# Target Points Tracking Control for Autonomous Cleaning Vehicle Based on the LSTM Network
## 2 Technical Background
### Vehicle Model and Stability Condition Analysis  
When deriving the target points tracking control algorithm, the kinematic vehicle model [21] is used. The kinematic vehicle model is based on the Ackermann steering geometry that assumes the line perpendicular to each wheel should intersect at the center of the turning arc of the vehicle with a turning radius of R. Vehicle tires are regarded as rigid wheels and the lateral sliding between the tires and the ground is not considered. This model is usually used to obtain the steering control law as it can approximate the motion of the vehicle reasonably well at moderate steering angles and low velocities. The kinematics model of the two-wheel vehicle is simplified as shown in Figure 1.  
Figure 1: The simplified two-wheel vehicle kinematic model.  
The vehicle is described with two sets of Cartesian coordinates, local (x-y) and global (X-Y). Local coordinate is regarded as a moving coordinate with respect to the global coordinate because the vehicle is moving constantly. Supposing the speed of the vehicle (\(v\)) in the body frame is purely in x-direction. The relationship between the front wheel steering angle \(\omega\) and the vehicle's yaw \(\theta\) is established as below. The L is the wheelbase, the \(\theta^{\prime}(t)\) is the yaw rate, \(x^{\prime}(t)\) is the X-direction speed and \(y^{\prime}(t)\) is the Y-direction speed. The following relations can be obtained by the kinematic analysis.  
\[x^{\prime}(t)=v\sin\theta \tag{1}\]  
\[y^{\prime}(t)=v\cos\theta \tag{2}\]  
\[\theta^{\prime}(t)=\frac{v\tan\omega}{L} \tag{3}\]  
The steering wheels of the cleaning vehicle is driven by stepping motors, whose speeds are proportional to the output of the controllers, so the whole steering system of the cleaning vehicle can be regarded as the first order inertia link, its formula is  
\[\dot{\omega}=-\frac{1}{T}(\omega-\omega_{U}). \tag{4}\]  
The \(\omega_{U}\) is the target angle of front wheel, \(T\) is the time constant in the first order inertia link. The relationship between the curvature of turning and front wheel angle \(\omega\) is as follows  
\[\gamma=\frac{\tan\omega}{l}. \tag{5}\]  
The \(\gamma\) is the actual turning curvature. When the front wheel angle is small, the above formula can be linearized as \(\gamma=\frac{\omega}{l}\), so the changes of the turning curvature for the cleaning vehicle also be regarded as the first order inertia link.  
\[\dot{\gamma}=-\frac{1}{T}(\gamma-\gamma u) \tag{6}\]  
The \(\gamma u\) is the target turning curvature, the \(\theta^{\prime}(t)\) in the kinematics formula can be regard as \(\theta^{\prime}(t)=v\gamma\). According to the method adopted in reference [22], when the model is in equilibrium \(y^{\prime}(t)=0\) and \(\omega=2/\pi\), the system characteristic polynomial is  
\[f(\lambda)=\lambda^{3}-\frac{1}{T}(\varphi_{\gamma}-1)\lambda^{2}-\frac{v}{T} \varphi_{\theta}\lambda+\frac{v^{2}}{T}\varphi_{y}. \tag{7}\]  
According to Routh Criterion and the parameter \(L_{d}\) in pure pursuit algorithm [23], we can get the condition of the system is in a stable state.  
\[L_{d}\geq Tv \tag{8}\]