Identification of the equivalent linear dynamics and controller design for an unmanned underwater vehicle

Ocean Engineering 139 (2017) 152–168

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Identification of the equivalent linear dynamics and controller design for an unmanned underwater vehicle

MARK

⁎

Afshin Banazadeha, , Mohammad Saeed Seif b, Mohammad Javad Khodaeib, Milad Rezaieb a b

Sharif University of Technology, Department of Aerospace Engineering, PO Box: 11365-11155, Tehran, Iran Sharif University of Technology, Department of Mechanical Engineering, Tehran, Iran

A R T I C L E I N F O

A BS T RAC T

Keywords: Frequency domain identification Autonomous underwater vehicle Equivalent linear dynamics Control design CIFER®

This paper investigates the applicability of frequency-domain system identification technique to achieve the equivalent linear dynamics of an autonomous underwater vehicle for control design purposes. Frequency response analysis is performed on the nonlinear and coupled dynamics of the vehicle, utilizing the CIFER® software to extract a reduced-order model in the form of equivalent transfer functions. Advanced features such as chirp-z transform, composite window optimization, and conditioning are employed to achieve high quality and accurate frequency responses. A particular frequency-sweep input is implemented to the nonlinear simulation model to achieve pole-zero transfer functions for yaw and pitch motions that were previously developed by the perturbed equations of motion. To evaluate the accuracy of the identified models, zig-zag test data are compared with the predicted responses for both identified and linearized models in time domain. The results show that the identified models perform significantly well in the presence of noise and model uncertainties with the maximum error of 12%, thanks to the precise spectral analysis. Proportional-integral controllers are designed based on the extracted models and tracking performance is experimentally demonstrated by several test results that show the ability of the vehicle to navigate autonomously and follow the GPS waypoints with reasonable accuracy.

1. Introduction Marine industry utilizes new technologies to meet the growing needs for exploration or extraction of undersea resources, inspection, environmental data collection, and installation or maintenance of coastal structures. Due to some restrictions of deep-sea explorations, autonomous underwater vehicles are the most powerful means for subsurface studies that help researchers with simple, low-cost, and rapid response capabilities through appropriate underwater data collections. However, the dynamics of these vehicles are complex with several nonlinear and coupled terms, which make it a challenging task to perform identification process for dynamic stability analysis and control design. Several approaches are proposed to model underwater vehicles that contain hydrodynamic coefficients (Fossen, 1994; Yuh, 1990). Commonly-used methods for modeling of underwater vehicles can be classified in four major categories (Xu et al., 2013). These categories include captive model test with planar motion mechanism (PMM) (Rhee et al., 2000), estimation with empirical formulas (Silva et al., 2007), numerical calculation based on computational fluid dynamics (CFD) (Toxopeus, 2009), and system identification (SI) in ⁎

combination with free-running model. Identification of marine vehicles is a highly versatile procedure for rapidly and efficiently extracting accurate dynamic models of a marine vehicle from the measured response to specific control inputs (Tischler and Remple, 2006). Various identification techniques have been developed since the 1950s for system dynamic modeling, parameters and states estimation, using nonlinear simulation and measurement data. Neural network (NN) (Mahfouz and Haddara, 2003; Shafiei and Binazadeh, 2015), support vector machines (SVM) (Zhang and Zou, 2011; Xu et al., 2013), time domain identification (Naeem et al., 2003; Shi and Zhao, 2009, Avila et al., 2013) and frequency domain identification (Selvam et al., 2005; Bhattacharyya and Haddara, 2006; Perez and Fossen, 2011) are the most applicable identification techniques used for marine vehicles. However, most of the published studies have focused on parameter estimation and presenting some approaches to measure hydrodynamics coefficients of ships and AUVs. For the control purpose, it is desirable to simplify and reduce the nonlinear and coupled dynamics of marine vehicles to equivalent linear models (Tiano et al., 2007). However, little attention has been paid to the frequency domain characteristics despite its advantages and restrictions of time domain.

Corresponding author. E-mail addresses: [email protected] (A. Banazadeh), [email protected] (M.S. Seif), [email protected] (M.J. Khodaei), [email protected] (M. Rezaie). http://dx.doi.org/10.1016/j.oceaneng.2017.04.048 Received 25 August 2016; Received in revised form 14 March 2017; Accepted 29 April 2017 Available online 06 May 2017 0029-8018/ © 2017 Elsevier Ltd. All rights reserved.

