## 3 Methods
### KDDPG: Local Path Planning Method Based on Kinematic Analysis
#### 3.2.1 Kinematic Model and Environmental Perception Model  
The kinematic model and the environmental perception model of a two-wheeled differential mobile robot are shown in Figure 2, illustrating the robot's position and state at consecutive time points as well as the surrounding environmental information. The robot's state update equation based on its kinematic is:  
\[s_{t+1}=f(s_{t},\overrightarrow{a_{t}})+\overrightarrow{\delta_{t}}=\begin{bmatrix}x _{t}+vT\cos(\theta_{t}+\frac{\omega T}{2})+\delta_{x,t}\\ y_{t}+vT\sin(\theta_{t}+\frac{\omega T}{2})+\delta_{y,t}\\ \theta_{t}+\omega T+\delta_{\theta,t}\end{bmatrix} \tag{6}\]  
where \(T\) is the unit sampling period, state \(s_{t}=[x_{t},y_{t},\theta_{t}]^{\mathrm{T}}\) collectively represents the robot's position (\(x_{t},y_{t}\)) and orientation angle \(\theta_{t}\), and control variables \(a_{t}=[v,\omega]^{\mathrm{T}}\) updates robot's state via its linear velocity \(v\) and angular velocity \(\omega\), \(\delta_{x,t}\), \(\delta_{y,t}\), and \(\delta_{\theta,t}\) denotes Gaussian noises in \(x,y\), and \(\theta\), respectively, simulating the impact of real-world uncertainties on the robot's motion.  
For a robot running at speed \(v\), the corresponding minimum braking distance \(X_{v,min}\) is given by:  
\[X_{v,min}=\frac{v^{2}}{2a_{max}} \tag{7}\]  
where \(a_{max}\) is the robot's maximum braking deceleration. Obviously, a higher running speed results in a longer minimum braking distance. Thus, the current work denotes the robot's minimum braking distance at its maximum speed \(v_{max}\) as the buffer distance \(X_{min}\), and defines the robot's safety radius \(r_{s}\) accordingly, such as,  
\[\begin{array}{l}r_{s}=X_{min}+R\\ X_{min}=\frac{v_{max}^{2}}{2a_{max}}\end{array} \tag{8}\]  
where \(R\) is the robot's minimum bounding radius. Furthermore, by comparing \(r_{s}\) with the distance between the robot and the obstacle, we can determine whether the robot is in a collision zone.  
For environmental perception, we set the goal attractive potential field \(U_{goal}(s_{t})\) and the obstacle repulsive potential field \(U_{obs}(s_{t})\) as:  
\[\begin{array}{l}U_{goal}(s_{t})=\frac{1}{2}{\rm{d}}{d_{goal}}^{2}\\ U_{obs}(s_{t})=\sum\limits_{i=1}^{n}{e_{i}\cdot(d_{i}-r_{s})}\end{array} \tag{9}\]  
where \(k\) is the coefficient of the attractive potential field, \(d_{goal}\) is the distance between the robot's current position and the goal point, \(d_{i}\) is the scanning distance of the \(i\)-th laser beam, and \(e_{i}\) is the weight of \(d_{i}\). With such notions, \(d_{i}-r_{s}\) represents the difference between the obstacle's distance and the safety distance, and a smaller (\(d_{i}-r_{s}\)) indicates a higher collision risk. The size of the obstacle is measured by the number of laser beams that sense the obstacle. \(U_{goal}(s_{t})\) indicates the robot's proximity to the goal point, while \(U_{obs}(s_{t})\) comprehensively considers the distance, relative direction, and size of the obstacle,  
Figure 2: The kinematic model and the environmental perception model of a two-wheel mobile robot.  
indicating the robot's proximity to the obstacle. We then refer to the gradients of \(U_{goal}(s_{t})\) and \(U_{obs}(s_{t})\) as,  
\[F_{goal,t}=-\
abla U_{goal}(s_{t}) \tag{10}\] \[F_{obs,t}=\
abla U_{obs}(s_{t})\]  
where \(F_{goal,t}\) represents a vector pointing towards the goal, and \(F_{goal,t}\) represents a vector pointing away from the obstacles. Therefore, \(F_{goal,t}\) directs the robot to move towards the goal, and \(F_{goal,t}\) guides the robot to move away from obstacles.  
Based on the above two models, we will design a reward function to optimize the robot's path-planning performance in the following section.