ROBUST TRAJECTORY TRACKING CONTROL OF UNDERACTUATED SURFACE VEHICLES WITH PRESCRIBED PERFORMANCE

In this paper, a robust sliding mode tracking controller with prescribed performance is developed for an underactuated surface vehicle (USV) with time-varying external disturbances. Firstly, to guarantee the transient and steadystate performance of the closed-loop system, the error transformation technique is presented. Further, the design of the prescribed performance function implements predefined tracking performance constraints, which eliminate the requirement for prior knowledge about the initial errors. Then, a Lyapunov stability synthesis shows that all closed-loop signals remain bounded and the tracking errors remain strictly within the predefined bounds. Finally, simulations and a comparative study are performed to illustrate the robustness and effectiveness of the proposed robust sliding mode control scheme.


INTRODUCTION
Tracking control of underactuated surface vehicles (USVs) has been widely used in military and civil fields such as coastal patrols, mine countermeasures, oceanographic sampling, military reconnaissance, and so on [1], [2]. In the case of tracking problems, trajectory tracking is more difficult than path following, since the control laws require the USV to be driven to reach and track a time-varying trajectory on time. The challenge comes mainly from the nonlinearity and underactuation of USV dynamical systems. At the same time, USVs inevitably suffer from uncertainties, complex hydrodynamics, and time-varying external disturbances such as wind, waves, and ocean currents [3]. Therefore, an actual mathematical model of a USV is hard to obtain precisely using current modeling methods. The other challenge, however, is that the number of USV control inputs is less than the degrees of freedom, which means that its system causes non-integral constraints [4].
On the one hand, it is clear that a USV is an underactuated system with nonholonomic constraints that cannot be stabilized by continuous time-invariant feedback [5]. Many nonlinear control works have been conducted to solve the tracking problem of underactuated vehicles, such as Lyapanov-based techniques [6], robust adaptive control [7], the backstepping technique [8], sliding mode control [9], neural networks [10], and disturbance-observer-based control [11]. Considering external perturbations, a full state feedback control algorithm based on a backstepping scheme was presented to track a straight line in [8]. Reference [12] proposes a point-to-point navigation technique combined with an adaptive and backstepping method to improve the robustness under the model uncertainties. However, the repeated differential of virtual control laws in the backstepping method increases the complexity of calculations. Moreover, due to the use of the adaptive technique and neutral networks, some parameters need to be adjusted online, which makes the computing process and parameter selection harder. Nevertheless, a sliding mode control structure is simple and provides excellent tracking performance even when uncertainty and bounded disturbance are acting on the underactuated vehicle.
On the other hand, it is necessary to consider the prescribed transient and steady-state control performance in practical controller design. More recently, the prescribed performance control (PPC) method was introduced in [13]. Furthermore, PPC was used in the tracking control for several classes of nonlinear systems [14] and robots [15] and for autonomous underwater vehicles [16] and surface vehicles [17]. However, logarithmic error mapping functions used in previous studies have led to the potential singularity problem of the designed control law. For tracking control of an underactuated vehicle with ensured transient performance [18], a robust adaptive controller under environmental disturbances was put forward, but it only solved the path-following problem. In [19], a robust fault-tolerant controller was proposed to ensure that the tracking errors of a USV were within predefined performance bounds. Reference [20] investigated the distributed containment control of networked surface vessels, which achieved the prescribed transient and steady-state performance. PPC is common in fully actuated nonlinear systems but has not been innovatively extended to the control of underactuated vehicles [21].
Motivated by the above observations, to guarantee the predefined performance of USVs under time-varying external disturbances, a robust sliding mode technology is used to design a control scheme. The main contributions of this article include the following: (1) unlike the existing literature, the proposed controller can guarantee that tracking errors will not exceed a prescribed performance bound; (2) the proposed method eliminates the requirement for prior knowledge about the initial errors.

MATHEMATICAL MODELS
Before presenting the design procedure, mathematical models of the USV are described. Figure 1 shows the threedegrees-of-freedom motions of the USV with body-fixed and earth-fixed frames. It should be noted that the kinematics and dynamics of the USV with time-varying external disturbances in the horizontal plane (surge, sway, and yaw) are as follows [22]:     Control objective: The control goal of the USV system is to design a robust sliding mode tracking controller with prescribed performance under time-varying external disturbances, ensuring that the system output can track the arbitrary smooth reference trajectory d Ω at the corresponding time (refer to Figure 1). All signals in the closed-loop system remain bounded. are bounded by unknown constants, that is, and its first order derivative are bounded.
Assumption 3: All the states are measurable and available for feedback.

PRESCRIBED PERFORMANCE
As shown in Figure 2, to guarantee the prescribed transient performance and ensure that the steady-state tracking error is bounded, the following definition is introduced first.  The prescribed performance can be achieved if the following conditions hold: where ( ) e t is the system state error, is the designed parameter, and [0, ) t ∈ ∞ . Because of the convergence performance in exponential form, the prescribed performance function in this paper is defined as: To achieve the control performance (2) and (3), the error transformation method is employed to eliminate the requirement for prior knowledge about the initial errors, and the error transformation function is constructed as: where σ is the transformation error, and the transformation function ( ) S σ is strictly increasing and satisfies: From the definition of ( ) S σ and , we can get: where ( ) F ⋅ is the homeomorphic mapping function, and the derivative of ( ) t σ is: (5) and (8), it can be seen that the bounded transformation error ( ) t σ can guarantee that the prescribed performance constraint is met. Meanwhile, the two systems are the homeomorphic mapping.

