biology-inspired robust dive plane control of non-linear...

Applied Bionics and BiomechanicsVol. 7, No. 2, June 2010, 153–168

Biology-inspired robust dive plane control of non-linear AUV using pectoral-like fins

Subramanian Ramasamy and Sahjendra N. Singh∗

Department of Electrical and Computer Engineering, University of Nevada Las Vegas Las Vegas, NV, USA

(Received 12 March 2009; final version received 4 February 2010)

The development of a control system for the dive plane control of non-linear biorobotic autonomous underwater vehicles,equipped with pectoral-like fins, is the subject of this paper. Marine animals use pectoral fins for swimming smoothly. Thefins are assumed to be oscillating with a combined pitch and heave motion and therefore produce unsteady control forces.The objective is to control the depth of the vehicle. The mean angle of pitch motion of the fin is used as a control variable.A computational-fluid-dynamics-based parameterisation of the fin forces is used for control system design. A robust servoregulator for the control of the depth of the vehicle, based on the non-linear internal model principle, is derived. For thecontrol law derivation, an exosystem of third order is introduced, and the non-linear time-varying biorobotic autonomousunderwater vehicle model, including the fin forces, is represented as a non-linear autonomous system in an extended statespace. The control system includes the internal model of a k-fold exosystem, where k is a positive integer chosen by thedesigner. It is shown that in the closed-loop system, all the harmonic components of order up to k of the tracking error aresuppressed. Simulation results are presented which show that the servo regulator accomplishes accurate depth control despiteuncertainties in the model parameters.

Keywords: pectoral fin control system; non-linear biorobotic autonomous underwater vehicle control; robust output regula-tion; non-linear servo regulation; non-linear internal model principle

1. IntroductionAquatic animals swim smoothly through water using avariety of- oscillating fins, and birds and insects fly usingflapping wings (Azuma 1992; Luca 1999; Sfakiotakis etal. 1999; Fish 2004; Lauder and Drucker 2004; Kato andKamimura 2008) . The extraordinary maneuverabilityof marine animals, birds and insects is the result oftheir ability to generate and control large forces fromunsteady hydrodynamics and aerodynamics, respectively.Presently, researchers are involved in developing bioroboticautonomous underwater vehicles (BAUVs) which havethe ability to swim like marine animals (Bandyopadhyay2005). In literature, mulitiple oscillating fins mountedon AUVs have been proposed to generate control forcesfor propulsion and manoeuvering (Triantafyllou andTriantafyllou 1995; Kato 2000, 2002; Triantafyllou et al.2004; Bandyopadhyay et al. 2008). A robotic turtle-likeunderwater vehicle, a robotic bat and a flying insect havebeen also developed by researchers (Luca 1999; Seo et al.2008; Chung et al. 2009). Laboratory experiments havebeen performed to obtain fin forces of oscillating fins(Triantafyllou and Triantafyllou 1995; Triantafyllou et al.2003, 2004; Bandyopadhyay 2008). Computational fluiddynamics (CFD) methods have been also used to derivethe fin forces (Singh et al. 2004; Narasimhan et al. 2006).The unsteady forces are complex periodic functions of the

∗Corresponding author. Email: [email protected]

oscillation parameters (bias angle, amplitude, frequency ofoscillation, relative phase angle, etc.). The mathematicalmodels of BAUVs, including the fin forces, are non-linearand time-varying.

In literature, methods of averaging and discretisation oftime-periodic systems have been proposed for the control ofBAUVs. By averaging method, one obtains an approximateaverage time-invariant representation of the BAUV modelfor simplicity in control law design (Luca 1999). But thecontrol system designed based on the time-invariant aver-age model ignores the effect of time-varying fin forces onthe vehicle motion. As such in the closed-loop system, thetracking error responses exhibit fluctuations caused by theharmonic components of the fin forces in the steady state.Based on exact discrete-time models of BAUVs (Singhet al. 2004; Narasimhan et al. 2006), sampled-data controlsystems have been developed. For BAUV models with para-metric uncertainties, discrete-time adaptive laws have beenalso designed (Naik and Singh 2007; Naik et al. 2009). Ofcourse, exact discretisation is not possible for non-linearmodels of BAUVs. Moreover, discrete-time controllers cangive zero tracking error only at the sampling instants, and inthe closed-loop system, large inter-sample excursions mayexist. Besides these approaches, fuzzy and neural controlof BAUVs have been considered (Yamamoto 1995; Kato2000, 2002). Open-loop control of a BAUV equipped with

ISSN: 1176-2322 print / 1754-2103 onlineCopyright C© 2010 Taylor & FrancisDOI: 10.1080/11762321003760936http://www.informaworld.com

154 S. Ramasamy and S.N. Singh

six oscillating fins using a cluster of inferior olive neuronshas been also attempted (Bandyopadhyay et al. 2008).

In this paper, a new approach for the dive plane controlof a non-linear AUV equipped with oscillating pectoral-likefins is presented. The method is based on the non-linearservo regulation theory (Huang 1995, 2004). The objectiveis to control the depth of the BAUV. The fins are assumed tooscillate harmonically and have a combined pitch and heavemotion. The pitch bias angle of the fin is treated as a controlinput. Oscillating fins produce time-periodic unsteady con-trol forces. A CFD-based parameterisation of the fin forcesis used for the design of control system. An exosystem ofthird order is introduced to model the periodic forces, andthe time-varying non-linear model of the BAUV is repre-sented as an autonomous non-linear system in an extendedstate space. For the depth control, based on the non-linearservo regulation theory, an internal model of k-fold exosys-tem driven by the tracking error is constructed, where k is apositive integer chosen to give desirable tracking accuracy.The k-fold exosystem has ability to produce monomials ofdegree up to k of the state variables of the exosystem. Thenthe composite system including the linearised model of theBAUV and the internal model of the k-fold exosystem isstabilised to obtain a robust state feedback control law forthe depth control. It is shown that the controller, includingthe internal model in the loop, suppresses harmonic fluc-tuations of degree up to k in the tracking error responses.This desirable closed-loop property is not possible using themethod of averaging (Luca 1999) or discretisation (Singhet al. 2004; Narasimhan et al. 2006; Naik and Singh 2007).Simulation results are presented which show that the servoregulator accomplishes set point control of the depth pre-cisely in spite of large parameter uncertainties in fin forces.

