2.1. Electrical Impedance Tomography

The operating principle of the EIT technique adopted by E-skins is illustrated in Figure 1a. An electrically conductive film is attached with eight electrodes around its boundary. The interior of the film is denoted as Ω, and its boundary is donated as δΩ. An electric current is injected into the film via a selected electrode and sunk via another. The injected current produces an electric field that generates an electric potential distribution over the film. Meanwhile, the voltage at each electrode is measured with respect to a reference electrode. Then the current injection is switched to the next pair of electrodes. The electrode pairing typically follows either an adjacent or an opposite pattern [13]. Then voltage measurements are taken again from the new potential distribution. This procedure is repeated several times until the selected current injection pattern is completed. The voltage data recorded throughout the process is taken as the input to a reconstruction algorithm that estimates the conductivity distribution within the Ω.

The working mechanism of EIT-based touch sensors. (a) The basic working principle of EIT: voltage measurements are taken at every electrode for each current injection. (b) Discretizing the domain of the sensing material into a collection of a finite number of elements and nodes using the finite element method. (c) The forward problem and the inverse problem.

The reconstruction algorithm consists of the forward problem and the inverse problem. The objective of the forward problem is to acquire a mathematical model for predicting the unknown electric potential distribution using a known current injection pattern and a known electrical conductivity distribution. The forward model will then serve as the foundation for the inverse problem, which allows the unknown conductivity distribution to be inversely calculated using measured voltage data. Derived from Maxwell’s equations, the governing equation of the forward problem is expressed in the form of an elliptical partial differential equation (PDE) [14], expressed as

with a boundary condition (BC) described by

where σ denotes the point-wise conductivity distribution of the medium, g is the total amount of current that enters and leaves a point, and n^ is a unit normal vector pointing towards the exterior of the boundary. The physical meaning of the PDE and its BC is that the net current flowing into and out of any unique point within Ω is zero, and the net current can only be non-zero at points on δΩ where electrodes are attached. The problem is solved numerically using the finite element method (FEM), which transforms the PDE from its continuous form into a discrete form constructed with a collection of M elements connected by N nodes, as illustrated in Figure 1b. For each element, the PDE and the BC become a combined and discretized form [15], expressed as

where σe denotes the element-wise conductivity, Φi and Φj are basis functions used in the FEM, i and j are node indices for the element, A is the area of the element, and S is the length of the boundary edge where it is applicable. Imposing a suitable electrode model and combining the amended local equations for all the elements results in a system of linear equations, which is expressed in its simplified form [14,15] as

where un represents nodal voltage data, ul contains the voltage at each electrode, I is the current injection pattern, and the A matrices contain the information on element conductivity, the finite element model described by the left side of (3), and the contact impedance of the electrodes. Using (4), the electrode voltages ul, as the solution to the forward problem, can thus be determined by solving the system of linear equations with a known current injection pattern I.

The inverse problem follows the opposite course by taking the measured electrode voltage data, ul, as its input and inversely calculating the element-wise conductivity data contained in the A matrices. This is typically achieved by deriving a Jacobian matrix, H, which takes the derivative of the ul with respect to σ [14]. In other words, H is a mapping function that predicts the perturbation in ul, caused by variations in σ. This relation is expressed as

In practice, △ul is taken as the difference in electrode voltage data measured between a time interval ΔT, and the inversely calculated △σ is the relative variation in conductivity distribution between the period ΔT. This method is known as time difference imaging. Notably, un is a part of the forward solution in (4), but it is not associated with H in the inverse problem; therefore, the system of equations described by (5) is underdetermined. A unique solution cannot be determined. To accommodate this problem, Tikhonov-styled regularization [16] is typically employed by minimizing the least square error described by the cost function described by

where eLS denotes the least square error, λ is the hyperparameter that controls the weight of the regularization, and R is a selected regularization matrix. Finally, a unique solution for which eLS is minimized can be obtained as

The summarized operating principle of EIT-based E-skins for touch sensing is illustrated in Figure 1c. Briefly, when a pressure or strain is applied to the sensing material, it induces a change in the conductivity distribution of the material. This change results in a difference in electric potential distribution and therefore causes a variation in the voltage data measured from the electrodes. This process is numerically modeled by the forward problem using FEM. Conversely, the inverse problem seeks a best-fitted conductivity solution by mapping the electrode voltage data backward using the Jacobian matrix and the method of regularization [17]. Finally, the conductivity solution reconstructed from the inverse problem can be used to interpret the presence of the applied mechanical stimuli in terms of planar spatial distribution.