Section S2- Actuation method
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 \cite{sitti2016,nelson2010,sitti2014}. 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 \(\vec{B}\) \cite{sitti2016,Xu2021}:
\[\begin{equation}
\begin{matrix}\frac{\partial B_{x}}{\partial x}+\frac{\partial B_{y}}{\partial
y}+\frac{\partial B_{z}}{\partial z}=0,\left(S2.1A\right)\\
\end{matrix}\nonumber \\
\end{equation}\]\[\begin{equation}
\begin{matrix}\frac{\partial B_{z}}{\partial x}=\frac{\partial B_{x}}{\partial
z},\ \ \frac{\partial B_{z}}{\partial y}=\frac{\partial B_{y}}{\partial z},\ \ \frac{\partial B_{y}}{\partial x}=\frac{\partial B_{x}}{\partial y}.\left(S2.1B\right)\\
\end{matrix}\nonumber \\
\end{equation}\]
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:
\[\begin{equation}
\begin{matrix}{\vec{B}}_{\text{grad}}=\left[\frac{\partial B_{z}}{\partial x}\text{\ \ \ \ }\frac{\partial B_{z}}{\partial y}\text{\ \ \ \ }\frac{\partial B_{z}}{\partial z}\text{\ \ \ \ }\frac{\partial B_{y}}{\partial
y}\text{\ \ \ \ }\frac{\partial B_{x}}{\partial y}\right]^{T}.\left(S2.2\right)\\
\end{matrix}\nonumber \\
\end{equation}\]
A. Deformation mechanism
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:
\[\begin{equation}
\begin{matrix}{\vec{M}}_{\left\{M\right\}}\left(s\right)=\left|{\vec{M}}_{\left\{M\right\}}\right|\begin{pmatrix}0,&\cos\left(\mathbf{-}\frac{2\pi}{L}s-\frac{\pi}{2}\right)\mathbf{,}&\sin\left(\mathbf{-}\frac{2\pi}{L}s-\frac{\pi}{2}\right)\\
\end{pmatrix}^{T},\left(S2.3\right)\\
\end{matrix}\nonumber \\
\end{equation}\]
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 \cite{Xu2021}. 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,\(\text{\ γ}\left(s\right)\) (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 \cite{Lum2016} :
\[\begin{equation}
\tau_{x,\{L\}}\left(s\right)=\begin{bmatrix}1&0&0\\
\end{bmatrix}\left\{\left(\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\left\{M\right\}}\left(s\right)\right)\times{\vec{B}}_{\left\{L\right\}}\right\},\nonumber \\
\end{equation}\]
where
\[\begin{equation}
\begin{matrix}\mathbf{R}_{x}\left(\gamma\right)=\begin{pmatrix}1&0&0\\
0&\cos\gamma&-\sin\gamma\\
0&\sin\gamma&\cos\gamma\\
\end{pmatrix}.\left(S2.5\right)\\
\end{matrix}\nonumber \\
\end{equation}\]
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 \cite{Lum2016}. Based on the Euler-Bernoulli equation, we can establish
the relationship between \(M_{b}\) and \(\gamma\) as \cite{Lum2016}:
\[\begin{equation}
\begin{matrix}M_{b}=EI\frac{\partial\gamma}{\partial s},\left(S2.6\right)\\
\end{matrix}\nonumber \\
\end{equation}\]
where E and 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:
\[\begin{equation}
\begin{matrix}-\begin{bmatrix}1&0&0\\
\end{bmatrix}\left\{\left(\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\left\{M\right\}}\left(s\right)\right)\times{\vec{B}}_{\left\{L\right\}}\right\}A=EI\frac{\partial^{2}\gamma}{\partial s^{2}}\left(s\right).\left(S2.7\right)\\
\end{matrix}\nonumber \\
\end{equation}\]
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\(\times\)104 A m-1 and 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 \cite{Xu2021,zhao2020}. 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.