2. Autonomous underwater vehicle model andcontrol problem

Figure 1 shows the schematic of a typical BAUV. Two finsresembling the pectoral fins of fish are symmetrically at-tached to the vehicle. The vehicle moves in the dive plane(XI − ZI plane), where OIXIZI is an inertial coordinatesystem. OBXBZB is body-fixed coordinate system with itsorigin at the centre of buoyancy. XB is in the forward di-rection, and ZB points down. Each fin has two degrees offreedom (pitch and heave) and oscillates harmonically. Weassume that the combined pitch–heave motion of the fin isdescribed as follows:

h(t) = hmsin(ωf t), (1)

ψ(t) = β + ψmsin(ωf t + ν1), (2)

where h and ψ correspond to heave and pitch angle of thefin, hm and ψm are the amplitudes of linear and angularoscillations, β is the pitch bias angle, ωf (in radians per

Figure 1. Model of BAUV.

second) is the frequency of oscillations of fins and ν1 is thephase difference between the pitching and heaving motion.

A CFD-based parameterisation of the fin force and mo-ment is used in this paper. (Readers may find the parame-terised fin force and moment in the paper by Narasimhanet al. 2006.) For simulating the flow past oscillating foils, afinite-difference-based, Cartesian grid immersed boundarysolver was used (Nazzar et al. 2003). The Eulerian formof the incompressible Navier–Stokes equations was discre-tised on a Cartesian mesh (Udaykumar et al. 2001). Forincluding the effect of the small sub-grid flow scales on thelarge resolved scales, the large-eddy-simulation approachwas used to develop the code. A Lagrangian dynamic model(Meneveau et al. 1996) was used to estimate the sub-grid-scale eddy viscosity. The details of the numerical methodand validation of the code can be found in Bozcurttas et al.(2005).

Based on the numerical results obtained by the CFDcode, the lift component (fp) and pitching moment (mp)generated by the oscillating fin can be described by theFourier series given by (Narasimhan et al. 2006)

fp(t) =M∑

n=0

[f sn (β)sin(nωf t) + f c

n (β)cos(nωf t)],

mp(t) =M∑

n=0

[msn(β) sin(nωf t) + f c

n (β)cos(nωf t)], (3)

where f an (β) and mb

n(β), a ∈ {s, c}, are the Fourier coeffi-cients, and M is an integer such that the neglected harmonicshave insignificant effect.

The Fourier coefficients are non-linear functions ofthe pitch bias angle. The net lateral force because of twofins is given by fpv = −2fp and mpv = 2(dcgf .fp + mp),

Applied Bionics and Biomechanics 155

respectively, where dcgf is the moment arm because of finlocation.

We assume that vehicle’s forward speed u is held con-stant by some control mechanism. The equations of motionof a neutrally buoyant vehicle are described by (Fossen1994)

m(w − uq − zGq2 − xGq)

= 0.5ρl4z′q q + 0.5ρl3(z′

ww + z′qqu)

+ 0.5ρl2z′wwu + fpv,

Iyq + mzG(u + wq) − mxG(w − uq)

= 0.5ρl5M ′q q + 0.5ρl4(M ′

w + M ′qqu)

+ 0.5ρl3M ′wwu − xGBWcosθ − zGBWsinθ + mpv,

z = −u sin(θ ) + w cos(θ ), (4)

where Iy is the moment of inertia, m is the mass, w is theheave velocity, θ is the pitch angle, q = θ , xGB = xG − xB ,zGB = zG − zB , l = body length, ρ = density and z is thedepth. fpv and mpv are the net force and moment actingon the vehicle because of pectoral fins. The primed vari-ables (M ′

q , z′q , M ′

w, etc.) are the non-dimensionalised hy-drodynamic coefficients. Here ((xB, zB) = 0) and (xG, zG)denote the coordinates of the centre of buoyancy and centreof gravity (cg) respectively.

Let zr = z∗ be a constant reference signal. Definingthe state vector x = (x1, x2, x3, x4)T = (w, q, z, θ )T ∈ R4,solving (4) and substituting for fpv and mpv from (3) givesthe state variable representation of the BAUV of the form

x = Ax + B1

[fp

mp

]+ nl(x),

e = z − zr , (5)

where e is the tracking error and nl(x) denotes the vectorbecause of the non-linear functions of (4). In view of (3),the state Equation (5) is a non-linear time-varying system.

We are interested in representing (5) as a time-invariantsystem. For this, we select an exosystem

⎛⎝ v0

v1

v2

⎞⎠ =

⎛⎝ 0 0 0

0 0 −ωf

0 ωf 0

⎞⎠

⎛⎝ v0

v1

v2

⎞⎠ = Avv, (6)

where v = (v0, v1, v2)T ∈ R3. Define vp = (v1, v2)T . Us-ing (6), one can generate any constant and sinusoidal sig-nals sin(nωf t) and cos(nωf t) for any integer n. This can beverified easily. Let v0 = 1, v1 = cos ωf t and v2 = sin ωf t ,then v1 and v2 satisfy (6), and one can easily show that

sin 2ωf t = 2v1v2,

cos 2ωf t = (v2

1 − v22

),

sin 3ωf t = 2v21v2 + (

v21 − v2

2

)v2,

cos 3ωf t = (v2

1 − v22

)v1 − 2v1v

22 . (7)

Continuing this process, one can easily show that sin(nωf t)and cos(nωf t) can be expressed as homogeneous polyno-mials, whose each term is monomial in variable v1 and v2 ofdegree n. As such one can express the fin force and momentas functions of state vector v in the form

(fp(t, β)mp(t, β)

)= γ0(β) +

M∑n=1

γ sn (β)πs

n(vp) + γ cn (β)πc

n(vp)

= gf (v, β) (8)

and πsn(vp) = sin nωf t and πc

n(vp) = cos nωf t are homo-geneous polynomials in variables v1 and v2 of degree n,and

γ0(β) =(

f c0 (β)

mc0(β)

),

γ sn (β) =

(f s

n (β)ms

n(β)

),

γ cn (β) =

(f c

n (β)mc

n(β)

).

