Abstract—Autonomous Underwater Vehicle (AUV) play an important role in underwater inspection mission. However, there are external disturbances and parameter uncertainties which degrade the performance of an AUV. A robust control is needed to minimize the effects of external influences on AUV’s system behaviour, subjects to the constraint of not having a complete representation of the AUV system. This paper proposed a time invariant tracking control method for AUV using robust filter approach. The proposed controller is able to achieve robustness against parameter uncertainties, model nonlinearities, and unexpected external disturbances with only rigid-body system inertia matrix information of AUV. Simulation results are presented to illustrate the performance of designed robust tracking control.
Index Terms—autonomous underwater vehicle, robust control, tracking control.
I. INTRODUCTION
Underwater mission often require a high stability Autonomous Underwater Vehicle (AUV) which is able to follow predefined trajectory with high accuracy. However, there are a lot of unpredictable disturbances in the underwater environment which have adverse effects on performance of AUV.
The control problem of underwater robot becomes more challenging when the precise mathematical representation of an AUV is very hard to achieve. It is extremely difficult to find all hydrodynamic parameters that occur in the interaction between the robot and the fluid with reasonable accuracy because of their variations against different manoeuvring conditions.
Therefore, a reliable underwater control method is needed to minimize the effects of external influences on system behaviour of an AUV, subjects to the constraint of not having its complete mathematical representation, i.e. robust control technique.
Sliding Mode Control (SMC) is one of the most powerful robust control techniques which many researchers used in tracking control of underwater vehicle. Strategy of SMC is to alter the dynamics of underwater robot by applying a discontinuous control signal that drive the system state error toward a specified surface called sliding surface and maintain the trajectory of the system state error on this surface [1].
However, in the presence of switching imperfections, there is chattering phenomena in the control input of SMC. Chattering results in high wear of thrusters and degrades performance of the system. In order to avoid chattering, some researchers change the dynamics in a small vicinity of the discontinuity surface using smoothing function such as saturation function and hyperbolic tangent function [2-3]. However, this approach only ensures the convergence to a boundary layer of the sliding surface. Thus, the accuracy and robustness of the controller are partially lost.
Second order SMC controller has been proposed to overcome the chattering effect [4-7]. Second order SMC controller can lead to continuous control signal without involving any smoothing function. This approach allows for finite-time convergence to zero of the first time derivative of sliding surface. However, the error of second order SMC controller converges to zero in a longer time than conventional SMC controller.
On the other hand, Time Delay Control (TDC) is a relatively new robust controller used in underwater environment. TDC controller is first designed by [8]. TDC controller assumes that a continuous signal remains the same during a small enough time and hence the past observation of uncertainties and disturbances can be used in the control action directly. TDC controller is able to achieve good performance for underwater robot tracking control even in the presence of sensor noise and ocean current disturbance [9-10].
However, TDC controller is unable to eliminate estimation error that arise due to the introduced delay. TDC controller requires the feedback data acquisition rate to be fast so that the delay time is short. Otherwise, TDC controller will have significant estimation error which critically affects the stability and performance of the system.
In this paper, a time invariant tracking control method using robust filter approach is designed to control the position of AUV. It is first proposed by [11]. It is able to achieve robustness against parameter uncertainties, model nonlinearities, and unexpected external disturbances with only inertia matrix information.
This paper is organized as follows: Dynamic model of an AUV is presented in Section II. Design of proposed robust control is shown in Section III. Robust analysis of designed
control system is given in Section IV. Simulation result of proposed control law is discussed in section V. Finally, conclusion of this paper is given in Section VI.
II. MATHEMATICAL MODELLING OF AUV
In this section, dynamic equation of an AUV is derived. First, two reference frames are defined, i.e. Earth-fixed frame and Body-fixed frame. Origin of Body-fixed frame is coincidence with the centre of gravity of the AUV.
For Earth-fixed frame, the x-axis points towards North, the y-axis points towards East, and the z-axis points downwards normal to the surface of Earth. For Body-fixed frame, the x-axis points from aft to fore, the y-axis points towards starboard, and the z-axis points downwards. Standard notation that complies with the Society of Naval Architects and Marine Engineers (SNAME) is used in this paper.
Normally, AUV used in inspection mission is passively stable in roll and pitch direction. Therefore, all the elements corresponding to roll and pitch directions are neglected during the derivation of dynamic equation.
