Section S3- Analysis of sixth-DOF torque
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 \cite{Hu2018,Ren2021,sitti2021,Culha2020,sitti2014a,Zhang2018,diller2015,diller2016}. 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 \cite{Xu2021}.
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 \(\times\)104 A m-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\(\times\)10-7 N m2 T-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 \cite{Hu2018,Ren2021,sitti2021,Culha2020,sitti2014a,Zhang2018,diller2015,diller2016} 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 \(\times\)104 A m-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:
\[\begin{equation}
{\vec{M}}_{\varphi,\ \ \left\{M\right\}}\left(s\right)=\left|{\vec{M}}_{\ \left\{M\right\}}\left(s\right)\right|\begin{pmatrix}0,&\cos\left(\mathbf{-}\frac{2\pi}{L}s+\varphi\right)\mathbf{,}&\sin\left(\mathbf{-}\frac{2\pi}{L}s+\varphi\right)\\
\end{pmatrix}^{T}. (S3.2)\nonumber \\
\end{equation}\]
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 \cite{Hu2018,Ren2021}. 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° \cite{Hu2018,Ren2021,sitti2021,Culha2020} and 0° ( \cite{sitti2014a,Zhang2018,diller2015,diller2016}, 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° \cite{Hu2018,Ren2021,sitti2021,Culha2020} and 0° \cite{sitti2014a,Zhang2018,diller2015,diller2016} (Fig. S6B). These simulation results indicate that the average \(\left|d_{3}\right|\) of our proposed MMR
( 5.77\(\times\)10-8 A m3 ) is 1.41-63.9 folds larger than
those that have \(\varphi\) of 45° (4.10 \(\times\)10-8 A m3 ) \cite{Hu2018,Ren2021,sitti2021,Culha2020} and 0° (9.03 \(\times\)10-10 A m3 ) \cite{sitti2014a,Zhang2018,diller2015,diller2016}. 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° \cite{Hu2018,Ren2021,sitti2021,Culha2020} and 0° \cite{sitti2014a,Zhang2018,diller2015,diller2016}.
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
\cite{Hu2018,Ren2021,sitti2021,Culha2020,sitti2014a,Zhang2018,diller2015,diller2016}. To prove this hypothesis mathematically, we first reanalyze the
generic magnetization profile in Eq. (S3.2) and arrange it to the
following format:
\[\begin{equation}
{{\vec{M}}_{\varphi,\ \ \left\{M\right\}}\left(s\right)=\mathbf{R}_{x}\left(\varphi\right)\left|{\vec{M}}_{\ \left\{M\right\}}\left(s\right)\right|\begin{pmatrix}0,&\cos\left(\mathbf{-}\frac{2\pi}{L}s\right)\mathbf{,}&\sin\left(\mathbf{-}\frac{2\pi}{L}s\right)\\
\end{pmatrix}^{T}} \nonumber \\
\end{equation}\]\[\begin{equation}
{=\mathbf{R}_{x}\left(\varphi\right){\vec{M}}_{\varphi 0,\ \ \left\{M\right\}}\left(s\right),}\nonumber \\
\end{equation}\]
where
\(\begin{equation}
\par
\begin{matrix}{\vec{M}}_{\varphi 0,\ \ \left\{M\right\}}\left(s\right)=\left|{\vec{M}}_{\left\{M\right\}}\left(s\right)\right|\par
\begin{pmatrix}0,&\cos\left(\mathbf{-}\frac{2\pi}{L}s\right)\mathbf{,}&\sin\left(\mathbf{-}\frac{2\pi}{L}s\right)\\
\end{pmatrix}^{T},\left(S3.3\right)\\
\end{matrix} \nonumber \\
\end{equation}\)
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 \(B_{z,\left\{L\right\}}\) \cite{Hu2018}. 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 \(\left({\vec{m}}_{\varphi}\right)\) \cite{Hu2018}. 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:
\[\begin{equation}
\par
\begin{matrix}{\vec{m}}_{\varphi }=\iiint{\mathbf{R}_{x}\left(\gamma\right)\mathbf{R}_{x}\left(\varphi\right){\vec{M}}_{\varphi 0,\ \ \left\{M\right\}}}dV
\end{matrix} \nonumber \\
\end{equation}\]
\[\begin{equation}
\par
\begin{matrix}=\iiint{\mathbf{R}_{x}\left(\varphi\right)\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\varphi 0,\ \ \left\{M\right\}}}dV
\end{matrix} \nonumber \\
\end{equation}\]
\[\begin{equation}
\par
\begin{matrix}=\mathbf{R}_{x}\left(\varphi\right)\iiint{\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\varphi 0,\ \ \left\{M\right\}}}dV
\end{matrix} \nonumber \\
\end{equation}\]
\[\begin{equation}
{=\mathbf{R}_{x}\left(\varphi\right){\vec{m}}_{\varphi 0,}} \nonumber \\
\end{equation}\]