## 2 Dynamics Model of Forklift EPS System  
According to the mechanical and electrical characteristics and parameters of the EPS system of the TFC35 forklift provided by Hefei Banyitong Science and Technology Development Co., Ltd., the EPS dynamic model and vehicle model are built to simulate and control the electric forklift 3-DOF model.  
The dynamics model of EPS establishes the relationship between the steering mechanism, the electrodynamics of the motor and the tire/road contact force. A steering mechanism model equipped with a brush permanent magnet DC motor is shown in Figure 1.  
Figure 1: EPS mechanism model.  
According to [PERSON]'s laws of motion, the equation of motion can be written as follows (Formulas (1)-(3) and Table 1):  
\[J_{c}\ddot{\theta}_{c}+B_{c}\dot{\theta}_{c}=T_{d}+K_{c}\frac{\theta_{m}}{g_{l}}- K_{c}\theta_{c} \tag{1}\]  
\[\left(J_{m}+\frac{R_{P}^{2}}{g_{l}{}^{2}}M_{r}\right)\ddot{\theta}_{m}+\left(B_ {m}+\frac{R_{P}^{2}}{g_{l}{}^{2}}B_{r}\right)\dot{\theta}_{m}=K_{l}I_{m}+K_{c} \frac{\theta_{c}}{g_{l}}-\left(\frac{K_{c}}{g_{l}{}^{2}}+\frac{K_{r}R_{p}^{2}} {g_{l}{}^{2}}B_{r}\right)\theta_{m}-\frac{R_{P}}{g_{l}}F_{r} \tag{2}\]  
\[U_{m}=\dot{I}_{m}L_{m}+I_{m}R_{m}+K_{m}\dot{\theta}_{m} \tag{3}\]  
Let \(\theta_{c},\ \dot{\theta}_{c},\ \theta_{m},\ \dot{\theta}_{m},\ \text{and}\ I_{m}\) represent the state variables in the motion process of the EPS system of the forklift, let \(\varepsilon(t)=\left[T_{d}\ F_{r}\right]^{T}\), and \(u=\left[U_{m}\right]\), \(\theta_{c},\ \dot{\theta}_{c},\ T_{c},\ \dot{\theta}_{m},\ I_{m}\) represent the output of the EPS system of the forklift in the process of movement; the following is the form of state space of the EPS system of the forklift:  
\[\left\{\begin{array}{c}\dot{x}(t)=Ax(t)+Bu(t)+M\varepsilon(t)\\ y(t)=Cx(t)\end{array}\right. \tag{4}\]  
where  
\[A=\left[\begin{array}{cccc}0&1&0&0&0\\ -\frac{K_{c}}{J_{c}}&-\frac{B_{c}}{J_{c}}&\frac{K_{c}}{J_{c}J_{c}}&0&0\\ 0&0&0&1&0\\ \frac{g_{l}K_{c}}{g_{l}^{2}l_{m}+R_{p}^{2}M_{r}}&0&-\frac{B_{r}R_{p}^{2}K_{r}+ K_{c}}{g_{l}^{2}l_{m}+R_{p}^{2}M_{r}}&-\frac{R_{p}^{2}B_{r}+g_{l}^{2}B_{m}}{g_{l}^{2} l_{m}+R_{p}^{2}M_{r}}&\frac{g_{l}^{2}K_{r}}{g_{l}^{2}l_{m}+R_{p}^{2}M_{r}}\\ 0&0&0&-\frac{K_{m}}{L_{m}}&-\frac{R_{m}}{L_{m}}\end{array}\right]\]  
\[B=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ \frac{1}{L_{m}}\end{array}\right]\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Symbol** & **Description** & **Values or Units** \\ \hline \(\theta_{c}\) & Steering column angle & rad \\ \(\theta_{m}\) & Motor angle & rad \\ \(K_{t}\) & DC motor torque constant & 0.0506 (N\(\cdot\)m)/A \\ \(K_{m}\) & Back electromotive force constant & 0.051 (V\(\cdot\)s/rad) \\ \(M_{r}\) & The mass of the steering rack & 32 kg \\ \(B_{r}\) & The damping of the steering rack & 3620 (N\(\cdot\)m)/rad \\ \(K_{r}\) & The stiffness of the steering rack & 46,000 N/m \\ \(J_{c}\) & The rotational inertia of the steering column & 0.089 kg \\ \(B_{c}\) & The damping of the steering column & 0.361 (N\(\cdot\)m\(\cdot\)s/rad) \\ \(K_{c}\) & The stiffness of the steering column & 115 (N\(\cdot\)m)/rad \\ \(J_{m}\) & The rotational inertia of the DC motor & 0.00045 kg\(\cdot\)m\({}^{2}\) \\ \(B_{m}\) & The damping of the DC motor & 0.0033 (N\(\cdot\)m\(\cdot\)s)/rad \\ \(I_{m}\) & The current of the DC motor & A \\ \(R_{m}\) & The resistance of the DC motor & 0.345 \(\Omega\) \\ \(L_{m}\) & The inductance of the DC motor & 0.00023 H \\ \(U_{m}\) & DC motor voltage & V \\ \(T_{d}\) & Steering wheel torque & N\(\cdot\)m \\ \(F_{r}\) & Steering resistance moment & N\(\cdot\)m \\ \(R_{P}\) & The radius of steering gear wheel & 0.4 m \\ \(g_{l}\) & The transmission ratio of deceleration mechanism & 16.5 \\ \hline \end{tabular}
\end{table}
Table 1: The variables of Equations (1)–(3).  
\[M=\left[\begin{array}{cc}0&0\\ \frac{1}{I_{c}}&0\\ 0&0\\ 0&\frac{S_{l}R_{p}}{S_{l}^{*}I_{m}+R_{p}^{*}M_{r}}\\ 0&0\end{array}\right]\]  
\[C=\left[\begin{array}{cccc}1&0&0&0&0\\ 0&1&0&0&0\\ K_{c}&0&-\frac{K_{c}}{S_{l}}&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\end{array}\right]\]