Safety-Guaranteed Trajectory Tracking Control for the Underactuated Hovercraft with State and Input Constraints

This paper develops a safety-guaranteed trajectory tracking controller for hovercraft by using a safety-guaranteed auxiliary dynamic system, an integral sliding mode control, and an adaptive neural network method.The safety-guaranteed auxiliary dynamic system is designed to implement system state and input constraints. By considering the relationship of velocity and resistance hump, the velocity of hovercraft is constrained to eliminate the effect of resistance hump and obtain better stability. And the safety limit of drift angle is well performed to guarantee the light safe maneuvers of hovercraft tracking with high velocities. In view of the natural capabilities of actuators, the control input is constrained. High nonlinearity andmodel uncertainties of hovercraft are approximated by employing adaptive radical basis function neural networks. The proposed controller guarantees the boundedness of all the closed-loop signals. Specifically, the tracking errors are uniformly ultimately bounded. Numerical simulations are implemented to demonstrate the efficacy of the designed controller.


Introduction
A hovercraft (Figure 1) is supported totally by its air cushion, with a flexible skirt system around its periphery to seal the cushion air [1].The hovercraft 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 trajectory tracking control of underactuated hovercraft moving with high velocities is meaningful and challenging to reduce the burden of pilot.
From a detailed review of the available literatures about the trajectory tracking control of hovercraft [2][3][4][5][6][7][8], only position errors were considered and the velocities of hovercraft were not controlled.However, the velocity is related to the resistance hump of hovercraft.From [9], the resistance hump occurs in the vicinity of Froude number   = 1 which can be calculated by   = /√  , where  is the velocity of hovercraft and   is the cushion length.From [10], two resistance humps (mainly caused by wave-making drag) are encountered for hovercraft during the acceleration process.It is shown in [1] that the resistance hump will be crossed as   increases and the craft will travel with better course stability and transverse stability.Hence, the velocity of hovercraft needs to be large enough, corresponding to large   , to avoid the resistance hump and hold the better stability.
Moreover, drift angle plays a key role in the high-speed moving process of hovercraft [11,12].If the drift angle exceeds the angle of drift which corresponds to the maximum of hydrodynamic forces, the behavior of hovercraft will be nonstable [13].The dangers caused by the drift angle include stern kickoff, plough-in, and great heeling [11].Hence, safety limit of drift angle must be strictly obeyed in the highspeed tracking process to ensure safe maneuvers of hovercraft [12].For instance, safety limit of a hovercraft shows if speed exceeds 40 knots, turning is not allowed; if speed is in the range of 25 knots∼35 knots, drift angle needs to be within the limits of 7.5 ∘ ∼2 ∘ .
Besides, from a practical viewpoint, the control input is restrained to prevent the actuators from going beyond their natural capabilities [14,15].And radical basis function neural networks (RBFNNs) are used to stabilize complex nonlinear dynamic systems and deal with model uncertainties [16][17][18].
The contributions of this paper are as follows: (i) The velocity of hovercraft is controlled within a reasonable range to avoid the effect of resistance hump and keep better stability.
(ii) The safety limit of drift angle is obeyed to get light safe maneuvers of hovercraft moving with high velocities.
(iii) The control input is constrained to handle input saturation.
This paper is organized as follows.Section 2 establishes a nonlinear model of underactuated hovercraft and the transformation of it.Section 3 proposes a safety-guaranteed auxiliary dynamic system for state and input constraints.Moreover, the controller is designed and analyzed in this section.Numerical simulation results are shown in Section 4, and the conclusion is discussed in Section 5.

Problem Formulation
2.1.Hovercraft Model Description.In general, only air propellers and rudders are available for hovercraft as shown in Figure 2. It means only the surge and yaw can be regulated directly, but without any actuators for their sway motion [19,20].
The nonlinear model of hovercraft is obtained by neglecting the roll and pitch motions. where where the drag's suffix  is the aerodynamic profile drag, wm is the wave-making drag,  is the air momentum drag, sk is the skirt drag,   ,  V ,   ,   ,  wm , and  sk are the drag coefficients,   is the cushion beam,   is the cushion length,  is the weight,  PP ,  LP , and  HP are the positive, lateral, and horizontal projection areas,  is the drift angle,  is the fan flow rate of cushion fan, ℎ is the distance between baffle and the bottom of skirt's finger,  sk is the total peripheral length of the skirts,  hov is the height of hovercraft,   and   are air and water density, and (  ,   ), (  ,   ), ( wm ,  wm ), and ( sk ,  sk ) are the coordinates of force's acting points.
and   in (3) can be obtained by in which   and   are absolute wind speed and direction.More details can be found in [1,9,21].
Remark 1.When a hovercraft is moving on a calm water surface, cushion pressure   varies within a narrow range and the heave motion is stable.This paper is the research about the horizontal motion of the hovercraft.Hence, the heave motion is not discussed and the cushion pressure   is assumed to be a constant.From Figure 2, we have In order to make  be the system state and more convenient for the constraint and control of , an improved model is derived from (1) and (5); that is,

State and Input Constraints. Saturation nonlinearities of actuators can be described by
where  max and  min are the maximum and minimum limitations of actuators,   are the designed control laws, and sat(⋅) is a generalized saturation function with the following form: Assumption 2. All position, orientation, velocity, and acceleration values of hovercraft are available for feedback.Safety limit of  and hump speed of hovercraft need to be obtained from model and real ship tests [11,12].In this paper, they are assumed to be known and available for the state constraint.Then the safe constraints of system state are defined as

