# Bio-Inspired Model-Based Design and Control of Bipedal Robot
## 3 Modeling and Simulation of Biped Robot
### Design of a Simulation Model of a Bipedal Robot
Convention states that the axis of rotation is the z-axis. Based on the positive direction of rotation, the direction of the axis was determined. For some opposing joints, the direction of the z-axis was chosen differently to preserve the natural direction of rotation. Subsequently, two different origins of zero coordinate systems were chosen, specifically for the right and left leg. In both cases, the origin was chosen at the joint that connects the leg to the frame. Unlike the rest of the centers of the coordinate systems, these were not numbered. The center of the coordinate system in the joint connecting the left leg and the frame was marked O\({}_{0}\), and the center in the joint connecting the right leg and the frame was marked O\({}_{0}\)'. A different zero-system naming convention for walking robots led to this decision. After finding the remaining origins of the coordinate systems, the \(x\) and \(y\) axes were determined by convention according to relations (3) and (4). \[x=z_{i-1}\times z_{0} \tag{3}\]  
\[y=z\times y \tag{4}\]  
In effector coordinate systems, the z-axis is generally oriented in the same way as the axis of the last arm. Thus, the two non-controlled joints at the ends of the kinematic chain will rotate along the y-axis, not along the z-axis like the controlled joints. In both cases, the origins of the effectors coordinate systems were chosen at the point created by the intersection of the axis perpendicular to the ground, passing through the center of gravity,  
Figure 5: Kinematic model of the robot (left), visualization of LIPM and ZMP principles on the kinematic model (right). and the straight line created when the given effector touches the ground when the robot is in the starting position. Under ideal conditions, this point is identical to the ZMP. The analysis of the DH parameters of the robot's legs is shown in Figure 6. The specific values of the DH parameters of the kinematic model of the robot are listed in Table 3.  
In the DH parameterization, the centers of the coordinate systems and their relative positions were found. Despite the fact that the effectors seem to fulfill the role of joints in the kinematic chain, this phenomenon is not included in the DH parameters, since they  
Figure 6: DH parameterization of the right (R) and the left (L) legs. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Left a\({}_{\text{i}}\) [mm] 1** & \(\alpha_{\text{i}}\) [deg] 2  & \(\text{d}_{\text{i}}\) [mm] 3  & \(\text{o}_{\text{i}}\) [deg] 4  \\ \hline
0 & 90 & L\({}_{1}\) & \(\text{o}_{1}\) \\
0 & \(-\)90 & L\({}_{2}\) & \(\text{o}_{2}+90\) \\ L\({}_{3}\) & 180 & 0 & \(\text{o}_{3}\) \\ L\({}_{4}\) & 0 & L\({}_{5}\) & \(\text{o}_{4}\) \\ L\({}_{6}\) & 0 & 0 & \(\text{o}_{5}\) \\ \hline
**Right a\({}_{\text{i}}\) [mm] 1** & \(\alpha_{\text{i}}\)[deg] 2  & \(\text{d}_{\text{i}}\) [mm] 3  & \(\text{o}_{\text{i}}\)[deg] 4  \\ \hline
0 & 90 & L\({}_{1}\) & \(\text{o}_{6}\) \\
0 & 90 & L\({}_{2}\) & \(\text{o}_{7}\)—90 \\ L\({}_{3}\) & 180 & 0 & \(\text{o}_{8}\) \\ L\({}_{4}\) & 0 & L\({}_{5}\) & \(\text{o}_{9}\) \\ L\({}_{6}\) & 0 & 0 & \(\text{o}_{10}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: DH parameters of the robot kinematic model. are uncontrolled \"joints\" between the ground and the robot. The parameterization of the effector was developed by creating a three-dimensional model of the entire kinetic chain based on the analyzed kinematic model. When modeling the effector, knowledge from the materials for the construction of the foot of a passive walking robot was used. In the starting position, when in contact with the ground, the effectors create support in the form of two parallel lines. Since the position of the effector itself is controlled, the length of the foot was designed to be as small as possible in order to maintain relative stability in the controlled direction of rotation. The foot width was determined to be as large as possible for the longest arch of its lower part.