rolling locomotion of the MMR: (i) Rolling along
the length of the MMR by continuously rotating it about
the~\(x_{\{L\}}\)-axis. (ii) Rolling along the width of the MMR by
continuously rotating it about the~\(y_{\{L\}}\)-axis. This mode of
rolling could be accomplished with gentler curvatures compared to
rolling along the MMR's length. (B) The MMR's sixth-DOF rotation
(rotating about the~\(z_{\{L\}}\)-axis). This rotation enabled the
MMR to steer its rolling locomotion. A red line was added to one of the
MMR's free ends to better illustrate this rotation. (C) Translating the
MMR in a paraffin oil reservoir. The MMR was commanded to translate
along its~\(x_{\{L\}}\)-,~\(y_{\{L\}}\)-
and~\(z_{\{L\}}\)-axes as shown in (i), (ii), and (iii),
respectively. Together with the rolling locomotion, these translations
indicated that the proposed MMR had six-DOF, i.e., having three types of
rotations and three types of translations. (D) By controlling the
angular displacements about its~\(x_{\{L\}}\)-axis, the MMR could
use the two-anchor crawling locomotion to ascend an inclined slope of
20°. (E) Using the two-anchor crawling locomotion, the MMR could move
across a narrow path with strict shape constraints. This was possible
because the MMR could precisely control its angular displacement about
the~\(y_{\{L\}}\)-axis to adopt a favorable shape for crossing this
narrow passage. Since the MMR could also precisely control its sixth-DOF
angular displacement about the \(z_{\{L\}}\)-axis, it could also
execute a sharp turn at the junction of a confined `L'-shaped path. The
legend showed the local reference frame of the MMR as well as the types
of arrows representing rotations about different axes and the MMR's
trajectory. Scale bars: 2 mm.
{\label{820217}}%
}}
\end{center}
\end{figure}\selectlanguage{ngerman}

\section*{}

{\label{468546}}

\section*{\texorpdfstring{\emph{SI Video
S1}}{SI Video S1}}

{\label{468546}}

\subsection*{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~Rolling locomotion and
translations~}

{\label{581784}}

Rich media available at \url{https://drive.google.com/file/d/1FpdxaOB0psp6XWmRz3YF9MhgvVChZS1X/view?usp=sharing}

In addition to rolling and translations, our proposed MMR could also
execute two-anchor crawling locomotion. A significant advantage of this
terrestrial locomotion was that it could allow the MMR to position
itself more precisely in unstructured environments compared to rolling
\hyperref[csl:10]{(Hu et al., 2018)}. This is because this mode of locomotion allowed the
MMR to precisely tune its stride length~\hyperref[csl:10]{(Hu et al., 2018)}. To implement
the two-anchor crawling locomotion, we used the free ends of the MMR to
serve as the two contact points with the substrate. This locomotion was
cyclic, and each cycle began by first allowing the MMR to assume a
`U'-shaped configuration and allowing both of its free ends to be in
contact with the substrate (Fig. S12D(i), the applied~\(\vec{B}\)
was along~\(z_{\{L\}}\)-axis with a magnitude of 18 mT).
Subsequently, we would anchor the MMR with its rear end so that it could
lift the front end of the body upwards via a rotation about
the~\(x_{\{L\}}\)-axis (Fig. S12D(ii)). This could be achieved by
rotating the applied~\(\vec{B}\) counterclockwise (15-30°) in
the~\(y_{\{L\}}z_{\{L\}}\) plane. Once the front end of the MMR was lifted,
we rotated~\(\vec{B}\) clockwise back to its original direction
and reduced its magnitude to 4 mT so that the actuator could extend and
produce a net displacement forward (Fig. S12D(iii)). Following this
extension, the front end of the MMR would resume contact with the
substrate and serve as the new anchor point. By
rotating~\(\vec{B}\) clockwise in the~\(y_{\{L\}}z_{\{L\}}\) plane
(15-30°) while increasing its magnitude to 18 mT, we could lift and
contract the rear end of the MMR (Fig. S12D(iv)). Thereafter, we
rotated~\(\vec{B}\) counterclockwise back to its original
direction to complete one cycle and produce a net displacement (Fig.
S12D(v)). Since the proposed MMR possessed six-DOF, we could control
this actuator's sixth-DOF angular displacement about
its~\(z_{\{L\}}\)-axis to steer it along the desired direction. In
addition, we could also rotate the MMR about
its~\(y_{\{L\}}\)-axis and tilt its body at an angle as it
executed the two-anchor crawling locomotion. By tilting the MMR, this
actuator could better conform to the strict shape constraints of a
narrow passage so that it could easily cross this passage with the
two-anchor crawling locomotion (Fig. 3E and SI Video S2). With six-DOF
control, the MMR could also control the angular displacements about
its~\(x_{\{L\}}\)-axis and perform the two-anchor crawling
locomotion to ascend an inclined slope of 20° (Fig. 3D and SI Video S2),
and successfully execute sharp turns by rotating about
its~\(z_{\{L\}}\)-axis at the junction of a confined `L'-shaped
path (Fig. 3E and SI Video S2). Although existing MMRs were steerable
when they executed the two-anchor crawling locomotion on obstacle-free
flat terrains~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:79]{Demir et al., 2021}; \hyperref[csl:80]{Culha et al., 2020)}, they could not use this locomotion to
concurrently accomplish the following tasks: (i) climb an inclined
slope; (ii) perform effective turns on sharp corners; and (iii) tilt at
an angle to better conform and thus crossing a narrow passage with
strict shape constraints. The proposed MMR therefore demonstrated
significantly higher dexterity than existing MMRs that could perform
two-anchor crawling locomotion~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:49]{Kim et al., 2011}; \hyperref[csl:73]{Wu et al., 2020}; \hyperref[csl:79]{Demir et al., 2021}; \hyperref[csl:80]{Culha et al., 2020}; \hyperref[csl:82]{Yang et al., 2020}; \hyperref[csl:84]{Ju et al., 2021)}.

