## 3 Double Sky-Hook Control Algorithm
### Double Sky-Hook Algorithm  
Following the basic idea of the ideal sky-hook damper, if an additional sky-hook damper is attached to the unsprung mass in order to reduce the wheel hop resonance, the force transmitted to the unsprung mass through an ideal sky-hook damper is expressed as  
\[F_{unsprung}=B2_{sky}\cdot\dot{x}_{u} \tag{13}\]  
where \(B2_{sky}\) is the sky-hook damping coefficient. In the real suspension configuration, however, the damping force is a function of a relative speed between a chassis and a wheel and is represented as  
\[F_{unsprung}=B2_{d}\cdot\left(\dot{x}_{u}-\dot{x}_{s}\right) \tag{14}\]  
The desired damping coefficient \(B2_{d}\) that generates the same damping force as that of the ideal sky-hook damper is determined from Equations (4) and (5) as  
\[B2_{d}=\frac{B2_{sky}\cdot\ddot{x}_{u}}{\dot{x}_{u}-\dot{x}_{s}} \tag{15}\]  
If we use the sky-hook damper only to the wheel, in other words, a real damping coefficient is adjusted by \(B2_{d}\), then the road-holding performance, which in turn, will greatly improve the vehicle dynamic stability. Since a real semi-active suspension system is capable of controlling a single damping coefficient, however, we need to combine two damping coefficients \(B1_{d}\) and \(B2_{d}\) in such a way that both sprung and unsprung mass resonances are controlled effectively in each frequency region. For a simple combination, we propose the following combination as  
\[B_{d}=B1_{d}+B2_{d} \tag{16}\]  
Note that for a constant magnitude of relative velocity \(\left(\dot{x}_{u}-\dot{x}_{s}\right)\), \(B1_{d}\) in Equation (12) takes a large value in the sprung mass resonance frequency where the sprung mass vertical velocity is magnified dynamically while \(B2_{d}\) in Equation (15) takes a large value in the unsprung mass resonance frequency where the unsprung mass vertical velocity is magnified. Therefore, each damping coefficient is considered to play an effective role in each frequency region of interest. In the case of Equation (16), the damping force transmitted is represented as  
\[F_{d}=B1_{sky}\cdot\dot{x}_{s}-B2_{sky}\cdot\dot{x}_{u} \tag{17}\]  
Then, the following four cases should be noted:  
**Case I. ** If \(B2_{sky}=0\), the algorithm behaves like a conventional ideal sky-hook algorithm where a sky-hook damper with a damping coefficient \(B1_{sky}\) is attached to a sprung mass. **Case II. ** If \(0<B2_{sky}<B1_{sky}\), as the value of \(B2_{sky}\) increases from zero when the value of \(B1_{sky}\) is fixed, we expect better unsprung mass resonance control over the wheel hop frequency region. This will result in a decrease in the tire contact load variation (thus tire deflection). In this case, however, average damping force transmitted to the sprung mass will be increased due to the larger value of \(B_{d}\), which in turn, will increase the sprung mass acceleration levels. Therefore, the selection of \(B2_{sky}\) for the fixed value of \(B1_{sky}\) should be made based on the relative importance between the sprung mass acceleration (ride quality) and the tire contact load variation (road-holding property: stability). **Case III. ** If \(B2_{sky}=B1_{sky}\), then \(F_{d}=B1_{sky}\cdot\left(\dot{x}_{s}-\dot{x}_{u}\right)\) from Equation (17). This case is equivalent to a passive damper with damping coefficient of \(B1_{sky}\). **Case IV. ** If \(B1_{sky}=0\) or if \(B2_{sky}>B1_{sky}\), then too large a damping force will be generated due to the fact that the vertical speed of the unsprung mass is usually far larger than that of the sprung mass. These cases will therefore degrade the ride quality significantly. **Step I. ** For a fixed trial value of \(B1_{sky}\), which may be determined according to the conventional sky-hook theory, adjust \(B2_{sky}\) from zero up to \(B1_{sky}\) until the tire contact load variation in the vicinity of the wheel hop frequency is similar to that of a passive suspension.