## 2 Optimal Control Model for PMCU Path Planning
### Optimal Control Model  
The PMCU path planning problem refers to finding a path between a start state and an end state that meets the constraints of the transport process. Typically, the constraints involved in this problem are time-dependent. To provide a complete and direct description of the PMCU path planning problem, we use optimal control for modeling.  
The optimal control model can be described as follows: for a controlled system, finding the optimal change of control variables to minimize the optimization objective following the state of the system while satisfying the constraints. Generally, for a common optimal control problem, it consists of the following parts:  
\[J=\Psi\Big{(}x(t_{0}),t_{0},x\Big{(}t_{f}\Big{)},t_{f}\Big{)}+\int_{0}^{t_{f}} L(x(t),u(t),t)dt \tag{1}\]  
\[\dot{x}(t)=f(x(t),u(t),t),t\in\Big{[}t_{0},\ t_{f}\Big{]} \tag{2}\]  
\[\Phi\Big{(}x(t_{0}),t_{0},x\Big{(}t_{f}\Big{)},t_{f}\Big{)}=0 \tag{3}\]  
\[h\ (x(t),u(t),t)\leq 0 \tag{4}\]  
Equations (1)-(4) represent a common optimal control problem, where \(x(t)\) denotes the state variables, \(u(t)\) denotes the control variables, \(t_{0}\) denotes the initial time, and \(t_{f}\) denotes the terminal time. Equation (1) represents the optimization objective of the optimal control problem, which usually consists of two parts: system terminal state objective and system process objectives, respectively. Equation (2) represents the differential equation constraint of the controlled system, describing the change of state variables and control variables of the controlled system. Equation (3) shows the boundary conditions of the controlled system, which are used to describe the range of states at a point in time within the control time. Equation (4) represents the path constraints, indicating the constraints imposed on the state and control of the system throughout the controlled time.  
We can completely define an optimal control problem by Equations (1)-(4). This means that when we model the optimal control of the PMCU path planning problem, we need to define Equations (1)-(4) separately.  
In general path planning problems, the optimization objective of path planning methods is usually the shortest moving path. However, in actual production, the time of the cabin pushing process is a key factor affecting the installation of PMCU, and the shortest path does not necessarily meet the requirements of optimal production efficiency. So, this paper considers using the cabin pushing operation time as the optimization objective, which is:  
\[J=\int_{t_{0}}^{t_{f}}dt=t_{f} \tag{5}\]  
Equation (5) represents the optimization objective function, \(t_{0}\) is the starting time and \(t_{0}=0\), \(t_{f}\) is the required time for cabin pushing to the target, and \(J\) represents the optimization objective. Compared with Equation (1), we do not require the optimization of the terminal state of PMCU path planning, so we only consider the process objective to minimize the terminal time to achieve maximum efficiency.  
The differential equation constraints of the PMCU path planning optimal control model correspond to its kinematic model. The boundary conditions correspond to the range of state and control variations of the PMCU pushing process. The path constraints need to restrict the PMCU pushing process from colliding with obstacles. The above three parts will be introduced in Section 2.2, Section 2.3, Section 2.4, respectively.