## 2 Enhanced Dynamic Model  
In this section, we examine a dynamic model of a dual-axis manipulator, which can be expressed as follows:  
\[M(q)\ddot{q}+C\big{(}q,\dot{q}\big{)}\dot{q}+G(q)=U \tag{1}\]  
where \(M(\theta)\) refers to the inertia matrix, \(C\big{(}\theta,\dot{\theta}\big{)}\) is the centripetal and Coriolis force vector, \(G(\theta)\) is the gravity vector, \(U\) is control torque, and \(q\), \(\dot{q},\dot{q}\) respectively indicate the angle, angular velocity, and angular acceleration of the joint. We first consider friction [10], which can be divided into static friction, viscous friction, and Coulomb friction. In Figure 1, \(\mathcal{B}\), \(\mathcal{Q}_{C}\), and \(\mathcal{Q}_{s}\), respectively, describe the slope of viscous friction, static friction, and Coulomb friction. Note that when the velocity of the manipulator is increased, static friction decreases rapidly, such that viscous friction dominates. Thus, we consider only viscous friction and Coulomb friction in the current study, the formulas of which can be expressed as follow  
\[f\big{(}\dot{q}\big{)}=\dot{B}\dot{q}+\mathcal{Q}_{C} \tag{2}\]  
and  
\[\mathcal{Q}_{C}=\left\{ \begin{array}{ll}0&\dot{q}=0\\ \mathcal{Q}_{C}^{+}&\dot{q}>0\\ \mathcal{Q}_{C}^{-}&\dot{q}<0\end{array} \right. \tag{3}\]  
Thus, the dynamic model of the manipulator can be rewritten as:  
\[M(q)\ddot{q}+C\big{(}q,\dot{q}\big{)}\dot{q}+G(q)+f\big{(}\dot{q}\big{)}=U \tag{4}\]  
Next, we consider the external disturbance \(d(t)\) to which the manipulator is most susceptible in practical applications. There are numerous forms of external disturbance, and they can be very difficult to model; therefore, we used wh  
Figure 1: Relationship between the velocity of the manipulator and corresponding friction in the joint. bound to represent external disturbances acting on the two joints of the manipulator, which can be expressed as follows:  
\[|d(t)|\leq\overline{d}(t) \tag{5}\]  
where \(\overline{d}(t)\) is an unknown positive constant vector. Thus, the dynamic model of the manipulator can be rewritten as  
\[M(q)\ddot{q}+C\big{(}q,\dot{q}\big{)}\dot{q}+G(q)+f\big{(}\dot{q}\big{)}+d(t)=U \tag{6}\]  
We also considered the effects of payload properties on the manipulator (a common issue in practical applications). Consider the example in Figure 2, illustrating the process of grasping the payload, moving it to the destination, and putting it down. During this process, the payload is converted into forces and torques acting on the joints and links, such that the dynamic model of the manipulator can be rewritten as follows:  
\[M(q)\ddot{q}+C\big{(}q,\dot{q}\big{)}\dot{q}+G(q)+f\big{(}\dot{q}\big{)}+d(t)+U _{p}=U \tag{7}\]  
Next, we considered the uncertainties in the manipulator model associated with system parameters. The basic dynamic model outlined above includes three main parameters: inertia matrix \(M(q)\), the centripetal force/Coriolis force matrix \(C\big{(}q,\dot{q}\big{)}\) and the gravity vector \(G(q)\). In the process of establishing a dynamic model, these parameters are generally derived empirically. Thus, we decompose the system parameters into two classes (known items and model uncertainties), which can be expressed as follows:  
\[M(q)=M_{0}(q)+\Delta M(q) \tag{8}\]  
\[C\big{(}q,\dot{q}\big{)}=C_{0}\big{(}q,\dot{q}\big{)}+\Delta C\big{(}q,\dot{q }\big{)} \tag{9}\]  
\[G(q)=G_{0}(q)+\Delta G(q) \tag{10}\]  
Thus, the dynamic model of the manipulator can be expressed as follows:  
\[M_{0}(q)\ddot{q}+C_{0}\big{(}q,\dot{q}\big{)}\dot{q}+G_{0}(q)+ \Delta M(q)\ddot{q}+\Delta C\big{(}q,\dot{q}\big{)}\dot{q} \tag{11}\] \[+\Delta G(q)+f\big{(}\dot{q}\big{)}+d(t)+U_{p}=U\]  
Finally, we consider backlash [11, 12], which refers to the combined time delay effects of occlusions among gears, chains, and belts used in the transmission of power from the motor to other components (see Figure 3). This phenomenon can be described using a class of nonlinear equations as follows [13]:  
\[V(U)=\left\{