Using (8) in (5) gives a time-invariant representation of (5)of the form

x = A(p)x + g(vp, β, p) + nl(x, p)�= gx(x, v, p), (9)

e = x3 − z∗v0,

where g(v, β, p) = B1gf (v, β) and p = Rm denotes thevector consisting of all the unknown parameters of theBAUV model including fin forces. For example, p includesthe uncertain parameters of the Fourier coefficients.

Expanding the non-linear terms of (9) in Taylor series,one can represent (9) in the form

x = A(p)x + B(p)uc + E(p)v + nl2(uc, v, x, p), (10)

e�= H1x + H2v

�= h(x, v0),

where uc = β, H1 = (0, 0, 1, 0), H2 = (−z∗, 0, 0), nl2 de-notes non-linear vector functions of second- and higher-order terms in β, v1, v2 and x, and

B(p) = ∂g

∂β(0, 0, p),

E(p) =[g(0, 0, p),

∂g

∂vp

(0, 0, p)

].

Note that E1v0 ( v0 = 1) is a constant vector and E1 is thefirst column of E. For the choice of v0 = 1, v1 = cos ωf t ,v2 = sin ωf t , the system (5) and (10) are equivalent.


For the purpose of control law derivation, we embed thesystem (5) in a larger class of system (10) in which we allowv ∈ V , an open set in R3. Of course, unknown coefficientsof two sinusoids sin(ωf t) and cos(ωf t) can be merged withvp and remaining unknown parameters are elements of p.We have set the goal for approximate tracking for practicalreasons. It will be seen that the design of control law suchthat e tends to zero is a difficult problem because of time-varying periodic fin forces. The class of control laws ofinterest is of the form uc = kc(x, xs), where xs ∈ Rnc isstate vector of a dynamic system

xs = gs(xs, e) (11)

for an appropriate choice of vector functions gs(xs, e). Weobserve that the tracking error is an input signal to thedynamical system (11).

Define xc = (xT , xTs ) ∈ R4+nc . Then the closed-loop

system can be written as

xc =⎡⎣A(p)x + Bkc(x, xs) + E(p)v

+ nl2(kc(x, xs), v, x, p)gs(xs, h(x, v0))

⎤⎦ (12)

�= gc(xc, v, p).

Let the nominal value of the unknown parameter vector p

be p∗ and p = p − p∗ be the perturbation from the nominalvalue. We assume that p ∈ p, an open set surrounding p =0. We introduce the following definition to be used later.

Definition. Let V be an open neighborhood of the originR3. A sufficiently smooth function ok : V → R is said tobe zero up to the kth order if ok(0) = 0 and its all partialderivatives of order less than or equal to k vanish at v = 0.

We are interested in the design of a kth-order non-linearrobust control system (termed kth-order servo regulator)such that the closed-loop system (12) has the followingproperties.

Property 1: All the eigenvalues of the matrix ∂gc

∂xc(0, 0, p∗)

have negative real parts.

Property 2: For all sufficiently small xc(0), v(0) and p, thetrajectory (xc(t), v(t)) of the composite system (12) and (6)satisfies

limt→∞(e(t) − ok(v(t))

= limt→∞(h(x(t), v(t)) − ok(v(t)) = 0, (13)

where k is the chosen positive integer.The Property 2 implies that steady-state tracking er-

ror of the closed-loop system is zero up to kth order. By

choosing k large enough, designer can accomplish desiredtracking error accuracy in the steady state.

3. Control law

In this section, the question of existence of a solution ofthe posed kth-order output regulation is considered. Basedon the work of Huang (1995, 2004), the following result isstated.

Theorem 1. Suppose that in the closed-loop system (12),Property 1 holds. Then the closed-loop system also sat-isfies Property 2 if and only if there exists sufficientlysmooth functions Xc(v, p) = [XT (v, p), XT

s (v, p)]T withXc(0, p∗) = 0 which satisfies for v ∈ V and p ∈ p

∂Xc(v, p)

∂vAvv = gc(Xc(v, p), v, p),

e(v, p) = h(X(v, p), v) = ok(v). (14)

It is possible to synthesise a control law to solve theproblem of the kth-order regulation using the solution of(14). However, it is not easy to solve the partial differentialEquation (14), and moreover, the solution depends on theunknown parameter p.

3.1. Internal model

In this sub-section, we now seek a solution for the reg-ulation problem based on the non-linear internal modelprinciple (Huang 2004). This approach avoids the com-putation of Xc(v, p). To motivate the construction of aninternal model, we look into an approximate solution of(14). One can attempt to obtain a solution of (14) by se-lecting Xc(v, p) = (XT (v, p), XT

s (v, p))T and Uc(v, p) =kc(X(v, p), Xs(v, p)) as polynomial functions of variablesv0, v1 and v2 given by

X(v, p) =k∑

l=1

Xlpv[l] + ok(v),

Xs(v, p) =k∑

l=1

Xslpv[l] + ok(v),

Uc(v, p) =k∑

l=1

Uclpv[l] + ok(v), (15)

where v[1] = [v0, v1, v2]T and v[l] = v[l]p = [vl

1, vl−11

v2, vl−21 v2

2, . . . , vl2]T , for l = 2, 3, . . ., and

e(v, p) = h(X(v, p), v) =k∑

l=1

Ylpv[l] + ok(v). (16)


