Twenty-Fourth Symposium on Naval Hydrodynamics (2003)

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24~ Symposium on Naval Hydrodynamics Fukuoka, JAPAN, 8-13 July 2002 Using Recursive Neural Networks for Blind Predictions of Submarine Maneuvers D. Hess (Naval Surface Warfare Center, USA) and W. Faller (Anolied Simulation Technologies, USA) - x--rr ABSTRACT The use of a Recursive Neural Network (RNN) maneuvering simulation tool for the prediction of blind submarine maneuvers is described. Inputs to the simulation are the controls acting on the vehicle such as propeller rotation speed, rudder and sternplane deflection time histories and the initial conditions. The outputs are time histories of three linear and three angular velocity components. These output data can be integrated to recover trajectory and attitude, and differentiated to determine the accelerations acting on the vehicle. This simulation effort is in response to the ONR Maneuvering Simulation Challenge in which several participants from Government and private organizations provided predictions of the maneuvering behavior of a Radio-Controlled Model (RCM) submarine. The RCM conducted a variety of maneuvers in a specially equipped basin at the Naval Surface Warfare Center, Carderock Division, in Bethesda, MD, USA (NSWCCD). A large amount of the experimental data was provided to each participant, then each was asked to provide predictions for a separate series of 34 blind maneuvers. A blind maneuver is one for which only the initial conditions and the controls directing the vehicle are provided, and each participant was required to provide predictions of the trajectory, attitude, velocities and accelerations for the ensuing maneuver. An independent arbitrator graded each of the blind maneuvers and scores of Excellent, Good, Fair and Poor were issued. The RNN simulation effort received 32 Good grades and 2 grades of Fair. This was the highest total score obtained by any participant for the blind predictions. This paper describes the RNN simulation effort and a sampling of the results. INTRODUCTION The U.S. Office of Naval Research (ONR) sponsored a series of experiments in March & May 2000 using a free-running, radio-controlled submarine model (ONR Body 11. The purpose of the experimental program was to support the continuing development and validation of various codes (CFD and otherwise) for submarine maneuvering simulation. The existence of a large experimental database permitted the invitation of several groups from Government research facilities and other selected agencies to participate in a Maneuvering Simulation Challenge. The challenge was an effort to quantify the current state-of-the-art in submarine maneuvering simulation and to categorize the strengths and weaknesses of current codes with the intent to guide future research efforts. Each participant was provided the geometric and physical properties of the model along with a large database of 78 experimental maneuvers. For each maneuver, the data set contained complete time histories of the controls acting on the vehicle: rudder and sternplane positions and propeller rotation speed; and the response of the vehicle: trajectory and attitude, linear and angular velocities and accelerations. Each participant was able to use these 78 data sets in any manner deemed necessary to prepare their method to make predictions for a second set of 34 maneuvers. Specifically, all participants were asked to provide trajectory, attitude, velocity and acceleration predictions for a set of 34 maneuvers for which only the initial conditions and the time histories of control surface positions and propeller rotation speed were made available. Both data sets included constant heading runs, vertical and horizontal overshoots and controlled and fixed plane turns (many with combined deflection of multiple appendages), and thereby demonstrated a wide range of the maneuvering capabilities of the vehicle. The 34 blind maneuver predictions represented the crux of the challenge, and an independent arbitrator, not affiliated with any of the participants, graded the results from each contributor. Several error metrics were applied to each predicted variable for a given maneuver, and the results were summarized into a single grade of Excellent, Good, Fair or Poor for each maneuver. The participants used such techniques as: unsteady Reynolds-Averaged Navier Stokes (RANS) codes, methods employing

2 lifting line theory, modified potential flow codes and neural networks. This paper describes the methods that were used and the predictions that were submitted by the Neural Network Development Laboratory (NNDL) at the Naval Surface Warfare Center (NSWCCD) using Recursive Neural Networks (RNNs). The NNDL submitted predictions for all 78 of the known maneuvers and all 34 blind maneuvers. Only the blind maneuvers were graded, and the results were grades of Good for 32 of the maneuvers and Fair for the other 2 maneuvers. The predictions that received a grade of Fair were both constant heading runs, for which the submarine traveled in a straight path at constant speed and did not maneuver. Thus, from a maneuvering point of view, these are the least interesting. None of the participants obtained the top grade of Excellent for any maneuvers, and the overall performance by neural networks proved to be oustanding relative to that of the other contributors. Because the efforts of each of the participants were not funded, the NNDL devoted a modest amount of time, 3 man-weeks, to develop the 112 maneuver predictions, and no additional time was spent in an effort to optimize the results. The solutions were developed on PC-based platforms requiring approximately 200 CPU hours to train; however. the trained neural network produced the predictions faster than real time requiring a fraction of a second per prediction. The Neural Network Development Laboratory was established at the Naval Surface Warfare Center (NSWC) in 1995 with the directive to apply neural network technology as a predictive tool to problems of interest to the Navy. The subsequent development of an RNN-based simulation tool for submarine maneuvering was documented in (Faller, et al., 1997) and (Faller, et al., 1998a). RNN simulations have been created using data from both model and full-scale submarine maneuvers. In the latter case, incomplete data measured on the full-scale vehicle was augmented by using feedforward neural networks as virtual sensors to intelligently estimate the missing data (Hess, et al., 1999~. The creation of simulations at both scales permitted the exploration of scaling differences between the two vehicles which is described in (Falter, et al., 1998b). These techniques were also extended to the development of simulations of ship maneuvers. An initial formulation of the problem using an RNN model for use with ships is described in (Hess, et al., 1998), and the technique was further developed for accurate predictions of tactical circle and horizontal overshoot maneuvers (Hess, et al., 20~)~. An RNN is a computational technique for developing time-dependent nonlinear equation systems that relate input control variables to output state variables. A recursive network is one that employs feedback; namely, the information stream issuing from the outputs is redirected to form additional inputs to the network. For this application, the RNNs are used to predict the time histories of maneuvering variables of a model submarine executing submerged maneuvers. Of the 78 data files provided for development, 67 were used to train the neural network and the remaining 11 were used for validation. Upon completion of training, data from maneuvers not included in the set of training maneuvers are input into the simulation and predictions of the motion of the vehicle are obtained. The input data consist of the initial conditions of the vehicle and time histories of the control variables: propeller rotation speed and rudder and sternplane deflection angles. As the simulation proceeds, these inputs are combined with past predicted values of the state variables (outputs) to estimate the forces that are acting on the vehicle. The resulting outputs are predictions of the time histories of the state variables: linear and angular velocity components which can then be used to recover the remaining hydrodynamic variables required to describe the motion of the vehicle. A schematic representation of the technique is shown in Fig. 1. Further details of the implementation follow in subsequent sections, beginning first with a description of the submarine model and the maneuvering data used for training and validation. RNN -, . .j~,j...: ~ j~j. Control ~ ~~ £omp~nant ~~ Hi ~> ~~ ~ my_ 1— Signals ~~-f:~:Modt~ be ~ _K" _ , j ~ ~ j ~ ~ ~ ~ _ Fig. 1 RNN submarine simulation. 6 DOF Trajectory State Variables & Angles u(t) v(t) w(t) P(t) q(t) r(t) SUBMARINE MODEL & EXPERIMENTAL DATA ONR funded the construction and operation of a submarine radio-controlled model: ONR Body 1. The vehicle has a length of approximately 6 m with an axisymmetric hull and appendage arrangements as shown in Figs. 2, 3 and 4. The motion of the submarine is controlled by: a single centerline mounted 3-bladed propeller, upper and lower rudders, and port and starboard sternplanes. The interior of the model is flooded with the exception of a center mounted electronics section, and the space that is occupied by the motor and control surface servo-controllers. Additionally, Styrofoam and lead are installed within the hull as necessary to make the RCM neutrally buoyant with no static pitch or roll angles. All of the appendages mounted on the ONR Body 1 RCM are NACA sections, and the sail is rigidly mounted to the hull. The rudders and sternplanes have no splay angle, so that at zero angle of deflection they

