*2.2. Brush Tire Model*

The longitudinal and lateral tire forces in the bicycle model are computed using a brush tire model, which constrains the maximum amount of tire force (the combined longitudinal and lateral forces) within the elliptical circle in Figure 4. A tire force curve versus the tire slip angle is illustrated in Figure 4, where the red area indicates the saturated area and the blue area denotes the unsaturated area. The brush tire model was employed using Equation (2).

Under normal driving conditions, the combined force acting on a tire remains within the elliptic region and the tire model remains in the unsaturated state. Conversely, when the magnitude of the combined force acting on the tire reaches the elliptic circle, the tire model moves to the saturated state and a large amount of slip occurs. This situation is dangerous because the vehicle can lock its wheels or skid, which increases the difficulty of controlling the vehicle.

$$F = \begin{cases} \gamma - \frac{1}{3\mu F\_z} \gamma^2 + \frac{1}{2\mu F\_x^2} \gamma^3, & \gamma \le 3\mu F\_z \\\\ \mu\_x F\_{zr} & \gamma > 3\mu F\_z \end{cases}$$

$$F\_x = \frac{C\_x}{\gamma} \left(\frac{\mathbf{x}}{1+\mathbf{x}}\right) F\_r$$

$$F\_y = \frac{C\_y}{\gamma} \left(\frac{\mathbf{x}\,\mathbf{n}}{1+\mathbf{x}}\right) F\_r$$

$$\gamma = \sqrt{C\_x^2 \left(\frac{\mathbf{x}}{1+\mathbf{x}}\right)^2 - C\_x^2 \left(\frac{\mathbf{x}\,\mathbf{n}}{1+\mathbf{x}}\right)^2},\tag{2}$$

$$\alpha = \begin{cases} \alpha\_f = \arctan\left(\frac{v\_f + l\_f \ast r}{v\_x}\right) - \delta \approx \arctan\left(\beta + \frac{l\_f}{v\_x} \ast r\right) - \delta\\ \alpha\_I = \arctan\left(\frac{v\_y - l\_s \ast r}{v\_x}\right) \approx \tan\left(\beta - \frac{l\_s}{v\_x} \ast r\right) \end{cases},$$

$$\kappa = \frac{v\_x - v\_y}{v\_x}.$$

**Figure 4.** Saturation conditions of the brush tire model: (**a**) longitudinal and lateral forces and (**b**) slip VS tire force.

#### *2.3. Drift Equilibrium State Analysis*

The vehicle's trajectory was predicted using the bicycle model defined in Equation (1) with speed and steering angle as the control inputs. To analyze the motion and stability of the vehicle, the bicycle model was combined with the brush tire model under specific conditions (Equation (2)). When the tire slip angle remains within a specific range and the tire force is unsaturated, the vehicle's motion will remain stable. However, when the tire slip angle increases and the resulting tire force becomes saturated, the vehicle's motion will destabilize and even a slight disturbance will divert its states from equilibrium.

To maintain the drift maneuver, the vehicle must be controlled in an unstable equilibrium state. Especially on a slippery road, maintaining a drift maneuver requires a precise and agile controller.

In this study, the equilibrium states were established using Equations (1) and (2) when the time derivatives of the vehicle's states were all zero.

Figure 5 plots the *β*, *r*, and *vx* equilibrium points according to the steering angle at a longitudinal speed of 1.7 m/s. Plotted are the equilibrium states during a normal driving maneuver (\*) and during a drift maneuver (o, Δ) in the clockwise and counterclockwise directions.

**Figure 5.** Equilibrium states of (**a**) sideslip angle (*β*), (**b**) yaw rate (*r*), and (**c**) rear wheel force (*Fxr*) at a vehicle speed of 1.7 m/s.

#### **3. Design of the Nonlinear Model Predictive Controller**

#### *3.1. Vehicle State Prediction Model*

Based on the dynamics of the controlled system, the NMPC method predicts the future motions of a vehicle over a fixed time horizon. In this study, the future trajectory was predicted by discretizing the model of the vehicle's dynamics (Equation (1) in Section 2). The vehicle states (*X*) comprise the sideslip angle (*β*), yaw rate (*r*), and speed (*vx*) of the vehicle as follows:

$$X = [\beta, r, v\_x]. \tag{3}$$

The control input vector (*u*) comprises the rear-wheel speed (*vw*) and the steering angle (*δ*) of the vehicle.

$$
\mu = [v\_{w\prime} \delta]. \tag{4}
$$

In terms of the rear-wheel speed (*vw*), the rear-wheel tire force in Equation (1) is given by the following simplified tire force relation:

$$F\_{xr} = \frac{C\_{xr}(v\_w - v\_x)}{v\_x}.\tag{5}$$

#### *3.2. Nonlinear Model Predictive Controller Cost Function*

The cost function for the optimization process of the NMPC method is the error vector (*X<sup>e</sup> k*) between the current vehicle state vector (*Xk*) and the target state vector (*Xref <sup>k</sup>* ).

$$\begin{aligned} \mathbf{X}\_k^{\varepsilon} &= \mathbf{X}\_k^{ref} - \mathbf{X}\_k\\ = \begin{bmatrix} \boldsymbol{\beta}\_k^{ref} - \boldsymbol{\beta}\_{k\prime} \ \boldsymbol{r}\_k^{ref} - \boldsymbol{r}\_{k\prime} \ \boldsymbol{v}\_{\mathbf{x}\_k}^{ref} - \boldsymbol{v}\_{\mathbf{x}\_k} \end{bmatrix}. \end{aligned} \tag{6}$$

The cost function (*J*) is defined in terms of the state error vectors and the control inputs.

$$J = \frac{1}{2} \left( X\_{k+N}^{\varepsilon} \right)^{\mathrm{T}} \ast P \ast X\_{k+N}^{\varepsilon} + \frac{1}{2} \sum\_{j=k}^{k+N-1} \left( X\_{j}^{\varepsilon} \right)^{\mathrm{T}} \ast Q \ast X\_{j}^{\varepsilon} + u\_{j}^{\mathrm{T}} R u\_{j}. \tag{7}$$

Note that the cost function comprises a quadratic term of the final *N*th step error (*X<sup>e</sup> N*), the sum of the quadratic terms of errors (*X<sup>e</sup> <sup>k</sup>*), and the quadratic terms of the control input (*uk*) in future steps from *k* to *k+N–* 1, with weight matrices of *P, Q,* and *R*, respectively. The inputs that minimize the cost function given by Equation (7) are determined by numerical optimization based on a conjugate gradient method.
