# Reducing Safety Risks in Construction Tower Crane Operations: A Dynamic Path Planning Model
## 2 A Dynamic Path Planning Model for Tower Crane Operations
### Path Planning Module  
The Dynamic Window Approach (DWA) is a commonly used dynamic path planning algorithm in robotics [42]. The core idea of DWA is to use the current state of the robot (position, speed, etc.) and surrounding obstacles to generate a dynamic feasible region within a velocity space, known as the velocity window. By dynamically adjusting the velocity window and selecting optimal velocity commands, mobile robots can safely avoid  
Figure 4: Construction site grid points. obstacles while moving quickly toward their targets. Due to its excellent performance, [PERSON] et al. [43] have continuously improved DWA for dynamic path planning. [PERSON] et al. [44] proposed a fuzzy logic improved dynamic windows approach for automatic navigation of mobile robots. [PERSON] et al. [45] dynamically modified the weight coefficient combination of the dynamic window method to cope with the constantly changing dynamic environment and suboptimal global planning path problems that arise after local obstacle avoidance. The dynamic window method is continuously improving to adapt to more complex path planning problems [46; 47; 48]. Therefore, this study attempts to apply an improved version of DWA to tower crane operation path planning for the first time. In solving the path planning problem for horizontal tower crane operations, the real-world environment can be simplified to a two-dimensional space. A working hazard radius is defined around the hook's center, creating a circular area, while workers in the environment are modeled as rectangular obstacles, with their top-left and bottom-right corners marked, as shown in Figure 5.  
Assuming the tower crane's base coordinate is at the origin \(O(0,0)\), the starting point of the construction task is denoted as S, and the endpoint as E. The solution space for path planning is defined as shown in Equation (2):  
\[P=[0,X]\times[0,Y]\cup[-X,0]\times[0,Y] \tag{2}\]  
where \(X,Y\in\mathbb{R}^{+},P\subset\mathbb{R}^{2}\). \(X,Y\) denotes the upper boundary of the construction environment. Each obstacle in the environment is represented as \(ob_{i}\), with the set of obstacles denoted as \(ob=\{ob_{1},ob_{2},\ldots,ob_{n}\}\), where n is the total number of obstacles. \(P_{obs}=ob_{1}\cup ob_{2}\cup\ldots\cup ob_{n}\) represent the infeasible regions in the environment; thus, each obstacle area satisfies the following condition: \(P_{obs}=\big{\{}(x,y)\mid x_{i_{1}}\leq x\leq x_{i_{2}},y_{i_{1}}\leq y\leq y _{i_{2}},i=1,2,\ldots,N\big{\}}\). The feasible solution region can be expressed as follows: \(P_{frae}=\big{\{}(x,y)\mid 0\leq x\leq X,0\leq y\leq Y,\ (x,y)\
otin ob_{i},i=1,2, \ldots,n\big{\}}\). If there exists a feasible path point represented as \(P_{j}(x_{j},y_{j})\), where \(P_{j}\in P_{free}\), then a safe working path for the tower crane will consist of \(P_{T}=\{P_{0},P_{1},P_{2},\ldots,P_{n},P_{n+1}\}\). The tower crane operation can be divided into two motion modes: the rotation of the boom and the trolley's amplitude adjustment. Therefore, the following physical constraints must also be satisfied during the motion:  
1. Rotation Radius:\(r\), \(r_{min}\leq r\leq r_{max}\), where \(r_{j}=\sqrt{x_{j}^{2}+y_{j}^{2}},j=0,1,2,\ldots,n+1\). 2. Velocity Constraints: \(V_{s}\), \(V_{s}=\big{\{}(v,\omega)|v\in[v_{\min},v_{\max}],\omega\in[\omega_{\min}, \omega_{\max}]\big{\}}\), where \(v\), \(\omega\) are linear and angular velocities, respectively. 3. Acceleration Constraints: \(V_{a}\), where \(v_{c}\), \(\omega_{c}\) are the current linear and angular accelerations and are their respective maximum values. 4. Braking Distance Constraint: \(V_{ob}=\left\{ \begin{array}{l}(v,\omega)\Big{|}v\leq\sqrt{2\times dist(v, \omega)\times\dot{v}},\\ \omega\leq\sqrt{2\times dist(v,\omega)\times\dot{\omega}}\end{array} \right\}\), where \(dist(v,\omega)\) is the distance from the simulated trajectory to the nearest obstacle. Figure 5: Modeling of potential drop zone and workers.