3 c=~ < ~~ ~ ~ ~ ::: ~~ ~ iS~ ~ ~: I: ~:~ $ _, ~,~ . ~ ~ ~ :: ~:~ ~ ~ ~ ~ ~ ' :i: :.: .~:'~-~ i ' ' a"''.-:..: ~ ,,§< ~~ ___ suitable platform for use in verifying the performance of computational tools intended to predict submarine maneuvering characteristics. The ONR Body 1 RCM is operated in the Maneuvering and Seakeeping (MASK) Basin at NSWCCD. A sketch of this facility (not to scale) is shown in Fig. 5. Fig. 2 ONR Body 1: View looking aft. Fig. 3 ONR Body 1: Astern view. _ . l 51 1 _~- e_ _N ~~ _ r it_ ~ _ _ ~ ''a' fir_ ~ _ ~ _ ._ _ _ - 1 ===:=== ~:;~ Fig. 4 ONR Body 1: View of propeller assembly. are aligned with the centerline of the axisymmetric hull. The propeller used to drive the ONR Body 1 RCM is a modification of a commercially available 3- bladed right-handed motorboat propeller. While not containing any specific attributes of actual U.S. submarines, the ONR Body 1 RCM was designed to present computational model developers with similar types of maneuvering challenges. Hence, it is a Fig. 5 MASK Basin at NSWCCD. This 110m long by 73m wide basin is 6.1 m deep except for a 110 m by 15.25 m portion which is 10.7 m deep. The basin is equipped with a large number of hydrophores that serve as a tracking system to give the longitudinal and lateral position of the vehicle within 0.5 cm anywhere within the basin. Further details of the basin, hydrophore array and RCM operation may be found in an NSWCCD report (Nigon, 1994~. Submarine motion is described in terms of a body coordinate system attached at a reference point (the center of gravity here) of the vehicle. A diagram describing the relationship of this system to a fixed coordinate system is given in Fig. 6. - ^~; _ Fig. 6 Standard submarine coordinate system. In this diagram, x is the longitudinal axis and is positive towards the bow, y is the transverse axis and is positive to starboard and the vertical axis z is positive downwards. Linear velocities in this coordinate system are indicated by u, v and w, and angular velocities by p, q and r. Accelerations are denoted similarly but with a dot above the letter to indicate a time derivative.

4 Angles of roll, pitch and yaw are given by A, ~ and By, respectively. The total speed of the vehicle is referred to by U. The vessel's controls are a propeller rotation speed, n, a rudder deflection angle, Or, and a sternplane deflection angle, &. These are the data that define the motion of the maneuvering vehicle and the controls that propel and direct the vessel. They are summarized below in Table 1. The shading for each variable is added to indicate data that are directly measured, variables that may be derived from directly measured data and information not measured. For RCM operation, all maneuvering data are available. Position information, x and y, is acquired directly from the tracking system, and depth, z, is determined from an onboard pressure gage. The velocities, u, v and w, and the accelerations, u, v and w are derived by differentiating position information. I inner Angular Trajectory / Angles Velocities Accelerations Speed (Rel to fluid) Controls ~ . 1 ~~ _ ~~—W CEIL Table 1 Required and measured data. Key EN ~[~3 ~ Measured Derived Unknown An inertial navigation system on the vehicle directly measures p, q and r and then automatically infers attitude information from integration and angular accelerations by differentiation. Encoders record propeller rotation speed and appendage deflection angles, and a small paddlewheel mounted on the hull gives the total speed, U. of the vehicle. In addition, all appendages, the sail and the propeller are fitted with six degree-of-freedom dynamometers that record three force and three moment components for each. Time histories for all of the data specified in Table 1 as well as the time variable (data were acquired equally spaced at 25 Hz) were stored in a single data file for each maneuver. The maneuvers included constant heading runs, vertical and horizontal overshoots and turns. Seventy-eight such data files spanning all maneuver types were provided to each participant. A second set of 34 data files, representing blind maneuvers, contained the full time history for each of the control variables and the first data point only for each of the remaining variables described in Table 1. These 34 maneuvers were chosen such that the range of each of the variables lay within the bounds of the parameter space defined by the 78-maneuver database. For these 34 maneuvers, each participant was asked to predict the missing data in the file, and this was an effort requiring interpolation as opposed to extrapolation. The trained RNN simulation was used to make predictions for each of the maneuver types described above. Only a sampling of the results will be shown in a later section. To give the reader a clear picture of the overall performance of the simulation, a validation run and two blind predictions, one graded Good and one graded Good (but close to the Fair/Good boundary) will be shown. Each of these maneuvers are horizontal overshoots in order to permit comparisons of relative quality. Therefore, before continuing to a discussion of the RNN architecture, an explanation of a Horizontal Overshoot maneuver is provided. Horizontal overshoot maneuvers are performed to characterize the handling response and rudder effectiveness of the vehicle, and such quantities as the heading overshoot angle, overshoot time, reach and period are used to establish this behavior. For this maneuver, the vehicle first travels in a straight line at constant speed in order to establish a period of steady initial conditions. This is followed by an order to EXECUTE the maneuver. At EXECUTE, the rudder is deflected to a predetermined entrance angle and maintained in this position. The heading of the vehicle changes in response to the rudder deflection. When the heading has changed by a desired amount (typically equal to the entrance angle), the rudder is reversed and set to the rudder checking angle (usually equal to the entrance angle). Because the vessel does not respond instantly, the heading continues in the same direction for a period of time before slowing and then reversing. This overshoot angle measures the inherent ability of the submarine to change direction. Figure 7 shows superimposed plots of rudder deflection angle and heading during a typical overshoot maneuver and depicts useful terms. A description of the structure of the neural networks follows next. RNN ARCHITECTURE The architecture of the neural network is illustrated schematically in Fig. 8. The network consists of four layers (groupings of nodes): an input layer, two hidden layers and an output layer. Within each layer are nodes, which contain a nonlinear transfer function that operates on the inputs to the node and produces a smoothly varying output. The binary sigmoid function was used for this work; for input x ranging from -or to or, it produces the output y that varies from 0 to 1 and is defined by y (x) =— l+e x (1)

is : : :. a:! :~a Ad: d. Input Vector 16 ~ ~ ~ ~ ~ 1~. (fFr' ~ Or HE - US Fig. 7 Horizontal overshoot maneuver. Taken from (Stepson & Hundley, 1989) and used with permission. Feed Forward I Hidden Layer(s) Connections I r Output Layer Recursive (Recurrent) Connections Fig. 8 Recursive neural network. Note that the nodes in the input layer simply serve as a means to couple the inputs to the network; no computations are performed within these nodes. The nodes in each layer are fully connected to those in the next layer by weighted links. As data travels along a link to a node in the next layer it is multiplied by the weight associated with that link. The weighted data on all links terminating at a given node is then summed and forms the input to the transfer function within that node. The output of the transfer function then travels along multiple links to all the nodes in the next layer, and so on. So, as shown in Fig. 8, an input vector at a given time step travels from left to right through the network where it is operated on many times before it finally produces an output vector on the output side of the network. Not shown in Fig. 8 is the fact that most nodes have a bias; this is implemented in the form of an extra weighted link to the node. The input to the bias link is the constant 1, which is multiplied by the weight associated with the link and then summed along with the other inputs to the node. For further details concerning the operation of neural networks, the reader is directed to (Haykin, 1994), and for recursive neural networks to (Faller, et al., 1997~. A recursive neural network has feedback; the output vector is used as additional inputs to the network at the next time step. For the first time step, when no outputs are available, these inputs are filled with initial conditions. The time step at each iteration represents a step in dimensionless time, /\t'. Time is rendered dimensionless using the vehicle's length and its speed computed from the preceding iteration; thus, the dimensionless time step represents a fraction of the time required for the flow to travel the length of the hull. The neural network is stepped at a constant rate in dimensionless time through each maneuver. Thus, an input vector at the dimensionless time, t', produces the output vector at t'+ ~ t', where t'+ /`t'= t'+ ~ ~ and /\t'= 0.05 (2) Because submarine speed, U(t'), varies while length, L, is constant, the spacing between samples, ~ t, must vary in order that the dimensionless time step, tat', remain constant at the chosen value of 0.05. The network described here has 80 inputs. Each hidden layer contains 60 nodes, and each of these nodes uses a bias. The output layer consists of 6 nodes, and does not use bias units. The network contains 126 computational nodes and a total of 8880 weights and biases. The input vector, described in detail below, consists of a series of forces and moments that act on the vehicle. The network then predicts at each time step dimensionless forms of the six state variables: three linear velocity components, u, v, and w, and three angular velocity components, p, q and r. Specifically, the outputs are defined as u'(t'+At') = ~ , ), Hand w' similar 'it'+ t'' pat'+ t')L ~ d ' i it U(l ~ (3) These velocity predictions are then used to compute at each time step the remaining kinematic variables described in Table 1: trajectory components, Euler angles and accelerations. The 80 contributions that form the input vector are described as follows. Seven basic force and moment terms describe the influence of the control inputs and of time-dependent flow field effects: thrust

