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