## 2 Materials and Methods
### Prediction of Rockfall Trajectories Induced by Rock Mass Instability  
#### 2.4.1 Theoretical Framework of the Rockfall Simulation Model  
The RAMMS:ROCKFALL module was jointly developed by the Center for Mechanics at ETH Zurich and the Swiss Federal Institute for Forest, Snow, and Landscape Research (WSL). It is based on rigid-body Lagrangian mechanics and employs an explicit time integration scheme to solve the six-degree-of-freedom dynamic response of rock blocks on 3D terrain surfaces, accounting for both translational and rotational motions. A key feature of the model is its implementation of full 3D rigid-body dynamics, incorporating non-smooth contact mechanics and hard-contact laws to accurately simulate transitions among bouncing, rolling, and sliding. This allows for a realistic representation of energy dissipation and dynamic coupling throughout the rockfall process [28]. Compared to traditional simplified models, this module integrates contact algorithms and a scarring mechanism, allowing it to accommodate complex block geometries and soft ground conditions and to comprehensively simulate multiple motion modes including bouncing, rolling, and sliding [29]. The system supports both single-trajectory analysis and multi-trajectory statistical simulations and provides output parameters including bounce height, velocity, rotational energy, and total kinetic energy. It provides high-precision, three-dimensional dynamic simulation support for rockfall hazard assessment and the design of protective engineering measures. During the free-flight phase, the motion of the rock block is governed by the following set of coupled dynamic equations:  
\[Mu-h(q,u)=0 \tag{6}\]  
\[h(q,u)= \begin{bmatrix}F_{g}+D\\ G\end{bmatrix}  \tag{7}\]  
Here, \(M\) is a constant, diagonal mass matrix that includes the total mass and the three principal moments of inertia (\(I\)). The vector \(u\) contains the three translational and three rotational velocity components of the rock block, while \(q\) denotes the generalized coordinates, including both the centroid position and orientation parameters. The motion of the rock block is governed by multiple forces that collectively determine its trajectory. The gravitational force Fg acts on the entire mass of the rock block. The resistance force (\(D\)) accounts for the effects of vegetation (e.g., trees and shrubs) and soil deformation. Additionally, the gyroscopic force (\(G\)) enables irregularly shaped rock blocks to maintain \begin{table}
\begin{tabular}{l} Stable \\ \end{tabular} & Stable & Stable \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Stability Analysis Results of Rockfall-Prone Areas. upright orientation and rotate around their rolling axis. In the RAMMS::ROCKFALL module, the rock-ground interaction is modeled using a non-smooth contact mechanics framework under hard-contact assumptions. The equations governing the motion of the rock block can be expressed as follows:  
\[Mu-h(q,u)=\lambda W(q) \tag{8}\]  
In the equation, \(M\) denotes the mass-inertia matrix. \(u\) represents the velocity vector, comprising three translational and three rotational components. \(h(q,\mu)\) denotes the combined external forces acting on the rock block, including gravity, resistance, and gyroscopic effects. \(\lambda\) denotes the Lagrange multiplier associated with the contact force. \(W(q)\) denotes the contact force direction matrix. RAMMS::ROCKFALL employs a rigid-body non-smooth dynamics model to explicitly simulate the rockfall process under multiple contact interactions between the rock and the terrain surface. The model allows contact forces to act at locations other than the centroid and fully incorporates angular velocity and its direction to generate realistic torques that drive rotational and rebound motions. This enables accurate characterization of complex response mechanisms, including impact, sliding, and rolling, and represents a significant improvement over conventional models based solely on restitution coefficients.