B. Six-DOF motion analysis
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 \cite{sitti2016,sitti2014,Xu2021}, we express
the net magnetic torque (\(\vec{T}\)) and force
(\(\vec{F}\)) applied on the MMR based on this reference
frame \cite{Xu2021}:
\[\begin{equation}
\begin{pmatrix}{\vec{T}}_{\left\{L\right\}}\\
{\vec{F}}_{\left\{L\right\}}\\
\end{pmatrix}
=\text{\ }\begin{pmatrix}\iiint{\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\left\{M\right\}}\times{\vec{B}}_{\left\{L\right\}}\text{\ d}V}+\iiint{{\vec{r}}_{\{L\}}\times\left(\left[\frac{\partial{\vec{B}}_{\left\{L\right\}}}{\partial x_{\left\{L\right\}}}\text{\ \ \ }\frac{\partial{\vec{B}}_{\left\{L\right\}}}{\partial y_{\left\{L\right\}}}\text{\ \ \ }\frac{\partial{\vec{B}}_{\left\{L\right\}}}{\partial z_{\left\{L\right\}}}\right]^{T}\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\left\{M\right\}}\right)\ dV}\\
\iiint{\left(\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\left\{M\right\}}\bullet\nabla\right){\vec{B}}_{\left\{L\right\}}\ dV}
\end{pmatrix}\nonumber
\end{equation}\]\[\begin{equation}
\begin{matrix}\mathbf{=D}\begin{pmatrix}{\vec{B}}_{\left\{L\right\}}\\
{\vec{B}}_{grad,\left\{L\right\}}\\
\end{pmatrix},\mathbf{}\left(S2.8\right)\\
\end{matrix}\nonumber \\
\end{equation}\]
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 \cite{sitti2016,Xu2021}. The design matrix \(\mathbf{D}\) of
our soft MMR can be explicitly expressed as:
\[\begin{equation}
\mathbf{D}=\begin{pmatrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}0\\
\left|\vec{m}\right|\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}&\begin{matrix}\begin{matrix}-\left|\vec{m}\right|\\
0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}&\begin{matrix}\begin{matrix}\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}&\begin{matrix}\begin{matrix}0\\
d_{2}\\
0\\
\end{matrix}\\
\begin{matrix}\left|\vec{m}\right|\\
0\\
0\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}&\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}d_{1}\\
0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
\left|\vec{m}\right|\\
0\\
\end{matrix}\\
\end{matrix}&\begin{matrix}\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
\left|\vec{m}\right|\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}&\begin{matrix}\begin{matrix}\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}&\begin{matrix}\begin{matrix}0\\
0\\
d_{3}\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\\
\end{pmatrix},\ \nonumber \\
\end{equation}\]\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 Cartesiany - and z -axes components of \({\vec{r}}_{\{L\}}\). Likewise, the variables, \(M_{y}\) and \(M_{z}\), represent the Cartesian y - and 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 \cite{sitti2016,Xu2021}.
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 \cite{Xu2021}. This is because the
local reference frame lacks the information of the MMR’s sixth-DOF
angular displacement (\(\theta\)) \cite{Xu2021}. 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:
\[\begin{equation}
\begin{pmatrix}{\vec{T}}_{\left\{I\right\}}\\
{\vec{F}}_{\left\{I\right\}}\\
\end{pmatrix}=\begin{pmatrix}\mathbf{R}_{z}\left(\theta\right)&\mathbf{0}_{3\times 3}\\
\mathbf{0}_{3\times 3}&\mathbf{R}_{z}\left(\theta\right)\\
\end{pmatrix}\begin{pmatrix}{\vec{T}}_{\left\{L\right\}}\\
{\vec{F}}_{\left\{L\right\}}\\
\end{pmatrix},\nonumber \\
\end{equation}\]
where
\[\begin{equation}
\begin{matrix}\mathbf{R}_{z}\left(\theta\right)=\begin{pmatrix}\cos\theta&-\sin\theta&0\\
\sin\theta&\cos\theta&0\\
0&0&1\\
\end{pmatrix}.\left(S2.10\right)\\
\end{matrix}\nonumber \\
\end{equation}\]
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}&\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}&\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}&\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}&\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}&\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}&\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}&\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:
\[\begin{equation}
\begin{pmatrix}{\vec{T}}_{\left\{I\right\}}\\
{\vec{F}}_{\left\{I\right\}}\\
\end{pmatrix}=\begin{pmatrix}\mathbf{R}_{z}\left(\theta\right)&\mathbf{0}_{3\times 3}\\
\mathbf{0}_{3\times 3}&\mathbf{R}_{z}\left(\theta\right)\\
\end{pmatrix}\begin{pmatrix}{\vec{T}}_{\left\{L\right\}}\\
{\vec{F}}_{\left\{L\right\}}\\
\end{pmatrix}\nonumber \\
\end{equation}\]\[\begin{equation}