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Nomenclature

Yr ̇ Yur Ywp Ypq Yuv Yδr ZHS Zw w Zq q Z ẇ Z q̇ Z uq Z vp Z rp Z uw Zδs KHS K ṗ KThrust MHS Mw w Mq q M ẇ Muq Mvp Mrp Muw Mδs NHS Nv v Nr r Nv̇ Nr ̇ Nur Nwp Npq Nuv Nδr KI KP R (s ) E (s ) Gc (s ) G Y (s ) GP (s ) Y (s ) u (t )

XYZ Earth-fixed frame xyz Body-fixed frame X, Y, Z External forces in the body-fixed frame K, M, N Moments of external forces in the body-fixed frame u, v, w Linear velocities in body-fixed frame p, q, r Rotational angular velocities ϕ, θ, ψ Euler angles Ix, Iy, Iz Mass moments of inertia in the body-fixed frame xG, yG, zG Center of gravity in the body-fixed frame xB, yB, zB Center of buoyancy in the body-fixed frame CB Center of buoyancy CG Center of gravity Mt Inertia matrix MA Added inertia matrix MRB Rigid-body inertia matrix CRB(V) Rigid-body Coriolis and centripetal inertia matrix CA(V) Hydrodynamic Coriolis and centripetal matrix D(v) Hydrodynamic damping matrix g(η) Restoring forces and moments matrix τ Propulsion forces and moments matrix δr Rudder deflection δs Elevator deflection KT Thrust force coefficient KQ Thrust torque coefficient n Propeller revolution J0 Advanced number W Weight B Buoyancy g Gravitational Acceleration m Mass ρ Density D Propeller diameter X u̇ Drag contribution in the longitudinal X direction due to time rate of change of u Xqq Drag force due to square of pitch rate of body Xrr Drag force due to square of yaw rate of body Xwq Drag force due to q and w Xvr Drag force due to r and v Xu u Drag force due to square of surge rate of body XHS Restoring force in the longitudinal X direction XThrust Propulsion force in the longitudinal X direction YHS Restoring force in the longitudinal Y direction Yv v Sway force due to square of sway of body Yr r Sway force due to square of yaw rate of body Yv̇ Sway force due to time rate of change of v

Sway force due to time rate of change of r Sway force due to r and u Sway force due to p and w Sway force due to q and p Sway force due to v and u Sway force due to rudder deflection Restoring force in the longitudinal Y direction Heave force due square of w Heave force due square of q Heave force due to time rate of change of w Heave force due to time rate of change of q Heave force due to u and q Heave force due to p and v Heave force due to p and r Heave force due to w and u Heave force due to elevator deflection Restoring moment around X direction Roll moment due to time rate of change of p Propulsion moment around X direction Restoring moment around Y direction Pitch moment due to square of w Pitch moment due to square of q Pitch moment due to time rate of change of w Pitch moment due to q and u Pitch moment due to p and v Pitch moment due to p and r Pitch moment due to w and u Pitch moment due elevator deflection Restoring moment around Z direction Yaw moment due to square of v Yaw moment due to square of r Yaw Moment due to time rate of change of v Yaw Moment due to time rate of change of r Yaw moment due to r and u Yaw Moment due to p and w Yaw Moment due to p and q Yaw Moment due to v and u Yaw Moment due to rudder deflection Integral controller gain Proportion controller gain Desired input Error signal Transfer function of controller Transfer function of actuator Transfer function of the system Output vector Control signal

process, particularly to adapt and evaluate frequency responses. It is notable that, identification methods are partially dependent on the maneuvers performed and subsequent data sets. Therefore, development of a nonlinear simulation model would help to understand the dynamics and generating the required data for the identification purpose. Finally, the best linear approximation of a nonlinear and coupled system is derived that will be used for non-parametric or parametric modeling to capture the key dynamic features of the system. The aim of this paper is to evaluate the applicability of frequency domain identification to be used for a subsurface vehicle and to achieve the best reduced order equivalent linear model with the purpose of designing a proper controller. Designing an optimal excitation signal, data conditioning and proposing dynamic model of actuators are also considered. In this paper, by using the well-known Fossen's model (Fossen, 1994), a nonlinear simulation model of an underwater vehicle is

