Generating rotating magnetic fields with a single permanent magnet for the Propulsion of untethered

本文是14年的T-RO。主要阐述了在旋转磁场下，探究主动旋转磁铁的位姿与被动旋转磁铁的位姿之间的关系。推导并总结了，给定主动磁铁和被动磁铁间的相对位置关系和期望的被动旋转磁铁的姿态，计算所需的主动磁铁姿态；或给定期望的被动旋转磁铁的姿态和主动磁铁姿态，计算所需的主动磁铁和被动磁铁之间的位置关系。

用一个单一永磁铁生成旋转磁场用于在一个腔道中的无线的磁设备的驱动
Generating rotating magnetic fields with a single permanent magnet for the Propulsion of untethered magnetic devices in a lumen [1]
Paper Link
Authors: Arthur W. Mahoney, etc.
2013，IEEE Transactions on Robotics

如果磁场$\mathbf{h}$被用一个单独驱动器磁铁生成，然后它的在相对于驱动器磁铁中心的无线的磁设备的位置$\mathbf{p}$的场，能被近似通过点-极子模型：
If the magnetic field $\mathbf{h}$ is generated using a single actuator magnet, then its field at the UMD’s position $\mathbf{p}$, relative to the actuator-magnet center, can be approximated by the point-dipole model:

$$\mathbf{h}=\frac{1}{4\pi {\Vert \mathbf{p}\Vert }^3}(3\hat{\mathbf{p}}{\hat{\mathbf{p}}}^T-\mathbf{I}){\mathbf{m}}_a$$

选择驱动磁铁旋转轴（$\widehat{{\mathbf{\omega }}_a}$）给定胶囊旋转期望轴（$\widehat{{\mathbf{\omega }}_h}$）和两者之间的相对位置（$\hat{\mathbf{p}}$）：对于一个由一个绕轴$\widehat{{\mathbf{\omega }}_a}$旋转的驱动器磁铁产生的极子场，来使在任意位置$\mathbf{p}$的磁场$\mathbf{h}$绕一个期望轴$\widehat{{\mathbf{\omega }}_h}$旋转的需要的旋转轴$\widehat{{\mathbf{\omega }}_a}$是不容易可见的。如果驱动器磁铁被旋转，这样它的极子磁矩${\mathbf{m}}_a$与$\widehat{{\mathbf{\omega }}_a}$垂直（i.e.，${\mathbf{m}}_a^T\widehat{{\mathbf{\omega }}_a}=0$）并且这是可以期望的，场$\mathbf{h}$与$\widehat{{\mathbf{\omega }}_h}$垂直（i.e.，${\mathbf{h}}^T\widehat{{\mathbf{\omega }}_h}=0$），然后给定一个期望的$\widehat{{\mathbf{\omega }}_h}$下的必要的$\widehat{{\mathbf{\omega }}_a}$能够被找到：
Choosing $\widehat{{\mathbf{\omega }}_a}$ Given $\widehat{{\mathbf{\omega }}_h}$ and $\hat{\mathbf{p}}$ : for a dipole field generated by an actuator magnet that rotates around the axis $\widehat{{\mathbf{\omega }}_a}$, the required rotation axis $\widehat{{\mathbf{\omega }}_a}$ to make the magnetic field $\mathbf{h}$ at any desired position $\mathbf{p}$ rotate around a desired axis $\widehat{{\mathbf{\omega }}_h}$. If the actuator magnet is rotated such that its dipole moment ${\mathbf{m}}_a$ is perpendicular to $\widehat{{\mathbf{\omega }}_a}$ (i.e.，${\mathbf{m}}_a^T\widehat{{\mathbf{\omega }}_a}=0$), and it is desired, the field $\mathbf{h}$ is perpendicular to $\widehat{{\mathbf{\omega }}_h}$（i.e.，${\mathbf{h}}^T\widehat{{\mathbf{\omega }}_h}=0$）， then the necessary $\widehat{{\mathbf{\omega }}_a}$ given a desired $\widehat{{\mathbf{\omega }}_h}$ can be found:

$${\mathbf{h}}^T\widehat{{\mathbf{\omega }}_h}=\frac{1}{4\pi {\Vert \mathbf{p}\Vert }^3}{\mathbf{m}}_a^T(3\hat{\mathbf{p}}{\hat{\mathbf{p}}}^T-\mathbf{I})\widehat{{\mathbf{\omega }}_h}=0$$

$$⟹0={\mathbf{m}}_a^T(3\hat{\mathbf{p}}{\hat{\mathbf{p}}}^T-\mathbf{I})\widehat{{\mathbf{\omega }}_h}={\mathbf{m}}_a^T\widehat{{\mathbf{\omega }}_a}$$

$$⟹\widehat{{\mathbf{\omega }}_a}=\hat{(3\hat{\mathbf{p}}{\hat{\mathbf{p}}}^T-\mathbf{I})\widehat{{\mathbf{\omega }}_h}}$$

[1]: Mahoney, Arthur W., and Jake J. Abbott. “Generating rotating magnetic fields with a single permanent magnet for propulsion of untethered magnetic devices in a lumen.” IEEE Transactions on Robotics 30.2 (2013): 411-420.