OctoMag: An Electromagnetic System for 5-DOF Wireless Micromanipulation

用于5D控制的电磁线圈组驱动系统，非常典型的设计，有详细的推导/建模，设计/制造过程。主要是突出5D控制，在使用过程中的位置闭环是靠视觉直接观察（人工闭环）。这篇文章提出的framework非常有借鉴意义，是很多类似电磁驱动系统的参考。

OctoMag: 用于5自由度无线微操作的一个电磁铁系统
OctoMag: An Electromagnetic System for 5-DOF Wireless Micromanipulation [1]
Paper Link
Authors: J. J. Abbott, etc.
2010, TRO IEEE Transactions on Robotics

1. 介绍 Introduction

OctoMag的设计有如下一些基本目标。
（1）固定电磁铁因它们安全被选择：它们不需要移动部件来控制场强度，它们是惰性的当断电，并且它们是故障安全的，在发生断电的情况下，微机器人在自身重力的作用下简单漂移下滑。软磁核心被选择而不是空气核心因为它们创造大约20倍强的磁场。与空气核心电磁铁相反的是，它们的单独磁场不是线性叠加的，这使建模和控制复杂化了。但是，正如我们将展示的，如果它们在它们的线性磁化区域内被操作的话，由高性能软磁材料制作的核心在建模和控制上强加了只有一个非常小的限制，基于实际能量限制来说，这最够大了。
（2）微机器人控制通常依赖于完全环绕工作区域的系统，比如电磁线圈组的正交排列，这在技术上很难按比例放大到体内设备控制所需的尺寸。
（3）这需要一个可用工作空间尺寸为25mm直径的球形，与一个人类眼睛的内体积一致，并且它需要一个更大的开放空间在电磁铁之间来安放一个小型动物头部。最终决定，一个直径为130mm的球形就足够了。
（4）工作空间应是几乎各向同性的，有能力来生成足够的磁力在任何微机器人位姿的任何方向。
The design of OctoMag began with a few goals as follows.
(1) Stationary electromagnets were chosen for their safety: they do not require moving parts to control field strength, they are inert when powered down, and their are fail-safe, in that the microrobo simply drifts down under its own weight in the event of power failure. Soft-magnetic cores were chosen over air cores because they create a field that is approximately 20 times stronger. As opposed to air-core electromagnets, their individual fields do not linearly superimpose, wich complicates modeling and control. However, as we will show, cores made of high-performance soft-magnetic material impose only a very minor constraint on modeling and control if they are operated within their linear magnetization region, which is sufficiently large based on practical power limits.
(2) Microrobot control has typically relied on systems that fully surround the workspace, such as orthgonal arrangments of electromagnetic coil pairs, which are technically difficult to scale up to the size that wound be required for control of in-vivo devices.
(3) This required a usable workspace the size of a 25-mm-diameter sphere, corresponding to the interior volume of a human eye, and it required a larger open volume between the electromagnets to accodamate a small animal head. It was determined that a sphere of diameter130 mm would suffice.
(4) The workspace should be nearly isotropic, with the ability to generate sufficient magnetic forces in any direction with any microrobot pose.

这是开放性设计问题被解决通过首先解决控制系统问题：给定一个任意数量的固定电磁铁在一个任意配置下，用于磁性微型机器人的5D控制的一个可行控制系统是什么？这是SectionII的主题。使用这最终控制系统，结果系统的表现被考虑来优化电磁铁配置的设计；这在SectionIII中被讨论。在完成电磁铁配置和考虑期望的工作区域尺寸后，机电部件被设计；这在SectionIV中是详细的。系统被表征在SectionV中。SectionVI表现在自动化和远程运行控制模式下的OctoMag的能力。在SectionVII,系统被用来做了一次动物实验。
This open-ended design problem was approached by first solving the control system problem: given an arbitrary number of stationary electromagnets in an arbitrary configuration, what is a viable control system for 5D control of magnetic microrobot? This is the topic of Section II. Using this final control system, the performance of the resulting system is considered to optimize the design of the electromagnet configuration; this is discussed in Section III. After finalizing the electromagnet configuration, and considering the desired workspace size, the mechatronic components were designed; this is detailed in Section IV. The system is characterized in Section V. Dection VI demonstrates the capabilities of OctoMag in both autonomous and teleoperation control modes. In Section VII, the system is used to conduct an animal experiment.

