Magnetic Methods in Robotics

As the author say, this is a thorough review, the paper is perfect. 本文非常非常详细地讲明了用于机器人领域的一些磁基础理论和推导整理过程。以及一些应用的推导整理过程。

在机器人技术领域中的磁理论
Magnetic Methods in Robotics [1]
Paper Link
Authors: Jack J. Abbott, etc.
2020, Annual Review of Control, Robotics, and Autonomous Systems

0. 摘要 Abstract

这篇文章的目标是对用于机器人技术中远程操作和无线驱动任务最新的磁理论提供一个彻底的介绍。这篇文章使用统一符号合成之前的工作，实现在机器人技术中的直接应用。文章以由磁材料和电磁铁产生的磁场的讨论开篇，磁材料如何在一个施加场中变磁化，和在磁物体上产生力和力矩。然后描述被用于产生和控制施加磁场的系统，包括电磁和永磁系统。最后，文章调查来自大量机器人应用领域的工作，研究人员在这些领域中已经使用磁理论，包括微机器人技术，医疗机器人技术，触觉技术和航天技术。
The goal of this article is to provide a thorough introduction to the state of the art in magnetic methods for remote-manipulation and wireless actuation tasks in robotics. This paper synthesizes prior work using a unified notation, enabling straightforward application in robotics. The paper begins with a discussion of the magnetic fields generated by magnetic materials and electromagnets, how the magnetic materials become magnetized in an applied field, and the forces and torques generated on magnetic objects. It then describes systems used to generate and control applied magnetic fields, including electromagnetic and permanenet-magnet systems. Finally, it surveys work from a variety of robotic application areas in which researchers have utilized magnetic methods, including microrobotics, medical robotics, haptics and aerospace.

1. 介绍 Introduction

我们主要的动力是提供一个坚实的在这个主题的指导，而不是一个文献的详细回顾。在这个方面，我们引用那些已经创造关键贡献的人。在这篇文章中，我们使用一个一致表示惯例来帮助读者快速辨别被表示量的类型。我们的惯例有时与先前工作中使用的惯例背道而驰。
Our primary motivation is to provide a solid tutorial on this topic rather than an exhaustive review of the literature. Along the way, we reference those who have made key contributions. Throughout the article, we use a consistent notation convention to help readers quickly discern the types of quantities being represented. Our conventions sometimes run counter to the conventions used in prior work.

然而，磁交互是极端复杂的，使得在除了最简单的所有系统中，闭形式的分析是挑战性的。例如磁阻和磁链的概念已经被开发在设计中提供帮助，但是这种系统的理解仍然严重依赖于有限元分析理论。在那些理论中的模型假设与这篇文章中的操纵或驱动问题的假设完全不同，这篇文章中的系统典型得拥有相对远隔开的部件。我们不进一步讨论传统磁设备，我们也不使用它们的专门设计和分析工具。
However, the magnetic interaction is extremely complex, making close-form analysis challenging in all but the simplest of systems. The concepts such as reluctance and flux linkage have been developed to aid in design, but the understanding of such systems still heavily relies on finite-element-analysis (FEA) methods. The modeling assumptions made in those methods are significantly different from those of the manipulation or actuation problems of interest in this paper, which typically have components spaced relatively far apart. We do not discuss traditional magnetic devices further, nor do we make use of their specialized design and analysis tools.

2. 由磁物体产生的磁场 Magnetic fields generated by magentic objects

在对maxwell方程组的简化后，在这个topic的应用中，有两个方程是成立的：

$$▽\cdot \mathbf{b}=0\to \frac{\partial b_x}{\partial x}+\frac{\partial b_y}{\partial y}+\frac{\partial b_z}{\partial z}=0.$$

$$▽\times \mathbf{b}=0\to \frac{\partial b_z}{\partial y}=\frac{\partial b_y}{\partial z},\frac{\partial b_x}{\partial z}=\frac{\partial b_z}{\partial x},\frac{\partial b_y}{\partial x}=\frac{\partial b_x}{\partial y}.$$

其中，$▽={\left[\frac{\partial }{\partial x}\frac{\partial }{\partial y}\frac{\partial }{\partial z}\right]}^T$。

2.1 磁偶极子模型 和 多极子拓展 Magnetic dipole model (approximation) and multipole expansion

磁偶极子模型：

b { r , m } = μ 0 4 π ∣ ∣ r ∣ ∣ 3 ( 3 r ^ r ^ T − I 3 ) m \mathbf{b}\{\mathbf{r},\mathbf{m}\}=\frac{\mu_{0}}{4\pi ||\mathbf{r}||^{3}}\left(3\hat{\mathbf{r}}\hat{\mathbf{r}}^{T}-\mathbf{I}_{3}\right)\mathbf{m} b{ r,m}=4π∣∣r∣∣3μ0​​(3r^r^T−I3​)m

有一个魔法角度，当$\mathbf{r}$和$\mathbf{m}$夹角大约为$54.7^{\circ }$时，$\mathbf{b}\perp \mathbf{m}$。

他们通常具有在他们的体积上相当均匀分布的磁强度。甚至项数很少的多极子扩展可以是相对于质心测量的磁场的精确近似。
They typically have a magnetic strength that is distributed fairly uniformly over the volume. A multipole expansion with even a small number of terms can be an accurate approximation of the magnetic field measured with respective to the center of mass.

b { r , m } = ( μ 0 4 π ∣ ∣ r ∣ ∣ 3 Γ 1 { r ^ } + μ 0 4 π ∣ ∣ r ∣ ∣ 5 Γ 2 { r ^ } + μ 0 4 π ∣ ∣ r ∣ ∣ 7 Γ 3 { r ^ } + ⋯   ) m \mathbf{b}\{\mathbf{r},\mathbf{m}\}=\left( \frac{\mu_{0}}{4\pi ||\mathbf{r}||^{3}} \Gamma_{1}\{\hat{\mathbf{r}}\} + \frac{\mu_{0}}{4\pi ||\mathbf{r}||^{5}} \Gamma_{2}\{\hat{\mathbf{r}}\} + \frac{\mu_{0}}{4\pi ||\mathbf{r}||^{7}} \Gamma_{3}\{\hat{\mathbf{r}}\} + \cdots \right)\mathbf{m} b{ r,m}=(4π∣∣r∣∣3μ0​​Γ1​{ r^}+4π∣∣r∣∣5μ0​​Γ2​{ r^}+4π∣∣r∣∣7μ0​​Γ3​{ r^}+⋯)m

注意这$\Gamma$矩阵是形状函数，与离源的距离无关。

2.2 来自于磁化物体的磁场 Magnetic fields from magnetized objects

由一个均匀磁化的球体产生的场被磁偶极子模型完美建模。任何其他几何形状需要多极子扩展。
The fild generated by a uniformly magnetizd sphere is modeled perfectly by the dipole model. Any other geometry requires the multipole expansion.

