Modular modeling and control for autonomous underwater vehicle (AUV)

MODULAR MODELING AND CONTROL FOR AUTONOMOUS UNDERWATER VEHICLE (AUV) CHEN YANG (B Eng.) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2007 Acknowledge Acknowledgements I would like to take this opportunity to thank my supervisor, Hong Geok Soon for his guidance and care for both my research work and life The numerous discussions in the past two years have been most fulfilling and have given me a deeper insight in modeling and control engineering I am also grateful to my friends, Wang Jiankui, Zhu Kunpeng for their advice and help for my research project I would like to thank members of my family, especially my parents, who believe and have faith in me, and supported me throughout my nineteen years of academic education i Table of Contents Table of Contents Acknowledgements i Table of Contents ii Summary v List of Tables vi List of Figures vii List of Symbols ix Chapter Introduction 1.1 Development and application of AUV 1.1.1 AUV development 1.1.2 Applications 1.2 Motivation 1.3 Objectives 1.4 Organization of the thesis Chapter Literature Review 2.1 Modeling method 2.2 Control schemes 10 Chapter AUV Dynamics 13 3.1 Coordinate Systems 13 3.1.1 Two coordinate systems 13 3.1.2 Coordinates transformation 15 3.1.2.1 Linear velocity transformation 15 3.1.2.2 Angular velocity transformation 16 3.2 Equations of AUV motion 18 3.2.1 The general equations of motion 18 3.2.2 The terms in motion equations 24 Chapter Hull Profile 27 ii Table of Contents 4.1 Myring hull profile 27 4.2 The essential data of each module 28 4.3 The fin 31 Chapter Modular Modeling 32 5.1 Computation of matrices in motion equations 32 5.2 Hydrostatic forces 36 5.2.1 The component of gravity 36 5.2.2 The component of buoyancy 37 5.2.3 Combining two components 38 5.3 Hydrodynamic forces 39 5.3.1 Drag 39 5.3.1.1 Axial drag coefficient 40 5.3.1.2 Crossflow drag coefficients 42 5.3.2 Added mass 43 5.4 Lift 49 5.4.1 Body lift 50 5.4.2 Fin lift 51 5.5 Thrust force 54 5.6 The whole model 55 5.6.1 Combining the coefficients 55 5.6.2 The total force and moment 56 5.6.3 The whole model 56 5.7 Comparing the simulation results 57 Chapter Control Design 61 6.1 PID controllers 61 6.1.1 Speed controller 62 6.1.2 Depth controller 63 6.1.2.1 Depth control law 65 6.1.3 Steering controller 68 6.2 State feedback controllers using LQR method 71 iii Table of Contents 6.2.1 Speed controller 72 6.2.2 Depth controller 73 6.2.3 Steering controller 76 6.3 Feedback linearization controllers 78 6.3.1 Speed controller 79 6.3.2 Depth controller 81 6.3.3 Steering controller 84 Chapter Conclusion 87 Bibiography 89 iv Table of Contents Summary Ocean exploration is becoming increasingly important and with it, the need for sophisticated Autonomous Underwater Vehicles Dynamic models of the AUV are the basis of controller design for the AUV This thesis proposes a new modular modeling method for AUVs with Myring hull profile This method divides the AUV into basic modules: the nose section, the middle section and the tail section with four control fins It is based on the essential data of each module and enables flexible derivation of dynamic models of different configuration By the use of basic geometrical parameters of modules, the essential data of each module can be calculated From the derived essential data, the hydrodynamic coefficients for the dynamic model are determined according to fluidics and empirical formulas When some component of the AUV is changed for different functional requirements, the new dynamic model can re-derived quickly from the given basic data of the new module For completeness, three control schemes are adopted and the specific controllers are designed to realize maneuverability of the AUV: forward speed control, steering control and depth control The simulation by using these controllers is given to demonstrate the performance of the proposed control scheme The results of simulation show that the performance of controllers is acceptable and the three types of controllers can be useful for application in AUV control v List of Tables List of Tables Table 3.1 The notation of SNAME for marine vehicles 13 Table 4.1 Parameters of the fin 31 Table 5.1 Empirical parameter α 46 vi List of Figures List of Figures Fig 3.1 Earth-fixed coordinates and body-fixed coordinates 14 Fig 3.2 The rotation sequence for transformation 16 Fig 3.3 The earth-fixed non-rotating reference frame and body-fixed rotating reference frame 18 Fig 4.1 Myring hull profile and modules 27 Fig 4.2 The middle section and its own coordinates 29 Fig 4.3 The tail section 31 Fig 5.1 The profile of tail section 48 Fig 5.2 Effective rudder angle of attack 52 Fig 5.3 Effective stern plane angle of attack 53 Fig 5.4-a Track in x-y plane by use of Prestero’s model 59 Fig 5.4-b Track in x-y plane by use of modular model 59 Fig 5.5-a Track in x-z plane by use of Prestero’s model 60 Fig 5.5-b Track in x-z plane by use of modular model 60 Fig 6.1 The speed response for proportional controller 63 Fig 6.2 Depth control system block