# A Novel Inverse Simulation Method of Helicopter Maneuvering Flight
## 4 Case Studies and Results
### Case II: Slalom Maneuver  
The slalom maneuver is described in the ADS-33 requirements document [37]. It is an instantaneous right-to-left or left-to-right turning maneuver in the horizontal plane. It also initiates level, unaccelerated flight, and lines up with the centerline, and is followed by a smooth turn to the left, followed by one to the right. According to the ADS33 requirements, the turns will be at least 50 ft (15.24 m), with a maximum lateral error of 50 ft (15.24 m), and the flight velocity through the maneuver must not be less than 40 knots or 60 knots (74.08 km/h or 111.12 km/h), respectively. In this case study, an idealized slalom maneuver proposed by [PERSON] [7] was conducted, to illustrate the degeneration method, but we used the UH-60A helicopter, instead of the Westland Lynx helicopter mentioned in [PERSON]'s article. The maneuver lasted for 10 s, the total distance was 300 m (\(s=100\) m, Figure 7), the lateral displacement, \(h\), was 15 m, and the flight velocity, \(V_{\rm P}\), was 60 knots (111.12 km/h). Figure 7: Idealized slalom maneuver in the literature. The slalom maneuver only involves the elements \(\psi_{\text{PE}}\) and \(V_{\text{P}}\) in the transitional vector \(\mathbf{\eta}(t)\). The remaining elements \(\theta_{\text{PE}}\) and \(\dot{\psi}_{\text{BP}}\) can be set to zero. Because the flight velocity, \(V_{\text{P}}\), is a constant in the whole process of this maneuver, it is not necessary to prescribe its boundary conditions. The boundary conditions of the yaw angle, \(\psi_{\text{PE}}\), are shown in Table 3. In detail, four time points are needed according to the nature of this idealized slalom maneuver. Time points 1 and 4 are the respective start and end, and time points 2 and 3 indicate the specific locations of the negative and positive maximum angular rates, respectively. The values of these maximum angular rates must match the maximum lateral displacement. The primary results of the inverse simulation are shown in Figures 8-11, and the simulation time step is selected as 0.02 s. Because, at present, there is a lack of flight test data and reference data of the UH\(-\)60A flying this idealized slalom maneuver, some of our results were compared with the inverse simulation results concerning the Westland Lynx helicopter flying the idealized slalom maneuver in [PERSON]'s literature [7]. Figure 8 shows the flight path, which indicates that the results are in good agreement with each other, although they involve different helicopter systems. The results of the prescribed variable, \(\psi_{\text{PE}}\), are shown in Figure 9. In the process of the slalom maneuvering flight, the maximum lateral displacement reached 15 m, which met the requirement of this idealized slalom maneuver. The positive maximum yaw angle was above 15 deg, and the negative maximum yaw angle was below \(-25\) deg. The time histories of the normal load factor, roll angle, and rotor thrust are shown in Figure 10. In particular, the roll angle is compared with [PERSON]'s inverse simulation results. These also match well with each other. In the process of the slalom maneuvering flight, the maximum normal load reached approximately 1.7 g, and the maximum rotor thrust reached approximately 130 kN. Figure 11 shows the time histories of the cockpit controls and the stabilator angle during the slalom maneuver. The controls (i.e., collective pitch \(\theta_{0}\), longitudinal cyclic pitch \(\theta_{\text{1s}}\), lateral cyclic pitch \(\theta_{\text{1c}}\), and tail rotor collective pitch \(\theta_{\text{0 tr}}\)) of [PERSON]'s inverse simulation results are also shown in Figure 11, to discuss the fidelity of our inverse simulation results. The changing trends of the controls are consistent with each other, although the helicopter system was different. More importantly, because our results for the time histories of the cockpit controls, which can be seen clearly in Figure 11a-d, have no high-frequency oscillations (HFOs), our method seems to be more robust than [PERSON]'s method [7].