磁物体的场是同质的，这在系统设计中可以是有用的。如果我们考虑由给定几何形状和给定磁化量的磁物体产生的场，我们只缩放物体的尺寸，我们发现场映射随放缩的物体一起收缩或扩大。
The fields of magnetic objects are homothetic, this can be useful in system design. If we consider the field generated by a magnetic object of a given geometry and a given magnetization, and we scale only the size of the object, we can find that the field map shrinks or stretches with the scaled object.

同质性质的一个结果是，大磁铁投射它们的磁场在空间中比小磁铁更远，这结果是相当直观的。另一个结果是，如果我们考虑给定大小的磁场中的一个位置，我们发现在这场中的空间导数对于更大磁铁来说是更小的；这种情况是因为，用一个更大的磁铁的话，场在空间中变化更不快由于同质性质。这确实导致对于更大的磁铁来说，对放置在一个给定强度的场中的一个磁物体上产生更小的力，这也是违反直觉的。
One consequence of the homothetic property is that larger magnets project their fields farther into space than do small magnets, this results are quite intuitive. Another consequence is that if we consider a location in the magnetic field with some given magnitude, we find that the spatial derivatives in the field are smaller for larger magnets; this is the case because, with a larger magnet, the field is changing less rapidly in space due to the homothetic property. This actually leads to smaller forces being generated on a magnetic object placed in the field at a given strength for the larger magnet, which is counterintuitive.

2.3 来自于电流的磁场 Magnetic fields from electric currents

计算在位置${\mathbf{p}}_b$的磁场$\mathbf{b}$，电流$i$（单位为$A$），场微元differential field $d\mathbf{b}$，长度微元$d\mathbf{l}$，特定位置${\mathbf{p}}_{d\mathbf{l}}$：

b { p b , i } = ∫ d b = ∫ μ 0 4 π i d l × ( p b − p d l ) ∣ ∣ p b − p d l ∣ ∣ 3 = μ 0 i 4 π ∫ S { p d l − p b } ∣ ∣ p b − p d l ∣ ∣ 3 d l \mathbf{b}\{\mathbf{p}_{b},i\}=\int d\mathbf{b}=\int \frac{\mu_{0}}{4\pi} \frac{i d\mathbf{l} \times (\mathbf{p}_{b}-\mathbf{p}_{d\mathbf{l}})}{||\mathbf{p}_{b}-\mathbf{p}_{d\mathbf{l}}||^{3}}=\frac{\mu_{0}i}{4\pi} \int \frac{\mathbb{S}\{\mathbf{p}_{d\mathbf{l}}-\mathbf{p}_{b}\}}{||\mathbf{p}_{b}-\mathbf{p}_{d\mathbf{l}}||^{3}}d\mathbf{l} b{ pb​,i}=∫db=∫4πμ0​​∣∣pb​−pdl​∣∣3idl×(pb​−pdl​)​=4πμ0​i​∫∣∣pb​−pdl​∣∣3S{ pdl​−pb​}​dl

where S { v } = [ 0 − v z v y v z 0 − v x − v y v x 0 ] \mathbb{S}\{\mathbf{v}\}=\left[\begin{matrix} 0 & -v_{z} & v_{y} \\ v_{z} & 0 & -v_{x} \\ -v_{y} & v_{x} & 0 \end{matrix}\right] S{ v}=⎣⎡​0vz​−vy​​−vz​0vx​​vy​−vx​0​⎦⎤​。

任何能被多极子模型描述的物体的磁场，将有一个磁偶极子磁矩$\mathbf{m}$，

m { i } = i 2 ∫ S { p d l − p r e f } d l \mathbf{m}\{i\}=\frac{i}{2} \int \mathbb{S}\{\mathbf{p}_{d\mathbf{l}}-\mathbf{p}_{ref}\} d\mathbf{l} m{ i}=2i​∫S{ pdl​−pref​}dl

其中，${\mathbf{p}}_{ref}$是任意固定的参考点。对于一个半径为$r$的细线圆环， m { i } = π r 2 i a ^ \mathbf{m}\{i\}=\pi r^{2} i \hat{\mathbf{a}} m{ i}=πr2ia^，$\hat{\mathbf{a}}$是缠绕轴的方向。对于一个内径为$r_i$，厚度为$t$的圆柱线圈， m { i } = π 3 ( 3 r i 2 + 3 r i t + t 2 ) i a ^ \mathbf{m}\{i\}=\frac{\pi}{3} (3 r_{i}^{2}+3 r_{i} t + t^{2}) i \hat{\mathbf{a}} m{ i}=3π​(3ri2​+3ri​t+t2)ia^。对于非圆柱线圈，偶极子可以近似为总循环电流$i$和环路面积的乘积。

在电磁铁设计中，要利用电流密度$j$（单位为$A/m^2$）。穿过截面积cross-sectional area $a$（单位为$m^2$）的电流为 $i=ja$。电流密度方面的思考有助于电磁铁的设计，在显性选择特定的线径，放大器，电源之前。

在磁化物体的磁场中可见的同质缩放在电磁铁中是不可见的。当缩放一个电磁铁的尺寸时（例如，回路的半径和截面积，用相同的比例），既不保持一个恒定的电流，也不保持一个恒定的电流密度，将导致场按同比例放缩。
The homothetic scaling seen in the fields of magnetized objects is not seen in electromagnets. When scaling the size of an electromagnet (e.g., the radius of the loop, as well as the cross-sectional area, by the same ratio), neither maintaining a constant current nor maintaining a constant current density will result in a field that scales homothetically.

