# Adaptive Trajectory Tracking Control for
## 3 Trajectory Tracking Control Design
### Yaw Rate Controller  
In this subsection, a second-order sliding mode will be employed to design yaw control law \(\tau_{r}\) to make \(v_{e}\) converge. Define the second sliding surface  
\[s_{v}\ =\ \dot{v}_{e}(t)+\lambda_{2}v_{e}(t)+\lambda_{3}\int_{0}^{t}v_{e}(\mu)d\mu \tag{19}\]  
where \(\lambda_{2}\) and \(\lambda_{3}\) are two positive design parameters. Taking the time derivative of \(s_{v}\) along (13) produces  
\[\dot{s}_{v}\ =\ \ddot{v}-\ddot{v}_{d}+\lambda_{2}(\dot{v}-\dot{v}_{d})+ \lambda_{3}v_{e} \tag{20}\]  
where  
\[\dot{v} =\ (-\frac{m_{11}}{m_{22}}ur-\frac{d_{22}}{m_{22}}v-\frac{1}{m_{22} }\Delta_{v}+\frac{b_{v}}{m_{22}})^{\prime}\] \[=\ (-\frac{m_{11}}{m_{22}}ir-\frac{m_{11}}{m_{22}}u(\frac{m_{11}-m _{22}}{m_{33}}uv-\frac{d_{33}}{m_{33}}r+\frac{1}{m_{33}}\tau_{r}-\frac{1}{m_{3 3}}\Delta_{r}+\frac{1}{m_{33}}b_{r})-\frac{d_{22}}{m_{22}}\dot{v}-\frac{1}{m_ {22}}\dot{\Delta}_{v}+\frac{\dot{b}_{v}}{m_{22}})\] \[=\ \frac{1}{m_{22}m_{33}}(-m_{11}m_{33}ir-m_{11}(m_{11}-m_{22})u^ {2}v+m_{11}d_{22}ur-m_{11}u\tau_{r}+m_{11}u(\Delta_{r}-b_{r})\] \[-\ m_{33}d_{22}\dot{v}-m_{33}(\dot{\Delta}_{v}-b_{v})) \tag{21}\]  
\[\dot{v}_{d}\ =\ \ v_{m}-v_{n} \tag{22}\]  
\[v_{m} =\ -r[\cos(\psi)\ddot{x}_{d}+\sin(\psi)\dot{y}_{d}]-\sin(\psi)_{d} +\cos(\psi)_{d}-r[\cos(\psi)\ddot{x}_{d}+\sin(\psi)\ddot{y}_{d}+v_{d}r\] \[-k(\cos(\psi)\dot{x}_{e}+\sin(\psi)\dot{y}_{e})]+k[r\cos(\psi) \dot{x}_{e}+\sin(\psi)\dot{x}_{e}+r\sin(\psi)\dot{y}_{e}-\cos(\psi)\dot{y}_{e}] \tag{23}\]  
\[v_{n}\ =\ u_{d}\frac{\tau_{r}-d_{33}r+(m_{11}-m_{22})uv}{m_{33}} \tag{24}\]  
Based on the above analysis, (20) can be rerepresented as  
\[\dot{s}_{v}\ =\ \ddot{v}-\ddot{v}_{d}+\lambda_{2}(\dot{v}-\dot{v}_{d})+ \lambda_{3}v_{e} \tag{20}\]0  
where \(v_{b}\ =\ \frac{(m_{22}u_{d}-m_{11}u)}{m_{22}m_{33}}\), \(v_{v}\ =\ \frac{-m_{11}m_{33}ir-m_{11}(m_{11}-m_{22})u^{2}v_{2}+m_{11}d_{22}ur-m_{33}d_{ 22}v-m_{22}d_{33}ru_{d}+m_{22}(m_{11}-m_{22})u_{d}uv}{m_{22}m_{33}}\) and \(v_{f}\ =\ \frac{m_{11}u(\Delta_{r}-b_{r})-m_{33}(\Delta_{v}-b_{v})}{m_{22}m_{33}}\). Similarly, in order to ensure that the second-order sliding mode surface can converge in a limited time, one can define that  
\[\dot{s}_{v}\ =\ \ddot{v}-\ddot{v}_{d}+\lambda_{2}(\dot{v}-\dot{v}_{d})+ \lambda_{3}v_{e} \tag{20}\]1  
where \(k_{v}\)e and \(\eta_{v}\) are two positive design parameters. Therefore, the corresponding yaw control law is selected as  
\[\dot{s}_{v}\ =\ \ddot{v}-\ddot{v}_{d}+\lambda_{2}(\dot{v}-\dot{v}_{d})+ \lambda_{3}v_{e} \tag{20}\]2  
where \(\tau_{re}\ =\ \frac{1}{v_{b}}(-\frac{1}{2}s_{v}\dot{\phi}_{v}h^{T}h-v_{v}+v_{m}- \lambda_{2}(\dot{v}-\dot{v}_{d})-\lambda_{3}v_{e})\) and \(\tau_{rw}\ =\ \frac{1}{v_{b}}(-k_{v}s_{v}-\eta_{v}\text{sgn}(s_{v}))\). The update law for the neural network minimum learning parameter method is taken to be  
\[\dot{s}_{v}\ =\ \ddot{v}-\ddot{v}_{d}+\lambda_{2}(\dot{v}-\dot{v}_{d})+ \lambda_{3}v_{e} \tag{20}\]3  
where \(\gamma_{v}\) and \(\kappa_{v}\) are positive design parameters. **Remark 1**. : _The reason that sliding mode control theory has strong robust ability is that the symbol function can resist unknown dynamics and time-varing disturbances better. However, with the increase of unknown dynamics and time-varing disturbances, this undoubtedly increases the degree of chattering of the control signals. If more than a critical value is exceeded, it will lead to the instability and even collapse of the control system. Therefore, neural network minimum learning parameter method is hired to compensate for unknown dynamics and time-varing disturbances, which undoubtedly reduces the chattering of control signals under the premise of ensuring system robustness._  
**Remark 2**. : _Neural network minimum learning parameter method is used instead of the RBF neural network to reduce the computing burden of the controller [22]. At the same time, in the previous literatures [39; 40; 41; 42], unknown dynamics and external disturbances are compensated separately.