An Autonomous Underwater Vehicle As an Underwater Glider and Its Depth Control

International Journal of Control, Automation, and Systems (2015) 13(5):1212-1220 ISSN:1598-6446 eISSN:2005-4092 DOI 10.1007/s12555-014-0252-8 http://www.springer.com/12555 An Autonomous Underwater Vehicle as an Underwater Glider and Its Depth Control

Moon G. Joo* and Zhihua Qu

Abstract: To provide a conventional autonomous underwater vehicle with gliding capability, we as- sume a moving battery and a buoyancy bag installed in a torpedo shaped autonomous underwater vehicle. We develop a mathematical model for the underwater vehicle and derive a stable gliding condition for it. Then an LQR controller is designed to control the zigzag depth of the vehicle, where the derived gliding condition is used as set-points of the control system. For control efforts in the gliding movement, the changes in the center of gravity and the net buoyancy are used, but neither thruster nor rudders are used. By using the gliding capability, the underwater vehicle can move to a farther location silently with less energy consumption and then start operating as a normal autonomous underwater vehicle. We show the feasibility of the proposed method by simulations using Matlab/Simulink.

Keywords: Autonomous underwater vehicle, depth control, LQR controller, underwater glider.

1. INTRODUCTION winged airplane, and it performs a vertical zigzag movement to go forward by changing net buoyancy and Unmanned underwater vehicles have been developed the position of center of gravity [6-10]. UWG has been in many countries. They can be roughly categorized into developed as a moving device for measuring temperature, autonomous underwater vehicle (AUV) [1-4] and salinity, water currents, etc. Since a very small amount of remotely operated vehicle (ROV) depending on the energy is required to inflate/deflate buoyancy bag or to existence of tether that determines the autonomy level. push/pull the position of the battery position to change AUV has on-board power and an on-board navigation the center of gravity, UWG has a long operation time system free from a tether, whereas ROV needs a tether to that lasts over a month and thus offers a long range be supplied with power and steering control from the coverage wider than 1000 km. However, it has mother ship on the surface. A kind of hybrid AUV has disadvantages from the large turning radius which makes also been developed which has the appearance of a it difficult for precise position control. conventional ROV having several thrusters to move to Combining the good aspects of the conventional AUV any direction but having no tether to transport power and and UWG is so natural that some of the combined control signals from the mother ship [5]. underwater vehicles have been developed including an The conventional AUV has a torpedo shape. It has at UWG without wings [11-13], an UWG with thruster and least one thruster to go forward and rudders to change rudders [14], and an AUV with buoyancy bag and direction. The main area of application is military moving mass [15]. It can be another type of hybrid AUV operation such as mine detection, terrain search, in a sense that it can be used either as an AUV or as an surveillance mission, and so on. Its advantages include a UWG when necessary. rapid response and a small turning radius. Since propeller From the viewpoint of modeling for control purpose, a thrust needs much energy, however, its operating time is REMUS as a conventional AUV has been modeled in [1], limited to a few hours. On the other hand, the underwater where the AUV was assumed to be a rigid body and the glider (UWG) is a kind of AUV, which is shaped like a external forces were mathematically obtained by the mechanical shapes of the hull, rudder, and sterns. Then ______the higher order terms of the external forces were Manuscript received June 23, 2014; revised September 26, ignored for simplification. Control inputs are the thruster 2014; accepted November 1, 2014. Recommended by Associate Editor Sung Jin Yoo under the direction of Editor Hyouk Ryeol velocity and the angle of sterns and rudders. There have Choi. been many control schemes reported such as PID, state This work was supported by Basic Science Research Program feedback, sliding mode controller, neuro-fuzzy system, through the National Research Foundation of Korea(NRF) funded and so on [16-20]. The modeling for UWG was also by the Ministry of Education, Science and Technology (2013 conducted in [21,22] assuming the vehicle as a multi R1A1A4A01005930). body system consisting of moving masses, a hull, and a Moon G. Joo is with the Department of Information and Communications Engineering, Pukyong National University, variable mass buoyancy. External forces are not modeled Busan 608-737, Korea (e-mail: [email protected]). directly, but obtained by the experiment or Zhihua Qu is with the Department of Electrical Engineering computational fluid dynamics software. They are and Computer Science, University of Central Florida, FL 32816, described as functions of lift and drag depending on the USA (e-mail: [email protected]). angle of attack. Control inputs are the position of moving

