Robust input design for nonlinear dynamic modeling of AUV

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Research article

Robust input design for nonlinear dynamic modeling of AUV Nowrouz Mohammad Nouri n, Mehrdad Valadi Department of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 24 July 2016 Received in revised form 8 November 2016 Accepted 7 February 2017

Input design has a dominant role in developing the dynamic model of autonomous underwater vehicles (AUVs) through system identification. Optimal input design is the process of generating informative inputs that can be used to generate the good quality dynamic model of AUVs. In a problem with optimal input design, the desired input signal depends on the unknown system which is intended to be identified. In this paper, the input design approach which is robust to uncertainties in model parameters is used. The Bayesian robust design strategy is applied to design input signals for dynamic modeling of AUVs. The employed approach can design multiple inputs and apply constraints on an AUV system’s inputs and outputs. Particle swarm optimization (PSO) is employed to solve the constraint robust optimization problem. The presented algorithm is used for designing the input signals for an AUV, and the estimate obtained by robust input design is compared with that of the optimal input design. According to the results, proposed input design can satisfy both robustness of constraints and optimality. & 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Robust input design nonlinear dynamic model Constraint Robust Optimization PSO AUV

1. Introduction Nowadays, AUVs are widely utilized in commercial, research, and military applications because of their economic advantages and maneuvering capability in dangerous environments [1]. Researchers model the dynamic behavior of AUVs to evaluate their performances [2] and also design suitable controllers for AUVs using these models [3–5]. Dynamic modeling of AUVs through system identification has been studied extensively during the past several decades [4,6–8]. Often focus of system identification studies are on developing identification algorithms to accurately estimate the unknown parameters of the model from the measured data. Developing identification algorithms to accurately estimate the model parameters is the last step of system identification and numerous works need to be done beforehand. No algorithm could achieve accurate modeling results If the measured input and output data are poor from identification point of view [9]. So, input signals that provide measured data have a major impact on the quality of data for modeling purposes. In the literature available on AUVs, the input signals of vehicles are not often designed for dynamic modeling, but for evaluating the performance of such vehicles [10]. The turning circle, spiral and zigzag maneuvers are usually designed and used to evaluate the performance of AUVs [11,12]. Kim et al. [13] used spiral and zigzag n

Corresponding author. E-mail addresses: [email protected] (N.M. Nouri), [email protected] (M. Valadi).

maneuvers to model linear AUV system. In Refs. [4,14] zigzag maneuver is used to estimate the model parameters of a ZS-AUV and a Pirajuba AUV respectively. Rentschler et al. [8] used different stepped inputs for dynamic modeling of Caribou AUV. None of these research works used the excitation input signals designed for system identification. The purpose of designing an inputs for AUV system identification is to excite an AUV system in a way that achieves the most information on its dynamic behavior through every measurement and leads to minimum uncertainty in estimating the model parameters. So designing an excitation inputs for system identification can be formulate as an optimal input problem. The excitation signals designed for estimating the parameters of linear systems should have a wide frequency bandwidth in order to excite all the dynamic modes of a system. Among the common excitation signals, the pseudo random binary signal (PRBS) is frequency-rich and could be used in practical applications. This kind of input excitation is also used in modeling of marine vehicles [6,15,16]. Considering the nonlinearity of the mathematical model that governs the motion of AUVs, the PBRS excitation is not suitable for estimating the parameters of this model; because it does not cover the nonlinear behavior of a system over the entire amplitude range [17]. The input signals for nonlinear systems should cover all the amplitude and frequency ranges of such systems; and these conditions could be satisfied through the excitation of the amplitude-modulated pseudo random binary signal (APRBS) [17,18]. Therefore, APRBS shape signal can be used to design optimal input signal for nonlinear dynamic modeling purpose.

http://dx.doi.org/10.1016/j.isatra.2017.02.006 0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Nouri NM, Valadi M. Robust input design for nonlinear dynamic modeling of AUV. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.006i