6 from the propeller, Tprop, lift from a deflected rudder, L,, lift from a deflected sternplane, Lstp,, two restoring moments resulting from disturbances in roll and pitch, Kr and Mr. and two Munk moments acting on the hull, MMU,~k and NMUnk. These terms are developed from knowledge of the controls (propeller rotation speed, sternplane and rudder deflection angles), geometry of the vehicle, and from output variables which are recursed and made available to the inputs. A detailed description of these seven inputs is reserved for the next section. Additional inputs are obtained by retaining past values of the seven basic inputs. This gives the network memory of the force and moment history acting on the vehicle and permits the network to learn of any delay that can occur between the application of the force or moment and the response of the vehicle. One past value from the propeller thrust term is retained to provide one additional input. For each of the remaining 6 basic inputs, 10 past values are retained as additional inputs. The number of past values to keep is chosen empirically and appears to be a function of the frequency response of the vehicle. In this case the network is given information about past events for a period of time required for the flow about the vehicle to travel a distance of 0.5L. Recursed outputs from the prior time step are used as six additional contributions to the input vector. Furthermore, the output vector from one previous time step is also retained and made available as six additional inputs. Knowledge of the output velocities for two successive time steps permits the network to implicitly learn about the accelerations of the vehicle. A summary of the various contributions that make up the input vector is provided in Table 2, and attention is next directed to a detailed explanation of the seven basic force and moment inputs. FORCE AND MOMENT INPUTS Neural networks have an amazing ability to identify and track nonlinear behavior linking a set of inputs to a set of outputs. This innate ability can be further augmented, however, by carefully constructing physically motivated input and output variables that form a well-posed problem. The expressions defining these variables need not be complex, but they must define inputs that are well correlated with the outputs. For this task, inputs to the neural network were cast in the form of forces and moments acting on the vehicle: thrust from the propeller, lift from deflected sternplanes and rudders, restoring moments resulting from disturbances in pitch and roll, and pitching and yawing Munk moments acting on the hull. These Input Description Tp,op (t') ~ Tprop (t'—~ t') Laud (t ), Lrud (t ~ t ), . . . , L,Ud (t 1 0A t ) Lstp, (t') ~ Lstpl (t' - ~ t'), . . . , Ls,p, (t' -10/` t') K,(t'), K,(t'- At'), . . . , K,(t'- l OAt') M,(t'), M, (t'- 1\ t'), . . . , M,(t'- 10/\ t') MMUnk (t ), MMU,~ (t /\ t ) ,. . ., Mount (t —1 0A t3 NMU,,.. (t ), NMunk (t /\ t ), . . ., Noun,,. (t —1 0A t ) u'(t'), v'(t'), w'(t'), p'(t'), q'(t'), r'(t') u'(t'- lit t'), v'(t'- /\ t'), w'(t'- ~ t'), p (t'- /\ t'), q'(t'- ~ t'), r'(t'- ~ t') Table 2 Summary of network inputs. To T: 11 _ 6 6 80 terms were fashioned from the basic control variables: propeller rotation speed, n; sternplane and rudder deflection angles, As and Or. Also available for the definition of the input terms are output variables from the previous time step, which are recursed and made accessible to the input side of the network. In this manner a true simulation is preserved as only the control histories and initial conditions of the vehicle are required to run the simulation. The following paragraphs describe the creation of each of the input terms. The first input to the network is a quantity proportional to thrust from the propeller. A dimensional analysis (Lewis, 1988) relates propeller thrust coefficient, CT, to Froude number, Fr, advance ratio, J. cavitation number, a, and Reynolds number, Re, as follows: CT = f (Fr, J. a, Re) , (4) where CT = T/0.5PD2UA, Fr = gD/UA, J = UA/nD, ~ = (P—PV )/0.5PUA, Re = UA D/V, UA is the advance speed of the propeller, D is the propeller diameter and Pv is the vapor pressure of the fluid. An approximation to this relationship may be found by considering only the primary dependence of the thrust coefficient upon the advance ratio. Accordingly, the first input to the network is the reciprocal of the advance ratio with U used as an approximation to UA .

7 1 nD = J U The propeller rotation speed from the current time step and ship speed from the previous time step are used in Eq. 5 to generate the input term. An approximation to the lift force generated by the rudder forms the next input to the network. Expressing the lift coefficient as Cal =L,Ud/O.SpUA2 Ar,,d, the neural network is instructed that lift varies with attack angle according to Cat ~ sin Ctrud ~ where A,Ud is rudder area and Crud iS the local angle of attack at the rudder. The latter quantity is calculated from ax = fir - tang ( v - r L; ) ~7' When a transverse velocity component, v, or a yaw angular velocity, r, is present, the simple result that angle of attack equals rudder angle must be corrected. The quantity L, is the axial distance from the center of gravity to the rudder pivot. The rudder lift input is then L,Ud ~ C,ud U sin (/rud ~ where C,ud was set to one for this work. An input proportional to sternplane lift is formed in the same fashion. The local angle of attack at the sternplane is determined from and the sternplane lift input becomes Ls~p' ~ Cs~p' U 2 sin ors,p' , ( 10) where Cs,p' was also set to one. Summarizing, at a given time step, the local angles of attack are computed from Eqs. 7&9 requiring the control signals: rudder angle and sternplane angle, and recursed outputs: u,v,w,q end r . Rudder and sternplane inputs are then computed from Eqs. 8&10, using U formed from recursed outputs u, v and w . In addition to the propulsion and steering inputs two righting moment inputs are provided to account for recovery from disturbances in roll and pitch. A study of metacentric stability reveals that the righting arm in each case is proportional to the distance from the center of gravity to a point known as the transverse or longitudinal metacenter. For a submerged submarine both of these quantities coincide (5) with the center of buoyancy and the moment arm is the distance BG . The product of the moment arm and the weight of the vehicle creates a couple which acts to restore the vehicle to its undisturbed orientation. These moments may be approximated by K, = -p g VBGsin~ _ , M, =-p g VBGsin~ where V is volumetric displacement and ~ and ~ are angles of roll and pitch, respectively, and the subscript (6) r is meant to denote restoring. The righting moment inputs that were used are simpler versions of Eq. 11, namely, K, =BGsin~ and M, =BGsin~ where the neural network is allowed to determine the leading constants. To implement these restoring moments, roll and pitch angles at the current time step were obtained by advancing previous roll and pitch angles using . (At'L) ~ =~i-t +~L U ) (8) (At'~) (13) The derivatives ~ and ~ are obtained by using recursed angular velocities p, q and r and previously computed roll and pitch angles from the equations , w+q4 -p+ an ,,(rcos ,+qsin `,) of,, = Us + tan ( ) , (9) o - q C°s ~i_~ - , sin of-. The final basic inputs to the neural network are moments which act on the hull of the vehicle (Munk, 1920~. Consideration of a body moving through a perfect fluid and creating a potential flow reveals that fluid pressure acting on the hull will produce a net moment on the vehicle. A real fluid with viscosity and deviation from a potential flow introduces modifications but does not substantially change the resulting moment. The magnitude and direction of the moment depend on the square of the velocity of the vehicle and on the position of the vessel relative to the direction of motion. The expressions used to approximate these moments are MMunk = U sin (x cos NMunk = U sin ,l] cos ,3 where ax and ,l] are vehicle attack and drift angles which are computed from