DESIGN OF PRESCRIBED PERFORMANCE TRACKING CONTROL
In this section, a robust sliding mode tracking controller based on prescribed performance (robust sliding mode with prescribed performance -RSMPP) will be designed for the system (1) with time-varying external disturbances. The whole control architecture can be seen in Figure 3. The detailed design procedure is as follows: Step 1. Tracking error transformation Define the position and velocity error variables as: where , d d x y and , u v α α are the desired position and velocity, respectively.
x y e e x y     Fig. 3. The whole control architecture Considering (1) and (10), the time derivative of the variable , x y e e can be calculated as: cos sin sin cos Here, the design purpose is to make the position vector track the desired position vector with prescribed performance. As mentioned above in Section 2.2, the error transformation function is designed as: Then, we have: Step 2. Design of the desired velocities To avoid design flaws in Remark 2 [24], the desired surge and sway velocities, that is, the virtual control law, are chosen as: where the choice of the controller gains 0, 0 x y k k > > can be tuned and determines how fast the tracking errors converge to zero so that the performance of the USV is robust against uncertainties. Based on (1) and (17) Remark 2. Avoiding large tracking errors causes the virtual control law to exceed the maximum velocity of the USV and thus affects the control effect. So the virtual control law selects (17), not the following: Step 3. Prescribed performance tracking controller design Combined with (11), the sliding surfaces are chosen such that: To ensure the finite-time convergence of 1 L and 2 L to zero, the design is as follows.    (26) and (27) with the virtual control law (17) guarantees the stability of the USV closed-loop system. All states remain bounded and it can be ensured that the tracking errors remain within the given prescribed performance bounds for 0 t ∀ ≥ .
Proof: Firstly, the complete Lyapunov function is assigned: In virtue of (22) and (23) Integrating (26)   Substituting (17) into (16), the time derivative of 2 V can be written as follows: Since 0 x k > and 0 y k > , it is clear from (32) that 2 0 So, it can be found that both x e and y e converge asymptotically to zero. Furthermore, consider the following Lyapunov function candidate: V is a decreasing function, which means that r is decreasing as well from .
Consequently, the control laws proposed in (26) and (27) ensure the convergence of position and velocity errors while the yaw motion remains bounded. In accordance with the above depiction, all states of the USV closed-loop control system remain bounded.

SIMULATION RESULTS AND ANALYSIS
In this section, simulation studies are performed to illustrate the effectiveness and robustness of the proposed control scheme. A USV from the Norwegian University of Science and Technology [22] is adopted and the parameters used for it are given in Table 1 λ is a proportional coefficient, which can speed up the reaction speed of the system and reduce the steady-state error. The coefficient selected should be neither too large nor too small: a large coefficient may make the system unstable and a small coefficient can reduce the reaction speed of the system. Therefore, 1 2 , λ λ , and 3 λ should be selected carefully. Then, it can be found from the theoretical analysis of Theorem 1 that the remaining control parameters 1 2 , , x y K K k and k should be designed to be positive. As with the performance function parameters, it is obvious that the parameter 0 ρ denotes the bound of the overshoot, ρ ∞ represents the maximum allowable steady-state tracking error, and l influences the convergence rate of the tracking error. Based on the above analysis and the debugging results, the control parameters are chosen as 1     Figure 4 gives the change of positions and heading. Whether or not the impact of time-varying external disturbances is taken into account, the controller performs the tracking mission well. In comparison, the proposed controller with prescribed performance converges faster. It is shown that the proposed controller has strong robustness to time-varying external disturbances. The tracking errors are shown in Figure 5, where the convergence time of the method without prescribed performance is more than 50 s. Besides, it will lead to undesirable transient-state performance when the large overshoot exceeds the boundary. As can be seen from Figure 6, the constraints for the case without the prescribed performance method where the tracking error is greater than 0.1 m, clearly exceed the performance boundary. On the contrary, the proposed method can quickly converge to the bounded sets and will always be maintained between the prescribed upper bound and lower bound with small overshoots. The prescribed performance function is designed to ensure the performance of the tracking control without any prior knowledge about the initial errors. The yaw rate of trajectory tracking for the USV is shown in Figure 7. Compared with the proposed controller, the method without prescribed performance requires greater control forces, which results in larger overshoots. The control inputs [ ] 0 u r τ τ τ Τ = are depicted in Figure 8, where only surge force and yaw moments exist. It is shown that the control force and moment can guarantee the stability of the closed-loop system under time-varying external disturbances. It can be concluded that the designed controller ensures that the tracking errors are within the predefined bounds and produce better transient and steady-state performance compared to the one without prescribed performance.

CONCLUSION
In the presence of time-varying external disturbances, the problem of trajectory tracking control for a USV with prescribed performance has been solved in this paper. Based on error transformation technique, a robust prescribed performance tracking control law is designed in combination with sliding mode technology. Meanwhile, the prescribed performance bound approach, which is combined with Lyapunov theory, successfully ensures that the tracking errors of the USV remain within the required performance constraints. The results of the simulation and comparison show that the tracking errors of the proposed controller remain strictly within the predefined bounds, while the one without prescribed performance cannot guarantee this.