# Albatross-Like Utilization of Wind Gradient for Unpowered Flight of Fixed-Wing Aircraft
## 5 Simulation Results and Discussion
### Rayleigh Cycle Simulation Results  
The parameters of the albatross in Table 2 were used for the simulation of its Raleigh cycle. The wind shear model, which was in accordance with Equation (1), was obtained from MIL-F-8785C [40]. \(W_{6}\) was set to 15 m/s, which is a common value on Earth [48]. A dynamic model block was built and used to perform the simulation with the aid of the Simulink and Stateflow software (MathWorks, Natick, MA, USA). Figure 14 shows a snapshot of the Simulink model, while Figure 15 shows the Stateflow chart corresponding to the Mode Switch block in Figure 14.  
Figure 14: Snapshot of the Simulink model. Figure 13: Variations in the air density and gravitational acceleration with the altitude on Mars. The Stateflow chart in Figure 15 realizes the switch logic in Figure 6, with the Controller block particularly realizing the algorithm described in Section 4.3. In Figure 15, the transition logic between the climbing state and the high-altitude turn is \(v_{z}<1\), because 1 is a margin that ensures that the aircraft enters the high-altitude turn state before it begins to dive. To minimize the energy loss, the discount rate \(\lambda\) should be tuned. When the discount rate is set to 0, the low-altitude turn requires about 10 s, and the final velocity is so small that the aircraft is incapable of connecting to the next upwind climb phase. When the discount rate \(\lambda\) is set to 1, although the time required for the turn is significantly shortened, the energy consumption also increases. The bisection method was used to iteratively tune the discount rate in the present work, by which a value of 0.9 was found to be suitable. For a given aircraft, the lift coefficient \(C_{L}\) varies with the angle of attack, which can be approximated as the angle between the total velocity vector and the horizontal plane of the aircraft. The angle of attack is thus the actual control input. Owing to the availability of very limited data on the aerodynamics of albatrosses, the specific relationship between their angle of attack and lift coefficient is unclear. In the present study, an albatross was simply considered as an aircraft, and its lift coefficient was assumed to be proportional to the angle of attack for small values of the latter. Although this is consistent with regular aerodynamics theory [43], it was still difficult to determine the proportionality. Nevertheless, with the assumption, the lift coefficient could be directly controlled by the angle of attack in the simulation. Figures 16-18 compares the simulation results for the proposed bio-inspired strategy with those for the traditional GPOPS optimized trajectory. The figures cover the same period in the entire time series for the two methods, coinciding with a full cycle of the GPOPS method, and a little longer than a cycle generated by the bio-inspired strategy. The yellow background in Figure 16 exactly covers a cycle of the bio-inspired strategy. The figure shows the variations in the energy and altitude with time. Figure 17 shows the variations in the control inputs with time. As can be observed, the computed control inputs for the GPOPS method form smooth signals that also comprise four phases corresponding to those of the Rayleigh cycle, although the phase switch boundaries are not as obvious as those of the bio-inspired strategy. In the upwind climb and downwind dive segments, the bank angles for both methods are very small, sometimes approaching zero. Conversely, the bank angles during the high-altitude and low-altitude turns are large. In addition, the lift coefficients for both methods are maximum during the high-altitude turn, and minimum during the upwind climb and downwind dive. In the low-altitude turn segment, both methods visibly compromise on the assigned lift coefficient, opting for medium values. Figure 15: Snapshot of the Stateflow chart. In Figure 16, the indicated boundaries of the four phases (upwind climb, high-altitude turn, downwind dive, and low-altitude turn) are for the bio-inspired strategy.