8 = tank (—) and ,6 = sine (—) (16) The implementation of these expressions as inputs to the network requires ship speed and recursed values of uandv. The seven basic inputs to the network have now been defined. They consist of the thrust from the propeller, lift forces from deflected rudders and sternplanes, restoring moments to disturbances in roll and pitch and Munk moments acting on the submarine hull. With the description of the architecture of the neural network now complete, attention is directed to the procedure by which the network was trained. TRAINING PROCEDURE As discussed in an earlier section, information presented to the inputs of a neural network is modified as it flows through the network by the presence of the weights and by the nonlinear outputs of each of the various nodes until it arrives at the output layer of the network. Thus, at each time step, an input vector produces a predicted output vector; this is then compared to the actual (target) output vector determined from the data. The difference between the target and predicted output vectors is a measure of the error of the prediction. The process by which the network is iteratively presented with an input vector in order to produce outputs that are then compared with a desired output vector is known as training. The purpose of training is to gradually modify the weights between the nodes in order to reduce the error on subsequent iterations. In other words, the neural network learns how to reproduce the correct answers. When the error has been minimized, training is halted, and the resultant collection of weights that have been established among the many connections in the network represent the knowledge stored in the trained neural net. Therefore, a training algorithm is required to determine the errors between the predicted outputs and the desired target values and to act on this information to modify the weights until the error is reduced to a minimum. The most commonly used training algorithm, and the one employed here, is called backpropagation which is a gradient descent algorithm. The collection of input and corresponding target output vectors comprise a training set, and these data are required to prepare the network for further use. Data files containing time histories of all of the variables described in Table 1 for such maneuvers as constant heading runs, vertical and horizontal overshoots and turns formed the training sets. After the neural network has been successfully trained, the weights are no longer modified and remain fixed. At this reprint the network may be presented with an input vector similar to the input vectors in the training set (that is, drawn from the same parameter space), and it will then produce a predicted output vector. This ability to generalize, that is, to produce reasonable outputs for inputs not encountered in training is what allows neural networks to be used as simulation tools. To test the ability of the network to generalize, a subset of the available data files must be set aside and not used for training. These validation data files then demonstrate the predictive capabilities of the network. A recursive neural network was trained in this manner to predict the various maneuvers. The 78 data files made available to each participant were subdivided into a set of 67 training files and 11 validation data files. All maneuver types were represented in each set. The network was initially trained for 100,000 epochs, where an epoch is defined as the presentation of the time series for all inputs and outputs for all files in the training set. During this training process, training is paused every 10 epochs, and the network is tested for its ability to generalize. To carry this out, all of the files in the training set are combined with the validation files set aside earlier for this purpose, and the entire set is presented to the network. During this generalization phase, the weights are not modified; the data from the files simply go through the network to produce predicted outputs and these are then compared with the measured outputs. Use of the word output in this context implies any of the kinematic variables described in Table 1 which are computed at every time step. The errors are quantified by computing three error measures at each time step for each output. These errors are averaged over all of the outputs and then further averaged over all of the time steps in the file to produce measures of the generalization error for each file in this testing set. The errors are then further averaged over all of the files in the testing set (training files and validation files) to produce a measure of the generalization error over the entire set at this stage of training. The error measures are the absolute error and the dimensionless quantities: average angle measure and correlation coefficient. The equation for absolute error is Abserr Nfile Npts(Ifile) Nvars . -r -- x-.~ _ ~ NJ;le Npts ( IJile ) Nvars ~ ~ ~ | mj(n)-pj(n) | Ifile=l n=l j=l (17) where mj (n) and pj (n) are the measured and predicted values for the jth output at time step n.

9 The average angle measure was developed by the Maneuvering Certification Action Team at NSWCCD and is similar to a correlation coefficient in that a value of one represents a perfect prediction and a zero value denotes a poor prediction. More will be said about this quantity in the next section. The corresponding equations for average angle measure and correlation coefficient are given by ~ Noble Nears Nile . Nvars ,~ ~ AAM j, where Npts ~ If lie ~ ~ Dj(n) | cajun) | AAM j = 1-— Liz . . . Npts ( IJile ) 9 ~ Dj(n) ~ = 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 o o ~j(n) = cos~'LI i,, (a, 1~, | 085 S 0.8 3 o.7s 0.7 _ D j(n) = in) + p, (n) and R . = - 1 Nfile Nears R= ~ ~ R. where Nile · Nvars ~le=l j=} £ (m (n) - m ) ( p (n) - p ) n=l 1 Npts ( Iffily ) 2 _ Npts ( [file ) ~ _ ~ (mi(n) - mj) ~ (pj(n) - pj) n=1 n=1 1 Npts ( Idle ) m. = ~ m (n) ' Npts(Ifile) n=1 1 Npts ( 1,file ) Npts(Ifile) ~~ (19) After training has concluded, one examines the error measures as a function of the number of epochs that have elapsed. The curves reveal an optimum number of epochs at which training should have ceased and where minimum absolute errors and maximums in the dimensionless measures occur. Periodically during training, every 100 epochs, the weights are written to a data file. Neural network training is then restarted, at the closest epoch for which a weight set was saved, and continued to the desired optimum stopping epoch. An example of the evolution of the generalization errors over the 100,000 epochs of the training phase for the network simulation may be found in Fig. 9. Of the 18 possible output variables described in Table 1 for which generalization errors may be computed, 3 were chosen as critical variables: u, z, and '; .S ~ _ · . ~ =. ~ S , · .. _ s! 20000 40000 60000 Epochs AAM vs Epochs 80000 100000 ., it_ ·~ 0 20000 0.95 - o.s 1 0.85 - 0.8 - 0.75 2 0.7 o 40000 60000 80000 100000 E Poe h 8 R vs Epochs in_ -~ it_ 4. _~ ~ ~ ~ ~ · .* 20000 40000 60000 Epochs 80000 100000 Fig. 9 Evolution of generalization errors. B. The generalization performance of the network was judged on the basis of these critical variables for purposes of choosing the optimum stopping epoch. The stopping epoch for this network was selected as epoch 97,390 (vertical lines in Fig. 9~. The network was restarted and then halted at this epoch. The absolute error was 0.161, the average angle measure was 0.964 and the correlation coefficient was 0.945. At this epoch the average angle measure reached its maximum, and the absolute error and the correlation coefficient were very close to their minimum and maximum, respectively. Figure 9 shows that after the first 15,000 epochs most of the points on each plot are clustered in a relatively thin band. This indicates that the solution is relatively smooth with small changes to the 8880 weights and biases at each epoch causing only minor fluctuations in errors.

