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\begin{document}

\title{Supplementary Information for ``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 26, 2024}


\begingroup
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\selectlanguage{english}
\begin{abstract}
\textbf{This Supplementary Information includes:}

Section S1- Fabrication method~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Section S2- Actuation method~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~

Section S3- Analysis of sixth-DOF torque~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Section S4- Experiments~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~~~~~

Figures S1-S31~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ ~ ~ ~

Supporting Table~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ ~

References
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Other supplementary materials for this manuscript include the following:

Supporting SI Videos S1-S10

\par\null

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



\sloppy




\section*{Section S1- Fabrication
method}

{\label{829307}}

In this section, we would first elaborate on the fabrication process of
our proposed soft MMR (section S1A). Subsequently, we discussed about
the experiments that evaluated the proposed MMR's material properties
(section S1B).

\subsection*{\texorpdfstring{\emph{A. Fabrication of six-DOF soft
MMR}}{A. Fabrication of six-DOF soft MMR}}

{\label{101278}}

The detailed dimensions of our six-DOF MMR could be found in Fig. S1. To
fabricate the proposed MMR, we first constructed its magnetic soft beam
component. This soft component was molded by embedding hard magnetic
microparticles (NdFeB, average diameter: 5 \selectlanguage{greek}µ\selectlanguage{english}m) into the liquid polymer
matrix (Ecoflex 00-10) with a 4:1 mass ratio. The mixture was then cured
in an oven for two hours at 80 \selectlanguage{ngerman}°C. Once the beam was molded (Fig. S2A),
it was rolled in a cylindrical jig and subsequently magnetized by a
strong, uniform magnetic field of 1.1 T (Fig. S2B). After this
magnetization process, the magnetized beam would be endowed with its
desired harmonic magnetization profile
(\(\vec{M}\),~\(\varphi=\) 90° ) after it recovered to its
original configuration (Fig. S2C). Two identical buoyant components,
made from Styrofoam and cut to their desired geometry, were each
adhesively bonded to the free ends of the magnetic soft beam to form the
proposed MMR (Fig. S1). For the experiment in Fig. 6D, the buoyant
components were also coated with a thin layer of Ecoflex 00-10 to make
their surfaces hydrophobic.

\subsection*{\texorpdfstring{\emph{B. Characterization of the magnetic
soft beam's material
properties}}{B. Characterization of the magnetic soft beam's material properties}}

{\label{829391}}

Here we conducted a simple experiment to evaluate the\(\left|\vec{M}\right|\)
of our magnetic soft beam. Five cuboid samples were constructed by the
same material used for fabricating our magnetic soft beam, and these
samples had the following dimensions: 6 mm length (\(L_{\text{sp}}\)),
4 mm width (\(w_{\text{sp}}\)) and 1 mm thickness (\(t_{\text{sp}}\))
(Fig. S3A). The samples were magnetized by a strong, uniform magnetic
field of 1.1 T along their length, and their weights
(\(W\)) were measured by a high-precision weighing scale
(A\&D GH-120). Different rotary equilibrium states could be established
when the samples were subjected to a~\(\vec{B}\), which was
inclined at 45° from the horizontal plane, with varying magnitudes from
4 mT to 6 mT (Fig. S3B). As the rigid-body torque induced by the
samples' weight would make the net magnetic moment of the samples
(\({\vec{m}}_{\text{sp}}\)) misalign with the applied~\(\vec{B}\), we
could use a camera to record the misalignment
angle,~\(\varnothing\), between~\({\vec{m}}_{\text{sp}}\)
and~\(\vec{B}\) (Fig. S3B). By performing a torque equilibrium
analysis on the samples (Fig. S3B), the following governing equation
could be obtained:

\[\begin{equation}
\left|{\vec{m}}_{\text{sp}}\right|\left|\vec{B}\right|\sin\varnothing=Wr_{0}\cos{\Gamma,}\nonumber \\
 \end{equation}\]

where

\[\begin{equation}
\begin{matrix}r_{0}=\frac{\sqrt{{L_{\text{sp}}}^{2}+{t_{\text{sp}}}^{2}}}{2}\ .\left(S1.1\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

The parameter \(r_{0}\) represented the distance between the
samples' center of mass and anchor point, and \(\Gamma\)
represented the angle between the inclined samples and the substrate
(Fig. S3B). Since the samples' \(\left|\vec{M}\right|\) could be computed
as~\(\frac{\left|{\vec{m}}_{\text{sp}}\right|}{V_{\text{sp}}}\), where \(V_{\text{sp}}\) represented the volume
of the cuboid samples, we could rearrange Eq. (S1.1) to
evaluate\(\left|\vec{M}\right|\):

\[\begin{equation}
\begin{matrix}\frac{Wr_{0}\cos\Gamma}{V_{\text{sp}}\sin\varnothing}=\left|\vec{M}\right|\left|\vec{B}\right|.\left(S1.2\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

Using Eq. (S1.2), we plotted~\(\frac{Wr_{0}\cos\Gamma}{V_{\text{sp}}\sin\varnothing}\)
against\(\left|\vec{B}\right|\) to determine the experimentally
obtained~\(\left|\vec{M}\right|\) via the gradient of the best fit line of
these data (Fig. S3C). Using this analysis, we evaluated
the~\(\left|\vec{M}\right|\) of our material to be~
9.40\(\times\)10\textsuperscript{4}~A m\textsuperscript{-1}
when it was magnetized by a 1.1 T magnetic field.

In addition, we also evaluated the Young's modulus of the magnetic soft
beam to be 271~\(\pm\) 11.7 kPa via a standard compression
test (SHIMADZU AG-X plus, 10 kN).

\section*{Section S2- Actuation
method}

{\label{679766}}

In this section, we will discuss the actuation principles of our
proposed MMR. Specifically, we will derive the deformation mechanism of
the MMR (section S2A), analyze its six-DOF motions (section S2B), and
include additional discussions (section S2C). For these analyses, we
assume that the applied~\(\vec{B}\) and its spatial gradients are
uniform across the MMR's body as it is difficult to spatially vary these
control signals at small scale~\hyperref[csl:1]{(Diller et al., 2016}; \hyperref[csl:2]{Kummer et al., 2010}; \hyperref[csl:3]{Diller et al., 2014)}. Furthermore, due to
Gauss's law and Ampere's law (assuming no electric currents flowing in
the workspace), we also include the following constraints on the spatial
gradients of~\(\vec{B}\)~\hyperref[csl:1]{(Diller et al., 2016}; \hyperref[csl:4]{Xu et al., 2021)}:

\[\begin{equation}
\begin{matrix}\frac{\partial B_{x}}{\partial x}+\frac{\partial B_{y}}{\partial
y}+\frac{\partial B_{z}}{\partial z}=0,\left(S2.1A\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]\[\begin{equation}
\begin{matrix}\frac{\partial B_{z}}{\partial x}=\frac{\partial B_{x}}{\partial
z},\ \ \frac{\partial B_{z}}{\partial y}=\frac{\partial B_{y}}{\partial z},\ \ \frac{\partial B_{y}}{\partial x}=\frac{\partial B_{x}}{\partial y}.\left(S2.1B\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

Equation (S2.1) is valid across all the reference frames, and it
dictates that there are only five independent spatial gradients
of~\(\vec{B}\). To facilitate our subsequent discussions, we will
represent \({\vec{B}}_{\text{grad}}\ \) with the following format:

\[\begin{equation}
\begin{matrix}{\vec{B}}_{\text{grad}}=\left[\frac{\partial B_{z}}{\partial x}\text{\ \ \ \ }\frac{\partial B_{z}}{\partial y}\text{\ \ \ \ }\frac{\partial B_{z}}{\partial z}\text{\ \ \ \ }\frac{\partial B_{y}}{\partial
y}\text{\ \ \ \ }\frac{\partial B_{x}}{\partial y}\right]^{T}.\left(S2.2\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

\subsection*{\texorpdfstring{\emph{A. Deformation
mechanism}}{A. Deformation mechanism}}

{\label{501243}}

In comparison, the buoyant components are much more rigid than the
magnetic beam component, and thus the deformation mechanism of the
proposed MMR is mainly contributed by the magnetic beam. As a result,
here we would only derive the theoretical quasi-static model that
describes the deformation characteristics of the beam, and we assume
that the buoyant components are rigid.

According to the materials reference frame (Fig. 1A(ii)), the harmonic
magnetization profile (\({\vec{M}}_{\left\{M\right\}}\)~) of the beam along its
length, \(s\), can be expressed mathematically as:

\[\begin{equation}
\begin{matrix}{\vec{M}}_{\left\{M\right\}}\left(s\right)=\left|{\vec{M}}_{\left\{M\right\}}\right|\begin{pmatrix}0,&amp;\cos\left(\mathbf{-}\frac{2\pi}{L}s-\frac{\pi}{2}\right)\mathbf{,}&amp;\sin\left(\mathbf{-}\frac{2\pi}{L}s-\frac{\pi}{2}\right)\\
\end{pmatrix}^{T},\left(S2.3\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

where~\(L\) represents the total length of the beam. In
general,~\({\vec{M}}_{\left\{M\right\}}\) can be approximated as a collection of
magnetic dipoles that are embedded in the MMR's magnetic beam component
~\hyperref[csl:4]{(Xu et al., 2021)}. By applying~\(\vec{B}\) along
the~\(z_{\{L\}}\)-axis, the interaction of~\({\vec{M}}_{\left\{M\right\}}(s)\) and
the applied magnetic field will generate a distribution of magnetic
torque (per unit volume) along the magnetic beam,\(\ \tau_{x,\{L\}}\left(s\right)\) ,
which will in turn deform the MMR. The deformation of the magnetic soft
beam can be represented mathematically by its rotational
deflection,\(\selectlanguage{greek}\text{\ γ}\selectlanguage{english}\left(s\right)\) (Fig. S4).