Using an equivalent linear model to study system dynamics is very popular due to the ease of analysis and maturity of controller design for the linear models. This technique has been used for more than 30 years for complicated systems like aircraft and marine vessels (Marchand and Fu, 1985; Källström and Åström, 1981; Selvam et al., 2005). It is much more convenient to provide a linear model of an underwater vehicle in the frequency domain rather than time domain due to unique features such as elimination of possible bias, sensor and process disturbances as well as less computational cost. Banazadeh and Ghorbani (2013), derived a linear dynamic for a surface ship in frequency domain by representing transfer functions to design proper PID controllers using CIFER® (Comprehensive Identification from FrEquency Responses) software. Nikusokhan and Banazadeh (2014), introduced the frequency domain identification of a servo mechanism in the presence of backlash and friction, using the same software. In the current study, this software is utilized in the identification 153

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global coordinates. According to the Newton's second law of motion along with Euler's Law, these equations are as follows:

created in Matlab-Simulink environment. The frequency sweep data is then used to identify the system dynamics in the frequency range of applicability. Spectral analysis is performed using Chirp-z transform and CIFER®. Here, windowing technique that is a convenient tool in data processing is also used to eliminate the induced random noises and to increase the value of the coherence function. Finally, with the help of this analysis, equivalent linear transfer functions are extracted. In addition, a relevant experimental setup is designed for the validation purposes, employing SUT-2 underwater vehicle. Moreover, in order to further comparison, Fossen's linear model is used. The results show that the identified equivalent dynamic model from the frequency responses of the complex analytical model is much better and more accurate than the linearized model demonstrating a very good matching with the experimental data. Finally, according to validated models a PI and an Integral controllers are designed for the vehicle tracking purpose. Finally, their performance are evaluated and compared by different experimental tests. Fig. 1 shows the detailed overview of the research.

MRB v ̇ + CRB (v ) v = τRB

(1)

MRB is the inertia matrix of the rigid-body and CRB (v ) denotes to the Coriolis and centripetal inertia matrix. The right side of Eq. (1) is the sum of the external forces and moments that can be written as: τRB = τ − MA v ̇ − CA (v ) v − D(v ) v − g(η)

(2)

Where, τ is the propulsion forces and moments, MA signifies the added mass matrix, CA (v ) represents the matrix of centrifugal and Coriolis terms, D(v ) denotes to the hydrodynamic damping, and g(η) is the restoring forces and moments. Therefore, 6DoF equations are (Prestero, 2001):

m (u ̇ − vr + wq − xG (q 2 + r 2 ) + yG ( pq − r )̇ + zG ( pr + q )̇ ) = m (v ̇ − wp + ur − yG

2. SUT-2 AUV description

( p2

m (ẇ − uq + vp − zG

+

r 2)

+ zG (qr − p ̇ ) + xG ( pq + r )̇ ) =

+

q 2)

+ xG (rp − q )̇ + yG (rq + p ̇ ) ) =

( p2

∑X ∑Y ∑Z

Ixx p ̇ + (Izz Z − Iyy ) qr − (r ̇ + pq ) Ixz + (r 2 − q 2 ) Iyz + ( pr − q )̇ Ixy

SUT-2 is a type of cruising underwater vehicle, which has been designed by a group of graduate students in the marine engineering group at Sharif University of Technology (Rahimian et al., 2009; Sadeghzadeh et al., 2009). This vehicle has an elliptical nose, a cylindrical mid-section, and four tail fins as rudders to execute control commands. The fins were designed with a NACA 0015 cross-section. Body lines were designed based on the Myring approach, which is one of the best configurations for drag reduction (Myring, 1976; Jun et al., 2009). As shown in Fig. 2, to determine the body dimensions, a complex algorithm has been used to take into account all influential design parameters (Rahimian et al., 2009; Sadeghzadeh et al., 2009). These dimensions are shown in Fig. 3 and presented in Table 1. Body materials and their thickness were selected according to some effective parameters such as working depth, weigh and cooling capability. Therefore, the thickness was selected 1 cm, and the noise and tail sections were made from PVC materials while the mid-body was manufactured with Aluminum alloy. A single electric motor drives the vehicle in pusher configuration and three servo motors drive the control surfaces. Fig. 3 shows the external configuration of the SUT-2.