\par\null

\section*{\texorpdfstring{\emph{SI Video
S2}}{SI Video S2}}\label{si-video-s2}

\subsection*{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~Two-anchor crawling locomotion~}\label{two-anchor-crawling-locomotion}

Rich media available at \url{https://drive.google.com/file/d/15zbajXe3WNx11Ra0QKLjivvbiayIMF5o/view?usp=sharing}

Terrestrial locomotion like rolling and two-anchor crawling might not be
applicable if the MMRs had to negotiate across highly confined and
enclosed channels. In such scenarios, executing undulating crawling
would be an effective locomotion for MMRs to overcome such obstacles
\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:53]{Ren et al., 2021)}. When our proposed MMR entered such channels, we
could continuously rotate~\(\vec{B}\) (15-20 mT) in
the~\(y_{\{L\}}z_{\{L\}}\) plane so that the actuator could undulate and
generate a traveling wave along its body (Fig. 4 and SI Video S3). Due
to the generated traveling wave, the proposed MMR was able to induce a
net propulsive force from the surroundings to thrust itself forward.
Since our MMR had six-DOF, it could precisely control the angular
displacements about its~\(y_{\{L\}}\)- and~\(z_{\{L\}}\)-axes
to negotiate through a complex channel that required it to sequentially
undulating crawl across two perpendicular planes (Fig. 4A and SI Video
S3). The MMR could also execute undulating crawling to ascend an
inclined slope of 20° by controlling the angular displacement about
its~\(x_{\{L\}}\)-axis, and rotate about
its~\(z_{\{L\}}\)-axis to make a sharp turn at the junction of a
confined `L'-shaped channel (Fig. 4B and SI Video S3). As previous MMRs
could only demonstrate undulating crawling through straight or planar
channels~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:53]{Ren et al., 2021}; \hyperref[csl:72]{Du et al., 2020)}, the advancement of our proposed actuator
was therefore highly significant since it could use this mode of
locomotion to negotiate across a much more diverse range of confined,
enclosed channels that had convoluted 3D geometries.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.91\columnwidth]{figures/figure-4/figure-4}
\caption{\selectlanguage{ngerman}{A demonstration of the MMR's undulating crawling locomotion. The
sub-figures were created by superpositioning the snapshots of the MMR at
different timestamps in SI Video S3. (A) The MMR could use its
undulating crawling locomotion to negotiate through a complex channel by
using this gait to crawl across two perpendicular planes. During this
process, the MMR had to control its angular displacement about
the~\(y_{\{L\}}\)-axis to transit between the planes. (B) The MMR
could execute undulating crawling to ascend an inclined slope of 20° by
controlling its angular displacements about the~\(x_{\{L\}}\)-axis.
By rotating about its~\(z_{\{L\}}\)-axis, the MMR could perform a
sharp turn at the junction of a confined `L'-shaped channel. The legend
showed the local reference frame of the MMR as well as the types of
arrows representing rotations about different axes and the MMR's
trajectory. Scale bars: 2 mm.
{\label{241052}}%
}}
\end{center}
\end{figure}\selectlanguage{ngerman}

\par\null

\section*{\texorpdfstring{\emph{SI Video
S3}}{SI Video S3}}\label{si-video-s3}

\subsection*{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~Undulating crawling locomotion~}\label{undulating-crawling-locomotion}

Rich media available at \url{https://drive.google.com/file/d/1VGC3l\_9Abg8G6edJb9QDcpa\_pU3owgIM/view?usp=sharing}

