2014 14th International Conference on Control, Automation and Systems (ICCAS 2014) Oct. 22-25, 2014 in KINTEX, Gyeonggi-do, Korea Hierarchical Backstepping Control for Trajectory-Tracking of Autonomous Underwater Vehicles Subject to Uncertainties * Hsiu-Ming Wu and Mansour Karkoub
Department of Mechanical Engineering, Texas A&M University at Qatar, Doha, 23874, Qatar (
[email protected]) * Corresponding author Abstract: In this study, the hierarchical backstepping control (HBC) is used for the trajectory-tracking of autonomous underwater vehicles (AUV) subject to uncertainties (e.g., current disturbances). The proposed HBC utilizes hierarchical structures of the backstepping control based on the kinematic and dynamic models such that both the virtual velocity control and trajectory-tracking of the AUV lead to asymptotic stability. The robustness of the proposed control technique is demonstrated via injection of uncertainties into the closed-loop model. The overall closed-loop stability of the proposed control scheme is guaranteed to have uniformly ultimately bounded (UUB) performance based on the Lyapunov stability criteria. Finally, the feasibility and effectiveness of the proposed scheme are evaluated through computer simulations. Keywords: Hierarchical backstepping control, Autonomous underwater vehicles, Virtual reference velocity, Lyapunov stability criteria.
positlon the ROVs with slow velocity requirements. Yuh [12] presented a dynamic model of the AUV and an adaptive control strategy for such vehicles. The results show that the use of the adaptive control system can provide the high performance in the presence of unpredictable changes in the dynamics of the vehicle and its environment. Based on the above literature survey, the robustness and the relatively large number of degrees of freedom are the main challenges for AUVs. Accordingly, the backstepping controller with hierarchical structures is designed here based on simplified kinematic and dynamic models with uncertainties. Finally, the stability and robustness are respectively validated via Lyapunov stability criteria and computer simulations. The paper is organized as follows: In Section 2, the kinematic and dynamic models of the AUVs and problem formulations are given. The hierarchical backstepping control design is described in Section 3. The simulation results are presented and discussed in Section 4. Finally, some conclusions are made in Section 5.
1. INTRODUCTION
In recent years, control of AUVs has drawn a lot of attention in the research community. Many practical applications of AUVs such as route surveying, underwater searches as in the case of the Malaysian airlines, construction or pipeline inspection, etc. can be achieved by following a predefmed trajectory. Hence, trajectory-tracking of AUVs is a basic and essential function for such robotic system. In fact, many research efforts and different approaches to this problem have been proposed including adaptive control [\]-[2], fuzzy control [3], backstepping control [4]-[6], and robust control [7]. In this study, our proposed HBC with robustness is utilized to deal with simplified kinematic and dynamic models of an AUV subject to uncertainties. Some representative schemes available in the literatures are discussed herein about the control of AUVs. In 2000, Fossen and Blanke [8] proposed an output feedback controller that compensates for variations in thrust due to time variations in advance speed to reconstruct the axial flow velocity from vehicle speed measurements. A family of fixed and adaptive model-based controllers are proposed to compare with PD control in [9] for the low-speed maneuvering of fully actuated underwater vehicles. It was shown that the model-based controllers outperformed PD control over a wide range of operating conditions. Moreover, in [10], a self-adaptive recurrent neuro-fuzzy as a feedforward control and a PD control as a feedback controller are utilized for controlling an AUV in an unstructured environment. Additionally, issues concerning input tracking, disturbance rejection, and plant variations on ROV dynamic positioning are discussed in [\\] whose evaluations considered the use of linear PID feedback and feedforward variants, and a robust nonlinear control strategies to dynamically 978-89-93215-06-9 95560/14/$15 ©rCROS
2. MODEL OF AUVS AND PROBLEM FO RMULATION
In this section, simplified kinematic and dynamic models of an AUV including model uncertainties ar e laid out. The problem formulation for AUV syste m is discussed thereafter. 2.1 Kinematic model
The two coordinate frame systems for the AUV are illustrated in Fig. I including the inertial frame system { 0 - XYZ} and the body-fixed frame system {Oo - X0 YoZo} in a three-dimensional Cartesian
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workspace. The kinematic model of the AUY can be expressed as follows:
= q=
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that
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this implies that Tj is bounded. We may also conclude that e is UUB. Subsequently, this virtual control input is utilized as a virtual reference input in the next closed-loop. In the next subsection, true control input T is discussed so that the virtual reference input is still UUB. t �
00,
3.2 Velocity control design
In the dynamic model (2) of the AUV, the control input T is designed to stabilize the virtual reference velocity (i.e., velocity control) based on the backstepping control technique as r
= M (qc + Kleq)+Cq+ l5q+ g
(10) 4 4 . . . m x IS a posItlve and diagonal matrix. J\
where K I E Define a Lyapunov function V 2 as V 2 = Vj +2' eTqeq 1
Fig. 2 Overall control block diagram.
