## 4 Simulation Results
### The Mantuce R3 Model 2  
Mantuce R3 is an industrial robot with 6 degrees of freedom; it is said that the first 3 d.o.f are responsible for the position and the last 3 d.o.f for the orientation of the tool center point frame [27]. For simplicity, the Mantucc R3 model 2 (R3M2) focused on the first 3 d.o.f case. The relative physical parameters are displayed in [26], the robotic manipulator with no load is taken into consideration, and the terminal time \(t_{f}\) is set as a free variable to find the shortest working time. There are some conditions for this, as follows:  
The initial state conditions, which give the positions of the start point and the end point and angular velocities on both points, are:  
\[\mathbf{q}(0)= \begin{pmatrix}0.00\\ -1.50\\ 0.00\end{pmatrix} ,\mathbf{q}(t_{f})= \begin{pmatrix}1.00\\ -1.95\\ 1.00\end{pmatrix} , \tag{36}\]  
\[\dot{\mathbf{q}}(0)=\mathbf{0},\dot{\mathbf{q}}(t_{f})=\mathbf{0}. \tag{37}\]  
The boundary conditions give constant boundaries of the controls and the states. It is noted that the motors in the robot are modeled as ideal voltage-torque converters without any dynamic effects; in other words, there is a linear relationship between voltage and torque. The linear converter coefficient \(\mathbf{LC}\) can be found in [26]. The controls boundaries of voltage \(\mathbf{u}_{max}\), and the linear converter coefficient \(\mathbf{LC}\) are shown as  
\[u_{i,max}=7.5(\mathrm{V}),i=1,2,3. \tag{38}\]  
\[LC[3]=[-126,252,72]^{T}(\mathrm{N}\cdot\mathrm{m}/\mathrm{V}). \tag{39}\]  
The torques boundaries \(\mathbf{\tau}_{max}\) can be calculated by  
\[\tau_{i,max}=LC[i]\times u_{i,max}(\mathrm{N}\cdot\mathrm{m}),i=1,2,3. \tag{40}\]  
The states' boundary conditions including the boundaries of positions \(\mathbf{q}_{max}\) and angular velocities \(\dot{\mathbf{q}}_{max}\) are as follows:  
\[ \begin{cases}q_{1,max}=2.97(\mathrm{rad}),&\dot{q}_{1,max}=3.00(\mathrm{rad}/ \mathrm{s}),\\ q_{2,max}=2.01(\mathrm{rad}),&\dot{q}_{2,max}=1.50(\mathrm{rad}/\mathrm{s}),\\ q_{3,max}=2.86(\mathrm{rad}),&\dot{q}_{3,max}=5.20(\mathrm{rad}/\mathrm{s}), \end{cases}  \tag{41}\]  
The LPM method is initialized with 20 and 100 nodes, and the initialization of the state \(\widetilde{\mathbf{q}}_{n}\), \(\dot{\widetilde{\mathbf{q}}}_{n}\) on each node is set as:  
\[\widetilde{\mathbf{q}}_{n}=\mathbf{q}(0)+\frac{n-1}{N-1}(\mathbf{q}(t_{f})-\mathbf{q}(t_{0})), \tag{42}\]  
\[\dot{\widetilde{\mathbf{q}}}_{n}=\mathbf{q}(0)+\frac{n-1}{N-1}(\dot{\mathbf{q}}(t_{f})- \dot{\mathbf{q}}(t_{0})). \tag{43}\]  
The initial control \(\widetilde{\mathbf{u}}_{n}\) on each node is defined as zero,  
\[\widetilde{\mathbf{u}}_{n}(t)=\mathbf{0}. \tag{44}\]  
where \(n\) represents the sequence number of the node, \(N\) represents the number of all nodes, \(\widetilde{\mathbf{q}}_{n}(t)\) represents the initial angular position on the node \(n\), and \(\dot{\widetilde{\mathbf{q}}}_{n}(t)\) represents the initial angular velocity on the node \(n\). After setting the initialized conditions of the problem, the result solutions are shown in Figure 4 and Table 1. It should be noted that the REF model of R3M2 uses the analytic RHS equations, and the FMU model of R3M2 uses the black-box model of FMU to replace the analytic RHS equations. Figure 4: The solutions of the R3M2. Subgraphs (**a–c**) are the solutions using LPM method with 20 nodes; Subgraphs (**d–f**) are the solutions using LPM method with 100 nodes. Subgraphs (**a,d**) print the curve of the input voltage of joint actuators. Subgraph (**b,e**) print the curve of the joint positions. Subgraph (**c,f**) print the curve of the joint velocities. The red curves with triangles \((-\triangledown-)\) refer to the solutions of the improved method with FMU. The black curves with plus signs \((-+-)\) refer to the reference solution. Figure 4 displays the solution of the trajectory planning problem; the figures of desire actuator current, joint position and joint velocity are given in the time domain. A different number of time nodes are considered.