An Autonomous Underwater Vehicle as an Underwater Glider and Its Depth Control 1213 mass and the mass of buoyancy. Some control schemes have been also reported [23-27]. In modeling of the hybrid AUV combining AUV and UWG, no unified modeling to handle both aspects of AUV and UWG has been reported yet to the best of our knowledge. We, thus, propose a method of developing a unified model in order to add a gliding capability to the conventional AUV with which it can travel a long distance to a far location as an UWG and then operate as a normal AUV. The REMUS model in [1] is applied basically to use the already developed AUV control scheme in the AUV mode, and the work in [21] is adopted to derive the stable gliding condition in glider Fig. 2. State variables represented in the body fixed mode. This kind of hybrid AUV is advantageous from coordinate system and the earth fixed coordinate the aspect of quietness and energy saving under low system. current since it makes neither the noise nor the turbulence which an AUV inevitably makes because of equations using 12 state variables, (,xyzuvw ,,,, ,,φθ , the propeller thruster. ψ ,,,)pqr as shown in Fig. 2 [28,29]. The NED Section 2 develops a dynamic model for an AUV as an coordinate is used as the earth fixed coordinate system. UWG. Section 3 derives the desired value of several Generally, the center of buoyancy is used as the origin necessary state variables for a stable gliding. Section 4 of the body fixed coordinate, i.e., xyz===0. The develops a linearized depth model to design a controller bbb center of gravity in each direction is denoted as (,x y , for vertical plane movement. Section 5 gives the gg z ). The mass of the vehicle is mmmm=++ simulation results using Matlab. A brief summary is g vh b where mh is the mass of hull and its static components, followed in Section 6. m m is the moving mass such as battery pack, and b is 2. MODELING OF AUV AS UWG the point mass buoyancy. The mass of the water m displaced by the vehicle is denoted as . Then the net mmm=−. Because the controllers of AUV such as REMUS have buoyancy mass is ov I been widely studied, this paper focuses on controlling an Now we have the following 12 dynamic equations. xx, I I AUV as UWG under the assumption that the AUV has a yy and zz are moments of inertia in the body fixed buoyancy bag and a moving battery pack in its hull as coordinate. Among them, equations (1)-(6) are labeled to shown in Fig. 1. The purpose of modification is to endow depict the vertical plane movement of the robot for later the AUV with gliding capability which does not use any use. propeller propulsion. m[() u −+ vr wq − x q22 + r It is worth to mention that unwinged UWGs in [11-13] vg (1) +−++=ypqr()()] zprq X, are supposed to be operated as underwater gliders only. g g ∑ ext UWG in [14] focused on the change of parameter values mv[() −+− wpury r22 + p due to the thruster and rudders from only the viewpoint vg (2) of underwater glider. AUV in [15] is most similar and +−++=z ()()]qr p x qp r Y , gg∑ ext has a buoyancy bag and a moving battery in its hull, but I pI+−()[() Iqrmywuqvp + −+ it uses the buoyancy bag and the moving battery to xx zz yy v g achieve zero angle of attack at neutral buoyancy, for −−+zvwpur(), ] = K g ∑ ext efficient propeller propulsion to minimize drag effect and I qI+−()[() Irpmzuvrwq + −+ thus to lengthen the operational time and range. In other yy xx zz v g (3) words, the AUV operates always as a propeller-driven −−+xwuqvp() ],= M g ∑ ext AUV, even though the vertical zigzag trajectory is used I rI+−()[() Ipqmxvwpur + −+ during its operation. zz yy xx v g −−+yuvrwq(), ] = N It is well known that an underwater robot with 6 g ∑ ext degrees of freedom is described by nonlinear differential xu =+cosψθ cos v (cos ψθφ sin sin − sin ψ cos φ) (4) ++w(cosψθφ sin sin sin ψφ sin ), yu =+sinψθ cos v (sin ψθφ sin sin + cos ψφ cos ) +−w(sinψθφ sin cos cos ψφ sin ), (5) zu =−sinθθφθφ + v cos sin + w cos sin , φφθφθ =+pqsin tan + r cos tan , θφφ =−qrcos sin , (6) Fig. 1. AUV model with a moving mass and a buoyancy bag. ψφφθ=+(qr sin cos ) / cos .

