## 4 Numerical Simulation  
To validate the effectiveness of the integrated semi-active seat suspension, numerical simulations were performed for the different road profiles. The numerical simulations were computed using MATLAB. The ode45 was used as the solver with a step size of 0.04 s. The performance of the semi-active and passive seat suspension was compared on the bump profile and the ISO road profiles of the different grades such as the C, D and E-grade with the varying speed from 30 km/h, 60 km/h and 90 km/h. The bump profile applied to the vehicle's wheels were determined by using Equation (20):  
\[z_{r}(t)=\left\{ \begin{array}{cc}\frac{a}{2}\Big{(}1-\cos\Big{(}\frac{2\pi v _{0}}{l}t\Big{)}\Big{)},&0\leq t\leq\frac{l}{v_{0}}\\ 0,&t\geq\frac{l}{v_{0}}\end{array} \right. \tag{20}\]  
More information on Equation (19) can be found in Reference [30]. The bump profile applied to the vehicle's wheels are shown in Figure 5. The bump road profiles at 30 km/h, 60 km/h and 90 km/h are shown in Figure 5a,c,e, respectively. Figure 4: Semi-active control system for integrated seat suspension. Figure 5: _Cont._In Equation (20), \(a\) is the amplitude of the bump, which is 0.1 m, the velocity \((v_{0})\) of the vehicle which is taken as 30 km/h. The driver's head acceleration response for the bump input at different velocities for semi-active and passive seat suspension is shown in Figure 5b,d,f. In Figure 5, passive represents the passive seat suspension without any controlled damper, whereas semi-active represents the semi-active seat suspension with the MR damper controlled by FLST-PID. For instance, the semi-active seat suspension reduces the peak-to-peak value of the head acceleration by 7.68%, as compared to that of the passive seat suspension only, improving the ride comfort in case of a bump road profile at 30 kmph vehicle speed. Although this percentage reduction is value is not very large; such an observation is due to the inclusion of the vehicle suspension model in the proposed integrated seat suspension model. In this study, we give the bump profile with a 0.1 m amplitude road input to the wheels of the vehicle and suspension damper, suppressing the vibrations at first, and then the displacement of the sprung-mass \(Z_{s}\) actually serves the input to the seat suspension system. Much previous work reports road input directly applied to the seat suspension as the disturbance and this may lead to larger values of percentage reduction of the head acceleration of the semi-active seat suspension, as compared to that of thee passive system. Therefore, the proposed approach of the integrated seat suspension with a vehicle suspension and human model looks more pragmatic. [PERSON] et al. [31] proposed a semi-active MR seat suspension system with a human body model without considering the vehicle suspension in their model. They simulated their model for bump road input and used a fractional-order proportional-integral-derivative (FOPID) with a genetic algorithm controller for their semi-active seat suspension system. They reported that the peak-to-peak value of the human head acceleration came out at 7.9442 m/s\({}^{2}\). We simulated our model with the same input as that of [PERSON] et al. and found the peak-to-peak value of the head acceleration to be 3.523 m/s\({}^{2}\) (see Figure 6) and, thus, show a reduction of \(\sim\)55% (both studies use semi-active suspension models). This result shows the effectiveness of the proposed model as well as the controller implemented. The model parameters for both studies are same, however, they did not consider vehicle suspension in their study. This justifies the need to consider integrated seat suspension modelling.