## 5 Control Design  
The traditional skyhook control algorithm was first suggested by [PERSON] [22]. It subsequently became one of the most popular control methods used in vehicle suspensions to reduce vibration [11]. It has also been used to control landing gear systems with MR dampers [9; 23]. Figure 7 shows the conceptual diagram of the traditional skyhook controller for our target helicopter with three landing gears equipped with MR dampers. It applies three translational virtual dampers that connect the helicopter and the imaginary support fixed in the sky. The skyhook controller has to control the MR dampers that emulates the behavior of the virtual dampers. Each virtual damper is connected to the helicopter at the position where a landing gear is attached. Thus, the traditional skyhook controller only reduces the vibration of a local motion of a part of the helicopter around each landing gear. In a semi-active controller, whose controllable damping force can be applied only in the opposite direction of the stroke velocity, the skyhook control law that is used for the \(i\)th landing gear is given as  
\[u_{i}=\left\{ \begin{array}{cc}C_{i}v_{i},&v_{i}\dot{s}_{i}\geq 0\\ 0,&v_{i}\dot{s}_{i}<0\end{array} \right. \tag{7}\]  
where \(C_{i}>0\) is the skyhook gain, \(v_{i}\) is the velocity of the point where the \(i\)th landing gear is attached to the main body, and \(\dot{s}_{i}\) is the stroke velocity of the \(i\)th landing gear. It is assumed that no sensor can directly measure \(v_{i}\) due to the practical limitations on weight and cost. However, the velocities can be estimated by following the approximate equations below. \[ \begin{array}{c}v_{1}\approx v+l_{i}\dot{\theta}\\ v_{2}\approx v-l_{\dot{b}}\dot{\theta}+l_{\dot{t}}\dot{\phi}\\ v_{3}\approx v-l_{\dot{b}}\dot{\theta}-l_{\dot{r}}\dot{\phi}\end{array}  \tag{8}\]  
Note that the control input (\(u\)) is a function of the sink speed (\(v\)) of the main body, pitch rate (\(\dot{\theta}\)), and roll rate (\(\dot{\phi}\)). For each landing gear, there is only one parameter to adjust the control input in various landing situations. To extend this control strategy to have more parameters for tuning the control input, a extended control algorithm for skyhook is presented in this study. Figure 8 shows the concept of the skyhook extended controller. It sets three virtual dampers--i.e., a vertical translational damper and two rotational dampers--against the motion of the center of gravity and this connects the  
Figure 7: Concept of traditional skyhook control for the helicopter with 3 landing gears. helicopter center and imaginary translation/rotation-free supports fixed in the sky. The skyhook extended controller must control each MR damper which emulates the behavior of one of the three virtual dampers. The skyhook extended control law is given as follows. \[u_{i}=C_{i}^{\mathrm{p}}v+C_{i}^{\theta}\dot{\theta}+C_{i}^{\phi}\dot{\phi} \tag{9}\]  
where \(C_{i}^{\mathrm{p}}\), \(C_{i}^{\theta}\), and \(C_{i}^{\phi}\) are the skyhook gains, which can have a positive or negative value. However, the control input must follow the semi-active condition, which is given by  
\[F_{i}^{MR}=\left\{ \begin{array}{cc}\mathrm{sat}(u_{i}),&u_{i}\dot{s}_{i}\geq 0 \\ 0,&u_{i}\dot{s}_{i}<0\end{array} \right. \tag{10}\]  
\[\mathrm{sat}(x)=\left\{ \begin{array}{cc}1,&x\geq 1\\ x,&0\leq x\leq 1\\ 0,&x\leq 0\end{array} \right. \tag{11}\]  
As mentioned in Section 4, the main control objective is to reduce the helicopter acceleration at the center of gravity for differing initial pitch and roll angles. Therefore, the skyhook gains in the controllers (both traditional skyhook and the proposed extended version) must be determined in the various landing scenarios by using a look-up table method (refer to Figure 9). The most popular and traditional search methods to find an optimal parameter are gradient methods such as the steepest descent. However, these methods need to calculate the derivative of the cost function. In our system, it is difficult to calculate this derivative.