1214 Moon G. Joo and Zhihua Qu

External forces exerted on the AUV [1] are written as rudder angle, respectively. The hydrostatic force on the underwater vehicle ∑ X = X ;HydroStatic ext HS assuming a moving mass shifting only forward and ++Xuuw ZwqZqq + q (0,(.))y ==zZconst backward ggG becomes −−Yvrvr Yrr ;Added Mass,Coriolis XWBHS =−()sin, − θ + Xuu ;Axial Drag uu YWBHS =−()cossin,θφ

+ X prop ;Propeller Thrust, ZWBHS =−()coscos,θφ KyW=−cosθφ cos zW cos θφ sin ∑Yext = YHS ;HydroStatic HS g g = ZWcosθφ sin , +++Yvvr Yr X u ur G MzWxW=−sinθθφ − cos cos −−ZwpZpqwq ;Added Mass,Coriolis HS g g =−ZWsinθθφ − xW cos cos , ++YvvYrrvv rr ;Crossflow Drag Gg NxW=+cosθφ sin yW sin θ +Yuv ;Body Lift HS g g uvl 2 = xWcosθφ sin , +++YurYuvYuruuδ rδ uvf urf ;FinLift, g

∑ Zext = ZHS ;HydroStatic where ++−Zw Zq Xuq wqu Wmmmg=++(),hb

++Yvpvr Yrp ;Added Mass,Coriolis Bmg= . ++ZwwZqq ww qq ;Crossflow Drag g is the gravitational acceleration constant. + Zuw ;Body Lift uwl 3. STABLE GLIDING CONDITION 2 ++ZusZuwuuδ s δ uwf

+ Zuquqf ;FinLift, To find the stable gliding condition of the underwater vehicle, we adopt the method in [21], and then newly K = K ∑ ext HS ;HydroStatic derive the desired values of necessary variables in this + Kpp ;Added Mass,Coriolis section. + Kpp Under the stable gliding condition where the lateral pp ;Rolling Drag Moment movement does not occur, the following holds; + K prop ;Propeller Torque, uvw== =0, v = 0, pqr======pqr0,

∑ Mext = M HS ;HydroStatic φψ==0. ++Mw Mq wq We add the restriction of using neither propulsion nor −−(Zwu X )uw − Yr vp ;Added Mass,Coriolis torque by thruster. Rudder and stern are not used in the glider mode, either; +−()KNrpZuqpr − q ;Added Mass,Coriolis XKprop== prop 0, δδrs==0. ++MwwMqqww qq ;Crossflow Drag

+ Muwuwl ;Body Lift Moment Then the underwater vehicle suffers no external forces, 2 therefore, the following equations are true; ++MusMuwuuδ s δ uwf ∑ XWBXuuext =−( − )sinθd +uu d d =0, (7) + Muquqf ;FinLift, ZWB=−()cosθ + Zww ∑ N = N ∑ ext d ww d d ext HS ;HydroStatic (8) +=Zuw 0, ++−−Nvvr Nr() X uv Y uv uw d d + Zwp MZ=−(sinθθ + x cos) W q ;Added Mass,Coriolis ∑ ext G d g d (9) −−KMpqYur + ++=MwwMuw 0, ()pq r ;Added Mass,Coriolis ww dd uwdd

