# A Bio-Inspired Control Strategy for Locomotion of a Quadruped Robot
## 3 Improved CPG
### Improved CPG Control Model  
The CPG mechanism of creatures contains the coupling relationship of the time domain and the spatial domain at the same time. Similarly, the CPG control network is a distributed network system composed of multiple oscillator units through a certain topological relationship, which simulate the biological CPG mechanism commendably and is very suitable for multi-legged robot distributed control [10; 11]. Aiming at the control of a quadruped robot, four CPG oscillator units based on the modified Hopf oscillators are used to form a fully symmetric CPG control network, as shown in Figure 4.  
In Figure 4, the numbers 1, 2, 3 and 4 are the left front (LF), right front (RF), right hind (RH), and left hind (LH) legs of a quadruped robot, respectively. The dashed arrow points to the head  
Figure 4: Full symmetric central pattern generator (CPG) control network of a quadruped robot. while the solid arrows represent a certain coupling relationship among the CPG oscillator units. Considering the CPG-NN-WT-based control strategy (which will be introduced in detail in Section 4) presented in this paper, the CPG only needs to output 4 smooth control signals with a certain phase coupling relationship instead of controlling any joint, and its duty factor and gait period are adjustable. To realize the smooth gait planning of quadruped, the CPG based on Hopf oscillators presented in [29] is improved as follows:  
1. In order to adjust the gait period and duty factor directly and independently, the frequency \(\omega\) of the Hopf oscillator is modified as Equation (8). 2. Parameter \(\delta\) is introduced as coupling intensity coefficient among the CPG oscillator units to control the gait transition speed and waveform. 3. Multiple feedbacks with their corresponding reflex coefficients are simultaneously introduced into the two states (\(x\) and \(y\)) of the CPG oscillator unit. With the reflex information matrix and reflex coefficient vector, a clear way of expression and implementation is provided to realize the biological reflex modeling. Then the improved CPG is described as follows:  
\[\left[ \begin{array}{c}\dot{x}_{i}\\ \dot{y}_{i}\end{array} \right]=\left[ \begin{array}{cc}\alpha\big{(}\mu-{r_{i }}^{2}\big{)}&-\omega_{i}\\ \omega_{i}&\gamma\big{(}\mu-{r_{i}}^{2}\big{)}\end{array} \right]\left[ \begin{array}{c}x_{i}\\ y_{i}\end{array} \right]+\delta{\sum\limits_{j=1}^{4}{\mathbf{R}}\Big{(}\theta_{i} ^{j}\Big{)}}\left[ \begin{array}{c}x_{i}\\ y_{i}\end{array} \right]+\sum\limits_{k=1}^{m}{g_{k}\left[ \begin{array}{c}s_{ ik}^{x}\\ s_{ik}^{y}\end{array} \right]},\;i=1,\ldots,\;4 \tag{9}\]  
\[\left\{ \begin{array}{c}r_{i}=\sqrt{x_{i}^{2}+y_{i}^{2}}\\ \omega_{i}=\frac{\pi}{\beta T\left(\mathrm{e}^{-\mathrm{i}y_{i}+1}\right)}+ \frac{\pi}{\left(1-\beta\right)T\left(\mathrm{e}^{\mathrm{i}y_{i}+1}\right)} \end{array} \right. \tag{10}\]  
where \(x_{i}\) and \(y_{i}\) are the outputs of oscillator \(i\), and \(x_{i}\) is chosen as the main output of the CPG. The second and third terms on the right side of Equation (9) are the coupling term and feedback term, respectively; \(\delta\) is the coupling intensity coefficient among the CPG oscillator units; \(g_{k}\) is the reflex coefficient of the \(k\)-th feedback; \(s_{ik}^{x}\) and \(s_{ik}^{y}\) are the \(k\)-th feedback input of \(x_{i}\) and \(y_{i}\) of oscillator \(i\), respectively; \(m\) is the total number of feedback items; \(\theta_{i}^{j}\) is the relative phase between the oscillator \(i\) and oscillator \(j\); and \({\mathbf{R}}\Big{(}\theta_{i}^{j}\Big{)}\) is the rotation matrix that describes the phase coupling relationship among the CPG oscillator units. Other parameters are the same as described in Equations (6)-(8). Matrix \({\mathbf{R}}\Big{(}\theta_{i}^{j}\Big{)}\) is given by  
\[{\mathbf{R}}\Big{(}\theta_{i}^{j}\Big{)}={\mathbf{R}}_{ji}=\left[ \begin{array}{cc} cos\theta_{ji}&-sin\theta_{ji}\\ sin\theta_{ji}&cos\theta_{ji}\end{array}