# Robust Adaptive Control for an Aircraft Landing
## 2 Mathematical Model of a Landing Gear and MR Damper  
In this paper, the mathematical model of an aircraft landing gear is taken from the previous work [18; 19; 20]. Figure 1 depicts the structural and dynamic models of a single landing gear equipped with an MR damper. Following the movement of the piston, MR fluid flows through narrow orifices in the piston head from one chamber to the other one. And, it creates viscous damping force from the pressure difference between the chambers. In addition to this general oleo-pneumatic structure, the orifices are surrounded by two coils, of which the magnetic field is controlled by applying current. When an electric current is applied, micron-sized particles in the MR fluid are rearranged along the magnetic field around the coils. This phenomenon causes the MR fluid to transform into a viscoelastic solid in milliseconds resulting in additional pressure difference and damping force. Thus, the MR damping force can be controlled by changing the electric current applied to the damper.  
Typically, the total force of an MR damper can be expressed as a combination of hydraulic force \(F_{h}\), pneumatic force \(F_{a}\), and friction force \(F_{f}\):  
\[F_{d}=F_{a}+F_{h}+F_{f} \tag{1}\]  
The hydraulic force, which is the sum of the viscous force \(F_{v}\) and the controllable MR damping force \(F_{MR}\), can be given by:  
\[F_{h}=F_{v}+F_{MR}=C\dot{s}+u\ sign(\dot{s}) \tag{2}\]  
where \(s=z_{1}-z_{2}\geq 0\) denotes the piston stroke, which is the difference between the sprung and un-sprung masses' displacement \(z_{1}\), \(z_{2}\), respectively, and u is the control input. The air chamber is filled with pressurized gas to provide an elastic force when compressed. The gas undergoes a polytropic process, and thus, the pneumatic force can be expressed as:  
\[F_{a}=\left(p_{0}\left(\frac{V_{0}}{V_{0}-A_{p}s}\right)^{n}-p_{ATM}\right)A _{p} \tag{3}\]  
The friction force appears due to the contact of the piston and cylinder, which can be expressed by [20]:  
\[F_{f}=sign(\dot{s})F\tau\left(\mu\frac{D_{b}}{D_{v}+s}\right) \tag{4}\]  
\(F_{T}\) denotes the tire force by the deflection of the wheel tire, and is expressed as:  
\[F_{T}=k_{T}z_{2}^{b} \tag{5}\]  
where \(k_{t}\) is the tire stiffness, and \(b\) is the polynomial index of the tire elasticity. In Equation (5), the tire is assumed to always contact with the ground, and thus, the un-sprung mass's displacement \(z_{2}\), i.e., the tire deformation, is always positive: \(z_{2}\geq 0\).  
As an aircraft touches down on the ground with its landing gear, the upper and lower bodies of the landing gear move separately. The equations of motion are given as:  
\[m_{1}\ddot{z}_{1}=m_{1}g-F_{d} \tag{6}\]  
\[m_{2}\ddot{z}_{2}=m_{2}g+F_{d}-F_{T} \tag{7}\]  
Figure 1: Dynamic model of a single landing gear equipped with a magnetorheological (MR) damper: (a) structure of the landing gear; (b) schematic model of the landing gear.  
\[z_{1}(t=0)=z_{2}(t=0)=0,\dot{z}_{1}(t=0)=\dot{z}_{2}(t=0)=v \tag{8}\]  
where \(m_{1}\) is the sprung mass, including the part of the aircraft mass supported by the landing gear, crew, pilot, and the upper components of the landing gear, such as the cylinder. The un-sprung mass \(m_{2}\) includes the piston, wheel, and other components below the damper. \(v\) is the sink speed of aircraft at the touchdown. All parameter values of the landing gear used in this paper are given in Table 1.