# Control Design and Assessment for a Reversing Tractor-Trailer System Using a Cascade Controller
## 1 Introduction
Fuzzy controllers were used in [11; 12; 13] to control a tractor-trailer, a tractor and two trailers system and a mobile robot with three trailers, respectively, in reverse motion. A simple control law for backing a tractor-trailer used in path following was proposed in [14]. The control law is based on curvature commands calculated from a path following error. In [15; 16] a simple feedback PID controller was used to stabilize the articulated angle. The output of the path tracker is the input to the controller and the output is the steering commands. In [17] a feedback controller was developed using the combination of two action a tracking term and a corrective term. Using finite-time optimal controller and mixed logical dynamics [18] was able to control a backing tractor-trailer system. Optimal control-based methods were used in [19; 20]. The proposed methods use a tractor-trailer model and simulation to compute the best control commands to follow a given path. A genetic programming algorithm was used to find an approximately correct controller for a backing tractor-trailer system to a loading dock was proposed in [21]. In [22; 23] virtual dynamics were used to linearize a non-linearizable tractor and trailer to develop a backward controller. This paper focuses on developing a cascaded controller and a path tracking controller to control a tractor-trailer system moving in reverse for following a predefined path. The novelty of proposed controllers is in the following: (1) Utilizing the Frenet-Serret frame and differential geometry to develop a path tracking controller that tracks a predefined path. The path tracking algorithm is applied on a point that lies on the trailer instead of the tractor. (2) The use of a closed-loop cascade controller to stabilize the tractor-trailer system and perform the path-tracking commands. To the best of the authors' knowledge these methods have not been used in the field of tractor-trailer systems. The cascaded controller is an inner loop controller that is used to stabilize the tractor-trailer system especially the articulated angle to prevent the occurrence of jackknife. The outer-loop is used to control the kinematics of the tractor-trailer system by utilizing the Frenet-Serret frame. The path tracking control system will issue the trailer's heading rate command to the cascade controller to guide the tractor and trailer on a predefined path. The proposed control method eliminates the need to use a jackknife prevention system similar to the one proposed in [9] by developing a cascade controller that insures the stability of the articulated angle and angle's rate by design. The proposed concept in this paper has been developed and is used widely in unmanned aerial system as in [24,25] and to our knowledge, it has not been implemented in the field of tractor-trailer system yet, especially when the system is in reverse motion. The remainder of this paper is organized as follows. The second section reviews the mathematical model of the tractor-trailer system. Third section presents the control design scheme as well as the stability analysis of the system by using the Lyapunov method. Simulation results are discussed in the fourth section. Finally, the conclusion is presented in the final section of this paper.