ROBUST ADAPTIVE MOTION CONTROL WITH ENVIRONMENTAL DISTURBANCE REJECTION FOR PERTURBED UNDERWATER VEHICLES

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1 Journal of Marine Science and echnology DOI: 0.69/JMS ROBUS ADAPIVE MOION CONROL WIH ENVIRONMENAL DISURBANCE REJECION FOR PERURBED UNDERWAER VEHICLES Hamid Reza Koofigar his article has been eer reviewed and acceted for ublication in JMS but has not yet been coyediting, tyesetting, agination and roofreading rocess. Please note that the ublication version of this article may be different from this version. his article now can be cited as doi: 0.69/JMS Key words: Motion control, Robust adative tracking, Unmodelled dynamics, Hydrodynamic coefficients, Environmental disturbances. ABSRAC his aer addresses the motion control roblem of autonomous underwater vehicles (AUVs) erturbed by unknown hydrodynamic coefficients, unmodelled dynamics and environmental disturbances. he roosed algorithm consists of an adative subcontroller to tackle the arametric uncertainties and a robust term to vanish the effects of unstructured uncertainties and disturbances. he resulting robust adative controller ensures the convergence of tracking error, without any assumtion on the uer bound of erturbations in the design rocedure. he closed loo stability is shown, using the Lyaunov stability theorem, and verified by various simulations. NOMENCLAURE Symbol Descrition θ φ ψ q r u v w x y z Pitch angle Roll angle Yaw angle Pitch angular velocity Roll angular velocity Yaw angular velocity Surge velocity Sway velocity Heave velocity Position in x-direction Position in y-direction Position in z-direction Paer submitted 03/0/; revised 03/9/3; acceted 05//3. Author for corresondence: Hamid Reza Koofigar ( koofigar@eng.ui.ac.ir). Deartment of Electrical Engineering, University of Isfahan, Isfahan, Iran. I. INRODUCION Motion control of underwater vehicles is one of the main toics in oceanic research and engineering, due to various alications of such vehicles in scientific and military oerations [4, 8]. In ractice, the dynamic equations of such vehicles are highly nonlinear and the system arameters like hydrodynamic coefficients and added mass are not exactly known. Furthermore, such environmental disturbances as currents and wave's effects cause the desired erformance not be achieved. Hence, taking into account the external disturbances, arametric uncertainties, and unmodelled dynamics is necessary in the design rocedure. During the ast years, comlete, simlified, and linearized model of UVs have been used to develo motion control algorithms. Although using a linearized model facilitates the designing rocedure, the validity of control algorithms deends heavily on the accuracy of the vehicle dynamics. Adoting the linearized dynamics, various linear control techniques such as PID controller [8], LQR and LQG algorithms [6], and linear H controller [] have been develoed. Some more recent investigations have decouled the diving lane and steering system states, by ignoring the interaction between the system variables [3, 5]. he decouled controllers e.g., deth, itch, and yaw controller can be develoed [3, 4, ], without taking the variation of vehicle roll and yaw angular velocity into account. From a ractical viewoint, such simlicity is obtained at the exense of violating some degrees of freedom and imosing some variables to be zero or constant. Concerning with the comlete model, develoing some six-degree of freedom controller, based on robust control and intelligent algorithms, has been reorted to achieve the desired erformance [6, 8]. he revious works suffer from at least one of the following restrictions, (i) the conservative assumtions make the algorithm be alicable for a secial class of UVs or for a certain direction of motion, (ii) unmodelled dynamics, arameter variations and environmental disturbances are not incororated in dynamical equations altogether, (iii) the uer bound of uncertainties and disturbances are known in advance, (iv) robust closed-loo stability and erformance is not ensured analytically.

