# Mobile Robot Path Planning Considering Obstacle Gap Features
## 2 Path Planning Method with Clearance Characteristics
### Real-Time Robot Control Method for Optimization of Clearance Angle
#### 2.2.2 Optimization of the Optimal Global Piecewise Control Point  
The global optimal path searched by the existing algorithms is still difficult to balance the cooperativity between global planning and trajectory tracking. Therefore, it is necessary to set a virtual endpoint that has not yet reached the global path and is not within the non-navigable area by estimating the piecewise control stability. This point is usually an accurate position, but the trajectory tracking usually cannot be accurate to that point. So, it is necessary to set an appropriate threshold between the position of the robot and the point to avoid the stagnation phenomenon of the robot wandering around the point.  
The stability of global path planning can be represented by the curvature of the trajectory curve. The larger the trajectory jitter is, the greater the absolute value of the trajectory curvature is. In this study, the curvature is adopted to evaluate the stability of hybrid path planning, which can account for wobble and jitter in the process of motion and represent the deviation effectively when the robot executes the global path. In the existing methods, the midpoint of the grid is used as the global path control point, which will limit the path selection of the robot and increase the collision risk of the turning angle and through the gap.  
Therefore, the trajectory tracking method is adopted in this paper to introduce the robot motion model into the global path planning process and derives the following Equation (12) for the robot motion control model:  
\[\mathrm{f}(x(t),a(t))=\left[\begin{array}{c}\cdot\\ \left[y(t)\right]\\ \left[\theta(t)\right]\\ \left[\omega(t)\right]\\ \left[v(t)\right]\end{array}\right]=\left[\begin{array}{c}a(t)\cos\theta( t)\\ a(t)\sin\theta(t)\\ \omega(t)\\ a_{x}(t)\\ a_{y}(t)\end{array}\right] \tag{12}\]  
Figure 4: Schematic diagram of optimal angle search.  
The motion of the robot with the method of the dynamic window can be regarded as an arc, and the position of the robot and the curvature of the arc can be solved by Equations (13)-(16):  
\[r(t_{i})=\frac{v_{r}(i+1)}{w_{r}(i+1)} \tag{13}\]  
\[x_{r}(t_{i+k})=x_{r}(t_{i})+kv_{r}(t_{i})cos(\theta(t_{i}))\delta t \tag{14}\]  
\[y_{r}(t_{i+k})=y_{r}(t_{i})+kv_{r}(t_{i})sin(\theta(t_{i}))\delta t \tag{15}\]  
\[\theta(t_{i+k})=\theta(t_{i})+kw_{r}(t_{i})\delta t \tag{16}\]  
The dynamic window approach models robot motion as a circular arc. Equation (12) defines the robot's kinematic model, showing how its state evolves based on control inputs. Equations (13)-(16) are derived from this model to calculate the radius of curvature and predict the robot's position and orientation over time. These formulas collectively enable motion prediction and path planning.  
In order to further calculate the curvature and solve the optimal trajectory control points of the dynamic window method, the minimal curvature is adopted as the control objective, and each segment trajectory is fitted by the fifth-degree polynomial piecewise, which is shown in Equation (17):  
\[f(t)=\left\{\begin{array}{l}\sum\limits_{