+ m ( yG (ẇ − uq + vp ) − zG (v ̇ − wp + ur )) = Iyy q ̇ + (Ixx − Izz ) rp − ( ṗ + qr ) Ixy +

( p2

−

∑K

r 2 ) Izx

+ m (zG (u ̇ − vr + wq ) − xG (ẇ − uq + vp )) =

+ (qp − r )̇ Iyz

∑M

Izz r ̇ + (Iyy − Ixx ) pq − (q ̇ + rp ) Iyz + (q 2 − p 2 ) Ixy + (rp − p ̇ ) Izx + m (xG (v ̇ − wp + ur ) − yG (u ̇ − vr + wp )) =

∑N

(3)

The right side of the above equations are the external forces and moments that can be calculated as follows (Beverley, 2009):

∑ X = XHS + Xu u u u + X u ̇u ̇ + Xwq wq + Xqq qq + Xvr vr + Xrr rr + XThrust ∑ Y = YHS + Yv v v v + Yr r r r + Yv v̇ ̇ + Yr ṙ ̇ + Yur ur + Ywp wp + Ypq pq + Yuv uv + Yδr u2δr ∑ Z = ZHS + Z w w w w + Zq q q q + Z ẇ ẇ + Z q q̇ ̇ + Z uq uq + Z vp vp + Z rp rp + Z uw uw + Zδs u2δs ∑ K = KHS + K pṗ ̇ + KThrust ∑ M = MHS + Mw w w w + Mq q q q + M ẇ ẇ + M q ̇q ̇ + Muq uq + Mvp vp

3. Nonlinear simulation

+ Mrp rp + Muw uw + Mδs u2δs ∑ N = NHS + Nv v v v + Nr r r r + Nv v̇ ̇ + Nr ṙ ̇ + Nur ur + Nwp wp + Npq pq

Three basic steps are necessary to simulate the movement of the underwater vehicle. First, the formulation of the equations of motion that actually describe the behavior of the vehicle. Second, an estimation of the hydrodynamic coefficients in order to obtain forces and moments, and third, solving and verification of these equations. The 6DoF (six Degrees of Freedom) equations of motion are ordinary differential equations that can be solved simply by using standard calculation methods (Lambert, 1973). In this study, the equations of motion are written in the body-axis coordinate system. Typically, the center of the body-axis is considered to coincide with the center of buoyancy (CB). Fig. 4 shows the local and

+ Nuv uv + Nδr u2δr (4) The above equations include hydrodynamic coefficients that need to be obtained by numerical, experiments, or analytical and semi-empirical methods (Barros et al., 2008; Sarkar et al., 1997). Accordingly, SUT-2 coefficients are already obtained by Sadeghzadeh et al. (2009) to be used in the simulation model. Here, the terms with subscript “HS” represents the hydrostatic force and moment, which is calculated as follows:

Fig. 1. The detailed overview of the research.

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Fig. 2. The design algorithm of the SUT-2 (Sadeghzadeh et al., 2009).

Fig. 4. The body-axis and earth-fixed coordinate system definitions (adapted from Fossen, 1994).

g(h)

Fig. 3. The SUT-2 autonomous underwater vehicle a) Parts description and b) Geometric characteristics.

⎡ ⎤ (W −B ) sinθ ⎢ ⎥ − ( W − B ) cosθsinϕ ⎢ ⎥ ⎢ ⎥ − (W −B ) cosθcosϕ =⎢ − ( yG W −yB B ) cosθcosϕ + (zG W −zB B ) cosθsinϕ ⎥ ⎢ ⎥ ⎢ (zG W −zB B ) sinθ + (xG W −xB B ) cosθcosϕ ⎥ ⎢⎣ − (x W −x B ) cosθsinϕ−( y W −y B) sinθ ⎥⎦ G B G B

Propulsion forces and moments that arise from the rotation of the propeller that is mounted at the tail, are determined as follows:

Table 1 Dimensions of SUT-2.

τ = [ρD 4KT J0 n2, 0, 0, ρD 5KQ J0 n2 , 0, 0]T

Parameter

Value (mm)

L D a b c

1412 200 235 706 471

(5)

(6)

Where, ρ is density, D represents the propeller diameter, KT denotes to thrust force coefficient, J0 signifies the advanced number, n is the propeller revolution and KQ shows the propeller torque coefficient. By determining the coefficients in (4) and the isolating the acceleration terms, the final equations can be rearranged as follows (Beverley, 2009):

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Fig. 5. Simulation model of the aforementioned equations in Simulink environment.

Fig. 6. Key steps in frequency response system identification.

Fig. 7. Frequency progression of a) An exponential function and b) The relations 11 and 12.

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Fig. 8. The computer generated frequency-sweep input for rudder.

Fig. 9. The yaw angle over time in horizontal plane.

Fig. 10. The pitch angle over time in vertical plane.

Fig. 11. Identification result for the yaw angle frequency-response.

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Fig. 12. Identification result for the pitch angle frequency-response.

produces no forces and moments due to surface movements.

