A six degree of freedom (6-DOF) nonlinear mathematical model for a submarine is
proposed. Then, the problem of manoeuvrability control for this vehicle is formulated
as an unconstrained optimal control problem which is solved by using a gradient
descent method. Simulation results presented are rooted by two requirements which
are of a major importance in naval industry, and they are compared to a classical
linear model.Supported by projects MTM2007-62945 from Ministerio de Ciencia y Tecnología (Spain),
08720/PI/08 from Fundación Séneca , Agencia de Ciencia y Tecnología de la Región de
Murcia (Programa de Generación de Conocimiento Científico de Excelencia, IIPCTRM
2007-10) and 21131/07MAE from Navantia S.A

García Peláez, Javier

Ovalle, Diana

Periago Esparza, Francisco

English

Repositorio Digital de la Universidad Politécnica de Cartagena

XXI Congreso de Ecuaciones Diferenciales y AplicacionesXI Congreso de Matemática AplicadaCiudad Real, 21-25 septiembre 2009(pp. 1–8)Numerical simulation of the manoeuvrability of anunderwater vehicleD. Ovalle1, F. Periago2, J. Garćıa31 Dpto. de Matemática Aplicada y Estad́ıstica, ETSI Industriales, Universidad Politécnica de Cartagena,30202 Cartagena, Spain - E-mail: diana.ovalle@upct.es.2 Dpto. de Matemática Aplicada y Estad́ıstica, ETSI Industriales, Universidad Politécnica de Cartagena,30202 Cartagena, Spain - f.periago@upct.es.2Dirección de Ingenieŕıa, DICA, Navantia S. A., 30205 Cartagena (Spain) - jgpelaez@navantia.es.Palabras clave: Manoeuvrability control, submarine, optimal control, gradient descent method.ResumenA six degree of freedom (6-DOF) nonlinear mathematical model for a submarine isproposed. Then, the problem of manoeuvrability control for this vehicle is formulatedas an unconstrained optimal control problem which is solved by using a gradientdescent method. Simulation results presented are rooted by two requirements whichare of a major importance in naval industry, and they are compared to a classicallinear model.1. IntroductionIn the development of a naval architecture tool for the guidance and autopilot of a sub-marine is important to choose both an accurate mathematical model for the equations ofmotion and a suitable control strategy. In spite of the practical importance of this matter,there are not so much works available in the literature, probably because of its applica-tions in military technology. In fact, most of the results which are used by naval industryare based on simplified (meanly linear) versions of more accurate nonlinear models or onnonlinear models with lower degrees of freedom. Nowadays, the following two challengesdeserve an special attention at the practical level:(i) optimality of sizing rudders (decision to be taken during the design state), and(ii) design of an automatic submarine autopilot to be able to guarantee smooth manoeu-vres as physically admissible in order to reduce hydraulic oil consumption and soreduce noise generation.1D. Ovalle, F. Periago, J. GarćıaHaving this in mind, firstly, a vehicle dynamics model based on a combination oftheory and empirical data is proposed in this work. Indeed, taking as a starting pointthe standard DTNSRDC (David Taylor Naval Ship Research and Development Center)nonlinear equations of motion for an underwater vehicle [5, 7, 10] and adapting thesegeneral equations to the particular characteristics of a prototype developed by the companyNavantia S.A. Cartagena Shipyard (Spain), we propose a mathematical model composedof a highly coupled and nonlinear system of ordinary differential equations (ODEs) withsix degrees of freedom.The second aim of this work is to design numerically an appropriate nonlinear controlsystem which may be used for simulation of the usual depth and course keeping manoeuvres(among others) as well as to provide some useful information to deal with the above twopoints (i)-(ii). To this end, we describe the controller design as an unconstrained optimalcontrol problem and implement a gradient descent method to solve it. This is an iterativealgorithm which requires a good initialization to get a rapid convergence. To