diagram 65 Fig 6.3-a The depth change with time 67 Fig 6.3-b Moving track in x-z plane 68 Fig 6.3-c Input angle of stern planes 68 Fig 6.4-a Steering angle change with time 70 Fig 6.4-b Moving track on x-y plane 71 Fig 6.4-c Input angle of rudders 71 vii List of Figures Fig 6.5 State feedback control scheme 72 Fig 6.6 Forward speed response for LQR speed controller 73 Fig 6.7-a The depth change with time 75 Fig 6.7-b Moving track in x-z plane 75 Fig 6.7-c Input angle of rudder 76 Fig 6.8-a Steering angle change with time 77 Fig 6.8-b Moving track on x-y plane 77 Fig 6.8-c Input angle of rudder 78 Fig 6.9 Surge speed response 81 Fig 6.10-a The depth change with time 83 Fig 6.10-b Moving track in x-z plane 83 Fig 6.10-c Pitch angle during diving process 84 Fig 6.11-a Steering angle with time 86 Fig 6.11-b Moving track in x-y plane 86 viii List of Symbols List of Symbols AUV Autonomous Underwater Vehicle APL the Applied Physics Laboratory CFD Computational Fluid Dynamics DOF Degrees of Freedom LQR Linear-Quadratic Regulator PID Proportional-Integral-Derivative RHS Right –Hand Side SISO Single-Input, Single-Output SNAME the Society of Naval Architects and Marine Engineers SPURV the Self Propelled Underwater Research Vehicle η = [η1T ,η 2T ]T the position and orientation vector in the earth-fixed coordinates η1 = [ x, y, z ]T the linear position vector in the earth-fixed coordinates η2 = [φ ,θ ,ψ ]T the angular position vector in the earth-fixed coordinates v = [v1T , v2T ]T the linear and angular velocity vector in the body-fixed coordinates v1 = [u, v, w]T the linear velocity vector in the body-fixed coordinates v2 = [ p, q, r ]T the angular velocity vector in the body-fixed coordinates τ = [τ 1T ,τ 2T ]T the forces and moments acting on the vehicle in the body-fixed frame τ = [ X , Y , Z ]T the forces acting on the vehicle in the body-fixed frame τ = [ K , M , N ]T the moments acting on the vehicle in the body-fixed frame ix Chapter Control Design where v and η are the velocity and position states of the system to be controlled, h is a nonlinear function of the states The nonlinearity can be cancelled by the choice of a suitable control law: τ = mav + h (6.25) Substituting the control law (6.25) into (6.24), the close loop system term becomes mv + h = mav + h ⇒ m(v − av ) = (6.26) Then, we can get v − av = (6.27) The commanded acceleration can be selected using linear control analysis, for example, we can choose av = vd − 2λ (v − vd ) − λ (η − η d ) (6.28) where vd and ηd are desired velocity and position respectively Choosing λ > yields e + 2λ e + λ e = (6.29) with e = η − η d , and ensures exponential convergence of e(t ) to zero 6.3.1 Speed controller When we use feedback linearization to design controllers, the nonlinear equations of the DOF dynamics model of the vehicle will be used Considering the speed control, we only use the surge equation which is expressed as ⎧⎪m ⎡ −vr + wq − xG (q + r ) + yG ( pq − r) + zG ( pr + q ) ⎤ ⎫⎪ ⎦ =X (m − X u )u + ⎨ ⎣ ⎬ prop ⎪⎩ −( X res + X u|u|u | u | + X wq wq + X qq qq + X vr vr + X rr rr ) ⎪⎭ 79 Chapter Control Design ⇒ (m − X u )u + h1 = ρ D KT ( J ) ω p ω p (6.30) where h1 contains terms which are not functions of u , and the RHS of (6.30) is a term relating the propeller rotating rate ω p with the propulsive force along the surge direction Simplify (6.30) as m1u + h1 = n1 (6.31) where m1 is the mass which includes added mass term, n1 represents the input propulsive term on the RHS of (6.30) The command acceleration for the control law is chosen as a1 = ud − λ1 (u − ud ) (6.32) where ud is desired surge velocity According to (6.32), we have the desired input propulsive force as n1 = m1a1 + h1 (6.33) Substituting (6.32) and (6.33) into (6.31), gives u − ud + λ1 (u − ud ) = ⇒ e1 + λ1e1 = (6.34) where e1 = u − ud The global asymptotic stability can be assured if λ1 > Here we select λ1 = 0.15 to guarantee good closed-loop performance The speed increases smoothly and the vehicle is stable 80 Chapter Control Design Figure 6.9 surge speed response 6.3.2 Depth controller Consider the pitch dynamic equation of motion, that is the fifth equation of (5.66), we use the procedure described in 6.3.1, and rewrite the equation as: m2 q + h2 = (δ s ) (6.35) where m2 = I yy − M q , h2 contains all the other terms except the control input term of function δ s (δ s ) represents the moment generated by deflection of the stern planes for depth control In addition, we have used other equations: θ = q (6.36) z = −Uθ (6.37) The command acceleration is chosen as: a2 = qd − λ21 (q − qd ) − λ22 (θ − θ d ) − λ23 ( z − zd ) (6.38) 81 Chapter Control Design where qd is desired pitch angle rate, θ d is the desired pitch angle, and zd is desired depth of the AUV Therefore, the moment generated by the deflection of stern planes is: (δ s ) = m2 a2 + h2 (6.39) Substituting (6.38) and (6.39) into (6.35) and considering the relationship described in (6.36) and (6.37) yield: q − qd + λ21 (q − qd ) + λ22 (θ − θ d ) + λ23 ( z − zd ) = ⇒  e2 + λ21e2 + λ21e2 + λ23e2 = (6.40) where e2 = z − zd For