## 4 Example: Kinematic and Dynamic Modelling of a Real Robot
Figure 8 shows the curves of \(\mathbf{E}(t)=\left[x_{E}(t)\ y_{E}(t)\ z_{E}(t)\right]^{T}\). 2. Take the data \(\mathbf{E}(t)=\left[x_{E}(t)y_{E}(t)z_{E}(t)\right]^{T}\) obtained in Step 1 as the input for the inverse kinematic analysis, and calculate the displacement of the joints \(q^{\prime}_{1}(t)\), \(q^{\prime}_{12}(t)\) and \(q^{\prime}_{22}(t)\). The results are shown in Figure 9.  
It is clearly shown in Figure 9 that the curves \(q^{\prime}_{1}(t)\), \(q^{\prime}_{12}(t)\) and \(q^{\prime}_{22}(t)\) obtained in Step 2 match exactly the given data \(q_{1}(t)\), \(q_{12}(t)\) and \(q_{22}(t)\), respectively. This demonstrates and validates the proposed kinematic modelling method. _Dynamic Modelling and Analysis of the Robot_  
The dynamic equation of this robot can be derived directly by using the formulation Equation (17). In order to formulate \(\mathbf{\overline{M}}(\mathbf{s},t)\),\(\mathbf{\overline{C}}\big{(}\mathbf{s},\dot{\mathbf{s}},t\big{)}\),\(\mathbf{G}(\mathbf{s},t)\) and \(\tau_{q}(t)\), the matrices \(\mathbf{M}(\mathbf{s},t)\), \(\mathbf{C}\big{(}\mathbf{s},\dot{\mathbf{s}},t\big{)}\), \(\mathbf{g}(\mathbf{s},t)\) and \(\mathbf{R}\) must be computed. Figure 8: The trajectory of the end-effector. In this manner, \(\mathbf{M}(\mathbf{s},t)\) can be calculated with Equation (11), where the masses \(m_{j}\) are given in Table 2, and the inertias \(\mathbf{I}_{j}\) can be calculated based on the geometrical dimension of the links given in Table 2 as well. In Equation (11), the Jacobian matrices are calculated as \(\mathbf{J}_{T_{j}}=\frac{\partial\mathbf{r}_{Cj}}{\partial\mathbf{s}}\) and \(\mathbf{J}_{R_{j}}=\frac{\partial\mathbf{\omega}_{j}}{\partial\mathbf{s}}\), where the centre of mass \(\mathbf{r}_{Cj}\) and the angular velocity \(\mathbf{\omega}_{j}\) are calculated and presented in the Appendix A. Based on all the components of \(\mathbf{M}(\mathbf{s},t)\), the matrix \(\mathbf{C}\big{(}\mathbf{s},\dot{\mathbf{s}},t\big{)}\) is then calculated by using Cristoffel notation. The component \(\mathbf{g}(\mathbf{s},t)\) is calculated with Equation (13), where the total potential energy of the robot, \(\Pi\), is calculated and presented in the Appendix A also. The matrix \(\mathbf{R}\) is calculated with Equation (25), where the Jacobian matrices \(\mathbf{J}_{u}\) and \(\mathbf{J}_{z}\) are presented in the Appendix A, respectively. Finally, all the terms of Equation (17) can be formulated properly that can be used for the dynamic analysis and the control law computation. In the following example, we demonstrate an inverse dynamic solution \(\mathbf{\tau}_{q}=\left[ \begin{array}{cccc}\tau_{1}&F_{12}&F_{22}&\tau_{4}\\ \end{array} \right]^{T}\) that was calculated with respect to a input data given as follows: \(q_{1}(t)=\frac{\gamma\times t}{18}\), \(q_{12}(t)=0.1+0.01t\), \(q_{22}(t)=0.15+0.005t\) and \(q_{4}(t)=\frac{\gamma\times t}{36}\). The responses of \(\mathbf{\tau}_{q}=\left[ \begin{array}{cccc}\tau_{1}&F_{12}&F_{22}&\tau_{4}\\ \end{array} \right]^{T}\) are shown in Figure 10.  
It has shown that instead of the complex governing equations seen in Equation (10), the minimal form of the dynamic Equation (17) can be used directly for computation of the control (the inverse dynamic analysis) for the robot. In addition, analysing the dynamic model of a hybrid robot does not need to take into account the unknown Lagrang  
Figure 9: The displacement of the active joints.