Section S1- Fabrication method

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).

A. Fabrication of six-DOF soft MMR

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 µm) 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 °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.

B. Characterization of the magnetic soft beam’s material properties

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\(\times\)104 A m-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).