Similar to undulating crawling, jumping would be an effective locomotion
for the MMR to overcome difficult obstacles~\hyperref[csl:6]{(Ng et al., 2021}; \hyperref[csl:10]{Hu et al., 2018)}.
Specifically, MMRs could execute such locomotion to jump across barriers
that had heights greater than their body length ~\hyperref[csl:6]{(Ng et al., 2021}; \hyperref[csl:10]{Hu et al., 2018)} .
The jumping locomotion could be executed by first deforming the proposed
actuator into a `U'-shaped configuration (the applied~\(\vec{B}\)
was along~\(z_{\{L\}}\)-axis with a magnitude of 20 mT) and then
orientating the MMR's~\(y_{\{L\}}z_{\{L\}}\ \) plane such that
its~\(z_{\{L\}}\)-axis had a clockwise 5° angle away from the
vertical upright direction. Subsequently, we rotated~\(\vec{B}\)
clockwise in the~\(z_{\{L\}}x_{\{L\}}\) plane (175°) via a step change so
that a strong magnetic torque could be exerted on the MMR to enable a
rigid-body rotation while the actuator concurrently underwent a
deformation. During this rotation and shape-changing process, the free
ends of the MMR would eventually strike the substrate and gain a
momentum to jump against gravity (Fig. 5A and SI Video S4). To allow the
proposed MMR to jump higher, we also applied an upward magnetic force to
this actuator via~\({\vec{B}}_{\text{grad}}\) (jumping height: 6.8 mm/ 1.06 body
length, jumping range: 9.56-11.2 mm/ 1.49-1.75 body length). In general,
we could also vary the jumping direction by tuning the MMR's orientation
such that its~\(x_{\{L\}}\)-axis is parallel to the desired
horizontal projectile motion before initiating the jump. A notable
advantage of our proposed actuator over existing soft jumping MMRs
(i.e.,~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:85]{Zhong \& Wei, 2021}; \hyperref[csl:86]{Wang et al., 2021)}) was that it could precisely control its three
axes of angular displacements in mid-air after the jumping locomotion
was executed. For example, we had shown that the proposed MMR could
control its angular displacements about the~\(z_{\{L\}}\)-
and~\(x_{\{L\}}\)-axes while it executed the jumping locomotion in
Fig. 5A(i) and SI Video S4. Similarly, this MMR could also precisely
control its angular displacements about the~\(z_{\{L\}}\)-
and~\(y_{\{L\}}\)-axes while it performed the jump in Fig. 5A(ii)
and SI Video S4. By having the ability to precisely control its
orientation during a jump, our proposed MMR could therefore have the
potential to adjust itself into a favorable shape necessary for jumping
across barriers, which have openings with strict shape constraints. Such
an ability would be highly desirable, but it had not been achieved by
existing jumping MMRs yet~\hyperref[csl:6]{(Ng et al., 2021}; \hyperref[csl:10]{Hu et al., 2018}; \hyperref[csl:85]{Zhong \& Wei, 2021}; \hyperref[csl:86]{Wang et al., 2021)}.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.91\columnwidth]{figures/figure-5/figure-5}
\caption{{Demonstrating the jumping and jellyfish-like swimming locomotion of the
proposed MMR. The sub-figures were created by superpositioning the
snapshots of the MMR at different timestamps in SI Videos S4-5. (A) The
MMR's jumping locomotion: (i) During the jump, the MMR could control its
angular displacements about the~\(z_{\{L\}}\)-
and~\(x_{\{L\}}\)-axes. (ii) During this jump, the MMR showed that
it could control its angular displacements about the~\(z_{\{L\}}\)-
and~\(y_{\{L\}}\)-axes. (B-D) The jellyfish-like swimming
locomotion of the MMR. (B) As the MMR swam, it could control its angular
displacement about the~\(x_{\{L\}}\)-axis to steer the swimming
direction. (C) Similar to (B), the MMR could steer the swimming
direction by controlling its angular displacement about
the~\(y_{\{L\}}\)-axis. (D) The MMR could also continuously rotate
about its sixth-DOF axis (the \(z_{\{L\}}\)-axis) as it executed
this swimming locomotion. The legend showed the local reference frame of
the MMR as well as the types of arrows for representing rotations about
different axes and the MMR's trajectory. Scale bars: 2 mm.
{\label{941502}}%
}}
\end{center}
\end{figure}

\par\null

\section*{\texorpdfstring{\emph{SI Video
S4}}{SI Video S4}}\label{si-video-s4}

\subsection*{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ ~Jumping locomotion~}\label{jumping-locomotion}

Rich media available at \url{https://drive.google.com/file/d/1btoz2Q53bhX530tsKiGaSlpl8LxhZ7ai/view?usp=sharing}