Here Xlp,Xslp, Uclp and Ylp are constant matrices of appro-priate dimensions depending, perhaps, on p. Each compo-nent of the vector polynomial v

[l]p is monomial in variables

v1 and v2 of degree l. Essentially the elements of v[l]p form a

basis for homogeneous polynomials of degree l in variablesv1 and v2. In view of (15), to satisfy (13) it is essentialto design a control law which can cancel all the terms ofv[l], l = 1, . . . , k, occurring in e(v, p) = h(X(v, p), v) insteady state. This requires construction of a dynamic system(termed internal model of k-fold exosystem) which can pro-duce signals v[l], l = 1, . . . , k. The k-fold exosystem canbe constructed as follows. First of all, v[1] = (v0, v1, v2)T

and v[l]p satisfy

v[1] = Avv[1] �= A[1]v[1],

v[l]p = A[l]

p v[l]p , l = 2, 3, . . . , k, (17)

where

Ap =[

0 −ωf

ωf 0

]

and A[j ]p are appropriate matrices. (The expressions for A

[l]p

and their characteristic polynomials, for l = 1, . . . , 4, arecollected in the appendix.)

Define a state vector

vkf =

⎡⎢⎢⎣

v[1]

.

.

v[k]p

⎤⎥⎥⎦ . (18)

The vector vkf satisfies the differential equation

vkf = diag[A[1], A[2]p , . . . , A[k]

p ]vkf (19)

�= Akf vkf ,

where Akf = diag(A[1], A[2]p , . . . , A

[k]p ). The system (20) is

the k-fold exosystem which generates not only the exoge-nous signal v, but also the higher-order terms of the ex-ogenous signal vp up to order k. According to the inter-nal model principle, for kth-order robust regulator design(Huang 2004), one introduces an internal model of k-foldexosystem (19).

The roots of the minimum polynomial of Akf are pre-cisely given by all the distinct members of the followingset:

�k = {λ | λ = 0 and jωf (l1 − l2);

l1 + l2 = l; l1, l2 = 0, 1, . . . , l; l = 1, 2, . . . , k}. (20)

Now the internal model is constructed using the mini-mum polynomial of Akf of the form

xs = G1xs + G2e, (21)

where xs ∈ Rc and

G1 = diag

(0,

[0 −ωf

ωf 0

],

[0 −2ωf

2ωf 0

], . . . ,

[0 −kωf

kωf 0

]). (22)

The vector G2 is chosen such that the pair (G1,G2) iscontrollable. It can be verified that G2 is given by

G2 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

b0

0b1

0b2

.

.

.

.

0bk

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(23)

which satisfies the controllability property of pair (G1,G2)as long as bi �= 0, i = 0, . . . , k.

3.2. Stabiliser design

For completing the design, all one now has to do is tostabilise the closed-loop system. For this purpose, considerthe augmented system (10) and (21) given by

d

dt

[x

xs

]=

[A(p)

G2H1

0

G1

] (x

xs

)+

[B(p)

0

]uc

+[

nl2(uc, x, v, p)

G2H2v

]+

[E(p)

0

]v. (24)

Now one needs to find a control law such that (x = 0,xs = 0) of the non-linear system (24) is exponentially stablefor v = 0. For exponential stabilization of the origin, it issufficient to stabilise the linearised model obtained from(24), which for v = 0, is given by

dxc

dt=

[A(p)

G2H1

0

G1

]xc +

[B(p)

0

]uc

�= Ac(p)xc + Bc(p)uc, (25)

where Ac and Bc are defined in (25).


For the system (25), a stabilising control law exists if(Huang 2004).

rank

[A(p) − λI

H1

B(p)

0

]= 4 (26)

for all λ, which are the roots of the minimal polynomial ofAkf (that is, for λ = 0 and λ = ±jωf l, l = 1, 2, 3, . . . , k).Here I denotes an identity matrix. Of course, the character-istic roots of Akf are the eigenvalues of the internal modelmatrix G1. The matrices A and B of the augmented system(25) depend on the unknown parameter vector p. As sucha feedback control law is obtained by the stabilisation of(25) at the chosen nominal (known) parameter value p∗. Astabilising feedback law takes the form

uc = −K1x − K2xs�= −Kxc, (27)

where the gain vector K can be computed using eitherpole assignment technique or the linear optimal controltheory such that the closed-loop matrix Acl = [Ac(p∗) −Bc(p∗)K] is Hurwitz.

Here, we design the control law using the optimal con-trol theory. For optimal control, a quadratic performanceindex

J =∫ ∞

0

(xT

c Qxc + ru2c

)dt (28)

is chosen, where the weighting matrix Q is a positive defi-nite symmetric matrix and r > 0. The optimal gain vectoris given by

K = r−1BTc (p∗)P, (29)

where P is the positive definite symmetric matrix, whichsatisfies the algebraic Riccati equation (Kailath 1980)

ATc (p∗)P + PAc(p∗)

−PB c(p∗)r−1BTc (p∗)P + Q = 0. (30)

The weighting matrix Q and r can be chosen to shape thetransient responses.

Although the gain vector K is computed for a knownnominal value of w, it follows that the closed-loop matrixAcl(p) remains Hurwitz for perturbations p ∈ p, wherep is a sufficiently small open set. Thus it follows thatthe origin xc = 0 of the non-linear system (24) is expo-nentially stable for v = 0 and for sufficiently small p. Inthe closed-loop system including the internal model of thek-fold exosystem, one can show that Ylp in the Taylor seriesexpansion of the tracking error (16) are null vectors forsmall p and l = 1, . . . , k. Thus the tracking error e(t) satis-

fies (13) as t → ∞, and therefore, the steady-state trackingerror is zero up to kth order.

