## 2 System Modelling and Problem Formulation
### Active Seat Suspension  
An active seat suspension prototype is built based on a modification of a normal passive seat suspension for heavy duty vehicles, as shown in Figure 1a. The seat is supported by a pair of scissor mechanisms. An actuator is installed on the center of the left side scissor structure, which comprises a servo motor and a reduction gear. Figure 1b exhibits the schematic of the active seat suspension from which we can see that the vertical active force is realized through the motor torque \(T_{a}\), which is given as  
\[F_{a}=\frac{2r_{g}}{\sqrt{L^{2}-h^{2}}}T_{a}, \tag{1}\]  
where \(L=a+b\) is the length of the scissor arm. h is the suspension height, respectively. \(r_{g}>1\) is the ratio of the reduction gear. Because the control scheme of the servo motor is beyond the research of this study, we assume that the control gain from control signals to the active torque is roughly linear:  
\[T_{a}=\kappa_{m}u+\Delta T_{a}, \tag{2}\]  
Figure 1: Active seat suspension: (**a**) seat suspension and (**b**) schematic. where \(u\) is the control input signal, \(\kappa_{m}\) is the linearized control gain, and \(\Delta T_{a}\) is the error of linearization. Combining Equations (1) and (2), the relationship between the equivalent vertical actuating force and control signal can be given as  
\[F_{a}=k_{a}u+\Delta F_{a}, \tag{3}\]  
where \(k_{a}\) is the equivalent control gain from the control signal to vertical active force, which will change with the variation of the suspension deformation. \(\Delta F_{a}\) is an equivalent disturbance term. \[ \begin{split} k_{a}=\frac{2r_{g}}{\sqrt{L^{2}-h^{2}}}\\ \Delta F_{a}=\frac{2r_{g}}{\sqrt{L^{2}-h^{2}}}\Delta T_{a}.\end{split}  \tag{4}\]  
In order to achieve better control performance, the original damper is removed, and the vertical stiffness is provided by a horizontally installed spring and the scissor mechanism. According to the geometry shown in Figure 1b, the vertical elastic force is calculated as  
\[ \begin{split} F_{s}=\frac{bhF_{sp}}{L\sqrt{L^{2}-h^{2}}}\\ F_{sp}=\frac{b}{L}\Big{(}\sqrt{L^{2}-h^{2}}-\sqrt{L^{2}-h_{0}^{ 2}}\Big{)}\kappa_{s},\end{split}  \tag{5}\]  
where \(F_{sp}\) is the elastic force of the spring with \(\kappa_{s}\) the spring rate, and h\({}_{0}\) is the suspension height in the equilibrium position. Taking the partial differential of the above equation, we can obtain the equivalent vertical stiffness as  
\[k_{s}=\frac{\partial F_{s}}{\partial h}=\frac{b^{2}\kappa_{s}}{L^{2}}\bigg{(} 1-\frac{L_{sp}}{L_{s}}-\frac{L_{sp}h^{2}}{L_{s}}\bigg{)}, \tag{6}\]  
where \(L_{s}=\sqrt{L^{2}-h^{2}}\) and \(L_{s0}=\sqrt{L^{2}-h_{0}^{2}}\) are the current and equilibrium length of the spring, respectively. It can be seen from Equation (4) that the equivalent vertical stiffness will change with the variation of the suspension height. For small vibration conditions, the stiffness can be linearized at the equilibrium position. Like the other mechatronic systems, the friction is inevitably encountered in the seat suspension. Considering that the friction force mainly results from the dry friction between the metal components, the Coulomb model is employed to depict the generalized dry friction within the system:  
\[F_{f}=-F_{c}\text{sign}(v), \tag{7}\]  
where \(F_{c}>0\) is the Coulomb friction force, \(\text{sign}(\cdot)\) is the signum function, and \(v\) is the deformation rate of seat suspension. Additionally, due to the installation of the reduction gear, external damping is generated. Unlike the dry friction force, the damping force shows obvious hysteresis characteristic, which can be described using the extended Bouc-Wen differential model [42], given as follows:  
\[ \begin{split} F_{b}&=-\kappa_{n}v+\kappa_{h}z_{h} \\ \hat{z}_{h}&=-\eta|v||z_{h}|^{n-1}z_{h}-\lambda v|z_ {h}|^{n}+\gamma v,\end{split}  \tag{8}\]  
where \(\kappa_{n}\) and \(\kappa_{h}\) are the non-hysteresis and hysteresis coefficients, respectively.