## 4 Experiment and Analysis
### Analysis of Experiments
#### 4.2.3 Correlation Analysis  
In this section, the experimental data are analyzed to identify the important input and output parameters for the low-speed stability of the motorcycle.  
The avg and SD for the lead time and the MCC, for the output parameters with steering angle and steering torque are shown in Figures 9 and 10 respectively using error bars. The MCC is calculated for the leading side of the output parameters such that it can be used for estimating the steering input to achieve the low-speed stability. The results of the analysis are examined in two steps: Firstly, the MCCs between the input and the output parameters are compared to identify the output parameters which can estimate the input parameter. Secondly, the lead time between useful output parameters and input parameters are compared to assess their usability by the riders.  
Figure 8: Validation of roll rate gain values based on theory and experiments.  
Figure 7: Validation of theoretical results from experiments.  
Figure 10: Maximum correlation coefficient (MCC) (R) and lead time for output parameters with steering torque.  
Figure 9: MCC (R) and lead time for output parameters with the steering angle.  
Figure 9 shows that the roll angle and the roll rate are the output parameters that can estimate the steering angle because the MCCs for these parameters with steering angle are strong; and, the lead time is also sufficient to be used by the riders. Whereas, the correlation between the yaw angle and steering angle is weak. The correlation between steering angle and yaw rate was strong (above 0.9) at 3 and 5 km/h. Although, it was not the parameter useful for the riders because its lead time was too low and beyond the human limit to use it for estimating the steering angle.  
Similarly, Figure 10 shows that the steering torque can be estimated from roll angle, roll rate and yaw rate because the MCCs between the steering torque and these parameters are strong. Whereas, the MCC between the yaw angle and steering torque was weak. The lead time for the yaw rate with the steering torque is below 0.2 s, which was lower than typical human response time. Thus, the yaw rate cannot be used by the rider for estimating steering torque, and it is a result of the change in other parameters. Therefore, the roll angle and the roll rate are important output parameters used by the riders to estimate the steering input to balance the motorcycle.  
Further, the appropriate input parameter between the steering torque and the steering angle is selected based on two requirements. Firstly, the multiple correlations between the input parameter and output parameters i.e., roll angle and roll rate should be strong. Secondly, it is preferred to select the input that can be applied as late as possible to accommodate the delay in the system.  
The MRA was performed individually for steering angle and steering torque as a dependent parameter using Equation (19). Figure 10(a) shows that the regression correlation between the identified output parameters (i.e., the roll angle and the roll rate) and the steering angle was stronger than the same with the steering torque. This shows that the steering angle can be estimated more accurately than the steering torque. The lead time and the MCC for the steering torque with the steering angle are shown in Figure 10(b). The MCC between the parameters is strong. The negative lead time (approximately \(-0.15\) s) shows that the steering torque had a delay with respect to the steering angle, which made it a suitable parameter for accommodating the delay in the system. Although, the steering angle was selected as an appropriate input parameter as the priority is given to the accurate estimation of the input parameter from the output parameters.  
Figure 11: Analyses to select an appropriate input parameter between the steering angle and the steering torque. (**a**) Regression correlation for the steering angle and the steering torque; (**b**) Lead time and MCC (R) for the steering torque with the steering angle.