2. 用固定电磁铁控制 Control with stationary electromagnets

目标是进行相对于一个参考的静止世界系的一个磁微机器人的5D无线控制。
The goal is to perform 5D wireless control of a magnetic microrobot with respect to a stationary world frame of reference.

The magnetic torque:

$$\mathbf{T}=\mathbf{M}\times \mathbf{B}==Sk(\mathbf{M})\mathbf{B}\text{\hspace{1em}(1)}$$

where,

$$\mathbf{M}=Sk\left(\left[\begin{array}{c}{m}_x\\ m_y\\ m_z\end{array}\right]\right)=\left[\begin{array}{ccc}0& -m_z& m_y\\ m_z& 0& -m_x\\ -m_y& m_x& 0\end{array}\right]\text{\hspace{1em}(2)}$$

The magnetic force:

$$\mathbf{F}=▽(\mathbf{M}\cdot \mathbf{B})\text{\hspace{1em}(3)}$$

以一个更直观的形式：
In a more intuitive form:

$$\mathbf{F}=\left[\begin{array}{c}\frac{\partial \mathbf{B}}{\partial x}\\ \frac{\partial \mathbf{B}}{\partial y}\\ \frac{\partial \mathbf{B}}{\partial z}\end{array}\right]\mathbf{M}\text{\hspace{1em}(4)}$$

在一个电磁铁的给定静态排布下，每个电磁铁创造一个在整个工作空间中的可以被预先计算的磁场。在工作空间中的任何给定点$\mathbf{P}$，由于驱动一个给定电磁铁的磁场能被表述为向量${\mathbf{B}}_e(\mathbf{P})$，它的大小随穿过电磁铁的电流线性变化，并且，因此，可以被描述为一个单位电流向量被一个标量电流值相乘：
With a given static arrangement of electromagnets, each electromagnet creates a magnetic field throughout the workspace, which can be precomputed. At any given point $\mathbf{P}$ in the workspace, the magnetic field due to actuating a given electromagnet can be expressed by the vector ${\mathbf{B}}_e(\mathbf{P})$, whose magnitude varies linearly with the current through the electromagnet, and, as such, can be described as a unit-current vector multiplied by a scalar current value:

$${\mathbf{B}}_e(\mathbf{P})=\widehat{{\mathbf{B}}_e(\mathbf{P})}{i}_e\text{\hspace{1em}(5)}$$

含有空气核心的电磁铁，分立的场贡献是解耦的，并且场能被分立地预计算然后被线性地叠加。这不是软磁核心电磁铁的情形。但是,如果假设一个有可忽略磁滞的理想的软磁铁材料，并且系统运行在核心在它们线性磁化区域内，假设还是有效的，分立电流的场贡献线性叠加。
With air-core electromagnets, the individual field contributions are decoupled, and the fields can be individually precomputed and then superimposed linearly. This is not the case with soft-magnetic-core electromagnets. However, if an ideal soft-magnetic material with negligible hysteresis is assumed, and the system is operated with the cores in their linear magnetization region, the assumption is still valid that the field contributions of individual currents superimpose linearly.

$$\mathbf{B}(\mathbf{P})=\sum _{e=1}^n{\mathbf{B}}_e(\mathbf{P})=\sum _{e=1}^n\widehat{{\mathbf{B}}_e(\mathbf{P})}{i}_e\text{\hspace{1em}(6)}$$

$$\mathbf{B}(\mathbf{P})=\left[\begin{array}{ccc}\widehat{{\mathbf{B}}_1(\mathbf{P})}& \cdots & \widehat{{\mathbf{B}}_n(\mathbf{P})}\end{array}\right]\left[\begin{array}{c}{i}_1\\ ⋮\\ i_n\end{array}\right]=\beta (\mathbf{P})\mathbf{I}\text{\hspace{1em}(7)}$$

