## 2 Problem Formulation  
In a global \(XOY\) coordinate system, the schematic diagram of the motion of a TMR is presented in Figure 1. Some notations mentioned in Figure 1 are listed in Table 1. Considering the skidding case, the TMR satisfies the following constraint [32]:  
\[A(q)\dot{q}=\xi, \tag{1}\]  
where \(q=\left[ \begin{array}{cc}x&y&\theta\end{array} \right]^{T}\) denotes the position and orientation of the TMR, \(A(q)=\left[ \begin{array}{cc}-\sin(\theta)&-\cos(\theta)&0\end{array} \right]\) is a vector of nonholonomic constraints, \(\xi\) is the lateral skidding velocity. The TMR kinematic model subject to skidding disturbance can be expressed as  
\[\dot{q}=J(q)z+\varphi(q,\xi), \tag{2}\]  
where \(\varphi(q,\xi)=\left[ \begin{array}{cc}\rho_{1}&\rho_{2}\end{array} \right]^{T}\) is the vector of the disturbance caused by the skidding velocity, with \(\rho_{1}=-\xi\sin(\theta),\rho_{1}=-\xi\sin(\theta),J(q)=\left[ \begin{array}[] {cc}\cos(\theta)&\sin(\theta)&0\\ 0&0&1\end{array} \right]^{T}\), \(z=\left[ \begin{array}{cc}v&\omega\end{array} \right]^{T}\), with \(v,\omega\) denoting the linear and the angular velocities, respectively. **Assumption 1**. : _The perturbation \(\varphi(q,\xi)\) is bounded because of \(\left\|[-\sin(\theta),\cos(\theta)]^{T}\right\|=1\), where \(\left\|\cdot\right\|\) is the Euclidean norm of the vector. Its first derivatives is also bounded._ \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Meaning** \\ \hline \((X,O,Y)\) & The global coordinate system \\ \((X_{1},O_{1},Y_{1})\) & The coordinate system attached to the TMR \\ \(q(x,y,\theta)\) & The actual position \\ \(q(x_{d},y_{d},\theta_{d})\) & The desired position \\ \((x_{e},y_{e},\theta_{e})\) & The tracking error \\ \(v_{c}\) & The linear velocity generated by the kinematic controller \\ \(\omega_{c}\) & The angular velocity generated by the kinematic controller \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations in Figure 1.  
Figure 1: Schematic diagram of TMR motion. The dynamic model of the TMR can be described by the following equation:  
\[M(q)\ddot{q}+C(q,\dot{q})\dot{q}+G(q)=B(q)\tau+A^{T}(q)\kappa+d. \tag{3}\]  
where \(M(q)\) denotes a symmetric and positive defined inertia matrix; \(C(q,\dot{q})\) is a Coriolis-centripetal matrix; \(G(q)\) represents a gravity vector; \(B(q)\) is an input transformation matrix; \(\tau=\left[ \begin{array}{cc}\tau_{1}&\tau_{r}\end{array} \right]^{T}\) are the control inputs, with \(\tau_{l}\) and \(\tau_{r}\) denoting torques of the left and right sides, respectively; \(\kappa=-m(\dot{x}\cos(\theta)+\dot{y}\sin(\theta))\dot{\theta}\) represents the vector of the Lagrange multiplier; \(d\) is the bounded external environmental disturbance. The matrices \(M(q)\), \(B(q)\) above are defined as follows:  
\[M(q)=\left[ \begin{array}{ccc}m&0&0\\ 0&m&0\\ 0&0&I\end{array} \right],B(q)=\frac{1}{r}\left[ \begin{array}{cc}\cos(\theta )&\sin(\theta)\\ \sin(\theta)&\sin(\theta)\\ b&-b\end{array} \right],\]  
where \(m\) is the overall mass of the TMR, \(I\) is the inertia, \(r\) represents the radius of the wheel and \(h\) represents half of the distance between the track wheels. The differential form of (2) is shown below:  
\[\ddot{q}=\dot{J}(q)z+J(q)\dot{z}+\dot{\varphi}(q,\tilde{\varsigma}). \tag{4}\]  
Notice that \(A(q)J(q)=0\), and it is assumed that the distance between the center of the TMR form and its center of mass is zero, so the effect of \(C(q,\dot{q})\) can be eliminated from (3). By multiplying both sides by \(J^{T}(q)\) simultaneously, with (4) being substituted into Equation (3), one can obtain  
\[\tilde{M}\dot{z}=\bar{B}\tau+\bar{d}, \tag{5}\]  
where \(\bar{M}=J^{T}MJ\), \(\bar{B}=J^{T}B\), \(\bar{d}=J^{T}[d-M\dot{\varphi}-M\dot{J}z-G]=\left[ \begin{array}{cc}\rho_{3} &\rho_{4}\end{array} \right]^{T}\). Evidently, the term \(\bar{d}\) contains the information of the unmeasured velocities and other external disturbances. Therefore, it is considered to be the lumped disturbance in the model. **Assumption 2**.