stability, the values of λ21 , λ21 and λ23 have to be greater than zero The final values selected for the three coefficients are: λ21 = , λ22 = , λ23 = 0.6 Applying the controller to the whole nonlinear model, the control performance can be observed in following figures The surge speed is 1.5m/s, and desired depth is m Observing Figure 6.10-a, the overshoot is less than 10%, and the AUV achieve the depth of m at 25s The pitch angle during the diving process is less than 27° 82 Chapter Control Design Figure 6.10-a The depth change with time Figure 6.10-b Moving track in x-z plane 83 Chapter Control Design Figure 6.10-c pitch angle during diving process 6.3.3 Steering controller A similar approach used in section 6.3.2 is applied to the design of steering controller As previously mentioned, the steering control is related to two state variables: the yaw angle rate r and the yaw angleψ Recall the yaw dynamic equation of motion from (5.66) and rewrite is as: m3 r + h3 = (δ r )3 (6.41) where m3 = I zz − N r , h3 contains all the other terms, (δ r )3 represents the moment generated by the deflection of the rudders In addition, we need to consider the relationship: ψ = r (6.42) For the control of direction heading, the command acceleration should be chosen as: a3 = rd − λ31 (r − rd ) − λ32 (ψ −ψ d ) (6.43) 84 Chapter Control Design where rd is the desired yaw angle rate, ψ d is desired direction heading Therefore, the moment generated by the rudder is given by (δ r )3 = m3 a3 + h3 (6.44) and then, r − rd + λ31 (r − rd ) + λ32 (ψ −ψ d ) = ⇒ e3 + λ31e3 + λ32 e3 = (6.45) where e3 = ψ −ψ d Choosing λ31 , λ32 >0 will guarantee global asymptotic stability The values for the two terms are: λ31 = 1.2 ， λ32 = 0.36 Applying this controller into the whole nonlinear model, the control performance can be observed in following pictures The forward speed of the AUV is 1.5m/s and the desired target is to turn left 90° in horizontal plane Observing Figure 6.11-a, the AUV turns left smoothly and finally turns 90° at 30s Figure 6.11-b shows the track in x-y plane The results accord with what we desire and the control performance is good 85 Chapter Control Design Figure 6.11-a Steering angle with time Figure 6.11-b Moving track in x-y plane 86 Chapter Conclusion Chapter Conclusion This thesis proposes a new modular modeling method for AUVs adopting Myring hull profile This method divides the AUV into basic modules: the nose section, the middle section and the tail section with control fins By use of the basic geometrical parameters, the essential data of each module for the modular method are calculated by relevant empirical formulas Based on these essential data, the hydrodynamic coefficients for the dynamic model are determined according to fluidics theories and empirical formulas By comparison with Prestero’s model in Section 5.7, it is verified that the accuracy of the model generated by the modular method is good and acceptable In addition, the essential data of every module which has been calculated are kept in files When the modules are used next time, we can load relevant data directly from these files This modular modeling method is based on the basic modules which constitute the whole AUV and makes modeling process quite flexible When some component of the AUV is changed for loading different functional requirements, we only need to calculate the essential data of new modules for dynamic modeling Given the data of the new modules, this method can combine them with the data of remaining components and build the new dynamic model quickly Therefore, this method improves the efficiency of data use and realizes flexible modeling Based on the nonlinear dynamic model with DOF, this thesis uses three control laws for controller design and presents the simulation results for each controller respectively These controllers aim to realize the forward speed control, steering control and depth control The three control laws are PID control, state feedback control with LQR 87 Chapter Conclusion method and feedback linearization control The PID controllers and LQR are designed based on the linearized model from the whole model, while the feedback linearization controller is designed based on the origin nonlinear model The performances of different kinds of controllers are all good By proper parameter setting, the types of controllers can be useful for application in AUV control However, we should point out that the modular modeling method assumes that the AUV with Myring hull profile has smooth hull shape That is to say the AUV has not any bulge or components outside the main hull except fins and propellers In practice, there are often some sensors fixed outside the hull which will generate additional hydrodynamic forces Thus the modeling for these bulges will be the next step work for perfecting 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.. important and with it, the need for sophisticated Autonomous Underwater Vehicles Dynamic models of the AUV are the basis of controller design for the AUV This thesis proposes a new modular modeling