:~e :~ ~ - ~ ~ - ~ l~ Ilk A: - ~ ~ - : A: he A: A: Hi: = Ail. ~ ~ -en her ~ ~~ of/ .r OF' 'I'm or .~ Summarizing, a recursive neural network was trained to predict constant heading runs, vertical and horizontal overshoots and turns using the procedure described in this section for the 78 data files made available to each participant. The trained neural network was then used to make predictions for all 34 blind maneuvers. A sampling of these results are provided next. RESULTS Predictions for the 78 known maneuvers and the 34 blind maneuvers were delivered to the independent arbitrator: the Technical Director of the Hydrodynamic/Hydroacoustic Technology Center (H/HTC), located at NSWCCD. The arbitrator graded only the blind maneuvers obtained from each participant, and used a grading system that relied on the Average Angle Measure. The Average Angle Measure was developed by the Maneuvering Certification Action Team at NSWCCD in 1993-1994 (see July and August 1994 reports). This metric was created in order to quantify (with a single number) the accuracy of a predicted time series when compared with the actual measured time series. The measure had to satisfy certain criteria; it had to be symmetric, linear, bounded, have low sensitivity to noise and agree qualitatively with a visual comparison of the data. The definition is given in Eq. 18 and is described as follows. Given a predicted value, p, and an experimentally measured value, s, one can plot a point in p-s space as shown in Fig. 10. :- - - r P. 1 P~:I=TED =~w VAR1~6 ~ Fig. 10 Definition of the Average Angle Measure. 10 If the prediction is perfect, then the point will fall on a 45° line extended from the origin; the distance from the origin will depend upon the magnitude of s. If p ~ s, the point will fall on one side or the other of the 45° line. If one extends a line from the origin such that it passes through this point, one can consider the angle between this new line and the 45° line, measured from the 45° line. This angle is a measure of the error of the prediction. To extend this error metric to a set of points, one computes the average angle of the set. A problem arises, however. When s is small and p is relatively close to s, one may still obtain a comparatively large angle. On the other hand, when s is large and p is relatively far from s, one may obtain a relatively small angle. To correct this, the averaging process is weighted by the distance of each point from the origin. The statistic is then normalized to give a value between -1 and 1. A value of 1 corresponds to perfect magnitude and phase correlation, -1 implies perfect magnitude correlation but 180° out of phase and zero indicates no magnitude or phase correlation. This metric is not perfect; it gives a questionable response for maneuvers with flat responses, predictions with small constant offsets and small magnitude signals. Nevertheless, it is in most cases an excellent quantitative measure of agreement. For a given maneuver, the arbitrator computed an average angle measure and a correlation coefficient for the following variables: x- x0, y - ye, z - z0, roll, pitch, u - u0, v - v0, w- we, U - U0, p, q, and r. The average angle results for each variable were then averaged to yield a single number for the maneuver. A grade was then assigned to that maneuver according to Table 3. Excellent > 0.9 Good 0.7 to 0.9 Fair 0.5 to 0.7 Poor < 0.5 Table 3 Grades based on the Average Angle Measure. The predictions for the training maneuvers, not graded by the arbitrator, were uniformly Good to Excellent using the criteria given in Table 3. This is the first test that the network must pass. If a relationship between the force and moment inputs and the velocity outputs exists, the network must determine this connection. If the experimental data is poor, or the network is improperly formulated, then the network's performance on the training data will be correspondingly poor. Neither is the case here; the trained network has learned how the submarine performs the various maneuvers. That this is true is demonstrated by the performance of the network on the

11 40 30 - -0.8 -0.6 -0.4 -0.2 0.2 1R _ _ ~ _/ 0.11 1 O - O -0.1 -0.2 _ _ -0.3 - o 0.2 01 t—- - - — o -0. 1 n 2 of 0.04- o -0.04 - _ ~ ~- 0.1 1 - -------------------- -------- —-O. 1 -0.o . ~ \__________________ ' ----- 1 2t __ ~ ________________ ___ ~ ______ 2s P v° ~ ~------ _ _ _ _ _ _~ _ --- k : ,;~ ------ l 0 5 10 15 Time (s) 2- 25 5 10 15 20 25 o ~- . 5 10 15 20 2s --------------1 -2 ~ ~ ~~~ -6 20 25 0 5 10 15 20 25 Tine (s) Fig. 11 Horizontal overshoot: Validation file 0021. Entrance Angle= 10°, Speed= 1.8 m/s Predictions: solid black lines, Measured: dashed red lines.

30 E O 20 1n o r, - . -5 -20- __ -25- 0 2 4 6 8 10 12 14 16 -0.6 -041 - - - - - - - - - - _ _ _ _ _ _ _ _ _ _ _ _ o2 ._ o - ~ _ _ E 4 6 12 14 16 W --------________ ~ _____________ > . I I I '-'- -' T-----— 8 10 12 14 16 .. _ -0.6 o 0.6 - 0.4 0.2 - E o - -0.2 _ -0.4 - _ -0.6 - 0.3 — ____________ ~ ,_____ __ ________. __________ ~ __________. _____________________ ~ ___ 0.2 - . `~ 0.1 - O- ~ 40 20 10 - O 2 -10 ~n 0.2 o O -0 2 -0.4 -0.6 6 8 ____________________f_____ ~ ____ 12 14 16 ____________________________ >,__ 0 2 4 6 8 10 12 14 16 s ~ ~-- -10 0 2 4 6 8 10 12 14 16 s 4 3 2 1 o ____________ ~ ___ __ ~ _________ ~ l 12 14 16 4 6 8 10 nme (8) Fig. 12 Horizontal overshoot: Blind file 4019. Entrance Angle = 15°, Speed = 3.0 m/s Predictions: solid black lines, Measured: dashed red lines. 12 , ~ ~f ~ ? 8 10 12 14

13 3OI 20 - _ _ E o 1C E — o - o~ 0.3 0.21 -0.4 0.4 02 -0.2 -A °4 -0.08 4 - ~ ? 14 1R 1R /k -0.4 0 2 4 6 8 10 12 14 16 18 ~_~~__V\~\ __ , , , , , , , , .. __1 0 2 4 6 8 10 12 14 16 18 nme ($) 20 - ~ D ~ ~ o. ,.~ ~ -10 -20 o 2 4 Fig. 13 Horizontal Overshoot: Blindfile4004. Entrance Angle = 35°, Speed= 1.8 m/s Predictions: solid black lines, Measured: dashed red lines. 8 4 ~ O - -4 o O.. O ~ E ~ -0: -0.4 - n ~ ._ ~ it______________ ________\~_____________ .____________ ~ __ 2 4 6 8 10 12 14 16 18 i----_ _ _ _ _ . - ________ ~ ~ ~ ___ ~ ________ | - ~ - Exp | | RNN| __ _____ ____

