# 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
\begin{array}{l}Y_{0,3}=I_{1}\cdot\boldsymbol{\eta}_{1}\\ X_{0,3}=0.5I_{0,1}+I_{1}\cdot\boldsymbol{\eta}_{2}\\ \end{array} \right. \tag{10}\]  
\[\left\{ \begin{array}{l}Y_{0,5}=I_{1}\cdot\boldsymbol{\eta}_{1}+I_{2}\cdot \boldsymbol{\eta}_{3}\\ X_{0,5}=0.5I_{0,1}+I_{1}\cdot\boldsymbol{\eta}_{2}+I_{2}\cdot\boldsymbol{\eta }_{4}\\ \end{array} \right. \tag{11}\]  
where the expressions of \(I_{1}\), \(I_{2}\), \(\boldsymbol{\eta}_{1}\), \(\boldsymbol{\eta}_{2}\), \(\boldsymbol{\eta}_{3}\), and \(\boldsymbol{\eta}_{4}\) can be found in Appendix A. Since a functional relationship between the motor's rotational angle \(\varphi_{i}\) (\(i=1,2,3,4\)) exists, at the time \(t\), we can learn that the velocities at the center-of-mass positions of racks \(L_{0,3}\) and racks \(L_{0,5}\) can be expressed as  
\[\left\{ \begin{array}{l}v_{Y_{0,3}}=I_{1}\cdot\boldsymbol{Q}_{1}\\ v_{X_{0,3}}=I_{1}\cdot\boldsymbol{P}_{1}\\ \end{array} \right. \tag{12}\]  
\[\left\{ \begin{array}{l}v_{Y_{0,5}}=I_{1}\cdot\boldsymbol{Q}_{1}+I_{2}\cdot \boldsymbol{Q}_{2}\\ v_{X_{0,5}}=I_{1}\cdot\boldsymbol{P}_{1}+I_{2}\cdot\boldsymbol{P}_{2}\\ \end{array} \right. \tag{13}\]  
where the expressions of \(\boldsymbol{Q}_{1}\), \(\boldsymbol{Q}_{2}\), \(\boldsymbol{P}_{1}\), and \(\boldsymbol{P}_{2}\) can be found in Appendix A. The acceleration to the center of mass position of rack \(L_{0,3}\) and rack \(L_{0,5}\) at the same time can be expressed as  
\[\left\{ \begin{array}{l}a_{Y_{0,3}}=I_{1}\cdot(\boldsymbol{Q}_{3}+\boldsymbol {H}_{1})\\ a_{X_{0,3}}=I_{1}\cdot(\boldsymbol{P}_{3}+\boldsymbol{I}_{1})\\ \end{array} \right. \tag{14}\]  
\[\left\{ \begin{array}{l}a_{Y_{0,5}}=I_{1}\cdot(\boldsymbol{Q}_{3}+\boldsymbol {H}_{1})+\boldsymbol{I}_{2}\cdot(\boldsymbol{Q}_{4}+\boldsymbol{H}_{2})\\ a_{X_{0,5}}=I_{1}\cdot(\boldsymbol{P}_{3}+\boldsymbol{I}_{1})+I_{1}\cdot( \boldsymbol{P}_{4}+\boldsymbol{I}_{2})\\ \end{array} \right. \tag{15}\]  
where the expressions of \(\boldsymbol{Q}_{3}\), \(\boldsymbol{Q}_{4}\), \(\boldsymbol{P}_{3}\), \(\boldsymbol{P}_In our study, we conducted in-depth kinematic analysis to examine the mathematical relationship between the input angle \(\varphi_{i}\) (\(i\) = 1, 2, 3, 4) of the four drive motors of the tensioning mobile robot and the position, velocity, and acceleration of the center of mass of the key rods. The establishment of this mathematical model not only provides us with the possibility to accurately describe the robot attitude but also provides an important basis for the subsequent motion plan and control strategy formulation of the robot. Through this model, we are able to understand the motion characteristics of the robot under different input angles more deeply, predict its trajectory and velocity changes, and then optimize its performance and improve its work efficiency. On the basis of robot kinematic modeling, we further carried out the dynamic modeling of the tensegrity jumping robot. The construction of this model aims to deeply study the dynamic characteristic parameters such as kinetic energy and potential energy of the robot in the process of movement. From Equations (12) and (13), we can learn that the total kinetic energy \(E\) of the system during the jumping process is  
\[E=0.5{\sum\limits_{i=1}^{2}{{m_{0,2i+1}}}}\left({{v_{{Y_{0,2i+1}}}^{2}}+{v_{{X _{0,2i+1}}}^{2}}}\right) \tag{17}\]  
The potential energy of the system, on the other hand, can be expressed as  
\[U={\sum\limits_{i=1}^{2}{{\left({{m_{0,2i+1}}gY_{0,2i+1}}+0.5k_{2i}\Delta l_{0,2i}^{2}}\right)}} \tag{18}\]  
where \(U\) represents the total potential energy of the system, \(k_{2}\) denotes the stiffness of the spring \(L_{0,2}\), and \(k_{4}\) denotes the stiffness of the spring \(L_{0,4}\). Thus, the tensegrity jumping robot Lagrangian quantity can be expressed as  
\[