By performing a quasi-static analysis on an arbitrary infinitesimal
element of the magnetic beam (Fig. S4), the torque equilibrium equation,
according to the local reference frame, can be derived as:

\[\begin{equation}
\begin{matrix}-\tau_{x,\{L\}}\left(s\right)Ads=\frac{\partial M_{b}}{\partial s}\left(s\right)ds,\left(S2.4\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

where~\(M_{b}\) and~\(A\) represent the bending
moment applied at~\(s\)and the corresponding
cross-sectional area of the beam, respectively. The
variable,~\(\tau_{x,\{L\}}\left(s\right),\) on the left side of Eq. (S2.4) can be
expanded into ~\hyperref[csl:5]{(Lum et al., 2016)} :

\[\begin{equation}
\tau_{x,\{L\}}\left(s\right)=\begin{bmatrix}1&amp;0&amp;0\\
\end{bmatrix}\left\{\left(\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\left\{M\right\}}\left(s\right)\right)\times{\vec{B}}_{\left\{L\right\}}\right\},\nonumber \\
 \end{equation}\]

where

\[\begin{equation}
\begin{matrix}\mathbf{R}_{x}\left(\gamma\right)=\begin{pmatrix}1&amp;0&amp;0\\
0&amp;\cos\gamma&amp;-\sin\gamma\\
0&amp;\sin\gamma&amp;\cos\gamma\\
\end{pmatrix}.\left(S2.5\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

The matrix,~\(\mathbf{R}_{x}\), is the standard rotational matrix about
the~\(x\)-axis, and it is used to account for the change
in~\(\vec{M}\) after the magnetic beam undergoes a large
deformation~\hyperref[csl:5]{(Lum et al., 2016)}. Based on the Euler-Bernoulli equation,
we can establish the relationship between~\(M_{b}\)
and~\(\gamma\) as~\hyperref[csl:5]{(Lum et al., 2016)}:

\[\begin{equation}
\begin{matrix}M_{b}=EI\frac{\partial\gamma}{\partial s},\left(S2.6\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

where \emph{E} and \emph{I} represent the Young's modulus and second
moment of inertia of the magnetic beam, respectively. By substituting
Eq.s (S2.5) and (S2.6) into Eq. (S2.4), the governing equation that
dictates the deformation characteristics of the MMR can be expressed as:

\[\begin{equation}
\begin{matrix}-\begin{bmatrix}1&amp;0&amp;0\\
\end{bmatrix}\left\{\left(\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\left\{M\right\}}\left(s\right)\right)\times{\vec{B}}_{\left\{L\right\}}\right\}A=EI\frac{\partial^{2}\gamma}{\partial s^{2}}\left(s\right).\left(S2.7\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

The deformation of the beam,~\(\gamma\left(s\right)\), in Eq. (S2.7) can be
solved numerically by using the following free-free boundary
conditions:\(\ \frac{\partial\gamma}{\partial s}\left(s=0\right)\ =\ \frac{\partial\gamma}{\partial s}\left(s=L\right)=0\). Physically, these boundary conditions
imply that the MMR will experience zero bending moments at its free
ends. Once~\(\gamma\) has been solved, the deformed
configuration of our MMR will be revealed. For these simulations, Eq.
(S2.7) is solved by using the MMR geometries shown in Fig. S1 as well as
the following material properties that have been obtained via
experimental means (SI section S1B):\(\left|{\vec{M}}_{\left\{M\right\}}\right|\) =
9.40\(\times\)10\textsuperscript{4}~A m\textsuperscript{-1}
and~\emph{E} =271 kPa.

Equations (S2.4-S2.7) indicate that the magnitude of~\({\vec{B}}_{\left\{L\right\}}\)
has a linear relationship with the bending moment experienced by the
magnetic beam. Hence, stronger magnitudes of~\({\vec{B}}_{\left\{L\right\}}\) can
generally allow our MMR to adopt sharper curvatures (Fig. S5). In
general, our numerical solution is able to predict the `U'- and inverted
`V'-shaped configurations produced by the MMR well. Because the
predicted shapes from the simulations agree well with the experimental
data (e.g., Fig. 1B), this suggests that the presented derivations can
describe the deformation physics of our MMR accurately. This is an
important criterion as the curvature of the proposed MMR must be
precisely controlled to enable its locomotion. In our derivations, we
assume that the distributed magnetic forces along the MMR's magnetic
beam component have negligible effects on its deformation. This is
because the mechanical stresses induced by the magnetic torques are
generally much larger than those generated by the magnetic
forces~\hyperref[csl:4]{(Xu et al., 2021}; \hyperref[csl:6]{Wang et al., 2020)}. Indeed, we did not observe noticeable
deformations on the proposed MMR when magnetic forces were applied to it
during the experiments. It is also noteworthy that if the direction
of~\({\vec{B}}_{\left\{L\right\}}\) is not aligned along
the~\(z_{\{L\}}\)-axis, the MMR will assume the `U'- or inverted
`V'-shaped configuration and produce a rigid-body rotation until its net
magnetic moment,~\({\vec{m}}_{\left\{L\right\}}\), is aligned
with~\({\vec{B}}_{\left\{L\right\}}\), where~\({\vec{m}}_{\left\{L\right\}}=\iiint{\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\left\{M\right\}}}dV\). The
variable,~\(V\), represents the volume of the beam
component of the deformed MMR. We will elaborate on the producible
rotations and translations of the MMR in the subsequent sub-section.

\subsection*{\texorpdfstring{\emph{B. Six-DOF motion
analysis}}{B. Six-DOF motion analysis}}

{\label{481222}}

Based on Eq.s (S2.3) and (S2.7), our undeformed MMR will have a null net
magnetic moment, i.e., \({\vec{m}}_{\left\{M\right\}}\)
=\(\iiint{\vec{M}}_{\left\{M\right\}}dV\ \)=\(\ {\vec{0}}_{\left\{M\right\}}\). However, once the proposed MMR
has assumed its `U'- or inverted `V'-shaped configuration, it will
possess an effective~\(\vec{m}\) necessary for implementing our
six-DOF control, i.e., allowing the MMR to rotate about three axes and
translate along three axes. The key concept of our six-DOF actuation
strategy is to control the actuating magnetic signals such that the
desired orientation of the MMR can become a minimum potential energy
configuration. Based on this control strategy, the MMR will constantly
experience three axes of restoring torques until it self-aligns to the
desired orientation. The restoring torques will also allow the proposed
MMR to reject mechanical disturbances such that its desired orientation
can be maintained.

When the proposed MMR assumes its deformed `U'- or inverted `V'-shaped
configuration, we can apply~\(\vec{B}\) and~\({\vec{B}}_{\text{grad}}\)
to exert magnetic torques and forces on it. Because it will be intuitive
to perform such analysis according to the local reference frame of the
MMR~\hyperref[csl:1]{(Diller et al., 2016}; \hyperref[csl:3]{Diller et al., 2014}; \hyperref[csl:4]{Xu et al., 2021)}, we express the net magnetic torque
(\(\vec{T}\)) and force (\(\vec{F}\)) applied on the MMR
based on this reference frame~\hyperref[csl:4]{(Xu et al., 2021)}:

\[\begin{equation}
\begin{pmatrix}{\vec{T}}_{\left\{L\right\}}\\
{\vec{F}}_{\left\{L\right\}}\\
\end{pmatrix}
=\text{\  }\begin{pmatrix}\iiint{\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\left\{M\right\}}\times{\vec{B}}_{\left\{L\right\}}\text{\ d}V}+\iiint{{\vec{r}}_{\{L\}}\times\left(\left[\frac{\partial{\vec{B}}_{\left\{L\right\}}}{\partial x_{\left\{L\right\}}}\text{\ \ \ }\frac{\partial{\vec{B}}_{\left\{L\right\}}}{\partial y_{\left\{L\right\}}}\text{\ \ \ }\frac{\partial{\vec{B}}_{\left\{L\right\}}}{\partial z_{\left\{L\right\}}}\right]^{T}\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\left\{M\right\}}\right)\ dV}\\
\iiint{\left(\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\left\{M\right\}}\bullet\nabla\right){\vec{B}}_{\left\{L\right\}}\ dV}
\end{pmatrix}\nonumber
 \end{equation}\]\[\begin{equation}
\begin{matrix}\mathbf{=D}\begin{pmatrix}{\vec{B}}_{\left\{L\right\}}\\
{\vec{B}}_{grad,\left\{L\right\}}\\
\end{pmatrix},\mathbf{}\left(S2.8\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

where~\(\vec{r}\) represents the displacement vector from the
MMR's center of mass to a point of interest in its body. The
matrix~\(\mathbf{D}\) is known as the design matrix, and it has a
6\(\times\)8~dimension ~\hyperref[csl:1]{(Diller et al., 2016}; \hyperref[csl:4]{Xu et al., 2021)}. The design
matrix~\(\mathbf{D}\) of our soft MMR can be explicitly expressed
as:

\[\begin{equation}
\mathbf{D}=\begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}0\\
\left|\vec{m}\right|\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}-\left|\vec{m}\right|\\
0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}0\\
d_{2}\\
0\\
\end{matrix}\\
\begin{matrix}\left|\vec{m}\right|\\
0\\
0\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}d_{1}\\
0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
\left|\vec{m}\right|\\
0\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
\left|\vec{m}\right|\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}0\\
0\\
d_{3}\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\\
\end{pmatrix},\ \nonumber \\
 \end{equation}\]\begin{equation}
d_{1}=\iiint\left(r_{y}M_{y}-r_{z}M_{z}\right)dV,\ \text{\ \ \ d}_{2}=\iiint{r_{z}M_{z}}dV,\ \text{\ \ \ d}_{3}=\iiint{-r_{y}M_{y}}dV.\ \ \ \ \ \ (S2.9)\nonumber \\
 \end{equation}

The variables,~\(r_{y}\) and~\(r_{z}\), represent
the Cartesian\emph{y} - and~\emph{z} -axes components
of~\({\vec{r}}_{\{L\}}\). Likewise, the variables,~\(M_{y}\)
and~\(M_{z}\), represent the Cartesian~\emph{y} - and~\emph{z}
-axes components of~\(\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\left\{M\right\}}\). A notable feature of
the~\(\mathbf{D}\) matrix is that its rank is six (full rank), and
this is an important criterion to achieve six-DOF \hyperref[csl:1]{(Diller et al., 2016}; \hyperref[csl:4]{Xu et al., 2021)}.