As an amphibious robot, our proposed MMR could also mimic the swimming
locomotion of a biological jellyfish to maneuver in water bulk. This
soft locomotive gait could be realized by oscillating~\(\vec{B}\)
along the~\(z_{\{L\}}\)-axis of the MMR so that the actuator could
alternate between its `U'- and inverted `V'-shaped configurations (SI
Video S5). By oscillating~\(\vec{B}\) at a frequency of 6-8 Hz
(maximum\(\left|\vec{B}\right|\): 24 mT), the proposed MMR could induce a net
propulsive force from its surroundings to swim against gravity (average
velocity: 7.65-10.4 mm s\textsuperscript{-1}). It was possible to swim
with this reciprocal gait because the MMR was able to attain an
adequately high Reynolds number of 54.9-74.6~to gain sufficient inertia
effects (SI Section S4D). The swimming direction of the MMR could also
be steered by controlling its angular displacements about
the~\(x_{\{L\}}\)- and~\(y_{\{L\}}\)-axes to dictate the
propulsion direction (Fig. 5B-C and SI Video S5). As it swam, the
proposed MMR could also rotate about its sixth-DOF axis
(the~\(z_{\{L\}}\)-axis) via controlling~\({\vec{B}}_{\text{grad}}\) (Fig.
5D and SI Video S5). Because we could precisely control the MMR's three
axes of angular displacements, we could potentially command this
actuator to adjust its orientation and adopt a favorable shape to easily
swim through barriers with openings that had strict shape constraints.
Having such dexterity would be a significant advancement over existing
five-DOF MMRs that could perform jellyfish-like swimming
locomotion~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:30]{Lum et al., 2016}; \hyperref[csl:64]{Ren et al., 2019)}. This is because although existing
five-DOF, jellyfish-like MMRs could also steer their swimming direction,
they would lack the required sixth-DOF dexterity to swim across
complicated barriers ~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:30]{Lum et al., 2016}; \hyperref[csl:64]{Ren et al., 2019)}. While the six-DOF soft MMR in
our previous work could also fully control its orientation to swim
across difficult barriers, it was restricted to having one soft-bodied
locomotion~\hyperref[csl:43]{(Xu et al., 2021)}. Thus, our previous actuator
in~\hyperref[csl:43]{(Xu et al., 2021)} would not be able to perform the other types of
gaits demonstrated by our proposed MMR.

\par\null

\section*{\texorpdfstring{\emph{SI Video
S5}}{SI Video S5}}\label{si-video-s5}

\subsection*{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~Jellyfish-like swimming
locomotion~}\label{jellyfish-like-swimming-locomotion}

Rich media available at \url{https://drive.google.com/file/d/1jr6sykD4mAAvEuvVFFcBA7oBI\_87rQ53/view?usp=sharing}

