# Fuzzy Logic-Based Controller for Bipedal Robot
## 2 Kinematics of Robot and Motion Trajectory Planning
### Robot Motion Trajectory Planning
The length of each step of the foot, also of the ankle joint, is denoted by \(2L\), and that of the hip joint is denoted by \(L\). Particularly for the starting step from the resting state, which with the initial position is vertical, the movement of the ankle joint is \(L\). Similarly, the final step back to rest, in an upright position, also has lengths \(L\) for the movement of the ankle joint. To create a database for the problem of dynamics and control, some numerical calculations and simulations of the motion trajectory of the robot were performed. A proposed robot model with kinematic and dynamic parameters is listed in Table 1. The process of starting the robot usually moves at an increasing speed, while the motion slows down to a state of rest. In the steady-state of motion, the horizontal velocity of the robot body was assumed to be constant. Calculations and simulations were performed with the typical steps for the above three processes as follows:  
* The first step of the acceleration process, called the \"starting step\" from now on;
* The step of the steady motion process, called the \"steady step\";
* The last step of deceleration to rest, called the \"final step\". The kinematic parameters in Table 1 were used to calculate and design motion trajectories for the robot corresponding to these three steps, and the results are shown in Figures 3-8. Figure 2: The walking steps of bipedal robot: (**a**) the starting steps; (**b**) the steady steps; (**c**) the final steps; (**d**) the robot’s gait; (**e**) posture of robot body when moving: [[https://youtu.be/8A3849](https://youtu.be/8A3849) PUCyM]([https://youtu.be/8A3849](https://youtu.be/8A3849) PUCyM) (accessed on 21 September 2021). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**The Geometrical, Kinematical, and Motion Trajectory Parameters**} \\ \hline
**Geometrical** & **Values** & **Dimensional** & **Kinematical Parameters** & **Values** & **Dimensional** \\ \hline foot size \(h_{f}\) & 0.08 & m & normal step length & \(2L=0.76\) & m \\ foot size \(d_{f}\) & 0.132 & m & starting step length & \(L=0.38\) & m \\ foot size \(d_{h}\) & 0.06 & m & final step length & \(L=0.38\) & m \\ thigh length \(a_{t}\) & 0.38 & m & steady velocity & \(Vc=4275\) & m/h (hour) \\ calf length \(a_{c}\) & 0.38 & m & average speed of starting step & \(Vs=1487\) & m/h \\ body height \(h_{b}\) & 0.45 & m & average speed of final step & \(Vf=1368\) & m/h \\ \hline  & \multicolumn{6}{c}{**The dynamical parameters**} \\ \hline
**Links** & **Mass** & \multicolumn{6}{c}{**Link’s center of mass**} \\  & \multicolumn{6}{c}{**coordinates (m)**} \\ \cline{2-9}  & \multicolumn{1}{c}{(kg)} & \multicolumn{1}{c}{\(x_{ci}\)} & \(y_{ci}\) & \(z_{ci}\) & \(lxx\) & \(lyy\) & \(lzz\) & \(lxy\) & \(lxz\) & \(lyz\) \\ \hline
1 & 1.22803 & \(-0.06792\) & 0.02140 & 0 & 0.00138 & 0.24390 & 0.0055 & 0.00129 & 0 & 0 \\
2 & 3.25665 & \(-0.18377\) & 0 & 0 & 0.00122 & 0.04695 & 0.04714 & 0 & 0 & 0 \\
3 & 5.86021 & \(-0.16274\) & 0 & 0 & 0.00279 & 0.10417 & 0.10467 & 0 & 0 \\
4 & 5.86021 & \(-0.21726\) & 0 & 0 & 0.00279 & 0.10417 & 0.10467 & 0 & 0 \\
5 & 3.25665 & \(-0.19623\) & 0 & 0 & 0.00122 & 0.04695 & 0.04714 & 0 & 0 & 0 \\
6 & 1.22803 & 0.05350 & 0 & 0 & 0.00490 & 0.00111 & 0.0055 & 0.000851 & 0 & 0 \\
7 & 38.8 & 0.36870 & 0 & 0 & 0.24390 & 2.21146 & 2.02948 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The kinematic, dynamic, and motion trajectory parameters of the proposed robot model. Figure 4: The motion trajectory of ankle joint in the starting (**a**), steady (**b**), and final (**c**) steps. Figure 3: The motion trajectory of the hip joint in the starting (**a**), steady (**b**), and final (**c**) steps. Figure 3a-c shows the motion trajectory of the hip joint in the starting, steady, and final steps, respectively. Similarly, the movement of the ankle joint is shown in Figure 4a-c.