# Design and Experimentation of Tensegrity Jumping Robots
## 3 Dynamic Modeling and Simulations Analysis  
### Dynamic Modeling  
#### 3.1.1 Dynamic Modeling of Two Modules' Simultaneous Jump  
Since our tensegrity jumping robot has a symmetrical layout, we can simplify the configuration diagram of our robot as shown in Figure 3. The dimensions of the mechanism of the tensegrity jumping robot remain unchanged after the equivalence. At the same time, the mass and the moment of inertia of the mechanism after the equivalence show a superposition property, i.e., their values are equal to the sum of the corresponding parameters before the equivalence. We establish the plane rectangular coordinate system \(O\)-\(XY\) in Figure 3 and set the midpoint of the rack \(A_{1,1}A_{2,1}\) as the origin. The X-axis is horizontally to the right and the Y-axis is vertically up. The X-axis is horizontally to the right and the Y-axis is vertically up. Given that the main masses of the tensegrity jumping robot are concentrated on the geometric centers of the racks \(L_{01}(A_{1,1}A_{2,1})\), \(L_{03}(A_{1,3}A_{2,3})\), and \(L_{0,5}(A_{1,5}A_{2,5})\), we focus the masses mainly on these three key components to simplify the model and to highlight the main influencing factors during the kinematic modeling. In addition, considering that the friction between the sliding rope and each node has a relatively small effect on the overall motion, we chose to ignore this factor during the modeling process to ensure the accuracy and computational efficiency of the model. It is worth noting that due to the uniqueness of the structural design of the tensegrity jumping robot, the magnitude of the axial force exerted on each portion of the broken ropes at nodes \(A_{1,2}\), \(A_{1,4}\), \(A_{2,2}\), and \(A_{2,4}\) remains the same. This property makes the lengths of the ropes on both sides of the above nodes equal, thus simplifying our analytical process. Thus, the equal lengths of the ropes on both sides of the above node can be expressed by us as  
\[s_{k,1}=s_{k,2}=s_{k,1}^{0}+\frac{\varphi_{k}R_{k}}{4},k=1\,\sim 4 \tag{1}\]  
Figure 2: Three-dimensional models. (**a**) Initial state. (**b**) Compressed state. (**c**) Step-up state. (**d**) Trigger device. where \(\mathbf{s}_{i,j}\) (\(i\) = 1-4 and \(j\) = 1-2) are the lengths between the critical points of the two phases, respectively, and \(s_{i,j}^{0}\) is the length of the sliding rope in the initial state. \(\varphi_{i}\) (\(i\) = 1-4) denotes the input angles of the four main drive motors and \(R_{i}\) (\(i\) = 1-4) denotes the radii of the four winding disks of the tensegrip  
Figure 3: Diagram of the simultaneous jump. As shown in Figure 3, the structure formed by the rods, springs, and four sections of ropes in the tensegrity is taken as the basic unit for kinematic analysis and the several closed loops contained in it can be expressed as follows  
\[\left\{ \begin{array}{l}I_{1,1}+I_{0,i}+s_{3-i,1}=\mathbf{0}\\ I_{1,2}+I_{0,i+1}+s_{3-i,2}=\mathbf{0}\\ I_{3,1}+I_{0,i+2}+s_{5-i,1}=\mathbf{0}\\ I_{3,2}+I_{0,i+3}+s_{5-i,2}=\mathbf{0}\\ \end{array} \right. (i=1,2) \tag{8}\]  
According to Formula (8) and the geometric relationship in the figure, we can obtain the angle of each rod and spring with the \(X\)-axis  
\[\left\{ \begin{array}{l}\theta_{i}=\arccos\frac{I_{1,2}^{2}+I_{0,i}^{2}-s_{3- i,1}^{2}}{2I_{1,1}I_{0,i}}-\theta_{i-1}^{\prime}\\ \theta_{i+2}=\arccos\frac{I_{1,2}^{2}+I_{0,i+1}^{2}-s_{3-i,2}^{2}}{2I_{0,2}I_{ 1,0}}-\theta_{i+1}^{\prime}\\ \theta_{i+4}=\arccos\frac{I_{1,2}^{2}+I_{0,i+2}^{2}-s_{3-i,1}^{2}}{2I_{3,1}I_{ 0,i}}-\theta_{i+3}^{\prime}\\ \theta_{i+6}=\arccos\frac{I_{2,2}^{2}+I_{0,i+3}^{2}-s_{3-i,2}^{2}}{2I_{3,2}I_{ 0,i}}-\theta_{i+5}^{\prime}\\ \end{array} \right. (i=1,2) \tag{9}\]  
where we let \(\theta_{1}^{\prime}=0\). From the geometric relation, we know that \(\theta_{i}=\theta_{i}^{\prime}\) (\(i=1\)-\(7\)). From this, we know that the center of mass positions \(X_{0,3}\), \(Y_{0,3}\), and \(X_{0,5}\), \(Y_{0,5}\) of rack \(L_{0,3}\), and rack \(L_{0,5}\), respectively, can be denoted as  
\[\left\{