achieve a goodinitialization and also for comparison purposes, we derive from the original nonlinear modela simplified (4-DOF) linear mathematical model and solve it by using classical results oncontrollability theory [8].Finally, numerical results are given to show the resulting nonlinear manoeuvres com-paring to the linear ones.2. Vehicle modelingThe three-dimensional equations of motion for a marine vehicle are usually describedby using two coordinate frames: the moving coordinate frame which is fixed to the vehicleand is called the body-fixed reference frame, and the earth-fixed reference frame whichis called the world reference frame. The origin of body coordinates is at the half pointalong the symmetric longitudinal axis. Typically, this point is not so far from the centerof buoyancy (CB) and from the center of gravity (CG). The body axes are longitudinalpointing in the nominal forward direction of the vehicle, lateral pointing through the righthand side of the level vehicle, and downward through the nominal bottom of the vehicle.The world coordinate system is defined by three orthogonal axes originating at an arbitrarylocal point at the ocean surface. North corresponds to x−axis, East corresponds to y−axisand increasing depth corresponds to z−axis. The position and orientation of the vehicleare described in the world system while the linear and angular velocities are expressed inthe body-fixed coordinate system. The kinematic equations which relate the body-fixedreference frame to the world reference system can be found in [7], [9].The dynamic equations of motion can be derived from a Newton-Euler formulationwhich is based on Newton’s second law, analyzing the forces and movements acting ineach of the submarine’s axis one can obtain those equations. The development of thefunctional form of the hydrodynamic forces and moments is by now well-known [5, 7, 10].Specific values of the particular hydrodynamic coefficients which appear in the equationsdepend on the specific vehicle and, therefore, would require modification if applied toother vehicles. The values of the coefficients used in this work have been experimentallyobtained in deeply submerged conditions by using a scaling model. A detailed study ofthe mentioned model as well as several captive manoeuvring tests appear in the internal2Numerical simulation of the manoeuvrability of an underwater vehiclereport [2]. Those equations include the functional form of mechanical and hydrodynamicadded mass, terms like:ρ2l3[X′u̇u̇ + X′vrvr + X′wqwq],coriolis and centripetal forces,terms like:m[u̇− vr + wq − xG(q2 + r2) + yG(pq − ṙ) + zG(pr + q̇)],damping forces and moments due to skin friction and vortex shedding, terms like:ρ2l4[X′qqq2 + X′rrr2 + X′rprp + X′q|q|q|q|] +ρ2l2[X′uuu2 + X′vvv2 + X′www2 + X′w|w|w|w|],restoring (i.e., gravitation and buoyant) forces and moments, terms like:(W −B) sin(θ),propeller forces and moments, terms like:(KT0 + KTJ(1− wf )nDu + KTJ2(1− wf )2(nD)2u2 + KTJ3(1− wf )3(nD)3u3 + KTJ4(1− wf )4(nD)4u4)n2D4,and, rudders control forces, terms like:ρ2l2[X′δrδru2δ2r + X′δsδsu2δ2s + X′δbδbu2δ2b ].We refer to [7, 11] for a detailed description and analysis of these forces and moments,but for a better understanding of the problem we indicate here that the state variable isdenoted byx (t) = [x (t) , y (t) , z (t) , φ (t) , θ (t) , ψ (t) , u (t) , v (t) , w (t) , p (t) , q (t) , r (t)] .The model for a submarine proposed by us is a set of 12 first order ODEs, that is a sixdegree of freedom (6-DOF) model, which can be found in [9]. There the first 6 equations,called kinematic equations, relate the body fixed reference frame to the word referencesystem, where the second member or the ODEs is mainly composed for trigonometricalfunctions of the submarine position variables. The other 6 equations, called dynamicalequations, are related to the dynamical analysis of the movement of the submarine, wherewe can find terms like the previous mentioned.For the specific vehicle considered in this work, control inputs come in the form ofthruster forces and moments and may be expressed as a vectoru (t) = [δb (t) , δs (t) , δr (t)] , (1)where δb is deflection of bow plane, δs is deflection of stern plane, and δr is deflection ofrudder.From a pure mathematical point of view, it is important to point out the followingfacts in the proposed model:The high nonlinear character of the equations and the high order of the system (12equations).3D. Ovalle, F. Periago, J. GarćıaThe lack of differentiability of the system which is caused for several terms involvingnon-differentiable functions such as the absolute value and squared roots.The controls appear in nonlinear (quadratic) form.Because of these three difficulties, it is certainly hard to study the model at the puremathematical level (existence, uniqueness of solutions, etc.). Indeed, very few is known inthe mathematical literature concerning to the control of non-differentiable systems withcontrols appearing in nonlinear form. We refer to [3] for a recent related paper. Since thiswork is mainly addressed to numerical simulations and application to a real-life engineeringproblem, we do not enter here in the above theoretical question which, however, will beanalyzed in a future work.Both kinematic equations and dynamic equations are the equations of motion for thesubmarine. In a compact form they can be written as:·x (t) = f (x (t) ,u (t)) . (2)3. Nonlinear controller designThe control problem can be formulated as follows: given an initial state x (0) = x0 ∈R12 and a desired final target xT , the goal is to calculate the vector of control u (t) which isable to draw our system from the initial state x0 to (or near to) the final one xT in a giventime T . In mathematical terms, this problem may be formulated as the unconstrainedoptimal control problemMinimize in u : J (u) = Φ(x (T ) ,xT)+∫ T0 F (x (t) ,u (t)) dtsubject to·x (t) = f (x (t) ,u (t))x (0) = x0(3)Here Φ(x (T ) ,xT)and F (x (t) ,u (t)) are two generic functions which can be chosen asdesired. At this point, a pair of remarks is in order:It is evident that the original problem includes some constraints on both the con-trol inputs and the state variables. This is very important in order to choose anappropriate numerical control method but, at the practical point of view, all ofthese restrictions can be easily satisfied by taking the final time T large enough.For this reason, in this preliminary work we will not consider the above mentionedconstraints.As for the cost function J (u) , we have written it in a general format because themethod we plan to develop in the remaining can be applied in this general set-ting. Concerning to the particular problem of control of the manoeuvrability for anunderwater vehicle, which is the main goal of this work, it is quite natural to takeΦ(x (T ) ,xT)=12∑j=1αj(xj (T )− xTj)2(4)4Numerical simulation of the manoeuvrability of an underwater vehiclewith αj > 0 penalty parameters, andF (x (t) ,u (t)) =3∑j=1βj (uj (t))2 (5)with βj > 0 another weight parameters. Therefore, the cost function J (u) is acommitment between reaching the final target and a minimal expense of control onhaving done the corresponding manoeuvre.There are several optimization methods which can be applied to solve (3). Due to thecomplexity of the state law and the large number of variables involved in the problem, itis quite reasonable to use a gradient descent method. We refer to [1, 4] for more detailson this method.4. Numerical simulationsIn this section, we show some numerical results obtained by implementing the approachdescribed in the preceding sections in Matlabr for a depth change. At each iteration thestate and adjoint state equations have been solved by using the ODE45 Matlab function,which is a one-step solver based on an explicit Runge-Kutta (4,5) formula (see [13]). Ourgoal is to provide some useful numerical information to address challenges (i)-(ii) describedin the introduction.The depth change we are going to consider is defined through a change between aninitial state and a final state, during a particular T and with the parameters λ and ε asfollows: λ = 0,001 ε = 10−6x (0) = (0, 0, 400, 0, 0, 0, 2,5, 0, 0, 0, 0, 0)xT = (0, 0, 50, 0, 0, 0, 0, 0, 0, 0, 0, 0) , T = 200.