We may point out that although this paper treats pectoralfin control of a BAUV, the design methodology developedhere is applicable to other swimming and flying roboticsystems, which employ oscillating control surfaces (flap-ping fins and wings) to produce propulsive and manoeu-vering forces. The control system designed here can besynthesised using the state variables of the robotic vehi-cle, and the linear servo compensator forms a part of thecontroller. Although, the stabiliser is designed based onthe linearised model, the linear quadratic design does pro-vide robustness to parametric uncertainties by the choiceof suitable performance index. This is of practical im-portance because system parameters are not preciselyknown.

4. Simulation results

In this section, simulation results for the closed-loop sys-tem (4), (21) and (27) using MATLAB/SIMULINK are pre-sented. The parameters of the vehicle model in Equation (4)are taken from Singh et al. (2004). The BAUV is assumedto be moving with a constant forward velocity of 0.8 m/s.The vehicle parameters are l = 1.282 m, m = 4.1548 kg,Iy = 0.5732 kg·m2, xG = 0 and ZG = 0.578802 × 10−8

m. The hydrodynamic parameters for the forward veloc-ity of 0.8 m/s are z′

q = −0.825 × 10−5, z′w = −0.825 ×

10−5, z′q = −0.238 × 10−2, z′

w = −0.738 × 10−2, M ′q =

−0.16 × 10−3, M ′w = −0.825 × 10−5, M ′

q = −0.117 ×10−2 and M ′

w = 0.314 × 10−2. The pectoral fins are at-tached at a distance of dcgf = 0.15 m. The simulation re-sults are obtained for fin oscillation frequencies of f = 6Hz and f = 8 Hz. Of course, the controller design is ap-plicable for any choice of frequency of oscillation of thefins.

Using CFD analysis, the fin forces and the momentscoefficients have been obtained for a fixed Strouhal numberSt = 0.6, where St = cf

U∞, U∞ is 0.8 m/s, f is the frequency

of the fin oscillation and c is the chord of the foil. Theparameter vectors fa , fb, ma and mb used for simulationsare

fa = (f c

0 (0), f s1 (0), f c

1 (0), . . . , f sM (0), f c

M (0))T

,

fb=(

∂f c0

∂β(0),

∂f s1

∂β(0),

∂f c1

∂β(0), . . . ,

∂f Ms

∂β(0),

∂f Mc

∂β(0)

)T

,

ma = (mc0(0),ms

1(0),mc1(0), . . . , ms

M (0),mcM (0))T ,

mb=(

∂mc0

∂β(0),

∂ms1

∂β(0),

∂mc1

∂β(0), . . . ,

∂mMs

∂β(0),

∂mMc

∂β(0)

)T

,

where fa, fb,ma,mb ∈ R2M+1. In the Fourier expansion,four harmonic components, which are dominant, are


0 10 20 30 40 50−20

0

20

40

Time (s)

z,zr (

m)

0 5 10 15 20 25 30−4

−2

0

2

Time (s)

Pit

ch

an

gle

, θ

(d

eg

)

0 5 10 15−20

0

20

Late

ral

force (

N)

Time (s)0 5 10 15

−0.2

0

0.2

Mom

en

t (N

m)

Time (s)

0 5 10 15 20 25−10

0

10

Time (s)

Bia

s a

ngle

(d

eg)

0 10 20 30 40 50−20

−10

0

10

Time (s)

Track

ing e

rror, e (

m)

147 147.5 148 148.5 149 149.5 150−0.1

−0.05

0

0.05

0.1

Time (s)

Track

ing e

rror, e (

m)

(a) (b)

(c)

(e) (f)

(g)

(d)

Figure 2. BAUV control using first-order servo compensator – ωf = 6 Hz and the nominal parameters: (a) dive plane depth,z, and reference depth, zr (in metres), (b) pitch angle, θ (in degrees), (c) lateral force (in Newtons), (d) moment (in New-ton metres), (e) bias angle (in degrees), (f) tracking error, e (in metres), (g) tracking error plotted for smaller time interval, e(in metres).

retained; that is, M = 4. Then the values of fa, fb,ma,mb

for M = 4 are

fa = (0,−40.0893,−43.6632,−0.3885, 0.6215, 6.2154,

− 10.1777,−0.1554, 0.6992),

fb = (68.9975, 0.4451,−16.4704, 64.1009,−19.5864,

− 0.8903,−2.2257, 2.2257, 4.8966),

ma = (0, 0.6037, 0.4895, 0,−0.0054, 0,

− 0.0925, 0,−0.0054),

mb = (−0.4986,−0.3739,−0.0935,−0.2493, 0.1246,

0.0312,−0.0312, 0.0935, 0).

(Readers may refer to Narasimhan et al. [2006] for the de-tails.) It is pointed out that these parameters are obtained us-ing the Fourier decomposition of the fin force and moment,


0 10 20 30 40 50−20

0

20

40

Time (s)

z,z

r (

m)

0 5 10 15 20 25 30−4

−2

0

2

Time (s)

Pit

ch

an

gle

, θ

(d

eg

)

0 5 10 15−20

0

20

La

tera

l fo

rce (

N)

Time (s)0 5 10 15

−0.2

0

0.2

Mo

men

t (N

m)

Time (s)

0 5 10 15 20 25−10

0

10

Time (s)

Bia

s a

ngle

(d

eg)

0 10 20 30 40 50−20

−10

0

10

Time (s)

Track

ing e

rror, e (

m)

147 147.5 148 148.5 149 149.5 150−0.05

0

0.05

Time (s)

Track

ing e

rror, e (

m)

(a) (b)

(c) (d)

(e) (f)

(g)

Figure 3. BAUV control using first-order servo compensator – ωf = 6 Hz, −25% uncertainty: (a) dive plane depth, z, and referencedepth, zr (in metres), (b) pitch angle, θ (in degrees), (c) lateral force (in Newtons), (d) moment (in Newton metres), (e) bias angle (indegrees), (f) tracking error, e (in metres), (g) tracking error plotted for smaller time interval, e (in metres).

and are computed by multiplying the Fourier coefficients by12ρWaU∞2 and 1

2ρWacU∞2, respectively, where Wa is thesurface area of the foil. For simulation, the initial conditionsof the vehicle are assumed to be x(0) = 0 and xs(0) = 0.