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The design of the optimal inputs signal is performed by optimizing a cost function related to the quality of dynamic model. Most objective metrics use some norm of the information matrix as the basis for optimal experiment design [19–23]. The information matrix depends on the true parameter values. Consequently, complicating issue is that an optimal input design will depend on the quantities that their true values are not known at the time of design [24–27]. Moreover, dynamic models of AUVs are nonlinear and complex, so they are often subject to uncertainties that make the modeling task difficult. These uncertainties can be structured when the equations are known, but not the value of their parameters, which is the result of incomplete or imprecise knowledge about model parameters [28,29]. In this case, it is particularly difficult to design suitable inputs for accurate dynamic modeling. One approach to get around these difficulties is to make the input design robust or insensitive to the true parameter values [30]. Bayesian and minimax approach are two approach that have been used in robust input design. Both approach must transform parameters uncertainties to uncertainties in information matrix. Bayesian approach uses expected operator to incorporate parameters uncertainties and requires the pre-specified prior probability distributions of the uncertain model parameters. The minimax approach optimizes the worst possible performance for all possible values of model parameters. Therefore, minimax approach does not require prior information about the parameters probability distribution, but are computationally very heavy. Detailed comparison of these approach can be found in Refs. [27,31]. To the best knowledge of the present authors, the robust input design has not been used for the dynamic modeling of AUVs thus far. The robust input design problem that satisfies constraints of AUV motion is a constraint optimization problem. The gradientbased and meta-heuristic methods are two general methods that can be used for solving such optimization problem. In general, gradient-based methods can obtain higher quality solutions and converge faster as compared to meta-heuristic approaches [32– 34]. But the constraint optimization problem comes from input design is nonlinear and non-convex [35,36] and has a large number of design variables. Also this problem is robust which increases the complexity of optimization problem. For such a moderate or large scale robust optimization problem the metaheuristic methods are more appropriate selection than gradientbased methods [37,38]. In this case, socially inspired meta-heuristic algorithm such as particle swarm optimization (PSO), which finds the optimum point globally can be used. This method lightens the need for continuous cost functions and variables used for gradient-based methods [39] and does not require the derivatives of the cost function and constraints. PSO have been applied in various fields of the constrained optimization problems [32,40,41]. Moreover, researchers use PSO to solve robust optimization [42,43]. Therefore, in the case of input design, the PSO could be good candidate to solve the constraint robust optimization problem. The objective of this research is to develop an algorithm to enable the robust input design of AUV system identification. The Bayesian robust strategy is used to design input signals for the dynamic modeling of AUVs. The employed approach can design multiple inputs and apply constraints on an AUV system’s inputs and outputs. Particle swarm optimization algorithm is used to find robust optima that satisfy the constraints. The main contribution of this paper is the formulation of an AUV input design that is robust to parameters uncertainties and can extract informationrich data to accurately estimate nonlinear model parameters of AUV. This paper is organized as follows. In Section 2, the developed input design approach is detailed, which includes optimal and robust input design formulation, input parameterizing and

optimization methodology. In Section 3, proposed approach is applied to an AUV case. Some concluding remarks and future works are proposed in Section 4.

2. The input design problem To present our approach the AUV nonlinear model is expressed as

⎧ ẋ(t ) = f (x, θ , u) ⎪ ⎨ y(t ) = g (x, θ , u) ⎪ ⎩ z(ti ) = y(ti, θ ) + ν(ti ),

i = 1, 2, ... , N

(1)

where x(̇ t ) = f (x, θ , u) represents a state space dynamic model, y(t ) = g (x, θ , u) is the mathematical model of the system’s output vector, f , g are the nonlinear functions of the system’s state vector x (includes the linear velocity (m/s), linear position (m), rotational velocity (rad/s) and rotational position (rad)), control input vector u (includes propulsion (N) and control surfaces displacement (rad)), and parameter vectors ( θ ). Also, because of the presence of noise in a measured output, the output is modeled by z (ti ) = y(ti, θ ) + ν(ti ) , i = 1, 2, ... , N , where ν is the measured noise, which is assumed to be a white noise with Gaussian distribution.

E[ν(ti )] = 0 and E⎡⎣ ν(ti )ν(ti )T ⎤⎦ = Rδij

(2)

R is the covariance matrix of the measured noise; which is assumed to be known. Although the purpose of designing the control inputs is to get the maximum amount of information from the behavior of a system through the measured outputs, it should be noted that the inputs and outputs of system are somewhat constrained due to practical considerations. In the case of AUV The amplitude (range of movement) of an AUV’s control surfaces may be constrained by mechanical restrictions, and in order to achieve an effective linear control. Also, some of the system outputs may be constrained in order to keep the AUV on the desired course of travel. These constraints apply to all the control inputs and some of the selected outputs, as follows:

u(t ) − u0 ≤ κ , ∀ t ∈ [0, tmax ], z(ti ) − y0 ≤ λ , i = 1, 2, ... , N

(3)

