# Adaptive Optimal Control of UAV Brake Based on Parameter Estimation
## 2 Dynamic Model  
### Braking Dynamics of UAV  
The dynamics model of the UAV braking system generally includes the body dynamics model, braking wheel dynamics model, tire-runway friction, and brake actuator. Brake control design should be based on a reasonable and simplified mathematical model. According to [35], consider the system formulated as  
\[\dot{\lambda}=-\frac{1}{v_{x}}\bigg{(}\frac{R^{2}}{f_{\rm w}}+\frac{ n(1-\lambda)}{m_{\rm A}}\bigg{)}F_{\rm Z}\mu+\frac{B_{\rm w}R\omega}{v_{x}f_{\rm w}} -\frac{1-\lambda}{v_{x}}\frac{F_{\rm X}}{m_{\rm A}}+\frac{K_{\rm b}R}{v_{x}J_{ \rm w}}P_{A},\] \[\dot{P}_{\rm A}=c_{\rm b}L_{0}\omega_{\rm m}/2\pi+\Delta_{1},\] \[\dot{\omega}_{\rm m}=-\frac{1}{J}\frac{L_{0}}{2\pi}P_{\rm A}-\frac {B_{\rm w}}{J}\omega_{\rm m}+\frac{K_{\rm T}}{J}i+\Delta_{2}. \tag{1}\]  
where \(\Delta_{1}\) and \(\Delta_{2}\) are unknown bounded disturbances. The above parameters are described in Table 1. The slip rate in the system is defined as  
\[\lambda=\frac{v_{x}-R\omega}{v_{x}} \tag{2}\]  
The slip rate represents the ratio of the sliding motion of the wheel to the runway; thus, \(0\leq\lambda\leq 1\). The braking process depends on the binding force between the brake wheel and the ground, which is defined as  
\[F_{z}=\mu F_{\rm f} \tag{3}\]  
It is affected by many factors and cannot be directly measured or accurately calculated.  
We define the state of the system as \([x_{1},x_{2},x_{3}]=[\lambda,P_{A},\omega_{m}]^{T}\), and then the system is described as  
\[\begin{cases}\dot{x}_{1}=f(x_{1})+K_{1}x_{2}\\ \dot{x}_{2}=K_{2}x_{3}+\Delta_{1}\\ \dot{x}_{3}=K_{3}u-K_{4}x_{2}-K_{5}x_{3}+\Delta_{2}\\ y=\lambda\end{cases} \tag{4}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Description** & **Value** \\ \hline \(m_{A}\) & Weight of the UAV & 5000 kg \\ \(v_{x}\) & Longitudinal velocity & - \\ \(\omega_{m}\) & Wheel speed & - \\ \(F_{X}\) & Aerodynamic drag & - \\ \(n\) & Number of brake wheels & 2 \\ \(F_{f}\) & The binding force & - \\ \(c_{\rm b}\) & Brake disc stiffness coefficient & \(5.0\times 10^{6}\) N/m \\ \(L_{0}\) & Ball screw travel & 0.008 m \\ \(F_{z}\) & Main wheel support force & - \\ \(\mu\) & Coefficient of static friction & - \\ \(K_{T}\) & Motor torque constant & 0.4 Nm/A \\ \(B_{v}\) & Viscous damping coefficient & 0.02 Nm/rad \\ \(J_{w}\) & Moment of wheel & 0.95 kg \(\cdot\) m\({}^{2}\) \\ \(R\) & Effective radius of main wheel & 0.3 m \\ \(J\) & Motor moment of inertia & \(6.38\times 10^{4}\) kg \(\cdot\) m\({}^{2}\) \\ \(K_{\rm b}\) & Braking torque and pressure & \(3.7\times 10^{-4}\) \\ \(B_{w}\) & Friction coefficient of axle & 0.01 Nm/rad \\ \(P_{A}\) & The pressure acting on the & - \\  & brake disc & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of UAV and EMA.  
where \(u=i\), \(K_{1}=\frac{Rk_{b}}{v_{x}l_{w}}\), \(K_{2}=\frac{\alpha_{1}L_{0}}{2\pi}\), \(K_{3}=\frac{k_{T}}{l_{m}}\), \(K_{4}=\frac{L_{0}}{2\pi l_{m}}\), and \(K_{5}=\frac{B_{2}}{l_{m}}\) are positive constants.  
Rewrite the derivative of \(f(x_{1})\) in (4) as \(\dot{f}(x_{1})=\Theta^{T}\Psi(x_{1})\), where \(\Theta\) is the unknown parameter vector, and \(\Psi(x_{1})\) is the known regressor vector.  
Let \(x_{1d}\) be the expected slip rate set, and define the tracking error \(e_{1}\) as  
\[e_{1}=x_{1}-x_{1d} \tag{5}\]  
**Assumption 1**: _The perturbation \(\Delta_{1},\Delta_{2}\) and its derivatives \(\dot{\Delta}_{1},\dot{\Delta}_{2}\), and \(\ddot{\Delta}_{1},\ddot{\Delta}_{2}\) are bounded._  
**Assumption 2**: _Linear velocity \(v_{x}\), angular velocity of the wheel \(\omega\), brake pressure \(P_{A}\), motor speed \(\omega_{m}\), and armature current \(i\) can be measured._  
The problems to be solved in this paper are as follows:  
* Online estimation of unknown parameters of UAV braking system to ensure convergence.
* When \(t>0\), tracking errors converge quickly and steady-state tracking errors remain within a small range.
* The system control input and filtering error meet the performance function.