++NvvNrr ;Crossflow Drag vv rr where + Nuvuvl ;Body Lift Moment ZZZuw=+ uwl uwf , 2 ++NurNuvuuδ r δ uvf MMuw=++ uwa MM uwl uwf , MZXuwa=−(), w − u + Nururf ;Fin Lift, and θd , ud , and wd are desired pitch angle, desired where X prop , K prop , δ s, and δr are the propeller surge, and the desired sway, respectively. propulsion, the propeller torque, the stern angle, and the The desired trajectory of the vehicle is normally

An Autonomous Underwater Vehicle as an Underwater Glider and Its Depth Control 1215

defined by the moving speed of the vehicle, Vd = Table 1. Desired values of parameters for stable gliding. uw22+ , ξ . dd and the angle of flight, d The desired angle downward upward of attack is denoted as αd, and it will be obtained soon. ξ d (deg) -30° -45° -60° 30° 45° 60° Given ξd and αd, we have the pitch angle of the robot as αd (deg) 12° 7° 4° -12° -7° -4° θξα=+. (10) ddd θ d (deg) -18° -38° -56° 18° 38° 56° α 1, Furthermore, if d then V d (m/s) 0.30 0.30 0.30 0.30 0.30 0.30 ⎛⎞ww u −1 dd d (m/s) 0.2936 0.2978 0.2993 0.2936 0.2978 0.2993 αd (rad)= tan ⎜⎟ . (11) ⎝⎠uudd wd (m/s) 0.0615 0.0364 0.0209 -0.0615 -0.0364 -0.0209 x Using (10) and (11), equations (7)-(9) are rewritten as gd (m) 0.0080 0.0167 0.0305 -0.0080 -0.0167 -0.0305 2 Z G (m) 0.02 0.02 0.02 0.02 0.02 0.02 ()sin()WB−+=ξαddXuuu d , (12) m 22bd (kg) 0.9436 0.8954 0.8778 0.7262 0.7743 0.7919 (WB−+=−− )cos(ξαdd ) Zww αα ddduwdd u Z α u,

(13) Along with the result of Section III, we design an LQR xWcos(ξα+=− )ZW sin( ξα + ) g dd G dd controller similar to the work in [21]. (14) ++MuMuαα22 α . Setting X prop = 0, δ s = 0 and assuming ww ddd uwdd pr==0, q 0, v = 0, w 0, (19) Dividing (12) by (13) gives (15), from which we obtain αd; we rewrite the vertical plane equations (1)-(3) as ()mXumzqWB−+ =−−+− ()sin(),θ Z mwq Xuu vu vg wv tan(ξα+= ) . (15) dd−−ZZαα α (20) ww dd uwd ()()mZwmxZq−− + vw vgq (21) Once αd is determined, from (12) and (13), the desired =−()cos(WBθ + m + Z ) uqZuw + , mass of variable buoyancy for the given trajectory is vuquw mzu−++−()() mx M w I M q calculated as vg vg w yy q =−(sincos)ZxWMuwθθ + + Gg uw (22) mmmmbd =−h − −+−mZ wq(), M mx uq 22 XZ++()αα Z α (16) vG uq vg uu ww dd uwd 2 ± ud . where g ZZZuw=+ uwl uwf,,, ZZZ uq =+ uqa uqf Z uqa =− X u Then, from (14), the desired center of mass for the MMuw=++ uwa MM uwl uwf,(), M uwa =−− ZX w u trajectory is calculated as MM=+ M,. M =− Z x = uq uqa uqf uqa q gd −++()sin(θααα + + )2 To get a constant inverse matrix later shown, we set ZmGh mmg bd d Mww dd M uwdd u mmmmmmm=++ += . vh bh v0 in the left side of ()cos++ θ mmmghbdd (20)-(22). Generally, it is true since mmmhb+ . Let- (,uw ,θ ,, qx , m ) (,uw , (17) ting an operating point gbop as dd V α θ ,,qx , m ), With d and d, the desired surge and heave are d d gd bd we can linearized the right side of calculated as equations around the operating point as ()−+ +− () Δ uVdd= cos(α d ), wVdd= sin(α d ). (18) mXumzqXZmwq00 vu vgopwvdd (23) −−()cossin,θθ Δ− θ Δ Table 1 shows the determined values for arbitrary WBdd g db m XZ/= 0.1364, ()()−− + + Δ chosen three flight angles when uu|| uw mZwmxZqZZwuvw00 vgq opuwd m 32.314. Z = 0.02 = G is assumed and that should be +Δ++Δ() Zuuw d w m vd Z uq u d q (24) determined when an AUV is implemented. Note that ξd −−()sincos,θθ Δ+ θ Δ and Vd are the design parameters and they must be WBdd g db m α 1 mzu−++−()() mx M w I M q M checked to give d from (15). vg00 vg w yy q op +Δ+Δ−ΔMwuMu wWcosθ x 4. DEPTH CONTROL OF THE VEHICLE uw d uw d d g (25) +−((mZw + M − mx )) u Δ q vd G d uq vd gd d Because many successful trajectory control methods +−(cossin),ZxWθθθ + Δ for the conventional AUV have been reported, we only Gdgdd consider the depth control regarding AUV as UWG. where

