## 4 Test Example
### Robot Arm  
This case is derived from a point-to-point control problem of a three-degree-of-freedom robot arm [20], which contains six state variables \([\theta_{1},\theta_{2},\theta_{3},\omega_{1},\omega_{2},\omega_{3}]\) and three control variables \([\tau_{1},\tau_{2},\tau_{3}]\). The structural diagram of this robot arm is shown in Figure 12. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Design Variable** & **Objective** & **Failure Probability** \\ \hline MDSDO & 0.0637 & \(-\)0.4461 & 49.67\% \\ C-MDSDO & 0.1293 & \(-\)0.9050 & 0.09\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The optimization result comparison for MDSDO and C-MDSDO. Figure 11: The comparison of control strategies for MDSDO and C-MDSDO. Figure 12: Structural diagram of the 3-DOF robot arm. In this paper, lengths of arms are treated as design variables \(\mathbf{d}=[L_{1},L_{2}]\) for reliability-based MDSDO. The reliability constraint requires that the probability of the error between the actual final position of the robot arm and the target position is no more than 0.05 under the influence of uncertainty, and the converted reliability index is \(\beta=3\). The mathematical expression of this optimization case is given in Formula (17). \[ \begin{array}{ll}\underset{d,u(t)}{min}&\phi{=}\;\mathrm{t}_{f}\\ \text{{subject to}}&\dot{x}=\mathrm{f}(d,x(t),u(t),t)\\ &x_{0}=[0,-1.5,0,0,0,0]\\ &x_{f_{MDSDO}}=[1,-1.95,1,0,0,0]\\ &prob\left[\left\|X-X_{obj}\right\|\geq 0.05\right]\leq\Phi(-\beta)\\ &\beta=3,\;\mathrm{L}_{1}\sim N\big{(}\mu_{L_{1}},0.006\big{)},\;\mathrm{L}_{2 }\sim N\big{(}\mu_{L_{1}},0.007\big{)}\\ &\mathbf{d}=[L_{1},L_{2}],\;\mathbf{x}=[\theta_{1},\theta_{2},\theta_{3}, \omega_{1},\omega_{2},\omega_{3}],\;\mathbf{u}=[\tau_{1},\tau_{2},\tau_{3}]\\ &d_{min}<\mathbf{d}\leq d_{max}\\ &x_{min}<\mathbf{x}\leq x_{max}\\ &u_{min}\leq\mathbf{u}\leq u_{max}\end{array}  \tag{17}\]  
where \(X\) represents the final position of the arm, and \(X_{obj}\) represents the objective position, which is the final position of the deterministic MDSDO solution. The boundary conditions of all the variables are listed in Table 5. In addition, the initial and the final conditions of the state and control variables are listed in the same table. Angle-related parameters were calculated using the radian system (rad). In addition, the arm was required to reach the target point within the specified time interval, and the time endpoint condition was \(t_{f}\in[0,0.5]\). In order to obtain the target position of the robot arm, it was necessary to convert the relative angular coordinates in Table 5 into the coordinates in the Cartesian coordinate system. The original arm length [0.5, 0.98] was substituted into the coordinate conversion (18) to obtain the target end position \(X_{t_{f}}\). Furthermore, during reliability-based MDSDO, the end position of the robot arm also needed to be converted and compared with \(X_{t_{f}}\), rather than directly using the relative angle \(x_{f_{MDSDO}}=[1,-1.95,1]\). \[\left\{ \begin{array}{ll}X_{t_{f}x}{=}\;L_{1}cos(\theta_{1})+L_{2}cos(\theta _{1}+\theta_{2})\\ \;X_{t_{f}y}{=}\;L_{1}sin(\theta_{1})+L_{2}sin(\theta_{1}+\theta_{2})\end{array} \right. \tag{18}\]  
The same state trajectory presentation method was used here to demonstrate the effect of uncertainty on the state trajectory of the robot arm in Figure 13. With the influence of uncertainty, the angle of the final state of the robot arm varied greatly. Moreover, the state variables \([\omega_{1},\omega_{2},\omega_{3}]\) represent the angular acceleration of the arm, which means that the arm was not stationary at the endpoint under nondeterministic conditions. When the end position was converted using Formula (18), approximately 30.8% of the MCS samples indicated that the end position error was greater than the threshold, which demonstrates the unreliability of the MDSDO solution and shows that reliability-based MDSDO is required.