*2.2. Tracking Error Dynamics Model*

To obtain the tracking error between the vehicle and the reference trajectory, we define the variable of tracking error:

$$q\_e = q - q\_r \tag{2}$$

where *q* denotes the position and orientation of the ATVs, and *qr* indicates the position and direction of the reference point on the desired path. Both *q* and *qr* are expressed in the earth-fixed frame. The tracking error *qe* should, however, be expressed in the vehicle-fixed frame to benefit from the computation of the kinematic controller. As a result, we use an orthogonal rotation matrix to translate the vehicle motion from the earth-fixed frame to a vehicle-fixed frame. Based on the kinematic parameter of ATVs, the transformation can be expressed as follows:

$$q\_{\mathcal{C}} = \begin{bmatrix} \varepsilon\_{\mathcal{X}} \\ \varepsilon\_{\mathcal{Y}} \\ \varepsilon\_{\theta} \\ \varepsilon\_{\psi} \end{bmatrix} = \begin{bmatrix} \cos \theta\_{r} & \sin \theta\_{r} & 0 & 0 \\ -\sin \theta\_{r} & \cos \theta\_{r} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \chi - \omega\_{r} \\ \mathcal{y} - \mathcal{y}\_{r} \\ \theta - \theta\_{r} \\ \psi - \psi\_{r} \end{bmatrix} \tag{3}$$

where *ex* and *ey* denote the position distance deviation projecting on the longitudinal and lateral directions, respectively. *e<sup>θ</sup>* and *e<sup>ψ</sup>* denote the orientation error of two units, respectively. This work presents a kinematic controller to propel the ATVs to follow a predefined or planned trajectory while maintaining the vehicle states within physical limits. In terms of the position deviation and orientation deviation, we propose the following differential equations as follows:

$$\begin{cases} \dot{e}\_{\lambda} = \dot{\theta}e\_{\theta} + \upsilon \cos e\_{\theta} - q\_{r} \\ \dot{e}\_{y} = -\theta e\_{\lambda} - \upsilon \sin e\_{\theta} \\ \dot{e}\_{\theta} = \dot{\theta} - \dot{\theta}\_{r} \\ \dot{e}\_{\psi} = \dot{\psi} - \dot{\psi}\_{r} \end{cases} \tag{4}$$