1216 Moon G. Joo and Zhihua Qu

Δ=uuu −,,, Δ= www − Δ= qqq − linearized as ddd Δ=−θθθ,, Δxxx = − Δ mmm = − ,θδ =+qq q, dgggdbbbd op (29) =++,, = mmmmWmgvd h bd d vd where XWB=−()sin, − θ op d d qop = 0. ZWB=−( )cosθ + Zuw, op d d uw d d In summary, the matrix version of (23)-(26), (28), and M =−(ZxWMuw sinθθ + cos ) + . op G d gd d d uw d d (29) is written as The actuator dynamics for the change in the center of MXAXBUF =Δ+ +, YCX=Δ, (30) gravity and the change in the mass of buoyancy bag are linearized as where xbux =+ bu, Δ=XXX − , Xuwqzxm= []′ θ T , g xx gop. xx d gb (26) mbum =+ bu, Xuw= [00],θ xmT Uuu= [],T bmbbopmb. d d d d gd bd xb M = where ⎡mX00− 00mZ 000⎤ xm==0, 0. ⎢ vu vG ⎥ gop.. bop 0()mZ−− mxZ +0000 ⎢ vw00 vgq ⎥ With (19), equations (4)-(5) representing the NED ⎢ mZ−+( mx M ) I − M 0000⎥ vG00 vg w yyq position of the robot in the vertical plane is written as ⎢ ⎥ ⎢ 00 01000,⎥ xu =+cosθθ w sin , ⎢ ⎥ ⎢ 00 00100⎥ zu =−sinθθ + w cos . ⎢ 00 00010⎥ ⎢ ⎥ A coordinate transform (27) is required to control with ⎢ 00 00001⎥ ease the deviated distance between the desired trajectory ⎣ ⎦ and the vehicle; ⎡ 00()0Zmwwvdd − ⎢ Zwuw d Zu uw d()0 m vd+ Z uq u d ⎡⎤xx′ ⎡⎤cosξξdd− sin ⎡⎤ ⎢ = ⎢⎥. (27) ⎢⎥zz′ sinξξ cos ⎢⎥ ⎢ M wMu a 0 ⎣⎦⎣⎦dd ⎣⎦ ⎢ uw d uw d 33 = A ⎢−sinααdd cos 0 0 The transformed variable z' means the perpendicular ⎢ 00 10 distance from the desired trajectory as shown in Fig. 3 ⎢ and the purpose of the control is to make z' as 0. x' is not ⎢ 00 00 ⎢ considered as a control purpose like as most of the UWG ⎣⎢ 00 00 does not. −−()cos0θθ− sin Differentiating (27), the z' part becomes WBdd g d⎤ −−()sin0θθ cos⎥ WBdd g d⎥ zxz′ =+sinξξdd cos − cosθ 0 ⎥ =−u(sinξθ cos cos ξθ sin ) aW35 dd ⎥ dd , a45 00⎥ ++w(sinξθdd sin cos ξ cos θ ) (28) 000⎥ zuw'sinop−Δ+Δαα d cos d ⎥ 000⎥ −+Δ(cosuwddddααθ sin), ⎥ 000⎦⎥ where aMmxumZw=−() − , 33 uq vd gd d vd G d zu'op=− d sinαα d + w d cos d = 0 (∵ tan αd =wu d / d ). aZ=−(cossin),θθ + x W 35 Gdgddd au=−cosαα − w sin , With (19), the pitch equation (6) of the robot is 45 dddd ⎡X ⎤ ⎡ 00⎤ ⎢ op ⎥ ⎢ ⎥ Z 00 ⎢ op ⎥ ⎡⎤1000000 ⎢ ⎥ ⎢ ⎥ ⎢ 00⎥ M ⎢⎥ ⎢ op ⎥ ⎢⎥0100000 ⎢ ⎥ ⎢ ⎥ B = ⎢ 00,⎥ Fq= , C = ⎢⎥0010000. ⎢ op ⎥ ⎢⎥ ⎢ 00⎥ ⎢z' ⎥ 0001000 ⎢ ⎥ op ⎢⎥ ⎢ ⎥ ⎢⎥ ⎢bx 0⎥ x ⎣⎦0000100 ⎢ gop. ⎥ ⎢ 0 b ⎥ ⎢ ⎥ ⎣⎢ m ⎦⎥ m Fig. 3. Coordinate transform. ⎣⎢ bop. ⎦⎥