(6)We are going to compare four different cases:Case 1. In order to check the algorithm with a typical manoeuvre, we take:{α3 = 1, αj = 0, j 6= 3βj = 0, j = 1, 2, 3(7)That is, we penalize only the final value for the depth state variable. We want thesubmarine goes from a depth of 400m to 50m, regardless the final value in the otherstate variables.Case 2. In order to study optimality of sizing rudders (which is a typical example of apre-contract navy requirement) we take:{α2 = α3 = α5 = 1, αj = 0, j 6= 2, 3, 5βj = 0, j = 1, 2, 3(8)In this case, we also penalize the final value for the state variables y (t) and θ (t). Wewant the submarine goes from a depth of 400m to 50m and, in addition, the finalpitch angle must be equal (or near) to zero in order to continue the movement withthe final value on the control variable without changes in depth.5D. Ovalle, F. Periago, J. GarćıaCase 3. In order to reduce hydraulic oil consumption (and therefore noise generation) wepenalize, in addition, the L2−norm of the control variables. Hence, now we take:{α2 = α3 = α5 = 1, αj = 0, j 6= 2, 3, 5,β1 = 100, β2 = β3 = 0(9)Case 4. The linear case which approaches this problem studied in (see [8]).0 50 100 150 200−8−7−6−5−4−3−2−101t (s)m  linearnonlinear z+nonlinear znonlinear u0 50 100 150 200050100150200250300350400450t (s)m  linearnonlinear z+nonlinear znonlinear uFigure 1. East movement(y(t)) Figure 2. Depth movement(z(t))0 50 100 150 20002468101214161820t (s)º(degrees)  linearnonlinear z+nonlinear znonlinear u0 50 100 150 200−0.12−0.1−0.08−0.06−0.04−0.0200.02t (s)º(degrees)  linearnonlinear z+nonlinear znonlinear uFigure 3. Pitch Angle (θ(t)) Figure 4. Deflection of rudder (δr(t))The results are shown in Figures 1-7. Dashed lines (- - -) display results for thegradient method and for the set of parameters (7),we refer to this case as to nonlinear z.Continuous lines (—) display the results for (8),we refer to this case as to nonlinear z+.Dash/dot lines (-·-·-·) show results for the gradient method and for the set of parametersas in (9) where the values for αj are replaced by α3 = 1 and αj = 0 for j 6= 3, i.e., wekeep the constrains in depth z (t), east movement y(t) and pitch angle θ(t), this last caseis named nonlinear u. Finally, dotted lines (· · · ) correspond to results obtained with thelinear model described in [8]. As shown in Figures 2, 3, 5 and 6 continuous, dash/dot anddashed lines are essentially the same. However, there is some differences between themin Figures 1 and 4. Obviously, the reason for these results is that in nonlinear z+ and innonlinear u we force the submarine to finish near to zero position on the y−axis, but innonlinear u we also constrain the movement of the rudder δr which gives the movement onthe y−axis. Therefore, δr needs to turn more in nonlinear z+ than in nonlinear u, where6Numerical simulation of the manoeuvrability of an underwater vehiclethis movement is restrictive, or in nonlinear z, where the constraint on y−axis is inactive.Finally, Figure 7 shows the evolution of the cost function (with respect to the number ofiterations) for the cases nonlinear z, nonlinear z+ and nonlinear u. An exponential decayis observed in all cases. As we expect, the minimal cost is reached for the less restrictiveproblem nonlinear z in less number of iterations, and the largest cost is reached for themore restrictive problem nonlinear u in a larger number of iterations. We do not includepictures for the rest of state variables because they are not relevant in this manoeuvre.0 50 100 150 200−4−3.5−3−2.5−2−1.5−1−0.500.51t (s)º(degrees)  linearnonlinear z+nonlinear znonlinear u0 50 100 150 200−0.500.511.52t (s)º(degrees)  linearnonlinear z+nonlinear znonlinear uFigure 5. Deflection of stern plane (δs(t)) Figure 6. Deflection of bow plane (δb(t))6 8 10 12 14 16020040060080010001200140016001800  nonlinear z+nonlinear znonlinear uFigure 7. Cost FunctionAs is typical in a gradient algorithm, results depend on the initialization. For instance,when we initiate our algorithm with zero value for u (t) , 0 ≤ t ≤ T, after convergence,we