# A Hybrid Control Path Planning Architecture Based on Traffic Equilibrium Assignment for Emergency
## 4 The Proposed Hybrid Control Architecture for Path Planning
### Path Planning Constrained by Equilibrium Traffic Assignment  
To make the path planning constrained by the equilibrium traffic assignment, we first train a mapping layer to obtain the equilibrium graph. The path planning makes the routing decision based on the road sections with potential traffic. Therefore, finding the equilibrium traffic graph in the period to guide path planning can induce the traffic network to equilibrium. We propose a data-driven deep learning solution to derive equilibrium traffic graphs, as shown in Figure 3.  
In the offline training phase, the SO-based traffic equilibrium sample generator uses the equilibrium graph obtained by the traffic assignment method to label the input traffic statuses, which provides data samples for the network training. The equilibrium mapping layer first obtains the joint traffic state distribution of ODs and background load, and then learns the mapping to the equilibrium graph based on the generated samples. The data-labeling process generates sufficient traffic distribution samples for the training of the equilibrium mapping layer. After offline training, the mapping module can perform online prediction and periodically update the parameter configuration to adapt the changes. The trained equilibrium mapping layer can rapidly derive traffic distribution from real-time ODs and background load, and map it to an optimal (or close-to-optimal) equilibrium graph. Equilibrium Mapping LearningTo build the neural network for traffic graph learning, the equilibrium mapping layer can be implemented with multiple fully connected layers. Figure 3: Hybrid control for path planning process based on data-driven deep learning. However, the mapping between traffic patterns and equilibrium distributions is complex, involving network traffic propagation and global implicit interaction. This requires a fully connected neural network with many layers of connection parameters, which are difficult to train in a reasonable time and limited samples [49]. The status of the traffic graph is complex and variable, so the interaction between sections should be dynamic as well. In particular, it has limitations in using the fixed adjacency matrix to model the spatial dependencies of ODs. Accordingly, we propose a multi-graph convolutional network that adopts an adaptive adjacency matrix [50] to capture the hidden spatial dependencies between traffic and topology configurations. As shown in Figure 4, the inputs to the model contain the OD's volume \(O\in\mathbb{R}^{n\times n}\) and the background load \(V\in\mathbb{R}^{n}\). They are fused after multi-graph convolution resulting in a unique traffic pattern that corresponds to an equilibrium traffic graph. The equilibrium mapping layer efficiently learns the patterns in traffic and topology to achieve SO with higher performance. The traffic load graph \(G_{v}(R,\Lambda_{v})\) is composed of the road sections \(R\) and topology relations \(\Lambda_{v}\). The OD's graph \(G_{o}(R,\Lambda_{o})\) consists of \(R\) and semantic relations \(\Lambda_{o}\) of ODs. We use two separate feature graphs as basic components, with each component focusing on its own feature extraction and finally fusing them through a concatenation operation. The spatial dependencies of the HCN and AdaptiveHCN are based on the adjacent neighborhood \(\Lambda_{v}\) and OD semantic neighborhood \(\Lambda_{o}\), respectively. The \(\Lambda_{o}\) is obtained by two learnable parameters \(E_{o},E_{d}\), where \(E_{o}\) is the origin embedding and \(E_{d}\) is the destination embedding. We obtain the OD semantic dependencies between the origin and destination, as shown in Equation (13). \[\Lambda_{o}=Softmax(Relu(E_{o}\times E_{d}^{\top})). \tag{13}\]  
The Softmax is a probability distribution mapping function. And ReLU is applied to eliminate minor dependencies. The output \(\Lambda_{o}\) can be considered as a transfer matrix for implicit propagation based on ODs.