An Autonomous Underwater Vehicle as an Underwater Glider and Its Depth Control 1217

Because M is a constant matrix, if it is invertible, the Hybrid Autonomous Underwater Vehicle : AUV + Glider At the end of the simulation, XY graph, XZ graph, depth graph, and XYZ graph appear. state equation (30) is rewritten as tested in Matlab R2012a. 2014.1 http://cal.pknu.ac.kr XMAXMBUMF =Δ++−−−111

x pos_t ref =Δ+AX BUF +, dS dR uX uB x x dS dR uX uB xG mB err prov e 9.25*0 dS dR ref Gen Xprop where controller xG mB Xprop Kprop uX uB −−−111 U_Glider (ZOH) A ===MABMB,, F MF . -0.543*0 Kprop Introduce the desired state equation, we set;

x.mat XAXBUFddd=Δ + +, To File T Ud = [0 0] . Fig. 4. Simulation using Matlab/simulink. Using the difference between above two matrix equations, we construct a regulator equation as follows u*180/pi X deltaS (radian) and the control purpose is to make ∆ →0; Fcn 2 1 deltaR (radian) Rate Limiter Saturation (radian) u*180/pi dS dR dS dR uX uB uX Fcn1 Δ=Δ+ΔXAXBU, uB Rate Limiter1 Saturation (radian)1 Δ=−UUU = U. 2 d Xprop SAUV_L1 1 x 3 ZOH REMUS AUV ZOH1 Pole placement or LQR can be used to determine the Kprop u : surge vel. v : sway vel. 1 xG w : heave vel. rateXg control input: s p : roll rate bX Integrator Saturation 3 q : pitch rate 1 mB xG mB r : yaw ate Δ=UU =−Δ KX. rateMb (31) s bM Integrator1 Saturation 1 xpos ypos zpos

5. SIMULATION 4 phi : roll angle uX uB theta ; pithch angle psi : yaw angle To verify and evaluate the feasibility of the proposed Fig. 5. Implementation of U_Glider(ZOH) block. method, simulations using Matlab/Simulink shown in Fig. 4 were performed, where the overall system was Table 2. AUV Parameters used for simulation (unit: considered as a discrete control system using ZOH with MKS). 1 second sampling time as shown in Fig. 5. = 30.4791 x = 0 I = 0.1770 In the simulation, the following reasonable measure- mh b xx = 1 y = 0 I = 3.4500 ment noises are added: m b yy = 32.3140 z = 0 m b I zz = 3.4500 uvwnn, , n : rand (−± 0.0051,0.0051) ; 0.01 knot (DVL), g = 9.8100 x = 0 pqr, , : rand (−± 0.2618,0.2618) ; 15° / sec(DVL), g nnn y = 0 g xyznnn, , : rand (−± 0.5000,0.5000) ; 0.5m (GPS), zg = 0.0200 φθϕ, , :rand (−± 0.0175,0.0175) ; 1° (IMU), nnn X =−0.9300 Y =−1310 Z =−131 u vv ww

