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:
  1. 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.
  2. 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}.
  3. 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.