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Control of oscillating foil for propulsion of biorobotic autonomous underwater vehicle (AUV

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Control of oscillating foil for propulsion of biorobotic autonomous underwater vehicle (AUV
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  Control of oscillating foil for propulsionof biorobotic autonomous underwatervehicle (AUV) doi:10.1533/abbi.2004.0044 S. N. Singh and S. Mani Department of Electrical and Computer Engineering, University of Nevada, Las Vegas, NV 89154-4026, USA Abstract:  The paper treats the question of control of a laterally and rotationally oscillating hydrofoil forthe propulsion of biologically inspired robotic (biorobotic) autonomous underwater vehicles (BAUVs).Sinusoidal oscillations of foils produce maneuvering and propulsive forces. The design is based onthe internal model principle. Two springs are used to transmit forces from the actuators to the foil.Oscillating fins produce periodic forces, which can be used for fish-like propulsion and control of autonomous underwater vehicles (AUVs). The equations of motion of the foil include hydrodynamiclift and moment based on linear, unsteady, aerodynamic theory. A control law is derived for the lateraland rotational sinusoidal oscillation of the foil. In the closed-loop system, the lateral displacement andthe rotational angle of the foil asymptotically follow sinusoidal trajectories of distinct frequencies andamplitudesindependently.Simulationresultsarepresentedtoshowthetrajectorytrackingperformanceof the foil for different freestream velocities and sinusoidal command trajectories. Key words:  Biorobotic AUV, hydrofoil oscillation control, servoregulator. INTRODUCTION Presently,thereisconsiderableinterestinthedevelopmentof biorobotic autonomous underwater vehicles (BAUVs)which can propel and maneuver themselves like marineanimals. Aquatic animals are excellent swimmers andhave ability to perform intricate maneuvers using flappingtails and fins. Fishes use a variety of fins (dorsal, caudal,pectoral, pelvic fins, etc.) for maneuvering and propulsion(Azuma 1992, Sfakiotakis et al 1999). Studies from fishand cetaceans has inspired researches to use oscillatoryfoils to produce propulsive and maneuvering forces for thecontrol of AUVs (Triantafyllou and Triantafyllou 1995,Harper et al 1998, Bandyopadhyay et al 1999a, 1999b,Kato 2002, Lee et al 2003). Readers may refer to a specialissue of   IEEE Journal of Oceanic Engineering , which hasseveralresearchpapersonbiologicallyinspiredscienceandtechnology (Bandyopadhyay 2002, Lauder and Drucker2004, Triantafyllou et al 2004, Walker 2004). The forcesand moments produced by the oscillating hydrofoils are Corresponding Author: S. N. SinghDepartment of Electrical and Computer EngineeringUniversity of NevadaLas Vegas, NV 89154-4026, USATel: (702) 895-3417; Fax: (702) 895-4075Email: sahaj@ee.unlv.edu complicated functions of mode of foil oscillation (lead-lag,feathering, and flapping motion) as well as the oscillationparameters (amplitude and frequency of oscillation, biasand phase angle). Fluid dynamicists are involved in usingexperimental and computational methods to characterizethe forces and moments produced by oscillating foils(Triantafyllou and Triantafyllou 1995, Bandyopadhyayet al 1999a, Lee et al 2003, Mittal 2004, Singh et al 2004).Although these results are extremely useful, it is especiallyimportant to develop analytical models of lift and momentobtainable from foil oscillation for the application of control system design for maneuvering AUVs.In an interesting work, motivated by the work of Lighthill (1970), Harper et al (1998) have developed alow-order model of a two-dimensional (2-D) oscillatingfoil based on the theory of unsteady aerodynamics, whichhas its srcins in the work of Theodorsen (1935). The foilhaslateralandrotationalmotionandrepresentsreasonablythe typical tail of thunniform swimmers which have highaspect ratio. Furthermore, in the model of Harper et al,springs are attached to the foil, and forces and momentsare transmitted to the foil (tail) by the springs from the ac-tuators.Thesespringsareanalogoustofishtendons,whichstore energy while transmitting forces from muscles.The forces and moments are functions of oscillating pa-rametersincludingfrequenciesofoscillations,phaseanglesand biases of the foil executing lateral and angular motion.Thus, for generating propulsive and control forces so that C  Woodhead Publishing Ltd  117  ABBI 2005 Vol. 2 No. 2 pp. 117–123   S. N. Singh and S. Mani the AUV can accomplish specific maneuvers, it is essen-tial to control the flapping and pitching motion of the foil.However, by a simple application of actuating signals tothe springs by the actuators, one cannot force the foil tooscillate with desirable oscillation parameters due to thecoupled lateral and rotational dynamics. Therefore, thereis a need to design control system so that the foil has anyspecified pattern of oscillatory motion and the lateral androtational displacements are independent of each other.The contribution of this paper lies in the design of acontrol system for the independent asymptotic control of the lateral and rotational motion of a 2-D hydrofoil basedon the internal model principle (servomechanism theory)(Davison 1976, Wonham 1985). The foil is spring drivenbytwoactuatingsignals andit experienceslateraldisplace-ment and the angular rotation in the free stream. The foilmodel includes hydrodynamic forces computed using thetheory of unsteady aerodynamics. A command generatoris used to generate specified command trajectories, whichare linear combinations of sinusoidal functions of distinctfrequencies, amplitudes, phase angles and average values.A feedback control law is designed so that plunge dis-placement and pitch angle of the foil asymptotically tracksthe command trajectories generated by the command gen-erator. The control system includes a servocompensator,which is fed by the lateral and rotational trajectory errors.Since the states associated with the Theodorsen functioncannot be measured, an observer is designed to obtain theestimates of the unavailable states. Then the controller issynthesizedusingtheestimatedstatevariables.Simulationresults are presented which show that in the closed-loopsystem, independent asymptotic control of the plunge dis-placement and pitch angle trajectories are accomplished.The presentation of this paper is as follows.Section “Mathematical model” presents the mathemati-cal model. A state variable representation of the hydrofoilmodel is obtained in Section “State variable representa-tion”. The design of the controller and the observer areconsidered in Section “Control law” and Section “Ob-server design”, respectively, and simulation results arepresented in Section “Simulation results”. MATHEMATICAL MODEL The spring-driven hydrofoil including a lateral spring isshown in Figure 1.  L  and  M   are the hydrodynamic lift andmoment and  F  a  and  τ  a  are the driving force and torqueapplied to the foil at the axis of rotation by the lateral androtational springs controlled by two independent actua-tors. (The rotational spring is not shown in the figure.)The complete equations of motion of the foil based on theunsteady aerodynamic theory has been derived in Harperet al (1998), which are given by m (¨ z t +  ¨ θ  t b ) =  L + F  a  (1)  J   ¨ θ  t  =  M  + τ  a − F  a b  (2) F a Z t K  z Z a w = bMw = − aw = aLU(t) τ a θ t Fish Motionw = o Figure 1  Spring-driven hydrofoil. where  z t  is the vertical position (plunge displacement) and θ  t  is the angular position (pitch angle) of the foil,  m  is themass,  J   is the moment of inertia, and  