Table 2 Hydrodynamic coefficients of the SUT-2 underwater vehicle.

4. Frequency domain identification

Parameter

Value

Parameter

Value

Parameter

Value

m

35.761 (Kg)

Yr

0.0107

Izz

Yv̇ xG Yr ̇ Nv̇ Nr ̇

−0.025 0 −0.0024 0.0024 −0.0014

Nv Nr Yδr Nδr Ixx

zG zB Mq̇ Mq

Yv

0.0107

Iyy

0.0085 0.0024 0.0316 −0.0122 0.1821 (Kg m2) 1.821 (Kg m2)

0.1821 (Kg m2) 0.015 (m) 0 −0.0014 −0.0024

Dynamics identification in the frequency domain is to extract a set of linear equations based on input-output observations. In this method, it is common to use Bode diagrams in order to check the compatibility of the real data with the predicted model. After collecting the maneuver data, spectral analysis is carried out to identify the frequency components. Fig. 6 shows the detailed flow chart to clarify the key steps followed in frequency response system identification (Tischler and Remple, 2006). Accordingly, the identification process consists of four major steps including generating the frequency sweep as input, data collection, consistency check and conditioning, spectral analysis and extracting the transfer functions or state space model. Optimal input design requires a knowledge of the structure of the model and its dynamic characteristics. The intention to obtain an optimal input is to acquire the maximum relevant information out of a given system in the shortest time. Design of such inputs for unconventional vehicles is very difficult. One of the best optimal input in frequency domain system identification is a sweep (Young and Patton, 1990). This input provides a fairly uniform spectral excitation over the frequency range of interest that is robust to uncertainties without the necessity of a detailed a priori knowledge of the system dynamics. Here, the minimum and maximum frequencies are defined by the frequency range associated with a given dynamics varying slow enough to guarantee the performance and operability in terms of safety and fast enough to minimize the required test time. Toward a perfect identification, the relation (8) should be considered between the maximum operating frequency of the system (ωmax ), filtering frequency (ωf ), and data sampling frequency (ωS ) (Tischler and Remple, 2006).

Fig. 13. a) CMPS10 tilt compensated compass. b) EP50S series shaft encoder to collect the experimental data. −1 ⎡ m − Xu ̇ mzG 0 0 0 0 ⎤ ⎡⎢ ∑ X ⎤⎥ ⎡ u̇ ⎤ ⎢ ⎥ m − Yv ̇ 0 − mzG 0 − Yr ̇ ⎥ ⎢ ∑ Y ⎥ ⎢ v̇⎥ ⎢ 0 ⎢ ẇ ⎥ ⎢ 0 m − Z ẇ 0 0 Zq̇ 0 ⎥ ⎢∑ Z ⎥ ⎢ ⎥ ⎢ ⎥=⎢ p ̇ Ixx − K p ̇ − mzG 0 0 0 ⎥⎥ ⎢ ∑ K ⎥ ⎢ ⎥ ⎢ 0 ⎢ q ̇ ⎥ ⎢ mzG Iyy − Mq ̇ 0 − M ẇ 0 0 ⎥ ⎢∑ M ⎥ ⎢⎣ r ̇ ⎥⎦ ⎢ ⎥ ⎥ ⎢ I Nr ̇⎦ ⎢⎣ ∑ N ⎥⎦ 0 − N 0 0 0 − ⎣ v̇ zz

ωs ≥ 5ωf ≥ 5ωmax

(8)

In this study, the system excitation frequency range lies between 0.2 and 30 rad/s, which is obtained by simulation studies. The following equations are used to produce the desired frequency sweep:

(7)

δSweep = A sin (θ (t ))

Based on the aforementioned equations, as shown in Fig. 5, a simulation model is developed in Simulink environment and the dynamic performance of the subsurface is studied. It is also assumed that the underwater vehicle is submerged in the water to a depth that

θ (t ) =

∫0

t

ω (τ ) dτ

ω (t ) = ωmin + K (ωmax − ωmin ) 158

(9) (10) (11)

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Fig. 14. a) Frequency sweep input to the servo in horizontal plane motion. b) Corresponding angles, recorded by the encoder for identification.

Fig. 15. a) Frequency sweep input to the servo in vertical plane motion. b) Corresponding angles, recorded by the encoder for identification.

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Fig. 16. a) Frequency response identification of the rudder servo system in horizontal and b) Vertical planes motion.