While it is intuitive to use the local reference frame to analyze the
net torque and force applied on the MMR, it is difficult to make the
desired orientation of the MMR into a minimum potential energy
configuration based on this reference frame~\hyperref[csl:4]{(Xu et al., 2021)}. This is
because the local reference frame lacks the information of the MMR's
sixth-DOF angular displacement (\(\theta\))
~\hyperref[csl:4]{(Xu et al., 2021)}. Therefore, we reanalyze Eq. (S2.8) based on the
intermediate reference frame. Specifically, we can use Eq. (2) to
reanalyze the left side of Eq. (S2.8) according to the intermediate
reference frame:

\[\begin{equation}
\begin{pmatrix}{\vec{T}}_{\left\{I\right\}}\\
{\vec{F}}_{\left\{I\right\}}\\
\end{pmatrix}=\begin{pmatrix}\mathbf{R}_{z}\left(\theta\right)&amp;\mathbf{0}_{3\times 3}\\
\mathbf{0}_{3\times 3}&amp;\mathbf{R}_{z}\left(\theta\right)\\
\end{pmatrix}\begin{pmatrix}{\vec{T}}_{\left\{L\right\}}\\
{\vec{F}}_{\left\{L\right\}}\\
\end{pmatrix},\nonumber \\
 \end{equation}\]

where

\[\begin{equation}
\begin{matrix}\mathbf{R}_{z}\left(\theta\right)=\begin{pmatrix}\cos\theta&amp;-\sin\theta&amp;0\\
\sin\theta&amp;\cos\theta&amp;0\\
0&amp;0&amp;1\\
\end{pmatrix}.\left(S2.10\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

In a similar way, we can reanalyze the actuating magnetic signals in Eq.
(S2.8), according to the intermediate frame, using the following
mapping:

\[\begin{equation}
\begin{pmatrix}{\vec{B}}_{\left\{L\right\}}\\
{\vec{B}}_{\text{grad},\left\{L\right\}}\\
\end{pmatrix}=\mathbf{A}\ \begin{pmatrix}{\vec{B}}_{\{I\}}\\
{\vec{B}}_{\text{grad},\{I\}}\\
\end{pmatrix},\nonumber \\
 \end{equation}\]

where

\[\begin{equation}
\begin{matrix}\mathbf{A=}\ \begin{bmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\cos\theta\\
-\sin\theta\\
\end{matrix}\\
\begin{matrix}0\\
0\\
\end{matrix}\\
\end{matrix}\\
\begin{matrix}\begin{matrix}0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}\begin{matrix}\sin\theta\\
\cos\theta\\
\end{matrix}\\
\begin{matrix}0\\
0\\
\end{matrix}\\
\end{matrix}\\
\begin{matrix}\begin{matrix}0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}\begin{matrix}0\\
0\\
\end{matrix}\\
\begin{matrix}1\\
0\\
\end{matrix}\\
\end{matrix}\\
\begin{matrix}\begin{matrix}0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
\cos\theta\\
\end{matrix}\\
\end{matrix}\\
\begin{matrix}\begin{matrix}-\sin\theta\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}\begin{matrix}0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
\sin\theta\\
\end{matrix}\\
\end{matrix}\\
\begin{matrix}\begin{matrix}\cos\theta\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
\end{matrix}\\
\end{matrix}\\
\begin{matrix}\begin{matrix}0\\
1\\
\end{matrix}\\
\begin{matrix}-\sin{{}^{2}\theta}\\
0.5\sin{2\theta}\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}\begin{matrix}0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
\end{matrix}\\
\end{matrix}\\
\begin{matrix}\begin{matrix}0\\
0\\
\end{matrix}\\
\begin{matrix}\cos{2\theta}\\
\sin{2\theta}\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}\begin{matrix}0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
\end{matrix}\\
\end{matrix}\\
\begin{matrix}\begin{matrix}0\\
0\\
\end{matrix}\\
\begin{matrix}-\sin{2\theta}\\
\cos{2\theta}\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\\
\end{bmatrix}.\mathbf{}\left(S2.11\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

By substituting Eq.s (S2.10) and (S2.11) into Eq. (S2.8), the
mathematical relationship between the wrench and the magnetic signals
can be expressed in the intermediate reference frame as:

\[\begin{equation}
\begin{pmatrix}{\vec{T}}_{\left\{I\right\}}\\
{\vec{F}}_{\left\{I\right\}}\\
\end{pmatrix}=\begin{pmatrix}\mathbf{R}_{z}\left(\theta\right)&amp;\mathbf{0}_{3\times 3}\\
\mathbf{0}_{3\times 3}&amp;\mathbf{R}_{z}\left(\theta\right)\\
\end{pmatrix}\begin{pmatrix}{\vec{T}}_{\left\{L\right\}}\\
{\vec{F}}_{\left\{L\right\}}\\
\end{pmatrix}\nonumber \\
 \end{equation}\]\[\begin{equation}
\mathbf{=}\begin{pmatrix}\mathbf{R}_{z}\left(\theta\right)&amp;\mathbf{0}_{3\times
3}\\
\mathbf{0}_{3\times 3}&amp;\mathbf{R}_{z}\left(\theta\right)\\
\end{pmatrix}\mathbf{D}\begin{pmatrix}{\vec{B}}_{\left\{L\right\}}\\
{\vec{B}}_{\text{grad},\left\{L\right\}}\\
\end{pmatrix}\text{\ }\nonumber \\
 \end{equation}\]\[\begin{equation}
\mathbf{=}\begin{pmatrix}\mathbf{R}_{z}\left(\theta\right)&amp;\mathbf{0}_{3\times
3}\\
\mathbf{0}_{3\times 3}&amp;\mathbf{R}_{z}\left(\theta\right)\\
\end{pmatrix}\mathbf{D}\mathbf{A}\begin{pmatrix}{\vec{B}}_{\left\{I\right\}}\\
{\vec{B}}_{\text{grad},\left\{I\right\}}\\
\end{pmatrix}\text{\}\nonumber \\
 \end{equation}\]\[\begin{equation}
=\mathbf{C}\left(\theta\right)\begin{pmatrix}{\vec{B}}_{\left\{I\right\}}\\
{\vec{B}}_{\text{grad},\left\{I\right\}}\\
\end{pmatrix},\ \ \nonumber \\
 \end{equation}\]

where

\[\begin{equation}
\begin{matrix}\mathbf{C}\left(\theta\right)=\begin{pmatrix}\mathbf{R}_{z}\left(\theta\right)&amp;\mathbf{0}_{3\times 3}\\
\mathbf{0}_{3\times 3}&amp;\mathbf{R}_{z}\left(\theta\right)\\
\end{pmatrix}\mathbf{D}\mathbf{A}.\left(S2.12\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

The matrix \(\mathbf{C}\left(\theta\right)\) is known as the control matrix
\hyperref[csl:4]{(Xu et al., 2021)}, and it can be expressed explicitly as:

\[\begin{equation}
\begin{matrix}\mathbf{C}\left(\theta\right)=\left(\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}0\\
\left|\vec{m}\right|\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}-\left|\vec{m}\right|\\
0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}d_{3}\sin\theta\cos\theta\\
d_{2}\cos^{2}\theta-d_{1}\sin^{2}\theta\\
0\\
\end{matrix}\\
\begin{matrix}\left|\vec{m}\right|\\
0\\
0\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}d_{1}\cos^{2}\theta-d_{2}\sin^{2}\theta\\
-d_{3}\sin\theta\cos\theta\\
0\\
\end{matrix}\\
\begin{matrix}0\\
\left|\vec{m}\right|\\
0\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\begin{matrix}\begin{matrix}\begin{matrix}0\\
0\\
d_{3}\sin\theta\cos\theta\\
\end{matrix}\\
\begin{matrix}0\\
0\\
\left|\vec{m}\right|\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}0\\
0\\
d_{3}\sin{2\theta}\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}0\\
0\\
d_{3}\cos{2\theta}\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\right).\left(S2.13\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

Since the rank of \(\mathbf{D}\) is six, \(\mathbf{C}\) would
be a full rank matrix too. To make the desired orientation of the MMR
into a minimum potential energy configuration,~\({\vec{T}}_{\left\{I\right\}}\) is
specified to be a null vector when it reaches the desired
\(\theta\). While the MMR will be in a rotational equilibrium
state at its desired orientation, a desired~\({\vec{F}}_{\left\{I\right\}}\) can
still be applied to the actuator. Based on the desired
\({\vec{F}}_{\left\{I\right\}}\), the required magnetic signals necessary for
implementing our six-DOF control can be derived by solving Eq. (S2.12):

\[\begin{equation}
\begin{matrix}\begin{pmatrix}{\vec{B}}_{\left\{I\right\}}\\
{\vec{B}}_{\text{grad,}\left\{I\right\}}\\
\end{pmatrix}=\ \mathbf{C}^{T}\left[\mathbf{C}\mathbf{C}^{T}\right]^{-1}\begin{pmatrix}{\vec{0}}_{\text{3x1,}\left\{I\right\}}\\
{\vec{F}}_{\text{desired,}\left\{I\right\}}\\
\end{pmatrix}+k_{1}\begin{pmatrix}\begin{matrix}0\\
0\\
1\\
\end{matrix}\\
\begin{matrix}\begin{matrix}0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}\\
\end{pmatrix}_{\left\{I\right\}}+k_{2}{\ \begin{pmatrix}\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\begin{matrix}\begin{matrix}0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
1\\
-\tan\left(2\theta\right)\\
\end{matrix}\\
\end{matrix}\\
\end{pmatrix}}_{\left\{I\right\}}.\left(S2.14\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

Equation (S2.14) is the general solution of the actuating magnetic
signals, and it includes a particular solution obtained via
pseudo-inverse (first right-hand component) and the homogeneous
solutions formed by the two null space vectors of \(\mathbf{C}\).
The variables, \(k_{1}\) and \(k_{2}\), are the
scale factors for the null space vectors. Each solution of Eq. (S2.14)
has a unique function:

\begin{enumerate}
\tightlist
\item
  The pseudo-inverse solution ensures that the MMR is in a rotational
  equilibrium state when it reaches the desired orientation, and the
  desired \({\vec{F}}_{\left\{I\right\}}\) can be applied to the actuator.
\item
  The first null space vector, \(\par
\begin{pmatrix}\par
\begin{matrix}\par
\begin{matrix}0&amp;0\\
\end{matrix}&amp;\par
\begin{matrix}1&amp;0\\
\end{matrix}\\
\end{matrix}&amp;\par
\begin{matrix}\par
\begin{matrix}0&amp;0\\
\end{matrix}&amp;\par
\begin{matrix}0&amp;0\\
\end{matrix}\\
\end{matrix}\\
\end{pmatrix}_{\{I\}}^{T}\), generates two axes
  of restoring torques for the MMR, but it cannot generate a restoring
  torque about the sixth-DOF axis of the actuator \hyperref[csl:1]{(Diller et al., 2016}; \hyperref[csl:2]{Kummer et al., 2010}; \hyperref[csl:3]{Diller et al., 2014)}.
\item
  By adjusting the magnetic actuating signals via the second null space
  vector, we can generate a restoring torque about the sixth-DOF axis of
  the MMR ~\hyperref[csl:4]{(Xu et al., 2021)}.~ ~ This restoring torque will in turn
  allow the MMR's sixth-DOF angular displacement to self-align into its
  desired~\(\theta\) and subsequently maintain this angle.
\end{enumerate}