(11)
Taking the time derivative of (11) along the dynamics (2) and substituting (10) into it, we have
3. HIERARCHICAL BACKSTEPPING CONTROL DESIGN
In this section, the virtual reference input and true control input are respectively discussed via the backstepping control strategy with a hierarchical structure so as to attain trajectory-tracking and velocity control for an AUV system subject to uncertainties. 1193
leads to excellent performance. Moreover, in the simulations, the following additive uncertainties (13) for the system functions are included to verifY the robustness of the proposed controller. The selected frequencies of the additive uncertainties range from low to high so that true uncertainties may be reflected and stimulate certain frequency of unmodeld dynamics. L1ill = ill [0.02cos(lOt)- 0.3sin(l00tx)] , i1i1 2 = il2 [0.02cos(1Ot) - 0.3sin(1OOtx)] , L1hl = h I [-0.03sin(2t)+OAcos(200t)] , L1h2 = h2 [-0.03sin(2t)+OAcos(200t)] ,
i1h3 =h3 [-0.Ssin(300t)] , i1i 44 =i44 [0.04 cos(O. It)] , L1mll = mll [-0.02cos(St) +O.2sin(SOtu)] , L1m22 =m22 [0.02sin(t) - 0.3 cos(40t)] , L1m33 =m33 [0.03cos(O.lt) - 0.4cos(60w)] , L1m44 = m 44 [-0.02 sin(O.Str) +0.3 cos(80t)] , L1C1 2 =c1 2 [-0.02 cos(St) +0.2 sin(SOtu)] , i1c21 =c2l [0.02sin(t)- 0.3cos(40t)] , i1c41 =c41 [0.03cos(O.lt) - 0.4cos(60w)] , L1C 44 =c 44 [-0.02 sin(O.Str) +0.3 cos(80t)] , MIl =dll [-0.02cos(St)+O.2sin(SOtu)] ,
rl = �"in { Kl +XMK1} > 0; r2 = max
{11-xiY!Cfc +( iY!-1 +X)( i1Cq +i1Dq +i1g )II}.
Since it satisfies
IIel12 fJ/a
Ileq112 r2/rl , V2 :s; 0
in the upper level and
as t � 00 This implies that V2 is bounded. We may conclude that e and eq are also UUB. .
Remark I: Tn the true control law of Eq. (10),
i1d22 =d22 [ 0.02 sin(t) - 0.3 cos(40t)] , M33 =d 33 [0.03cos(0.lt)-OAcos(60w)] , M44 = d 44 [-0.02sin(0.Str) +0.3cos(80t)] , i1g11 = gll [-0.02cos(St) +0.2 sin(SOtu)] , L1g2l =g 2l [0.02 sin(t)- 0.3cos(40t)] , L1g3l =g 3l [0.03cos(0.lt)- 0.4cos(60w)] , L1g4l = g4l [-0.02sin(0.Str)+0.3cos(80t)] .
qc can qc i.e.,
be obtained from the different forms of Cfc=[qc(kT,)-qc((k-l)T,)]/Ts where T, denotes the sampling time. Tn general, the smaller the sampling time T, is, the more accurate the time derivative is. 4. SIMULATION RESULTS AND DISCUSSI ON
(13)
The corresponding responses are shown in Fig. 4. It can be clearly seen that the tracking responses of trajectory and virtual reference velocity are still satisfactory which proves the high level of robustness in the closed-loop system. Actually, the appropriate selection of control parameter matrices K and Kl can enhance the performance and increase the relative closed-loop stability. Tn general, the larger the chosen values are, the better the performances will be. However, the control efforts would be larger. As a result, a compromise has to be set for acceptable performance and stability.
To evaluate the performance and verifY the robustness of the proposed control, the simulations for a planned trajectory-tracking of an AUV without uncertainties and subject to uncertainties are performed. Tn the simulations, the physical and suitable control parameters of an AUV [14] are given as 2 m=IOkg , I� =30kg ·m , Xzi =34, Yv =7S, Zw =33, Ny =62, Xu =6, Yv =10, Zw =7, Nr =14, Xuu =18, K =diag {2, 2, 2, S} , Zww = 4, N rr =14, Yvv = 4, Kl =d iag {2, 2, 2, 2} . The initial position and orientation of the AUV is {O,-I, O, O} .ln addition, the desired trajectory and orientation 17d is chosen as {2sin(0.St),-2 cos(O.St), O.lt, O.St} . The tracking responses of trajectory and virtual velocity for the AUV system without uncertainties are shown in Fig. 3. From these responses, it is clear that the proposed control 1194
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Fig. 3 Tracking responses of trajectory and virtual r eference velocity for an AUV without uncertainties
Fig. 4 Tracking responses of trajectory and virtual r eference velocity for an AUV subject to uncertainti es 5. CONCLUSIONS
In this paper, simplified kinematic and dynamic models (i.e., p = q = 0 ) including model uncertainties for describing main motion characteristics of an AUV are given. HBC has been designed such that tracking of the virtual reference input and trajectory-tracking exhibit UUB performance. The high level of robustness for our proposed control technique is assured in spite of 1195
the presence of large uncertamtles. The above two features are verified via Lyapunov stability criteria and simulation results. REFERENCES
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