# Tube-Based Robust Model Predictive Control for Autonomous Vehicle with Complex Road Scenarios
## 2 Problem Statement and Preliminaries
### Dynamic Vehicle Model  
The vehicle dynamics model is developed for model predictive control, prioritizing computational efficiency while retaining essential planar motion characteristics. Based on the prior knowledge of vehicle dynamics, a simplified three-degree-of-freedom vehicle model is developed under specific assumptions [31]. Key assumptions include: (1) planar motion constraints (longitudinal, lateral, yaw) under flat-road assumptions; (2) rigid-body suspension and chassis; (3) decoupled pure lateral tire slip properties; (4) single-track bicycle model eliminating load transfers; (5) quasi-static longitudinal dynamics; (6) neglected aerodynamic forces. These assumptions reduce the system to three degrees of freedom with front-wheel-drive actuation. The dynamic model of an autonomous vehicle is illustrated in Figure 1. Utilizing trigonometric relationships, the forces acting on the front and rear tires in the x- and y-directions can be expressed as follows:  
\[\left\{\begin{array}{l}F_{xf}=F_{lf}\cos\delta_{f}-F_{cf}\sin\delta_{f}\\ F_{xr}=F_{lr}\cos\delta_{r}-F_{cr}\sin\delta_{r}\\ F_{yf}=F_{lf}\sin\delta_{f}+F_{cf}\cos\delta_{f}\\ F_{yr}=F_{lr}\sin\delta_{r}+F_{cr}\cos\delta_{r}\end{array}\right., \tag{1}\]  
where \(F_{lf}\), \(F_{cf}\), and \(F_{cr}\) denote the longitudinal forces acting on the front and rear tires, respectively; \(F_{xf}\) and \(F_{yr}\) represent the combined forces acting on the front and rear tires, decomposed into longitudinal and lateral components; and \(\delta_{f}\) and \(\delta_{r}\) describe the steering angles of the front and rear tires.  
According to [PERSON]'s second law in Figure 2, the equilibrium equations of vehicle dynamics are obtained as follows:  
\[\left\{\begin{array}{l}m\ddot{y}=-m\dot{x}\dot{\varphi}+2F_{xf}+2F_{yf}\\ m\ddot{x}=m\dot{y}\dot{\varphi}+2F_{xf}+2F_{xr}\\ I_{z}\ddot{\varphi}=2\alpha F_{yf}-2 bF_{yr}\end{array}\right., \tag{2}\]  
where \(a\) and \(b\) denote the distances from the center of mass to the front and rear axles, respectively; \(m\) represents the mass of the autonomous vehicle; \(I_{z}\) corresponds to the moment of inertia; \(\varphi\) describes the heading angle; and \(\dot{x}\) and \(\dot{y}\) indicate the center-of-mass velocities along the x-axis and y-axis, respectively.  
Under the small-angle assumption derived from the Magic Tire model in [32], the lateral force is approximately linearly related to the sideslip angle of the tire, as shown below:  
\[\left\{\begin{array}{l}F_{l}\approx C_{l}S,F_{c}\approx C_{c}\alpha_{(f/r)} \\ \alpha_{f}=\frac{\dot{y}+a\dot{\varphi}}{\dot{x}}-\delta_{f},\alpha_{r}=\frac{ \dot{y}-b\dot{\varphi}}{\dot{x}}\end{array}\right., \tag{3}\]  
where \(C_{l}\) is the tire cornering stiffness, \(C_{c}\) refers to the lateral stiffness, \(s\) defines the slip ratio, and \(\alpha_{f}\) and \(\alpha_{r}\) are the front and rear wheel slip angles, respectively.  
Using (1)-(3), a nonlinear model of a dynamic vehicle system based on a small front wheel deflection angle and tire model assumption is obtained as follows:  
Figure 1: Three-degree-of-freedom vehicle model.  
\[\left\{\begin{array}{l}m\dot{v}_{y}=-mv_{x}\dot{\varphi}+2\left[C_{\rm cf}\left( \delta_{f}-\frac{v_{y}+a\dot{\varphi}}{v_{x}}\right)+C_{\rm cr}\frac{b\dot{ \varphi}-v_{y}}{v_{x}}\right]\\ m\dot{v}_{x}=mv_{y}\dot{\varphi}+2\left[C_{\rm f}s_{\rm f}+C_{\rm cf}\left( \delta_{f}-\frac{v_{y}+a\dot{\varphi}}{v_{x}}\right)\delta_{f}+C_{\rm lr}s_{\rm r }\right]\\ I_{z}\ddot{\varphi}=2\left[aC_{\rm cf}\left(\delta_{f}-\frac{v_{y}+a\dot{ \varphi}}{v_{x}}\right)-b\overline{C}_{cr}\frac{b\dot{\varphi}-v_{y}}{v_{x}} \right]\\ \dot{Y}=\dot{x}\sin\varphi+\dot{y}\cos\varphi\\ \dot{X}=\dot{x}\cos\varphi-\dot{y}\sin\varphi\end{array}\right., \tag{4}\]  
Combining the above equations, the state-space equation of nonlinear dynamics is written as  
\[\dot{\xi}_{dyn}=A\xi_{dyn}+Bu_{dyn}. \tag{5}\]  
with \(\left\{\begin{array}{l}\ddot{\xi}_{dyn}=\left[\dot{y},\dot{x},\varphi,\dot {\varphi},Y,X\right]^{T}.\\ u_{dyn}=\delta_{f}\end{array}\right.\)