Because our proposed MMR could concurrently possess six-DOF and execute
multimodal soft-bodied locomotion, it was able to demonstrate
significantly higher dexterity than existing MMRs
(e.g.,~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:43]{Xu et al., 2021}; \hyperref[csl:53]{Ren et al., 2021}; \hyperref[csl:64]{Ren et al., 2019}; \hyperref[csl:72]{Du et al., 2020)}). To demonstrate such capabilities, we
commanded our MMR to negotiate across three synthetic, unstructured
environments that had challenging obstacles. For the first unstructured
environment, the MMR began by jumping through a narrow slot to reach a
higher ground (height: 6.5 mm) (Fig. 6A and SI Video S6). This was
possible because the MMR was able to adjust its orientation during the
jump such that it could adopt a favorable shape to fit through the slot.
An additional video was taken in slow motion to better illustrate this
jumping process (Fig. 6B and SI Video S6). Once the actuator arrived at
the higher ground, it could use its two-anchor crawling locomotion to
precisely maneuver itself to reach the narrow opening of a wall on its
right (Fig. 6A and SI Video S6). In order to bypass this wall, the MMR
had to roll along its width (rotating about the~\(y_{\{L\}}\)-axis)
so that it could adopt a gentle curvature necessary for squeezing
through the wall's narrow opening (Fig. 6A and SI Video S6, average
speed: 0.37 mm s\textsuperscript{-1}); otherwise, it would not be able
to roll across this rigid barrier. After passing through this wall, the
MMR controlled its sixth-DOF angular displacement about
the~\(z_{\{L\}}\)-axis to steer the rolling direction through the
opening of the final wall. Because this opening was much larger than the
previous one, the MMR could pass through by rolling along its length
(rotating about the~\(x_{\{L\}}\)-axis) and achieved a relatively
fast rolling speed to reach the final destination (SI Video S6, average
speed: 0.46 mm s\textsuperscript{-1}). It is noteworthy that existing
five-DOF MMRs, which could execute multimodal soft-bodied locomotion,
would not be able to negotiate across this unstructured
environment~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:53]{Ren et al., 2021}; \hyperref[csl:72]{Du et al., 2020)}. This is because such existing soft MMRs
would neither be able to fully control their orientation to jump through
narrow slots nor roll along their width (shorter side).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.91\columnwidth]{figures/figure-6/figure-6}
\caption{\selectlanguage{ngerman}{The six-DOF MMR utilized its multimodal soft-bodied locomotion to
navigate through various synthetic, unstructured environments with
complex barriers and performed a 3D pick-and-place operation. Except for
(E), the sub-figures were created by superpositioning the snapshots of
the MMR at different timestamps in SI Videos S6-8. (A) The MMR navigated
through an unstructured terrestrial environment via the jumping and
rolling locomotion. The MMR first arrived at a higher ground by jumping
through a narrow slot via precisely controlling its orientation.
Subsequently, the MMR rolled along its width with a gentle curvature to
squeeze through the first wall's narrow opening and then rolled along
its length to bypass the second larger wall with a relatively fast
rolling speed. (B) The MMR jumped through a narrow slot, with frames
captured in slow motion. (C) The MMR navigated through an unstructured
terrestrial environment via the two-anchor and undulating crawling
locomotion. The MMR was first commanded to two-anchor crawl across a
path with strict shape constraints and subsequently climbed up a
stair-like structure and moved cross a gap. To negotiate across a
confined and enclosed channel with convoluted geometries, the MMR
activated its undulating crawling locomotion to ascend an upslope of
10°, perform a sharp turn at the junction of an `L'-shaped channel, and
transit to crawl across a vertical plane. To better illustrate the
crawling locomotion within the transparent complex channel, we included
rendered images of the undulating crawling MMR into this sub-figure. (D)
The MMR navigated in a hybrid aquatic-terrestrial environment. The MMR
first performed its jellyfish-like swimming locomotion with precise
orientation control to bypass a barrier and then rolled across an
air-water interface to transit from water to land. (E) The MMR performed
a 3D pick-and-place operation. Snapshots were taken from SI Video S9.
(i) This operation would require the MMR to grab, transport and stack
two objects at the desired placement location (highlighted by the dotted
red ellipse). (ii) The MMR rolled towards an object and grabbed it.
(iii)-(iv) The MMR held on to the object and transported it to the
desired placement location. (v)-(viii) The MMR repeated the same process
for the second object so that it could stack both objects on top of each
other. Scale bars: 2 mm.~
{\label{914655}}%
}}
\end{center}
\end{figure}\selectlanguage{ngerman}

\par\null

\section*{\texorpdfstring{\emph{SI Video
S6}}{SI Video S6}}\label{si-video-s6}

\subsection*{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~Multimodal
locomotion: Jumping and
rolling}\label{multimodal-locomotion-jumping-and-rolling}

Rich media available at \url{https://drive.google.com/file/d/1yf91BYVsvHMY5lNtFHmoOE1RPzfVJE9s/view?usp=sharing}

Next, we commanded the proposed MMR to overcome all the obstacles in the
second unstructured terrain (Fig. 6C and SI Video S7). The MMR began by
first rotating about its~\(y_{\{L\}}\)-axis so that it could tilt
and adopt a favorable orientation while executing the two-anchor
crawling locomotion. This tilting was necessary as it allowed the
actuator to two-anchor crawl across a path that had strict shape
constraints (Fig. 6C and SI Video S7). Subsequently, we steered the MMR
while it continued to use the two-anchor crawling locomotion to climb up
the stair-like structure and move across the gap to reach the left side
of the terrain (Fig. 6C and SI Video S7). The actuator was then
commanded to negotiate across a confined and enclosed channel, which had
convoluted geometries. To move across such a complex channel, the MMR
activated its undulating crawling locomotion to ascend an upslope of 10°
until it reached a horizontal plane parallel to the ground (Fig. 6C and
SI Video S7). Subsequently, it continued to use this locomotion in the
horizontal plane to perform a sharp turn at the junction of an
`L'-shaped channel and transited to crawl across a vertical plane to
reach its final destination (Fig. 6C and SI Video S7). We would like to
highlight that existing soft MMRs would not be able to navigate across
this unstructured terrain because they would not be able to tilt while
performing the two-anchor crawling locomotion~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:49]{Kim et al., 2011}; \hyperref[csl:73]{Wu et al., 2020}; \hyperref[csl:79]{Demir et al., 2021}; \hyperref[csl:80]{Culha et al., 2020}; \hyperref[csl:82]{Yang et al., 2020}; \hyperref[csl:84]{Ju et al., 2021)}. As a
result, they would not be able to cross the narrow path that had strict
shape constraints. Furthermore, existing soft MMRs would not have the
dexterity to undulating crawl across convoluted, enclosed channels with
non-planar geometries~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:53]{Ren et al., 2021}; \hyperref[csl:72]{Du et al., 2020)}.

