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 \(\times\)10 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 \(\times\)10 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:
\[\begin{equation} \par \begin{matrix}{\vec{m}}_{\varphi }=\iiint{\mathbf{R}_{x}\left(\gamma\right)\mathbf{R}_{x}\left(\varphi\right){\vec{M}}_{\varphi 0,\ \ \left\{M\right\}}}dV \end{matrix} \nonumber \\ \end{equation}\]
\[\begin{equation} \par \begin{matrix}=\iiint{\mathbf{R}_{x}\left(\varphi\right)\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\varphi 0,\ \ \left\{M\right\}}}dV \end{matrix} \nonumber \\ \end{equation}\]
\[\begin{equation} \par \begin{matrix}=\mathbf{R}_{x}\left(\varphi\right)\iiint{\mathbf{R}_{x}\left(\gamma\right){\vec{M}}_{\varphi 0,\ \ \left\{M\right\}}}dV \end{matrix} \nonumber \\ \end{equation}\]
\[\begin{equation} {=\mathbf{R}_{x}\left(\varphi\right){\vec{m}}_{\varphi 0,}} \nonumber \\ \end{equation}\]