In the above equation, κ , λ are constant and positive values and u0 , y0 are the nominal values of u, y. In these conditions, the aim is to design the control inputs for System (1) in the presence of Constraints (3). These designed inputs should produce maximum information from the collected data and ultimately lead to a proper estimation of the examined system’s parameters. The quality of the estimated parameters is evaluated by the level of uncertainty achieved in the estimation of parameters. Theoretically, the minimum uncertainty in the estimation of parameters is expressed as the inverse of information matrix M, which is identical to Cramer-Rao inequality:

cov(θ ) ≥ M −1

(4) −1

According to Eq. (4), M is the minimum covariance matrix available for asymptotically unbiased estimators [44]. The information matrix is a criterion for the information contents of the data obtained from the applied inputs; and it is expressed as follows: N

M=

T

⎛ ∂y(i) ⎞ −1⎛ ∂y(i) ⎞ ⎟ R ⎜ ⎟ ⎝ ∂θ ⎠ ∂θ ⎠ i=1

∑ ⎜⎝

(5)

Please cite this article as: Nouri NM, Valadi M. Robust input design for nonlinear dynamic modeling of AUV. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.006i

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In Eq. (5),

∂y ∂θ

is the output sensitivity to the variation of model

parameters; which is obtained by solving the following equation:

∂f d ⎛ ∂x ⎞ ∂f ∂x ⎜⎜ ⎟⎟ = + ∂θj dt ⎝ ∂θj ⎠ ∂x ∂θj ∂y ∂g ∂x ∂g = + ∂θj ∂x ∂θj ∂θj

(6)

Since the differentiation of nonlinear functions f and g is a complex task and differs for each distinct problem; the finite difference approximation is used instead of Eq. (6).

y(θ0 + δθj ) − y(θ0 − δθj ) ∂y = ∂θj 2 δθj

(7)

2.1. Optimal input design In order to design the optimal control inputs, a function of the information matrix or its inverse can be optimized. −1

J (Ξ ) = F (M (θ , Ξ ))

or

J (Ξ ) = F (M (θ , Ξ ))

(8)

Vector Ξ contains the input signal’s characteristic parameters, which have to be optimized, and it is defined by determining the shape of the input signal. θ is a vector that contains the hydrodynamic derivatives to be estimated; however, an initial estimation of these values is necessary for designing the input signal. This estimation could be based on previous experience in system identification or it could be achieved by other methods of estimating the hydrodynamic derivatives. Although the optimization problem can be simplified by using M (θ , Ξ ), it is preferred to design the input based on M−1(θ , Ξ ); because by doing so, a parameter uncertainty is optimized directly [18]. Function F (M−1) could be either tr (M−1), λ max(M−1) or det(M−1). Fig. 1 conceptually shows the uncertainty ellipsoid for the estimation of parameters. In this figure, tr (M−1) is the sum of the squares of parameter uncertainties, λ max(M−1) is the square of the uncertainty ellipsoid’s maximum radius, and det(M−1) is the volume of the uncertainty ellipsoid. For each of these functions the Ξ and θ need to be defined initially; and then, the optimization problem can be solved by applying a proper optimization algorithm. 2.2. Robust input design Since only a priori knowledge of model parameters is known, this implies a circulatory dependence of optimal input design on the true values of the parameters. this problem requires a robust

Fig. 1. Uncertainty ellipsoid for the estimation of a two-dimensional parameter vector.

3

design optimization approach to determine solutions which are insensitive with respect to sources of model uncertainty [26,45]. To do robust input design, consider the model parameters lie in a bounded region which is obtained from prior knowledge around the nominal values θN , Θ = ⎡⎣ θN − δθ , θN + δθ ⎤⎦. A robust input design formulation can be as:

min

Ξ ∈ Ω, θ ∈ Θ

Ψ (F (M −1(θ , Ξ )))

(9)

where Ψ can either be a worst-case operator or an expected average operator which corresponds to a minimax and a Bayesian experimental design strategy respectively. This research uses Bayesian approach to design input for AUV modeling. Bayesian implementation of the robust input design consider a prior distribution of the parameters, P (θ ), and then integrate the uncertainty effects in the design process. A Bayesian design criterion can be presented as:

min E {F (M −1(θ , Ξ ))}

Ξ ∈ Ω, θ ∈ Θ

=min Ξ∈Ω

∫Θ F (M −1(θ, Ξ )) P(θ ) dθ

(11)

where E{·} represents the expectation operator. The parametric uncertainty is represented by the prior distribution P (θ ). The integration of Eq. (11) can be calculated by Monte Carlo integration within the uncertainty region. If assume that P (θ ) is a uniform distribution function, then Eq. (11) can be approximated as:

min E {F (M −1(θ , Ξ ))}

Ξ ∈ Ω, θ ∈ Θ

⎛ 1⎞ ≈min ⎜ ⎟ ⎝ k⎠ Ξ ∈ Ω, θ ∈ Θ

k

∑ F (M −1(θi, Ξ )) i

(12)

Where k is the number of sampling points for θ in its distribution. Using more sampling point k increasing the quality of the Bayesian design and also computation cost.