In theory, it is ideal to make the strength of the restoring torques in
all axes stronger by increasing the magnitudes of~\(k_{1}\)
and~\(k_{2}\) in Eq. (S2.14)~\hyperref[csl:4]{(Xu et al., 2021)}. However, the
magnitudes of~\(k_{1}\) and~\(k_{2}\) are in
practice constrained by the capacity of the magnetic actuation systems
(e.g., the electromagnetic coil system described in SI section S4A).
Therefore, the magnetic actuating signals in Eq. (S2.14) are computed
based on the highest permissible magnitudes
of~\(k_{1}\)and~\(k_{2}\). Once the values
of~\(k_{1}\) and~\(k_{2}\) are determined, these
actuating signals will be specified according to the global reference
frame via this mapping:

\[\begin{equation}
\begin{pmatrix}{\vec{B}}_{\left\{G\right\}}\\
{\vec{B}}_{grad,\left\{G\right\}}\\
\end{pmatrix}=\begin{bmatrix}\mathbf{R}_{x}\left(\alpha\right)\mathbf{R}_{y}\left(\beta\right)&amp;\mathbf{0}_{3\times 5}\\
\mathbf{0}_{5\times 3}&amp;\mathbf{A}_{2}\left(\alpha,\beta\right)\\
\end{bmatrix}\ \begin{pmatrix}{\vec{B}}_{\left\{I\right\}}\\
{\vec{B}}_{grad,\left\{I\right\}}\\
\end{pmatrix},\nonumber \\
 \end{equation}\]

where

\begin{equation}
\mathbf{R}_{x}\left(\alpha\right)=\begin{pmatrix}1&amp;0&amp;0\\
0&amp;\cos\alpha&amp;-\sin\alpha\\
0&amp;\sin\alpha&amp;\cos\alpha\\
\end{pmatrix},\ \ \mathbf{R}_{y}\left(\beta\right)=\begin{pmatrix}\cos\beta&amp;0&amp;\sin\beta\\
0&amp;1&amp;0\\
-\sin\beta&amp;0&amp;\cos\beta\\
\end{pmatrix},\nonumber \\
 \end{equation}

and

\[\begin{equation}
\begin{matrix}\mathbf{A}_{2}\left(\alpha,\beta\right)=\ \begin{bmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\cos\left(\alpha\right)\cos\left(2\beta\right)\\
0.5sin\left(2\alpha\right)\sin\left(2\beta\right)\\
\end{matrix}\\
\begin{matrix}-\cos^{2}\left(\alpha\right)\sin\left(2\beta\right)\\
{-sin}^{2}\left(\alpha\right)\sin\left(2\beta\right)\\
-\sin\left(\alpha\right)\cos\left(2\beta\right)\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}\sin\left(\alpha\right)\sin\left(\beta\right)\\
\cos\left(2\alpha\right)\cos\left(\beta\right)\\
\end{matrix}\\
\begin{matrix}\sin\left(2\alpha\right)\cos\left(\beta\right)\\
-\sin\left(2\alpha\right)\cos\left(\beta\right)\\
\cos\left(\alpha\right)\sin\left(\beta\right)\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}\begin{matrix}\cos\left(\alpha\right)\sin\left(2\beta\right)\\
-0.5sin\ \left(2\alpha\right)\cos\left(2\beta\right)\\
\end{matrix}\\
\begin{matrix}\cos^{2}\left(\alpha\right)\cos\left(2\beta\right)\\
\sin^{2}\left(\alpha\right)\cos\left(2\beta\right)\\
-\sin\left(\alpha\right)\sin\left(2\beta\right)\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}0.5\cos\left(\alpha\right)\sin\left(2\beta\right)\\
0.5\sin{\left(2\alpha\right)\left(1+\sin^{2}\left(\beta\right)\right)}\\
\end{matrix}\\
\begin{matrix}\sin^{2}\left(\alpha\right)-\cos^{2}\left(\alpha\right)\sin^{2}\left(\beta\right)\\
\cos^{2}\left(\alpha\right)-\sin^{2}\left(\alpha\right)\sin^{2}\left(\beta\right)\\
-0.5\sin\left(\alpha\right)\sin\left(2\beta\right)\\
\end{matrix}\\
\end{matrix}&amp;\begin{matrix}\begin{matrix}\sin\left(\alpha\right)\cos\left(\beta\right)\\
-\cos\left(2\alpha\right)\sin\left(\beta\right)\\
\end{matrix}\\
\begin{matrix}-\sin\left(2\alpha\right)\sin\left(\beta\right)\\
\sin\left(2\alpha\right)\sin\left(\beta\right)\\
\cos\left(\alpha\right)\cos\left(\beta\right)\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\\
\end{bmatrix}\text{.\ }\left(S2.15\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

Equation (S2.15) concludes our proposed six-DOF control as it shows how
we can specify the required actuating signals (based on the global
reference frame) such that the MMR's desired orientation can become a
minimum potential energy configuration. Based on this actuation method,
our MMRs can also be controlled to follow a given angular trajectory.
This can be done by discretizing the trajectory into a sequence of
angular displacements and sequentially make all these orientations into
a minimum potential energy configuration. As these actuators have full
six-DOF motions, desired magnetic forces can also be applied on the MMRs
at any point along the angular trajectory.

\subsection*{\texorpdfstring{\emph{C. Additional
discussion}}{C. Additional discussion}}

{\label{403883}}

Based on the endowed \({\vec{M}}_{\left\{M\right\}}\), the proposed MMR has two
unique features which allow it to concurrently achieve six-DOF and
multimodal soft-bodied locomotion.

Although the MMR's \textbar{}\(\vec{m}\)\textbar{} will change
according to the strength of \(\vec{B}\) (due to having
different amounts of deformation), its \(\vec{m}\) will always
be aligned to \(\vec{B}\) for all the `U'- and inverted
`V'-shaped deformed configurations.

While the value of \(d_{3}\) in Eq. (S2.9) will change when
the proposed MMR undergoes different amounts of deformations, its value
will always remain negative across all the deformed configurations of
this actuator.

Having these features is highly advantageous because they ensure that
the direction of the null spaces in Eq. (S2.14) will remain the same
across all the deformed configurations of our proposed MMR. Thus, the
general solution of Eq. (S2.14) will be applicable for implementing
six-DOF control on our proposed MMR at all times. An important criterion
to compute the general solution of Eq. (S2.14) correctly is that the
changes in the MMR's \(\vec{m}\) and
\(d_{1}\)-\(d_{3}\)robotic parameters have been
fully accounted for when the actuator deforms. For instance, by using
the theoretical model in SI section S2A, we are able to predict the
deformation of the MMR \((\gamma)\) well. Using the computed
\(\gamma\), the deformed MMR's~\({\vec{m}}_{\left\{L\right\}}\) can be
continuously updated by using the following
equation:~\({\vec{m}}_{\left\{L\right\}}=\iiint{\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\left\{M\right\}}}dV.\) It is important to update the
MMR's~\({\vec{m}}_{\left\{L\right\}}\) so that the pseudo-inverse solution in Eq.
(S2.14) can be computed accurately. In SI section S3, we will elaborate
how the \(d_{1}\)-\(d_{3}\) robotic parameters of
our MMR will vary as the actuator undergoes different amounts of
deformation.

Equation (S2.14) suggests that the magnitude of the second null space
vector will approach infinity when \(\theta=\pm\frac{\pi}{4}\). Those angles of
\(\theta\) are known as the singularity angles (4). The values
of the singularity angles, however, can be altered via changing the
format of the second null space vector:

\[\begin{equation}
\begin{matrix}\begin{pmatrix}{\vec{B}}_{\left\{I\right\}}\\
{\vec{B}}_{grad,\left\{I\right\}}\\
\end{pmatrix}=\ \mathbf{C}^{T}\left[\mathbf{C}\mathbf{C}^{T}\right]^{-1}\begin{pmatrix}{\vec{0}}_{\text{3x1,}\left\{I\right\}}\\
{\vec{F}}_{\text{desired,}\left\{I\right\}}\\
\end{pmatrix}+k_{1}\begin{pmatrix}\begin{matrix}0\\
0\\
1\\
\end{matrix}\\
\begin{matrix}\begin{matrix}0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}\\
\end{pmatrix}_{\left\{I\right\}}+k_{2}{\ \begin{pmatrix}\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\begin{matrix}\begin{matrix}0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
-\cot\left(2\theta\right)\\
1\\
\end{matrix}\\
\end{matrix}\\
\end{pmatrix}}_{\left\{I\right\}}. \left(S2.16\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

By alternating the format of the second null space vector according to
the range of \(\theta\), the singularity issues can be
moderated because the MMR will be able to avoid all the singularity
points as it undergoes an angular trajectory.

\section*{Section S3- Analysis of sixth-DOF
torque}

{\label{673708}}

This section presents the numerical analysis for computing the
producible sixth-DOF torque of our MMR. In addition, we also compare
this torque to the producible sixth-DOF torque of existing MMRs that
have single-wavelength, harmonic magnetization
profiles~\hyperref[csl:7]{(Hu et al., 2018}; \hyperref[csl:8]{Ren et al., 2021}; \hyperref[csl:9]{Demir et al., 2021}; \hyperref[csl:10]{Culha et al., 2020}; \hyperref[csl:11]{Diller et al., 2014}; \hyperref[csl:12]{Zhang \& Diller, 2018}; \hyperref[csl:13]{Zhang \& Diller, 2015}; \hyperref[csl:14]{Zhang et al., 2016)}. We will also discuss how the other robotic
parameters of our MMR will vary as the actuator undergoes different
amounts of deformation.

Based on our derivations in SI section S2B, the sixth-DOF torque of our
MMR can be determined by analyzing the third row of Eq. (S2.8).
Specifically, the sixth-DOF torque of the MMR,~\(T_{z,\left\{L\right\}}\), can
be expressed mathematically according to its local reference frame as:

\[\begin{equation}
\begin{matrix}T_{z,\left\{L\right\}}=d_{3}\frac{\partial B_{y,\left\{L\right\}}}{\partial x_{\left\{L\right\}}}.\left(S3.1\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

According to Eq. (S3.1), the sixth-DOF torque of our MMR can be enhanced
by either increasing the magnitudes of~\(\left|d_{3}\right|\)
or~\(\frac{\partial B_{y,\left\{L\right\}}}{\partial x_{\left\{L\right\}}}\text{.\ }\)Since the magnitude of~\(\frac{\partial B_{y,\left\{L\right\}}}{\partial x_{\left\{L\right\}}}\) is
dependent on the capacity of the magnetic actuation systems (e.g., the
electromagnetic coil system described in SI section S4A), it is
therefore ideal for the MMR to maximize its~\(\left|d_{3}\right|\) so
that~\(T_{z,\left\{L\right\}}\) can be optimized. From the physical perspective,
the magnitude of~\(\left|d_{3}\right|\) represents the normalized sixth-DOF
torque of the MMR, i.e., the producible sixth-DOF torque of the MMR
after it has been normalized according to the strength of the actuating
signals. It is noteworthy that MMRs with larger\(\left|d_{3}\right|\) will
be able to produce larger sixth-DOF torque across all types of magnetic
actuation systems~\hyperref[csl:4]{(Xu et al., 2021)}.