\par\null

\section*{\texorpdfstring{\emph{SI Video
S7}}{SI Video S7}}\label{si-video-s7}

\subsection*{~ ~ ~ ~ ~ ~ ~ ~ ~ Multimodal locomotion: Two-anchor crawling
and undulating
crawling}\label{multimodal-locomotion-two-anchor-crawling-and-undulating-crawling}

Rich media available at \url{https://drive.google.com/file/d/179WATu4KafwUNRCuvyYwDQSa-cH0GAX8/view?usp=sharing}

The third unstructured environment, which the proposed MMR navigated in,
was a hybrid aquatic-terrestrial environment (Fig. 6D and SI Video S8).
The MMR began at the bed of the water reservoir, and it started to swim
against gravity by executing the jellyfish-like swimming locomotion
(Fig. 6D and SI Video S8, average speed: 9.37 mm s\textsuperscript{-1},
Reynolds number: 67.2). Because the proposed MMR was able to precisely
control its three axes of angular displacements, it was able to steer
towards the opening of a barrier and subsequently adopted a favorable
orientation which could allow it to negotiate across the opening (Fig.
6D and SI Video S8); otherwise, the MMR would not be able to cross this
rigid obstacle. Upon reaching the water surface, the proposed MMR
adopted a rolling locomotion to peel its surface off the air-water
interface and eventually transited from water to land (Fig. 6D and SI
Video S8). To allow the MMR to peel its surface off the air-water
interface more easily, we had coated the buoyant components of the MMR
with a thin layer of hydrophobic polymer (Ecoflex 00-10). Nonetheless,
because the surface tension of the air-water interface was strong, the
MMR had to perform several rounds of rolling before it could eventually
break off this surface tension (SI Video S8). Another limitation of the
MMR was that its free ends were found to be stuck to a water droplet
after it landed. Hence, the MMR was unable to recover to its flat
configuration and had to remain in the `U'-shaped configuration. In
theory, this issue could be mitigated if the surfaces of the MMR could
be coated with a thin layer of Teflon to become superhydrophobic because
this would prevent the water droplets from sticking to the actuator
after it emerged from water~\hyperref[csl:87]{(Im et al., 2010}; \hyperref[csl:88]{Van Der Wal \& Steiner, 2007)}. We aim to investigate
this approach more thoroughly in the future. Despite these limitations,
we would like to highlight that existing five-DOF, soft MMRs would not
have the dexterity to overcome the obstacle in the reservoir because
they could not fully control their orientation and adopt a favorable
shape to swim through such a tight opening with strict shape
constraints~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:30]{Lum et al., 2016}; \hyperref[csl:64]{Ren et al., 2019)}. Although the six-DOF, jellyfish-like MMR
in our previous work could theoretically overcome this barrier too, it
could not perform a similar rolling locomotion to transit from water to
land~\hyperref[csl:43]{(Xu et al., 2021)}. The experiments in Fig. 6 and SI Videos S6-8
therefore demonstrated that our proposed MMR had unprecedented dexterity
since it was able to negotiate across difficult barriers that are
impassable by existing small-scale devices (e.g.,~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:43]{Xu et al., 2021}; \hyperref[csl:53]{Ren et al., 2021}; \hyperref[csl:64]{Ren et al., 2019}; \hyperref[csl:72]{Du et al., 2020)}).

\par\null

\section*{\texorpdfstring{\emph{SI Video
S8}}{SI Video S8}}\label{si-video-s8}

\subsection*{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Multimodal locomotion:
Jellyfish-like swimming and
landing~}\label{multimodal-locomotion-jellyfish-like-swimming-and-landing}

Rich media available at \url{https://drive.google.com/file/d/1ohZekOJdFFliUTitkPYryV231dgpUkD9/view?usp=sharing}

