## 5 Practical Implementations in Engineering Applications
### Improvement of the Methodology Application  
The most obvious limitation of the methodology application lies in the complex iterative process and great computational workload of the iterative algorithm, which is also very time-consuming. In order to accelerate the iterative algorithm and make the C-HITL platform apply to a more real-time scenario, numerical solutions with compromised step sizes that are suitable for both rigid and non-rigid systems, with lower accuracy rates, can be considered for the purpose of engineering applications.  
During the simulation, we specified the integration order of MATLAB Solver. The Solver was set to an integration order of odeN [31] with a default setting of N\({}_{\text{ode}}\) = 3, that is, the ode3 Solver with a third-order accuracy. In other words, this solver was used to integrate states into each time step of a simulation that used the nonadaptive odeN variable-step solver, which achieved a compromise between the fast fixed-step-size solution and adaptive variable-step-size solution. Compared with the conventional four-order Runge-Kutta method, its accuracy was lower. However, it could greatly improve the operation speed in the iterative process. The iteration time was only 1/20 for the former, which was more suitable for engineering applications.  
Still, using the first case in Figure 4 as an example, Figure 9 shows the controlled two displacements, LSJ displacement \(x_{1}=s\) and yaw angle \(x_{2}=\theta\), by using the ode3 Solver. The two controlled displacements in Figures 4 and 9 show almost the same fluctuation amplitudes, except that the sampling step sizes are finite. However, the calculation speed greatly improved by only about 1/20 for the former. Figure 9 shows the changes in the maximum absolute values of error \(i(i=1,2,3)\) with iterative times during the 40 iterations for the three controlled displacements. The tracking errors of the LSJ vibration fluctuated between 0.085 and 0.095 after 25 iterations, which was still a very small fluctuation range, indicating that the LSJ vibration was basically stable and no large jump occurred, thus, in the same way, achieving the goal of chatter control.  
Figure 8: The C-HITL experiment plan illustrations: (**a**) C-HITL experiment plan based on OPC communication; (**b**) practical control inputs \((u_{1}\) and \(u_{2})\).  
Figure 9c shows the related, practical control inputs (\(u_{1}\) and \(u_{2}\)) by way of the C-HITL platform by using the odeN Solver. Compared with the C-HITL platform in Figure 8b, the  
Figure 9: Illustrations created by using odeN Solver. (**a**) The controlled two displacements: LSJ displacement \(x_{1}=s\) and yaw angle \(x_{2}=\theta\); (**b**) changes in the maximum absolute values of errors with iterative times; (**c**) practical control inputs displayed by HMI by using odeN Solver.  
fluctuation curves in Figure 9c appear smoother, which was the result for the large-step and fixed-step simulations and sampling. However, the corresponding speed of the C-HITL platform was faster, approximately 1/40-1/30 of the former. In summary, we can use fixed-step software simulation and large-step hardware sampling to make the proposed methodology more widely applicable for the repeatability of the research elsewhere.