\mathbf{=}\begin{pmatrix}\mathbf{R}_{z}\left(\theta\right)&\mathbf{0}_{3\times
3}\\
\mathbf{0}_{3\times 3}&\mathbf{R}_{z}\left(\theta\right)\\
\end{pmatrix}\mathbf{D}\begin{pmatrix}{\vec{B}}_{\left\{L\right\}}\\
{\vec{B}}_{\text{grad},\left\{L\right\}}\\
\end{pmatrix}\text{\ }\nonumber \\
\end{equation}\]\[\begin{equation}
\mathbf{=}\begin{pmatrix}\mathbf{R}_{z}\left(\theta\right)&\mathbf{0}_{3\times
3}\\
\mathbf{0}_{3\times 3}&\mathbf{R}_{z}\left(\theta\right)\\
\end{pmatrix}\mathbf{D}\mathbf{A}\begin{pmatrix}{\vec{B}}_{\left\{I\right\}}\\
{\vec{B}}_{\text{grad},\left\{I\right\}}\\
\end{pmatrix}\text{\}\nonumber \\
\end{equation}\]\[\begin{equation}
=\mathbf{C}\left(\theta\right)\begin{pmatrix}{\vec{B}}_{\left\{I\right\}}\\
{\vec{B}}_{\text{grad},\left\{I\right\}}\\
\end{pmatrix},\ \ \nonumber \\
\end{equation}\]
where
\[\begin{equation}
\begin{matrix}\mathbf{C}\left(\theta\right)=\begin{pmatrix}\mathbf{R}_{z}\left(\theta\right)&\mathbf{0}_{3\times 3}\\
\mathbf{0}_{3\times 3}&\mathbf{R}_{z}\left(\theta\right)\\
\end{pmatrix}\mathbf{D}\mathbf{A}.\left(S2.12\right)\\
\end{matrix}\nonumber \\
\end{equation}\]
The matrix \(\mathbf{C}\left(\theta\right)\) is known as the control
matrix \cite{Xu2021}, and it can be expressed explicitly as:
\[\begin{equation}
\begin{matrix}\mathbf{C}\left(\theta\right)=\left(\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}0\\
\left|\vec{m}\right|\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}&\begin{matrix}\begin{matrix}-\left|\vec{m}\right|\\
0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}&\begin{matrix}\begin{matrix}\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}&\begin{matrix}\begin{matrix}d_{3}\sin\theta\cos\theta\\
d_{2}\cos^{2}\theta-d_{1}\sin^{2}\theta\\
0\\
\end{matrix}\\
\begin{matrix}\left|\vec{m}\right|\\
0\\
0\\
\end{matrix}\\
\end{matrix}&\begin{matrix}\begin{matrix}d_{1}\cos^{2}\theta-d_{2}\sin^{2}\theta\\
-d_{3}\sin\theta\cos\theta\\
0\\
\end{matrix}\\
\begin{matrix}0\\
\left|\vec{m}\right|\\
0\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\begin{matrix}\begin{matrix}\begin{matrix}0\\
0\\
d_{3}\sin\theta\cos\theta\\
\end{matrix}\\
\begin{matrix}0\\
0\\
\left|\vec{m}\right|\\
\end{matrix}\\
\end{matrix}&\begin{matrix}\begin{matrix}0\\
0\\
d_{3}\sin{2\theta}\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}&\begin{matrix}\begin{matrix}0\\
0\\
d_{3}\cos{2\theta}\\
\end{matrix}\\
\begin{matrix}0\\
0\\
0\\
\end{matrix}\\
\end{matrix}\\
\end{matrix}\right).\left(S2.13\right)\\
\end{matrix}\nonumber \\
\end{equation}\]
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:
- 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.
- The first null space vector, \(\par
\begin{pmatrix}\par
\begin{matrix}\par
\begin{matrix}0&0\\
\end{matrix}&\par
\begin{matrix}1&0\\
\end{matrix}\\
\end{matrix}&\par
\begin{matrix}\par
\begin{matrix}0&0\\
\end{matrix}&\par
\begin{matrix}0&0\\
\end{matrix}\\
\end{matrix}\\
\end{pmatrix}_{\{I\}}^{T}\), generates two axes of restoring torques
for the MMR, but it cannot generate a restoring torque about the
sixth-DOF axis of the actuator \cite{sitti2016,nelson2010,sitti2014}.
- 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 \cite{Xu2021}. 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.
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) \cite{Xu2021}. 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:
\[\begin{equation}
\begin{pmatrix}{\vec{B}}_{\left\{G\right\}}\\
{\vec{B}}_{grad,\left\{G\right\}}\\
\end{pmatrix}=\begin{bmatrix}\mathbf{R}_{x}\left(\alpha\right)\mathbf{R}_{y}\left(\beta\right)&\mathbf{0}_{3\times 5}\\
\mathbf{0}_{5\times 3}&\mathbf{A}_{2}\left(\alpha,\beta\right)\\
\end{bmatrix}\ \begin{pmatrix}{\vec{B}}_{\left\{I\right\}}\\
{\vec{B}}_{grad,\left\{I\right\}}\\
\end{pmatrix},\nonumber \\
\end{equation}\]
where
\begin{equation}
\mathbf{R}_{x}\left(\alpha\right)=\begin{pmatrix}1&0&0\\
0&\cos\alpha&-\sin\alpha\\
0&\sin\alpha&\cos\alpha\\
\end{pmatrix},\ \ \mathbf{R}_{y}\left(\beta\right)=\begin{pmatrix}\cos\beta&0&\sin\beta\\
0&1&0\\
-\sin\beta&0&\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}&\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}&\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}&\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}&\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.
C. Additional discussion
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 |\(\vec{m}\)| 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.