A smooth reference trajectory zr (t) converging to z∗,the target depth, using a fourth-order filter

Gc(s) = λ1ω2nc

(s + λc1)(s + λc2)(s2 + 2ζcωncs + ω2

nc

)is generated, where ωnc = 4.95, ζc= 0.707, λc1 = 0.14 andλc2 = 3.5 are the real poles.

A simple servo compensator of first order providing in-tegral error feedback, as well as a compensator representingthe internal model of two-fold exosystem are designed. Thelatter servo compensator is of fifth order. For the first-ordercompensator, G1 = 0 and G2 = 0.5, and for the fifth order,one has

dG1 = diag

(0,

[0 −ωf

ωf 0

],

[0 −2ωf

2ωf 0

]),

G2 = [0.5, 0, 0.5, 0, 0.5]T .


0 10 20 30 40 50−20

0

20

40

Time (s)

z,z

r (

m)

0 5 10 15 20 25 30−4

−2

0

2

Time (s)

Pit

ch

an

gle

, θ

(d

eg

)

0 5 10 15−50

0

50

La

tera

l fo

rce (

N)

Time (s)0 5 10 15

−0.5

0

0.5

Mo

men

t (N

m)

Time (s)

0 5 10 15 20 25−50

0

50

Time (s)

Bia

s a

ngle

(d

eg)

0 10 20 30 40 50−10

−5

0

5

Time (s)

Track

ing e

rror, e (

m)

147 147.5 148 148.5 149 149.5 150−0.02

−0.01

0

0.01

0.02

Time (s)

Track

ing e

rror, e (

m)

(a) (b)

(c) (d)

(e) (f)

(g)

Figure 4. BAUV control using internal model of two-fold exosystem – ωf = 6 Hz and the nominal parameters: (a) dive plane depth, z,and reference depth, zr (in metres), (b) pitch angle, θ (in degrees), (c) lateral force (in Newtons), (d) moment (in Newton meter), (e) biasangle (in degrees), (f) tracking error, e (in metres), (g) tracking error plotted for smaller time interval, e (in metres).

If the input e to the servo compensator is zero, thenthe first-order system generates constant trajectory, butthe fifth-order servo compensator can generate trajecto-ries of the form c1 + c2sin(ωf t + θ1) + c3sin(2ωf t + θ2)by appropriate choice of initial conditions. For the con-trol law design, linear optimal control theory is used.For the first-order compensator, Q=1, and for the fifthorder, Q is an identity matrix of dimension 9 × 9, andr = 0.0001.

Case I: BAUV control using first-order servocompensator: ω f =6 Hz, nominal parameters

The complete closed-loop system including the nominalBAUV model and the first-order servo compensator is sim-ulated for a fin oscillation frequency of 6 Hz. A smoothreference trajectory zr (t) converging to z∗ = 25 m is gen-erated. Thus it is desired to steer the BAUV to a depth of25 m. The optimal controller gains are computed for thenominal BAUV model. Selected responses are shown in


0 10 20 30 40 50−20

0

20

40

Time (s)

z,z

r (

m)

0 5 10 15 20 25 30−2

−1

0

1

Time (s)

Pit

ch

an

gle

, θ

(d

eg

)

0 5 10 15−50

0

50

La

tera

l fo

rce (

N)

Time (s)0 5 10 15

−0.5

0

0.5

Mo

men

t (N

m)

Time (s)

0 5 10 15 20 25−50

0

50

Time (s)

Bia

s a

ngle

(d

eg)

0 10 20 30 40 50−10

−5

0

5

Time (s)

Track

ing e

rror, e (

m)

147 147.5 148 148.5 149 149.5 150−0.05

0

0.05

Time (s)

Track

ing e

rror, e (

m)

(a) (b)

(c) (d)

(e) (f)

(g)

Figure 5. BAUV control using internal model of two-fold exosystem – ωf = 6 Hz, +25% uncertainty: (a) dive plane depth, z, andreference depth, zr (in metres), (b) pitch angle, θ (in degrees), (c) lateral force (in Newtons), (d) moment (in Newton metres) (e) bias angle(in degrees) (f) tracking error, e (in metres) (g) tracking error plotted for smaller time interval, e (in metres).

Figure 2. It is observed that the BAUV attains the targetdepth in little over 30 s. In steady-state, fin forces, momentand bias angle exhibit bounded periodic oscillations. Notethat the chosen servo compensator has a simple pole atzero and as such it can only suppress any non-zero bias inthe tracking error. We observe that indeed average trackingerror is zero, but periodic oscillations including the fun-damental component (6 Hz) and higher harmonics persist.The magnitude of the tracking error e is observed to bearound 0.08 m. The maximum control magnitude is around8◦. The peak control force needed is 12 N and the controlmoment is 0.15 N·m.

Case II: BAUV control using first-order servocompensator: ω f = 6 Hz, −25% uncertainty

Now simulation is done to examine the effect of uncer-tainties in the control force and moment coefficients. Forthis purpose the elements of the vectors fa , fb, ma and mb

are perturbed by a factor of 0.75 for simulation; that is,the perturbed values of these vectors are 25% lesser thanthe nominal values. However, the controller gains used inFigure 2 computed for the nominal values are retained. Se-lected plots are provided in Figure 3. It is again noted thatwith the servo compensator of first order, the oscillatorycomponents of the tracking error are not suppressed. But,


0 10 20 30 40 50−20

0

20

40

Time (s)

z,zr (

m)

0 5 10 15 20 25 30−4

−2

0

2

Time (s)

Pit

ch

an

gle

, θ

(d

eg

)

0 5 10 15−10

0

10

Late

ral

force (

N)

Time (s)0 5 10 15

−0.1

0

0.1

Mo

men

t (N

m)

Time (s)

