# A Bio-Inspired Control Strategy for Locomotion of a Quadruped Robot
## 3 Improved CPG
### Improved CPG Control Model
\right] \tag{11}\]  
where \(\theta_{i}^{j}=\theta_{ji}=\varphi_{i}-\varphi_{j}\), and \(\varphi_{i}\) is the phase of the oscillator \(i\). Equation (9) can be rewritten as  
\[\dot{{\mathbf{Q}}}={\mathbf{F}}({\mathbf{Q}})+\delta{\mathbf{R}}{\mathbf{Q}}+{\mathbf{S}}{\mathbf{G}} \tag{12}\]  
where \({\mathbf{Q}}=\left[ \begin{array}{cccccc}x_{1}&y_{1}&x_{2}&y_{2}&x_{3}&y_{3}&x_{4 }&y_{4}\end{array} \right]^{\mathrm{T}}\), and \({\mathbf{R}}=\left[\begin{array}{cccccc}{\mathbf{R}}_{{\mathbf{11}}}&{  
and \(\mathbf{G}=\left[ \begin{array}{c}g_{1}\\ \vdots\\ g_{m}\end{array} \right]\) are the reflex information matrix and reflex coefficient vector respectively. The walk, trot, pace and gallop are the four typical gaits of quadruped robots and their phase relationships is shown in Table 2 respectively. After introducing the gait matrix into Equation (11), we can obtain the corresponding connection weight matrix \(\mathbf{R}\). Further, the connection weight matrices of the four typical gaits are given by Figure 5.  
An important feature of CPG control network is the output waveform of gaits and their mutual transitions. The CPG oscillation units can be coupled with any relative phase, since they are coupled to each other with its two output states by the rotation matrix. Besides, a gait can be transformed to another gait by replacing the corresponding target \(\mathbf{R}\) directly, and the gait transition point can be selected arbitrarily when expecting the gait transition. As the walk and trot gaits are the most representative, with the coupling intensity coefficient \(\delta=~{}1\) and \(\delta=~{}0.3\) respectively, the output waveform of CPG in the walk and trot gaits and their mutual transitions are shown in Figure 6, where \(\mu=1\), \(\alpha=\gamma=b=50\), \(T=1\) s, the third term on the right side of Equation (9) is 0 since there is no feedback to the CPG, and \(\beta=0.75\) in the walk gait while \(\beta~{}=~{}0.5\) in the trot gait. As can be seen from Figure 6, the gait transition speed and its output waveform are adjusted by \(\delta\) independently. Specifically, when \(\delta\) is relatively large (as shown in the upper part (a) of Figure 6, \(\delta=~{}1\)), a gait can be transformed to another gait quickly (the transition time \(\leq T\)). However, due to the relatively strong coupling, the output waveforms rise slowly in the front part of the ascending phase while rising violently in the rear part in walk gait with the duty factor \(\beta>0.5\), and the greater the duty factor \(\beta\) exceeds 0.5, the more obvious the over-coupling phenomenon is. However, when a smaller coupling intensity coefficient \(\delta\) is used, the gait transition speed is reduced relatively, but it can effectively avoid the over-coupling phenomenon and the whole transition process is smoother (as shown in the lower part (b) of Figure 6, \(\delta=0.3\)). Therefore, \(\delta\) can be used to control the gait transition speed in different situations. \begin{table}
\begin{tabular}{c c} \hline
**Gait** & **Gait Matrix (Phase Relationships)** \\ \hline walk & (0, \(\pi\), \(\pi\)/2, \(3\pi\)/2) \\ trot & (0, \(\pi\), \(0\), \(\pi\)) \\ pace & (0, \(\pi\), \(\pi\), \(0\)) \\ gallop & (0, \(0\), \(\pi\), \(\pi\)) \\ \hline \end{tabular}
\end{table} 
Table 2: Four typical gaits and their matrixes. Figure 5: Connection weight matrixes of the four typical gaits. (**a**) walk; (**b**) trot; (**c**) pace; (**d**) gallop.