3. 磁化 Magnetization

施加磁场和物体磁化之间的关系由表观磁化张量来给定：对于复杂的几何形状来说，关系最好用FEA理论来计算，而对于简单几何形状来说，解析模型存在的。
The relationship between the applied field and the magnetization of an object is given by the apparent susceptibility tensor: for complex geometries, the relationship is best calculated using FEA methods, whereas the analytical models exist for simple geometries.

表观磁化张量（apparent susceptibility tensor）是物体尺寸（size）的一个函数，我们分别考虑宏观和中尺度物体（macro- and mesoscale objects）。表观磁化张量也是物体材料种类（type of magnetic material）的一个函数，我们关注于铁磁和超级顺磁材料（ferromagnetic and superparamagnetic materials）。

3.1 宏观和中尺度物体的磁化 Magnetization of macro- and mesoscale objects

在一般铁磁材料（ferromagnetic material）的磁化曲线中，有一些特征需要注意。第一，有磁滞现象。第二，以渐进的方式逼近饱和。第三，在曲线靠近磁化为0的地方有特征坡度（名义磁化率nominal susceptibility）。第四，当施加场为0时，磁化并不会回归0；如果之前施加磁场使材料达到过饱和（saturation）${\psi }_{sat}$，那么减小磁场后，材料会有剩磁（remanent magnetization）${\psi }_r$，需要使用一个矫顽场（coercive field）$h_c$才能使材料磁化为0。有低矫顽力（coercivity）的材料被称为软磁材料（soft-magnetic material），e.g., paramagnetic material。有高矫顽力的材料被称为硬磁材料（hard-magnetic material），e.g., permanent-magnetic material。

机器人技术中的磁理论应用避免讨论复杂的磁滞性质，而是使用可以被假定为完全软磁或者硬磁的材料。

在使用的软磁材料中，在线性磁化曲线和饱和渐进线之间有一段平滑过渡（transition），误差主要在这里产生。软磁材料经常作为电磁铁磁芯（core）的选择。

在使用的硬磁材料中，只有超级强的外加磁场才能使硬磁材料受到影响，一般是不考虑硬磁材料的磁化的。

在几何形状长度较长的方向施加外场，容易磁化。

3.1.1 软磁球体和椭球体的磁化 Magnetization of soft-magnetic spheres and ellipsoids

球体是第一简单的。椭球体是第二简单的几何形状。

$$||\psi ||=\{\begin{array}{rl}3||\mathbf{h}||,& ||\mathbf{h}||\le \frac{{\psi }_{sat}}{3}\\ {\psi }_{sat},& ||\mathbf{h}||\ge \frac{{\psi }_{sat}}{3}\end{array}$$

3.1.2 非椭球体几何形状的磁化 Magnetization of nonellipsoidal geometries

对于机器人技术中的通常的非椭球形状，我们建议使用实验性的或有限元分析理论来直接确定所需要的响应。
For general nonellipsoidal shapes in robotics, we suggest using experimental or FEA methods to determine the required response directly.

3.2 微米和纳米颗粒的磁化 Magnetization of micro- and nanoparticles

一些在大尺度是铁磁性的材料用于微米或纳米机器人当在亚微米尺度时表现出一种特殊的磁响应类别，被称为超顺磁性。
Some materials used for micro- or nanorobotics that are ferromagnetic at large sizes exhibit a special class of magnetic response known as superparamagnetism when at submicrometer size.

4. 作用在磁物体上的力和力矩 Force and Torque on magnetic objects

4.1 在一个磁偶极子上的力和力矩 Force and torque on a magnetic dipole

磁偶极子被迫去移动和旋转来进行一次减少磁能的尝试。一个替代概念是力和力矩被生成来进行一次通过移动和旋转增加能量$\mathbf{b}\cdot \mathbf{m}$的尝试。
The magnetic dipole is compelled to translate and rotate in an attempt to minimize magnetic energy. An alternative conceptualization is that force and torque are generated in an attempt to increase the energy $\mathbf{b}\cdot \mathbf{m}$ through translation and rotation, respectively.

$$\mathbf{f}=▽(\mathbf{b}\cdot \mathbf{m})=\left[\begin{array}{c}\frac{\partial {\mathbf{b}}^T\mathbf{m}}{\partial x}\\ \frac{\partial {\mathbf{b}}^T\mathbf{m}}{\partial y}\\ \frac{\partial {\mathbf{b}}^T\mathbf{m}}{\partial z}\end{array}\right]={\left[\begin{array}{ccc}\frac{\partial \mathbf{b}}{\partial x}& \frac{\partial \mathbf{b}}{\partial y}& \frac{\partial \mathbf{b}}{\partial z}\end{array}\right]}^T\mathbf{m}=\left[\begin{array}{c}{m}_x\frac{\partial b_x}{\partial x}+m_y\frac{\partial b_y}{\partial x}+m_z\frac{\partial b_z}{\partial x}\\ m_x\frac{\partial b_x}{\partial y}+m_y\frac{\partial b_y}{\partial y}+m_z\frac{\partial b_z}{\partial y}\\ m_x\frac{\partial b_x}{\partial z}+m_y\frac{\partial b_y}{\partial z}+m_z\frac{\partial b_z}{\partial z}\end{array}\right]$$