3xn的$\beta (\mathbf{P})$被定义在工作空间内每个点\mathbf{P}，它要么能被在线解析计算，或者一个预计算或测量点的网格能被在线插值。
3xn $\beta (\mathbf{P})$ is defined at each point \mathbf{P} in the workspace, which can either be analytically computed online, or a grid of precomputed or measured points can be interpolated online.

$$\frac{\partial \mathbf{B}(\mathbf{P})}{\partial x}=\left[\begin{array}{ccc}\frac{\partial \widehat{{\mathbf{B}}_1(\mathbf{P})}}{\partial x}& \cdots & \frac{\partial \widehat{{\mathbf{B}}_n(\mathbf{P})}}{\partial x}\end{array}\right]\left[\begin{array}{c}{i}_1\\ ⋮\\ i_n\end{array}\right]={\beta }_x(\mathbf{P})\mathbf{I}\text{\hspace{1em}(8)}$$

The magnetic torque and magnetic force:

$$\left[\begin{array}{c}\mathbf{T}\\ \mathbf{F}\end{array}\right]=\left[\begin{array}{c}Sk(\mathbf{M})\beta (\mathbf{P})\\ {\mathbf{M}}^T{\beta }_x(\mathbf{P})\\ {\mathbf{M}}^T{\beta }_y(\mathbf{P})\\ {\mathbf{M}}^T{\beta }_z(\mathbf{P})\end{array}\right]\left[\begin{array}{c}{i}_1\\ ⋮\\ i_n\end{array}\right]={\alpha }_{T,F}(\mathbf{M},\mathbf{P})\mathbf{I}\text{\hspace{1em}(9)}$$

For desired torque and force:

$$\mathbf{I}={\alpha }_{T,F}(\mathbf{M},\mathbf{P}{)}^{†}\left[\begin{array}{c}{\mathbf{T}}_{des}\\ {\mathbf{F}}_{des}\end{array}\right]\text{\hspace{1em}(10)}$$

伪逆找到最小化电流向量的两个范数的解，这对于能量损耗和热量产生的最小化来说是期望的。$\alpha$的伪逆利用奇异值分解$\alpha =\mathbf{U}\mathbf{\Sigma }{\mathbf{V}}^T$，$\mathbf{\Sigma }$是6xn奇异值矩阵，左侧6x6的元素构成一个6阶奇异值${\sigma }_i$的对角矩阵，$\mathbf{U}$是一个6x6的正交矩阵，它的列是6个输出奇异向量，$\mathbf{V}$是nxn的正交矩阵，它的列是n个输入奇异向量。伪逆被计算为${\alpha }^{†}=\mathbf{U}{\mathbf{\Sigma }}^{†}{\mathbf{V}}^T$，这里${\mathbf{\Sigma }}^{†}$是一个nx6矩阵，上6x6元素构成一个对角矩阵，第j个对角元素一定为$1/{\sigma }_j$，0如果${\sigma }_j=0$，所有其他条目都是0. ${\sigma }_6=0$, 并且$\mathbf{U}$的第六列将总是：
The pseudoinverse finds the solution that minimizes the two-norm of the current vector, which is desirable for minimization of both power consumption and heat generation. The pseudoinverse of $\alpha$ makes use of singular value decomposition $\alpha =\mathbf{U}\mathbf{\Sigma }{\mathbf{V}}^T$, where $\mathbf{\Sigma }$ is 6xn singular-value matrix, where the leftmost 6x6 elements form a diagnoal matrix of 6-order singular values ${\sigma }_i$, $\mathbf{U}$ is a 6x6 orthonormal matrix, whose columns are 6 output singular vectors, and $\mathbf{V}$ is nxn orthonormal matrix, whose columns are n input singular vectors. The pseudoinverse is computed as ${\alpha }^{†}=\mathbf{U}{\mathbf{\Sigma }}^{†}{\mathbf{V}}^T$, where ${\mathbf{\Sigma }}^{†}$ is a nx6 matrix, where the uppermost 6x6 elements form a diagonal matrix with the j-th diagonal element defined as $1/{\sigma }_j$, and as 0 if ${\sigma }_j=0$, and all other entries being equal 0. ${\sigma }_6=0$, and the sixth column of $\mathbf{U}$ will always be:

