# C-Legged Hexapod Robot Design Guidelines Based on Energy Analysis
## 2 Kinematic and Energy Model of a Single C-Leg System  
### Motion Mechanics  
Let us start by considering a simplified system like the one shown in Figure 2, consisting of a square chassis represented in black and a semicircle leg (blue). The leg's rotating point or hip (position where the motor is connected to the leg, represented in green) will be placed at the chassis center. The red dot will represent the contact point between the robot's leg and the ground (black dotted line). The X-axis is the horizontal axis and is parallel to the ground and indicates the direction of forward and backward motion. The Y-axis is perpendicular to the ground line and is the direction in which gravity acts upon. As the motor rotates in the anticlockwise direction, from an initial position shown in Figure 2, the system starts pivoting around the contact point (red dot in Figure 2), generating a circular trajectory with a radius equal to the legs diameter. This corresponds to the red trajectory that is shown in Figure 3 (left image). The angle rotated by the motor starting from the initial position (Figure 2) will be denoted \(\theta\) and can be seen in Figure 3 (right image). In the case of a dimensionless chassis, this pivoting type of motion continues until the hip touches the ground, configuration 2 in Figure 3 (right image). From this point and for the next \(\frac{\pi}{2}\) radians the leg is rotating free, and thus, not moving the system. When \(\theta\) takes a value between \(\pi\) and \(2\pi\), the leg gets back in touch with the ground, and the system starts moving again, but following a different trajectory than during the first period. Figure 3: Trajectory is described by the system’s hip throughout a complete revolution (**left**) representation of angle \(\theta\) and the orientations at which the motion of the system changes (**right**). Figure 2: C-leg schematics positioned in the most upright position. The leg's orientation, in the limits of these intervals, can be seen in Figure 3 in the image on the right. Therefore, as it has been seen, the motion of the system can be split in to three different intervals in terms of the angle rotated by the motor \(\theta\):  
* \(\theta\in\left[ \begin{array}{cc}0,&\frac{\pi}{2}\end{array} \right]\): The system is pivoting around the contact point (Type I motion). It is important to note that this pivoting type of motion requires relatively slow rpms. Large rpm speeds will lead to the system to free fall, as the leg would be moving up in the vertical direction quicker than the chassis down. Hence, there would be no reaction force on the pivot point and no forward motion. * \(\theta\in\left[ \begin{array}{cc}\frac{\pi}{2},\pi\end{array} \right]\): The system is stationary. * \(\theta\in\left[ \begin{array}{cc}\pi,&2\pi\end{array} \right]\): The system is moving in a rolling type of motion (Type II motion). In the model, developed in the following subsection, the following assumptions have been assumed:
* **Massless leg:** This is an acceptable assumption as it is important to note that the C-leg mass of hexapod robots accounts for only 5% of the total robot mass [10]. Therefore, assuming a massless leg will not largely impact the motion mechanics of the system and will still allow us to develop an adequate energy and power consumption model. * **Rigid one-dimension semi-circle leg:** Compliant legs are an important feature in this kind of robot, as they enhance the robots' mobility by deforming the legs shape according to the load experienced, and thus, acting as a suspension [23]. Even so, the compliant legs do not affect the fundamental motion behavior of the system. This means that a C-shaped leg will lead to a pivoting and cycloid motion of the system (as it will later be seen) no matter whether the leg is compliant or not. Having said this, the exact motion and position of the robot will be affected by the deformation experienced by the leg.