# 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
The resulting shape of the foot design is shown in Figure 7.  
Due to the chosen ratio of the weights of the legs together with the effectors, with respect to the weight of the robot frame, the requirement for the use of the LIPM walking model for the designed three-dimensional model is fulfilled. The basic parameters of the robot are listed in Table 4.  
The resulting dimensions from Table 4 were used in motion modeling using the inverse pendulum model. In the modeling process, the center of gravity of the robot represented the head of the pendulum in the form of a mass point \(p\) defined in space by default as \(p=(x,y,z)\) according to (5)-(7). This position is described in this model as a system of state variables \(q=(\theta r1,\theta p1,z)\) in which \(r\) represents the length of pendulum arm, or robot leg in this specific case [22]. \[x=rS_{p};S_{p}\equiv sin\theta_{p} \tag{5}\]  
\[y=-rS_{r};S_{r}\equiv sin\theta_{r} \tag{6}\]  
\[z=rD;\ D\equiv\sqrt{1-Sr^{2}-Sp^{2}} \tag{7}\]  
\(\tau_{r}\), \(\tau_{p}\) and \(f\), as vectors acting on the center of gravity, arm and head of the pendulum in Formula (8), represent the inputs to the system of equations describing the movement of the inverse pendulum in space, in the Cartesian right-handed system [22] as  
\[m \begin{pmatrix}\ddot{x}\\ \ddot{y}\\ \ddot{z}\end{pmatrix} = \begin{pmatrix}J^{T}\end{pmatrix} ^{-1} \begin{pmatrix} \tau_{r}\\ \tau_{p}\\ f\end{pmatrix} + \begin{pmatrix}0\\ 0\\ -mg\end{pmatrix} , \tag{8}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Property** & **Value** & **Unit** \\ \hline the height of the robot’s center of gravity & 445 & mm \\ height of the center of gravity of the frame & 645 & mm \\ foot height & 30 & mm \\ the length of the foot arch & 144 & mm \\ the weight of the robot & 2.755 & kg \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of the three-dimensional model. Figure 7: Proposal of a robot effector. where \(m\) is the mass of the center of gravity. Since the arm of the pendulum is considered as a link, the variable m belongs to the mass point \(p\). The Jacobian \(J\) is an antisymmetric matrix dependent on the current configuration \(q(t)\) and for motion analysis, according to [21; 22], has the form  
\[J=\frac{\partial p}{\partial q}= \begin{pmatrix}0&rC_{p}&S_{p}\\ -rC_{r}&0&-S_{r}\\ \frac{-rC_{r}S_{r}}{D}&\frac{-rC_{p}S_{p}}{D}&D\end{pmatrix} ;C_{r}\equiv cos \theta_{r},C_{p}\equiv cos\theta_{p} \tag{9}\]  
A similar Jacobian was later used in the recalculation of joint velocities of the robot in order to determine the inverse kinematics. In this case, it was used to create the dynamic equations of the pendulum, and therefore it is multiplied by the vector \(\langle\tilde{x},\tilde{y},\tilde{z}\rangle\) from the left to create a mathematical model of the inverse pendulum in space [21; 22] according to the relation  
\[m \begin{pmatrix}0&rC_{p}&S_{p}\\ -rC_{r}&0&-S_{r}\\ \frac{-rC_{r}S_{r}}{D}&\frac{-rC_{p}S_{p}}{D}&D\end{pmatrix}\begin{pmatrix} \tilde{x}\\ \tilde{y}\\ \tilde{z}\end{pmatrix} = \begin{pmatrix}\tau_{r}\\ \tau_{p}\\ f\end{pmatrix} -mg \begin{pmatrix}-rC_{r}S_{r}\\ \frac{-rC_{p}S_{p}}{D}\end{pmatrix}  \tag{10}\]  
In order to use this model to generate step trajectories, restrictions on the movement of the pendulum along the z-axis were defined. These constraints ensure that the height of the pendulum head does not change under the influence of the inputs \(q(t)\). However, this height can be changed manually. This restriction is represented in a form of help vector \((k_{x},k_{y},-1)\) of controlled inputs of height of the robot and the intersection \(z_{c}\) of the z-axis. The step width was determined as the length of the arch of the foot; i.e., the maximum possible. The parameters for the design of the pendulum are listed in Table 5 [22]. Step generation took place in three stages. In the first stage, the movement of the center of gravity and the ZMP in space was identified (Figure 8). According to the limitations of the model, the height of the center of gravity was anchored.