# C-Legged Hexapod Robot Design Guidelines Based on Energy Analysis
## 2 Kinematic and Energy Model of a Single C-Leg System
### Energy Model
#### 2.3.5 System's Energy and Power Consumption Model  
To obtain the total energy consumption, and therefore, the total motor power demand associated with KE and GPE variations, Equations (16), (18) and (20) are simply summed together.  
\[P_{total}=P_{PE}+P_{KE_{y}}+P_{KE_{x}} \tag{20}\]  
Taking the example that has been used throughout this article, the total power demand, due to the kinematic motion of a single-leg is:  
Figure 11 shows the total power demand of the system throughout a complete revolution. During type I motion (0-0.25 s), one can observe that the power demand associated with the KE variations cancel out, this behavior might seem unexpected, but it can be explained by the mechanics of this type of motion. Under this type of motion, the system  
Figure 10: Kinetic energy in the X-axis of the system with w = 60 rpm, m = 1 kg and r = 0.2 m (**left**), and power consumption associated with kinetic energy variation, m = 1 kg, w = 60 rpm, r = 0.2 m (**right**).  
Figure 9: Kinetic energy in the Y-axis of the system with w = 60 rpm, m = 1 kg and r = 0.2 m (**left**), and power consumption associated with kinetic energy variation, m = 1 kg, w = 60 rpm, r = 0.2 m (**right**).  
is essentially pivoting around the contact point position and at a constant angular velocity. Therefore, if the magnitude of the hip's linear velocity vector \(\left(\|v\|=\sqrt{{v_{x}}^{2}+{v_{y}}^{2}}\;\right)\) is calculated, the value should be constant during this type of motion.  
\[\|v\|=\sqrt{\left(2\
u\omega cos(\omega t)\right)^{2}+\left(-2\
u\omega sin( \omega t)\right)^{2}}=2\tau\omega \tag{21}\]  
Indeed, the magnitude of the hip's linear velocity during Type I motion is constant and proportional to the leg's radius and to the angular velocity, but this is not the case for Type II motion where the system is essentially rolling.  
While the system is moving at a constant velocity, the total KE (sum of the KE in the X- and Y-direction) will be constant. For this reason, there is no power demand associated with KE variations in Type I motion.  
The fact that the systems KE is constant, makes the net power demand during Type I motion only be the rate of change of GPE, leading to a negative power demand. Contrary to an object in free fall, our system has to lose GPE, while keeping the same KE, this means that this energy has to be dissipated via de motor. This can give rise to problems, such as components like microcontrollers burning down, if precautions are not taken. To avoid the flow of current in the opposite direction, the use of diodes is recommended.  
Unlike Type I motion, during Type II motion, the net power demand is positive as the system must rise in height and accelerate, requiring energy to be supplied to the system.  
The total power demand is not only useful from an energetical analysis perspective, but it can be very useful to identify the right motor for the robot. From these graphs, the maximum power of the motor can be determined, as well as the torque required. The torque exerted by the motor can be calculated using the power and angular velocity, as they are related together by the following equation:  
\[\tau=\frac{P}{\omega} \tag{22}\]  
where \(P\) is the power of the motor, \(\tau\) the torque exerted by the motor and \(\omega\) the angular velocity.  
Therefore, the motor specifications when related to power and torque can be determined from the system's physical properties, such as the legs radius, the mass and \(l_{c}\).  
Figure 11: Total power demand for a single-leg system with r = 0.2, m = 1 kg and w = 60 rpm.