2 Motivated by the aforementioned drawbacks, this aer concerns with develoing an adative-based motion controller for UVs to ensure the rescribed robustness roerties. In fact, adative control is adoted here to tackle the arametric uncertainties, originated by the hydrodynamic coefficients' variations in different oerating conditions. Among the Lyaunov-based adative design algorithms, backsteing and assivity- based techniques may be commonly alied to secial classes of nonlinear systems [0, ]. Such methods have been develoed for secial cases of UV maneuverings, as in diving and steering lanes [3, 4]. Furthermore, such strategies may be jointed to some intelligent algorithms to comensate the effects of uncertainties [7]. In general, conventional adative control may fail in the resence of unstructured (non-arametric) uncertainties and environmental disturbances [, ]. In this aer, a robustifying mechanism is incororated to ensure the robustness roerties of the roosed algorithm. In the design rocedure, unstructured uncertainty is incororated into the nominal model to reflect the existence of unmodelled dynamics and the uncertainty in the matrix of inertia. Moreover, the hydrodynamic coefficients, as arametric uncertainties, are allowed to vary with unknown bound and no re-assumtions are made on the uer bound and eriodicity of environmental disturbances. Comared with the revious motion control algorithms, some secific roerties of the roosed method are: (i) dynamical equations may be erturbed by arameter variations, unstructured uncertainties, and external disturbances, altogether, (ii) neither the uer bound nor the eriodicity of erturbations is required in the design rocedure, (iii) the stability analysis is resented analytically based on the Lyaunov stability theorem. his aer is organized as follows. In section, introducing the system dynamics, the control roblem is formulated. Dealing with various kinds of erturbations, two robust adative motion control and the stability roofs are given in section 3. he simulation results are resented in section 4, to evaluate the erformance of the roosed methods. he concluding remarks are finally given in section 5. hroughout the aer, for an n vector V, V denotes the Euclidean vector norm, and V := Q V QV, with the weighting matrix Q. Furthermore, V L [0, ) if V <, t, and V L [ 0, ] for all [0 ) [0 ). if V ( t) dt <, II. PROBLEM FORMULAION In general, two coordinates including earth-fixed frame and body-fixed frame, are used to describe the AUV kinematics. In fact, the osition and orientation of the vehicle are described in the earth-fixed coordinate, and the linear and angular velocities are described in the body-fixed frame. he dynamic equations of motion for a six degrees-of-freedom underwater vehicle can 0 be reresented in a comact form as [8] M ν + C( ν ) ν + D( ν ) ν + g( η) = τ () η = J ( η) ν where ν = [ u, v, w,, q, r] is the vector of linear and angular velocities in body-fixed frame, η = [ x, y, z, φ, θ, ψ ] denotes the osition and orientation in earth-fixed frame, and 6 6 J ( η ) R reresents a transformation matrix between the body-fixed frame and the earth-fixed coordinate. he ositive 6 6 definite inertia matrix M R consists of rigid-body mass 6 6 and added mass, C( ν ) R denotes the coriolis and 6 6 centrietal matrix, and D( ν ) R is the matrix of hydrodynamic and daming forces. he vector of control 6 forces and moments is reresented by τ R, and g (η ) is the vector of buoyancy and gravitational terms. he equivalent of equation () can be reresented in body-fixed (global) frame as M η ( η) η + Cη ( η, ν ) η + Dη ( η, ν ) η + gη ( η) = τη + dη () where = M J MJ C = J ( C MJ J ) J DJ η, η, J Dη = J, gη ( η) = J g( η), τ η = τ, and d η reresents the external disturbance vector. In ractice, the arameters included in C (ν ) and D (ν ), e.g., hydrodynamic and daming coefficients, may be changed in various circumstances. On the other hand, in such a comlex nonlinear system, model uncertainty inevitably erturb the nominal model. Hence, by defining H ( η, ν ) = C ( η, ν ) η + D ( η, ν ) η, one can write η η H ( η, ν ) = H0( η, ν ) + ζ ( η, ν ) Θ (3) where H 0( η, ν ) is the known art, Θ denotes an unknown arameter vector, and ζ ( η, ν ) reresents a dimensionally comatible matrix. he uncertainties in the elements of inertia matrix are modelled here by decomosing M generally as Mη = M 0 + M Δ ( t) (4) in which M 0 denotes the known nominal art and M Δ (t) is a norm-bounded unstructured time-varying erturbation with bounded time-derivative, i.e., M Δ ( t ) α, and M Δ ( t ) α, where α, α are two unknown constants. Meanwhile, an uncertain arameter is defined as α = max { α, α}. he motion control objective is to track any given smooth state trajectory η d, in the resence of unstructured uncertainties, unknown hydrodynamic coefficients and environmental disturbances. III. ROBUS ADAPIVE MOION CONROL his section is devoted to design adative motion control

3 algorithms which guarantee the robust stability and erformance of UVs, in various ractical situations. Consider the dynamic equations of UVs given by (). o facilitate the design rocedure, take an equivalent disturbance vector as d = d η g η ( η) and rewrite the dynamic equation () as ( M 0 + M Δ ) η + H 0( η, ν ) + ζ ( η, ν ) Θ = τη + d (5) For a given desired vector η d, define the tracking error e = ηd η, and the error metric functions S ( t) = e ( t) + e( t) and Sr( t) = η d ( t) + e ( t). he alied control inut is roosed here as τ η = KS + M 0S r + H0 + τ a + τ s (6) where K is a ositive definite matrix, τ a denotes the adative subcontroller, and τ s resents the robust subcontroller. In the following, deending on the characteristics of disturbances, two novel control algorithms are roosed to design τ a and τ s. o this end, an adative-based H algorithm iss first develoed, assuming the external disturbance is belong to L sace. hen, removing such hyothesis, a robust adative motion control technique is roosed to tackle various kinds of bounded erturbations, without any conservative assumtions. heorem. For the underwater vehicles described by dynamical equation (5), the control inut (6), with SSr Sr τ Θ + + a = τ a ζ ˆ = ˆ α S ˆ α (7) Sr S ˆ α + δe, τ s = S ρ (8) and udate laws Θ ˆ = ΓSζ ( η, ν ) (9), ˆ α = α S S + Sr S (0) where Γ = Γ > 0 is the adatation matrix, and α > 0 denotes the adatation gain, solves the motion control roblem in the resence of system uncertainties and disturbances. Moreover, δ and σ are two (small) ositive constants to rovide the smoothness of control inut. Proof. ake a Lyaunov function candidate as V e e Θ e Ke + S M S + (,, α, ) = η α + Θ Γ Θ () α where α = α ˆ α, and Θ = Θ Θˆ denote the estimation errors. aking the time derivative of V gives V = e + S ( M e + M e) + S M η η η S () + αα + Θ Γ Θ α By e = ηd η and dynamical equation (5), and relacing control inut (6), one can obtain S ( Mηe + Mηe ) = S (( M 0 + M Δ ) ηd ( M 0 + M Δ ) η + ( M0 + M Δ) e ) (3) = S ( M 0Sr + M ΔSr + H0 + ζθ τη d ) = S ( KS + M ΔSr + ζθ τ a τ s d ) Substituting (3) and subcontrollers (7) and (8) in () gives V = e S KS + S M ΔSr + S ( ζθ ζθˆ ) S SSr Sr ˆ αs S ˆ α σ t Sr S ˆ α + δe (4) S S S d + S M ΔS ˆ Θˆ αα Γ Θ ρ α Alying the equivalence S S S d = ( S + ρd ) ( S + ρd ) + ρ d ρ ρ into (4) and some maniulations imly that V e + α S S + Sr S + S ζθ + δe ( S + ρd ) ( S + ρd ) + ρ d (5) ρ ˆ Θˆ αα Γ Θ α Now, incororate the adatation mechanisms (9) and (0) to obtain V e ( S + ρd ) ( S + ρd ) ρ (6) + ρ d + δe By omitting some strictly negative terms from right hand side of inequality (6), one can conclude t V σ e Ke + ρ d + δe (7) and t V σ e + ρ d + δe (8) which give the following consequents. (i) he boundedness of disturbance signal d imlies that there exists a d > 0 such that d d. By inequality (7), one can obtain V λ K e + ρ d + δ, where λ K is the ρd + δ minimum eigenvalue of K. Choosing λk > for any ε small ε > 0, there exists a κ > 0 such that V κ e < 0 for all e > ε. hus, there is a > 0 such that e ε for all t. his imlies that the tracking error e (t) is uniformly ultimately bounded [], and all the closed-loo signals are also bounded.