因为根据 章节 2. 中的 $▽\cdot \mathbf{b}=0\to \frac{\partial b_x}{\partial x}+\frac{\partial b_y}{\partial y}+\frac{\partial b_z}{\partial z}=0$ 和
$▽\times \mathbf{b}=0\to \frac{\partial b_z}{\partial y}=\frac{\partial b_y}{\partial z},\frac{\partial b_x}{\partial z}=\frac{\partial b_z}{\partial x},\frac{\partial b_y}{\partial x}=\frac{\partial b_x}{\partial y}$，可以知道$▽\mathbf{b}=\left[\begin{array}{ccc}\frac{\partial \mathbf{b}}{\partial x}& \frac{\partial \mathbf{b}}{\partial y}& \frac{\partial \mathbf{b}}{\partial z}\end{array}\right]$是一个对称（symmetric）矩阵，并且trace为0。所以只使用5自由度的空间导数，我们能够将$\mathbf{f}$表示为如下形式：

f = [ m x m y m z 0 0 0 m x 0 m y m z − m z 0 m x − m z m y ] [ ∂ b x ∂ x ∂ b x ∂ y ∂ b x ∂ z ∂ b y ∂ y ∂ b y ∂ z ] ↔ f = M G { m } G \mathbf{f}=\left[\begin{matrix}m_{x}&m_{y}&m_{z}&0&0\\0&m_{x}&0&m_{y}&m_{z}\\-m_{z}&0&m_{x}&-m_{z}&m_{y}\end{matrix}\right]\left[\begin{matrix}\frac{\partial b_{x}}{\partial x}\\\frac{\partial b_{x}}{\partial y}\\\frac{\partial b_{x}}{\partial z}\\\frac{\partial b_{y}}{\partial y}\\\frac{\partial b_{y}}{\partial z}\end{matrix}\right] \leftrightarrow \mathbf{f}=\mathbb{M}_{G}\{\mathbf{m}\}\mathbf{G} f=⎣⎡​mx​0−mz​​my​mx​0​mz​0mx​​0my​−mz​​0mz​my​​⎦⎤​⎣⎢⎢⎢⎢⎢⎡​∂x∂bx​​∂y∂bx​​∂z∂bx​​∂y∂by​​∂z∂by​​​⎦⎥⎥⎥⎥⎥⎤​↔f=MG​{ m}G

力矩较为简单，如下所示：

τ = m × b = S { m } b \mathbf{\tau}=\mathbf{m} \times \mathbf{b}=\mathbb{S}\{\mathbf{m}\}\mathbf{b} τ=m×b=S{ m}b

4.2 在磁偶极子之间的力和力矩 Force and torque between magnetic dipoles

如果驱动磁场可以被建模成一个磁偶极子的磁场，那么在该磁场中的一个小磁偶极子的所受的力和力矩就比较好计算了。
磁矩为${\mathbf{m}}_j$的位于${\mathbf{p}}_j$的偶极子$j$在磁矩为${\mathbf{m}}_i$的位于${\mathbf{p}}_i$的偶极子$i$产生的磁场中，偶极子$j$上受到偶极子$i$的力和力矩为：

$$\mathbf{f}=\frac{3{\mu }_0}{4\pi ||{\mathbf{r}}_{ij}|{|}^4}\left(({\widehat{{\mathbf{r}}_{ij}}}^T{\mathbf{m}}_j){\mathbf{m}}_i+({\widehat{{\mathbf{r}}_{ij}}}^T{\mathbf{m}}_i){\mathbf{m}}_j+({\mathbf{m}}_i^T{\mathbf{m}}_j-5({\widehat{{\mathbf{r}}_{ij}}}^T{\mathbf{m}}_i)({\widehat{{\mathbf{r}}_{ij}}}^T{\mathbf{m}}_j))\widehat{{\mathbf{r}}_{ij}}\right)$$

τ = S { m j } b { r i j , m i } \mathbf{\tau}=\mathbb{S}\{\mathbf{m}_{j}\}\mathbf{b}\{\mathbf{r}_{ij},\mathbf{m}_{i}\} τ=S{ mj​}b{ rij​,mi​}

$${\mathbf{r}}_{ij}={\mathbf{p}}_j-{\mathbf{p}}_i$$

这个等式的一个重要的结果是，$\tau$总是与$\mathbf{m}$和$\mathbf{b}$都垂直，所以对于这单个偶极子来说，这是不可能的去产生绕$\mathbf{m}$轴的力矩，不论外加磁场怎么样。这把一个偶极子上的力矩产生限制到了2自由度，并且限制力-力矩产生到了5自由度。另一个重要的结果是，在给定强度的磁场中，能在一个永磁铁$\mathbf{m}$上产生的最大力矩是当外加磁场被垂直施加到偶极子上时获得。
An important result of this equation is that, $\tau$ is always orthogonal to both $\mathbf{m}$ and $\mathbf{b}$, so for this single dipole, it is impossible to generate torque about the $\mathbf{m}$ axis, regardless of the magnetic field. This constrains torque generation on a dipole to 2 DoF, and thus force-torque generation to 5 DoF. Another important result is that the maximum torque that can be generated on a permanent magnet $\mathbf{m}$ in a field with a given strength is achieved when the field is applied orthogonal to the dipole.

4.3 软磁性物体上的力和力矩 Force and torque on soft-magnetic objects

对于软磁材料来说，先要计算在场中的软磁材料的磁矩，第二步再计算力和力矩。

虽然复杂的软磁形状经常在机器人技术中被避免，它们非均匀的磁化能有额外的能力，比如六自由度操纵。在这种情况中，最好使用有限元理论计算力和力矩来确定在磁物体内的磁化响应。力和力矩然后能被建模成一个多极子物体的连续分布的情况。
Although complex soft-magnetic shapes are often avoided in robotics, their nonuniform magnetization enables additional capabilities, such as 6 DoF manipulation. In this case, force and torque are best calculated using FEA methods to first determine the magnetization response within the magnetic body. The force and torque can then be modeled as a continuously distributed case of the multi-dipole object.