$${\mathbf{U}}_6=\left[\begin{array}{c}0\\ 0\\ 1\\ 0\\ 0\\ 0\end{array}\right]\text{\hspace{1em}(11)}$$

对应于关于磁化轴无力矩产生，这是不可能的。对于完整的5D控制，其他5个奇异值必须为非零。
corresponding to no-torque generation about the magnetization axis, which is never possible. For full 5D control, the other 5 singular values must be zero.

而不是显式地控制力矩，可以简单地控制磁场到达期望的朝向，微机器人将自然对齐这个朝向，
Rather than explicitly controlling the torque, one can simply control the magnetic field to the desired orientation, to which the microrobot will naturally align.

$$\left[\begin{array}{c}\mathbf{B}\\ \mathbf{F}\end{array}\right]=\left[\begin{array}{c}\beta (\mathbf{P})\\ {\mathbf{M}}^T{\beta }_x(\mathbf{P})\\ {\mathbf{M}}^T{\beta }_y(\mathbf{P})\\ {\mathbf{M}}^T{\beta }_z(\mathbf{P})\end{array}\right]\left[\begin{array}{c}{i}_1\\ ⋮\\ i_n\end{array}\right]={\alpha }_{B,F}(\mathbf{M},\mathbf{P})\mathbf{I}\text{\hspace{1em}(12)}$$

for desired magnetic field and force:

$$\mathbf{I}={\alpha }_{B,F}(\mathbf{M},\mathbf{P}{)}^{†}\left[\begin{array}{c}{\mathbf{B}}_{des}\\ {\mathbf{F}}_{des}\end{array}\right]\text{\hspace{1em}(13)}$$

完全5D控制，在前面(9)需要一个rank5的驱动矩阵，对应了这里(12)的rank6的驱动矩阵。

整篇文章中，在微机器人的位置一个恒定的$|\mathbf{B}|=15mT$被使用：在试验性质的测试后值被选择，因为它导致在电磁铁的通常运行中的低峰电流。通过保持$|\mathbf{B}|$恒定，控制系统等式被保持线性，也使(13)能在一个迭代中计算，但是是次优化的。为了允许$|\mathbf{B}|$去改变将导致稍微更好的性能但是以在实时优化中增加计算开销为代价。
Throughout this paper, a constant $|\mathbf{B}|=15mT$ at the location of the microrobot is used: the value is chosen after pilot testing because it results in low-peak currents in the electromagnets during typical operation. By keeping $|\mathbf{B}|=15mT$ constant, control system equations are kept linear, which enables (13) to be calculated in one iteration, but it is suboptimal. To allow $|\mathbf{B}|$ to vary would result in somewhat better performance but at the added computation cost of optimization in real time.

3. 电磁铁配置的设计Design of the electromagnet configuration

special equation:

$$\left[\begin{array}{c}\lambda \mathbf{B}\\ \mathbf{F}\end{array}\right]=\left[\begin{array}{c}\lambda \beta (\mathbf{P})\\ {\mathbf{M}}^T{\beta }_x(\mathbf{P})\\ {\mathbf{M}}^T{\beta }_y(\mathbf{P})\\ {\mathbf{M}}^T{\beta }_z(\mathbf{P})\end{array}\right]\left[\begin{array}{c}{i}_1\\ ⋮\\ i_n\end{array}\right]=\alpha (\mathbf{M},\mathbf{P})\mathbf{I}\text{\hspace{1em}(14)}$$