4 (ii) aking the inequality (8) into account and following a rocedure, similar to that given in (i), the boundedness of e is also concluded. (iii) Integrating the inequality (7) from t = 0 to t = yields e t dt V ( e e ( ) + ( ), ( ), α ( ), Θ ( )) 0 K t V ( e(0), e(0), (9) δ σ α (0), Θ (0)) + ( e ) + ρ d( t) dt σ 0 for all 0 <. his imlies that for any disturbance, belongs to L sace, e (t) is square-integrable which together with the boundedness roerty of e (t) and e, achieved in (i) and (ii), the Barbalat's lemma (see the aendix) [], ensures the convergence of tracking error e (t), desite the system uncertainties and environmental disturbances. Remark. he exonential terms, incororated in adative subcontroller (7), are to avoid chattering and discontinuity of control inut, without violating the convergence roerty of tracking error and closed-loo stability. Remark. Choosing a smaller ρ > 0 rovides the system with faster resonse. his may be obtained at the exense of larger control effort. In fact, there exists a trade-off between the value of subcontroller gain ρ and the magnitude of control inut τ s. In some ractical situations, belonging the wave disturbances to L sace, imlicitly assumed in heorem, may be not satisfied. On the other hand, the uncertainty in hydrodynamic and daming matrices, taken as arametric variations in (3), cannot reflect all asects of erturbations. Hence, equation (3) is modified as H ( η, ν ) = H0( η, ν ) + ζ ( η, ν ) Θ + H Δ ( η, ν ) (0) where H Δ denotes the unstructured uncertainty. he imortance of robust stability and erformance in various oeration conditions of UVs motivates develoing a more general motion control algorithm, relaxing any conservative assumtion. As a reliminary ste to design such a controller, the vector of lumed uncertainty is defined as Δ( η, ν ) = HΔ ( η, ν ) d () with Δ, where is an unknown arameter. heorem. Consider the dynamical equation of erturbed underwater vehicles, described by (5). he control law (6), formed by τ = 0 and adative subcontroller s S τ a τ a + ˆ = S ˆ () + δe with adatation mechanisms (9), (0), and ˆ = S (3) where ˆ denotes the estimated value of, and > 0 is the adatation gain, ensures the convergence of tracking error desite the erturbations. Proof. Choose the Lyaunov function U( e, e, α, Θ, ) = (,, V e e α, Θ ) + (4) where = ˆ is the arameter estimation error. Differentiating U imlies that U = V + (5) aking the equalities () and (0) into account, and then substituting τ η from (6), one can obtain e S ( M S + M S + H + ζθ τ + H d ) U = e + S 0 r Δ r + η αα ˆ Θˆ S M S Γ + S Δ S α KS + α S τ a + αs r ζθ ˆ ααˆ ˆ S Θ Γ α 0 Θ S + S Relacing adative subcontroller () yields U e + δe ˆ e S S S ˆ + δe + S + δe ˆ S Δ ˆ + δe η Θ + S Δ ˆ ˆ (6) (7) Now, incororate the udate law (3) to obtain t U e + δe σ (8) Following a rocedure, similar to the roof of heorem, imlies that the goal of motion control is achieved, desite the unstructured and arametric uncertainties, and environmental disturbances. Remark 3. In order to design a six-degree of freedom controller, decouling the equations of motion is avoided here to take into account all the ossible interactions between system variables in the design rocedure. In general, such decomosition restricts the alication of UVs to a certain oerating conditions such as moving in diving lane or travelling in a fixed deth [4, 5]. Remark 4. Unlike some revious works [, 5, 4], almost all kinds of uncertainties and disturbances due to the various alications of underwater vehicles in different conditions, are taken into account by theorems and, without any re-assumtion on the eriodicity or the bound of such erturbations. Remark 5. From a ractical oint of view, the limitations on choosing small samling time and imerfect imlementation of adatation mechanism (3) may cause the estimated value ˆ increase without bound. On the other hand, ˆ has direct imact on control law () and may cause instability. Hence, an effective modification is roosed here to alleviate this ractical drawback, as follows. Substitute the udate law (3) in theorem, with ˆ = η S (9)