4.4 超越点偶极子 Beyond point dipoles

一个物体内有多个偶极子${\mathbf{m}}_i$，物体的原点${\mathbf{p}}_0$上的总力${\mathbf{f}}_0$和总力矩${\tau }_0$为：

f 0 = ∑ i = 1 n f i , τ 0 = ∑ i = 1 n ( τ i + S { p i − p 0 } f i ) \mathbf{f}_{0}=\sum_{i=1}^{n}\mathbf{f}_{i}, \mathbf{\tau}_{0}=\sum_{i=1}^{n}(\mathbf{\tau}_{i}+\mathbb{S}\{\mathbf{p}_{i}-\mathbf{p}_{0}\}\mathbf{f}_{i}) f0​=i=1∑n​fi​,τ0​=i=1∑n​(τi​+S{ pi​−p0​}fi​)

因为一般来说一个独立的5自由度力-力矩能被施加到每一个偶极子上，所以施加一个6自由度的力-力矩到这个物体上是可能的。
Because an independent 5 DoF force-torque can be applied to each of the dipoles in general, it is possible to apply a 6 DoF force-torque to this object.

4.5 磁稳定性 Magnetic Stability

earnshaw定理告诉我们，在一个静态磁场中不能存在稳定的磁平衡点。
Earnshaw’s theorem tells us that there can be no stable magnetic equilibrium point in a static magnetic field.

反磁性材料能够稳定悬浮并且已经被使用在机器人操作器中，用铁磁体悬浮在反磁性石墨片上。
Diamagnetic materials can levitate stably and have been used in robotic manipulators that utilize ferromagnets levitating over diamagnetic graphite sheets.

5. 用于操纵和驱动的固定系统 Stationary systems for manipulation and actuation

5.1 用于操纵和驱动的磁场的电磁控制 Electromagnetic control of magnetic fields for manipulation and actuation

对于一个任意排布的电磁铁，磁场相对于流过电磁铁的电流是线性关系。施加电流与在工作空间中的每一点${\mathbf{p}}_b$的场和场梯度的线性映射能被表示为：
For an abitrary arrangement of electromagnets, the magnetic field is linear with respect to the currents flowing through the electromagnets. The linear mapping between the applied currents and the field and field gradient at each point ${\mathbf{p}}_b$ in the workspace can be represented by:

$$\left[\begin{array}{c}{b}_x\\ b_y\\ b_z\\ \frac{\partial b_x}{\partial x}\\ \frac{\partial b_x}{\partial y}\\ \frac{\partial b_x}{\partial z}\\ \frac{\partial b_y}{\partial y}\\ \frac{\partial b_y}{\partial z}\end{array}\right]=\left[\begin{array}{ccc}{b}_{1x}& \cdots & b_{nx}\\ b_{1y}& \cdots & b_{ny}\\ b_{1z}& \cdots & b_{nz}\\ \frac{\partial b_{1x}}{\partial x}& \cdots & \frac{\partial b_{nx}}{\partial x}\\ \frac{\partial b_{1x}}{\partial y}& \cdots & \frac{\partial b_{nx}}{\partial y}\\ \frac{\partial b_{1x}}{\partial z}& \cdots & \frac{\partial b_{nx}}{\partial z}\\ \frac{\partial b_{1y}}{\partial y}& \cdots & \frac{\partial b_{ny}}{\partial y}\\ \frac{\partial b_{1y}}{\partial z}& \cdots & \frac{\partial b_{ny}}{\partial z}\end{array}\right]\left[\begin{array}{c}{i}_1\\ ⋮\\ i_n\end{array}\right]\leftrightarrow \mathbf{F}=\left[\begin{array}{c}\mathbf{b}\\ \mathbf{G}\end{array}\right]=\left[\begin{array}{c}\mathbb{B}\\ \mathbb{G}\end{array}\right]\mathbf{I}={\mathbb{F}}_{\mathbf{I}}\mathbf{I}$$

${\mathbb{F}}_{\mathbf{I}}$矩阵的第k列被创建当设定电流$i_k$为1A而其他电流为0A。为了获得完整的力-力矩操纵，$\mathbf{F}$中的所有8个项必须是独立可控制的。至少8个源是必须的。
The k-th column of the ${\mathbb{F}}_{\mathbf{I}}$ matrix is created by setting current $i_k$ to 1A and all other currents to 0A. To achieve full force-torque manipulation, all 8 terms in the field array $\mathbf{F}$ must be independently controllable. At least 8 sources are necessary.

我们能够定义各种矩阵来推导出我们想要的控制量，建立与电流之间的关系，如下形式：

Y d e s = M Y d e s { m } F I { p b } I \mathbf{Y}_{des}=\mathbb{M}_{\mathbf{Y}_{des}}\{\mathbf{m}\}\mathbb{F}_{\mathbf{I}}\{\mathbf{p}_{b}\}\mathbf{I} Ydes​=MYdes​​{ m}FI​{ pb​}I

比如：

M b { m } = [ I 3 0 ] ; M τ , f { m } = [ S { m } 0 0 G G { m } ] ; M b , f { m } = [ I 3 0 0 M G { m } ] \mathbb{M}_{\mathbf{b}}\{\mathbf{m}\}=\left[\begin{matrix}\mathbb{I}_{3}&\mathbb{0}\end{matrix}\right]; \mathbb{M}_{\mathbf{\tau},\mathbf{f}}\{\mathbf{m}\}=\left[\begin{matrix}\mathbb{S}\{\mathbf{m}\}&\mathbb{0}\\ \mathbb{0}&\mathbb{G}_{\mathbf{G}}\{\mathbf{m}\}\end{matrix}\right]; \mathbb{M}_{\mathbf{b},\mathbf{f}}\{\mathbf{m}\}=\left[\begin{matrix}\mathbb{I}_{3}&\mathbb{0}\\ \mathbb{0}&\mathbb{M}_{\mathbf{G}}\{\mathbf{m}\}\end{matrix}\right] Mb​{ m}=[I3​​0​];Mτ,f​{ m}=[S{ m}0​0GG​{ m}​];Mb,f​{ m}=[I3​0​0MG​{ m}​]

给定一些期望输出${\mathbf{Y}}_{des}$，能够使用伪逆找到电流。
Given some desired outputs ${\mathbf{Y}}_{des}$, the currents can be found using pseudo-inverse.

