## 4 Results and Discussion
### Energy Management Results
#### 4.3.2 Comparison with Existing Energy Management Strategies  
To verify the effectiveness of the proposed EMS, the offline DP, rule-based energy management strategy, and conventional MPC energy management strategy with objective Equation (19) are compared and assessed. First, Figure 10 illustrates the HESS output power results of the proposed learning-based MPC energy management strategy and conventional MPC method. It can be observed that the output power of the conventional MPC is less than the power demand under high power demands, resulting in insufficient kinematic performance of the legged robot, which can reduce the success rate of high power demand moves such as obstacle crossing as shown in Figure 10a. The proposed learning-based MPC energy management strategy can fully meet the power demand, as shown in Figure 10b, which can guarantee the kinematic performance of the legged robot. Figure 8: Comparison of battery and supercapacitor power for the MPC with the smoothing term and the MPC without the battery power smoothing term. (**a**) Battery and supercapacitor power results for MPC allocation with the battery power smoothing term. (**b**) Battery and supercapacitor power results results for MPC allocation without the battery power smoothing term. Figure 9: Adaptive weighting factor results under different operation conditions. Regarding the SC SoC results, Figure 11 shows that the proposed learning-based MPC energy management strategy can maintain the SC SoC at a relatively high level, similar to offline DP, which is beneficial to fulfilling the load power demand. The rule-based method also maintains the SC SoC at a relatively high level but suffers from poor SC energy utilization, i.e., the SC SoC is in the range of [0.7, 0.9]. For the conventional MPC method, the SC SoC is gradually decreasing due to the fixed objective function, which results in insufficient movement of the legged robot. Figures 12-14 illustrate the power allocation results of different energy management strategies under general power demand (0-4 s), high feedback power demand (10.5-14.5 s), and high power demand (16-20 s), respectively. Figure 15 shows the battery capacity loss results of different energy management strategies for the whole operation process (0-30 s). Subfigure (a) of Figures 12-14 is the load power demand results, subfigure (b) is the battery power results, and subfigure (c) denotes the SC power. In Figure 11, the SC SoC of the rule-based method remains high enough to guarantee the kinematic performance of the legged robot. However, it lacks the flexibility of margin for high feedback power. In contrast, Figure 15 shows that the rule-based energy management strategy has a higher battery capacity loss, meaning the fixed rules lack flexibility and can be challenging to maintain good power allocation performance under diverse operation conditions. Figure 11: Supercapacitor SoC variations in different energy management strategies. Figure 10: Total output power of conventional MPC and the proposed method. **(a)** The total output power of conventional MPC. **(b)** The total output power of the proposed method. Figure 12: Comparison of battery power and SC power under general power demand (first 0–4 s). (**a**) Load power demand. (**b**) Battery power. (**c**) SC power. Figure 13: Comparison of battery power and SC power under high feedback power demand (10.5–14.5 s). (**a**) Load power demand. (**b**) Battery power. (**c**) SC power. During high feedback power demand conditions, as shown in Figure 13, the battery power of the proposed energy management strategy is smoother, the feedback energy is recovered by the supercapacitor, and the weighting factor of SC SoC is increased to reduce the battery capacity loss. Then, the supercapacitor usage increases, and the supercapacitor SoC decreases at this time to meet the demand for recovery margin. During high power demand conditions, as shown in Figure 14, the main objective of the proposed energy management is to reduce the battery capacity loss and smooth the current.