2.3. Shape of the input signals To design the input signal, the shape of the signal must be determined first, and then the design parameters of this input signal must be parameterized. The input signals for nonlinear systems should cover all the amplitude and frequency ranges of such systems; and these conditions could be satisfied through the excitation of the APRBSs [17,18]. Fig. 2 shows this square wave signal, which has different amplitudes at different sections. The APRBS is expressed by Eq. (13). The amplitudes and switching time are the parameters of the APRBS ( Ξ = (ai , ti, i = 0, ... , n)), which will be optimized in the optimization problem.

Fig. 2. APRBS input.

Please cite this article as: Nouri NM, Valadi M. Robust input design for nonlinear dynamic modeling of AUV. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.006i

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u = u0 +

∑ (ai − ai − 1) h(t − ti),

⎧ 1 t ≥ ti h(t − ti ) = ⎨ ⎩ 0 t < ti ⎪

Table 1 Constraint Robust PSO algorithm for Robust Input Design.

(13) Constraint Robust PSO for input design

2.4. Optimization problem As noted in Section 2.1, for designing the control inputs, the function of M−1 is minimized. With this objective function, the optimization problem becomes non-convex [35]; and the considered optimization algorithm must find the optimal solution globally. Also, the algorithm which is used to solve the robust optimization problem should be able to apply the existing constraints on the inputs and outputs of the system. Also the considered algorithm must have the ability to manage a large number of design variables. A particle swarm optimization is an algorithm which can be used for constraint robust optimization problem [42,43]. PSO algorithm is a population based optimization methodology introduced by Kennedy and Eberhart [46]. The general idea in the PSO algorithm is that there exists a swarm of particles and each particle is at a position x i and move with velocity vi in the search space. There is a cost function f (x i ) associated to each position. The → personal best position pi and the global best position gi influence the velocity of each particle. The degree of this influence is defined by a coefficient ϕ1 and ϕ2 as follows:

→ → vi : = χ . ( vi → → → + U (0, ϕ1) ⊗ ( pi − xi ) → → → + U (0, ϕ1) ⊗ (gi − xi ))

(14)

Input: control input space lower and upper bounds [lb, ub] Input: output constraint input: Personal best : ϕ1 input: Neighborhood best : ϕ2 input: Number of particles to use: n input: Number of samples from Θ region to use: k 1 Initialize particle position ( Ξ ∈ [lb, ub] ) → → → → → → → 2 Initialize particle velocity v1: = χ . ( U (0, ϕ1) ⊗ ( p1 − x1) + U (0, ϕ1) ⊗ ( g1 − x1)) 3 while termination condition not met do 4 for each particle i of n do → → → → → → → → 5 vi : = χ . ( vi + U (0, ϕ1) ⊗ ( pi − xi ) + U (0, ϕ1) ⊗ ( gi − xi )) → → → 6 xi : = xi + vi 7 end for

( [ θN − δθ, θN + δθ ], k)

8

Θ = {θ1, θ 2, ... , θk}: = sampler

9

for each sample j in uncertainty region Θ, do

10

Y:¼ Evaluate Fitness ( F (M −1(θj, Ξ )) ).

11 V: ¼ Constraint Violation criteria. 12 end do 13 calculate expected value of Fitness: ¼ 14 14 15 16 16 18

( ) ∑ F (M 1 k

k j

−1

(θj, Ξ ))

calculate expected value of constraint violation update expected value of fitness according to expected constraint Violation find best personal position find best global position end while output: best solution found

2.5. Summarizing robust input design algorithm

New position of particles is updated as follows:

→ → → xi : = xi + vi

(15)

After a certain amount of time the swarm particles converge towards optimal position [42]. This approach is used for unconstraint optimization problem. but AUV input design problem is transformed into constraint optimization problem. Therefore, proposed optimization algorithm must handle constraints. The most common approach for constraint handling, is transforming the constraint problem to the unconstraint problem by penalizing the constraints. The present study uses nonstationary penalty function which dynamically modified the penalty value as follows.

→ → → P (θ , x ) = F (θ , x ) + Φ(θ , x )

(16)

→ → where P (θ , x ) is new cost function, F (θ , x ) is cost function of → constraint problem and Φ(θ , x ) is penalty function which is calculated as following equation.