b  is the position of the axis of rotation along the chord.Acompletederivationofthehydrodynamicliftandmo-ment including the added mass and wake effect based onunsteady aerodynamic theory has been obtained in Harperet al (1998). The lift and moment are given by L = πρ  2 aU   − ˙ z t + U  θ  t +  a 2 − b   ˙ θ  t  C  (i ω ) + a 2 ( − ˙ z t + U   ˙ θ  t − b  ¨ θ  t )   (3)  M  =− 2 πρ aU   a 4 2   ˙ θ  t + πρ a 2 U   − ˙ z t + U  θ  t +  a 2 − b   ˙ θ  t  C  (i ω ) − π 8 ρ a 4 ¨ θ  t  (4)where ρ  is the density,  a  is the half chord length of the tail, U   is free stream velocity (or equivalently fish’s forwardmotion), and  C  (i ω ) is the theodorsen function. A third-order transfer function to obtain a good approximation of the Theodorsen function used in the study is C  (i ω ) = a 3 (i σ  ) 3 + a 2 (i σ  ) 2 + a 1 (i σ  ) + a 0 (i σ  ) 3 + b 2 (i σ  ) 2 + b 1 ( σ  ) + b 0 where  σ   =  ω aU   is the non-dimensional reduced frequencyand  a i   and  b i   are given by[ a 3 , a 2 , a 1 , a 0 ] = [0 . 500000 , 1 . 07610 , 0 . 524855 , 0 . 045133][ b 2 , b 1 , b 0 ] = [1 . 90221 , 0 . 699129 , 0 . 0455035]The force and moment,  F  a  and  τ  a  applied to the foil bysprings in series with the actuators are F  a  =  K  z ( z a − z t )(5) τ  a  =  K  θ  ( θ  a − θ  t ) 118 ABBI 2005 Vol. 2 No. 2   doi:10.1533/abbi.2004.0044  C  Woodhead Publishing Ltd  Control of oscillating foil for propulsion of biorobotic autonomous underwater vehicle (AUV) where  K  z  and  K  θ   are spring constants, and  z a  and  θ  a  arethe positions of the lateral and rotational actuators. Ac-cording to biomechanists, fish have compliances in theirtail tendons to reduce energy costs of muscles. Harper et al(1998) have shown that these springs can similarly reduceactuator energy. Oscillating foil produces periodic forcesandmoments,whichcanbeutilizedforthepropulsionandcontrol of AUVs.Suppose that Y  r ( t  ) =  z r ( t  ) θ  r ( t  )   (6)is the specified reference trajectory. These trajectories aregenerated by exosystems given by  z (  s )ˆ z r (  s ) = 0(7)  θ  (  s )ˆ θ  r (  s ) = 0where ˆ z r  and ˆ θ  r  denote Laplace transforms of   z r ( t  ) and θ  r ( t  ), respectively, and   z (  s ) and   θ  (  s ) are appropriatepolynomials of the form:  z (  s ) =  s  m z i  = 1   s 2 + ω 2 zi    (8)  θ  (  s ) =  s  m θ  i  = 1   s 2 + ω 2 θ  i   Inthesepolynomials, ω zi   and ω θ  i   arepositiverealnumbers.We are interested in deriving a control law  U  c  = ( z a ,θ  a ) T such that in the closed-loop system the outputvector  Y  = ( z t ,θ  t ) T asymptotically follows the refer-ence trajectory  Y  r ( t  ), that is, the tracking error ˜ Y  = [( z r − z t ) , ( θ  r − θ  t )] T converges to zero as  t   →∞ . Notethat by the choice of the initial conditions  z ( i  )r i   (0),where  i   = 0 , 1 , 2 , 3 ,...  ( m z − 1),  θ   j  r  j  (0), where  j   = 0 , 1 , 2 ,...  ( m θ   − 1) and frequencies  ω zi  ,  ω θ   j  , one cangenerate a linear combination of sinusoidal trajectories of desirable amplitude, phases, biases and frequencies, in or-der to produce required control force and moment forthe propulsion and control of AUV. Presently, there isconsiderable interest in exploring the relationships amongthe fin forces and moments and the modes of oscillation(feathering, lead-lag, and flapping motion) and oscillationparameters of the foils, and numerical and experimentalresults have been obtained. STATE VARIABLE REPRESENTATION For the purpose of control system design, it will beconvenient to obtain a state variable representation of the hydrofoil and the reference trajectory generator. TheTheodorsen function can be treated as a filter with input: V  f   =− ˙ z t + U  θ  t +  a 2 − b   ˙ θ  t △ = C  f  k  z t θ  t  + C  f  d   ˙ z t ˙ θ  t   (9)( △ = denotes equality by definition) and output  Y  f  ˆ Y  f  (  s ) = C  (  s ) V  f  (  s ) (10)where  C  (  s ) is obtained by substituting i σ   by  asU   in C  (i ω )The transfer function  C  (  s ) can be written as C  (  s ) = h c + h 2  s 2 + h 1  s  + h 0  s 3 +  p 2  s 2 +  p 1  s  +  p 0 (11)where h c  = a 3  and  h i   and  p i   areappropriaterealnumbers.Although it is possible to obtain infinitely many real-izations of   C  (  s ), one can obtain a minimal realization of dimension 3 in the form:˙ x f   =  0 1 00 0 1 −  p 0  −  p 1  −  p 2  x f   +  001  V  f  =  A f1  z t θ  t  +  A f2  ˙ z t ˙ θ  t  +  A f3 x f   (12) Y  f   = h c V  f   + h 0 x f1 + h 1 x f2 + h 2 x f3 = h c V  f   + ( h 0  h 1  h 2 ) x f  = h c C  f  k  z t θ  t  + C  f  d   ˙ z t ˙ θ  t  + C  f2 x f   (13)where  A f1  = (0 0 1) T C  f  k  A f2  = (0 0 1) T C  f  d  C  f2  = ( h 0  h 1  h 2 )Collecting the coefficients of of  ¨ z t  and ¨ θ  t , one obtainsfrom (1) and (2):  M   ¨ z t ¨ θ  t  =  D  ˙ z t ˙ θ  t  + K   z t θ  t  + K  f  Y  f   + B 0 U  c  (14)where  U  c  = ( z a ,θ  a ) T is the control input,  M  =  m  + πρ a 2 mb  + πρ a 2 b 0  J   +  π 8 ρ a 4  , D  =  0  πρ a 2 U  0  − 2 πρ a 3 4  U   K   =  − K  z  0 bK  z  − K  θ   ,  K  f   =  2 πρ aU  πρ a 2 U   B 0  =   K  z  0 − bK  z  K  θ   Substituting for  Y  f   from (13) and solving for (¨ z t  ¨ θ  t ) T gives:  ¨ z t ¨ θ  t  =  M  − 1 ( K  + K  f  h c C  f  k )  z t θ  t  +  M  − 1 ( D + K  f  h c C  f  d  )  ˙ z t ˙ θ  t  +  M  − 1 K  f  C  f2 x f   +  M  − 1 B 0 U  c =  A 1  z t θ  t  +  A 2  ˙ z t ˙ θ  t  +  A 3 x f   + B 1 U  c  (15) 119 C  Woodhead Publishing Ltd doi:10.1533/abbi.2004.0044  ABBI 2005 Vol. 2 No. 2   S. N. Singh and S. Mani Define the state vector x  = [ z t  θ  t  ˙ z t  ˙ θ  t  x f  ] T ∈ℜ 7 (16)A state variable representation of (12) and (15) takes theform:˙ x  =  0 2 × 2  I  2 × 2  0 2 × 3  A 1  A 2  A 3  A f1  A f2  A f3  x +  0 2 × 1 B 1 0 3 × 2  U  c =  Ax + BU  c  (17)where the system matrices are  A ∈ℜ 7 × 7 and  B ∈ℜ 7 × 2 .The output vector  Y  is then written as Y  = Cx  = [ I  2 × 2  0 2 × 5 ] x  (18) CONTROL LAW In this section, the state variable model (17) is used for thederivation of a control law such that  y ( t  ) asymptoticallytracks  y r ( t  ). The derivation of the control systems is basedon the internal model principle (Davison 1976, Wonham1985). The controller includes a servocompensator incor-porating the modes of the reference trajectories.Let   e (  s ) be a polynomial which is the least commonmultiple of the two polynomials   z (  s ) and   θ  (  s ) andsuppose that it has (2 m  + 1) distinct roots 0 and  ±  j  ω i  , i   = 1 ,..., m . Then   e (  s ) takes the form:  e (  s ) =  s   s 2 + ω 21  ···   s 2 + ω 2 m  △ =  s 2 m + 1 + a c2 m − 1  s 2 m − 1 +···+ a c3  s 3 + a c1  s (19)where the coefficients  a c k  are defined in (19). Note that  e (  s ) is an odd polynomial.For the derivation of control law based on the internalmodel principle (Davison 1976, Wonham 1985), considerservocompensatorsdrivenbytheerrorsignals( z r − z t )and( θ  r − θ  t ) of the form:   ˆ Y  c z ˆ Y  c θ   =   − 1 e  (  s ) 00   − 1 e  (  s )  ˆ z r (  s ) −  ˆ z t (  s )ˆ θ  r (  s ) −  ˆ θ  t (  s )   (20)A state variable representation of the system (20) can beeasily shown to be˙ x s z =  A s x s z + B s ( z r − c 1 x ) Y  c z = (1 0 1 × 2 m ) x s z = c s x s z ˙ x s θ   =  A s x s θ   + B s ( θ  r − c 2 x ) Y  c θ   = c s x s θ   (21)where  A s  =  0 1 0  ···  00 0 1  ···  0...............0  − a c1  0  ··· − a c2 m − 1  ,  B s  =  00...1  Define the augmented state vector x a  =  x T , x Ts z , x Ts θ   T ∈ℜ 7 + 2(2 m + 1) (22)Then the derivative of   x a  can be written as˙ x a  =   A  0 0 − B s c 1  A s  0 − B s c 2  0  A s  x a +  B 00  U  c +  0 0 B s  o 0  B s  z r θ  r  =  A a x a + B a U  c + B d  ( z r ,θ  r ) T (23)For the asymptotic trajectory tracking of   Y  r ( t  ), accord-ingtotheservomechanismtheory(Davison1976,Wonham1985), one