Abstract—This paper addresses trajectory tracking control of hovercraft vehicles in the presence of actuator constraints. A saturated tracking controller is designed by utilizing hyperbolic tangent functions to reduce the risk of actuator saturation.
Adaptive robust techniques are adopted to guarantee the robustness of the control system against unknown parameters and external disturbances. A Lyapunov-based stability analysis shows that all signals of the closed-loop system are bounded and tracking errors are semi-globally uniformly ultimately bounded. Finally, simulation results are provided to illustrate the effectiveness of the proposed controller.
Keywords—Actuator saturation; adaptive robust control;
trajectory tracking; underactuated hovercraft I. INTRODUCTION
The motion control of underactuated ocean vehicles including ships, hovercrafts and underwater robots has attracted a great deal of attention from the control and robotic communities over past years [1]-[6]. The main concern in the design of controllers for such systems is that number of their independent actuators is fewer than degrees of freedom. Motivated by the challenging nonlinear nature of such vehicles and their offshore applications, some of researchers have proposed attractive solutions for the motion control of hovercraft vehicles. Fig. 1 shows a commercially available hovercraft system. In [1] and [2], the stabilization problem of a hovercraft vehicle is addressed. In [3], higher order sliding mode technique has been used to solve tracking control of a hovercraft vessel. Aguiar and Hespanha [4] proposed a tracking controller for a nonlinear underactuated hovercraft using backstepping method. A quantitative feedback theory technique is used in [5] for the tracking control of hovercrafts. In [6], an input-output linearization controller has been also proposed for the vessel steering. Point-to-point navigation of underactuated ships has been addressed in [7]. Recently, most of results have been presented for cooperative and formation control of surface vehicles [8]-[12].
This Research work was supported by research and technology program funded by Najafabad branch, Islamic Azad University under the research project “Designing tracking controllers for the navigation of autonomous ocean vessels with limited information.”
978-1-4799-6743-8/14/$31.00 ©2014 IEEE
However, most of presented works including [1]-[14]
assume that vehicle actuators are able to accept every level of input signals and generate the necessary level of torque signals. From a practical viewpoint, the generated control signals may make the actuators go beyond their natural capabilities and the saturation may not be avoidable. This, in turn, may result in poor tracking performance of the proposed controller. Furthermore, long-term saturation may lead to serious physical damages, thermal or mechanical failures of the vehicle actuators. One solution to alleviate the mentioned problem is bounding of the closed-loop error variables by applying saturation functions to the design of the tracking controller which is subject of this paper.
To the best of the author knowledge, this problem has not been addressed for the motion control of a hovercraft system with limited torque. Thus, the main contribution of this paper with respect to the available literature is designing a saturated adaptive robust tracking controller by employing hyperbolic tangent function. For this purpose, a second- order dynamic error equation is developed which helps the designer take the advantage of saturation functions.
The rest of the paper is organized as follows. The problem formulation and control objectives are presented in the next section. In Section III, the controller design and a Lyapunov stability analysis are presented. In Section IV, simulation results are provided for a hovercraft vehicle to evaluate the proposed controller. Conclusions are drawn in Section V.
II. PROBLEM STATEMENT
Notations: The following notations are used throughout
Trajectory Tracking Control of Autonomous Underactuated Hovercraft Vehicles with Limited Torque
Khoshnam Shojaei Electrical Engineering Department,
Najafabad Branch, Islamic Azad University, Najafabad, Iran Email: [email protected]
Fig. 1. A commercially available hovercraft vehicle. Courtesy http://www.griffonhoverwork.com/products-services.
this paper. λmax() (λmin()) denotes the largest (smallest) eigenvalue of a matrix. ||x|| is used as Euclidean norm of a vector x n, while the norm of a matrix A is defined as the induced norm ||A||2 = λmax(ATA). The matrix In denotes n- dimensional identity matrix and diag[●] denotes a diagonal matrix. To facilitate the subsequent control design and stability analysis, the following notations are used:
1 2
Tanh( ) : [ tanh ( ), tanh ( ),x x x , tanh ( )]xn T and
Sech( ) : diag[sech( ), , sech( )]x x1 xn with
1 2
[ , , , n]T n
x x x x where tanh( ) and sech( ) are the hyperbolic tangent function and its derivative, respectively.