当软磁物体被操纵时，它们被施加磁场所磁化，所以磁操纵矩阵将依赖于外加磁场，在这情形下，一个线性解是不可能用于一些驱动矩阵的，而非线性理论必须被应用。解决方案能被构成一个控制或优化问题，比如局部线性化系统和控制输出改变，而不是直接控制输出。
When soft-magnetic objects are manipulated, they are magnetized by the applied magnetic field, so the magnetic manipulation matrix will depend on the applied magnetic field, in which case a linear solution is not possible for some actuation matrices, and nonlinear methods must be employed. Solutions can be framed as a control or optimization problem, such as locally linearizing the system and controlling the change in the output rather than controlling the output directly.

Y ˙ = J { m , p b , I } I ˙ \dot{\mathbf{Y}}=\mathbb{J}\{\mathbf{m},\mathbf{p}_{b},\mathbf{I}\}\dot{\mathbf{I}} Y˙=J{ m,pb​,I}I˙

$\mathbb{J}$是雅可比矩阵，映射电流改变量到输出改变量的关系。

5.2 基于特定电磁铁的磁正交系统 Magnetically orthogonal systems based on specialized electromagnets

这些电磁铁被组合的排列创造多自由度的能力，每一个电磁铁有一个从其他电磁铁中解耦出来的特定的任务。
These electromagnets are combined in arrangements that create multi-DoF capabilities, where each electromagnet has a specific task that is decoupled from others.

特定系统中第一个也是最重要的是亥姆霍兹线圈。如果两个等半径的环形线圈被同轴排布，有相同的电流以相同的方向流过，然后在线圈中心的场是局部均匀的而且对准轴方向，因为系统中的对称性。那是因为，场中的空间导数在每个方向上都是0。亥姆赫兹线圈能被用来在一个磁物体上产生一个真正的无力力矩。
The first and the most important of the specialized systems is Helmholtz coil. If two circular coils of equal radius are arranged coaxially, with the same current flowing with the same handedness, then the field at the common center of the coils will be locally uniform and aligned with the axial direction due to the symmetry in the system. That is, because the spatial derivative in the field is 0 in each direction. A Helmholtz coil can be used to generate an essentially force-free torque on a magnetic object.

如果我们回到等半径的两个同轴环形线圈的设置，但是不是原来的源，电流以相反方向穿过线圈，我们发现在中心场完美地消失到0，但是中心的场的空间导数不是0.一个麦克斯韦线圈就是这样一个排布，这在这个情形中是优化的，场的二阶导数在轴向是0，创造一个所谓均匀梯度。一个麦克斯韦线圈能被使用来在一个磁物体上创造一个真正的无力矩力。
If we return to our set of two circular coils of equal radius, but instead source the current through the coils with opposite handedness, we find that the fields perfectly cancel to 0 at the common center, but the spatial derivative of the field at the common center is not 0. A Maxwell coil is such an arrangment, which is optimal in the sense that the second derivative of the field in the axial direction is 0, creating a so-called uniform gradient. A Maxwell coil can be used to generate an essentially torque-free force on a magnetic object.

马鞍线圈从它们的特征形状中得到它们的名字，它们的几何形状被限制在一个圆柱的表面。马鞍线圈能被优化来生成一个均匀场或者一个垂直于圆柱轴线的梯度场。双马鞍线圈创造了一个横向梯度。
Saddle coils get their name from their charateristic shape, with their geometry constrained to the surface of a cylinder. Saddle coils can be optimized to generate a uniform field or a gradient field orthogonal to the axis of the cylinder. Double-Saddle Golay coils create a transverse gradient.

5.3 电磁铁的磁非正交系统 Magnetically non-orthogonal systems of electromagnets

已经有从磁正交系统到电磁铁系统，电磁铁在工作区域周围并且以一种耦合方式一起工作，就是一般来说所有电磁铁对所有的驱动指令都有效。这些配置中的每一个都有完整的8自由度场生成能力。许多使用超过8个电磁铁的系统已经被描述，创造一个冗余自由度。另一种模块化概念包含全磁铁。
There has been a move away from magnetically orthogonal systems to systems of electromagnets that surround a workspace and work together in a coupled fashion, where all electromagnets are active for all actuation commands in general. Each of these configurations is capable of full 8 DoF field generation. A number of systems have been described that use more than 8 electromagnets, creating a degree of redundancy. Another modular concept involves Omnimagnets.

5.4 电磁铁设计 Electromagnet design

电磁铁被通常地设计为包裹在一个铁磁芯外的绝缘铜线，铁芯用来放大场。这芯通常由一个理想软磁材料来制作，为了避免磁滞效应。在一些情况下，核心超出线圈，通常使相对小直径的核心接近工作区域同时保持相对大直径的线圈在一定距离。在一些情况中，电磁铁（不论是线圈还是核心）是渐缩的，也是因为更近靠近工作空间的目标。迄今为止，设计铁磁核心电磁铁已经是一个反复的过程，很大程度上依赖于FEA方法。
Electromagnets are typically designed as insulated copper wire wrapped around a ferromagnetic core, which serves to amplify the field. The core is typically made of an ideal soft-magnetic material to avoid effects of hysteresis. In some cases, the core extends beyond the coil, typically to enable the relatively small-diameter core to get closer to the workspace while keeping the relatively large-diameter coils at a distance. In some cases, the electromagnet is tapered, also with the goal of getting closer to the workspace. To date, designing ferromagnetic-core electromagnets has been an iterative process, heavily relying on FEA methods.

电磁铁也能被设计为不含任何核心，这有时被指为有一个空气核心。这一种设计选择的好处是一个独立电磁铁的场能被简单计算，并且多磁铁互相是准静态解耦的（这在移动电磁铁系统中尤其理想）。无核心电磁铁的最优设计，在一个给定点的最大化场强度来说，是已知的。无芯电磁铁很少被使用因为通常假定核心将使系统更强。如果被操纵或者驱动的物体相对于工作区域来说是小的，以致它与核心之间的距离是自身长度的许多倍，这个假设可能正确。但是，当物体变得更大时，它们对核心的自吸引可能使核心不像期望的那样。
Electromagnets can also be designed without any core, which is sometimes refered to as having an air core. The benefits of such a design choice are that the field of an individual electromagnet can be calculated simply, and that multiple electromagnets are quasi-statically decoupled from each other (which is particularly desirable in systems of moving electromagnets). The optimal design of coreless electromagnets, in terms of maximizing field strength at a given location, is known. Coreless electromagnets are rarely used because it is typically assumed that cores will enable stronger systems. If the object being manipulated or actuated is small relative to the workspace, such that it is many of its own body lengths away from the cores, the assumption is likely correct. However, as objects become larger, their self-attraction to the cores may make cores undesirable.