()⋅ Zw =−35.5000 Y = 0.6320 Z =−0.6320 where n denotes the noise to each variable, and rr qq rand() Z =−1.9300 denotes a random value within the given range. q Y =−18.9600 Z =−18.9600 The values are dictated from the specification of uvl uwl Yv =−35.5000 Yuuδ r = 9.6400 Zuuδ s =−9.6400 NavQuest 600 Micro by LinkQuest Inc. for DVL, Y = 1.9300 r Y =−9.6400 Z =−9.6400 AsteRx1 by Septentrio for GPS, and MTi by Xsens for uvf uwf X =−3.9000 uu IMU. Yurf = 6.1500 Zuqa = 0.9300 The change ratios in the buoyancy mass and the center Zuqf =−6.1500 of gravity are assumed as 1 Kg/min and 1 mm/sec, K =−0.0704 M w =−1.9300 N = 1.9300 respectively; bx = 0.0167, bm = 0.001. p v K =−0.1300 M =−4.8800 N =−4.8800 Table 2 lists the simulation parameters for AUV given pp q r M = 3.1800 N =−3.1800 in [1] except for the movable mass, that is arbitrarily ww vv chosen as m = 1 kg. M =−188 N =−94 To control the zigzag depth forcing the AUV go qq rr M =−4.4200 N = 4.4500 forward, an LQR controller with 1 second sampling time uwl uvl N =−6.1500 is designed using a built-in “dlqr” command in Matlab. M uuδ s =−6.1500 uuδ r N = 6.1500 The weighting factors are selected to give stress on depth, M uwf =−6.1500 uvf pitch, the center of gravity, and the buoyance mass and M =−3.9300 N =−3.9300 are shown as uqf urf

1218 Moon G. Joo and Zhihua Qu

Q = diag(10,10,10,0.001,10,100,100 ), The simulation demonstrated the AUV moving

R = diag(1000,10 ).00 forward by the controlled vertical zigzag movement as shown in Fig. 6(a). It showed that the AUV moved For the case when ξd =±30 ° from Table 1, feedback forward about 800 m by two laps of upward and gains K in (31) are obtained; downward movements, when the values of θ, u were controlled to the values θ , u in Table. 1. In Fig. 6(b), ⎡0.1275−−− 0.0477 0.0021 0.0009 d d K = ⎢ control efforts ux, ub were mainly occurred when the ⎣0.0685−−− 0.0195 0.0021 0.0004 flight direction was to be changed which is common to 0.0253 18.2924 0.4136⎤ the underwater glider, because it means low energy ⎥ consumption. It also showed that xg, mb were well 0.0110 6.9030 0.6220⎦ controlled to the values xgd, mbd by the proposed LQR for downward flight and controller.

