## 6 Simulation Results  
To verify the effectiveness of the proposed controller in off-normal landing conditions, numerical simulations were executed with various initial pitch and roll angles. All simulations were set up and processed based on the co-simulation framework linkedbetween MATLAB and RECURDYN (as can be seen in Figure 12). In this framework, Simulink-MATLAB is the host that runs the controller algorithm and environment for the co-simulation, while RECURDYN is the client that receives electrical current signals from the controller and then returns the state of the system and damping force to MATLAB. Figure 13 shows the simulation setup in the case of non-zero pitch and roll angles: \(\theta=7^{\circ}\) and \(\Phi=7^{\circ}\), respectively. At the outset, as the right main landing gear makes contact with the ground, the left main landing gear and nose landing gear are still in the air. The initial velocity of the helicopter body is set to be equal to the designed sink speed of 2.54 m/s, following the standards outlined in FAA Part 27 [14]. Therefore, the speeds of the latter landing gears at the moment they touch the ground would be significantly larger than the designed sink speed, with a negative effect on damping the landing impact. Figure 12: Co-simulation between MATLAB-RECURDYN. Figure 14 shows the simulation results of the landing gears using different control methods in the case of \(\theta=7^{\circ}\) and \(\Phi=7^{\circ}\). In this landing case, the right main landing gear makes contact with the ground first, and then soon after it slips due to the dynamic asymmetry. During the slip, this landing gear releases energy. As the left main landing gear touches the ground, the slipping ends, and both of the main landing gears start to absorb energy. This makes the shapes of the load-stroke curves in Figure 14f,g more complex than the typical curves produced by single landing gear tests. Because the initial sink speed of the nose landing gear is much higher than it was designed for, this landing gear with the passive damper cannot absorb all of the potential and kinetic energy during touchdown. Hence, the nose landing gear rebounds, and thus its damping force nearly decreases to zero during the release phase of the damper, which can be verified in the efficiency curve in Figure 14h. Both the traditional skyhook controller and the skyhook extended controller exhibited better performance than the passive damper in this off-normal landing attitude condition. All damping forces remained positive during the release phase, so there were no rebounds. In particular, the skyhook extended controller showed better performance than the passive damper and traditional skyhook controller. The damper using the proposed controller produced the smallest magnitude of peak values in displacement, velocity, and acceleration at the helicopter's center of gravity. This signifies that the proposed controller is able to mitigate more vibration for the simulated helicopter during touchdown, which may enhance pilot and crew comfort in adverse landing conditions. Figure 15 compares the performance of the control methods in mitigating acceleration with the cost defined as in Equation (6) in differing initial pitch and roll angles. As can be seen, both skyhook controllers produced a significantly smaller cost than the passive damper. For the cases where \(\Phi=0^{\circ}\) and \(\theta<3^{\circ}\), the traditional skyhook controller showed almost the same performance as the skyhook extended controller due to close-to-symmetric landing attitudes. In other cases, the skyhook extended controller reduced the cost to a greater extent than the traditional skyhook. The biggest improvement of the proposed controller over the traditional skyhook was about 10% at \(\theta=7^{\circ}\) and \(\Phi=7^{\circ}\). Generally, the skyhook extended controller exhibited better performance in reducing acceleration than both the traditional skyhook controller and passive damper in all ranges of initial pitch and roll angles.