14 validation maneuvers. Recall that the validation runs were never used to modify the weights during training, and in this sense, have never before been seen by the network. Fig. 11 gives plots of a selection of variables for a horizontal overshoot validation maneuver. A visual inspection shows that the agreement is very good. In particular, the peaks in p and q are captured quite accurately. This is reinforced quantitatively by listing the average angle and correlation coefficient metrics for each variable. These are found in Table 4 for each of the variables plotted in Figs 11-13. The table also gives the average computed across all of the variables in the table for each metric. For the validation maneuver shown in Fig. 11, the average angle measure averaged over all of the variables is 0.90 which is right on the boundary between Good and Excellent as defined by Table 3. Vars x-xO . Y-YO . z-zo Roll Pitch u-uO v-vO . w-wO . ~~U-Uo . . r Total Honz O/S Validation 0021 AAM 1.00 o.ss 0.90 0.94 0.93 0.84 0.94 0.80 0.83 0.79 0.86 0.94 0.90 CC 1.00 1.00 1.00 1.00 1.00 0.98 0.99 0.94 0.98 0.98 0.98 1.00 0.99 Honz O/S Blind 4019 AAM 1.00 1.00 0.81 0.93 0.82 0.89 0.91 0.78 0.88 0.82 0.83 0.95 0.88 CC .00 1.00 0.97 .00 . 0.97 0.98 0.99 093 0.98 0.97 0.96 1.00 098 Honz O/S Blind 4004 AAM Cc 0.99 0.98 0.56 0.72 0.43 0.85 0.92 0.36 0.84 0.68 0.63 0.94 0.74 1.00 1.00 0.83 0.96 0.80 0.97 0.99 0.76 0.97 0.88 0.89 1.00 0.92 Table 4 Average Angle and Correlation Coefficient Results for Selected Variables for Each Maneuver. Note that the oscillations evident in the measured velocities in Fig. 11 are noise that remains after slipping through the passband of a low-pass filter designed to remove the bulk of the noise. (The noise is only evident when the velocities are at relatively small values.) Rather than try to follow every oscillation, the RNN predictions tend to follow the low frequency trend in the data. However, this reduces the average angle values as can be seen in Table4; the actual agreement is better than suggested by the average angle metric. Referring to the correlation coefficient values for these variables supports this. Figure 12 shows overplots of predicted and measured variables for a blind horizontal overshoot maneuver. This particular blind maneuver received a grade of Good from the independent arbitrator. The only information provided to the trained RNN for this maneuver were the time histories of the controls and the initial conditions. The resulting predictions for each variable are Good to Excellent, and the average for all variables is again close to the Good/Excellent boundary. Thirty-two of the blind maneuver predictions were assigned a grade of Good by the arbitrator and were of similar quality. The blind horizontal overshoot plotted in Fig. 13 was also graded Good by the arbitrator. However, the grade was close to the Fair/Good boundary and gives the reader a lower bound to the quality of the RNN predictions for the runs in which the vehicle actually maneuvered. Visually, the curves do not compare quite as well as in the other two cases, and this is reinforced by the corresponding error metrics as given in the third column of Table 4. However, a closer inspection of the actual magnitudes of the errors is quite revealing. Errors in the trajectory variables are 0.2 m or less and in attitude 2 deg or less. Errors in the linear velocities are everywhere less than 0.1 m/s and less than 2 deg/s for the angular velocities. Finally, Fig. 14 is included to give the reader a guide to the performance of the RNN simulation on the blind predictions as a function of maneuver type. 0 O o Constant Horizontal Vertical O/S Controlbd Fixed-Plane Heading O/S Turns Turns 1 1 1 1 1 1 20 1 ,l~i ~ ~1~ I Fig. 14 Overall Challenge results by maneuver type. The figure plots an average angle measure averaged over all of the runs of a particular maneuver type for each of the participants in the Challenge. The horizontal lines on the plot delineate the Good region as defined by Table 3. Just below the plot is a bar chart showing the number of runs provided by each participant for evaluation in each category. (Only two participants provided all 34 runs for evaluation.) Accordingly, the averages in the upper chart are

15 created from the number of runs listed in the lower chart. The figure shows that the RNN simulation produced Good predictions in all maneuver categories with the exception of the Constant Heading runs where the vehicle did not maneuver. (Two of the three runs in this category were graded Fair.) Figure 14 also shows that the RNN simulation did very well for each maneuver type when compared to the other participants, especially when the number of runs provided in each category are considered. CONCLUSIONS A recursive neural network maneuvering simulation, trained on constant heading runs, vertical and horizontal overshoots and controlled and fixed plane turns, was used to produce predictions for 78 known runs and 34 blind maneuvers of a radio- controlled submarine. An independent arbitrator graded the 34 blind runs and assigned 32 grades of Good and 2 grades of Fair, and this was the highest total score obtained by any participant for the blind predictions. Good predictions were obtained for all maneuver types except Constant Heading runs, and the RNN simulation effort performed very well relative to the other participants in the Challenge. The simulation required 3 man-weeks to develop the 112 maneuver predictions, and no additional time was spent in an effort to optimize the results. The solutions were developed on PC-based platforms requiring approximately 200 CPU hours to train; however, the trained neural network produced the predictions faster than real time requiring a fraction of a second per . . prec action. The most difficult predictions for the simulation were clearly the constant heading runs for which the vehicle traveled in a straight line at constant speed. For these maneuvers, the RNN was required to make predictions for variables that oscillated about values very close to zero (with the exception of forward speed). In hindsight, better performance should be obtained for these maneuvers by training a second network to predict only this maneuver type. This work demonstrates that recursive neural networks exhibit an ability to be a robust and accurate maneuvering simulation tool for submarines. The ONR Maneuvering Challenge has provided a fair and rigorous test of the predictive capabilities of the . . slmu. .atlon. ACKNOWLEDGEMENTS The U.S. Office of Naval Research sponsors this work, and the program monitor is Dr. Patrick Purtell, Code 333. The support of Dr. Promode Bandyopadhyay, Code 342, is also gratefully acknowledged. The assistance of LCDR Craig Merrill, Technical Director, H/HTC, for the overplots of the experimental data in Figs. 11-13 and for providing Fig. 14 is greatly appreciated. REFERENCES Faller, W.E., Smith, W.E., and Huang, T.T. "Applied Dynamic System Modeling: Six Degree-Of-Freedom Simulation Of Forced Unsteady Maneuvers Using Recursive Neural Networks", 35th AIAA Aerospace Sciences Meeting, Paper 97-0336, 1997, pp. 1-46. Faller, W.E., Hess, D.E., Smith, W.E. and Huang, T.T., "Full-Scale Submarine Maneuver Simulation," 1 st Symposium on Marine Applications of Computational Fluid Dynamics, U.S. Navy Hydrodynamic / Hydroacoustic Technology Center, McLean, Va., May 1998a. Faller, W.E., Hess, D.E., Smith, W.E., and Huang, T.T. "Applications of Recursive Neural Network Technologies to Hydrodynamics", Proceedings of the Twenty-Second Symposium on Naval Hydrodynamics Washington, D.C., Vol. 3, August 1998b, pp. 1-15. Haykin, S. Neural Networks: A Comprehensive Foundation, Macmillan, New York, 1994. Hess, D.E., Faller, W.E., Smith, W.E., and Huang, T.T., "Simulation of Ship Tactical Circle Maneuvers Using Recursive Neural Networks," Proceedings of the Workshop on Artificial Intelligence and Optimization for Marine Applications Hamburg Germany , , , September 1998, pp. 19-22. Hess, D.E., Faller, W.E., Smith, W.E., and Huang, T.T. "Neural Networks as Virtual Sensors", 37th AIAA Aerospace Sciences Meeting, Paper 99-0259, 1999, pp. 1-10. Hess, D.E. and Faller, W.E. "Simulation of Ship Maneuvers Using Recursive Neural Networks," Proceedings of the Twenty-Third Symposium on Naval Hydrodynamics, Vat de Reuil, France, September 2000. Lewis, E.V., ea., Principles of Naval Architecture, Second Revision, Vol. 2, The Society of Naval Architects and Marine Engineers, Jersey City, 1988, pp. 127-153. Maneuvering Certification Action Team, "Evaluation of the Average Angle Measure for the SSN 688 Certification," July 1994, Copy resides at the H/HTC, NSWCCD. Maneuvering Certification Action Team, "Maneuvering Code Certification: George/TRJV Evaluation of the Average Angle Measure for the SSN

16 751 Certification," August 1994, Copy resides at the H/HTC, NSWCCD. Munk, M.M. "The Aerodynamic Forces on Airship Hulls," NACA TR-184, 1920, Reproduced in: Jones, R.T. "Classical Aerodynamic Theory," NASA Reference Publication 1050, Dec. 1979, pp. 111-126. Nigon, R.T. "Radio Controlled Model Fluid Research Capabilities," CRDKNSWC-HD-0386-137, Jan. 1994. Sten son, R.J . and Hundley, L. L. "Performance and Special Trials on U.S. NAVY Surface Ships," David Taylor Research Center Ship Hydromechanics Department Research and Development Report DTRC/SHD-1320-01, April 1989, pp. 60, 66.