A. Model Description
Consider a class of underactuated autonomous hovercraft vehicles whose mathematical models are described as follows [2]-[5]:
cos( ) sin( ), sin( ) cos( ), ,
x u v
y u v
r
(1) ( ),
( ), ( ),
u u wu
v wv
r r wr
mu mvr d u t
mv mur d v t
Jr d r t
(2) where x, y and denote the vehicle position and orientation (i.e. yaw angle), respectively, in the earth-fixed frame, the signals u,v and r represent the surge, sway and yaw velocities in the body-fixed frame, u and r are the torque signals which are provided by the actuators, wu(t), wv(t) and
wr(t) are bounded time-varying disturbances and unmodeled dynamics, m and J denote the mass and inertia parameters of the hovercraft vehicle, du, dv and dr are damping coefficients.
In order to facilitate controller design in the next subsection, the dynamic of the vehicle is re-written in the actuated directions as follows:
1 1( ) 1 w1( ) a( )
MC v D t t , (3) where M1 is a symmetric positive-definite matrix, C1(v) is the centripetal and Coriolis matrix, D1 is the hydrodynamic damping matrix which is also symmetric and positive- definite, w1 = [wu,wr]T is the vector of forces and moments induced by environmental disturbances, a = [u,r]T is the vector of actuators inputs, and
1 1
1 1
0 0
, ( ) ,
0 0 0
( )
0 , ( ) .
( ) 0
wu u
w
wr r
m mv
M C v
J
t
D d t
t d
Assumption 1: The sway velocity of the vehicle is passive bounded in the sense that supt0||v(t)|| < Bv where Bv is an unknown constant [7], [8], [13].
Remark 1: As reported in the literature [7], [8] and [13], it is easy to systematically analyze the passive-boundedness of sway velocity of ocean vehicles. Since in practice the hydrodynamic damping forces in (2) are dominant in the
sway direction and, as a result, the sway velocity is damped out by such forces, Assumption 1 is realistic. The interested reader is referred to [7] for a detailed discussion about this assumption.
Remark 2: In practice, the response of actuators and thrusters is much faster than the surface vehicle response.
Therefore, their dynamics is reasonably neglected in this paper and their trivial effects are considered as unmodeled dynamics. Otherwise, if the surface vessel is large-size and it is working at high speeds, one should take actuator dynamics into account. This normally complicates the controller design process.
Assumption 2: The disturbance signals wu(t), wv(t) and
wr(t) are bounded such that |wu(t)| wu, |wv(t)| wv and
|wr(t)| wr where wu, wv and wr are unknown positive constants.
Definition 1 (Control Problem): Given a smooth bounded reference trajectory (i.e. xd(t), yd(t) and d(t)) which is generated by an associate timing law or an open-loop path-planner, the control objective discussed in this paper is to design the surge force and yaw moment, i.e. a = [u,r]T, for a hovercraft vehicle such that the tracking errors xe = xd ‒ x, ye = yd ‒ y and e = d ‒ converge to a small ball containing the origin in the presence of uncertain dynamics, modeling errors and environmental disturbances which are induced by wave, wind and ocean currents. Furthermore, the risk of the actuator saturation can be reduced in order to provide feasible control signals and prevent a poor tracking performance in the transient response of the tracking control system.
In addition to Assumptions 1 and 2, following assumptions are also required to design the controller:
Assumption 3: The reference path of the target point is chosen such that x x x y y yd, , , , , , d d d d d d, d, and d are bounded signals.
Assumption 4: The position, orientation, velocity, and acceleration of the vehicle in all degrees of freedom are available for feedback in real-time.
