## 2 Methods and Materials
### CoT Optimization Using Red Fox Optimization (RFO) Algorithm
Optimization aims to find the minimum value of CoT. Therefore, the fitness function is defined by Equation (4). ## 3 Results

### Power Consumption Measurement Results

The hexapod robot step height and body elevation were set to 1 cm and 10 cm accordingly. These values are the standard configuration for hexapod robot movement and 1 cm is the minimum step height required to execute leg transfer. All experiments were carried out on flat terrain with a soft surface for better contact between the feet and the ground to minimize leg slippage. The first experiment was done without using any additional load on the robot. The average current dependence on robot movement speed and different gaits is shown in Figure 2(a). The speed at which the gait must be switched to sustain the minimal current usage is \(v\ =\ 0.046\) m/s (Figure 2(b)). Wave gait is the most energetically efficient for low speeds (0-0.046 m/s) and tripod gait is more efficient for faster movement (0.046-0.33 m/s). The power consumption at low speeds is \(\sim\)12-13 W (wave gait), and at faster speeds is \(\sim\)13-28 W (tripod gait). Using proposed gait switching between wave and tripod gaits, \(\sim\)7.7-21% of energy can be saved. To make sure that the recurrence is stable and does not change depending on conditions, we repeated the experiment with 1.16 kg and 2.9 kg loads placed on the top of the robot. Results are shown in Figure 4. All curves for different gaits with 1.16 kg load were arranged in the same way as with no load. The point of gait switching decreased (v = 0.028 m/s). Only with a 2.9 kg load does the arrangement of curves start to change. This shows that energy consumption depends on the robot's weight. With increased robot's weight, gait switching speed decreases, i.e., the tripod gait becomes more effective at slower speeds with increased load. Figure 3: Average power dependence on robot speed and gaits: (**a**) normal view, (**b**) enlarged view. Experiments were carried out without additional load on the robot. Each dot represents a separate experiment with a certain configuration. Lines show polynomial approximation. No gait type switching was used. ### Verifying Overall Energy Efficiency

To evaluate our hexapod robot's dependence of CoT on robot movement speed, we applied Equation (5) for each gait, speed, and each obtained power consumption result presented in Figure 5. As we can see from Figure 5a, CoT does not depend on gait type. Even if we take a particular speed, CoT remained the same for each gait. The only difference is that gaits vary in speed intervals. The results in Figure 5b show that the CoT value decreases when the movement speed increases. This means that to obtain lower CoT, it is necessary to switch to the gait with a higher speed interval (e.g., tripod gait). These results along with Equation (4) can be used to optimize the energy consumption according to robot speed. Using only the logarithmic scale, we cannot define the working space of our hexapod robot that is energetically efficient. This is the reason for using the linear scale also (Figure 5b). The robot should be programmed to move only at speeds higher than 0.04 m/s due to CoT having the lowest values in that interval. Between the speed of 0.04 m/s and 0.33 m/s, the value of CoT decreases from \(\sim\)25 to \(\sim\)5.75, whereas between the speed of 0.04 m/s down to 0.00042 m/s, the value of CoT rises from \(\sim\)25 up to \(\sim\)2000. The tripod gait was the most efficient because it used the least energy, while the robot can reach the highest speed. Figure 4: Average power dependence on robot speed and gaits with different robot loads: (**a**) 1.16 kg load, (**b**) 2.9 kg load. Each dot represents a separate experiment with a certain configuration. Lines show polynomial approximation. No gait switching was used. Figure 5: Cost of transport dependence on robot speed: (**a**) logarithmic scale, (**b**) linear scale.