# Control Design and Assessment for a Reversing Tractor-Trailer System Using a Cascade Controller
## 5 Conclusions  
In this paper, a control strategy has been developed and represented for controlling a tractor-trailer system while reversing. The controller consisted of two parts. The first part was a path tracking controller that uses a moving Fernet frame along the path. Frame transformation was used in the path tracking problem to transform the location of a point on the edge of the tractor-trailer system from a local reference frame centered at (0, 0) to a frame rotating with respect to the path angle and centered at a moving target location along the path. The rate at which the target advances has been developed and is related to both the speed of the system and the angle error between the path angle and the track heading. Many of the past researches in tractor and trailer systems used simple path tracking algorithms, while in this research differential geometry was utilized to track a virtual moving target on the path. The output of the path tracking controller, which is the required trailers track angle's rate, is a setpoint for the second controller. The second part of the proposed controller is a cascade controller with an objective to control the tractor-trailer dynamics and stabilize the articulated angle to prevent jackknifing event that will cause the system to be uncontrollable. The proposed controller was tested and evaluated through simulation on a predefined sinusoidal path with varying frequency and discontinuity. The controller achieved a good performance when compared to other controllers. The RMS value of the lateral error was around 0.22 m. Further development is needed to improve the controller to overcome path discontinuity by pre-smoothing the path. The stability of the proposed path tracking controller and cascade controller was tested using the Lyapunov method. It must be noted that the proposed controller does not control the speed of the system, and this will be a future work.  
Conceptualization, A.A. and Y.B.S.; methodology, A.A. and Y.B.S.; software, A.A. and Y.B.S.; validation, A.A., Y.B.S. and I.A.; formal analysis, A.A.; investigation, A.A.; resources, A.A. and Y.B.S.; data curation, Y.B.S.; writing--original draft preparation, A.A.; writing--review and editing, Y.B.S. and I.A.; visualization, A.A.; supervision, I.A.; project administration, Y.B.S.; funding acquisition, Y.B.S. All authors have read and agreed to the published version of the manuscript.  
Research was funded by King Saud University through Researchers Supporting Project number (RSP-2021/270), King Saud University, Riyadh, Saudi Arabia.  
The authors would like to acknowledge the authors of [10] for providing the source code that has been developed further to test the new controller.  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Criterion** & **Value** \\ \hline RMS of lateral distance Error (m) & 0.2290 \\ RMS of Heading Angle Error (Deg) & 4.7001 \\ Mean Absolute Steering (Deg) & 5.8138 \\ Mean Absolute Steer Jerk (Deg/s) & 0.7398 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Controller Metrics.