→ → Φ((θ , x )) = max [0, C (θ , x )]

(17)

→ where C (θ , x ) is constraints of problem. For Bayesian robust optimization both part of Eq. (16) must be calculated in Bayesian context.

⎛ ⎞ k 1 → → FBayesian(Θ, x ) = ⎜⎜ ⎟⎟ ∑ F (θi, x ) k ⎝ ⎠ i

⎛ ⎞ k 1 → → ΦBayesian(Θ, x ) = ⎜⎜ ⎟⎟ ∑ Φ(θi, x ) ⎝ k⎠ i

Following steps summarizing the main part of APRBS robust input design algorithm: 1. Parameterizing the input signals as APRBS input (Eq. (13)). 2. Using random sampling to choose the model parameters values from bounded region. 3. Using candidate inputs to solve dynamic model (Eq. (1)) for all chosen parameter sets. 4. Calculating the sensitivity matrix (Eq. (7)) and information matrix from resultant motions (Eq. (5)). 5. Using expected operator to calculating Bayesian cost function. 6. Checking constraints violation of problem (Eq. (3)) and using expected operator to calculating Bayesian constraints effects (Eq. (19)). 7. penalizing the Bayesian cost function based on Bayesian constraints effects (Eq. (16)). 8. Searching the best APRBS input which has minimum penalized cost function. In this paper, only the case of an AUV model is considered. Nevertheless, the proposed approach can be applied to most of the dynamical models with constraints on inputs and outputs.

3. Input design for an AUV case

(18)

(19)

Where Θ is the bounded region of model parameters values. Table 1 shows the robust optimization algorithm applied to find the robust inputs design.

To evaluate the proposed algorithm, robust inputs are designed for an AUV. At first, dynamic model of an AUV is presented in Section 3.1. This model includes the parameters (hydrodynamic derivatives) which characterize AUV hydrodynamic behavior. The aim of present paper is to design an input which can accurately estimates these parameters based on uncertain prior information about these parameters. Next, Section 3.2 applies the robust input design for Hydrolab300 AUV.

Please cite this article as: Nouri NM, Valadi M. Robust input design for nonlinear dynamic modeling of AUV. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.006i

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5

12

12

10

10 δr (degree)

δr (degree)

Fig. 3. AUV motion in the inertial and body coordinate systems.

8 6 4

8 6 4 2

2 0

10

20

30 Time (s)

40

50

0 0

60

10

20

a

30 Time (s)

40

50

60

b Fig. 4. Designed inputs a) Optimal b) Robust.

0.03 0.025

v (m/s)

0.02 0.015 0.01 0.005 0 0

10

20

30 Time (s)

40

50

60

a

0.025

v (m/s)

0.02 0.015 0.01 0.005 0 0

10

20

30 Time (s)

40

50

60

b

Fig. 5. Sway velocity and acceleration for nominal values of hydrodynamic derivatives. (a) Optimal design (b) Robust design.

3.1. Mathematical model governing the AUVs Based on the ITTC and SNAME’s offer, the motion of AUVs, with 6 degrees of freedom (6 DOF), can be expressed by two inertial and body-fixed coordinate systems, as shown in Fig. 3. The motion of an AUV in these coordinate systems is expressed

by vectors → η = [→ η1, → η2] , → ν = [→ ν1, → ν2] , → τ = [→ τ1, → τ2], which represent the position, velocity and force vectors, respectively. The linear and → angular components of these vectors are η1 = [x, y, z ], → → → → η = [φ , θ , ψ ], ν = [u, v, w ], ν = [p, q, r ], τ = [X , Y , Z ] and 2

1

1

1

→ τ2 = [K , M , N ], respectively. The mathematical equations of motion for an AUV in the body coordinate system are expressed as

Please cite this article as: Nouri NM, Valadi M. Robust input design for nonlinear dynamic modeling of AUV. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.006i

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Table 2 Monte Carlo simulation results at nominal values of parameters. Hydrodynamic Derivatives

Nominal value

Optimal Design

Optimal Standard Deviation

Robust Design

Robust Standard Deviation

Yv̇ (kg) Yv (kg/s) Yv v (kg/m)

 189.9  1936  2067

 187  1936  2071.2

4.31 3.11 151.1

 186.71 3.53  1936.03 3.22  2062.38 157.06

Fig. 7. Compare PDF of RU of Hydrodynamic Derivatives.