当$\lambda \to 0$，奇异值和输出奇异向量解耦为纯力和纯场方向。${\sigma }_4$，${\sigma }_5$，${\sigma }_6$由于$\lambda$变小，并且列${\mathbf{U}}_4$，${\mathbf{U}}_5$，${\mathbf{U}}_6$几乎倾向于纯场方向。这使${\sigma }_1$，${\sigma }_2$，${\sigma }_3$几乎对应于纯力方向，并且比例${\sigma }_3/{\sigma }_1$能被用来作为力条件数，当系统有各项同性力控制能力的时候，它接近1。保持$\lambda >0$，产生磁场，机器人的朝向可能是任何期望的方向。力条件数可能不是最好的评估准则来判断系统的性能，因为有力生成能力的系统，在每个方向上生成的力很差，也会返回一个很好的力条件数。结果是，${\sigma }_3$，作为在最糟糕方向的力生成的测量，被用来作为性能评估尺度。目标是在整个工作区域有足够的力生成能力，不论微机器人朝向，这样微型机器人控制主动全不会丢失。

两组m个电磁铁被考虑，被指定为upper和lower组，每一个电磁铁建模为一个单位强度的点极子，指向工作空间的中点，并位于离中心65mm远处。在upper组和通常轴之间的夹角为${\varphi }_{upper}$，同理还有${\varphi }_{lower}$。对于每个给定的电磁铁配置，电磁铁公用中点和17个有规律放置的点，定义一个半径为10mm的半球。在18个点中的每个点，一个单位强度的极子磁矩被考虑在26个基本方向。用于电磁铁配置的评估标准是468个机器人位姿中最低（最差）的${\sigma }_3$。最终，配置为，上层下层各4个电磁铁，上层${\varphi }_{upper}\approx 45^{\circ }$，下层${\varphi }_{lower}\approx 90^{\circ }$。

4. 系统实现 System implementation

每个磁铁包含内径为44mm，外径为63mm，长度为210mm的线圈，线圈共712匝，1.6mm线径，绝缘铜导线。电磁核心由VACOFLUX 50制成，是CoFe Alloy钴铁合金。核心直径42mm，长度210mm。集成电磁铁的电感为89mH，电阻1.3$\Omega$。系统电源为SM 70-90，它能提供6KW的能量，能同时提供8路20A的驱动电流。开关放大器的开关频率为150kHz。开关放大器的控制器为Sensoray 626 DAC card (14-bit resolution)。两个顶部相机提供视觉位置反馈。

5. 系统表征 System characterization

在这片文章中，对每一个单位电流贡献，一个解析模型，点极子模型，被拟合到从最终系统的FEM模型中获得的场数据。
In this paper, an analytical model, the point-dipole model, was fit to field data obtained from an FEM model of the final system for each of the unit-current contributions.
将点极子模型拟合到每个电磁铁的仿真数据中，可以求得，拟合极子的位置和磁矩方向。

标定被进行来考虑磁核心的不完美，在磁核心，线圈绕组，电磁铁对齐方面。在工作空间中心的场被测量当每个线圈上流经6A的电流时。在幅值上的误差被用来作为缩放尺度使生成场匹配控制算法计算的场，算法使用点极子模型。角度误差不需要补偿，并且不要在放缩后改变，但是在期望和实际场之间的不对齐是小的。
This calibration is performed to account for the imperfections in the magnetic cores, the coil wrapping, and the alignment of the electromagnets. The field at the center of the workspace is measured when for 6A of current flowing through each coil. The error in the amplitude is used as a scaling factor to make the generated field match that calculated by the control algorithm, which uses the point-dipole models. The angular error is not compensated for, and it does not change after scaling, but it can be seen that the disalignment misalignment between the desired and actual field is small.

[1]: Kummer, Michael P. , et al. “OctoMag: An Electromagnetic System for 5-DOF Wireless Micromanipulation.” IEEE Transactions on Robotics 26.6(2010):1006-1017.