THE PAPER | Trajectory Tracking Control for Underactuated USV by Using Backstepping Approach
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THE PAPER | Trajectory Tracking Control for Underactuated USV by Using Backstepping Approach

By Zaopeng Dong, Lei Wan, Yueming Li1, Guocheng Zhang  and Tao Liu

Unmanned Surface Vessel (USV) is attracting more and more attention from researchers all over the world because of its extensive applications in military reconnaissance, homeland security, shallow-water surveys, environmental monitoring and coordinating working with Autonomous Underwater Vehicle (AUV) (Campbell et al., 2012; Martin, 2013; Sharma et al., 2014).

On 4 June 2013, China's Ministry of Transport and Maritime Bureau  revealed a new engineering prototype unmanned surface vehicle (USV).

Image Attribute: On June 4, 2013, China's Ministry of Transport and Maritime Bureau  revealed a new engineering prototype unmanned surface vehicle (USV).  The fully enclosed vessel is designed for hydrographic surveys of shoals and reefs in the South China Sea.  The craft has a maximum speed of 18 knots and can be controlled remotely or navigate autonomously.  The USV features a commercial radar and forward looking sonar to assist in automatic surface and underwater obstacle avoidance, and a multi-beam side scan sonar for underwater survey work.

As USV is usually controller remotely by humans, an effective and reliable motion controller for its autonomous sailing is very important. A typical motion control problem for USV is trajectory tracking, which is concerned with the design of control laws that force USV to reach and follow a time parameterized reference trajectory. Most of the deployed and developing USVs are underactuated as they are not actuated in the sway axis for economic and practical considerations. We can see in (Sharma et al., 2014; Do et al., 2004; Fredriksen and Pettersen, 2006) that trajectory tracking controller design for fully actuated vehicle is not so hard while it is especially a challenging for underactuated USV because of its nonholonomic constraints and cannot be fully feedback linearized. 

Important Contributions in the Field of Underactuated Control System
In order to overcome the difficulties of trajectory tracking control of under-actuated USV, different nonlinear control methods have been proposed in last few years, such as 
  • Sliding Mode Control (SMC) (Cheng et al., 2007; Ashrafiuon et al., 2008; Soltan et al., 2009; Yu et al., 2012; Fahimi and Van Kleeck, 2013),
  • Back Stepping Technique (Do and Pan, 2005; Chen and Tan, 2013; Sonnenburg and Woolsey, 2013; Liao et al., 2014), 
  • Lyapunov’s Direct Method (Ma and Xie, 2013), 
  • Dynamic Surface Control (DSC) (Chwa, 2011), 
  • Robust Control Method (Gierusz et al., 2007; Yang et al., 2014), 
  • Intelligent Control Technology (Gierusz et al., 2007; Zhang et al., 2011) and 
  • Hybrid Control Technology (Liu et al., 2014), etc (Harmouche et al., 2014; Katayama and Aoki, 2014; Serrano et al., 2014; Wu et al., 2014).
SMC is one of the most widely used trajectory tracking control methods for USV. Considering the limitation of full state feedback linearization for trajectory tracking problem of USV, a multivariable SMC controller is designed in (Cheng et al., 2007), and stability of the control law is proved by Lyapunov theory. A first-order sliding surface in terms of surge tracking errors and a second-order surface in terms of lateral motion tracking errors are introduced into the SMC law in (Ashrafiuon et al., 2008), where it guarantees the position tracking errors of USV converge to zero and meanwhile the rotational motion remains bounded. Furthermore in paper (Yu et al., 2012), uncertainty associated with the hydrodynamic damping coefficients of the ship is discussed while controller design method is the same with that in (Ashrafiuon et al., 2008). Moreover in paper (Soltan et al., 2009), a set of two Ordinary Differential Equations (ODEs) in terms of the position state feedback is used for transitional trajectory between the USV’s initial condition and the desired trajectory such that the ODE solution converges to the desired trajectory path, which solve the limitations factor of SMC law designed that it can only guarantee position tracking as long as the USV’s initial conditions are on the desired trajectory. A nonlinear robust model-based sliding mode controller is designed in (Fahimi and Van Kleeck, 2013), where the concept of shifting the control point is first tried for trajectory tracking of under-actuated surface vessels.

Backsteeping technique is another frequently used nonlinear control method for trajectory tracking of USV. A back stepping technique based controller that forces position and orientation of under-actuated ship to globally track a reference trajectory is designed in (Do and Pan, 2005), which not required that the reference trajectory be generated by a ship model. A nonlinear back stepping controller which show excellent trajectory tracking performance even for aggressive and variable speed trajectories is proposed in (Sonnenburg and Woolsey, 2013), which is more reliable than the PD cascade approach. 

Block Diagram Attribute: Inner structure of the control system: first is a linear 
filter used as memory, followed by nonlinear backstepping control

An adaptive back stepping controller is proposed in (Chen and Tan, 2013) for fully actuated surface vessels with the option of high-gain observer for output feedback control, where the stability of the closed-loop systems is explored through Lyapunov theory. Moreover in (Liao et al., 2014), under the transformation of tracking control problem into stabilization problem of trajectory tracking error equation, a nonlinear state feedback controller based on backstepping technique is developed and the stability of the system is proved by Lyapunov theory.