0 5 10 15 20 25−10

0

10

Time (s)

Bia

s a

ngle

(d

eg)

0 10 20 30 40 50−20

−10

0

10

Time (s)

Track

ing e

rror, e (

m)

147 147.5 148 148.5 149 149.5 150−0.05

0

0.05

Time (s)

Track

ing e

rror, e (

m)

(a) (b)

(c) (d)

(e) (f)

(g)

Figure 6. BAUV control using first-order servo compensator – ωf = 8 Hz and the nominal parameters: (a) dive plane depth, z, andreference depth, zr (in metres), (b) pitch angle, θ (in degrees), (c) lateral force (in Newtons), (d) moment (in Newton metres), (e) biasangle (in degrees), (f) tracking error, e (in metres), (g) tracking error plotted for smaller time interval, e (in metres).

the average value of the tracking error is zero. The magni-tude of the tracking error is 0.05 m. The maximum controlmagnitude and the response time to attain the target depthare of the same order as in Figure 2.

Simulation is also done for fin force coefficients with+25% uncertainty (perturbed values are 1.25 times thenominal values) using the nominal controller. In this casealso, a smooth control to the desired depth is accom-plished. (In order to save space, the results are not shownhere).

Case III: BAUV control using internal model of2-fold exosystem: ω f = 6 Hz, nominal parameters

For attenuating the dominant oscillatory components of thetracking error, it is essential to synthesise servo compen-sator of higher order. For the purpose of illustration, a servocompensator of fifth order is deigned using internal modelof two-fold exosystem. Selected responses for the nominalBAUV model are shown in Figure 4. It is seen that desireddepth is smoothly attained. The magnitude of the trackingerror is significantly smaller compared with that of Figure 2.


0 10 20 30 40 50−20

0

20

40

Time (s)

z,z

r (

m)

0 5 10 15 20 25 30−2

−1

0

1

Time (s)

Pit

ch

an

gle

, θ

(d

eg

)

0 5 10 15−10

0

10

La

tera

l fo

rce (

N)

Time (s)0 5 10 15

−0.1

0

0.1

Mo

men

t (N

m)

Time (s)

0 5 10 15 20 25−10

0

10

Time (s)

Bia

s a

ngle

(d

eg)

0 10 20 30 40 50−20

−10

0

10

Time (s)

Track

ing e

rror, e (

m)

147 147.5 148 148.5 149 149.5 150−0.05

0

0.05

Time (s)

Track

ing e

rror, e (

m)

(a) (b)

(d)

(f)

(g)

(e)

(c)

Figure 7. BAUV control using first-order servo compensator ωf = 8 Hz, +25% uncertainty: (a) dive plane depth, z, and reference depth,zr (in metres), (b) pitch angle, θ (in degrees), (c) lateral force (in Newtons), (d) moment (in Newton metres), (e) bias angle (in degrees),(f) tracking error, e (in metres), (g) tracking error plotted for smaller time interval, e (in metres).

Interestingly, the designed servo compensator suppressesthe constant bias, fundamental and second harmonic com-ponents in the tracking error response, and only oscillationsof frequency 18 Hz and higher remain. Although one candesign a higher-order compensator, we observe that eventhis fifth-order servo compensator yields maximum errorlittle over 0.01 m, which is negligible for practical pur-poses. The maximum control magnitude is observed to bearound 30◦. The target depth is attained in a little over 30 sas in Figure 2.

Case IV: BAUV control using internal model oftwo-fold exosystem: ω f = 6 Hz, +25% uncertainty

Now simulation is done to examine the robustness of thecontrol system. It is assumed that the fin force coefficientsare 25% greater than the nominal values, but the nomi-nal control system used for Figure 4 is retained. Selectedresponses are shown in Figure 5. Similar to Figure 4, weobserve that the vehicle attains the desired depth and con-troller is able to suppress the bias, fundamental and secondharmonics in the tracking error response. Compared with


0 10 20 30 40 50−20

0

20

40

Time (s)

z,z

r (

m)

0 5 10 15 20 25 30−4

−2

0

2

Time (s)

Pit

ch

an

gle

, θ

(d

eg

)

0 5 10 15−20

0

20

40

La

tera

l fo

rce (

N)

Time (s)0 5 10 15

−0.2

0

0.2

Mo

men

t (N

m)

Time (s)

0 5 10 15 20 25−50

0

50

Time (s)

Bia

s a

ngle

(d

eg)

0 10 20 30 40 50−10

−5

0

5

Time (s)

Track

ing e

rror, e (

m)

147 147.5 148 148.5 149 149.5 150−5

0

5x 10

−3

Time (s)

Track

ing e

rror, e (

m)

(a) (b)

(c) (d)

(e) (f)

(g)

Figure 8. BAUV control using internal model of two-fold exosystem – ωf = 8 Hz and the nominal parameters: (a) dive plane depth, z,and reference depth, zr (in metres), (b) pitch angle, θ (in degrees), (c) lateral force (in Newtons), (d) moment (in Newton metres), (e) biasangle (in degrees), (f) tracking error, e (in metres), (g) tracking error plotted for smaller time interval, e (in metres).

Figure 4, the tracking error magnitude in Figure 5 is a littlehigher at 0.02 m, but the maximum control magnitude andthe target depth response time are of the same magnitude.