Table 3 Monte Carlo simulation results for 25% uncertainty about nominal values. Relative Uncertainty Of Hydrodynamic Derivatives

Optimal Design (%)

Optimal Standard Deviation (%)

Robust Design (%)

Robust Standard Deviation (%) 0.167

RUYv̇

2.341

0.160

1.845

RUYv

0.316

0.002

0.287

0.003

RUYv v

14.942

4.961

14.110

4.433

Fig. 6. Plot with the 95% confidence ellipsoids for input.

+ Yrṙ 2 + X uu + X u u u u X = X u̇u̇ − Z ẇ wq − Z qq̇ 2 + Yvvr ̇ −(W − B)sin θ + Xprop

m⎡⎣ u̇ − vr + wq − xG q2 + r 2 + yG (pq − r )̇ + zG(pr + q)̇ ⎤⎦ = X

( ) ⎡ ̇ m⎣ v − wp + ur − y ( r + p ) + z (qr − ṗ) + x (qp + r )̇ ⎤⎦ = Y m⎡⎣ ẇ − uq + vp − z ( p + q ) + x (rp − q)̇ + y (rq + ṗ)⎤⎦ = Z 2

2

G

G

2

G

− X u̇ur + Yvv + Yrr + Yv v v v Y = Yvv̇ ̇ + Yrṙ ̇ + Z ẇ wp + Z qqp ̇ +Yr r r r + (W − B)cos θ + Yuuδu2(δrT + δrB )

G

− Yrrp + X u̇uq + Z w w + Zqq + Z w w w w Z = Z ẇ ẇ + Z qq̇ ̇ + − Yvvp ̇ ̇

2

G

G

+Zq q q q + (W − B)cos θ cos φ + Z uuδu2(δsR + δsL )

Ixṗ + (Iz − Iy)qr + m[yG (ẇ − uq + vp) − zG(v ̇ − wp + ur )] = K

− Z qqv + (Yv ̇ − Z ẇ )vw + Kpp + Kp p p p K = K pp ̇ ̇ + Yr rw ̇ ̇

Iyq ̇ + (Ix − Iz )rp + m[zG(u̇ − vr + wq) − xG (ẇ − uq + vp)] = M Izr ̇ + (Iy − Ix)pq + m[xG (v ̇ − wp + ur ) − yG (u̇ − vr + wq)] = N

(20)

where [xG , yG , zG] is the center of mass in terms of the bodyfixed coordinate system, m is the AUV’s mass and Ix, Iy, Iz are the AUV’s moments of inertia. The right hand sides of these equations are the external forces and moments applied to the vehicle; which include the restoring forces (weight and buoyancy), control forces (control surfaces and propulsion) and hydrodynamic forces (inertia and viscous effects):

+(yG W − yB B)cos θ cos φ − (z GW − zBB)cos θ sin φ +K uuδu2(δrT − δrB ) + K uuδu2(δrT − δrB ) + Nrrp + (Z ẇ − X u̇)uw + (Yr ̇ − K p )̇ vp M = Mẇ ẇ + Mqq̇ ̇ + Z qqu ̇ ̇ +Mww + Mqq + Mw w w w + Mq q q q + (z GW − zBB)sin θ −(x G W − xB B)cos θ cos φ + Muuδu2(δsR + δsL ) − Yrur + (X u̇ − Yv )̇ uv + (k p ̇ − Mq)̇ pq N = Nvv̇ ̇ + Nrṙ ̇ − Z qwp ̇ ̇ +Nvv + Nrr + Nv v v v + Nr r r r + (x G W − xB B)cos θ sin φ −( yG W − yB B)sin θ + Nuuδu2(δrT + δrB )

(21)

Please cite this article as: Nouri NM, Valadi M. Robust input design for nonlinear dynamic modeling of AUV. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.006i

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Fig. 8. Robust inputs design for 6 DOF motion.

Table 4 Monte Carlo simulation results at nominal values of hydrodynamic derivatives. Added mass

Estimated values

Standard deviation

Added mass

Xu̇ (kg)

-4.1832

4.5712

K ṗ (kg m2/rad)

0.1354

0.0003

Yv̇ (kg) Yr ̇ (kg m/rad)

-186.9171 12.8865

0.8768 2.7607

Mẇ (kg m)

-12.8864 -92.5554

1.624 0.0827

Z ẇ (kg) Zq̇ (kg m/rad)

-186.9175 -12.8862

7.2018 0.6080

12.8861 -92.5554 Nr ̇ (kg m2/rad) Linear damping Estimated values

Nv̇ (kg m)

Standard deviation

0.2197 0.4784

Linear damping

Estimated values

Xu (kg/s)

-176.7630

4.1851

Kp (kg m2 /(s rad))

