## 4 Experiments and Results
### Open Loop Control Results  
Open-loop experiment operation process. First, preset the motion trajectory, then reversely solve the current sequence according to the mapping relationship between the current and the magnetic field distribution, and finally input the current sequence to record the motion trajectory of the milli/microrobot.  
We initially demonstrated that the magnetic carrier can be manipulated by fixed-point rotating under the open-loop control which can visually characterize the control accuracy of the system in the direction of the magnetic field \(B\). The origin of the global coordinate system (\(X=0\) mm, \(Y=0\) mm, \(Z=0\) mm) was selected for performing experiments and fixed-point rotation experiments around the \(Z\)-axis are performed as shown in Figure 8. The purpose of this experimental demonstration is mainly to show that the system allows us to decouple degrees of freedom for rotation and translation. The orientation of the milli/microrobot converges at one point in the trajectory, and at the same time, the trajectory of its centroid point is a circle in the plane. Therefore, the two-DOF orientation motion and three-DOF translational motion of the milli/microbot are decoupled.  
Table 2 illustrates the experimental results of the fixed-point rotation (a) and the errors in each situation.  
In the second experiment, as shown in Figure 9 we verify that the system has the ability to provide a stable direction-determining force and the magnetic carrier moving along two different diagonal lines are performed which can intuitively characterize the accuracy of the magnetic field gradient direction control.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline
**Data** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** \\ \hline Desired Angle (\({}^{\circ}\)) & 15 & 30 & 45 & 60 & 75 & 90 & 105 & 120 & 135 & 150 & 165 & 180 \\ Measured Angle (\({}^{\circ}\)) & 13.9 & 29.3 & 44.7 & 57.8 & 76.2 & 89.4 & 105.8 & 119.7 & 133.7 & 149.7 & 164.3 & 179 \\ Error (\({}^{\circ}\)) & 1.1 & 0.7 & 0.3 & 2.2 & -1.2 & 0.6 & -0.8 & 0.3 & 1.3 & 0.3 & 0.7 & 1.0 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Experimental Results.  
Figure 8: Demonstration of rotation control. (**a**) The magnetic carrier rotates around the \(Z\)-axis at the coordinate point (0, 0, 0), and the rotation amplitude is 15 degrees each time. (**b**) The magnetic carrier enable more complex spatial rotational motion. The direction of the magnetic moment of the magnetic carrier is 45 degrees from the \(Z\)-axis, and the circular rotation around the \(Z\)-axis is performed at coordinates (0, 0, 0).  
Figure 9: Demonstration of displacement control. (**a**) The milli/microbot moves from the upper-right corner of the workspace along with the \(XOY\) projection to the lower-left corner while descending stepwise in the \(Z\)-axis direction. Average trajectory-completion time: 10.83 s. (**b**) The milli/microbot moves from the lower-right corner of the workspace along with the \(XOY\) projection to the upper-left corner while rising stepwise in the \(Z\)-axis direction. Average trajectory-completion time: 14.4 s. The left figures are trajectories in the 3D workspace. The right figures show the errors between actual and desired trajectories.