## 2 Materials and Methods
### Neural Network Modelling of Specific Energy Requirements  
A multilayer feed-forward artificial neural network (ANN) model was used to predict SPE requirement of the rotary weeding unit [28; 39]. The model inputs included DO (mm), CI (kPa), and \(u/v\) ratio. The trials generated a total of 81 sets of data that were averaged and divided into training (70%) and testing groups (30%). The ANN architecture in this study consisted of three layers of interconnected neurons: the input layer, the hidden layers, and the output layer (Figure 5). The network outputs were obtained by applying activation functions, denoted as _fh_ (.) and _fo_ (. ), to the weighted sum of inputs from the previous layer. The mathematical expression for the \(k\)th network output was defined using Equation (7). \[y_{k}=f_{o}\left[\sum_{j=1}^{N_{h}}W_{jk}\cdot f_{h}\left(\sum_{i=1}^{N_{i}}W_{ ij}x_{i}+\left(b_{h}\right)_{j}\right)+\left(b_{o}\right)_{k}\right], \tag{7}\]  
where \(y_{k}\) is the \(k\)th output variable; \(x_{i}\) is the ith input variable; \(f_{h}\) and \(f_{o}\) are the activation functions in the hidden and the output layers, respectively. \(i\), \(j\), and \(k\) denote the input layer, the hidden layer, and the output layer neurons, respectively, and are given as \(i\) = 1, 2... N\({}_{i}\); \(j\) = 1, 2... N\({}_{h}\) and \(k\) = 1, 2... N\({}_{0}\). \(W_{ij}\) represents the connection weight between the \(i\)th input neuron and the \(j\)th hidden neuron. \(W_{jk}\) represents the connection weight between the \(j\)th hidden neuron and the \(k\)th output neuron. \(\left(b_{h}\right)_{j}\) denotes the bias for the \(j\)th neuron in the hidden layer and \(\left(b_{o}\right)_{k}\) is the bias for the \(k\)th neuron in the output layer. The entire modeling was implemented on a MATLAB R2014b package. An iterative trial and error approach was used to fine-tune the model configuration for optimal neurons. The number of neurons in the input and output layers was determined based on the respective input and output parameters. Consequently, the input layer consisted of three neurons, while the output layer had one neuron. To determine the optimal number of neurons in the hidden layer, a range of one to seven neurons was tested. The criterion for selecting the optimal configuration was based on the minimum mean squared error (_MSE_). By iteratively evaluating different configurations, it was determined that the hidden layer should have five neurons. Hence, the resulting optimal neuron configuration for the ANN model was 3-5-1. The log-sigmoid transfer function (logsig) was used as the activation function in the hidden layer (Equation (8)). For the output layer, the linear transfer function (purelin) was utilized (Equation (9)). The topology of the developed ANN is visually represented in Figure 5. To expedite the training process and ensure fair treatment of all variables during model training, experimental data were normalized using a linear transformation (Equation (10)) [40] within a standardized range of 0.1-0.9 before training. \[f_{h}(x)=\frac{1}{(1+e^{-x})}\,\ 0\leq f_{h}(x)\leq 1, \tag{8}\]  
Figure 4: (a) Test of vertical axis rotor; (b–d) Soil profiles after operations in soil bin. \[f_{o}(x)=x,-\infty\leq f_{o}(x)\leq+\infty, \tag{9}\]  
\[x_{n}=0.1+0.8\times\left(\frac{x_{o}-x_{min}}{x_{max}-x_{min}}\right), \tag{10}\]  
where \(x_{n}\) and \(x_{o}\) are the normalized and the original values of the input parameters, and \(x_{max}\) and \(x_{min}\) are the maximum and the minimum values of the input parameters, respectively. During model training, network weights and biases were chosen based on the training dataset by employing the Levenberg-Marquardt backpropagation learning algorithm. To ensure optimal convergence, the maximum number of epochs was set at 1000, while the minimum performance gradient was set at \(10^{-7}\). Additionally, a damping factor (\(\upmu\)) of 0.001 was applied, and the training process was monitored with a maximum of 6 validation checks before termination.