# Neural Approximation Enhanced Predictive Tracking Control of a Novel Designed Four-Wheeled Rollator
## 3 Predictive Controller Design of the Rollator Based on Neural Approximation
### The State Model of the Elderly Rollator  
The kinematical model of the assisted elderly rollator is displayed in Figure 9. \(\left(X_{r},Y_{r}\right)\) and \(\left(X_{f},Y_{f}\right)\) denote the coordinate of the rear axis and front axis, respectively. \(\delta_{f}\) and \(\varphi\) represent the steering angle and yaw angle, respectively. \(P\) denotes the circle center, \(R\) is the steering radius, and \(L\) and \(M\) represent the wheel track. In addition, the speed of the rear and front are the \(v_{r}\) and \(v_{f}\), respectively.  
Considering the same conditions of the steering radius and road curvature [47], we can obtain the rear wheel speed as  
\[v_{r}=\dot{X}_{r}\cos\varphi+\dot{Y}_{r}\sin\varphi. \tag{8}\]  
Figure 9: Model of the assistant elderly walker system.  
Then, we can obtain the kinematic model of the assistant system as:  
\[\begin{split}&\dot{X}_{f}\sin\left(\varphi+\delta_{f}\right)-\dot{Y }_{f}\cos\left(\varphi+\delta_{f}\right)=0,\\ &\dot{X}_{r}\cos\varphi-\dot{Y}_{r}\sin\varphi=0.\end{split} \tag{9}\]  
According to Equation (9), we can define:  
\[\begin{split}&\dot{X}_{r}=v_{r}\cos\varphi,\\ &\dot{Y}_{r}=v_{r}\sin\varphi.\end{split} \tag{10}\]  
According to the steering relation of the rollator, we can achieve the rollator state function as  
\[\begin{split}& X_{f}=X_{r}+L\cos\varphi,\\ & Y_{f}=Y_{r}+L\sin\varphi.\end{split} \tag{11}\]  
Meanwhile, the following yaw angular velocity \(\omega\) can be described as:  
\[\omega=\frac{v_{r}\tan\delta_{f}}{L}. \tag{12}\]  
Thus, the trajectory tracking control of the assistant walker system can be represented as:  
\[\left[\begin{array}{c}\dot{X}_{r}\\ \dot{Y}_{r}\\ \dot{\varphi}\end{array}\right]=\left[\begin{array}{c}\cos\varphi\\ \sin\varphi\\ 0\end{array}\right]v_{r}+\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\omega, \tag{13}\]  
where \(\xi_{s}=\left[X_{r},Y_{r},\varphi\right]^{T}\) is the system state and \(u_{S}=\left[v_{r},\omega\right]^{T}\) is the control state.  
To further address the dynamic model of the proposed rollator system, according to the [53; 54; 55], the rollator dynamic model is as follows:  
\[\begin{split}& m\ddot{x}=m\dot{y}\omega_{c}+F_{a1}\cos\delta_{f}+F_{ a2}\cos\delta_{f}+F_{a3}+F_{a4},\\ & m\dot{y}=-m\dot{x}\omega_{c}+F_{b1}\cos\delta_{f}+F_{b2}\cos\delta_{f}+F_{ b3}+F_{b4},\\ & I_{z}\ddot{\varphi}=A\left(F_{b1}\cos\delta_{f}+F_{b2}\cos\delta_{f} \right)-B\left(F_{b3}+F_{b4}\right)+M\left(-F_{a1}\cos\delta_{f}+F_{a2}\cos \delta_{f}-F_{a3}+F_{a4}\right),\end{split} \tag{14}\]  
where \(F_{a1}\), \(F_{a2}\), \(F_{a3}\) and \(F_{a4}\) are the wheel force of left front, right front, left rear and right front, respectively, in the \(OX\) direction. \(I_{Z}\) is the rotational inertia and \(\omega_{c}\) the center yaw velocity.  
In order to calculate the lateral force of the tire, the following assumptions can be addressed as:  
\[\begin{split}& F_{b1}=\psi_{\delta F}\Gamma_{\delta F},\\ & F_{b2}=\psi_{\delta B}\Gamma_{\delta B},\\ &\psi_{\delta F}=\beta+\frac{M\omega_{r}}{v_{x}}-\delta_{f},\\ &\psi_{\delta B}=\beta+\frac{M\omega_{r}}{v_{x}},\end{split} \tag{15}\]  
where \(\psi_{\delta F}\) and \(\psi_{\delta B}\) are tire cornering angle, \(\Gamma_{\delta F}\) and \(\Gamma_{\delta B}\) are cornering stiffness, and \(\beta\) is the slip angle.  
At the same time, to manage stable tracking control, the slip ratio of the rollator should be constrained as  
\[\begin{split}& J\dot{\omega}=RF_{x}-RF_{\omega}-T_{b},\\ & M\dot{\omega}=-F_{x}-F_{y}.\end{split} \tag{16}\]We assume that the system state is \(x_{1}=\frac{v}{R},\quad x_{2}=\omega\) and \(x_{3}=S\), then the tire function can be transformed as:  
\[\dot{x}_{1} =-\frac{F_{v}+F_{2}\mu}{MR}, \tag{17}\] \[\dot{x}_{2} =\frac{F_{2}R\mu-F_{w}R-T_{b}}{J},\] \[\dot{x}_{3} =\frac{1}{v}\left[\frac{\left(S-1\right)\left(F_{v}+F_{2}+\mu F_{ 2}\right)}{M}+\frac{F_{z}R^{2}\left(T_{b}-\mu\right)}{J}\right].\]