⎡ ⎤ ⎛ c t ⎞ K = c1 ⎢exp ⎜ 2 ⎟ − 1⎥ ⎝ TRecord ⎠ ⎣ ⎦

noise is added to the sweep. The excitation signal is shown in Fig. 8. (12)

δExcitation = δSweep + δNoise

Where, A is the amplitude of the input signal and TRecord represents the duration of stimulation. In the process of the optimal input design, the simple exponential progression function of the frequency is a common way but incorrect according to Fig. 7a. It increases frequency sharply, which is dangerous for the system and is not applicable in the applications. While, as illustrated in Fig. 7b, Eqs. (11) and (12) propose a progression function of the frequency, which is more convenient and desirable. In the relation 12, c1 and c2 are calculated 0.0187 and 4 to produce ωmin and ωmax at t = 0 and t = TRecord . The signal amplitude is steadily increased in 60 s to prevent from sudden increase in the yaw response and to keep it symmetric around the trim point. The final amplitude is chosen to be 0.087 rad (5 degrees). Also, to attain a rich spectral content, a band-limited white-

(13)

The frequency response analysis is performed by applying the fast Fourier transform to the time history data, to decompose input and output signals into a series of frequencies, magnitudes and phases. Figs. 9 and 10 show the corresponding responses after applying the ideal frequency sweep input to the rudders. Considering the subsurface vehicle as an input-output system and relating the control inputs to the vehicle dynamic response by data processing and spectral analysis in CIFER®, the system identification results are presented in Figs. 11 and 12. As seen in these figures, the pitch and yaw responses are more accurate in the frequency range 0.4–24 rad/s and less accurate at higher frequencies according to the coherence function (more than 0.6). It is notable that, the order of the transfer functions are initially 160

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Fig. 17. a) Multi-step input for validation of the rudder servo system in horizontal and b) Vertical planes.

Fig. 18. a) Output deflection angles for the identified model and the real servo in horizontal and b) Vertical plane motions.

guesstimated based on what is found from the linearized model in Section 5 (Fossen, 1994) and then it is finalized according to the optimized cost function in CIFER®. The pitch and yaw transfer functions are obtained as follows:

θ −0.8417 = 2 δs s + 0.8978s + 0.9615

(14)

ψ −0.7345s − 0.3341 = 3 δs s + 1.304s 2 + 0.7533s

(15)

These transfer functions are validated in Section 6.

Fig. 19. Test of the SUT-2 underwater vehicle at the university swimming pool.

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Fig. 20. a) Doublet input to the rudder to excite the yaw and b) Pitch angle.

Fig. 21. a) Yaw-response comparison of the experimental data with the identified, linear and nonlinear models. b) Pitch-response comparison of the experimental data with the identified, linear and nonlinear models.

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Fig. 22. General block diagram of a complete feedback loop with Proportional Integral (PI) controller.

⎡ (Mq − mxG ) u 0 − BGZ W 0 ⎤ ⎡ q ⎤ ⎡ M ⎤ ⎡ Iyy − Mq ̇ 0 0 ⎤ ⎡ q ⎤̇ δ ⎥ ⎢ ⎥ ⎢ ⎥̇ = ⎢ ⎢ ⎥ 2 1 0 0 ⎥ ⎢ θ ⎥ + ⎢ 0 ⎥ u 0 δs 0 1 0⎥ ⎢θ ⎥ ⎢ ⎢ ⎢ ⎥ ⎣ 0 − u0 0 ⎥⎦ ⎣ z ⎦ ⎣ 0 ⎦ 0 0 1 ⎦ ⎣ z ̇⎦ ⎢⎣

Table 3 Dynamics of the proposed controllers. Pitch controller

Yaw controller

(20) Gc =

0.27 s

Gc =

1.045s + 0.013 s

where, BGZ is zG − zB , which denotes the vertical distance between the center of buoyancy and the center of gravity. The transfer function for the pitch motion is found to be:

Table 4 Transient response characteristics for the designed controllers. Parameter

Pitch controller

Rise time (s) Overshoot Peak time (s) Settling time (s)