Because our proposed MMR had adequate stiffness, it could also perform
3D pick-and-place operations. For instance, our MMR could roll towards
an object and control its `U'-shaped curvature to grip that object (Fig.
6E(i)-(ii) and SI Video S9). Subsequently, the MMR could hold on to the
object while transporting it to the desired placement location via the
rolling locomotion (Fig. 6E(iii) and SI Video S9). Upon arrival, the MMR
could reduce its gripping force and release the object (Fig. 6E(iv) and
SI Video S9). By repeating this process for the second object (Fig.
6E(v)-(vii)), the MMR could successfully stack these objects on top of
each other (Fig. 6E(viii) and SI Video S9). To evaluate the payload of
our soft MMR, we had commanded our MMR to grab various objects and use
the rolling locomotion to transport them to a desired placement location
(Fig. S13). Our experiments indicated that the proposed MMR was able to
carry objects that had 0.5 to 3 folds of its weight (29.4 \selectlanguage{greek}µ\selectlanguage{english}N). We also
observed that the MMR would not be able to grab an object firmly when
the object's weight was increased to 3.5 times folds of its weight. In
addition, we had computed the theoretical positioning resolution of our
MMR during the pick-and-place operation. Assuming that the MMR was
operating on an obstacle-free flat terrain, the angular resolution of
the MMR when it was rotating about the \(x_{\{L\}}\)-
and~\(y_{\{L\}}\)-axes would be 0.57\selectlanguage{ngerman}° (SI section S4F). The
sixth-DOF angular resolution of the MMR would be 0.36° (SI section S4F).
These angular resolutions were computed based on the resolution of our
actuating magnetic signals (SI section S4A). During the pick-and-place
operation, the shape of the rolling MMR could be approximated as a
circle, and the radius of this best fit circle was about 0.95 mm (Fig.
S14). Under no-slip conditions, the theoretical translational resolution
of the MMR when it rolled along its length could be computed as 9.5 \selectlanguage{greek}µ\selectlanguage{english}m
via the product of its radius and \(y_{\{L\}}\)-axis angular
resolution (SI section S4F). The positioning resolution, however, might
become coarser if the no-slip condition was violated. In the future, we
aimed to explore the feasibility of performing 3D pick-and-place
operations in conjunction with the MMR's other modes of locomotion. As
it was challenging to draw the magnetic actuating signals applied to our
MMR in most of the figures and videos, we had instead presented these
experimental signals separately in Fig.s S15--S31.

\par\null

\section*{\texorpdfstring{\emph{SI Video
S9}}{SI Video S9}}\label{si-video-s9}

\subsection*{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 3D pick-and-place
operation~}\label{d-pick-and-place-operation}

Rich media available at \url{https://drive.google.com/file/d/12ERhoxl121qVGSd14zNdemUiNgfsi7nN/view?usp=sharing}

\section*{Discussion}

{\label{189801}}

In this section, we will provide additional discussions on the
locomotion of our proposed MMR. Relevant comparisons between the
proposed MMR and other similar existing devices will also be made here.
To make our discussions more complete, the limitations of our MMR will
also be explicitly discussed. Furthermore, we will also suggest feasible
future works that may potentially address or mitigate these issues.

Similar to the five-DOF soft MMRs presented by Hu et
al.~\hyperref[csl:10]{(Hu et al., 2018)}, our proposed actuator can also perform
undulating swimming and meniscus-climbing on an air-water interface (SI
section S4E). Hence, the proposed MMR is able to execute a total of
seven types of soft-bodied locomotion. However, possessing six-DOF will
not enhance the dexterity of the MMR when it is executing the undulating
swimming and meniscus-climbing gaits. This is because these two modes of
locomotion require the MMRs to be constrained on an air-water
interface~\hyperref[csl:10]{(Hu et al., 2018)}. Therefore, the motions of the MMR are
restricted to 2D for these gaits, and this has in turn eliminated the
benefits of having the additional sixth-DOF rotation. In the future, we
aim to endow the proposed MMR with more modes of soft-bodied locomotion,
which can benefit from having six-DOF. We will also investigate feasible
gaits that can eventually allow our proposed MMR to reliably transit
from an air-liquid interface to the liquid bulk.

\par\null

\section*{\texorpdfstring{\emph{SI Video
S10}}{SI Video S10}}

\subsection*{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Undulating swimming locomotion}

Rich media available at \url{https://drive.google.com/file/d/1s0ShtycGSgpD22xpXM4mzKi9Db-yT6oS/view?usp=sharing}

