# A Review of Recent Advances in Roll Stability Control in On-Road and Off-Road Vehicles
## 3 Theoretical Foundations of Roll Stability Control
### Rollover Prediction Index  
The design and application of vehicle rollover prediction indicators is one of the core tasks in rollover prevention control research. A reasonable rollover prediction indicator can help vehicle safety systems to identify potential rollover risks and take appropriate proactive or passive safety measures. To date, researchers have proposed various rollover prediction indicators to accommodate different vehicle types and operating conditions. These indicators are typically classified based on their physical significance, calculation methods, and applicable scenarios, including roll angle [76], time to rollover (TTR) [77; 78], and load transfer ratio (LTR) [79], among others. The roll angle represents the vehicle's lateral tilt around its longitudinal axis, and is commonly used in real-time stability control due to its intuitive physical meaning and ease of measurement via IMUs or dynamic models. While widely applied in active suspension and ESP systems, it cannot directly quantify rollover risk, and is sensitive to vehicle design and load conditions. To improve its predictive value, [PERSON] et al. [76] analyzed roll angle behavior under varying speeds and steering inputs, revealing drastic directional changes in low-friction, high-speed scenarios. [PERSON] et al. [80] developed a high-precision roll angle estimation method using an extended Kalman filter and a dual-mass model combined with IMU and steering sensor data. TTR refers to the time required for the vehicle to reach a critical rollover condition from its current state. It is an important indicator for assessing the dynamic variation of rollover risk. Its calculation is based on the vehicle dynamic model, and uses the current roll angle, roll rate, and lateral acceleration to predict when the vehicle will reach the critical rollover state. The formula for TTR is as follows:  
\[TTR=\frac{\varphi_{\text{crit}}-\varphi}{\dot{\varphi}} \tag{5}\]  
TTR has clear physical significance and can directly quantify the rollover risk, making it widely used in autonomous driving systems and intelligent vehicle control, especially when combined with MPC to optimize rollover prevention strategies. However, since TTR calculation relies on an accurate dynamic model, its application is influenced by vehicle parameter errors and sensor noise. [PERSON] et al. [81] proposed an improved rollover time prediction method based on a multi-layer neural network. By developing a vehicle rollover dynamic model that accounts for the battery box structure and mass, they classified \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**References** & **Modeling Approach** & **Application** & **Analysis Perspective** & **Simulated Motion** & **Remarks** \\ \hline \multirow{4}{*}{[69; 70]} & Vehicle stability analysis model with tire parameter estimation & Influence analysis of rollover factors & Effects of tire parameters & Roll, lateral, vertical motion & Verified the effects of tire parameters on rollover stability \\ \hline \multirow{4}{*}{[71; 72; 73]} & Multi-body dynamic model & Heavy-duty vehicle rollover prevention & Structural deformation effects & Roll, yaw, lateral motion & Investigated rollover risk under high-speed cornering with multi-body dynamics \\ \hline \multirow{4}{*}{[74; 75]} & Integrated roll & Intelligent roll & Data-driven approach for active roll control & Roll, yaw, lateral motion & Utilized neural networks to enhance active roll \\  & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cont. the rollover indicators containing hyperparameters into five categories, simplifying the algorithm structure using a multi-layer neural network and real-time calculation of variable-step operational-state parameters. [PERSON] et al. [78] introduced a TTR early-warning algorithm based on a BP neural network, which corrects the TTR indicator online by integrating multiple state parameters, such as the roll angle and yaw rate. LTR measures the imbalance of vertical loads between the left and right wheels of the vehicle.