DISCUSSION C. Merrill Naval Sea Systems Command, USA You point out that due to funding and time con- straints, "no additional time was spent to optimize" the ONR Body 1 RNN simulation. However, assuming that a reasonable optimization effort was possible, what additional training techniques might you have employed to improve your blind predictions? Since one will typically not have a set of blind data with which to clearly evaluate the simulation's predictions, it is necessary that we find methods for estimating the uncertainty of the predictions. Are there such techniques for RNNs? For example, could you have used the eleven validation sets as a means of predicting the uncertainty? Generally speaking, factors other than pure hydrodynamic considerations drive the hull shape and appendage location and configuration for submarines. Hence, maneuvering and control designers are given a limited design space in which they may alter the ship's exterior configuration. Therefore, what is needed is a design tool that can rapidly evaluate a number of design "deviations" from an initial baseline, in order to optimize within the available design space. It would appear that if a more complete form of the force and moment equations were built-in to an RNN (so that it could evaluate changes in geometry), it might provide an ability to perform just such a design evaluation with less reliance on experimental modeling. Do you The currently used force and moment inputs to the agree? If not, why, and if so, how could this be done neural network help to form a well-posed problem. and what level of effort would it take? Therefore, these equations need not be exceedingly accurate. Adding geometry to these equations would require that training data with the proposed geometry changes be available, else one is relying on the predictive capability of the equations vis a vis the network itself. A new neural network design tool is currently under development. This tool uses a grid generation routine to accurately define geometry and a RAN S code to predict forces and moments for a matrix of steady flow configurations. The force and moment database is then used as training data for a recursive neural network equation-of-motion solver. Design changes to the vehicle will require a new series of RAN S runs to compute a new force and moment database. However. the Previously trained AUTHORS' REPLY An attempt to further improve the neural network solution would proceed in several ways. The choice of the number of processing units in the two hidden layers is guided not by theory, but by prior experience. Similarly, the choice of the number of past values to retain for each input (see Table 2) can be varied. To explore these changes, one should perform a matrix of simulations to examine the sensitivity of the solution to such changes. The dimensionless time step, fit', could also be adjusted. The quality of the solutions would suggest that no ~ ~ first-order input terms were omitted; however one RNN that maps applied forces and moments to vehicle motions would not need to be retrained. It prediction of all maneuver types. This need not be the case. A series of neural networks could be trained such that each was used to predict just one maneuver type. The trained networks could then be combined with appropriate switching logic into one simulation. Unfortunately, these avenues could not be explored with an unfunded task. Validation data sets can indeed be used as the blind data sets were used to give an indication of the quality of the simulation as defined by the error measures. A quantitative measure of uncertainty might be obtained somewhat differently. What one would like to know is this: If the uncertainty in each of the inputs is specified, how does that uncertainty propagate through the trained neural network to appear as uncertainty in each of the outputs? Propagation of specified uncertainty is treated quite extensively in Coleman and Steele (1999~. One would have to determine a matrix of partial derivatives that define the sensitivities of each output to small changes in each input. For a trained network, the weights are fixed and known, and the relationship of each output to each input is specified. Thus, the partial derivative matrix can be determined analytically but is somewhat tedious. Alternatively, one may use a jitter routine which applies small changes to each input in turn and observes changes in each output. Either way the propagation of uncertainty through the network can be determined. The bias uncertainty in the network can only be determined through quality measures as defined by appropriate error measures and as used for the validation and blind data sets. could consider additional input terms that might further refine the solution. Finally, the simulation was designed to employ one neural network for the would determine the resulting vehicle motion using the updated force and moment database. The speed

of this design tool would then be limited by the grid generation time and the required number of RANS calculations as well as the available computational hardware. Coleman, Hugh W. and W. Glenn Steele, Experimentation and Uncertainty Analysis for Engineers, Second edition, John Wiley and Sons, New York (1999~. DISCUSSION T. Sedler Northrop Grumman Newport News, USA I would like to start off by congratulating Drs. Hess and Falter on what is a major advancement in our ability to perform maneuvering analysis. I will divide my discussion into three parts in order to impart to the readers and authors why I believe this work is so important. I will describe why accurate submarine maneuvering predictions are important, why I believe this maneuvering simulation method is superior to other approaches, and finally give my recommendation. The United States Navy has learned from painful experience to be prepared for submarine maneuvering system component failures. This is particularly important during lead ship trials for a new class. Maneuvering analysis is used to develop operational envelopes that are intended to ensure that under any operating condition the submarine can recover prior to exceeding its collapse depth. These operating envelopes depend upon estimates of the submarine's response to casualty recovery efforts. Accurate estimates of the submarine's normal and casually maneuvering responses are critical for its operation. Our current submarine maneuvering analysis practice is to use a combination of semi-empirical simulation codes and radio-controlled model (RCM) experiments. Lead ships have no precedence and hence no historical data and judgment is used in the semi-empirical analysis and in scaling the RCM results. The U.S. Navy has pursued the development of improved computational methods in an effort to obtain) better full-scale simulations. Better simulations will yield safer submarine operation. The Office of Naval Research (ONR) has pursued two approaches for solving the problem. The first has been the development of Reynolds-Averaged Navier-Stokes (RANS) codes. The second approach has been to pursue recursive neural network (RNN) nonlinear modeling. It has been this discussor's experience that viscous effects account for 90 percent of the unpredicted submarine maneuvering behavior observed. Adverse viscous effects include upstream generated vortices landing on control surfaces, interactions between appendages and body shed vortices, separation on control surfaces and body, and appendage/turbulent boundary layer interaction. These effects are highly nonlinear, involve large volumes, and their nonlinearity varies as a function of speed, geometry, and maneuver type. Any natural process (differential equation) modeled on a computer must use a numerical approximation scheme for a solution. In general, nonlinear effects are best modeled using nonlinear numerical approximation techniques, such as that presented here. Numerical approximation of highly nonlinear differential equations is not "physics-based." RANS codes, which typically use linear shape functions, try to capture nonlinear behavior by finely descritizing the boundary layer. The more nonlinear the behavior the more grid cells required, and hence more storate space and CPU time. The problems with RANS- based codes are that (1) the models must be huge to capture the nonlinear behavior and (2) with a change in a problem's parameters the "ridding and turbulence model that worked in the previous case may no longer capture the new nonlinear behavior. These problems prevent RANS maneuvering codes from being an effective tool for the ship designer. Use of RNN code ensures that the submarinie's nonlinear behavior is fully captured within the parameter and training set space. RANS and semi- empiricl methods cannot do this. Capturing nonlinear behavior for a nontraditional submarine design is critical. New submarine geometry can be modeled using an RCM and by developing RNN training sets from the measured data. Drs. Hess and Falter have developed a method to scale this data appropriately thus enabling accurate full-scale estimates. New geometry is outside the purview of semi-empirical methods. In examining the data presented in the paper, I noted that high-rudder angles incurred more error. I believe this is due to limitations on the input instruction that C, ~ sin (a), which does not hold at high angles of attack. Would the authors please comment on this? The discussion on the constant heading runs is intriguing. It would appear that the RNN needs additional initial conditions for angular accelerations

and/or the rate of change in angular acceleration at the initiation of the maneuver. The paper is missing a discussion on how experimental measurement error banding impacts the training of the RNN and the resulting accuracy of the simulation. Do the authors think including error banding would have improved their scores? I would like to see from the authors a paper in the future that covers how RNN maneuvering codes could be incorporated into preliminary design. I would also like to hear their opinions on the possibilities of coupling RNN and RANS codes. For example, would it be possible to use a RNN code for modeling the turbulent boundary layer within a RAN S code? I would like to again thank the authors for an excellent paper and project. AUTHORS' REPLY Regarding the question on the use of simple force and moment expressions, and in particular appendage lift, I would reiterate that these expressions are designed to help the neural network understand how to map hydrodynamic forces and moments to vehicle motions. We are relying on the inherent predictive capabilities of the neural network, and not on the completeness of the expressions (used to form a well- posed problem). The quality of the results would indicate that this is a useful way to proceed. That said, efforts to refine the expressions may indeed produce some improvement. The free-running sub- marine used for this work was equipped with six- degree-of-freedom dynamometers on each appen- dage, the sail and the propeller. Thus, it is possible to acquire precise Cat vex information on each appendage. A fit to this data could then be used in place of the simple analytical expressions and may indeed result in better performance at high rudder or sternplane angles. Constant heading data were runs for which nearly all variables were held at a constant value or were zero. The overwhelming majority of training data, on the other hand, was maneuvering runs in which all variables were changing and usually by large amounts. Thus, a more successful approach for the prediction of constant heading runs might have been to train a second network to predict these runs only, and to employ appropriate switching logic within the simulation. The accuracy of the neural network can be no better than that of the data used to train it. The discusser is referring to this with his comments on error banding. If specific inputs have substantial amounts of error (uncertainty) associated with them, this will have an adverse effect on the simulation. The experiment was a high quality test designed and executed precisely to minimize uncertainty. Secondly, the experimental uncertainty that remained likely affected all participants more or less to the same extent. A discussion of the manner in which a quantitative measure of uncertainty propagation through the network may be obtained has been provided in the Reply to Craig Merrill's discussion. The authors have work under way in which a recursive neural network can be used not only as a predictive tool but also as a design tool. This next generation code forms a partnership with grid generation tools and RANS codes to effectively incorporate geometry information and to construct a force and moment database to be used as training data for the neural network. Some additional comments about this approach may be found in the Reply to Craig Merrill's discussion.