In theory, it is also possible to implement six-DOF control on previous
soft MMRs with harmonic magnetization profiles~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:53]{Ren et al., 2021}; \hyperref[csl:63]{Diller et al., 2014}; \hyperref[csl:66]{Zhang \& Diller, 2015}; \hyperref[csl:67]{Zhang et al., 2016}; \hyperref[csl:78]{Zhang \& Diller, 2018}; \hyperref[csl:79]{Demir et al., 2021}; \hyperref[csl:80]{Culha et al., 2020)}. The
key reason why such MMRs can only achieve five-DOF motions previously is
because their orientation is controlled solely by~\(\vec{B}\)
~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:53]{Ren et al., 2021}; \hyperref[csl:63]{Diller et al., 2014}; \hyperref[csl:66]{Zhang \& Diller, 2015}; \hyperref[csl:67]{Zhang et al., 2016}; \hyperref[csl:78]{Zhang \& Diller, 2018}; \hyperref[csl:79]{Demir et al., 2021}; \hyperref[csl:80]{Culha et al., 2020)}.~Unlike these previous works, here we use
both~\(\vec{B}\) and~\({\vec{B}}_{\text{grad}}\) to control the
orientation of the proposed MMR, and this enables us to generate a
sixth-DOF restoring torque for this soft actuator, effectively allowing
it to realize six-DOF motions. Although it is possible to implement
six-DOF control on previous soft MMRs with harmonic magnetization
profiles ~\hyperref[csl:10]{(Hu et al., 2018}; \hyperref[csl:53]{Ren et al., 2021}; \hyperref[csl:63]{Diller et al., 2014}; \hyperref[csl:66]{Zhang \& Diller, 2015}; \hyperref[csl:67]{Zhang et al., 2016}; \hyperref[csl:78]{Zhang \& Diller, 2018}; \hyperref[csl:79]{Demir et al., 2021}; \hyperref[csl:80]{Culha et al., 2020)}, their producible sixth-DOF torque will be
1.41-63.9 folds lower than our proposed actuator.

While the proposed MMR had demonstrated unprecedented dexterity, it was
controlled manually by a human operator in this first study. In the
subsequent studies, we aim to increase the speed of the MMR by
implementing closed-loop position control with visual feedback. For
lab-on-chip applications, such visual feedback for the MMRs can be
obtained via standard cameras~\hyperref[csl:22]{(Chung et al., 2015)}. In the future, when
MMRs are deployed for their targeted biomedical applications, ultrasound
imaging technology would be a promising candidate for providing such
visual feedback for these actuators~\hyperref[csl:10]{(Hu et al., 2018)}. This is because
recent studies have shown that the actuating magnetic fields for MMRs
have minimal interference with ultrasound imaging
systems~\hyperref[csl:10]{(Hu et al., 2018)}. To make our proposed MMRs more favorable for
potential medical applications, we aim to further scale down the size of
these actuators in the future.

For this first study, our main objective is to investigate the
enhancement in dexterity when a soft MMR is able to concurrently possess
six-DOF motions and multimodal locomotion. Hence, we did not perform
detailed characterization on the proposed MMR to evaluate the maximum
speed achievable by each locomotion. We also did not evaluate the
maximum height or range that the MMR could jump. Such in-depth
experimental investigations can potentially be pursued in subsequent
studies. In the future, we also aim to investigate if six-DOF control
can be applied to soft MMRs with 3D magnetization profiles. This can
potentially allow us to discover new MMRs that can realize higher
sixth-DOF torque and yet capable of performing multimodal soft-bodied
locomotion. The feasibility of optimizing the MMR's geometries via
numerical methods~\hyperref[csl:89]{(Lum et al., 2015}; \hyperref[csl:90]{Lum et al., 2015}; \hyperref[csl:91]{Lum et al., 2013}; \hyperref[csl:92]{Lum et al., 2017)} to further enhance its dexterity
can similarly be explored in such further studies too.

While our proposed MMR can jump across complex barriers, it is difficult
to accurately control the initial jumping velocity. This is because it
is very challenging to model the dynamic interaction between the MMR and
the substrate when the former strikes the latter to gain the necessary
momentum to jump~\hyperref[csl:6]{(Ng et al., 2021}; \hyperref[csl:10]{Hu et al., 2018)}. In the future, we aim to
investigate whether deep learning techniques can be employed to overcome
this limitation \hyperref[csl:93]{(Silver et al., 2017)}. By obtaining experimental data to
train the weights of the neural networks, it may be possible to develop
numerical models that can accurately predict the MMR's initial jumping
velocity. Using such numerical models, we can potentially allow the
jumping locomotion to be more deterministic.

In summary, this work proposes a six-DOF MMR that can execute seven
modes of soft-bodied locomotion and perform 3D pick-and-place
operations. Based on the proposed control strategy, our MMR has
demonstrated unprecedented dexterity by negotiating across barriers
impassable by existing small-scale devices. We envision that this work
can potentially inspire future MMRs to be significantly more dexterous,
and this would be a critical step towards making these actuators
practical for their targeted biomedical applications.

\par\null

\section*{Acknowledgments}\label{acknowledgments}

G.Z.L. is funded by the start-up grant awarded by Nanyang Technological
University. The authors would like to thank C. S. X. Ng and N. Y. W. Foo
from Nanyang Technological University for their help in revising the
paper.

\section*{}\label{section}

\section*{Conflict of Interest}\label{conflict-of-interest}

The authors declare no conflict of interest.

\selectlanguage{english}
\FloatBarrier
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\end{document}

