# Analysis and Comparison of Electromagnetic Microrobotic Platforms for Biomedical Applications
## 2 Theoretical Background  
### Magnetic Manipulation  
EMA systems consisting of several electromagnets allow generating a magnetic field and/or a gradient field in a given workspace, as shown in Figure 2. These fields induce a magnetic torque and a force on the magnetized materials of the untethered microrobots. The expression of the magnetic field generated by an electromagnetic coil is derived from a single wire and the magnetic dipole. The magnetic field from any electromagnetic coil \(c\) can be approximated as a magnetic dipole characterized by its magnetic moment \(\mathbf{m}_{c}\), and the point-dipole model is proposed. It can be shown that the magnetic field and its gradient are proportional to the electric current \(i_{c}\) flowing through the coil \(c\). The overall magnetic field generated by the \(n\)-coils is the superposition of each field. The magnetic field and its gradient are then expressed as:  
Figure 1: The concept of EMA system applying for various biomedical applications. \[\mathbf{B}(\mathbf{p}) =\mathcal{B}(\mathbf{p})\mathbf{i}=\left(\mathcal{B}_{x}\quad B_{y} \quad B_{z}\right)^{\intercal}\mathbf{i} \tag{1}\] \[\
abla\mathbf{B}(\mathbf{p}) =\mathcal{G}(\mathbf{p})\mathbf{i}=\left(\frac{\partial\mathcal{B} _{x}}{\partial x}\quad\frac{\partial\mathcal{B}_{x}}{\partial y}\quad\frac{ \partial\mathcal{B}_{x}}{\partial z}\quad\frac{\partial\mathcal{B}_{y}}{ \partial y}\quad\frac{\partial\mathcal{B}_{y}}{\partial z}\right)^{\intercal} \mathbf{i} \tag{2}\]  
where \(\mathbf{i}=(i_{1},\dots,i_{n})^{t}\) is the electric current vector. When the magnetic dipole moment and the magnetic field are given, the induced magnetic torque and force can be easily obtained by the Maxwell's equations. The induced magnetic field aligns the microrobot to a desired direction, and the magnetic force provides the propulsion force to move the microrobot to complete the task. Through the mathematical transformation, and for more convenient investigations of the magnetic actuation properties, the equations of torque and force can be rearranged into the following expression:  
\[ \begin{pmatrix}\mathbf{t}_{\pi n}\\ \mathbf{t}_{\pi m}\end{pmatrix} = \begin{pmatrix}\mathbf{m}\times\mathcal{B}( \mathbf{p})\\ \mathbf{m}^{\intercal}\mathcal{G}(\mathbf{p})\end{pmatrix} \mathbf{i}= \mathcal{A}(\mathbf{m},\mathbf{p})\mathbf{i} \tag{3}\]  
where \(\mathcal{A}(\mathbf{m},\mathbf{p})\in\mathbb{R}^{6\times n}\) is an _actuation matrix_ mapping the current to the applied magnetic wrench. This magnetic actuation matrix is a function of both the position \(\mathbf{p}\in\Omega\), and the magnetic moment \(\mathbf{m}\) of the microrobot. Therefore, substituting Equations (1) and (2) into Equation (3), the equations of torque and force can be presented by the actuation matrix \(\mathcal{A}(\mathbf{m},\mathbf{p})\) in the further details as:  
\[\mathcal{A}(\mathbf{m},\mathbf{p})(\mathbf{i})= \begin{pmatrix}m_{x} &m_{y}&m_{z}&0&0&0&0&0\\ 0&m_{x}&0&m_{y}&m_{z}&0&0&0\\ -m_{z}&0&m_{x}&-m_{z}&m_{y}&0&0&0\\ 0&0&0&0&0&0&-m_{z}&m_{y}\\ 0&0&0&0&0&m_{z}&0&-m_{x}\\ 0&0&0&0&0&-m_{y}&m_{x}&0\end{pmatrix}\begin{pmatrix}\frac{\partial\mathcal{B} _{x}}{\partial x}\\ \frac{\partial\mathcal{B}_{x}}{\partial y}\\ \frac{\partial\mathcal{B}_{x}}{\partial z}\\ \frac{\partial\mathcal{B}_{y}}{\partial y}\\ \frac{\partial\mathcal{B}_{y}}{\partial z}\\ \frac{\partial\mathcal{B}_{y}}{\partial z}\\ \mathcal{B}_{x}\\ \mathcal{B}_{y}\\ \mathcal{B}_{z}\end{pmatrix} \mathbf{i} \tag{4}\]  
Each column of the matrix \(\mathcal{A}(\mathbf{m},\mathbf{p})\) represents the wrench on the force and torque per unit current created by each electromagnet. If there are greater than \(n>6\) electromagnets, the actuation matrix \(\mathcal{A}(\mathbf{m},\mathbf{p})\) leads to a better conditioned matrix, a more  
Figure 2: Illustration of the use of the magnetic manipulation of untethered microrobots. isotropic workspace \(\Omega\), a reduction of singularity configurations, and lower current requirements [72; 73; 81].