B. Open-Loop Error Dynamic Equation
Motivated by [7], if the controller is designed such that the following error variables converege to zero, the above control objectives in Definition 1 are satisfied:
2 2
, ,
e d e xe ye
(4) where d denote the desired yaw angle and are defined as follows:
atan2( , ),
d y xe e
(5) where atan y x2( , ) returns the arc tangent of y x/ within
( , ]. In order to stabilize the tracking error e, the following definitions can be easily obtained from (4) and (5):
cos( ), sin( )
e e d e e d
x y (6)
By differentiating e in (4), one gets ee x xe e y yee. By replacing (1), (4) and (6) in the recent equation, one obtains
cos( ) sin( ) , ,
e e e d
e d
u v
r
(7) where d xdcos( )d ydsin( )d . Now, by defining
: [ , ]T
e e e
z as the new state vector, one may write (7) in the following form:
( ) ( , , , )
e e e d d
z R z v (8) where
sin( ) cos( ) 0
( ) ,
0 1
e d
e e
d
R z v
Differentiating (8) yields ze RR which may be re-written as R z1eR R1R1 which together with
1 1
R ze R
, and (3) yield
( )e e ( , )e e e ( , )e e T a( )
M z z C z z z D z v z R t , (9) where
1 1
1 1
1 1
1
( 1
( ) ( ) ( ),
( , ) ( ) ( ) ( , ) ( ),
( , ) ( ) ) ( ),
T
e e e
T
e e e e e e e
T
e e v D e
M z R z M R z
C z z R z M R z R z z R z
D z v R z C R z
and M z( )e C z z( , )e e D z v( , )e RT( )ze w1( )t which incorporates parametric uncertainties, unmodeled dynamics and bounded time-varying disturbances induced by wave, wind, and ocean currents.
The term is bounded as G where , e , , T( )e T
G z R z
, (10) represents the regression matrix and 4denotes the parameters vector.
Property 1: Considering that R(ze) is a full rank matrix provided that e = / 2, and recalling Assumption 1 and the fact that M1 and D1 are symmetric positive-definite matrices, the following properties can be proven for model (9) [15]:
P11: M z( )e is a symmetric and positive-definite matrix which is upper and lower bounded such that
2 2 2
( ) ,
T
m x x M z xe M x x ze
, and
0mM where m and M are defined as
2 min
: min ( ( ))
e
m e
z
M z
and
2 max
: max ( ( ))
e
M e
z
M z
,
respectively.
P12: The Coriolis matrix satisfies the following propertiesx x y z z1, 2, , ,e e 2:
(i) x M z1T( ( ) 2 ( , ))e C z ze e x10; (ii) C z x x( , )e 1 2 C z x x( , )e 2 1;
(iii) C z x( ,e 1x y C z x y C z x y2) ( , )e 1 ( , )e 2 ;
(iv) C z x x( ,e 1) 2 C x1 x2 for some constant
C 0
.
P13: D z v( , )e is a symmetric and positive-definite matrix which is upper and lower bounded such that
2 2 2
( , ) , ,
T
d x x D z v xe D x x ze v
, and
0d D where d and D are defined as
min max
: min ( ( , )), : max ( ( , )).
d D z ve D D z ve
III. CONTROLLER DESIGN AND STABILITY ANALYSIS
In this section, a saturated tracking controller is designed based on the model (9) to solve the trajectory tracking problem of the hovercraft vessel which is defined in the previous section. By defining the following filtered error signal
( ) ( ) T anh ( )
f e p e
z t z t z t , (11) the following saturated tracking controller proposed in this paper:
( ) T Tanh ( ) Tanh ( )
a t R Kp ze Kv zf uR
, (12)
where p Tp 0, Kp k Ip 2 with kp 1 and Kv 2 2 are positive-definite symmetric gain matrices. The term uR is an adaptive robust control law which is given by
ˆTanh( ˆ ( ) / )
R f d
u G G z t , (13) where ˆ 4 is updated by the following adaptive rule
ˆ GT zf (ˆ 0)
, (14) where 4 4 denotes the adaptation gain, is a small positive design parameter, 0 4 is a priori estimate of the unknown parameter .
Remark 3: It should be noted that the robust control term (13) is a continuous approximation of sign function in nonlinear robust control theory and sliding-mode control technique [15], [16]. This control action is responsible for the compensation of the uncertain nonlinearity in the system dynamic model given by (9). The robust controller (13) may be replaced by every saturation function in order to prevent the chattering in control signals. The interested reader is referred to [15] and [16] for more discussion about this robust control action and its stability analysis issues.