5 where if e > ε η = 0 otherwise Such modification ensures that all the signals and states of the closed loo system are bounded and the norm of tracking error is robustly converged to a (small) rescribed bound ε > 0. In fact, this modification acts as a rojection algorithm and therefore the stability analysis can be followed, similar to that of conventional rojection methods in the literature [0]. IV. SIMULAION SUDY In order to illustrate the validity of the roosed motion control algorithms, the simulation results are resented here, using the nominal arameters of the vehicle, as given by Fossen [7]. In this study, the erformance of the methods, develoed by theorems and, are evaluated by incororating various kinds of uncertainties and disturbances into the six degree of freedom model. o this end, two cases are considered here as follows. Case I. Uncertain hydrodynamic arameters and white Gaussian noise. he erformance of the develoed algorithm in theorem is evaluated in the resence of the zero mean white noise, deicted in Fig.. Choosing ρ = and ρ = 0.3, Fig. show that the outut regulation is achieved, without steady state error. In other words, roll stabilization, regulation of the desired itch, corresonding to the desired deth z = 8 m, and ψ o = 0 as the desired yaw angle are all satisfied desite the arametric uncertainties and disturbance inut. Moreover, Fig. show that choosing a lower attenuation level ρ can more effectively override the effect of external disturbances. In order to illustrate the convergence of unknown system arameters, using the roosed adative control algorithm, the absolute values of the some hydrodynamic coefficients, normalized by density value, are shown in Fig. 3. In fact, the convergence of such arameters which corresond to diagonal elements of matrix, concisely satisfies the robust erformance of the algorithm. Case II. Mass uncertainty, Unmodelled dynamics, and biased sinusoidal disturbance. Although the nominal equations of motion have been determined in the literature, but robust erformance is obtained rovided that various kinds of erturbations in real world alications are taken into account. In this case, the variation of inertia matrix is also taken into account. o this end, consider the unstructured uncertainty about 0 % of the nominal arameters, in the resence of sinusoidal disturbance sin0. t, biased by 0.5, from t = 0 sec. As shown in Fig. 4, the desired regulation erformance goal is met by the roosed robust adative control algorithm, considering almost all kinds of ossible uncertainties and environmental disturbances. he convergence of system arameters is shown by Figs. 5 and Fig. 6 which resectively illustrates the norm of estimation error and the hydrodynamic arameter adjustment. Fig.. he zero mean white Gaussian noise. (c) Fig.. ime resonse of outut signals in case, when ρ = ( ) and ρ = 0.3 ( ), (a) Roll angle, (b) Pitch angle error, (c) Yaw tracking error. (a) (b)

6 (a) (e) (b) (f) Fig. 3. Convergence of hydrodynamic coefficients, using ρ = 0.3 in heorem, (a) X u, (b) Y v, (c) Z w, (d) K, (e) M q, (f) N r. (c) V. CONCLUSION Robust adative strategy is roosed here for motion control of a wide of class of underwater vehicles. Deending on the characteristics of the system uncertainties and disturbances, two control algorithms are develoed to ensure the robust stability and erformance. Using the Lyaunov stability theorem, the closed loo stability is guaranteed without any re-assumtion on the bound of erturbations. Various simulation results are also resented to verify the effectiveness of the methods. VI. APPENDIX One of the results of Barbalat's lemma, used in the stability roof in this aer, is as [9, ] Lemma: If e, e L and e L, then e 0 as t. (d) VII. ACKNOWLEDGEMEN his work was suorted by Research Affairs, University of Isfahan, under grant No

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