The 4 Degree of Freedom (DOF) nonlinear dynamic equations of motion of the AUV in Body-fixed frame can be conveniently expressed in a vectorial setting as shown in (1)-(7), where v is vector of linear and angular velocities expressed in Body-fixed frame, MRB is rigid-body system inertia matrix, MA is added mass system inertia matrix, DL is linear hydrodynamic damping matrix, DQ is quadratic hydrodynamic damping matrix, g is vector of gravitational forces, buoyancy forces and moments, B is buoyancy force in heave direction, Wa is weight in air, τ is vector of control input forces and moment expressed in Body-fixed frame, and w is vector of environmental disturbances expressed in Body-fixed frame [12]. wgDDMM QLARB +=++++ τννν )()( (1)
[ ]Trwvu=ν (2)
][ zRB ImmmdiagM = (3)
][ ,,,, rAwAvAuAA MMMMdiagM = (4)
][ ,,,, rLwLvLuLL DDDDdiagD = (5)
][ ,,,, rQwQvQuQQ DDDDdiagD = (6)
[ ]TaWBg 000 −= (7)
The linear and angular velocities expressed in the Body-fixed frame can be decomposed in the Earth-fixed frame by using Euler angle transformation, as shown in (8)-(10), where η is vector of position and attitude expressed in Earth-fixed frame, and J is Jacobian matrix. νψη )(J= (8)
Tzyx ][ ψη = (9)
−
=
1000
0100
00cossin
00sincos
)(ψψψψ
ψJ (10)
Dynamic equation of AUV in Earth-fixed frame is needed in control system design because normally the inspection trajectory is planned in Earth-fixed frame. From (1)-(10), dynamic equation of AUV expressed in Earth-fixed frame is obtained as shown in (11), where C* is Coriolis-centripetal matrix, F is vector of control input forces and moment expressed in Earth-fixed frame, and W is vector of environmental disturbances expressed in Earth-fixed frame.
WFgDCM +=+++ ηψνηψηψ ),()()( *** (11)
)())(()( 1* ψψψ −−+= JMJMM AT
RB (12)
)()()())(()( 11* ψψψψψ −−− += JJJMMJC RBAT (13)
)())((),( 1* ψνψψν −− += JDDJD QLT (14)
τψ )(TJF −= (15)
wJW T )(ψ−= (16)
III. ROBUST TRACKING CONTROL DESIGN
In this section, the derivation of robust tracking control law using robust filter approach is presented. The presented control system is a linear time invariant robust controller.
First, the dynamic model of AUV described in (11) is rearranged into (17). An artificial signal called equivalent disturbance is created to represent the effect of parametric perturbations, model nonlinearities, coupling effect, and external disturbances on the AUV system. As shown in (18), q is the artificial signal. Notice that the thruster dynamic of the HAUV is included in the equivalent disturbance, rather than assuming the thruster response is fast enough to be ignored. FqgMRB =++η (17)
WDCJMJq AT −++= −− ηψνηψηψψ ),()()())(( **1 (18)
As shown in Fig. 1, the proposed control system adopted two loop configuration: outer loop and inner loop. The outer loop is the nominal control loop whereas inner loop is the robust control loop.
Figure 1: Block diagram of proposed controller
The mentioned control system is designed in two steps. The
first step is to design a nominal controller to get exact output tracking. The second step is to design a robust compensator based on low pass filter to reduce the influence of uncertain parameter and disturbance on the tracking performance. Therefore, there are two components in the control signal, which are nominal control signal, uN and robust compensating signal, uR, as shown in (19).
The design of the nominal control signal are shown in (20)-(23), where ηd is vector of desired position and attitude, e is vector of tracking error, KD is Positive derivative gain matrix, and KP is Positive proportional gain matrix. geKeKMu PDdRBN +++= )( η (20)
ηη −= de (21)
][ ,,,, ψDzDyDxDD KKKKdiagK = (22)
][ ,,,, ψPzPyPxPP KKKKdiagK = (23)
The nominal signal is designed based on the assumption that the equivalent disturbance is completely cancelled by the robust compensator signal. Therefore, from Eq. (17)-(21), one can get the equation for tracking error dynamics as (24). It is a second order linear homogeneous differential equation.
