# 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
It is one of the key indicators for evaluating vehicle rollover risk, and the calculation formula is as follows:  
\[LTR=\frac{F_{xx}-F_{x1}}{F_{xx}+F_{x1}} \tag{6}\]  
where \(F_{xx}\) and \(F_{x1}\) represent the vertical load on the right and left wheels, respectively. The LTR value typically ranges from [\(-\)1, 1]. An LTR of 0 indicates that the load is evenly distributed between the left and right wheels, and the vehicle is in a stable state. An LTR of \(\pm 1\) indicates that one side of the vehicle has completely lost contact with the ground, and the vehicle is on the verge of rollover. The LTR is straightforward to calculate and can directly reflect rollover risk, making it widely used in rollover warning systems for commercial and heavy vehicles. It is also combined with active rollover control to optimize suspension parameters. However, the LTR is significantly affected by sensor accuracy and road conditions, and there is a risk of misjudgment. [PERSON] et al. [82] proposed a method using Kalman filtering to estimate the dynamic LTR in real time, overcoming the limitations of traditional static thresholds and reducing the vehicle stabilization time by approximately 2 s. [PERSON] et al. [83] introduced a rollover warning algorithm for heavy commercial vehicles based on the dynamic LTR. This method computes the LTR value in real time using a three-degree-of-freedom model and compares it with a preset safety threshold to trigger differential braking control. In addition to the main indicators mentioned above, researchers have also proposed supplementary rollover prediction metrics to enhance the accuracy of rollover risk assessment, as shown in Table 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Prediction Index** & **Evaluation Criteria** & **Equation** & **Rollover Threshold** & **Remarks** & **References** \\ \hline \multirow{4}{*}{Roll rate (\(\dot{\varphi}\))} & Rate of change in roll angle, used to assess roll dynamics & \(\dot{\varphi}=\frac{d\varphi}{dt}\) & No fixed threshold, usually combined with roll angle & Reflects roll tendency, but requires roll angle for rollover assessment & [84, 85] \\ \cline{1-1} \multirow{4}{*}{Roll rate (\(\dot{\varphi}\))} & Lateral acceleration of vehicle, indicating roll roller tendency & \(a_{y}=\frac{\varphi^{2}}{K}\) & Typically, indicates roll roll roller & \multirow{4}{*}{[84, 85]} \\  & & & rollover risk & single for rollover assessment & \\ \hline \multirow{4}{*}{Lateral acceleration (a\({}_{y}\))} & Lateral acceleration of vehicle, indicating roll roller tendency & \(a_{y}=\frac{\varphi^{2}}{K}\) & Typically, \(a_{y}>0.4\) g indicates roll roller & \multirow{4}{*}{[68]} \\  & & & rollover risk & suspension considerations & \\ \cline{1-1} \multirow{4}{*}{Roll energy} & Kinetic and potential energy of vehicle roll motion & \(E_{r}=\frac{1}{2}I_{y}\dot{\varphi}^{2}+\) dependent, no fixed threshold & \multirow{4}{*}{[14, 86]} \\  & & & & \\ \cline{1-1} \cline{4-4}  & & & & \\ \cline{1-1} \cline{4-4}  & & & & \\ \cline{1-1} \cline{4-4}  & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of other rollover prediction indexes.