3.59 0.26% 14 11.9

θ (s ) −0.9346 = 2 δs (s ) s + 0.4641s + 0.9152

Yaw controller 1.82 8.71% 3.83 22.6

6. Experimental setup The purpose of test setup is to compare the nonlinear simulation model, the identified linear model in frequency domain, and the linearized analytical model and to validate the extracted transfer functions. It is then necessary to study the effects of rudder deflections on the yaw and pitch angles by recording the output of each sensor. Using CMPS10 compass, it is possible to measure the yaw and pitch angles at the center of gravity with the measurement accuracy of 0.1 degree for yaw and 1 degree for pitch angles (Fig. 13). Also, it is possible to apply the control inputs to the control surfaces by a JR XG8 hand-held radio transmitter that provides a 2.4 GHz radio link with 8 output channels. As shown in Fig. 13, a shaft encoder with a resolution of 360 divisions (1 degree) is also used to measure and identify the servo actuator angular position. Due to the space limitation inside the body, it is not possible to directly connect the encoder to the shaft that is connected to the rudder. Therefore, pre-identification of the servo actuator dynamics and the mechanical transmission mechanism is carried out using the same method of identification. This is also called control rigging calibration. To do this, the encoder is connected to the shaft servo by a mechanical coupling and the frequency sweep input is applied to the servo by a radio controller and the angle is recorded. As mentioned, the SUT-2 has four rudders and three servos. Two rudders, which operate in on-axis horizontal motion are connected to one servo by a mechanical coupling and the other rudders are connected by electrical coupling. Figs. 14 and 15 show the inputoutput data, recorded by the receiver and encoder in horizontal and vertical planes. Using CIFER® and performing spectral analysis of the experimental data sets, it is evident from Fig. 16 that the results provide remarkable coherence over the frequency range of 0.4–5. Hence, the equivalent

5. Linearizing the equations of motion By eliminating the nonlinear terms and neglecting the heave, sway, roll, and pitch effects, the equations will be reduced to the following simple form:

m (v ̇ + ur + xG r )̇ = Yr ̇ r ̇ + Yv ̇ v ̇ + Yr ur + Yv uv + Yδr u2δr

(16)

Iz r ̇ + mxG v ̇ + mxG ur = Nr ̇ r ̇ + Nv ̇ v ̇ + Nr ur + Nv uv + Nδr u2δr

(17)

Assuming a constant surge velocity ( u 0 ), these equations are reorganized as: ⎡ m − Yv ̇ mxG − Yr ̇ 0 ⎤ ⎡ v ̇ ⎤ ⎡ Yv u 0 (Yr − m ) u 0 0 ⎤ ⎡ v ⎤ ⎡ Yδr ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥ r ⎢ ⎥ 2 ⎢ mxG − Nv ̇ Izz − Nr ̇ 0 ⎥ ⎢ r ̇ ⎥ = ⎢ Nv u 0 (Nr − mxG ) u 0 0 ⎥ ⎢⎣ ψ ⎥⎦ + ⎢ Nδr ⎥ u 0 δr ψ ̇ ⎣ ⎦ ⎣ 0 ⎦ ⎣ 0 ⎣ 0 1⎦ 0 0 1⎦

(18) By adjusting the engine speed to 600 rpm, the subsurface will achieve a velocity of up to 0.5594 m/s. Substituting this velocity into the above equations and using the coefficients presented in Table 2, the transfer function of yaw motion is obtained as follows:

ψ (s ) −0.7311s − 0.0000848 = 3 δ r (s ) s + 0.8734s 2 + 0.2874s

(21)

(19)

Likewise, by neglecting the yaw, heave, sway and roll effects and fixing the surge velocity (Fossen, 1994), the equations for the movement in the vertical plane are found to be:

Fig. 23. Subsurface trajectory in presence of the PI and Integral controllers.

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Fig. 24. a) Control commands generated by the PI and b) Integral Controllers.

Fig. 25. The applied measurement noise for the yaw angle.

Fig. 26. Rudder deflections in the presence of noise for yaw angle.

Fig. 27. Subsurface planar trajectory in the presence of noise.

As shown in Fig. 17, two multi-step inputs, which resemble zig-zag maneuvers, are applied to the servo motors by the radio controller to validate the identified transfer functions by comparison with the experimental data. Accordingly, corresponding output deflections for both the identified models and the real servo motors are presented in Fig. 18. The results are in good agreement, with less than 10% percent error. After ensuring the proper rigging identification and correct data recording, for comparison of the identified underwater vehicle model

transfer functions are derived in this frequency range as follows:

Rudderdeflectionangle 0.00009s − 0.0047 = Inputcommand s + 3.345

(22)

Rudderdeflectionangle 0.001s − 0.01 = Inputcommand s + 14.06

(23)

It is notable these dynamics represent the control rigging dynamics for the rudders in the horizontal and vertical planes, respectively. 164

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Fig. 28. The subsurface yaw angle in the presence of noise.

