Abstract
This paper develops a safety-guaranteed course controller for air cushion vehicle (ACV). As the safety criterion, the unique safety limit of ACV including turn rate (TR) and sideslip angle (SA) changes with the speed when ACV is turning. To be more intuitive to show the change of safety limit and more convenient for safety monitoring and control, dynamic safe space of ACV is proposed. If the work point is within the safe space during the manual operation or automatic control, the sailing of ACV is safe. Then, the safety-guaranteed controller is designed to keep TR and SA within the safe space during the course control process based on the dynamic safe space constraint, terminal sliding mode control, and adaptive mechanism. The adaptive mechanism can effectively estimate the system uncertainty and external disturbances online without the requirement of their upper bounds. The proposed controller guarantees the convergence of tracking error. Simulations are implemented to demonstrate the efficacy of the designed controller.
1. Introduction
An air cushion vehicle (ACV, as shown in Figure 1) is supported totally by its air cushion, with a flexible skirt system around its periphery to seal the cushion air [1]. The ACV is able to run at high speed over shallow water, rapids, ice, and swamp where no other craft can go. These “special abilities” have attracted many military and civil users with particular mission requirements. The study about the safety-guaranteed motion control of ACV moving with high velocities is meaningful to reduce the burden of drivers.
From a detailed review of the available literatures about the motion control of ACV such as Antonio Pedro Aguiar [2], Hebertt Sira-Ramirez [3], Tanaka K [4], Han B [5], Aranda [6], Zhao Jingbo [7], Morales R [8], Gerasimos G. Rigatos[9], and Shojaei Khoshnam [10, 11], the designs only concern the success of control task and the improvement of control performance. And the safety of ACV which is the necessary precondition to complete the control task is neglected by them.
The turn rate (TR) and sideslip angle (SA) play key roles for the safety of ACV in the high-speed moving process [12, 13]. If the SA exceeds the angle of drift which corresponds to the maximum of hydrodynamic forces, the behavior of hovercraft will be nonstable [14]. The dangers caused by the TR and SA include stern-kickoff, plough-in, and great heeling [12]. Hence, safety limit of TR and SA must be strictly obeyed in the high-speed moving process to ensure safe maneuvers of ACV [13]. For instance, safety limit of an ACV in [13] shows that if speed is 35 knots, the limits of TR and SA are and . If speed is 14 knots, the limits of TR and SA are and . The safety limit of TR and SA has been considered in the course controller and trajectory controller by Mingyu Fu [1, 15]. However, the constant safety limit is used in them. In practice, the safety limit changes with the speed of ACV. Hence in this paper, the changed safety limit is considered. And a dynamic safe space is established based on the relationship of the speed and the safety limit.
Moreover, sliding mode control (SMC) has attracted significant amount of interest due to its fast global convergence, simplicity of implementation, and high robustness to external disturbances [16, 17]. Compared to the conventional linear sliding mode (LSM) control, terminal sliding mode (TSM) control has superior properties such as fast and finite-time convergence and smaller steady-state tracking errors [18–22]. So a safety-guaranteed course controller is designed based on the dynamic safe space constraint and TSM method. The system uncertainty and external disturbances can be estimated online by the designed adaptive mechanism in this paper without the requirement of their upper bounds.
Remark 1. The control task in this paper is chosen as course control. It is an important and common control task for ACV due to the difference with heading control as shown in Figure 2. But it needs to be noted that the dynamic safe space constraint method is suitable to other control task of ACV for safety.
2. Problem Formulation
2.1. ACV Model Description
The three degrees of freedom (DOF) model iswhere is the inertia matrix of ACV, and are position/Euler angles and velocities, is the bounded input disturbance, , where , is the forces and moments provide by the actuators, and is the external environment forces written aswhere the suffixes , , and mean aerodynamic profile drags in three DOF, respectively. The suffixes , , and mean wave-making drag, air momentum drag, and skirt drag, respectively. and denote cushion length and beam, , , and are positive, lateral, and horizontal projection areas, means cushion area, means drift angle, is the average clearance for air leakage in static hovering mode, is the total peripheral length of the skirts, is the height of the hull, and are air and water density, and , ,, and are the coordinates of these acting points. can be obtained byin which and are absolute wind speed and direction. For more details can see [1, 17, 23].