As~\(d_{3}\) is dependent on the magnetization profile and
geometries of the MMR (Eq. (S2.9)),~\(\left|d_{3}\right|\) will vary as the
actuator undergoes different amounts of deformation. Using Eq.s (S2.7)
and (S2.9), we can compute~\(\left|d_{3}\right|\) of our MMR across all
possible `U'- and inverted `V'-shaped deformation configurations that
can be generated by our magnetic actuation system (Fig. S6B), i.e., by
adjusting~\(B_{z,\left\{L\right\}}\) from -30 mT to 30 mT. For these
simulations, Eq.s (S2.7) and (S2.9) are solved by using the MMR
geometries shown in Fig. S1 as well as the following material properties
that have been obtained via experimental means (SI section
S1B):~\(\left|{\vec{M}}_{\left\{M\right\}}\right|\) =
9.40~\(\times\)10\textsuperscript{4~}~A m\textsuperscript{-1}
and~\(E\) = 271 kPa. Based on these simulations, we can
conclude that while~\({|d}_{3}|\) is non-zero across all the
deformation configurations of our proposed MMR, it will be favorable
for~\(B_{z,\left\{L\right\}}\) to be applied within 4-6 mT~so that our actuator
can maximize its~\(\left|d_{3}\right|\) (Fig. S6B). In general, the
information in Fig. S6B is critical in our six-DOF control strategy
because it allows us to determine the required~\(\frac{\partial B_{y,\left\{L\right\}}}{\partial x_{\left\{L\right\}}}\) that
can generate the desired sixth-DOF torque for the proposed MMR. The
information in Fig. S6B is also important for computing the
pseudo-inverse solution in Eq. (S2.14). Based on the simulation results
in Fig. S6B, the predicted~\(\left|d_{3}\right|\) value of our MMR is
computed as 1.02\(\times\)10\textsuperscript{-7} N
m\textsuperscript{2}~T\textsuperscript{-1}~when the actuator is
undeformed, i.e., when~\(B_{z,\left\{L\right\}}\) = 0. We have compared this
predicted value to our MMR's normalized sixth-DOF torque obtained via
the experiments illustrated in Fig. 2. The simulation results from Fig.
S6B also suggest that the MMR may produce higher normalized sixth-DOF
torque if it does not remain perfectly in its undeformed configuration.

The key difference between our proposed MMR and other existing similar
MMRs with harmonic magnetization profile~\hyperref[csl:7]{(Hu et al., 2018}; \hyperref[csl:8]{Ren et al., 2021}; \hyperref[csl:9]{Demir et al., 2021}; \hyperref[csl:10]{Culha et al., 2020}; \hyperref[csl:11]{Diller et al., 2014}; \hyperref[csl:12]{Zhang \& Diller, 2018}; \hyperref[csl:13]{Zhang \& Diller, 2015}; \hyperref[csl:14]{Zhang et al., 2016)} is that our
phase shift angle (\(\varphi\)) is different. Here we report
that having different~\(\varphi\) in the harmonic magnetization
profile will significantly affect the MMR's
achievable~\(\left|d_{3}\right|\). To investigate the relationship
between~\(\varphi\) and~\(\left|d_{3}\right|\), numerical
simulations based on Eq.s (S2.7) and (S2.9) have been conducted. In
these simulations, we assume that all the MMRs have the geometries shown
in Fig. S1 and their material properties are equal to those of our
proposed MMR (SI section S1B): ~\(\left|{\vec{M}}_{\left\{M\right\}}\right|\) =
9.40~\(\times\)10\textsuperscript{4~}~A m\textsuperscript{-1}
and~\(E\) = 271 kPa . The magnetization profile of an MMR
with a generic~\(\varphi\),~\({\vec{M}}_{\varphi,\left\{M\right\}}\left(s\right)\), can be
mathematically expressed in the simulations as:

\[\begin{equation}
{\vec{M}}_{\varphi,\ \ \left\{M\right\}}\left(s\right)=\left|{\vec{M}}_{\ \left\{M\right\}}\left(s\right)\right|\begin{pmatrix}0,&amp;\cos\left(\mathbf{-}\frac{2\pi}{L}s+\varphi\right)\mathbf{,}&amp;\sin\left(\mathbf{-}\frac{2\pi}{L}s+\varphi\right)\\
\end{pmatrix}^{T}. (S3.2)\nonumber \\
 \end{equation}\]

In the simulations, we also assume that all the MMRs have a harmonic
magnetization profile that has a single wavelength because only such
harmonic profiles have so far allowed MMRs to achieve multimodal
soft-bodied locomotion ~\hyperref[csl:7]{(Hu et al., 2018}; \hyperref[csl:8]{Ren et al., 2021)}. The results from our
simulation are presented in Fig. S6A, where we plotted the highest
achievable~\(\left|d_{3}\right|\) of the MMRs,~\(\left|d_{3,highest}\right|\), against
different~φ. From these results, it can be seen that
the maximum~\(\left|d_{3,highest}\right|\) will be achieved
when~\(\varphi\) =\(\pm\) 90°. Hence, we have
selected~\(\varphi\) = -90° for our proposed MMR so that its
producible sixth-DOF torque can be maximized. In comparison to existing
MMRs with harmonic magnetization profiles that have~\(\varphi\)
of~ 45° ~\hyperref[csl:7]{(Hu et al., 2018}; \hyperref[csl:8]{Ren et al., 2021}; \hyperref[csl:9]{Demir et al., 2021}; \hyperref[csl:10]{Culha et al., 2020)} and~ 0° ( ~\hyperref[csl:11]{(Diller et al., 2014}; \hyperref[csl:12]{Zhang \& Diller, 2018}; \hyperref[csl:13]{Zhang \& Diller, 2015}; \hyperref[csl:14]{Zhang et al., 2016)},~ the
simulation results in Fig. S6A indicate that our proposed MMR can
produce 1.40-76.0 folds larger~\(\left|d_{3,highest}\right|\) than such existing
actuators. Because the achievable~\(|d_{3}|\) of the MMRs will
change as the actuators deform, we have also used Eq.s (S2.7) and (S2.9)
to compute how~\(\left|d_{3}\right|\) will vary with\(B_{z,\left\{L\right\}}\) for
existing MMRs that have a harmonic magnetization profile
with~\(\varphi\) of~ 45° ~\hyperref[csl:7]{(Hu et al., 2018}; \hyperref[csl:8]{Ren et al., 2021}; \hyperref[csl:9]{Demir et al., 2021}; \hyperref[csl:10]{Culha et al., 2020)} and~ 0°
~\hyperref[csl:11]{(Diller et al., 2014}; \hyperref[csl:12]{Zhang \& Diller, 2018}; \hyperref[csl:13]{Zhang \& Diller, 2015}; \hyperref[csl:14]{Zhang et al., 2016)} (Fig. S6B). These simulation results indicate that
the average~\(\left|d_{3}\right|\) of our proposed MMR (
5.77\(\times\)10\textsuperscript{-8} A m\textsuperscript{3} )
is 1.41-63.9 folds larger than those that have~\(\varphi\) of~
45° (4.10 ~\(\times\)10\textsuperscript{-8} A
m\textsuperscript{3}~)~\hyperref[csl:7]{(Hu et al., 2018}; \hyperref[csl:8]{Ren et al., 2021}; \hyperref[csl:9]{Demir et al., 2021}; \hyperref[csl:10]{Culha et al., 2020)} and~ 0° (9.03
~\(\times\)10\textsuperscript{-10} A
m\textsuperscript{3}~)~\hyperref[csl:11]{(Diller et al., 2014}; \hyperref[csl:12]{Zhang \& Diller, 2018}; \hyperref[csl:13]{Zhang \& Diller, 2015}; \hyperref[csl:14]{Zhang et al., 2016)}. Note that the
average~\(\left|d_{3}\right|\) of these MMRs is computed via the following
integral:~\(\frac{\int_{-30}^{30}{|d_{3}|}\text{\ d}B_{z,\left\{L\right\}}\ }{60}\), i.e., finding their corresponding area
under the curve in Fig. S6B and subsequently dividing those areas across
the entire domain of~\(B_{z,\left\{L\right\}}\). The numerical simulations in
Fig. S6B therefore suggest that the proposed MMR is able to produce much
higher sixth-DOF torques than existing similar actuators that
have~\(\varphi\) of~ 45°~\hyperref[csl:7]{(Hu et al., 2018}; \hyperref[csl:8]{Ren et al., 2021}; \hyperref[csl:9]{Demir et al., 2021}; \hyperref[csl:10]{Culha et al., 2020)} and~
0°~\hyperref[csl:11]{(Diller et al., 2014}; \hyperref[csl:12]{Zhang \& Diller, 2018}; \hyperref[csl:13]{Zhang \& Diller, 2015}; \hyperref[csl:14]{Zhang et al., 2016)}.

A notable advantage of our proposed MMR is that its sixth-DOF torque has
been optimized without compromising its actuation capabilities in the
traditional five-DOF motions. This is because the achievable net
magnetic moment of our proposed MMR has the same magnitude with all
other MMRS that have single-wavelength, harmonic magnetization profiles
\hyperref[csl:7]{(Hu et al., 2018}; \hyperref[csl:8]{Ren et al., 2021}; \hyperref[csl:9]{Demir et al., 2021}; \hyperref[csl:10]{Culha et al., 2020}; \hyperref[csl:11]{Diller et al., 2014}; \hyperref[csl:12]{Zhang \& Diller, 2018}; \hyperref[csl:13]{Zhang \& Diller, 2015}; \hyperref[csl:14]{Zhang et al., 2016)}. To prove this hypothesis mathematically, we first
reanalyze the generic magnetization profile in Eq. (S3.2) and arrange it
to the following format:

\[\begin{equation}
{{\vec{M}}_{\varphi,\ \ \left\{M\right\}}\left(s\right)=\mathbf{R}_{x}\left(\varphi\right)\left|{\vec{M}}_{\ \left\{M\right\}}\left(s\right)\right|\begin{pmatrix}0,&amp;\cos\left(\mathbf{-}\frac{2\pi}{L}s\right)\mathbf{,}&amp;\sin\left(\mathbf{-}\frac{2\pi}{L}s\right)\\
\end{pmatrix}^{T}}  \nonumber \\
 \end{equation}\]\[\begin{equation}
{=\mathbf{R}_{x}\left(\varphi\right){\vec{M}}_{\varphi 0,\ \ \left\{M\right\}}\left(s\right),}\nonumber \\
 \end{equation}\]

where

\(\begin{equation}
\par
\begin{matrix}{\vec{M}}_{\varphi 0,\ \ \left\{M\right\}}\left(s\right)=\left|{\vec{M}}_{\left\{M\right\}}\left(s\right)\right|\par
\begin{pmatrix}0,&amp;\cos\left(\mathbf{-}\frac{2\pi}{L}s\right)\mathbf{,}&amp;\sin\left(\mathbf{-}\frac{2\pi}{L}s\right)\\
\end{pmatrix}^{T},\left(S3.3\right)\\
\end{matrix}  \nonumber \\
\end{equation}\)

and it represents the harmonic magnetization profile of an MMR that has
a~\(\varphi\) value of~ 0°. When an MMR with a
generic~\(\varphi\) undergoes a spatially varying rotary
deformation,~\(\gamma\left(s\right)\), across its body, it can possess the
following net magnetic moment~\(({\vec{m}}_{\varphi})\):

\[\begin{equation}
\begin{matrix}{\vec{m}}_{\varphi}=\iiint{\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\varphi,\ \ \left\{M\right\}}}dV=\iiint{\mathbf{R}_{x}\left(\gamma\right)\mathbf{R}_{x}\left(\varphi\right){\vec{M}}_{\varphi 0,\ \ \left\{M\right\}}}d\text{V.\ }\left(S3.4\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

A unique characteristics of MMRs with harmonic magnetization profile is
that, regardless of~\(\varphi\), they will always be able to
produce the same amount of deformation,~\(\gamma\left(s\right)\), under the
same~\(B_{z,\left\{L\right\}}\)~\hyperref[csl:7]{(Hu et al., 2018)}. This is assuming that
the~\(z_{\left\{L\right\}}\)-axis of the MMR is always assigned to be
parallel to the actuators' net magnetic moment~\(\left({\vec{m}}_{\varphi}\right)\)
\hyperref[csl:7]{(Hu et al., 2018)}.~Due to this unique feature, our analysis in Eq.
(S3.4) can therefore be simplified because we do not need to account for
different deformation characteristics,~\(\mathbf{R}_{x}\left(\gamma\right)\), for MMRs
with different~\(\varphi\).