Desired tracking error dynamic can be achieved by choosing suitable value for both KP and KD matrixes. In this paper, all parameters in both KP and KD matrixes are chosen as (25). Therefore, the characteristic equation of (24) will having two roots matrix, α1 and α2, which have negative value in the diagonal elements and zero elsewhere. This guarantees that the tracking error will converge to zero as time goes infinity. 0=++ eKeKe PD (24)
04 ,2
, >> iPiD KK (25)
The design of robust compensating signal is given by (26) and (27), where s is Laplace operator and FLP is robust filter matrix. Robust filter is constructed as a second order low pass filter with unity gain as described by (28) and (29), where i represents the direction in surge, sway, heave, and yaw, and fs,i and fl, i are positive value parameters. )()( sqsFu LPR = (26)
If the fs,i and fl,i parameters in robust filter are sufficiently large, one can expect that the robust filter would have sufficiently wide frequency bandwidth such that the low frequency primary components of the interested signals can pass through [11]. Therefore, robust compensating signal will be able to cancel the effect of equivalent disturbance. As a result, the tracking error will converge to zero following (24).
However, the equivalent disturbance is an artificial signal which cannot be measured directly. From (17), (30) is used to get the value of equivalent disturbance. gMFq RB −−= η (30)
Noted that the output force and moments from proposed control methods is expressed in Earth-fixed frame. One needs to transform the Earth-fixed forces and moments into Body-fixed frame using Eq. (31) so that it can be used as control signal for thruster.
FJ T )(ψτ = (31)
IV. ROBUSTNESS PROPERTY ANALYSIS
In this section, the proposed closed-loop system will be proven to have robust tracking performance, i.e. for a given positive constant, ε and any bounded initial condition of e(t) and )(te , one can find a finite constant T* and sufficiently large
values of parameters fs,i and fl,i with fl,i is much larger than fs,i, such that *,)( Ttte ≥∀≤ ε .
From (17), (19), and (20), the closed-loop equation for the proposed controller is described by (32), where e(0) is initial value of tracking error , )0(e is initial value of derivative of
tracking error, I is 4 by 4 identity matrix, λe0 is a positive constant matrix which obey (33), and δF is a positive constant matrix.
∞∞
+≤ qe Fe δλ 0 (32)
0
212
1
121
2
)exp()0()0(
)exp()0()0(
e
teIe
teIe
λα
ααα
αααα
≤
−−+
−−
∞
(33)
If the parameters fs,i and fl,i have sufficiently large positive values with fl,i is much larger than fs,i, δF can be made as small as desired [11].
Furthermore, from (18), one can obtain positive constant matrixes, λq0, λq1, and λq2, such that described by (34). If δF is sufficiently small and satisfy condition stated by (35), (36) can be obtained from (32) and (34), where ςq is a positive constant matrix satisfy (37).
2
210 ∞∞∞++≤ eeq qqq λλλ (34)
FF
qq eδδ
λλ+
≤+∞
121 (35)
( )
F
q
F
qFFeqδς
δλδδλ
≤++
≤∞
00 (36)
( ) qqFFe ςλδδλ ≤++ 00 (37)
From (32), (33), and (36), one can obtain (38) where ck is a 1 by 4 matrix with zeros except one on the kth column.
qF
ktkkk
teIe
teIe
cte
ςδ
αααα
αααα
+
−−+
−−
≤ ≥
)exp()0()0(
)exp()0()0(
supmax)(max
212
1
121
2
0
(38)
Therefore, it is proved that for a given positive constant, ε and any bounded initial condition of e(t) and )(te , there is a
finite constant T* and sufficiently large values of parameters fs,i and fl,i with fl,i is much larger than fs,i, such that
Parameters of an open frame underwater robot named JHUROV are used for simulation purpose, they are shown in Table 1 [13-14]. JHUROV is a 140 kilogram ROV which is slightly negative buoyant. It is 1.5 meter long, 1 meter wide and 0.6 meter high. The parameters are identified using adaptive identification method and experimentally validated.
Table 1: Parameters of JHUROV
Direction Parameters
ARB MM + LD QD aWB −
Surge kg266 10 −kgs 11173 −kgm N0
Sway kg425 10 −kgs 12121 −kgm N0
Heave kg1603 10 −kgs 11767 −kgm N31−
Yaw 298kgm 120 −skgm kgm187 N0
There are two types of disturbance in the underwater
environment, which are current disturbance and wave disturbance. In the simulation, current disturbance is assumed to be a 10 Newton constant force whereas wave disturbance is assumed to be a sinusoidal force with 5 Newton constant amplitude and 0.1 Hertz constant frequency, as described by (39). Notice that the external disturbance is expressed in Earth-fixed frame in the North direction.
