# Gait Planning Research for an Electrically Driven Large-Load-Ratio Six-Legged Robot
## 4 Gait Planning of the Large-Load-Ratio Six-Legged Robot
### Motion Planning of the Triped Gait
#### 4.1.1 Motion Planning of the Transfer Phase  
The foot of the transfer phase swings along a curve from the PEP to the AEP. Due to the arbitrary initial state for the swing leg, the moving trajectory of leg \(i\) in the transfer phase is the curve from the arbitrary position to the AEP. Two constraint conditions need to be considered in the motion planning of the transfer phase. Firstly, the velocity value of the foot is zero at the AEP and the PEP. Secondly, the acceleration curve of the foot has no the jump in the swing process of the leg. Generally, the energy consumption of the support phase is much larger than that of the transfer phase when the walking speed of robot is not high. Based on the research results of [PERSON] [20] and [PERSON] [21], we find that there is little difference between the foot trajectory of the swing leg and the cosine curve. Hence, cosine programming is employed for the gait planning in this article.  
Owing to the initial lift height \(h_{\text{TB}}\), the foot trajectory of the leg \(i\) can be divided into three parts in the transfer phase. In the first part, the foot of the leg \(i\) chooses the velocity \(v_{\text{BT}}\) to uplift from 0 to \(h_{\text{TB}}\) along the \(Z_{\text{B}}\) direction during the time \(0{-}t_{1}\). In the second part, the foot of the leg \(i\) moves to the AEP along the cosine curve during the time \(t_{1}{-}t_{2}\). In the third part, the foot of the leg \(i\) selects the velocity \(v_{\text{BT}}\) to fall from \(h_{\text{TB}}\) to 0 along the \(Z_{\text{B}}\) direction during the time \(t_{2}{-}T/2\). Then, the leg \(i\) is completely turned into the support leg.  
In the first part, the mathematical model of the foot trajectory can be written for the leg \(i\) in a period of the tripod gait as follows,  
\[\left\{\begin{array}{l}{}^{\text{B}}X_{\text{T1}}={}^{\text{B}}P_{\text{F} \text{I}}^{(i)}\\ {}^{\text{B}}Y_{\text{T1}}={}^{\text{B}}P_{\text{F}\text{J}}^{(i)}\\ {}^{\text{B}}Z_{\text{T1}}=v_{\text{T}\text{B}}t+{}^{\text{B}}P_{\text{F} \text{I}}^{(i)}\\ 0\leq t\leq t_{1}\end{array}\right. \tag{20}\]  
In the second part, the mathematical model of the foot trajectory can be written for the leg \(i\) in a period of the tripod gait as follows,  
\[\left\{\begin{array}{l}{}^{\text{B}}X_{\text{T2}}=\frac{({}^{\text{B}}P_{ \text{F}\text{J}}^{(i)}+{}^{\text{B}}A_{\text{T}\text{-B}}P_{\text{F}\text{ I}}^{(i)})}{2}\times\left(\cos\left(\frac{\pi}{t_{2}-t_{1}}(t+t_{2}-2t_{1}) \right)+1\right)+{}^{\text{B}}X_{\text{T1}}\\ {}^{\text{B}}Y_{\text{T2}}=\frac{({}^{\text{B}}P_{\text{F}\text{J}}^{(i)}+{}^{ \text{B}}A_{\text{T}\text{-B}}P_{\text{F}\text{J}}^{(i)})}{2}\times\left(\cos \left(\frac{\pi}{t_{2}-t_{1}}(t+t_{2}-2t_{1})\right)+1\right)+{}^{\text{B}}Y_{ \text{T1}}\\ {}^{\text{B}}Z_{\text{T2}}=-\frac{(h_{\text{T}\text{-max}}-h_{\text{TB}})}{2} \times\left(\cos\left(\frac{2\pi}{t_{2}-t_{1}}(t+t_{2}-2t_{1})\right)-1\right) +{}^{\text{B}}Z_{\text{T1}}\\ t_{1}\leq t\leq t_{2}\end{array}\right. \tag{21}\]  
In the third part, the mathematical model of the foot trajectory can be written for the leg \(i\) in a period of the tripod gait as follows,  
\[\left\{\begin{array}{l}{}^{\text{B}}X_{\text{T3}}={}^{\text{B}}X_{\text{T2} }\\ {}^{\text{B}}Y_{\text{T3}}={}^{\text{B}}Y_{\text{T2}}\\ {}^{\text{B}}Z_{\text{T3}}={}^{\text{B}}Z_{\text{T2}}-v_{\text{TB}}t\\ t_{2}\leq t\leq T/2\end{array}\right. \tag{22}\]