In addition, A trajectory tracking controller which achieve global k-exponential convergence of state to the desired reference trajectory is designed based on Lyapunov’s direct method in (Ma and Xie, 2013), where Persistent Exciting (PE) conditions is needed. A global trajectory tracking controller based on DSC for under-actuated ship is proposed in (Chwa, 2011), where the controller is designed using the linearization of kinematic and dynamic systems similarly as in the back stepping technique. A complex trajectory tracking control system based on two different controllers connected in parallel, one is robust controller and the other is fuzzy logic controller, is presented in (Gierusz et al., 2007) for autonomous model of the Very Large Crude Carrier (VLCC). A trajectory tracking robust control law has been designed for fully actuated surface vessels in the presence of uncertain time-variant disturbances in (Yang et al., 2014), where vectorial back stepping technique based disturbance observer is employed to compensate disturbance uncertainties. Considering the coupling interactions among forces from each Degree of Freedom (DOF) and nonlinear characteristics of the hydrodynamic damping, a Neural Network Feedback Feedforward Compensator (NNFFC) controller is designed in (Zhang et al., 2011) for trajectory tracking control of a surface ship.

A hybrid controller based on adaptive technique and hierarchical SMC is presented in (Liu et al., 2014), where adaptive technique is employed to deal with the uncertainties of the mathematical model while hierarchical SMC is used to deal with the under-actuation of surface vessels. In paper (Wu et al., 2014), a novel finite-time switching controller based on the inherent cascaded interconnected structure of the ship dynamics is developed for the ship tracking a reference trajectory generated by a virtual ship. A global tracking controller based on saturated-state feedback control method for under-actuated ship is designed in (Harmouche et al., 2014), where the controller still works and remains stable with observers in the absence of velocity measurements. The problem of straight line trajectory tracking control of under actuated ships with both state and output feedback controllers is addressed and analyzed by introducing a nonlinear sampled-data control theory in (Katayama and Aoki, 2014), where state feedback controllers and reduced-order observers based on Euler approximate models are combined to obtain output feedback controllers. A trajectory tracking controller for under actuated ships is designed based on searching for conditions under which the system of linear equations had exact solution in (Serrano et al., 2014), where the method proposed does not need a coordinate transformation and the algorithm is implemented directly on the ship’s micro-controller.

Though researchers have made a lot of contributions and proposed many pioneering methods for trajectory tracking control of USV in the literature mentioned above, we can still see some limitations in them. In paper (Cheng et al., 2007; Chen and Tan, 2013; Yang et al., 2014), only fully actuated controller is designed for trajectory tracking, it is not suitable for most USV as they are under-actuated. In addition in paper (Soltan et al., 2009; Ma et al., 2013; Harmouche et al., 2014; Liao et al., 2014), the well know assumption that PE conditions of yaw velocity is needed, so a straight line reference cannot be tracked, while in paper (Katayama and Aoki, 2014), only straight line trajectory tracking can be achieved. Moreover in paper (Fahimi and Van Kleeck, 2013; Sonnenburg and Woolsey, 2013), trajectory tracking problem is decomposed into several sub-problems, such as separately considering of course control and position control, thus it would lead to the loss of global stability of the overall system and then the system would only be stable in some certain conditions. Though Wu Y. Q. and Zhang Z. C. try to relax the PE conditions by designing finite-time switching controller in paper (Wu et al., 2014), the control performance is not good enough as steady position tracking errors appears.

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Motivated by the above considerations, the researchers of this article aimed to provide and prove a nonlinear back stepping trajectory tracking method which can track an arbitrary reference trajectory for underactuated USV.

For most USV in the horizontal plane, only the yaw and surge are directly actuated while the sway axis is not actuated, so a challenging under-actuated problem has been studied here. The well know assumption that PE conditions of yaw velocity is completely relaxed for trajectory tracking control of USV in this paper, thus a controller that can track both curve trajectory and straight line trajectory with high accuracy is designed.

Controller for trajectory tracking of USV is designed based on the overall system and the global stability is proved by Lyapunov theory and Barbalat’s Lemma, which is improved and derived from separate controller design in (Chen and Tan, 2013; Fahimi and Van Kleeck, 2013). In order to enhance the steady state performance and precision of the trajectory tracking controller for USV, an integral action is added into the backstepping control law. Simulation results (in the Paper) are presented to demonstrate the effectiveness of the proposed control schemes.


We would like to acknowledge the support of the National High Technology Research and Development Program 863 of PR China (Nos. 2012AA09A304 and 2014AA09A509) and National Natural Science Foundation of China (Nos. 51209025, 51409054, 51409059, 51409061 and 51579022).

About The Authors:

Zaopeng Dong, Lei Wan, Yueming Li, Guocheng Zhang - National Key Laboratory of Science and Technology on Autonomous Underwater Vehicle, Harbin Engineering University, Harbin, China

Tao Liu - National Key Laboratory of Science and Technology on Autonomous Underwater Vehicle, Harbin Engineering University, Harbin, China  and College of Automation, Harbin Engineering University, Harbin, China

Publication Details:

Int. J. Nav. Archit. Ocean Eng. (2015) 7:817~832 SNAK, 2015 pISSN: 2092-6782, eISSN: 2092-6790

© Society of Naval Architects of Korea. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)