Propagation of Uncertainty through a Neural Network David E. Hess - 2 Oct 2002 This note will describe a technique that may be used to determine the uncertainties propagated into the outputs of a neural network if the uncertainties in the inputs are known. In general, if a result, r, is a function of N variables xi r = r(X~,x2, ·~. AN) then, the uncertainty in the result, Ur, is a function of the uncertainties in each of the xi, denoted by Ux;, and is given by u ~ ar u J + ~ ar u ~ + + Ha UXN`) . (2) This equation describing the propagation of uncertainty into a result may be found in (Coleman and Steele, 1999~. To apply Eq. 2 for the determination of neural network output uncertainties, one must derive the matrix of partial derivatives from the equations relating the outputs to the inputs. Before proceeding further, a critical assumption must be made; namely, assume that the weights and biases in the trained network are constants with zero uncertainty. Clearly, if experimental data with associated uncertainty are used, then the weight set obtained from the training process will be affected. However, the uncertainty in the weights does not grow without bound as training proceeds because the backpropagation algorithm continually corrects the weight set to minimize error. The degree to which the training data may be reproduced by the trained network will depend upon how effectively the backpropagation algorithm can refine the weight set. The remaining error in the trained network will be treated as a fossilized bias error. In other words, no matter how many times the trained network is executed, it will always produce the same bias error for a given input vector. Expressing the output prediction error as a bias is a more straightforward approach than assigning uncertainties to each weight and bias in the network. The applicable feedforward equations for a network with two hidden layers are as follows. The notation used considers i = 1, . . ., Ninput input nodes, j = 1, . . ., Chid l and k = 1, . . ., Chid 2 nodes in the two hidden layers and ~ = I, . . ., Noutput output nodes. Input layer to first hidden layer Ninput Vhinl j = Bhl j + ~Wghtih'; Finput 1 + e-V7nin~j (j= 1,...,Nhidl) (3)

2 First hidden layer to second hidden layer Nhid 1 Vhin2k = Bh2k + ~Wghthhjk Vhoutl j=1 V7'out2k = ~ + nm2, Second hidden layer to output layer Nhid 2 Voutinl = ~ Wghthokl Vhout2k k=1 Voutout = ~ ~ + e-Vou~in~ (k = 1,...,Nhidl2) (l = 1,. . .,Noutput) (4) (5) Substituting Eqs. 3 into Eqs. 4 and then Eqs. 4 into Eqs. 5 will yield a set of equations relating Voutout~ to Vinputi of the form of Eq. 1. The set of partial derivatives that are required are ' and will be derived step by step using repeated application of the OKlnput Chain Rule. Input layer to first hidden layer a Vhoutl . e-Vhin1 j J = = Vhoutl . (1 + -Vhinl j )2 J OVhin Ov~in1 . ~ = Wghtihij OF`nputi OVhout1 . aVinput = Wghtihi; Vhoutlj (l-Vhoutlj) . (l -Vhoutlj) First hidden layer to second hidden layer ;3Vh 2 ~ = Vhout2k ( 1- Vhout2,, ;3 Vhin2~ W hthh · ~ h' ) Wghthh ik V77out2 k ( 1 Vhout2 k ) OVhout1 j h (6) (7)

Second hidden layer to output layer av0UtOUtl = Voutout, (~-Voutoutl) Shouting · e Ovoutin aVhout2k Wghtho~1 · jVh 2'=Wghtho~1Voutoutl(1-Voutoutl) Now, we need to apply the results from Eqs. 6-8 to obtain I aFinputi backward, we first want to determine I OVhoutl that Voutoutl = fI (shouts, Vhout2 2, ... Vhout2 Nhi42 ~ and Vhout2k = gk (VhoutI i, VhoutI 2, . . . Mouth Nail ~ . Working . By referring to Eqs. 5 and 4, we find . (9) Application of the chain rule for the case of multiple independent variables requires a . . sum giving avoutoutl Shiv avoutout, avhout2` OVhout! k=! Mouths bailout! j Chid 2 = ~ [Wghtho~l Voutout~ ( ~ - Voutoutl )] , k=} . [Wghthhjk Vhout2k ( ~—Vh°Ut2k )] (10) where the second line follows from substituting Eqs. ~ and 7. Finally, we can compute avOUtoUtl as follows OFinputi avoutouti Nz} jVoutOUtl aVhOUti j OKinputi j=l OVhoutlj OFinput = ~~[Wghtihij Shout! j ( ~—Shout! j )] j=! ; 7iwshthhjk Vhout2k ( ~—Vhout2k )] 1~ · [Wghtho;~ Voutoutl ( ~ - Voutoutl )]J where the results of Eqs. 10 have been used. Examination of Eqs. ~ ~ reveals that the sensitivity derivatives depend upon the inputs and the weights and biases; namely,

4 = fly (Vinput,, Wghtihjj, Wghthh jk, Wghtho~d, Bhl j, Bh2k ) . ( 12) OUl~putj For a specified input vector and weight set, a computer subroutine implementing the feedforward equations (Eqs. 3-5) can be easily modified to also compute ' by OF`nput implementing Eqs. 11. Furthermore, if the vector of input uncertainties is also specified, one may then compute the uncertainty propagated into any of the outputs. For example, the calculation for the uncertainty propagated into the first output is given by 2 UVoutout' ~tavOUtoUt1U :2+(3VOutO~tlU ~ ~< OFinputl ~ ) ~< OFinput Finput2 ) +...+r aVOutoutl U ~ a VinPUtNinput with similar equations for the other outputs. The analytic expressions for the sensitivity derivatives, Eqs. 11, can be checked for accuracy using a numerical procedure referred to as a jitter program in the literature. For Ni,tpu' ~ , (13) example, to check the quantity 1, one proceeds as follows. Add a small OFinput1 increment to Vinputl, call it I\ vinput1 = l x l O-6 · Finputl, say. Compute Voutoutl(Vinputl +AVinputl) by using Eqs. 3-5 with Vinput1 replaced by Vinput1 +i\Vinput~. Then, compute Voutout~(Finput1 -/\Finput1) by recalculating Eqs. 3-5 with Finput1 replaced by Vinput~-l\Vinput~. A central difference i ti t th t I f OVoutout1 th b bt i df OF'nput1 a Voutout1 Voutout~ (Vinput1 + ~ Vinput~ ) - Voutout1 (Vinput~ - /\ Vinput~ ) OFinput1 2 /\Finput1 (14) By keeping /\Finput1 a very small value, the approximation of Eq. 14 wil1 compare wel1 to the exact value computed by Eqs. 11. Reference Coleman, Hugh W. and W. Glenn Steele, Experimentation and Uncertainty Analysis for Engineers, Second edition, John Wiley and Sons, New York ( 1999).