## 2 System Description and Preliminaries  
### Physical Model of Flexible Robotic Arm  
In this paper, a single link manipulator driven by \(DC\) motor with flexible joint is used as the controlled object (as shown in Figure 1a). The dynamic characteristics of a flexible joint can be approximated by a linear torsion spring with an elastic coefficient \(k\). Inspired by [27], the structure is shown in Figure 1b. The ideal dynamic equations of the flexible robotic arm systems are given by  
\[I\dot{q}(t)+MgH\sin q(t) = k(\theta(t)-q(t))+\mu_{1}(t) \tag{1}\] \[J\ddot{\theta}(t)+k(\theta(t)-q(t)) = u_{0}(t)+\mu_{2}(t)\]  
where \(\theta(t)\) is the rotation angle, \(J\) is the equivalent moment, \(\mu_{1}(t)\) represents external disturbance of motor; \(q(t)\), \(I\), \(\mu_{2}(t)\) represent the same parameters of the connecting rod. \(k\) represents the equivalent spring elastic coefficient; \(u_{0}(t)\) is the input torque of the motor; \(M\) is the weight of the connecting rod; \(H\) is the center of mass and the distance between the axis of rotation, the acceleration of gravity is represented by \(g\). Figure 1: Model Overview. (**a**) Physical model of flexible robotic arm; (**b**) Dynamical model of flexible joint. Let us define the following state variables:  
\[x_{1}(t)=q(t),\ x_{2}(t)=\dot{q}(t),\ x_{3}(t)=\theta(t),\ x_{4}(t)=\dot{\theta}(t) \tag{2}\]  
where \(x_{1}(t)\) denotes the rotation angle of the DC motor, \(x_{2}(t)\) is the angular velocity of the motor, \(x_{3}(t)\) is the rotation angle of connecting rod, and \(x_{4}(t)\) denotes the angular velocity of the connecting rod. **Assumption 1**. : _The nonlinear term \(\sin q(t)\) is presented in the dynamic equation mentioned above. We make the assumption that \(\sin q(t)=q(t)\), \(q(t)\in(-\frac{\pi}{2},\frac{\pi}{2})\) to remove the nonlinear term's influence. In other words, the connecting rod's accessible range is partially constrained._  
It is well known that the moment of inertia depends on: (a) the shape of the object, (b) the position of the axis of rotation, and (c) the distribution of mass. When grasping objects of different masses, where (a,b) remain unchanged and (c) changes with the end-effector grasping different objects, the moment of inertia of the connecting rod has different modes in Figure 2. A flexible joint manipulator system is a complex system with continuous time and discrete state. Take into consideration the subsequent continuous-time stochastic systems over a probability space (\(\hat{\lambda},\mathcal{F},\mathcal{P}_{r}\)) with Semi-Markov properties, add the dynamical equation as shown below. \[\left\{ \begin{array}{ll}\dot{x}(t)=\hat{A}(t,\hat{r}(t))x(t)+\hat{B}_{1}(\hat {r}(t))\omega(t)+B_{2}u(t),\\ z(t)=Cx(t),\end{array} \right. \tag{3}\]  
where \(x(t)=[x_{1}(t)\ \ x_{2}(t)\ \ x_{3}(t)\ \ x_{4}(t)]^{T}\), \(\omega(t)\) means the external disturbance and \(\omega^{T}(t)\omega(t)\)\(\leq I\), \(u(t)\) represents the input signal of the closed-loop system, and  
Figure 2: Moment of inertia under different working conditions. (**a**) \(I_{1}\) of State 1; (**b**) \(I_{1}\) of State 2; (**c**) \(I_{1}\) of State 3; (**d**) \(I_{1}\) of State 4. \[\hat{A}(t,\hat{r}(t))= \begin{bmatrix}0&1&0&0\\ -\frac{M_{\mathcal{G}}H+k}{I(t,\hat{r}(t))}&0&\frac{k}{I(t,\hat{r}(t))}&0\\ 0&0&0&1\\ \frac{k}{\hat{r}}&0&-\frac{k}{\hat{r}}&0\end{bmatrix} ,\quad\hat{b}_{1}(t,\hat{ r}(t))= \begin{bmatrix}0\\ 1\\ 0\\ 1\end{bmatrix} ,\quad B_{2}= \begin{bmatrix}0\\ 0\\ 0\\ 1\end{bmatrix} ,\]  
and \(C=[1\;0\;0\;0]\). \(\hat{r}(t)\) represents a finite state semi-Markov jump process, which accepts discrete values inside the specified finite set \(\{1,2,\cdots,m+1\}\), the state \(m+1\) is absorbing and other states are transient. Abbreviated symbols, \(\hat{A}_{i}\) for \(\hat{A}(t,r(t))\), \(\hat{b}_{1i}\) for \(\hat{B}_{1}(t,r(t))\). The infinitesimal generator is  
\[\Xi\;=\;\left[ \begin{array}{cc}\mathbf{T}_{m\times m}&\mathbf{T}_{m\times 1 }^{0}\\ \mathbf{T}_{1\times m}&0\end{array}