## 2 Problem Formulation
### Actual Limitations  
To obtain the actual physical limitations of the mobile platform, some constraints on the input variables and output variables must be described by formulas before performing the trajectory tracking task. Therefore, we specifically express various constraints as follows:  
(1) Structure parameter limitations: The measurement error cannot be ignored when obtaining the radius of the driving wheels and the width of the mobile platform, which may be caused by the limitation of the measurement tool and the surveyor's experience. Therefore, the measurement error must be in a particular range, which can be expressed as follows:  
\[\left\{\begin{array}{c}\max\left\|Bc-\tilde{B}c\right\|<b_{Bc}\\ \max\left\|Rc-\tilde{R}c\right\|<b_{Bc}\end{array}\right.\Rightarrow\max\left\| B(q_{k})\right\|<b_{B} \tag{12}\]  
where \(b_{Bc},b_{Rc}\) is the maximum error of the actual estimated parameters, and \(b_{B}\) is the maximum error of the control object.  
(2) Signal disturbance limitations: The external environment is one of the key factors that cause other disturbances in the system. Therefore, the random disturbance of the state parameters and the output variable of the system are also within a specific constraint range, defined as follows:  
\[\left\{\begin{array}{c}\max\left\|\beta(k)\right\|<b_{\beta}\\ \max\left\|\alpha(k)\right\|<b_{\alpha}\end{array}\right.\Rightarrow\left\{ \begin{array}{c}\max\left\|\beta(k)+d_{k}(\tilde{B}(q_{k})-B(q_{k}))\cdot u (k)\right\|<\tilde{b}_{\beta}\\ \max\left\|\tilde{\alpha}_{i}(k)\right\|=\max\left\|\alpha(k)\right\|<\tilde {b}_{\alpha}\end{array}\right. \tag{13}\]  
where \(b_{\beta}\) is the maximum disturbance of the system state variable, and \(b_{\alpha}\) is the maximum disturbance of the output signal.  
(3) Motor physical limitations: Due to the limitations of the motor and the voltage of the driving wheels, the rotation speed of the driving wheels must have a maximum speed limitation, expressed as follows:  
\[\left\{\begin{array}{c}\max\left\|\dot{q}_{cl}(k)\right\|\leq b_{w}\\ \max\left\|\dot{q}_{cl}(k)\right\|\leq b_{w}\end{array}\Leftrightarrow\max \left\|u_{i}(k)\right\|\leq b_{u_{d}} \tag{14}\]  
where \(b_{w}\) is the maximum speed of the two driving wheels, and \(b_{u_{d}}\) is representing the maximum norm of the two-wheel drive speed matrix. The driving wheels were produced as one model to improve the control accuracy of the mobile platform.  
(4) Initial error limitations: During the trajectory tracking task, the initial position of the robot and the starting position of the given trajectory are kept within a certain error range, which can be expressed as follows  
\[\left\|q_{d}(0)-q_{k}(0)\right\|\leq C_{q_{0}} \tag{15}\]  
where, \(q_{k}(0)\) represents the initial position and posture of the robot in each iteration learning, \(q_{d}(0)\) represents the starting position and posture of a given trajectory, and \(C_{q_{0}}\) is the maximum norm of the initial error matrix of the system.  
(5) Kinematics limitations: In the robot state Equation (11), the B matrix needs to meet \(\max\|B(q_{k})\|<b_{B}\). Meanwhile, the function needs to satisfy the global Lipschitz condition  
\[\|d_{k}\cdot B(q_{i+1}(k))-d_{k}\cdot B(q_{i}(k))\|\leq C_{B}\|q_{i+1}(k)-q_{i}( k)\| \tag{16}\]  
where \(d_{k}\) is a constant that represents the sampling time, and \(C_{B}\) is the maximum value.