## 4 Theoretical Works  
The specified prototype of the wheeled vehicle is depicted in Figure 3. It belongs to the Differential-Driving-Wheel (DDW) type which is actively driven by two side wheels and set free for two front and back ones. Consider that \(2b\), \(2r\), \(\theta\), and \(\theta_{d}\) are the length between two wheels, diameter of wheel, differences between actual, desired heading angle and \(Ox\), respectively. When the mobile vehicle moves on planar ground, there is no variation in \(Oz\). Perfectly, the distribution of the robot's weight is uniform and its central mass is located in the middle of the active driven wheels. We assume that the vector of system state \(q^{T}=\left[ \begin{array}{cc}x&y&q\end{array} \right]\) represents location and driving angle, \(\delta^{T}=\left[ \begin{array}{cc}v&\omega\end{array} \right]\) defines the kinematic information. Figure 3: Modeling of wheeled autonomous robot. Figure 2: Experimental platform of AGR. The relationship amongst linear velocity and angular velocity with full rank matrix \(S(q)\) is expressed as  
\[\dot{q}=S(q)\vartheta=\left[ \begin{array}{cc}\cos\theta&0\\ \sin\theta&0\\ 0&1\end{array} \right]\left[ \begin{array}{c}v\\ \omega\end{array} \right] \tag{1}\]  
A nonholonomic mobile vehicle system having an n-dimensional configuration space C with generalized coordinates \((q1,\ldots,qn)\) can be described by:  
\[M(q)\ddot{q}+F\dot{(q)}+\tau_{d}=B(q)\tau-A^{T}(q)\lambda \tag{2}\]  
where  
\(\lambda\): Lagrange multiplier constraint  
\(F(\dot{q})\): friction factors  
\(M(q)\): symmetric and positive-definite inertia matrix  
\(A(q)\): matrix associated with the constraints  
\(B(q)\): input transformation matrix  
\(\tau_{d}\): disturbances torque  
\(\tau_{1},\tau_{2}\): driving torques of two wheels, \(\tau^{T}=\left[ \begin{array}{cc}\tau_{1}&\tau_{2}\end{array} \right]\)  
The Euler-Lagrange equations of motion are used to derive the dynamics of the mobile vehicle systems. The dynamical equations of the mobile vehicle system can be expressed as:  
\[M(q)=\left[ \begin{array}{ccc}m&0&0\\ 0&m&0\\ 0&0&J\end{array} \right],\;B(q)=\frac{1}{r}\left[ \begin{array}{cc}\cos \theta&\cos\theta\\ \sin\theta&\sin\theta\\ b&-b\end{array} \right]A^{T}(q)=\left[ \begin{array}{c}-\sin\theta\\ \cos\theta\\ 0\end{array} \right]\]  
where  
\(J\): inertia moment  
\(m\): total weight of system  
**Assumption 1:** The vehicle system has three-dimensional configuration space with generalized coordinate vector \(q\), subject to nonslipping and pure rolling. \[A(q)\dot{q}=0 \tag{3}\]  
**Assumption 2:**\(S\in\Re^{2\times 3}\) is a full rank matrix satisfying  
\[S^{T}(q)A^{T}(q)=0 \tag{4}\]  
**Assumption 3:** It is assumed that the disturbance vector is unknown but bounded. \[|\tau_{d}|\leq\partial \tag{5}\]  
Remember that the constraint imposed on \(\partial\) is to ensure that the level of uncertain factors is not so large that the stable state of system can be reversed. Taking derivative of Equation (1), we have  
\[\ddot{q}=S(q)\dot{\vartheta}+\dot{S}(q)\vartheta \tag{6}\]  
Substituting Equation (6) into Equation (2)  
\[M(q)\big{(}S(q)\dot{\vartheta}+\dot{S}(q)\vartheta\big{)}+F\dot{(q)}+\tau_{d} =B(q)\tau-A^{T}(q)\lambda \tag{7}\]Multiplied by \(S^{T}(q)\) in both right and left side  
\[S^{T}(q)M(q)\Big{(}S(q)\dot{\delta}+\dot{S}(q)\delta\Big{)}+S^{T}(q)\Big{(}F\! \left(\dot{q}\right)+\tau_{d}\Big{)}=S^{T}(q)B(q)\tau-S^{T}(q)A^{T}(q)\lambda \tag{8}\]  
From Assumption 2, it can be  
\[S^{T}(q)A^{T}(q)=0,\ S^{T}(q)M(q)\dot{S}(q)=0 \tag{9}\]  
The dynamic of mobile system becomes  
\[\overline{M}\dot{\delta}=\overline{B}\tau+d(t) \tag{10}\]  
with  
\[\overline{M}=\left[ \begin{array}{cc}m&0\\ 0&J\end{array} \right],\ \overline{B}=\frac{1}{r}\left[ \begin{array}{cc}1&1\\ b&-b\end{array} \right],\ d(t)=-S^{T}(q)\Big{(}F\!\left(\dot{q}\right)+\tau_{d} \Big{)}\]  
As a result, the system state variables are rewritten as  
\[\dot{q}=\left[ \begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\theta}\end{array} \right]=\left[