6. 用于操纵和驱动的移动系统 Moving systems for manipulation and actuation

6.1 移动永磁铁 moving permanent magnets

永磁铁能产生强大的场-没有电流或相关的热量的产生-这能通过移动和/或转动这些源来控制。虽然对场和场的导数的移动和旋转的影响是非线性的，这些系统的控制能通过非线性解算理论来获得。这些源通常被建模为点偶极子，这是甚至对于非球形磁铁来说都是相当精确的只要交互距离超过两倍的最小边界球的半径。如果需要更近的距离，能应用更复杂的场模型。因为它们的磁化对它们经受的磁场是不敏感的，所以叠加理论应用，并且每一个源能被独立地对待。因而，在位置${\mathbf{p}}_b$来自于$n$个驱动永磁铁的场和它的梯度是所有单独贡献的总和：
Permanent magnets can generate strong fields - with no electrical currents or the associated generation of heat - which can be controlled by translating and/or rotating these sources. Although the effects of translation and rotation on the field and field derivatives are nonlinear, control of these systems can be achieved using nonlinear solution methods. These sources are commonly modeled as point dipoles, which is quite accurate even for nonspherical magnets as long as the interaction distances exceed two minimun-bounding-sphere radius. If closer distances are required, more complex field models can be emploied. Since their magnetization is not sensitive to the field they experience, superposition applies, and each source can be treated independently. Thus, the magnetic field and its gradient at location ${\mathbf{p}}_b$ from $n$ actuating permanent magnets are the summation of all of the individual contributions:

b { p b } = ∑ i = 1 n b { p b , m i , p i } ; G { p b } = ∑ i = 1 n G { p b , m i , p i } ; F { p b } = [ b { p b } G { p b } ] \mathbf{b}\{\mathbf{p}_{b}\}=\sum_{i=1}^{n}\mathbf{b}\{\mathbf{p}_{b},\mathbf{m}_{i},\mathbf{p}_{i}\}; \mathbf{G}\{\mathbf{p}_{b}\}=\sum_{i=1}^{n}\mathbf{G}\{\mathbf{p}_{b},\mathbf{m}_{i},\mathbf{p}_{i}\}; \mathbf{F}\{\mathbf{p}_{b}\}=\left[\begin{matrix} \mathbf{b}\{\mathbf{p}_{b}\} \\ \mathbf{G}\{\mathbf{p}_{b}\} \end{matrix}\right] b{ pb​}=i=1∑n​b{ pb​,mi​,pi​};G{ pb​}=i=1∑n​G{ pb​,mi​,pi​};F{ pb​}=[b{ pb​}G{ pb​}​]

6.2 移动电磁铁 moving electromagnets

电磁铁也能被移动穿过空间来改变对机器人系统的输入。还是应用非线性理论，$\mathbf{I}$和$\mathbf{q}$是完整的输入设置。计算这些耦合雅可比对于含有软磁耦合的系统来说通常不是不重要的，但是对于无核系统来说可以是直截了当的因为由单个线圈生成的场是独立的。
Electromagnets can also be moved through spave to vary the inputs to the robotic system. Nonlinear methods still apply, with $\mathbf{I}$ and $\mathbf{q}$ as the complete input set. Calculating these coupled Jacobians is generally not trivial for systems with soft-magntic coupling, but can be straightforward for coreless systems because the fields generated by individual coils are independent.

7. 系统表征和标定 System characterization and calibration

在前面章节描述的理论和分析假定对系统产生的场有准确的理解，这可以从表征（场映射）或者标定（拟合参数到一个结构化模型）中获得。对于使用或不使用软磁耦合的固定的电磁铁系统，在所有电磁铁都在原位的情况下，会在工作空间框架中产生场映射或模型。
The methods and analysis described in the previous sections assume an accurate understanding of the field generated by the system, which can be achieved through either characterization (field mapping) or calibration (fitting parameters to a structured model). For stationary electromagnetic systems, with or without soft-magnetic coupling, field maps or models are generated in the workspace frame with all electromagnets in situ.

表征依赖于场的测量，而不是物理模型。来自每个线圈的场贡献的测量从FEA仿真或者直接实验测量中获得。场数据被记录在一个查询表中并且之后用标准理论被插值。虽然表征理论已经被成功应用在文献中，它们不能检测和校正在传感器阅读，朝向或位置的不确定。更进一步，因为只有磁场（而不是场导数）传感器是现在可获得的，用来计算场导数矩阵的数值差分法导致不符合麦克斯韦等式的数值。成功使用表征理论主要因为反馈控制的鲁棒性。
Characterization depends on field measurements, with no physics-based model. Measurements of the fild contribution from each coil are obtained from either FEA simulations or direct experimental measurements. The field data are recorded in a look-up table and are later interpolated using standard methods. Although characterization methods have been successfully emploied in the literature, they can not detect and correct for uncertainties in the sensor readings, orientations or positions. Furthermore, since only magnetic field (not field derivative) sensors are presently available, the numerical differentiation used to calculate the field gradient matrix results in values that do not conform maxwell’s equations. Success using characterization methods is due primarily to the robustness of feed-back control.