[ ]TNmNNNtW 000)2.0sin(510 π+= (39)
Simulation is done using Simulink, Matlab. Fourth order Runge-Kutta method with 0.01 second fixed step size is used. The simulations are divided into two parts. First part is to verify the station keeping capability of the proposed control system whereas second part is to verify the path following capability. For both part of simulations, the control parameters of robust filter are chosen as shown in (40) and (41). ψ,,,6, zyxif is =∀= (40)
ψ,,,30, zyxif il =∀= (41)
First part simulation is to shown the capability of proposed control system to maintain its position and attitude at desired value in the presence of external disturbances and large initial position error and small initial attitude error. Parameters in both KP and KD matrixes are chosen as (42) and (43). They are chosen in the way that the control inputs generated do not exceed the upper limit of the thrusters used, which is assumed to be 200 Newton. The desired position and attitude is shown in (44) whereas the initial position and attitude error is shown in (45).
[ ]TDK 2.22.22.22.2= (42)
[ ]TPK 4.004.004.004.0= (43)
[ ]Td radmmm 1101010=η (44)
[ ]Tradmmme 1101010= (45)
The simulation results of the first part simulation are shown in Fig. 2 – Fig. 7. Overall, the performance of proposed control system in station keeping is satisfied. The AUV is able to
maintain its position and attitude at desired values without obvious steady state position and attitude errors.
From Fig. 2, the AUV is able to reach the desired position within 250 second in the present of wave and current disturbances. Due to the limitation of the thrust force provided by an underwater thruster and large drag force faced by a box-shaped AUV, the maximum speed of the AUV is slow, i.e. approximate 0.2 meter per second. Therefore, 250 second settling time is acceptable for response of AUV position with large 10 meter initial errors. Due to the small initial yaw angle error, the settling time of the attitude response is only 25 second, which is shorter than the settling time of position response, as shown in Fig. 3.
The position and attitude tracking errors of the AUV are converged to zero without overshoot and oscillation, as shown in Fig. 4 and Fig. 5. There are no obvious steady state error in the response of tracking error. Although the parameters of AUV in surge, sway and heave direction are different from each other, the response of their tracking error in these three direction are almost the same. This is because the tracking error dynamic is following (24) regardless the AUV parameters.
Fig. 6 and Fig. 7 show that the thruster force and moment provided by the proposed control system is practicable. Furthermore, there are no high frequency oscillation in the thruster’s control input signals.
Second part simulation is to shown the capability of proposed control system to follow a predefined trajectory in the presence of external disturbances. Parameters in both KP and KD matrixes are chosen as (46) and (47) whereas the predefined trajectory is shown in (48) and (49). It is a helical path with 10 meter radius. The AUV accelerates for the initial 5s until reaches a constant speed. There is no initial position and attitude error for this simulation.
[ ]TDK 2.22.22.22.2= (46)
[ ]TPK 4.004.004.004.0= (47)
T
d radmmm
= ψ
πψψψη 5.2
sin5cos5 (48)
≥−<
=stt
stt
51.004.0
5004.0 2
ψ (49)
The simulation result is shown in Fig. 8 – Fig. 13. Overall, the performance of proposed control system is satisfied. From Fig. 8 and Fig. 9, the AUV is able to follow the predefined helical path without obvious tracking error even in the presence of external wave and current disturbances.
The position tracking error and attitude tracking error of the AUV is bounded within 0.01 meter and 0.005 radian respectively as shown in Fig. 10 and Fig. 11.
Besides, Fig. 12 shows that the thruster force response is smooth without high frequency oscillation. There is a sharp jump observed in the response of thruster moment. It is due to the sharp speed change in the desired angular velocity in yaw direction at time equal to 5 second, as shown in (63). This problem can be solved easily by providing smooth input signal to the control system.
Figure 10: Response of AUV position tracking error in path
following
Figure 11: Response of AUV attitude tracking error in path
following
Figure 12: Response of thruster force in path following
Figure 13: Response of thruster moment in path following
VI. .CONCLUSION
In this paper, an underwater tracking control method using robust filter approach is proposed. The designed controller is able to minimize the effects of external influences on AUV’s system behaviour, subjects to the constraint of not having a complete representation of the AUV system. Simulation results show that the proposed control system has great performance in both station keeping and path following tasks. The tracking error of the AUV system is bounded within small value and the control input signals generated for thrusters are practicable without high frequency oscillation.
ACKNOWLEDGMENT
The authors would like to thank RUI Grant from Universiti Sains Malaysia (USM) (Grant number: 1001/PELECT/814234) for supporting the research.
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