Safety-Guaranteed Auxiliary Dynamic System
Proposition 3. A constraint error function is designed as follows: Δ =  1 (sat (,   ,   ) − ) where  1 > 0,  2 > 0,  is the system state, and   is the designed control input.Then an auxiliary dynamic system is designed by where  is the state of the auxiliary dynamic system,  1 ,  2 are positive constants, ,  are positive small design constants, (⋅) can be derived from the stability analysis, and deadzone(Δ, ) is a dead zone function given by

Design of the Desired States.
Desired reference trajectory is generated by a virtual ship described in the following form: Then the trajectory tracking errors are defined as For the position tracking, the desired states are designed by where   > 0 and   > 0 are control gains.
Remark 4. To guarantee the traceability of the reference trajectory under state constraints, the desired reference trajectory needs to satisfy the following conditions: (1)   , η  , and η  are all bounded, in which   = {  ,   ,   }.

Controller Design.
State tracking errors are defined as Then two integral sliding mode manifolds are given by where  1 ,  2 , and  3 are positive constants.Using ( 6), (17), and (18), the time derivatives of   and   are expressed as where To deal with high nonlinearity and model uncertainties,   (x  ) and   (x  ) are approximated by RBFNNs.
where x  ∈   is the input vector and W *  ∈   is the ideal weight vector.H  (x  ) :   →   is the basis function vector with element ℎ  (x  ) shown as follows: where   is the center of the receptive field and   is the width of the Gaussian function.The approximation error   satisfies |  | ≤   .Using (10) and (11), constraint error functions are designed as follows: where  1 > 0,  2 > 0,  1 > 0, and  2 > 0.
Then the auxiliary dynamic system is designed as where Finally, the control laws are given by where  1 ,  2 ,  1 ,  2 ,   , and   are positive constants, Ŵ = W + W *  , Ŵ = W + W *  , and   is defined in (20).And the adaptive laws are where   is the adaptive coefficient.
The position and desired state tracking errors can be made arbitrarily small by appropriately selecting design parameters.And the yaw motion will remain bounded.
Proof.The following Lyapunov function is defined: From ( 19), (20), and (29), V can be expressed as By defining   = Δ  +   and  = ,  and using ( 22), (26), and (27), we have Substituting adaptive laws (28) into it yields The process of stability analysis is, respectively, discussed in the following two cases.
It implies that there exists  > 0 such that, for all  > , where √/ can be made arbitrarily small by appropriately selecting the design parameters.Further, the following dynamics are obtained from ( 18) and (44): where   and   are arbitrarily small errors.
To prove that   and   are UUB, the Lyapunov function is given by The time derivative of this Lyapunov function along the dynamics in (45) is such that From Young's inequality, the following fact is obtained: (48) Then where Similar to the analysis in (42)∼( 44), there exists   > 0 such that, for all  >   , where   = {∫  0   ,   }.It is obvious that ∫  0    and   are UUB and will be arbitrarily small by choosing suitable parameters.
From   =   +  1 ∫  0   , we know   will be arbitrarily small.From ( 6) and ( 15), we have where It is clear that || = 1 which indicates that it is nonsingular.Then we have where   ,   are arbitrarily small errors.Furthermore, consider the following Lyapunov function candidate: The time derivative of   along the dynamics in (54) is such that where Also similar to the analysis in (42)∼(44), there exists   > 0 such that, for all  >   , where   = {  ,   }.
From the reason that   and   are arbitrarily small errors, we know that   and   are UUB and will be arbitrarily small.Also,  is continuously differentiable in the moving process of hovercraft.Hence, u is bounded.From (6), we have From (15), Remark 4, and the boundedness of   and   , we have that   and   are bounded.Then , V, and  are bounded from ( 5) and (17).Furthermore,   is bounded from (3).Therefore, it can be concluded that  will remain bounded from ( 7) and (59).This concludes the proof.
Remark 6.In order to avoid the well-known chattering problem, the sign function used in the control laws ( 26) and ( 27) can be replaced by hyperbolic tangent function which is continuous such that sign() = tanh(  ), where   is a positive scalar which can be chosen to get a very good approximation.

Simulations
Two different cases are implemented to verify the effectiveness and superiority of the proposed controller.In simulations, the main particulars and constraints of hovercraft are shown in Tables 1 and 2. The water surface is calm without waves.The comparisons of three different methods are carried out in each case.The legend "Method A" means the method in [14]; the legend "Method B" means the method without state and input saturation constraints.
Saturation coefficients are process of Methods A and B. These are absolutely not allowed for hovercraft.In contrast, you can see from Figure 20 that surge speed can still be positive, even large enough to avoid the influence of the resistance humps under the control of the proposed controller.And drift angle is also within the safety limit from Figure 21.From Figures 22 and 23, we know that the input constraint ability of the proposed controller is effective.Figures 13 and 24 show the heave position of hovercraft.

Conclusion
A safety-guaranteed trajectory tracking controller has been proposed for underactuated hovercraft in this paper.The safety-guaranteed auxiliary dynamic system is designed to deal with state and input constraints.The velocity of hovercraft is constrained to eliminate the effect of resistance  presented to illustrate the effectiveness of the proposed controller.

Figure 1 :Figure 2 :
Figure 1: A 3D model of hovercraft.Photo is from the international cooperation project described in Acknowledgments.

Table 1 :
Main particulars of hovercraft.

Table 2 :
State and input constraints.