Case V: BAUV control using first-order servocompensator: ω f = 8 Hz, nominal parameters

The frequency of the pectoral fin oscillation is set at 8 Hzand the performance of the first-order servo compensatoris evaluated. Of course, for 8 Hz the fin force and mo-

ment coefficients have changed, and the feedback gains ofthe controller are redesigned using the same values of Qand r. The responses are shown in Figure 6. It is observedthat the average tracking error is zero, but the oscillatorycomponents in the tracking error are still present. The sim-ulation results shown in Figure 6 are some what similar tothe results obtained in Figure 2. Though the tracking errorpattern in Figures 2 and 6 are the same, the magnitude ofthe tracking error in Figure 6 is 0.025 m, lesser than that inFigure 2. The maximum control magnitude and the target


0 10 20 30 40 50−20

0

20

40

Time (s)

z,z

r (

m)

0 5 10 15 20 25 30−5

0

5

Time (s)

Pit

ch

an

gle

, θ

(d

eg

)

0 5 10 15−20

0

20

40

La

tera

l fo

rce (

N)

Time (s)0 5 10 15

−0.5

0

0.5

Mo

men

t (N

m)

Time (s)

0 5 10 15 20 25−50

0

50

Time (s)

Bia

s a

ngle

(d

eg)

0 10 20 30 40 50−10

−5

0

5

Time (s)

Track

ing e

rror, e (

m)

147 147.5 148 148.5 149 149.5 150−5

0

5x 10

−3

Time (s)

Track

ing e

rror, e (

m)

(a) (b)

(c) (d)

(e) (f)

(g)

Figure 9. BAUV control using internal model of two-fold exosystem – ωf = 8 Hz, −25% uncertainty: (a) dive plane depth, z, andreference depth, zr (in metres), (b) pitch angle, θ (in degrees), (c) lateral force (in Newtons), (d) moment (in Newton metres), (e) biasangle (in degrees), (f) tracking error, e (in metres), (g) tracking error plotted for smaller time interval, e (in metres).

depth response time in both Figures 2 and Figure 6 arealmost the same.

Case VI: BAUV control using first-order servocompensator: ω f = 8 Hz, +25% uncertainty

An uncertainty of +25% is introduced in the fin force co-efficents for simulation. The results using the nominal con-troller of Figure 6 are shown in Figure 7. The magnitude ofthe tracking error in Figure 7 is 0.04 m, whereas in Figure

6 it was 0.025 m. The target depth response time is close to35 s. The maximum control magnitude is around 8◦.

Case VII: BAUV control using internal modelof two-fold exosystem: ω f = 8 Hz, nominalparameters

It is assumed that the fins are oscillating at 8 Hz. For obtain-ing improved responses, a fifth-order servo compensator forthe nominal values of the parameters is designed. Simula-tion results of the nominal BAUV are shown in Figure 8.


We observe that target depth is attained and oscillationsof fundamental and second harmonic in the tracking errorare suppressed. Compared with the case of first-order servocompensator in Figure 6, we observe significant reductionin the peak magnitude of the steady-state tracking error. Themagnitude of the tracking error is 0.004 m in Figure 8 whichis substantially lesser than the tracking error magnitude of0.025 m in Figure 6. The maximum control magnitude isaround 30◦ as in the case of Figure 4. The target depth isreached in little over 30 s.

Case VIII: BAUV control using internal model oftwo-fold exosystem: ω f = 8 Hz, −25% uncertainty

To examine the robustness of the designed controller anuncertainty of −25% is added to the nominal values ofthe vectors fa , fb, ma and mb for simulation. The selectedresponses are shown in Figure 9. The tracking error mag-nitude for the chosen uncertainty is around 0.002 m. Themaximum control magnitude and the target depth responsetime are similar in both Figures 8 and 9.

It is pointed out that although the controller is designedbased on the linearised model, these results show robustperformance even for ±25% parametric uncertainties inthe Fourier coefficients. These coefficients are used in thecomputation of the fin force and moment according to (3).It is noted that the input matrix (which is a function ofthe Fourier coefficients) plays a key role in shaping the re-sponses. The robustness of the controller depends on the theweighting matrix Q and scalar r appearing in the perfor-mance index. Of course, there does not exist a simple way todetermine their proper values. The values of Q and r havebeen selected here by observing the simulated responses inseveral trials.

Conclusion

Based on the non-linear servo regulation theory, a new de-sign methodology for the control of multi-input BAUVs us-ing pectoral-like oscillating fins was presented in this paper.For the dive plane control, a pair of pectoral-like harmon-ically heaving and pitching fins were used. The bias angleof pitch motion of the fin was treated as a control input.For the purpose of design, an exosystem of third order wasintroduced and the original time-varying non-linear systemwas embedded in a larger class of time-invariant non-linearsystem. For robust design, an internal model of k-fold ex-osystem was introduced. The augmented system, includingthe internal model, was stabilised using optimal controltheory. In the closed-loop system, including the internalmodel of the k-fold exosystem, harmonic components oforder up to k of the tracking error are suppressed. Thisspecial property is not possible using averaging methodor discretisation approach reported in literature. A simpleservo compensator using only integral error feedback and

a fifth-order servo compensator were designed. Simulationresults were obtained, which showed robust depth controlperformance. Interestingly, the internal model of two-foldexosystem was capable of attenuating the depth trackingerror to a negligible level in spite of uncertainties in thesystem parameters. It was seen that flexibility exists in thechoice of weighting matrices for shaping responses usingoptimal control theory.

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AppendixMatrices A[l]

p and minimum polynomials

A[2]p =

⎡⎣ 0 2ωf 0

−ωf 0 ωf

0 −2ωf 0

⎤⎦ ,

A[3]p =

⎡⎢⎣

0 3ωf 0 0−ωf 0 2ωf 0

0 −2ωf 0 ωf

0 0 −3ωf 0

⎤⎥⎦ ,

A[4]p =

⎡⎢⎢⎢⎣

0 4ωf 0 0 0−ωf 0 3ωf 0 0

0 −2ωf 0 2ωf 00 0 −3ωf 0 ωf

0 0 0 −ωf 0

⎤⎥⎥⎥⎦ .

Computing the minimum polynomials p[1](λ) of A[1] and p[k](λ)of A[k]

p (k = 2, 3, 4), one finds that

p[1] = λ(λ2 + ω2f ),

p[2] = λ(λ2 + (2ωf )2),

p[3] = (λ2 + ω2f )(λ2 + (3ωf )2),

p[4] = λ(λ2 + (2ωf )2)(λ2 + (4ωf )2).

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Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

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