# Hybrid Model-Based Analysis of Underground Articulated Vehicles Steering Characteristics
## 4 Analysis of Steering Characteristics  
Owing to the mobility of the two frames, the different rotated angles of the frames cause changes of the steering center, as shown in Figure 14. Notably, these results are obtained by the verified dynamic model. For the situation in Figure 14a, the front frame rotates with a large angle when it comes to steer, and the center of rotation is lower than the articulation point. In contrast, the rear frame rotates with a large angle in Figure 14b, and the center of rotation is higher than the articulation point. The above situations are similar with oversteering and understeering situations, while it often appears as Figure 14c during normal working conditions. However, the quantitative relationship between the motion of the two frames is ambiguous. After the validation in the above sections, the dynamic model is used to analyze the steering characteristics of UAVs and determine the quantitative relationship, in which different load conditions are involved according to the working characteristics of the LHD, with the velocity set as 2 m/s. The angular velocity response results are shown in Figure 15. As we can see from Figure 15, when the load of the front frame increases, the peak angular velocity of the front frame decays; in contrast, the rear frame has an obvious rise. According to the results in Figure 16, the trajectories of the front frame at different loads are different, although the articulation angle is the same. Moreover, the maximum error exceeds 0.3 m in the lateral direction, which is greater than the accuracy of some control systems. Therefore, the increasing load of the front frame leads to a decrease in the corresponding angular velocity, which increases the 'oversteer'. Thus, these characteristics should be involved in the control of UAVs.  
Figure 14: Steering characteristic of UAVs. (**a**) Front frame rotates with larger scale; (**b**) rear frame rotates with larger scale; (**c**) two frames rotate with comparative scales.  
Figure 13: Comparing frequency results of articulation angle by short-time Fourier transform (SFT). (**a**) SFT results of field test; (**b**) SFT results of simulation results.  
In many studies of autonomous UAVs, the front frame is selected as the representative motion and trajectory of the UAV. The motion and trajectory of the frames are the same according to the geometric characteristics when the vehicle goes through the process of steady steering (Figure 14), while the transient process of steering is ignored. Therefore, the validated model is used to analyze the difference between the two trajectories, in which the steady and transient processes are considered. The model-based analysis results of the two trajectories with a constant steering radius are shown in Figure 17, in which the velocity is set as the common velocity range in the longitudinal (x axis) direction (i.e., 2 m/s, 3 m/s, 4 m/s, respectively) and the articulation angle was set as 15 degrees under different velocity conditions.  
Indeed, the two trajectories coincide completely after the steering process according to Figure 17a, thus, it is reasonable to represent the vehicle by the front frame. However, it is worth noting that the rear frame trajectory deviates from the front frame according to the detailed information of trajectories in Figure 17b. The deviation occurs in the transient process of steering according to the time mark. The maximum error of the deviation is 0.21 m when the velocity is 2 m/s; this value is greater than the errors in some control and positioning systems used in UAVs [11,19]. The deviation decreases when the velocity increases. However, the errors are not negligible during the common velocity range according to the results shown in Figure 17b. Therefore, the deviation of the two frames cannot be neglected when it comes to the control, navigation, and positioning of autonomous UAVs.  
Figure 16: Trajectories of front frame at different load conditions.  
Figure 15: Steering response of frames at different load conditions.