Remark 4: It should be noted that one may compromise between final tracking accuracy and smoothness of the control signal by parameter d in (13). As discussed in [16], smaller values of d increase the chattering in control signals and improve final tracking accuracy. Therefore, one may choose larger values for d to provide smoother and chattering-free control signals regarding limited bandwidth of vehicle actuators.
By substituting (11) and (12) into (9) and using items (ii) and (iii) of Property P12, the following closed-loop error dynamics is obtained:
( ) ( , ) ( , ) Tanh ( ) Tanh ( )
,
e f e f f e f
p e v f
R
M z z C z z z D z v z
K z K z
u
(15) where
( ) Sech ( )2 ( , ) Tanh( ) ( , Tanh( )) Tanh( )
( , Tanh( )) ( , ) Tanh( )
e p e e e p e
e p e p e
e p e f e f p e
M z z z D z v z
C z z z
C z z z C z z z
which is bounded as follows by employing item (iv) of Property P12 and Properties P11 and P13:
2
1 x 2 x
, (16) where x = [TanhT(ze), zfT]T, 1 and 2 are unknown positive constants. Finally, the stability of the proposed tracking control system is summarized by the following theorem:
Theorem 1: Consider the kinematic and dynamic models of autonomous hovercraft vehicles which are given by (1) and (2). Given a bounded continuous desired trajectory, and under Assumptions 1-4, the proposed saturated tracking controller (11)-(14) guarantees that all signals in the closed- loop system are bounded and the tracking errors are semi- globally uniformly ultimately bounded (SGUUB) and converge to a small ball containing the origin.
Proof: Consider the following Lyapunov function
2 1
1
ln cosh( ) 0.5 T 0.5 T
pi ei f f
i
V k z z M z
, (17)where : ˆ . Differentiating (17) along (11), (15) and applying item (i) of Property P12 yields
1
1
Tanh ( ) ( )
0.5 ( )
Tanh ( ) Tanh( ) Tanh( )
,
T T
e p e f e f
T T
f e f
T T
e p p e f f
T T T T
f v f f R f f
T
V z K z z M z z
z M z z
z K z z Dz
z K z z u z z
, (18)
Then, by considering (14), (18) is expressed as follows:
2 2 min
0
{ } Tanh( )
Tanh( )
(ˆ ),
p p e d f
T T T T
f v f f R f f
T T T
f
V K z z
z K z z u z G z
G z
, (19)
Then, by recalling that ( )t G, 1 x 2 x 2, considering that z KTf vTanh( )zf 0, recalling the fact
T Tanh( / d) d
h x x h h x n , x n, h , t 0 from reference [17] and the Young’s inequality
2 2
( ) / 2
ab a b , one gets
2 min
2
1 2
2 2 4
1 2
{ } Tanh( )
0.5 0.5 )
0.5 0.5 ,
(
p p e
d f
V K z
z
c x x
(20)
where
2 2 2
(1 0.5 / ) , 0.5 0 2 d
c . Provided that d 0.510.52, (20) is written as
2 2
1 2
2
( ) ( 0.5 0.5 )
( ),
V t m x x
c t
(21) where
min 1 2
min{ { }, ( 0.5 0.5 )}
m Kp p d
Hence, if m is chosen such that
2
1 2
0.5 0.5
m x
, (22) one gets
2 2
( ) m ( ),
V t c x c t (23) Thus, provided that the condition (22) is satisfied, ( )V t is strictly negative outside the compact set
( )| 0 ( ) /
x x tt x tt c
,
where xt [xT,T]T. This means that V(t) is decreasing outside the set x. Hence, one concludes that ||xt(t)|| is semi- globally uniformly ultimately bounded by considering (22).
By taking the properties of saturation functions into account, this result implies that z ze, f,L. This result completes the proof. □
IV. NUMERICAL SIMULATIONS
Some numerical simulations have been performed to illustrate the effectiveness of the proposed controller. The simulation results are depicted in this section. All of simulations are carried out using MATLAB software platform. A Gaussian white noise is also added to the measured signals including position, orientation and velocities using randn(●) function to simulate real sensors.