obtain different results: the costs for the case nonlinear z are the same and very closeto zero, but the optimal controls are slightly different; for the case nonlinear z+, thefinal optimal cost is higher than in the case tested before, and the optimal controls arealso different. These results seem to indicate the existence of several local minima and/ornon-uniqueness of solutions.5. ConclusionsThe problem of manoeuvrability control for a submarine is addressed. The vehicledynamics is modelled as a simplified version of the classical DTNSRDC equations of motionthat adapts to the specific type of submarine under consideration. The manoeuvrability7D. Ovalle, F. Periago, J. Garćıacontrol problem is then formulated as a nonlinear unconstrained optimal control problemwhich is numerically solved by using a gradient descent method.Numerical results show that, although the nonlinear model proposed in this work fol-lows, in some sense, the same tendency as the more usual linear model, however, importantchanges in some state and control variables are present. As a conclusion, the nonlinearitieswhich appear in the kinematic and dynamic equations of motion cannot be removed inorder to have more accurate simulation results.In what concerns the optimization method proposed, it allows us to obtain satisfactorysimulation results in a model which presents serious mathematical difficulties.As for the computational cost of the algorithm, for a depth change manoeuvre it takesabout 60 seconds in a PC for the test cases 1, 2 and 3.Finally, we emphasize that at the practical level it is interesting to include the propelleras a control variable and to impose some (a priori) constraints on the control variables.We plan to address these two issues as well as the more theoretical questions concerningexistence and uniqueness of solutions in a future work.AcknowledgementsSupported by projects MTM2007-62945 from Ministerio de Ciencia y Tecnoloǵıa (Spain),08720/PI/08 from Fundación Séneca , Agencia de Ciencia y Tecnoloǵıa de la Región deMurcia (Programa de Generación de Conocimiento Cient́ıfico de Excelencia, IIPCTRM2007-10) and 21131/07MAE from Navantia S.A.Referencias[1] G. Allaire, Analyse numérique et optimisation, Les Éditions de l’École Polytechnique, Paris 2005.[2] L. Byström, SSPA report no 2000 0101-4, for E. N. Bazan de CNM, SA, 2001.[3] T. Chambrion and M. Sigalotti, Tracking control for an ellipsoidal submarine driven by Kirchoof’slaws, IEEE Transactions on Automatic Control, 33 (1) (2008), 339-349.[4] J. C. Culioli, Introduction à l’optimisation, Éditions Ellipses, Paris 1994.[5] J. Felman, Revised standard submarine equations of motion. Report DTNSRDC/SPD-0393-09,David W. Taylor Naval Ship Research and Development Center, Washington D.C., 1979.[6] E. Fernández-Cara and E. Zuazua, Control theory: History, mathematical achievements and per-spectives, Bol. Soc. Esp. Mat. Apl. 26 (2003), 79-140.[7] T. I. Fossen, Guidance and control of ocean vehicles, John Wiley and sons, 1994.[8] J. Garćıa-Peláez, J. A. Murillo, I. A. Nieto, D. Pardo and F. Periago, Numerical simulation of a depthcontroller for an underwater vehicle, MARTECH’07, International Workshop on Marine Technology,Villanova i la Geltrú, 2007.[9] J. Garćıa-Peláez, D. Ovalle and F. Periago, Optimal control design for the nonlinear manoeuvrabilityof a submarine, pre-print available at http://filemon.upct.es/∼fperiago/archivos investigacion/submarine paper.pdf, 2009.[10] M. Gertler and G. R. Hagen, Standard equations of motionfor submarine simulations, NSRDC Rep.2510, 1967.[11] J. N. Newman, Marine hydrodynamics, The MIT Press, Cambridge, Massachusetts, London, 1977.[12] K. Ogata, Modern Control Engineering. Prentice-Hall, fourth edition edition, 2001.[13] L. F. Shampine, M. W. Reichelt, The Matlab ODE Suit. SIAM Journal on Scientific Computing,vol. 18, 1997, pp 1-22.8

Numerical simulation of the manoeuvrability of an underwater vehicle

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Numerical simulation of the manoeuvrability of an underwater vehicle

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