must find a control law of the form: U  c  =− K  a x a  (24)such that the closed-loop matrix  A ac  = (  A a − B a K  a ) (25)isHurwitz(i.e.,allitseigenvalueshavenegativerealparts).There exists feedback law of the form (24) for stabilizationif the transmission zeros of the system (17) and (18) donot coincide with the roots of the polynomials   z (  s ) and  θ  (  s ). For the computation of   K  a , one can use optimalcontrol theory or pole assignment technique. In this study K  a  has been obtained assigning eigenvalues of   A ac  in astable region of the complex plane. OBSERVER DESIGN For the synthesis of control law (24), the measurementof the state vector  x  is essential. However, the state of the filter associated with the Theodorsen function cannotbe measured. As such it becomes necessary to obtain anestimate of the state subvector  x f  .For the state estimation, consider an observer given by˙ˆ x  =  A ˆ x + BU  c + F  ( Y  − C   ˆ x ) (26)where ˆ x  denotes the estimate of   x  and  F   is the feedbackmatrix. The dynamics of state error ˜ x  = x −  ˆ x  is of theform:˙˜ x  = (  A − FC  )˜ x  =  A 0  ˜ x  (27)For the convergence of the state estimation to zero,one selects  F   such that the eigenvalues of   A 0  are in theleft half of the complex plane. Again, one can use theoptimal control theory or the pole assignment techniquefor the computation of   F  . In this study, we have usedpole placement design approach for the computation of the feedback matrix  F  .For the synthesis of control law, the estimated states aresubstituted in (21) to yield U  c  =− K  a  ˆ x T , x Ts z , x Ts θ   T (28)In the closed-loop system, for any reference trajectory( z r ,θ  r ) T which satisfies (7), it follows from the servomech-anism theory that ( z t ( t  ) ,θ  t ( t  )) tends to ( z r ( t  ) ,θ  r ( t  )) as t   →∞ . 120 ABBI 2005 Vol. 2 No. 2   doi:10.1533/abbi.2004.0044  C  Woodhead Publishing Ltd  Control of oscillating foil for propulsion of biorobotic autonomous underwater vehicle (AUV) 0 0.5 1 1.5 20 − 0.06 − 0.04 − 0.020.020.040.06Time (sec)    L  a   t  e  r  a   l   t  a   i   l  p  o  s   i   t   i  o  n ,  z    t    (  m   ) 0 0.5 1 1.5 2 − 505Time (sec)    P   i   t  c   h  a  n  g   l  e ,         θ    t    (   d  e  g   ) 0 0.5 1 1.5 20 − 0.04 − 0.020.020.040.06Time (sec)    T  a   i   l  p  o  s   i   t   i  o  n  e  r  r  o  r   (  m   ) 0 0.5 1 1.5 2 − 6 − 4 − 2024Time (sec)    P   i   t  c   h  a  n  g   l  e  e  r  r  o  r   (   d  e  g   ) a bc d 0 0.5 1 1.5 2 − 0.0500.05Time (sec)   z   r    (  m   ) 0 0.5 1 1.5 2 − 505Time (sec)         θ   r    (   d  e  g   ) 0 0.5 1 1.5 2 − 0.2 − 0.15 − 0.1 − 0.0500.050.10.15Time (sec)   z   a    (  m   ) 0 0.5 1 1.5 2 − 6 − 4 − 20246Time (sec)         θ   a    (   d  e  g   ) e fg h Figure 2  Oscillation control:  ω z = 4,  ω θ   = 6 rad/s. (a) Lateral position,  z t  (m), (b) pitch angle,  θ  t  ( ◦ ), (c) tail position error, z t − z r  (m), (d) pitch angle error,  θ  t − θ  r  ( ◦ ), (e) reference trajectory,  z r  (m), (f ) reference trajectory,  θ  r  ( ◦ ), (g) control input, z a  (m), (h) control input,  θ  a  ( ◦ ), (i) state,  x f  1 , (j) state,  x f  2 , (k) state,  x f  3 . SIMULATION RESULTS This section presents the results of digital simulation.The hydrodynamic parameters are taken from Harperet al (1998) and are collected in the appendix. For anillustration, reference trajectories of the form  y r ( t  ) = [  A z  sin( ω z t  ) ,  A θ   sin( ω θ  t  )] T are selected, where  A z = 0 . 05 m and  A θ   = 5 ◦ . For the given  y r ,   e (  s ) =   s 2 + ω 2 z   s 2 + ω 2 θ   , and one has  x s z , x s θ   ∈  R 4 . The aug-mented matrix  A a  has seven stable and eight imaginary 121 C  Woodhead Publishing Ltd doi:10.1533/abbi.2004.0044  ABBI 2005 Vol. 2 No. 2 
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