Yv (kg/s) Yr (kg m/(s rad))

-1936.2365 1984.5123

25.8660 49.0000

935.9383 Mq (kg m2 /(s rad)) -2322.8120

16.9185 6.0746

-1936.1235 Z w (kg/s) Zq (kg m/(s rad)) -1984.5153

39.3984 -41.8916

-935.9406 Nr (kg m2 /(s rad)) -2322.7853 Nonlinear Estimated damping values

19.9342 11.1071

Nonlinear damping

Estimated values

Standard deviation

Mq̇ (kg m2/rad)

Estimated values

Standard deviation

-33.0368

Mw (kg m/s) Nv (kg m/s)

Standard deviation 0.0046

Standard deviation

Xu u (kg/m)

-4.4191

0.1066

Kp p (kg m2 / rad2 )

0.0661

0.0257

Yv v (kg/m)

-2066.8810

68.2901

Mw w (kg)

2807.2135

10.4591

Yr r (kg m/ rad2 )

1.7

0.5254

Mq q (kg m2 / rad2 ) -160.5280

Z w w (kg/m)

-2066.7215

42.0113

Nv v (kg)

-2805.8783

16.6919

0.2654

Nr r (kg m2 / rad2 )

-160.4761

12.6974

Zq q (kg m/ rad2 ) -1.6997

6.5657

Fig. 9. Input signals of zigzag maneuver.

Please cite this article as: Nouri NM, Valadi M. Robust input design for nonlinear dynamic modeling of AUV. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.006i

N.M. Nouri, M. Valadi / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

8

Table 5 Comparison of relative Cramer-Rao lower bounds (%). Parameter

Xu̇

Yv̇

Yr ̇

Nv̇

Nr ̇

Xu

Yv

Yr

Zigzag Designed Robust input

10.7198 1.0563

0.4860 0.0048

1.8185 0.2147

3.8829 0.0160

0.1000 0.0051

0.2321 0.0229

0.3157 0.0132

0.2441 0.0243

Parameter

Nv

Nr

Xu u

Yv v

Yr r

Nv v

Nr r

Zigzag Designed Robust input

0.4798 0.0207

0.0613 0.0047

0.278 0.0232

0.4879 0.3651

131.6082 32.7524

0.0775 0.0588

0.3687 0.0761

Fig. 10. Comparison of probability density functions for zigzag maneuver and robust APRBS input.

In Eq. (21), the parameters showing the derivatives of forces and moments with respect to velocity and acceleration are considered as the hydrodynamic derivatives. If the derivation is with respect to acceleration, the resulting derivative will be called the added mass hydrodynamic derivative; and if the derivation is with respect to velocity, the resulting derivative will be called the damping hydrodynamic derivative. δrT , δrB are the control inputs applied by the top and bottom rudders, and δsR , δsL are the control inputs applied by the left and right hydroplanes. Xprop , W , B are the propulsion, weight and buoyancy forces, and [xB , yB , zB] is the center of buoyancy in the body-fixed coordinate system.

3.2. Input design for Hydrolab300 AUV To evaluate the proposed algorithm, APRBS inputs are designed for the Hydrolab300 AUV. The nominal values of hydrodynamic derivatives of Hydrolab300 have been extracted via analytical and semi-empirical methods. At first, input for sway degree of freedom is designed considering nonlinear dynamic model for sway as follows:

(m − Yv )̇ v ̇ = Yvv + Yv v v v + YuudrU2δr

(21)

Please cite this article as: Nouri NM, Valadi M. Robust input design for nonlinear dynamic modeling of AUV. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.006i

N.M. Nouri, M. Valadi / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

where Yv,̇ Y , Yv v are added mass, linear, and nonlinear damping hydrodynamic derivatives, respectively. The objective is to design a rudder input δr . Nominal values of hydrodynamic derivatives are -189.9, -1936, and -2067, respectively. The optimal APRBS inputs are designed according to the nominal values of hydrodynamic derivatives considering Trace(M−1) as the optimization criterion. The maximum time of 60 s is established for the optimized inputs. The main constraint is imposed on the sway acceleration ( v ̇). The deflection of the rudder is also constrained to 10º at a time. The constraints on the inputs and outputs of the optimized APRBS inputs are as follows:

δr (t ) ≤ 10, ∀ t ∈ [0, 60], v(̇ ti ) ≤ 0.05, i = 1, 2, ... , N

(22)

