# 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 starting point for the design of the robot was the kinematic model of the robot, which can be obtained in different ways depending on the principles used [19; 20]. In this case, the modeling process was dependent on the analyzed walking model LIPM and ZMP, which was adapted to the number of degrees of freedom of the robot. Since it is a mobile robot, no part of the robot is firmly attached to the ground, so we cannot talk about a mechanism, but about a kinematic chain. The kinematic chain was divided into three main parts. * **The frame** is a mobile base on which two manipulators are attached. The center of gravity of the robot is also located in this part and forms the material part (the head) of the inverse pendulum. The frame represents the human pelvis. * **The left leg** is the manipulator of the robot, one end of which is attached to the left side of the frame, and the other end is terminated by an effector. The manipulator consists of four arms and six joints. The manipulator represents the arm of the inverted pendulum and the human leg, where the effector fulfills the role of the foot. * **The right leg** is a manipulator functionally identical to the left leg, but it is fixed on the right side of the frame. Based on the movement analysis, it follows that there are two degrees of freedom in the ankle and up to three in the hip joint. The implementation of several degrees of freedom in one place--such as in a spherical kinetic pair--was realized by dividing the joint into three rotary kinematic pairs with the smallest possible arms. These rotation pairs rotate around different axes of the coordinate system. In this way, the three degrees of freedom of the spherical pair were preserved with a simpler implementation in practice. One manipulator has six degrees of freedom, just like the analyzed template. Five degrees of freedom are controlled and one degree of freedom at both ends of the chain is uncontrolled. The robot has two arms on a frame with six degrees of freedom. The additional six degrees of freedom were granted because it is a mobile robot which can freely move and rotate in space in all directions. The number of freedom degrees of the kinetic chain  
\[n_{robot}=6+\sum_{i=1}^{j}n_{i}=6+12=18\geq 6, \tag{1}\]  
where \(n_{i}\) represents the number of degrees of freedom of the \(i\)-th kinematic pair and \(j\) is the total number of kinematic pairs, is greater than the minimum required number for the LIPM model. The graph of the robot in Figure 4 shows the relationships between individual kinematic pairs. In the vertices of the graph, the number of degrees of freedom in each pair is indicated. The mobility of the robot is represented by the absence of a link between the ground and the kinematic chain. Furthermore, it can be observed that the graph of the robot is acyclic; therefore, we speak of an open kinematic chain. The kinematic model of the robot was derived from the robot graph (Figure 5). Kinematic pairs were defined in space and therefore in this model we are already talking about rotational joints with one degree of freedom. Relationships between joints are now represented by arms. This is a model with a combined type of walking (passive and controlled). Figure 4: Robot graph based on robot analysis (left), robot graph adapted for robot construction (right). Denavit-Hartenberg (DH) parameters were determined in the kinematic model (Figure 6). Conventional DH parameterization uses a uniform method for marking the axes of coordinate systems in individual joints. It uses two rules:  
\[\text{DH1}:x_{i}\bot z_{i-1}\,;\,\text{DH2}:|x_{i}\cap z_{i-1}\lor 1. \tag{2}\]  
Therefore, in DH, the representation of the homogeneous transformation contains two parameters less than the usual homogeneous transformation, which facilitates the analysis of the n-arm manipulator [21]. In the process of identifying the DH parameters, the axes of rotation were first marked depending on the types of joints.