Considering the modeling errors and the variations of parameters, and can be rewritten as follows:whereSubstituting (5) into (3) yields where is defined as follows:Then the following assumptions are made.
Assumption 2. The norm of inertia matrix is upper bounded by a positive number .
Assumption 3. The vector is upper bounded by a positive functionwhere , , and are positive numbers.
Then the system uncertainty will be bounded in the following form:This bounded property has been used by some researchers of [24–26].
2.2. Dynamic Safe Space of ACV
Lemma 4. Assuming that is any positive value, then the open interval is neighborhood of , write for , that is,
Definition 5. From Lemma 4, we define the following expression:Defining the interval of speed of ACV as , for any , the safety limit values of TR and SA are written as and , and then the safety area of ACV at the speed isThen the sailing is safe if the following condition is satisfied.The green limit values of TR and SA at the speed of are defined as and which satisfy and .
From Definition 5, we know thatThen the dynamic safe space is obtained as follows:The dynamic green space is defined as follows:Then the safe buffer space is gotten byThe relationships of , , and are shown in Figure 3. If the speed of ACV exceeds , the motion of turning is not allowed. If the speed is lower than , the TR and SA are not required to be limited. The inside space of green boundary is green space. The inside space of yellow boundary is safe space. The space between green boundary and yellow boundary is buffer space.
Theorem 6. For any , if or , the sailing of ACV is safe.
Proof. or means .
It is obvious from (18) that , where and are the safety limit of TR and SA at the speed . Hence, the sailing of ACV is safe and stability.
3. Dynamic Safe Space Constraint Controller
3.1. Controller Design
For course safe space constraint controller design, define and thenChoose the sliding mode surface and approaching law as follows based on the fast terminal sliding mode approach:where are odd numbers and . are odd numbers, and .
After considering the safe space constraint, the control law is given bywhere and
The adaptive laws arewhere , , and are arbitrary positive constants.
3.2. Stability Analysis
Theorem 1. If the control law (25), the sliding mode surface (23), approaching law (24), and adaptive laws (27) are applied to ACV, the controller is stable and the course error will converge to zero in finite time.
Proof. The following Lyapunov function is defined:where .
Then Substituting (25) into (29) yields whereUsing (27), we rewrite (30) asFrom , we haveFrom [27], we know that . It is obvious from (26), (31)-(32), and (34) that . Hence, the closed-loop system is stable.
Integrating both sides of (34), we haveThenWe can see thatThereforeAccording to Barbalat’s Lemma, we haveFrom (23), we haveThis concludes the proof.
4. Simulations
Simulations are implemented to verify the effectiveness and superiority of the proposed controller. In simulations, the main particulars of ACV are shown in Table 1.
The initial course angle is , and the desired course angle is . The bounded input disturbance is chosen as .
You can see from Figures 4 and 6–8 that the proposed dynamic safe space can effectively reflect the variations of safety limit following the speed. The course controller is verified to make the error converge to zero from Figure 5. From the comparison with general unsafe controller without considering the safety limit of ACV in Figures 6–8, the proposed safety-guaranteed course controller can effectively keep TR and SA in the safe space during the course control process.
5. Conclusion
In this paper, dynamic safe space of ACV is proposed to be more intuitive to show the change of safety limit and more convenient for safety monitoring and control. For the safety of ACV, a safety-guaranteed course controller is designed based on the dynamic safe space constraint. In the controller, the adaptive mechanism is designed to estimate the system uncertainty and external disturbances online without the requirement of their upper bounds. The proposed controller guarantees the safe space constraint and the convergence of the tracking error.
Data Availability
The model used in the simulations is one of the research achievements of our team. If the simulation data is provided, it may reveal information about the characteristics of our model and affect the protection of our research achievements. So we are sorry that we cannot support the data.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The supports from the National Natural Science Foundation of China (Grant no. 51309062) and the project “Research on Maneuverability of High Speed Hovercraft” (Project no. 2007DFR80320) are gratefully acknowledged.