All of simulations are performed based on Euler approximation with a time step of 10 msec. It is assumed that the hovercraft vessel is equipped with propellers to provide the surge force and yaw moment.
0 5 10 15 20 25 30 35
-10 -5 0 5 10 15
X(meter)
Y(meter)
vehicle trajectory reference trajectory
Fig. 2. x-y plot of hovercraft trajectory for the proposed controller
For this simulation, the system parameters are selected as 25 kg
m , J 2.5 kgm2and di 5kgms ,-1 iu v r, , . In this simulation, the performance of the proposed tracking controller is evaluated for large initial posture errors. It is assumed that the hovercraft actuators are saturated such that
100 Nm
au and au 100 Nm. The reference trajectory is selected as follows:
( ) cos( ),
( ) sin( ),
d f f f
d f f f
x t x R t
y t y R t
(24) where xf = 15m, yf = 0m, Rf 15 mand f 0.02 rad/s. The control parameters are selected as p diag[0.25, 2.5],
p 10 n
K I , Kv 25In, diag[10,10,10, 0.1], and 0.0001
. It is assumed that vehicle parameters are not exactly known. The following disturbance signals are considered to evaluate the robustness of the controller:
( ) 0.55 sin( / 20), ( ) 0.55 cos( / 20), ( ) 0.3 sin( / 20),
wu wv wr
t t
t t
t t
(25) Fig. 2 shows that the hovercraft vehicle successfully tracks its desired trajectory in x-y plane by the proposed controller.
Tracking errors are depicted by Fig. 3. Fig. 4 illustrates control signals which are not breaching the actuator limits.
The estimated parameters of the upper bounding function
0 50 100 150
-20 0 20
Time(sec.) x e (m)
0 50 100 150
-50 0 50
Time(sec.) y e (m)
0 50 100 150
-10 0 10
Time(sec.)
e (rad)
Fig. 7. Tracking errors for controller (26)
0 50 100 150
-10 -5 0 5x 104
Time(sec.)
u (Nm)
0 50 100 150
-10 -5 0 5x 104
Time(sec.)
r (Nm)
Fig. 8. Torque signals for controller (26)
0 10 20 30
-10 -5 0 5 10 15
X(meter)
Y(meter)
Fig. 6. x-y plot of hovercraft for controller (26)
0 50 100 150
0 5 10 15 20 25
Time(sec.)
estimates
1
2
3
4
Fig. 5. Estimation of upper bounding function parameters
0 50 100 150
-20 -10 0 10
Time(sec.)
u (Nm)
0 50 100 150
-20 0 20 40
r (Nm)
Fig. 4. Torque signals for the proposed controller
0 50 100 150
-20 0 20
Time(sec.) xe (m)
0 50 100 150
-50 0 50
Time(sec.) ye (m)
0 50 100 150
-5 0 5
Time(sec.)
e (rad)
Fig. 3. Tracking errors for the proposed controller
are also shown by Fig. 5. A shown by these figures, all signals are bounded in the closed-loop system, tracking errors converge to a neighborhood of the origin, and subsequently, control objectives are satisfied.
In order to show the performance of the proposed saturated controller, an adaptive robust PD controller is designed as follows:
( ) T
a t R K zp e K zv f uR
, (26) where zf and uR are defined by (11) and (13), respectively.
Simulation results are provided by Fig. 6 through Fig. 8. By comparing simulation results for both controller, one definitely find that the proposed controller (12) provide a smoother transient response than the controller (26). In addition, the proposed controller generates amplitude- limited control signals which are feasible for real applications. This feature is clearly verified by simulation results even for very large initial tracking errors. Further simulation results are omitted here and are left to interested readers.
