## 2 Forklift Model  
### 3-DOF Forklift Model  
Considering the complexity of modeling, the forklift dynamics model is simplified to a 3-DOF vehicle model, i.e., the roll motion around the x-axle, the lateral motion along the y-axle, and the yaw motion around the z-axle, with the horizon as the x-o-y plane, and the vertical direction as the z-axis. The 3-DOF forklift model is shown in Figure 1.  
The dynamic forklift model is given as follows [19]:  
\[\left\{\begin{array}{l}I_{x}\dot{p}-I_{xz}\dot{\omega}_{r}=M_{s}gh_{l}\psi-K_{ \psi}\psi-C_{\psi}p+M_{s}h_{l}v_{x}(\dot{\beta}+\omega_{r})\\ Mv_{x}(\dot{p}+\omega_{r})-M_{s}h_{l}\dot{p}=F_{ylf}+F_{yrf}+F_{ylr}+F_{yrr}\\ I_{z}\dot{\omega}_{r}-I_{xz}\dot{p}=-b(F_{ylf}+F_{yrf})+a(F_{ylr}+F_{yrr})\\ F_{y}=k\alpha\end{array}\right. \tag{1}\]  
where \(M\) is the total mass of the vehicle, \(M_{s}\) is the sprung mass of the vehicle, \(h_{l}\) is the distance between the original center of the forklift and the roll center, \(v_{y}\) and \(v_{x}\) are the longitudinal velocity and the lateral velocity at the center of gravity of the vehicle, respectively, \(\omega_{r}\) is the yaw rate of the forklift, \(\beta\) is the sideslip angle of the forklift, \(\psi\) is the body roll angle, \(p\) is the roll velocity, \(\dot{p}\) is the roll acceleration, \(K_{\psi}\) is the total roll rate, \(C_{\psi}\) is the roll damping, \(I_{x}\) and \(I_{z}\) are the moment of inertia around the \(x\)-axis and the \(z\)-axis, respectively, \(I_{xz}\) is the product of \(I_{x}\) and \(I_{z}\), \(k_{1}\), and \(k_{2}\) are the front and the rear tire cornering stiffness values, respectively; \(a\) and \(b\) represent the distances from the center of gravity to the front and the rear axles, respectively. Additionally, \(F_{yli}\) and \(F_{yri}\), denote the left and right lateral tire forces, respectively, and \(i\) represents the front and the rear. \(\alpha\), which is directly proportional to \(F\) (\(F_{ylf}\), \(F_{ylr}\), \(F_{yrf}\), \(F_{yrr}\), \(F_{yi}\)), denotes the sideslip angle of a tire.  
As \(\beta\) is always small enough, it can be approximated as  
\[\beta\approx\tan\beta=\frac{v_{y}}{v_{x}} \tag{2}\]  
Based on this, sideslip angles of the front and rear tires can be approximated as  
\[\alpha_{f}=\beta+\frac{a}{v_{x}}\omega_{r}-\delta_{f}-R_{f}\psi \tag{3}\]  
\[\alpha_{r}=\beta-\frac{b}{v_{x}}\omega_{r}-R_{r}\psi \tag{4}\]  
where \(\delta_{f}\) denotes the front steering angle, and \(R_{f}\) and \(R_{r}\)are the roll steer coefficients of the front and rear wheels, respectively.  
According to (1)-(4), the state space of a forklift can be derived as follows by considering \([\omega_{r},\beta,\psi,p]\) as state vectors and \(u(t)=\delta_{f}\) as the input vector [5]:  
\[\left\{\begin{array}{c}\dot{x}(t)=Ax(t)+Bu(t)\\ y(t)=Cx(t)\end{array}\right. \tag{5}\]  
Figure 1: Three-degree-of-freedom (3-DOF) forklift model: (**a**) Top view; (**b**) Rear view.  
where  
\[A=M_{1}^{-1}M_{2}\ B=M_{1}^{-1}M_{3}\ C=\left[\begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right]\] \[M_{1}=\left[\begin{array}{cccc}0&Mv_{x}&0&-M_{s}h_{l}\\ I_{z}&0&0&-I_{xz}\\ -I_{xz}&-M_{s}h_{l}v_{x}&0&I_{x}\\ 0&0&1&0\end{array}\right]M_{3}=\left[\begin{array}{c}k_{1}\\ k_{1}a\\ 0\\ 0\end{array}\right]\] \[M_{2}=\left[\begin{array}{cccc}\frac{-k_{1}a+k_{2}b}{v_{x}}-Mv_ {x}&-k_{1}-k_{2}&(k_{1}R_{f}+k_{2}R_{r})&0\\ -\frac{k_{1}a^{2}+k_{2}b^{2}}{v_{x}}&-ak_{1}+bk_{2}&(k_{1}aR_{f}-k_{2}bR_{r})&0 \\ M_{s}h_{l}v_{x}&0&-(K_{\psi}-M_{s}gh_{l})&-C_{\psi}\\ 0&0&0&1\end{array}\right]\]