一个磁系统的标定没有分享表征的限制，因为它拟合一个场的物理模型到收集到的数据上。基于模型的标定起作用，通过用一个一个场源的单独或多极子拓展来近似每一个场源，标定过程定义每个源的位置，朝向和强度。举一个例子，在一个有8铁磁核心电磁铁的系统中，每个电流输入不同地磁化了每个核心；因而，系统应该潜在地考虑64个源。一个含有唯一偶极子场项，考虑的源少于以上描述的l理想情况，通常是足够的。但是，包含所有潜在源和多极子拓展中的前三项，在实际中导致低于1%的误差。给定一个与$n_s$个磁源位置相关联的$n_c$电流输入的系统，即一组一个$n_c\times n_s$偶极子源，位于${\mathbf{p}}_{i,j}$处，单位电流源强度${\mathbf{m}}_{i,j}$与$I(i)=1A$相关，用于工作空间中场的建模。对于在位置${\mathbf{p}}_k$的给定场测量${\mathbf{b}}_k$，采用输入电流阵列$I_k$，期望的场测量为：
Calibration of a magnetic system does not share the limitations of characterization, as it fits a physics-based model of the field to the data collected. Model-based calibration works by approximating each field source with a single- or multidipole expansion of a field source, where the calibration process defines the locations, orientations and strengths of each source. For example, in a system with 8 ferromagnetic-core electromagnets, each current input differently magnetizes each of the cores; thus, the system should potentially consider 64 sources. A model comprising only dipole-field terms, with fewer sources considered than ideal described above, is often sufficient. However, including all potential sources, as well as the first three terms in the multipole expansion, results in less than 1% error in practice. Given a system with $n_c$ current inputs associated with $n_s$ source positions, a set of $n_c\times n_s$ dipole sources, located at positions ${\mathbf{p}}_{i,j}$ with unit current source strengths ${\mathbf{m}}_{i,j}$ associated with $I(i)=1A$, are used to model the field in the workspace. For a given field measurement ${\mathbf{b}}_k$ at location ${\mathbf{p}}_k$, taken with input current array $I_k$, the expected field measurement is:

b ~ k = b e + ∑ i = 1 n c ∑ j = 1 n s b { p k , m i , j , I k ( i ) , p i , j } \tilde{\mathbf{b}}_{k}=\mathbf{b}_{e}+\sum_{i=1}^{n_{c}}\sum_{j=1}^{n_{s}}\mathbf{b}\{\mathbf{p}_{k},\mathbf{m}_{i,j},I_{k}(i),\mathbf{p}_{i,j}\} b~k​=be​+i=1∑nc​​j=1∑ns​​b{ pk​,mi,j​,Ik​(i),pi,j​}

这里${\mathbf{b}}_e$是一个恒定背景项。给定$n_m$个场测量${\mathbf{b}}_k$，标定流程发现单位电流源强度${\mathbf{m}}_{i,j}$和源位置${\mathbf{p}}_{i,j}$的组合和背景场，来满足最小化问题：
where ${\mathbf{b}}_e$ is a constant background term. Given $n_m$ field measurements ${\mathbf{b}}_k$, the calibration process finds the set of unit current source strengths ${\mathbf{m}}_{i,j}$ and source locations ${\mathbf{p}}_{i,j}$, as well as the background field, to satisfy the minimization problem

$$arg\underset{{\mathbf{m}}_{i,j},{\mathbf{p}}_{i,j},{\mathbf{b}}_e}{\min }\sum _{k=1}^{n_m}||{\mathbf{b}}_k-{\tilde{\mathbf{b}}}_k|{|}^2$$

对于可移动的磁源，包含永磁铁和无芯电磁铁，场映射或模型在每个单独的源的坐标系产生，使用一个上面等式的适当简化版：对于无芯电磁铁，$n_s=1$；对于永磁铁，${\mathbf{m}}_{i,j}$是源强度，没有电流的概念。
For movable magnetic sources, including permanent magnets and coreless electromagnets, field maps or models are generated in each individual source’s coordinate frame, using an appropriate simplification of the above equation: for coreless electromagnets, $n_s=1$; for permanent magnets, ${\mathbf{m}}_{i,j}$ are source strengths, with no notion of current.

8. 拓展和应用 Extensions and applications

8.1 拓展 Extensions

8.1.1 旋转磁场 Rotating fields

电磁场和永磁体场源都已被开发来产生连续旋转的偶极子场。组合多个旋转偶极子场可以生成更均匀的旋转场。
Both electromagnetic and permanent-magnet field sources have been developed to generate continuously rotating dipole fields. Combining multiple rotating dipole fields can generate rotating fields that are more uniform.

8.1.2 多自由度磁机理 Multi-DoF magnetic mechanisms

8.2 应用 Applications

8.2.1 微型机器人 Microrobotics

通常在光学显微镜的引导下以及在体内医学应用中，磁操纵通常用于微型机器人的驱动。
Magnetic manipulation is commonly used in the actuation of microrobots, often under the guidance of an optical microscope and for in vivo medical applications.

8.2.2 医疗机器人 Medical robotics

8.2.2.1 胶囊内窥镜 Capsule endoscopes

不同的解决方案着重于胃肠道内的各个器官，例如胃，小肠和大肠，但许多技术只需很少的改动即可转化为其他器官。
Different solutions have focused on individual organs within the gastrointestinal tract - stomach, small intestine, and big intestine - but many techniques could be translated to other organs with minimal modifications.

8.2.2.2 导管和其他连续装置 Catheter and other continuum devices

自从这些最初的努力以来，对磁性连续装置的控制已经分为两个区域：类导管装置和可控针装置。
Since these initial efforts, control of magnetic continuum devices has branched into two areas: catheter-like devices and steerable-needle devices.

8.2.2.3 核磁共振成像扫描仪 MRI scanner

8.2.3 其他 Others

8.2.3.1 无绳磁触觉接口 Untethered magnetic haptic interfaces

UMHI消除了对机械联动装置及其相关惯性和摩擦的需要。
UMHIs eliminate the need for a mechanical linkage, with its associated inertia and firction.

8.2.3.2 航天系统 Aerospace systems

一个牢固地嵌入在飞机模型中的球体使其能够支撑在一个风洞中而没有机械附件会干扰气流。
A sphere rigidly embedded in an airframe model enabled its support in a wind tunnel with no mechanical attachments that would disturb the airflow.

[1]: Abbott, Jake J., Eric Diller, and Andrew J. Petruska. “Magnetic methods in robotics.” Annual Review of Control, Robotics, and Autonomous Systems 3 (2020).