Because \(\mathbf{R}_{x}\left(\varphi\right)\) is constant across the entire MMR, Eq.
(S3.4) can be rearranged into the following format:

\[\begin{equation}
\par
\begin{matrix}{\vec{m}}_{\varphi }=\iiint{\mathbf{R}_{x}\left(\gamma\right)\mathbf{R}_{x}\left(\varphi\right){\vec{M}}_{\varphi 0,\ \ \left\{M\right\}}}dV
\end{matrix}  \nonumber \\
\end{equation}\]

\[\begin{equation}
\par
\begin{matrix}=\iiint{\mathbf{R}_{x}\left(\varphi\right)\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\varphi 0,\ \ \left\{M\right\}}}dV
\end{matrix}  \nonumber \\
\end{equation}\]

\[\begin{equation}
\par
\begin{matrix}=\mathbf{R}_{x}\left(\varphi\right)\iiint{\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\varphi 0,\ \ \left\{M\right\}}}dV
\end{matrix}  \nonumber \\
\end{equation}\]

\[\begin{equation}
{=\mathbf{R}_{x}\left(\varphi\right){\vec{m}}_{\varphi 0,}}  \nonumber \\
 \end{equation}\]

\par\null

where \(\begin{equation}
\par
\begin{matrix}{\vec{m}}_{\varphi 0}=\iiint{\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\varphi 0,\ \ \left\{M\right\}}}dV,\left(S3.5\right)\\
\end{matrix}  \nonumber \\
\end{equation}\)

and it represents the net magnetic moment produced by a deformed MMR
that has a magnetization profile of~\({\vec{M}}_{\varphi 0,\ \ \left\{M\right\}}\left(s\right)\). Since the
rotational matrix,~\(\mathbf{R}_{x}\left(\varphi\right)\), is an orthonormal matrix, Eq.
(S3.5) concludes that the magnitude of~\({\vec{m}}_{\varphi}\) is equal
to~\({\vec{m}}_{\varphi 0}\) for all values of~\(\varphi\), including
our proposed~\(\vec{M}\) that has a~\(\varphi\) value of~
-90°. As a result, this implies that our proposed MMR does possess
a~\(\overrightarrow{m}\) that has the same magnitude with all other MMRS
that have single-wavelength, harmonic magnetization profiles
\hyperref[csl:7]{(Hu et al., 2018}; \hyperref[csl:8]{Ren et al., 2021}; \hyperref[csl:9]{Demir et al., 2021}; \hyperref[csl:10]{Culha et al., 2020}; \hyperref[csl:11]{Diller et al., 2014}; \hyperref[csl:12]{Zhang \& Diller, 2018}; \hyperref[csl:13]{Zhang \& Diller, 2015}; \hyperref[csl:14]{Zhang et al., 2016)}. Therefore, our proposed MMR is able to maximize its
sixth-DOF torque without compromising its actuation capabilities for the
traditional five-DOF motions.

To successfully implement six-DOF control on our proposed MMR, we have
also analyzed how its \(d_{1}\) and \(d_{2}\) robotic
parameters in the~\(\mathbf{D}\) matrix of Eq. (S2.9) will vary as
it undergoes different amounts of deformation. These simulation results
can be found in Fig. S7. While the values of \(d_{1}\) and
\(d_{2}\) do not affect the producible sixth-DOF torque of our
MMR, it is still important to model these parameters so as to compute
the pseudo-inverse solution in Eq. (S2.14) accurately.

\section*{Section S4- Experiments}

{\label{492396}}

In this section, we would describe our magnetic actuation system in
section S4A. Subsequently, we would elaborate on the experiments that
evaluated the producible sixth-DOF torque of the proposed MMR (section
S4B). We would also include additional discussion pertaining to the
rolling (section S4C) and jellyfish-like swimming locomotion (section
S4D). Finally, we would conclude this section by discussing the
undulating swimming and meniscus-climbing locomotion of our MMR in
section S4E.

\subsection*{\texorpdfstring{\emph{A. Magnetic actuation
system}}{A. Magnetic actuation system}}

{\label{289066}}

Our magnetic actuation system was a customized electromagnetic coil
system, which had a nine-coil configuration (Fig. S8). For all the
experiments, the proposed MMR was placed at the center of the coil
system, which had a workspace of 16 mm\(\ \times\ \)16
mm\(\ \times\ \)16 mm that could produce a 90\% homogeneous field.
By controlling the electrical current in the coils, we could generate
the desired actuating magnetic signals for our proposed MMR. This system
could produce a maximum~\(\left|\vec{B}\right|\) of 30 mT, and the highest
producible spatial gradient of the coil system was 0.4 T
m\textsuperscript{-1}. The smallest angular change achievable by our
applied~\(\vec{B}\) was 0.57° while the resolution of the
field's spatial gradients was 0.01 T/m.

\subsection*{\texorpdfstring{\emph{B. Experimental investigation on the
MMR's sixth-DOF
torque}}{B. Experimental investigation on the MMR's sixth-DOF torque}}

{\label{109890}}

In the experiment illustrated in Fig. 2, we could deduce the sixth-DOF
torque generated by our undeformed MMR via measuring the angular
deflection at the free end of the fixed-free beam,~\(\theta_{\text{tip}}\),
and subsequently applying the Euler-Bernoulli equation. Based on the
Euler-Bernoulli equation, the governing equation for describing the
deformation of the larger fixed-free beam could be expressed
as~\hyperref[csl:5]{(Lum et al., 2016)}:

\[\begin{equation}
\begin{matrix}M_{\text{bt}}\left(s_{\text{bt}}\right)=E_{\text{bt}}I_{\text{bt}}\frac{\partial\theta_{\text{bt}}}{\partial s_{\text{bt}}}\left(s_{\text{bt}}\right),\left(S4.1\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

where~\(M_{\text{bt}}\) and~\(\theta_{\text{bt}}\) represented the
bending moment and the angular deflections along the length of the
fixed-free beam,~\(s_{\text{bt}}\), respectively (see Fig. S9
for~\(\theta_{\text{bt}}\) and~\(s_{\text{bt}}\)). The
variable~\(E_{\text{bt}}\) represented the Young's modulus of the
fixed-free beam, and this parameter was evaluated to be
787\(\pm\) 8.18 kPa via a standard compression test
(SHIMADZU AG-X plus, 10 kN). The remaining
variable,~\(I_{\text{bt}}\), represented the second moment of area of
the fixed-free beam, and it could be computed as
3.90\(\times\)10\textsuperscript{-15} m\textsuperscript{4}
based on the beam's dimensions.

In theory, the sixth-DOF torque generated by our magnetic
beam,~\(T_{z,\{L\}}\), would induce a constant \(M_{\text{bt}}\)
across the larger fixed-free beam, and the magnitudes of
\(T_{z,\{L\}}\) and~\(M_{\text{bt}}\) would be equal. By applying
such loading conditions and the fixed-free boundary conditions to Eq.
(S4.1), \(T_{z,\{L\}}\) could be deduced by the following equation:

\[\begin{equation}
\begin{matrix}T_{z,\left\{L\right\}}=\frac{E_{\text{bt}}I_{\text{bt}}}{L_{\text{bt}}}\theta_{\text{tip}},\left(S4.2\right)\\
\end{matrix}\nonumber \\
 \end{equation}\]

where \(L_{\text{bt}}\) = 16\(\times\)10\textsuperscript{-3}
m and it represented the total length of the fixed-free beam (Fig. S9).
Using Eq. (S4.2), we could therefore execute the experiments in Fig. 2A
and establish the relationship between \(T_{z,\{L\}}\) and the
actuating magnetic signals (Fig. 2B).

\subsection*{\texorpdfstring{\emph{C. Rolling
locomotion}}{C. Rolling locomotion}}

{\label{949433}}

By having six-DOF, our soft MMR could choose to roll along its length
(rotating about the~\(x_{\{L\}}\)-axis) or along its width
(rotating about the~\(y_{\{L\}}\)-axis). Assuming no-slip
conditions, the achievable speed for both types of these rolling
locomotion would theoretically be linearly proportional to the frequency
of the applied rotating~\(\vec{B}\) ~\hyperref[csl:7]{(Hu et al., 2018)}. To
validate this hypothesis, experiments were conducted to establish the
relationship between the rolling speed of our MMR with respect to
different rotating frequencies of~\(\vec{B}\). For the
experiments pertaining to rolling along the length of the MMR, the
magnitude of~\(\vec{B}\) was held constant at 15 mT while the
frequency of~\(\vec{B}\) varied between 0.25 Hz to 10 Hz. The
data of these experimental results were presented in Fig. S10. In
general, the experimental data agreed that there was a linear
relationship between the rolling speed of the MMR and the frequency of
the applied~\(\vec{B}\).

For the experiments pertaining to rolling along the width of the MMR,
the magnitude of~\(\vec{B}\) was held constant at 6 mT~so that
this actuator could produce a gentler curvature that would be favorable
for negotiating across narrow barriers (Fig. S11). The rolling speeds
for this type of rolling were measured as the frequency of the
rotating~\(\vec{B}\) was varied from 0.25 Hz to 3 Hz (Fig. S10).
In general, the relationship between the rolling speed of the MMR and
the frequency of the applied~\(\vec{B}\) was linear, and
therefore the experiments agreed with the theoretical prediction. As
predicted by the theory, the speed achievable by rolling along the width
of the MMR was slower than the speed of the MMR when it rolled along its
length (Fig. S10).