The robust APRBS inputs are designed by considering 25% parametric uncertainty around the nominal parameters. The optimization criterion, time, and constraints are the same as the ones in the optimal input design. Figs. 4 and 5 show the optimal and robust APRBS inputs for the rudder and resultant motion of the Hyrolab300 AUV at nominal values of hydrodynamic derivatives. Obviously, both optimal and robust designed APRBS inputs satisfy the constraint for the inputs and output. For comparison purposes, initially, the Monte Carlo simulation for optimal and robust designed inputs is done at the nominal values of Yv,̇ Y , Yv v . A Monte Carlo analysis is performed by applying white Gaussian noise sequences (with the magnitude resulting in the signal-to-noise ratio of 30) to the simulation outputs. Table 2 shows a comparison of the parameter estimation results of 1000 runs in Monte Carlo simulation. The standard errors are computed from the scatter in parameter estimates as:

σ^θ = 1 θ¯ = n

1 (n − 1)

n

∑ i=1

2

( θ^ − θ¯) i

n

∑ θ^i

δθ × 100 θ^

(24)

According to Fig. 6, output constraint of all 1000 Monte Carlo runs is satisfied for robust input design, but an optimal input just satisfies the constraint for 667 Monte Carlo runs. Also, Fig. 7 and Table 3 demonstrate that overall smaller mean values and scatter for robust input compared to optimal input. Further, we can use the proposed algorithm to design inputs for 6 DOF motion of Hydrolab300. The constraints on the inputs and outputs of the optimized APRBS inputs are as follows:

δrT (t ) ,

δrB(t ) ,

Fig. 8 shows the designed robust inputs for the Hydrolab300 AUV. Also Table 4 indicates the estimated values and their standard deviations from these designed inputs at nominal values. Also we do comparative study to show the superiority of designed robust input rather than conventional maneuver to estimate AUV hydrodynamic derivatives. The conventional excitation signals analyzed were inputs of zigzag maneuver. Fig. 9 shows the excitation signals of the zigzag maneuver for Hydrolab300 AUV. For comparison purposes, the relative Cramer-Rao lower bounds of the designed and zigzag maneuvers’ excitation signals have been presented in Table 5. The Table 5 shows that the designed input reduce the estimate uncertainty for all of hydrodynamic derivatives. Fig. 10 compares probability density function of the Monte Carlo parameter estimation results for nonlinear damping hydrodynamic derivatives. The parameter estimation results from the Monte Carlo analysis demonstrate that more accurate modeling results can be obtained by using the robust designed inputs.

4. Conclusion In this paper, the Bayesian experiment design and its application to AUV input design were presented for dynamic modeling through system identification. Hydrodynamic derivative distributions were the knowledge about the model that was used to design robust inputs. To design the inputs, the expected values of the trace of lower bound Cramer-Rao inequality were minimized and the practical constraints were applied to input and output variables. Constraint robust particle swarm optimization was employed to solve this optimization problem. A comparative study was conducted to evaluate the proposed approach and the resulting inputs were used in the estimation of Hydrolab300 derivatives. According to the result, the proposed robust input design can satisfy both robustness of constraints and optimality. The approach presented in this work had the following advantages:

(23)

i=1

where n is the total number of Monte Carlo runs. The computed σ^θ quantifies the scatter in repeated parameter estimates. The results, as shown in Table 2, indicate that optimal input generally has a better estimation result than the robust one at nominal values. Below, a Monte Carlo analysis is performed to evaluate the robustness of the designed inputs. Also, 25% parameter uncertainty around nominal values is considered for sampling hydrodynamic derivatives in Monte Carlo simulations. The estimated values and their uncertainties are the results obtained from each Monte Carlo run. RU variable is used to compare robust and optimal input design, which is the percent of parameter uncertainty to its estimated value.

RUθ =

9

δsR(t ) and δsL(t ) ≤ 10, ∀ t ∈ [0, 60],

°

θ (ti ) ≤ 25 , i = 1, 2, ... , N φ(ti ) ≤ 25° , i = 1, 2, ... , N

(25)

1. It could provide an input design which can extract rich-information data set to accurately estimates AUV hydrodynamic derivatives. 2. It could apply constraints to the input and output variables, resulting in optimized inputs that stay within the operational envelope of AUVs. 3. It uses a trackable method to do input design that is insensitive to parameter values. The current study takes into account the parameter uncertainty, which is caused by imprecise knowledge about model parameters, to design the robust input. The future research will take into account the model structure uncertainty, which is caused by simplification of a system under study and/or incomplete understanding of physical processes.

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Please cite this article as: Nouri NM, Valadi M. Robust input design for nonlinear dynamic modeling of AUV. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.02.006i