V. CONCLUSION
This work has proposed a trajectory tracking controller for an underactuated hovercraft system with limited torque input. The proposed controller utilizes hyperbolic tangent function to bound closed-loop error variables in order to alleviate the risk of actuator saturation. It makes the tracking errors converge to a small ball containing the origin in the presence of uncertain model parameters, unmodeled dynamics and environmental disturbances. A Lyapunov- based stability analysis is employed to show semi-global uniform ultimate boundedness of tracking errors. Simulation results are provided to illustrate the efficacy of the proposed control law. Further results on the saturated tracking control of surface vehicles are devoted to the future works.
REFERENCES
[1] K. Tanaka, M. Iwasaki, H. O. Wang, “Switching Control of an R/C Hovercraft: Stabilization and Smooth Switching,” IEEE Trans.
Systems, Man, And Cybernetics-Part B: Cybernetics, vol. 31, no. 6, pp. 853-863, 2001.
[2] I. Fantoni, R. Lozano, “Stabilization of a nonlinear underactuated hovercraft,” International Journal of Robust and Nonlinear Control, vol. 10, pp. 645-654, 2000.
[3] M. Defoort, T. Floquet, A. Kokosy, W. Perruquetti, “A novel higher order sliding mode control scheme,” Systems & Control Letters, vol.
58, pp. 102-108, 2009.
[4] A. Aguiar, J. P. Hespanha, “Trajectory-Tracking and Path-Following of Underactuated Autonomous Vehicles With Parametric Modeling Uncertainty,” IEEE Trans. Automatic Control, vol. 52, no. 8, pp.
1362-1379, 2007.
[5] R. Munoz-Mansilla, D. Chaos, J. Aranda, J. M. Diaz, “Application of quantitative feedback theory techniques for the control of a non- holonomic underactuated hovercraft,” IET Control Theory and Applications, vol. 6, no. 14, pp. 2188-2197, 2012.
[6] L.P. Perera, C.G. Soares, “Lyapunov and Hurwitz based controls for input-output linearisation applied to nonlinear vessel steering,” Ocean Engineering, vol. 66, pp. 58-68, , 2013.
[7] J.H. Li, P. M. Lee, B. H. Jun, Y. K. Lim, “Point-to-point navigation of underactuated ships,” Automatica, vol. 44, no .12, pp. 3201-3205, 2008.
[8] Z. Peng, D. Wang, Z. Chen, X. Hu, W. Lan, “Adaptive dynamic surface control for formations of autonomous surface vehicles with
uncertain dynamics,” IEEE Trans. on Control Systems Technology vol. 21, no. 2, pp. 513-520, 2013.
[9] R. Cui, S.S. Ge, B. Voon Ee How, Y. Sang Choo, “Leader-follower formation control of underactuated autonomous underwater vehicles,”
Ocean Engineering, vol. 37, no. 17, pp. 1491-1502, 2010.
[10] K.D. Do, “Formation Control of underactuated ships with elliptical shape approximation and limited communication ranges,” Automatica, vol. 48, pp. 1380-1388, 2012.
[11] E. Borhaug, A. Pavlov, E. Panteley, K.Y. Pettersen, “Straight line path following for formations of underactuated marine surface vessels,”
IEEE Transactions on Control Systems Technology, vol. 19, no.3, pp.
493-506, 2011.
[12] J. Ghommam, F. Mnif, “Coordinated path-following control for a group of underactuated surface vessels,” IEEE Transactions on Industrial Elctronics, vol. 56, no. 10, pp. 3951-3963, 2009.
[13] T. I. Fossen, Marine Control Systems. Marine Cybernetics, Trondheim, Norway, 2002.
[14] K. D. Do , J. Pan, Control of Ships and Underwater Vehicles: Design for Underactuated and Nonlinear Marine Systems. Springer, 2009.
[15] F. L. Lewis, D. M. Dawson, C. T. Abdallah. Robot Manipulator Control Theory and Practice. 2nd ed. Revised and Expanded, Marcel Dekker: New York, 2004.
[16] J. J. Slotine, and W. Li, Applied Nonlinear Control, Prentice-Hall, NJ, 1991.
[17] M. M. Polycarpou, “Stable Adaptive Neural Control Scheme for Nonlinear Systems,” IEEE Trans. on Automatic Control, vol. 41, no.
3, pp. 447-451, 1996.