\subsection*{\texorpdfstring{\emph{D. Jellyfish-like swimming
locomotion}}{D. Jellyfish-like swimming locomotion}}

{\label{857764}}

For the jellyfish-like swimming locomotion, the average speeds
(\(V_{R}\)) of our MMR were
7.65\(\times\)10\textsuperscript{-3} m
s\textsuperscript{-1},~10.4\(\times\)10\textsuperscript{-3} m
s\textsuperscript{-1} and 9.49\(\times\)10\textsuperscript{-3}
m s\textsuperscript{-1} when it was rotating about
the~\(x_{\{L\}}\)-,~\(y_{\{L\}}\)-
and~\(z_{\{L\}}\)-axes, respectively. Therefore, the Reynolds
number~\((\text{Re})\) in these experiments could be computed as
54.9, 74.6 and 68.1, respectively. Note that~\(Re=\frac{V_{R}L_{R}}{v}\), where
the length of our MMR,~\(L_{R}\), was 6.4 mm (Fig. S1) and the
kinematic viscosity of water at 25°C,~\(v\), was
8.92\(\times\)10\textsuperscript{-7~}m\textsuperscript{2}
s\textsuperscript{-1~}\hyperref[csl:15]{(Kestin et al., 1978)}. In a similar way, the Reynolds
number of our MMR could be calculated to be 67.2~when it executed the
experiments in SI Video S8 (average swimming speed:
9.37\(\times\)10\textsuperscript{-3} m s\textsuperscript{-1}).

\subsection*{\texorpdfstring{\emph{E. Undulating swimming and meniscus
climbing
locomotion}}{E. Undulating swimming and meniscus climbing locomotion}}

{\label{659303}}

On an air-water interface, our soft MMR could swim via an undulating
locomotion. This gait could be activated by rotating~\(\vec{B}\)
continuously in the MMR's~\(y_{\{L\}}z_{\{L\}}\) plane so that a traveling
wave could be generated along the soft body of the MMR (Fig. S12A and SI
Video S10) \hyperref[csl:7]{(Hu et al., 2018}; \hyperref[csl:11]{Diller et al., 2014}; \hyperref[csl:12]{Zhang \& Diller, 2018}; \hyperref[csl:13]{Zhang \& Diller, 2015}; \hyperref[csl:14]{Zhang et al., 2016)}. As this is a non-reciprocal swimming
gait, our proposed MMR could produce a net propulsion in low Reynolds
number regimes (\emph{Re} : 3.45). By controlling the MMR's sixth-DOF
angular displacement, we could also steer this actuator to follow an
`L'-shaped trajectory on the air-water interface (Fig. S12B and SI Video
S10). Our experiments indicated that the proposed MMR could swim at a
speed of 5.80 mm s\textsuperscript{-1} when the
applied~\(\vec{B}\) of strength 20 mT was rotating at a
frequency of 10 Hz.

Alternatively, our soft MMR could also choose to climb up the meniscus
of an air-water interface. By applying~\(\vec{B}\) (25 mT) along
the~\(z_{\{L\}}\)-axis of the MMR, we could deform this actuator
into its `U'-shaped configuration (Fig. S12C(i) and (ii)). In this
deformed configuration, the MMR could displace more water and generate a
greater buoyancy force \hyperref[csl:7]{(Hu et al., 2018)}. By increasing the buoyance of
the MMR, we could rotate our actuator to perform the meniscus-climbing
locomotion (Fig. S12C(ii)-(iv)).

\subsection*{\texorpdfstring{\emph{F. 3D pick-and-place
operation}}{F. 3D pick-and-place operation}}

{\label{541641}}

Here we will compute the theoretical angular and linear resolutions of
our MMR when it was executing the pick-and-place operation. By
exploiting the phenomenon in which the MMR's \(\vec{m}\) will
tend to align with the applied \(\vec{B}\) \hyperref[csl:1]{(Diller et al., 2016}; \hyperref[csl:2]{Kummer et al., 2010)},
we can rotate the actuator about its \(x_{\{L\}}\)- and
\(y_{\{L\}}\)-axes via controlling the direction of
\(\vec{B}\) (SI Section S2B). Since the smallest angular change
in \(\vec{B}\) is 0.57° (SI Section S4A), this implied that the
angular resolution of the MMR about its \(x_{\{L\}}\)- and
\(y_{\{L\}}\)-axes would be 0.57° too. The sixth-DOF angular
resolution of the MMR can be computed based on the resolution of the
field's spatial gradients. Based on Eq. (S2.14), it can be seen that the
MMR's desired sixth-DOF angle, \(\theta\), can be controlled
via tuning the field's spatial gradients. Because there were no magnetic
forces applied to the MMR during the pick-and-place operation, the
pseudo-inverse solution in Eq. (S2.14) would be a null vector (SI
section S2B). Hence, the eighth row of Eq. (S2.14) can be simplified to
establish the relationship between\(\frac{\partial B_{x,\ \ \{I\}}}{\partial y_{\{I\}}}\)
and~\(\theta\):

\[\begin{equation}
\frac{\partial B_{x,\ \ \{I\}}}{\partial y_{\{I\}}}=-k_{2}\tan\left(2\theta\right). (S4.3)\nonumber \\
 \end{equation}\]

By differentiating both sides of Eq. (S4.3) with respect to
\(\theta\), we can establish the following relationship:

\[\begin{equation}
\frac{\partial B_{x,y\left\{I\right\}}}{\partial\theta}=-2k_{2}\sec{{}^{2}{\left(2\theta\right),}}\text{\ }\nonumber \\
 \end{equation}\]\[\begin{equation}
\text{where\ }B_{x,y\left\{I\right\}}=\ \frac{\partial B_{x,\ \ \left\{I\right\}}}{\partial y_{\left\{I\right\}}}\text{\ .\ }\left(S4.4\right)\nonumber \\
 \end{equation}\]

Equation (S4.4) can be approximated as:

\[\begin{equation}
\Delta B_{x,y\left\{I\right\}}=-2k_{2}\sec{{}^{2}{\left(2\theta\right)\Delta \theta},}\ (S4.5)\nonumber \\
 \end{equation}\]

where \(\Delta B_{x,y\left\{I\right\}}\) and \(\Delta\theta\) represent the
resolutions of the field's spatial gradient and the MMR's sixth-DOF,
respectively. Based on Eq. (S4.3), we can also express
\(k_{2}\) as:

\[\begin{equation}
k_{2}=-\frac{B_{x,y\left\{I\right\}}}{\tan\left(2\theta\right)}.\  (S4.6)\nonumber \\
 \end{equation}\]

By substituting Eq. (S4.6) into Eq. (S4.5), Eq. (S4.5) can then be
further simplified and rearranged into:

\[\begin{equation}
\Delta\theta=\frac{\Delta B_{x,y\left\{I\right\}}}{4B_{x,y\left\{I\right\}}}\sin\left(4\theta\right).\  (S4.7)\nonumber \\
 \end{equation}\]

To evaluate \(\Delta\theta\), we substitute the largest and smallest
value of~\(B_{x,y\left\{I\right\}}\) and \(\Delta B_{x,y\left\{I\right\}}\) respectively into
Eq. (S4.7) so that the magnitude of \(\Delta\theta\) can be minimized
(SI Section S4A):

\[\begin{equation}
\Delta\theta=6.25\times 10^{-3}\sin\left(4\theta\right).\ \  (S4.8)\nonumber \\
 \end{equation}\]

Although Eq. (S4.8) implies that the sixth-DOF angular resolution of the
MMR is dependent on~\(\theta\), here we
assign~\(\sin\left(4\theta\right)\) with the highest value of 1 to obtain the
coarsest~\(\Delta\theta\) for simplicity purposes. In this case, the
MMR's sixth angular resolution can therefore be computed via Eq. (S4.8)
as 6.25\(\times\)10\textsuperscript{-3} rad or 0.36°.~ ~ ~~

To compute the translational resolution of the MMR, the shape of the
rolling MMR is approximated to be a circle, which has a radius of 0.95
mm (Fig. S14). To identify the best fit circle, we only consider the
deformation of the magnetic beam component when\(\left|\vec{B}\right|\) = 20
mT because this is the applied magnetic field during the pick-and-place
operations. Assuming no-slip conditions, the translational resolution
achievable by our MMR can therefore be computed by the product of its
radius and the~\(y_{\{L\}}\)-axis angular resolution (9.5
\(\text{μm}\)).

