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
\cite{Hu2018,Ren2021,sitti2021,Culha2020,sitti2014a,Zhang2018,diller2015,diller2016}. 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 S4- Experiments
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.
A. Magnetic actuation system
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-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.
B. Experimental investigation on the MMR’s sixth-DOF torque
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 \cite{Lum2016}:
\[\begin{equation}
\begin{matrix}M_{\text{bt}}\left(s_{\text{bt}}\right)=E_{\text{bt}}I_{\text{bt}}\frac{\partial\theta_{\text{bt}}}{\partial s_{\text{bt}}}\left(s_{\text{bt}}\right),\left(S4.1\right)\\
\end{matrix}\nonumber \\
\end{equation}\]
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\(\times\)10-15 m4 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\(\times\)10-3 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).
C. Rolling locomotion
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 \(\vec{B}\) \cite{Hu2018}. 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).
D. Jellyfish-like swimming locomotion
For the jellyfish-like swimming locomotion, the average speeds
(\(V_{R}\)) of our MMR were 7.65\(\times\)10-3 m s-1, 10.4\(\times\)10-3 m s-1 and 9.49\(\times\)10-3 m s-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\(\times\)10-7 m2 s-1 \cite{wakeham1978}. 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\(\times\)10-3 m s-1).
E. Undulating swimming and meniscus climbing locomotion
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)
\cite{Hu2018,sitti2014a,Zhang2018,diller2015,diller2016}. As this is a non-reciprocal swimming gait, our proposed MMR
could produce a net propulsion in low Reynolds number regimes
(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-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 \cite{Hu2018}. By
increasing the buoyance of the MMR, we could rotate our actuator to
perform the meniscus-climbing locomotion (Fig. S12C(ii)-(iv)).
F. 3D pick-and-place operation
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 \(\vec{B}\) \cite{sitti2016,nelson2010}, 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\(\times\)10-3 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 \(\text{μm}\)).
Figures