# A Hybrid Soft Actuator Inspired by Grass-Spike: Design Approach, Dynamic Model, and Applications
## 3 Dynamic Models
### A Single HSA Model
From the geometric relationship, we can easily obtain the following:  
\[x = x_{1}+l\sin\phi+d\quad\Rightarrow\quad x_{1}=x-l\sin\phi-d \tag{9}\]  
Accordingly,  
\[x_{i} = x_{1}+b\cdot i = (x-l\sin\phi-d)+b\cdot i\quad\Rightarrow\quad\delta x_{i}=\delta x -l(\cos\phi)\delta\phi \tag{10}\]  
where \(d\) is the distance from the center of mass of the HSA \(x\) to the root of the first spike and \(b\) is the gap between the spikes. Substituting Equation (10) into Equation (8) yields:  
\[ \begin{array}{rcl}\delta W&=&Q_{x}\delta x+Q_{\phi}\delta\phi\\ &=&F_{1}\delta x_{1}+F_{2}\delta x_{2}+\cdots+F_{n}\delta x_{n}\\ &=&F_{1}(\delta x-l(\cos\phi)\delta\phi)+F_{2}(\delta x-l(\cos\phi)\delta\phi) +\cdots+F_{n}(\delta x-l(\cos\phi)\delta\phi)\end{array}  \tag{11}\]  
Since the external force only depends on \(x\) and \(\phi\), we have:  
\[Q_{x} = \frac{\partial(\delta W)}{\partial(\delta x)}\ =\ F_{1}+F_{2}+\cdots+F_{n} \tag{12}\] \[Q_{\phi} = \frac{\partial(\delta W)}{\partial(\delta\phi)}\ =\ -l(F_{1}+F_{2}+\cdots+F_{n})\cos\phi \tag{13}\]  
Substituting Equations (5), (6), (12), and (13) into Equation (7) yields:  
\[(m_{0}+m_{1})\ddot{x}+m_{1}\ddot{\xi}=F_{1}+F_{2}+\cdots+F_{n}, \quad\mbox{for $q_{i}=x$} \tag{14}\] \[(m_{0}+m_{2})l^{2}\ddot{\phi}\sin\phi-(m_{0}+m_{2})l^{2}\dot{\phi }^{2}\sin\phi\cos\phi+nc(\phi-\phi_{0})-m_{2}l\eta\sin\phi-(m_{0}+m_{1}+m_{2}) gl\sin\phi\] \[=-l(F_{1}+F_{2}+\cdots+F_{n})\cos\phi, \quad\mbox{for $q_{i}=\phi$} \tag{15}\]  
The friction force \(F_{i}\) can be calculated as:  
\[F_{i}=-\mu N_{i}\tanh(\dot{x}-l\dot{\phi}\cos\phi) \tag{16}\]  
where \(N_{i}\) and \(\mu\) are the normal force and the friction coefficient, respectively. From Figure 3, \(N_{i}\) can be obtained by considering the motion of the center of mass of the HSA; thus:  
\[(m_{0}+m_{2})\ddot{y}=N_{1}+N_{2}+\cdots+N_{n}-(m_{0}+m_{2})g \tag{17}\]  
and \(\ddot{y}\) can be easily calculated from Equation (4). We finally have:  
\[N_{1}+N_{2}+\cdots+N_{n}=(m_{0}+m_{2})(g-\ddot{\phi}lsin\phi-\dot{\phi}lcos \phi)+m_{2}\ddot{\eta} \tag{18}\]  
The friction coefficient \(\mu\) is anisotropic; therefore, it can vary as follows:  
\[\mu=\left\{ \begin{array}{ll}\mu_{+},&\mbox{if $\dot{x}-\dot{\phi}l\cos\phi<0$} \\ \mu_{-},&\mbox{if $\dot{x}-\dot{\phi}l\cos\phi>0$}\end{array} \right. \tag{19}\]  
In Table 1, most of the parameters can be determined by practical measurement; however, the friction coefficient will be identified later on by the system identification process based on the empirical data. The details of the system identification process will be discussed in Section 4.2. Finally, by substituting Equations (18) and (19) into Equations (14) and (15) and rearranging for state variable \([\phi,\dot{\phi},x,\dot{x}]=[\phi,\omega,x,v]\), we obtain the equation of motion for the HSA. For the numerical simulation, we can express it as a discrete time version with a sampling time \(\Delta t\); consequently,\[\phi_{k+1} = \phi_{k}+\Delta t\cdot\omega_{k} \tag{20}\] \[\omega_{k+1} = (\omega+\Delta t((m_{0}+m_{2})l^{2}\cos\phi_{k}(\sin\phi_{k}-\mu \cos\phi_{k}\cdot\tanh(v_{k}-l\omega_{k}\cos\phi_{k}))\omega_{k}^{2}\] (21) \[-gl((m_{0}+m_{1}+m_{2})\cdot\sin\phi_{k}-\mu(m_{0}+m_{2})\cos\phi _{k})\tanh(v_{k}-l\omega_{k}\cos\phi_{k})\] \[+nc(\phi_{k}-\phi_{0})+m_{2}l(\cos\phi_{k}\cdot\tanh(v_{k}-l\omega _{k}\cos\phi_{k})-\sin\phi_{k})\tilde{\eta}_{k}))(-(m_{0}\] \[+m_{2})l^{2}\sin\phi_{k}(\sin\phi_{k}-\mu\cos\phi_{k}\tanh(v_{k} -l\omega_{k}\cos\phi_{k})))\] \[x_{k+1} = x_{k}+\Delta t\cdot v_{k}\] (22) \[v_{k+1} = (v_{k}+\Delta t(m_{1}\tilde{\xi}+\mu((m_{0}+m_{2}))(g-l(\omega_{ k+1}-\omega_{k})cos\phi_{k}-l\cos\phi_{k}\omega_{k}^{2})\] (23) \[+m_{2}\tilde{\eta}_{k})\cdot\tanh(v_{k}-l\cos\phi_{k}\omega_{k} )))/(-(m_{0}+m_{1}))\]  
By assuming that \(\Omega\) is an input to the HSA to generate a harmonic oscillation \(\eta(t)=A\sin(\Omega t)\) and \(\xi=0\), Equations (20)-(23) become system the equation yielding \(\phi,\omega,x,v\) as the outputs.