\par\null

\section*{Figures}

{\label{750390}}

\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig1/SI-fig1}
\caption{Fig. S1. The dimensions of our proposed MMR when it is seen from the (A)
top view and (B) front view.
{\label{456570}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig2/SI-fig2}
\caption{Fig. S2. The magnetization process of the magnetic beam component. The
pre-magnetized beam in (A) was rolled in a cylindrical jig, and then
magnetized by a strong, uniform magnetic field of 1.1 T in (B). The free
ends of the beam were illustrated at the exaggerated gap in (B). (C)
Once the magnetic beam recovered back to its original shape, it would
possess the optimal harmonic magnetization profile that had a phase
shift angle (\(\varphi\)) of -90 ° .
{\label{892266}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig3/SI-fig3}
\caption{Fig. S3. An experiment for evaluating the achievable~\(\left|\vec{M}\right|\)
of the proposed MMR. (A) The dimensions and net magnetic
moment,~\(\vec{m}_{sp}\), of the molded cuboid samples. (B) A
free-body diagram analysis for this experiment. (C) The
deduced~\(\frac{Wr_{0}\cos\Gamma}{V_{\text{sp}}\sin\varnothing}\) was plotted against the magnitude of the
applied magnetic fields (\(\left|\vec{B}\right|\)). Each data point had five
trials and the error bars of the data points represented the standard
deviation of these points. The best fit line of this graph was
constrained to intersect the origin, and its gradient represented the
MMR's achievable~\(\left|\vec{M}\right|\). The gradient of this best fit line
was computed to be 9.40×10\textsuperscript{4} A m\textsuperscript{-1}.
{\label{788881}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig4/SI-fig4}
\caption{Fig. S4. Quasi-static analysis of the magnetic soft beam. The
variable,~\(\ \tau_{x,\{L\}}\left(s\right)\), represents the generated magnetic torque
(per unit volume), which is distributed along the entire beam. We only
show a segment of~\(\ \tau_{x,\{L\}}\left(s\right)\) for illustration purposes.
{\label{858728}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig5/SI-fig5}
\caption{Fig. S5. Simulation results of the magnetic beam's deformation when it
is subjected to a series of~\(B_{z,\{L\}}\). As the magnitudes
of~\(B_{z,\{L\}}\) increase, it can be seen that the curvatures of
the magnetic beam will become sharper for both the (A) `U'- and (B)
inverted `V'-shaped configurations.~
{\label{567386}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig6/SI-fig6}
\caption{Fig. S6. Numerical analysis for the normalized sixth-DOF
torque,~\(\left|d_{3}\right|\), of MMRs with different~\(\varphi\).
(A) Plotting the MMRs'~\(\left|d_{3,\text{highest}}\right|\) with respect to
different~\(\varphi\), whereby~\(\left|d_{3,\text{highest}}\right|\) is seen to be
maximized when~\(\varphi\) = ±90° (indicated by the red dots).
From the graph, the ~\(\left|d_{3,\text{highest}}\right|\) of our MMR is
1.27×10\textsuperscript{-7} A m\textsuperscript{3}. For existing MMRs
with a harmonic magnetization profile that have ~\(\varphi\) of
45° ~\protect\hyperref[csl:7]{(Hu et al., 2018}; \hyperref[csl:8]{Ren et al., 2021}; \hyperref[csl:9]{Demir et al., 2021}; \hyperref[csl:10]{Culha et al., 2020)} and 0°~\protect\hyperref[csl:11]{(Diller et al., 2014}; \hyperref[csl:12]{Zhang \& Diller, 2018}; \hyperref[csl:13]{Zhang \& Diller, 2015}; \hyperref[csl:14]{Zhang et al., 2016)}, their computed
~\(\left|d_{3,\text{highest}}\right|\) are 9.06 ×10\textsuperscript{-8} A
m\textsuperscript{3} and 1.67× ×10\textsuperscript{-9} A
m\textsuperscript{3} , respectively. (B) Evaluating the
MMRs'~\(\left|d_{3}\right|\) when they deform under
different~\(B_{z,\{L\}}\). Three different MMRs have been analyzed:
(i) our proposed MMR, and existing MMRs with a harmonic magnetization
profile that have ~\(\varphi\) of (ii) 45° ~\protect\hyperref[csl:7]{(Hu et al., 2018}; \hyperref[csl:8]{Ren et al., 2021}; \hyperref[csl:9]{Demir et al., 2021}; \hyperref[csl:10]{Culha et al., 2020)}
and (iii) 0°~\protect\hyperref[csl:11]{(Diller et al., 2014}; \hyperref[csl:12]{Zhang \& Diller, 2018}; \hyperref[csl:13]{Zhang \& Diller, 2015}; \hyperref[csl:14]{Zhang et al., 2016)}. The average ~\(\left|d_{3}\right|\) of
each MMR can be computed by finding their corresponding area under the
curve and subsequently dividing those areas across the entire domain
of~\(B_{z,\{L\}}\).
{\label{730484}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig7/SI-fig7}
\caption{Fig. S7. Simulation results of (A)~\(d_1\) and
(B)~\(d_2\) against different~\(B_{z,\{L\}}\).
{\label{605758}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/SI-fig8/SI-fig8}
\caption{Fig. S8. Our customized electromagnetic coil system in which the central
workspace of the system was highlighted.
{\label{358349}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/SI-fig9/SI-fig9}
\caption{Fig. S9. The variables required to deduce the MMR's producible sixth-DOF
torque in the experiment shown in Fig. 2 of the main text. Scale bar: 2
mm.~
{\label{665410}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig10/SI-fig10}
\caption{Fig. S10. The measured rolling speed plotted against different rotating
frequencies of~\(\vec{B}\). Each data point had five trials and
the error bars represented their corresponding standard deviation. The
solid red line and dashed blue line represented the best fit line for
the data points pertaining to rolling along the MMR's length and width,
respectively. Because higher speed could be achieved when the MMR was
rolling along its length than rolling along its width, the gradient of
the solid red line was steeper than its dashed blue line counterpart.
{\label{316882}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig11/SI-fig11}
\caption{Fig. S11. Rolling along the MMR's width with a gentle curvature
(\(\left|\vec{B}\right|\)=6 mT). Scale bar: 2 mm.
{\label{434937}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig12/SI-fig12}
\caption{Fig. S12. The undulating swimming, meniscus climbing and two-anchor
crawling locomotion of the proposed MMR. (A) The MMR could execute its
undulating swimming locomotion by continuously generating a traveling
wave along its body. (ii)-(vi) The MMR produced a net displacement after
generating one period of the traveling wave. (B) As the MMR swam via the
undulating swimming locomotion, we could steer it to follow an
`L'-shaped trajectory. (C) Meniscus-climbing could be executed by the
MMR when (i)-(ii) a strong upward~\(\vec{B}\)
(\(\left|\vec{B}\right|\)=25 mT) was applied to deform the actuator and
subsequently rotating~\(\vec{B}\) from (iii) 30° to (iv) 60°
(with respect to the vertical upright direction). (D) An illustration of
the MMR's two-anchor crawling locomotion. The dashed line represented
the initial location of the MMR's rear end, and the solid line
represented the final location of the MMR's rear end after it performed
one cycle of two-anchor crawling. The distance between the dashed and
solid lines represented the net displacement produced by the MMR in one
cycle. Scale bars: 2 mm.
{\label{978921}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig13/SI-fig13}
\caption{Fig. S13. Experiment for evaluating the payload of our soft MMR. As an
example, here we presented the experiment procedure of the MMR carrying
an object with 3 folds of its weight. The procedures for carrying
objects with other weights were the same. (A) The experiment would
require the MMR to grab, transport and place the object to the desired
placement location (highlighted by the dotted red ellipse). (B) The MMR
grabbed the object. (C)-(D) The MMR held on to the object and
transported it. (E)-(F) The MMR released the object at the desired
placement location. Scale bar: 2 mm.~
{\label{663873}}%
}
\end{center}
\end{figure}

\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig14/SI-fig14}
\caption{Fig. S14. The best fit circle of the rolling MMR during the
pick-and-place operations. The shape of the deformed magnetic beam
(black thick line) is simulated to determine the best fit circle (green
line). The radius of the best fit circle is 0.95 mm.~
{\label{906522}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig15/SI-fig15}
\caption{Fig. S15. The global reference frame in the experiments shown in SI
Video S1-S10. Scale bars: 2 mm.
{\label{641731}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig16/SI-fig16}
\caption{Fig. S16. The magnetic actuating signals for the experiment in SI Video
S1 (rolling).
{\label{443998}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig17/SI-fig17}
\caption{Fig. S17. The magnetic actuating signals for the experiment in SI Video
S1 (translations).
{\label{941429}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig18/SI-fig18}
\caption{Fig. S18. The magnetic actuating signals for the experiment in SI Video
S2 (using the two-anchor crawling locomotion to ascend an inclined slope
of 20°).
{\label{679264}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig19/SI-fig19}
\caption{Fig. S19. The magnetic actuating signals for the experiment in SI Video
S2 (two-anchor crawling across obstacles).
{\label{292234}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig20/SI-fig20}
\caption{Fig. S20. The magnetic actuating signals for the experiment in SI Video
S3 (undulating crawling across two perpendicular planes to negotiate a
complex channel).~
{\label{700842}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig21/SI-fig21}
\caption{Fig. S21. The magnetic actuating signals for the experiment in SI Video
S3 (undulating crawling across a complex channel that had an inclined
slope of 20° and a sharp turn).
{\label{983008}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig22/SI-fig22}
\caption{Fig. S22. The magnetic actuating signals for the experiment in SI Video
S4 (jumping while controlling the angular displacements about
the~\(z_{\{L\}}\)- and~\(x_{\{L\}}\)-axes of the MMR).
{\label{971377}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig23/SI-fig23}
\caption{Fig. S23. The magnetic actuating signals for the experiment in SI Video
S4 (jumping while controlling the angular displacements about
the~\(z_{\{L\}}\)- and~\(y_{\{L\}}\)-axes of the MMR).
{\label{776291}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig24/SI-fig24}
\caption{Fig. S24. The magnetic actuating signals for the experiment in SI Video
S5 (performing the jellyfish-like swimming locomotion while controlling
the MMR's~\(x_{\{L\}}\)-axis rotation).
{\label{906574}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig25/SI-fig25}
\caption{Fig. S25. The magnetic actuating signals for the experiment in SI Video
S5 (performing the jellyfish-like swimming locomotion while controlling
the MMR's~\(y_{\{L\}}\)-axis rotation).
{\label{354247}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig26/SI-fig26}
\caption{Fig. S26. The magnetic actuating signals for the experiment in SI Video
S5 (performing the jellyfish-like swimming locomotion while controlling
the MMR's~\(z_{\{L\}}\)-axis rotation).
{\label{406911}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig27/SI-fig27}
\caption{Fig. S27. The magnetic actuating signals for the experiment in SI Video
S6 (multimodal locomotion: jumping and rolling).
{\label{215907}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig28/SI-fig28}
\caption{Fig. S28. The magnetic actuating signals for the experiment in SI Video
S7 (multimodal locomotion: two-anchor crawling and undulating crawling).
{\label{677827}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig29/SI-fig29}
\caption{Fig. S29. The magnetic actuating signals for the experiment in SI Video
S8 (multimodal locomotion: jellyfish-like swimming and landing).
{\label{420155}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig30/SI-fig30}
\caption{Fig. S30. The magnetic actuating signals for the experiment in SI Video
S9 (3D pick-and-place operation).
{\label{797141}}%
}
\end{center}
\end{figure}



\begin{figure}[H]
\begin{center}
\includegraphics[width=0.84\columnwidth]{figures/SI-fig31/SI-fig31}
\caption{Fig. S31. The magnetic actuating signals for the experiment in SI Video
S10 (undulating swimming locomotion).
{\label{246522}}%
}
\end{center}
\end{figure}

\section*{~}

{\label{925260}}

\section*{Supporting Table}

{\label{162306}}\par\null

\textbf{Table S1.} Nomenclature.

\begin{figure}[H]
\begin{center}
\includegraphics[width=0.91\columnwidth]{figures/table/table}

\end{center}
\end{figure}

\section*{~ ~ SI Video S1}

{\label{950035}}

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

{\label{730375}}

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

\section*{~ ~ SI Video S2}

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

{\label{907969}}

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

\section*{SI Video S3}

{\label{198538}}

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

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

\section*{SI Video S4}

{\label{761826}}

\subsection*{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~~~~~~~~~ Jumping locomotion~}

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

\section*{SI Video S5}

{\label{386097}}

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

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

\section*{SI Video S6}

{\label{247790}}

\subsection*{~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~
Multimodal locomotion: Jumping and
Rolling~}

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

\section*{SI Video S7}

{\label{655088}}

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

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

\section*{SI Video S8}

{\label{823939}}

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

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

\section*{SI Video S9}

{\label{839695}}

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

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

\section*{SI Video S10}

{\label{789708}}

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

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

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

