A Critical Analysis of Eight-Electromagnet Manipulation Systems：The Role of Electromagnet Configurat

这篇文章详细的例举了非常著名的几个电磁操作系统，并提出了力矩/力产生能力、各向同性、空间可获得性的评估标准以及详细的分析，很好地总结了这几个系统的操纵特点。

八电磁铁操作系统的一个批判性分析：电磁铁配置对强度，各向同性，和可行性的作用
A Critical Analysis of Eight-Electromagnet Manipulation Systems：The Role of Electromagnet Configuration on Strength, Isotropy, and Access [1]
Paper Link
Authors: Pourkand Ashkan, etc.
2018, IEEE Robotics and Automation Letters (RA-L)

0. 摘要 Abstract

众所周知，在使用一组固定的电磁体来创建各向同性的电磁操纵系统中，八个电磁体是必要的以确保工作空间中没有奇异之处。迄今为止，已经提出了多种八电磁体配置。在这封信的第一个贡献中，我们对这些建议的构型进行了严格的比较，发现某些构型在磁场和扭矩产生方面是优越的，而其他构型则在力的产生方面是优越的，由于磁场的空间导数。所有现有配置都具有高度的对称性，但是在设计之间对工作空间的开放性差异很大。在这封信的第二个贡献中，我们提出了一种新配置，该配置不包含现有系统中看到的对称性，并且比以前的系统提供对操作工作区的更多开放性，但仍表现出各向同性的工作区，其性能可与先前的系统相媲美。我们得出的结论是，各向同性八电磁操纵系统的设计人员不应感到必须将对称性纳入其设计中。
It is known that in the creation of isotropic electromagnetic manipulation systems using a set of stationary electromagnets, eight electromagnets are necessary to ensure that there are no singularities in the workspace. A variety of eight-electromagnet configurations have been proposed to data. In the first contribution of this letter, we conduct the critical comparison of these proposed configurations, and find that certain configurations are superior in terms of field and torque generation, whereas others are superior in terms of force generation due to the spatial derivative of the field. All of the prior configurations comprise a high degree of symmetry, yet access to the workspace varies widely between designs. In the second contribution of this letter, we propose a new configuration that does not comprise the symmetry seen in prior systems, and also provides more open access to the manipulation workspace than prior systems, yet still exbihits an isotropic workspace with comparable performance to prior systems. We conclude that the designers of isotropic eight-electromagnet manipulation systems should not feel obliged to incorporate symmetry into their design.

3. 八电磁铁配置的批判性比较 Critical Comparison of eight-electromagnet configurations

前两个配置已经在文献中被广泛使用，第三个配置已经被使用在一个商用系统中。所有这三个系统含有一个高度的对称性。在第四个系统中，我们提出了一个新的配置，严重地违反了对称性的假设，并且我们发现它的表现和其他三个不相上下。在每一个配置中，我们以力矩和力生成能力（包括大小和各向同性）的名义来量化工作区域的表现，同时我们用一组未占用的立体角来量化工作区域的可获得性。在每一个所考虑的配置中，我们假设，以优先级顺序：（1）所有电磁铁是同直径的圆柱；（2）电磁铁的轴线都交于一个中点，它被假设为操纵工作空间的中心；（3）所有电磁铁距离中点相同的距离；（4）所有电磁铁尽量接近中心点，以至于在电磁铁间的碰撞已经发生。
The first two configurations have been highly utilized in the literature, and the third configuration has been utilized in a commercial system. All three of these systems comprise a high degree of symmetry. In the fourth system, we propose a new configuration, which sbstantially violates an assumption of symmetry, and we find that its performance is comparable to the other three. we quantify the performance of the workspace in terms of torque- and force-generation capability (both magnitude and isotropy), and we quantify accessibility of the workspace as a set of unoccupied solid angles. In each of the configurations considered, we assume, in order of priority: (1) that all the electromagnets are cylinders with the same diameter; (2) that the axes of the electromagnets all intersect at a common point, which is assumed to be the center of the workspace for manipulation; (3) that all electromagnets are the same distance from the common center; (4) and that all electromagnets are as close as possible to the common point, such that a collision between electromagnets has occured.

3.B. 用于数值比较的方法论 Methodology for Numerical Comparison

当评估每种配置的操纵性，我们考虑一个边长$0.5D_0$的正方形工作区域，中心位于电磁铁轴线的交点。我们考虑立方形工作区域的中心，和立方体的角落和它的每个棱边和面的中点，总计27个点贯穿工作区域。在每个点，我们计算3X8驱动矩阵和5X8驱动矩阵的奇异值。对于那些值，我们发现最小的奇异值，分别是${\sigma }_3$和${\sigma }_5$，这是我们期望的尽量大值来最大化系统的强度在保守的最坏的情况下。我们还计算了驱动矩阵的条件数，${\sigma }_3{\sigma }_1$和${\sigma }_5{\sigma }_1$，这是我们期望尽量统一的值来最大化各向同性。我们计算以上每一个矩阵在贯穿工作区域的27个点。

3.C. 结果 Results

在磁力矩和磁力方面，更大的电磁铁导致更大的值。但是，我们发现条件数与电磁铁的尺寸无关。我们还发现，除了一些例外，不同配置间的相对比较结果对电磁铁尺寸不敏感。
larger electromagnets lead to higher values in terms of both field (torque) and field derivative (force). However, we find the condition number is insensitive to the size of the electromagnets. We also find that, with few exceptions, the results of a relative comparison between configurations are insensitive to the size of the electromagnets.

在磁场产生/磁力矩产生能力上，square-antiprism和Magnetecs-CGCI是最强的，OctoMag和Open-asymmetric是最弱的。弱也只弱了22%。Open-asymmetric在工作区域条件上是最好的，square-antiprism在工作条件上第二好。

在磁场导数/磁力的产生能力上，OctoMag是最强的，接下来是Open-asymmetric（OctoMag的81%），然后是square-antiprism（OctoMag的63%），Magnetecs-CGCI最弱（OctoMag的9%）。Open-asymmetric有最好的工作条件。

[1]: Pourkand, Ashkan, and Jake J. Abbott. “A critical analysis of eight-electromagnet manipulation systems: The role of electromagnet configuration on strength, isotropy, and access.” IEEE Robotics and Automation Letters 3.4 (2018): 2957-2962.