# A Review of Recent Advances in Roll Stability Control in On-Road and Off-Road Vehicles
## 3 Theoretical Foundations of Roll Stability Control  
### Roll Dynamic Model  
Research on vehicle roll stability control is highly dependent on precise dynamic modeling to characterize the vehicle roll behavior under different operating conditions [52; 53]. Developing an accurate and widely applicable full-vehicle dynamic  
Figure 2: Research trends in vehicle roll stability control: (**a**) annual number of publication articles from 2000 to 2024; (**b**) keyword co-occurrence network in RSC field. model not only aids in developing a deeper understanding of roll and roller mechanisms, but also provides a theoretical foundation for the design and validation of control strategies. Therefore, this section will focus on the process of constructing a vehicle dynamic model, presenting its basic equations, and reviewing the advancements in precise modeling in existing research. During vehicle operation, multiple forces act on the vehicle, including gravity, the tire-ground interaction force, aerodynamic forces, and the supporting forces from the suspension system. The interaction of these forces determines the vehicle roll response. Particularly under high-speed cornering, emergency evasive maneuvers, or complex road conditions, factors such as changes in the center of mass height, increased tire lateral forces, or suspension deformation can lead to significant roll, and may even cause the tries to lose traction, resulting in rollover [54; 55; 56]. Therefore, the primary goal of dynamic modeling is to accurately describe the motion states of the entire vehicle and characterize the coupling relationships between the longitudinal, lateral, yaw, and roll motions, while also considering key influencing factors, such as suspension characteristics, tire mechanics, and changes in the center of mass. This improves the model applicability to various operating conditions, and provides the theoretical basis for deriving and applying linear and nonlinear control methods, thus supporting effective roll stability control design [57; 58]. To meet these objectives, full-vehicle dynamic models typically adopt multi-degree-of-freedom (DOF) system modeling approaches, where the roll motion is primarily described by the rotational motion equation around the longitudinal axis of the vehicle. This is then integrated with the vehicle yaw, lateral, and longitudinal dynamics for a comprehensive analysis. The model complexity can be adjusted according to research needs, ranging from simplified models (e.g., 2-DOF roll models) to higher-dimensional models (e.g., 14-DOF vehicle models). Figure 3 presents a common vehicle roll dynamic model, with its specific equations derived as follows [14]:  
Equations of motion in the longitudinal plane:  
\[m\big{(}\dot{v}_{x}-v_{y}\omega_{z}\big{)}=(F_{\rm{df}}+F_{\rm{xrf}})\cos \delta_{\rm{f}}-\Big{(}F_{\rm{ylf}}+F_{\rm{yrf}}\Big{)}\sin\delta_{\rm{f}}+F_ {\rm{xlr}}+F_{\rm{xrr}} \tag{1}\]  
\[m\big{(}\dot{v}_{y}-v_{x}\omega_{z}\big{)}=(F_{\rm{xlf}}+F_{\rm{xrf}})\sin \delta_{f}+\Big{(}F_{\rm{ylf}}+F_{\rm{yrf}}\Big{)}\cos\delta_{\rm{f}}+F_{\rm{ ylr}}+F_{\rm{yrr}} \tag{2}\]  
\[ \begin{array}{rl}I_{z}\dot{\omega}_{z}=&{\rm{a}}\Big{[}(F_{\rm{xlf}}+F_{\rm{ xrf}})\sin\delta_{\rm{f}}+\Big{(}F_{\rm{ylf}}+F_{\rm{yrf}}\Big{)}\cos\delta_{f} \Big{]}-{\rm{b}}\Big{(}F_{\rm{ylr}}+F_{\rm{yrr}}\Big{)}\\ &+0.5B_{\rm{f}}\Big{[}(F_{\rm{xlf}}-F_{\rm{xrf}})\cos\delta_{\rm{f}}+\Big{(}F_ {\rm{ylf}}-F_{\rm{yrf}}\Big{)}\sin\delta_{\rm{f}}\Big{]}+0.5B_{\rm{r}}(F_{\rm {xlr}}-F_{\rm{xrr}})\end{array}  \tag{3}\]  
Roll motion equation:  
\[I_{x}\ddot{\varphi}=mh_{x}\big{(}\dot{v}_{y}+v_{x}\omega_{z}\big{)}\cos\varphi +mgh_{x}\sin\varphi-K_{\varphi}\varphi-C_{\varphi}\dot{\varphi} \tag{4}\]  
where \(v_{x}\) and \(v_{y}\) represent the longitudinal and lateral velocities of the vehicle, respectively. \(\omega_{z}\) is the yaw angular velocity. \(m\) denotes the vehicle mass. \(F_{\rm{xlf}}\), \(F_{\rm{xrf}}\), \(F_{\rm{xlr}}\), and \(F_{\rm{xrr}}\) are the longitudinal forces at the four wheels, respectively.