⎡−−−−0.1278 0.0481 0.0074 0.0009 6. CONCLUSION K = ⎢ ⎣−−−−0.0683 0.0196 0.0049 0.0004 The conventional autonomous underwater vehicle uses 0.0253 18.2666 0.4104 ⎤ one or a few thrusters to go forward and consumes much ⎥ 0.0109 6.8504 0.6195⎦ energy, which limits the operation range of the vehicle. Conversely, the underwater glider uses the changes in the for upward flight. center of gravity and the net buoyancy to go forward, The simulation started from the surface and the target consuming much less energy. To combine the good depth was given as a sequence of (100 m, 0 m, 100 m, 0 aspects of both, we assumed an autonomous underwater m, …). When the AUV reached within 1 m from the vehicle with a moving battery and a buoyancy bag in its target depth, the controller changed feedback gain hull to add a gliding capability to the conventional according to the upward or downward flight and the autonomous underwater vehicle. target depth was changed to the next one. The stable gliding condition for the underwater vehicle was derived not by using the typical glider model but by ) d 0.5 ra the model of the conventional autonomous underwater ( 0 ta e vehicle. By using the resultant values as the set points, an h -0.5 t 0 500 1000 1500 2000 2500 3000 LQR controller was designed and its effectiveness was (sec.) demonstrated. ) 0.4 /s m0.2 Proposed modeling method is beneficial because the ( u 0 0 500 1000 1500 2000 2500 3000 parameters of the autonomous underwater vehicle are (sec.) available for both autonomous underwater vehicle ) 100 operation and underwater glider operation. Before this m ( 50 z research, the optimal condition for the gliding operation 0 0 500 1000 1500 2000 2500 3000 was investigated only by the underwater glider model, (sec.) 1000 because the set of the parameters was totally different ) m ( 500 from that of the autonomous underwater vehicle. x 0 0 500 1000 1500 2000 2500 3000 Future research is necessary to implement the (sec.) proposed underwater vehicle and to verify the (a) Attitude, surge, and position of the robot. experimental results based on the theoretical research.

0.2 X REFERENCES u 0 -0.2 0 500 1000 1500 2000 2500 3000 [1] T. Presto, Verification of a Six-degree of Freedom (sec.) Simulation Model for the REMUS Autonomous Un- 0.2 derwater Vehicle, M.S. thesis, Applied Ocean B u 0 -0.2 Science and Engineering, MIT & WHOI, 2001. 0 500 1000 1500 2000 2500 3000 [2] D. R. Yoerger, M. Jakuba, A. M. Bradley, and B. (sec.) ) 0.01 Bingham, “Techniques for deep sea near bottom m ( 0 survey using an autonomous underwater vehicle,” G x -0.01 0 500 1000 1500 2000 2500 3000 The International Journal of Robotics Research, (sec.) vol. 26, no. 1, pp. 41-54, 2007. ) 1 g k [3] P. E. Hagen, N. Størkersen, B.-E. Marthinsen, G. ( 0.8 B Sten, and K. Vestgård, “Rapid environmental as- m0.6 0 500 1000 1500 2000 2500 3000 sessment with autonomous underwater vehicles- (sec.) Examples from HUGIN operations,” Journal of (b) Control efforts and the changes in center of gravity Marine Systems, vol. 69, no. 1-2, pp. 137-145, 2008. and buoyancy mass. [4] B.-H. Jun, J.-Y. Park, F.-Y. Lee, P.-M. Lee, K. Kim, Fig. 6. Simulation results when ξd =±30 ° . Y.-K. Lim, and J.-H. Oh, “Development of the

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1220 Moon G. Joo and Zhihua Qu trial Science and Technology, Pohang, Korea from 1996 to 2003. Since 2003, he has been a professor of Pukyong National University, Busan, Korea. Recently, he has been with the Department of Electrical Engineering and Computer Science, University of Central Florida, FL, for his sabbatical leave. His research interests include factory automation, intelligent control, and underwater vehicles.

Zhihua Qu received his Ph.D. degree in Electrical Engineering from the Georgia Institute of Technology, Atlanta, in 1990. Since then, he has been with the Univer- sity of Central Florida, Orlando, where he is currently SAIC Endowed Professor and Chair of Electrical and Computer Engineering. He has published a number of papers in these areas and is the author of three books, Robust Control of Nonlinear Uncertain Systems (Wiley Interscience, 1998), Robust Tracking Control of Robotic Manipulators (IEEE Press, 1996), and Cooperative Control of Dynamical Systems (Springer-Verlag, 2009). He served or has been serving on the Board of Governors, IEEE CSS and as an Associate Editor for Automatica, IEEE ACCESS, IEEE Trans- actions on Automatic Control, and the International Journal of Robotics and Automation. His main research interests are nonlinear and networked systems, cooperative and robust controls, robotics and autonomous vehicle systems, and energy systems.