# Helicopter Safe Landing Trajectory after Main Rotor Actuator Failures
## 5 Results and Discussion  
This study takes the UH-60 as a sample helicopter. The geometry parameters of the swashplate can be found in [29; 30]. The take-off weight in this study is chosen as 7257 kg, and other modelling parameters of UH-60 can be found in [23]. As mentioned before, the swashplate actuator stuck failure is assumed to take place during level flights. Thus, the initial states and manipulations are given by trim solutions of steady level flights. For each actuator failure case, a specified actuator is assumed to be stuck in its trimmed position. Let \(\theta_{\text{trim}}\) and \(\theta_{\text{lstrim}}\) denote the initial control inputs \(\mathbf{u}_{0}\) during steady flights, which are manipulations of trim solutions. The blade control after the actuator failure can be described as  
\[\left\{ \begin{array}{c}\theta_{1s}=\theta_{1\text{lstrim}}+\Delta\theta_{1s }^{\prime}\\ \theta_{0}=\theta_{0\text{trim}}+\Delta\theta_{0}^{\prime}\end{array} \right. \tag{25}\]  
where \(\Delta\theta_{1s}^{\prime},\ \ \Delta\theta_{0}^{\prime}\) refers to the actual blade control input after the actuator failure defined in (4)-(9). To be specific, the key parameters of the swashplate [29] are \(\gamma=90^{\circ},\ r_{\text{r}}/r_{\text{nr}}=1.17\). Thus, an increase of the longitudinal cyclic pitch \(\Delta\theta_{1s}^{\prime}\) results in:  
Case 1 (forward actuator failure): a decrease of the collective pitch of \(\Delta\theta_{0}^{\prime}=-\Delta\theta_{1s}^{\prime}/1.17\). Case 2 (aft actuator failure): an increase of the collective pitch of \(\Delta\theta_{0}^{\prime}=\Delta\theta_{1s}^{\prime}/1.17\). Case 3 (lateral actuator failure): no influence on the collective pitch. It is shown from the above analysis that, because modern helicopters share similar swashplate structures, although a UH-60 helicopter is taken as a sample model, such method is also applicable for other helicopters, including large-scale helicopters and small-scale helicopters. The path constraints are chosen as  
\[z\geq 0\ \text{m},0\ \text{m}/\text{s}\leq v_{x}\leq 70\ \text{m}/\text{s},v_{z} \leq 10\ \text{m}/\text{s},\ |\theta|\leq 40^{\circ},\ |\eta|\leq 40^{\circ}/\text{s}\]  
where descent rate, pitch angle and pitch rate are limited mainly to avoid vortex ring state [21] and fierce maneuvers. The touchdown speed and attitude are limited by terminal constraints, chosen as:  
\[0\ \text{m}/\text{s}\leq v_{x}(t_{f})\leq 15\ \text{m}/\text{s}, 0\ \text{m}/\text{s}\leq v_{z}(t_{f})\leq 3\ \text{m}/\text{s},\] \[-10^{\circ}\leq\theta(t_{f})\leq 15^{\circ},\ -15^{\circ}/\text{s} \leq q(t_{f})\leq 15^{\circ}/\text{s}\]  
The weight factors of the terminal cost described in Equation (19) are chosen as \(W_{x}=0.7,\ W_{z}=1.0\). Sets of speed sweeps of the three actuator failure cases are conducted under the above optimal control constraints. The speed sweeps range from 5 m/s to 70 m/s. Successful safe landing trajectories of the three actuator failure cases are shown and discussed. To be specific, the trajectories are shown and discussed in two groups. The first group demonstrates the trajectories of the same initial conditions for each actuator failure case. The second group demonstrates typical trajectories with different initial states for each actuator failure case, due to the safe boundaries are different for each failure case, which will be discussed later. Figure 2 compares the three failure cases at the same flight speed of 20 m/s and the same altitude of 50 m.  
Apparently, when landing, while the vertical speeds are all within the safe landing upper limit of 3 m/s, the resulting forward landing speeds are different. In particular, the forward landing speeds are 12.7 m/s, 8.7 m/s and 0.0 m/s, respectively. The characteristic \"bucket profile\" of the collective pitch during level flight is shown in Figure 3, which means a steady flight of a lower forward speed near the hover point requires a higher collective pitch.