## 2 ANN-Based Vehicle Mass Classifier Using Perturbed Input  
This section describes the ANN classifier using perturbed input data of the vehicle and the training process. The proposed ANN consists of an input layer, multiple hidden layers, and an output layer, with each hidden layer learning complex data patterns and extracting features. While other study methods [20; 21; 22] generally utilized the speed, acceleration, engine torque, gear ratio, and gear efficiency as the input of ANN model (with slight pre-processing but without any significant manipulation), this work relies on perturbed input data (i.e., perturbed velocity, perturbed acceleration, and perturbed engine torque), where this selection enables us to track the complicate pattern change in the ratio between acceleration and traction force (roughly becomes the vehicle mass). Figure 1 depicts the proposed ANN structure for vehicle mass estimation, consisting of an input layer, three hidden layers, and an output layer together with pre-processing and averaging processes. Here, \(\widetilde{V}_{x}\), \(\widetilde{E}_{tx}\), and \(\widetilde{a}_{x}\) indicate a longitudinal velocity, an engine torque, and a longitudinal acceleration. The \"pre-process\" achieves the perturbed inputs (i.e., \(d\widetilde{V}_{x}=\left(\widetilde{V}_{x,t}-\widetilde{V}_{x,t-1}\right)/ \Delta t\), \(\mathrm{d}\widetilde{E}_{tx}=\left(\widetilde{E}_{tx,t}-\widetilde{E}_{tx,t- 1}\right)/\Delta t\), and \(d\widetilde{a}_{x}=\left(\widetilde{a}_{x,t}-\widetilde{a}_{x,t-1}\right)/ \Delta t\), where \(\Delta t\) is a sampling time) based on raw inputs \(\widetilde{V}_{x}\), \(\widetilde{E}_{tx}\), and \(\widetilde{a}_{x}\), while the \"averaging process\" generates the averaged inputs for the input of the ANN classifier, averaged perturbed velocity (\(d\widetilde{V}_{x,N}\)), averaged perturbed engine torque (\(d\widetilde{E}_{tx,N}\)), and averaged perturbed acceleration (\(d\widetilde{a}_{x,N}\)), as computed in Equations (4)-(6). Each hidden layer contains 25 neurons, chosen to capture complex nonlinear relationships between the input data and an output. The weight matrices (\(W_{1}\in\mathbb{R}^{n\times 4}\), \(W_{2}\in\mathbb{R}^{n\times n}\), \(W_{3}\in\mathbb{R}^{m\times n}\), \(W_{4}\in\mathbb{R}^{1\times m}\), where \(n=25\), \(m=5\)) represent the connections between the respective layers, while the vectors (\(b_{1}\in\mathbb{R}^{n\times 1}\), \(b_{2}\in\mathbb{R}^{n\times 1}\), \(b_{3}\in\mathbb{R}^{m\times 1}\), \(b_{4}\in\mathbb{R}\)) correspond to the bias of each neuron in layers. Figure 1: Proposed ANN structure with multiple hidden layers. The average value of the perturbed data (\(\widetilde{dV}_{x,N},d\widetilde{E}t_{x,N},d\widetilde{a}_{x,N}\)) was selected as input to the ANN classifier. Using the perturbed and averaged data, two major benefits can be found:  
(i) The perturbed data represent the difference between the current and previous data over the sampling time. Existing methods often rely on current data, making it difficult to track data changes or patterns. By utilizing the variables between the previous time \(t-1\) and the current time t, this approach benefits from remembering previous states, allowing for better tracking of data changes and patterns. This concept is mathematically supported by Equation (3), which is derived from Equations (1) and (2) based on the longitudinal vehicle dynamics. \[\frac{\gamma\widetilde{E}t_{x,t}}{r_{eff}}-F_{zero,x.t}=m\widetilde{a}_{x,t} \tag{1}\]  
\[\frac{\gamma\widetilde{E}t_{x,t-1}}{r_{eff}}-F_{zero,x.t-1}=m\widetilde{a}_{x, t-1} \tag{2}\]  
Here, \(\widetilde{E}t_{x,t}\) represents the engine torque at the current time step, and \(F_{zero,x.t}\) denotes the longitudinal aerodynamic force at the current time step; \(m\) is the vehicle mass, and \(\widetilde{a}_{x,t}\) represents the longitudinal acceleration. Additionally, \(\gamma\) and \(r_{eff}\) represent a gear ratio and an effective radius, respectively.