# Pace Running of a Quadruped Robot Driven by Pneumatic Muscle Actuators: An Experimental Study
## Appendix A Design of Two-Level CPG Model  
We describe our previously proposed two-level CPG model in this section. For more detailed information, refer to our previous studies [16, 18].  
In Figure 5, the membrane potential of the main neurons in the RG and PF parts and the six motoneurons is expressed as  
\[C\cdot\frac{dV}{dt}=-I_{\text{NaP}}-I_{\text{K}}-I_{\text{Leak}}-I_{\text{ SynE}}-I_{\text{SynI}} \tag{1}\]  
and that of the interneurons is expressed as  
\[C\cdot\frac{dV}{dt}=-I_{\text{Leak}}-I_{\text{SynE}}-I_{\text{SynI}} \tag{2}\]  
where \(C\) is a neuronal capacitance, \(V\) is the average membrane voltage of the neuron, \(I_{\text{NaP}}\), \(I_{\text{K}}\), and \(I_{\text{Leak}}\) are persistent sodium, potassium rectifier, and leakage currents, respectively. \(I_{\text{SynE}}\) and \(I_{\text{SynI}}\) are excitatory and inhibitory synaptic currents. The currents are expressed as  
\[\begin{split} I_{\text{NaP}}=\overline{g}_{\text{NaP}}\cdot m_{ \text{NaP}}\cdot h_{\text{NaP}}\cdot(V-E_{\text{Na}})\\ I_{\text{K}}=\overline{g}_{\text{K}}\cdot m_{\text{K}}^{\star} \cdot(V-E_{\text{K}})\\ I_{\text{Leak}}=\overline{g}_{\text{Leak}}\cdot(V-E_{\text{ Leak}})\\ I_{\text{SynE},i}=\overline{g}_{\text{SynE}}\cdot\big{(}V_{i}-E_{ \text{SynE}}\big{)}\cdot\Big{(}\sum_{j}a_{ji}\cdot f\big{(}V_{j}\big{)}+c_{i} \cdot d\Big{)}\\ I_{\text{SynI},i}=\overline{g}_{\text{SynI}}\cdot\big{(}V_{i}-E_{ \text{SynI}}\big{)}\cdot\Big{(}\sum_{j}b_{ji}\cdot f\big{(}V_{j}\big{)}+w_{i} \cdot feedback+\sum_{k}\alpha_{ki}\cdot f(V_{k})+\sum_{m}\beta_{mi}\cdot f(V_{ m})\Big{)}\end{split} \tag{3}\]  
where \(\overline{g}_{\{\text{NaP, K, Leak, SynE, SynI}\}}\) and \(E_{\{\text{Na, K, Leak, SynE, SynI}\}}\) are the maximal conductances of the ionic channels and the reversal potentials. \(a_{ji}\) and \(b_{ji}\) are the gains of the excitatory and inhibitory synaptic input from neuron \(j\) to neuron \(i\), and \(c_{i}\) is the gain of the tonic descending signal \(d\) to neuron \(i\). \(w_{i}\) is a feedback gain. \(\sum_{k}\alpha_{ki}\cdot f(V_{k})\) represents the mutual inhibitory connections of the RG neurons between the contralateral legs, as shown in Figure 5. \(\sum_{m}\beta_{mi}\cdot f(V_{m})\) represents the mutual excitatory connections of the RG neurons between the ipsilateral legs. The variables \(m_{\text{K}}\), \(m_{\text{NaP}}\), and \(h_{\text{NaP}}\) are described in our previouspaper [15]. The average population activity \(f\big{(}V_{j}\big{)}\) from neuron \(j\) is expressed by a sigmoid function as  
\[f(V)=\begin{cases}1/(1+\exp(-(V-V_{1/2})/k))&(\text{if $V\geq V_{\text{th}}$})\\ 0&(\text{otherwise})\end{cases} \tag{10}\]  
where \(V_{1/2}\) is a half-activation voltage, \(k\) is a sigmoid gain, and \(V_{\text{th}}\) is the threshold of each neuron.  
Leg loading feedback is input to \(feedback\) only for the RG-F neuron through a sigmoid function as  
\[feedback=\left\{\begin{array}{c}1/\Big{(}1+\exp\Big{(}-\big{(}load_{leg}-5 \big{)}\cdot k_{f}\Big{)}\Big{)}\\ 0\qquad\qquad\qquad(otherwise)\end{array}\right. \tag{11}\]  
where \(load_{foot}\) is the value of the force sensor attached to the tip of each leg to detect the footfall (if \(load_{foot}\geqq 5\ [\text{N}]\), the leg is in the stance phase), \(load_{leg}\) is leg load, which is estimated from the length of the two-joint muscle actuator for the shoulder and elbow joints or the knee and ankle joints, and \(k_{f}\) is a sigmoid gain.