Fig. 29. Instrumentation, microcontroller and servo motor.

Fig. 30. Subsurface undergoes a test at the pool.

with the linearized model and the nonlinear simulation model, a specific control command is applied to the vehicle. As seen in Fig. 19, this test is performed at the swimming pool with a known geographic position (305 degrees with respect to the North). As illustrated in Fig. 20, doublet inputs are applied to the vehicle control surfaces to excite the dynamics in vertical and horizontal planes in a zig-zag maneuver. The pitch and yaw angles are then measured by the compass. To obtain the absolute magnitude of the yaw angle, the geographic angle with respect to the North is deducted from the recorded data. After applying the aforementioned inputs to the nonlinear simulation model, the identified model as well as the linearized model and

Fig. 31. Subsurface trajectory in the XY plane.

processing the raw data, the validation of underwater dynamic system is carried out. The results are compared as shown in Fig. 21. Fig. 21 indicates that the experimental data and the identified ones are in good agreement with an error of less than 12% in horizontal plane motion. Also, it shows that the identified model for pitch 165

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response is in a very good agreement with the experimental results with an error lower than 9% in the entire domain. The key point to notice in both figures is that the greatest difference between the identified model and experimental data occurs at the beginning of the test. This is partly due to the assumption of a constant velocity (600 rpm and 0.55 m/s) for the identification that is not perfectly followed by the underwater vehicle. Nonetheless, as is evident from Fig. 21, this difference decreases over time after stabilization of the subsurface velocity. Although the results show that the equivalent linear identified models give the closest match, the nonlinear simulation model shows a relative reasonable accuracy as well. This arises partly because of the accurate hydrodynamic coefficients that have been determined by experiment. It is also obvious from these results that there is immense difference between the experimental and the linearized model due to the nonlinear terms and coupling factors in the equations of motion.

Table 5 The test initial conditions. Test number

(X0,Y0)

Yaw angle to desired point (deg)

1 2 3

(4,1.5) (6.5, 1) (2,0.5)

17 47 68

Brng angle (deg)

Yaw angle to

−80 −89 −71

−63 −42 −3

the North (deg)

7. Controller design Control design for unmanned underwater vehicles is not a new concept, which is widely discussed in literature using different approaches. Craven et al. (2000), provided a comprehensive review of modern control approaches and artificial intelligence techniques which have been applied to the autopilot design problem for unmanned underwater vehicles. Herman (2009), proposed a PD set-point controller for autonomous underwater vehicles (AUVs). The controller was expressed in transformed equations of motion with a diagonal inertia matrix. Xianbo et al. (2013), adapted a classic PID controller to simplify the path following control design by integrating the 3D guidance law for individual AUVs. Sarhadi et al. (2016), proposed an adaptive control for an autonomous underwater vehicle. Due to industrial and academic interests, the proposed method was embedded with a Proportional–Derivative–Integral (PID) controller. Accordingly, it is generally believed that PID controllers are applicable to most of the underwater vehicles and sufficient for many control problems. Congruently, PI controllers are the most popular applications of PID controllers that can properly satisfy many control requirements when the response time is not an issue and disturbances and noise are present in the process. Fig. 22, presents a general block diagram of a complete feedback loop with Proportional Integral (PI) controller.

Fig. 32. Subsurface trajectory with initial condition of 47 degrees to the desired waypoint.

⎛ KI ⎞ ⎟ Gc = K p ⎜1 + K pTi ⎠ ⎝

(24)

It is notable that some physical characteristics of the servo actuators, such as rotational saturation ( ± 14.3 degrees for both pitch and yaw motions) and applicable frequency range are considered in the controller design process. As well, the system responses are compared on the basis of transient behavior criteria, including overshoot (less than 10%) and rise time (less than 5 s). The dynamic models of the servos for yaw and pitch are considered and included in the dynamic model of the system. As shown, since the transfer function of pitch angle does not possess an integrator and the controller input is constant for pitch tracking, an integral controller is designed. Finally, regarding the order and type of the identified transfer functions following the aforementioned criteria, PI and Integral controllers are designed for the underwater vehicle as presented in Table 3. As seen in Table 4, the transient response characteristics, including rise time, overshoot, peak time, and settling time are remained in the desired ranges of interest. The proposed controllers are first verified against the nonlinear simulation model. Since the objective of the controller is to drive the subsurface to a desired point in the vertical plane, the difference between the subsurface yaw angle and the angle to the desired position is applied to the controller as an input. The desired position error is chosen to be less than 2%, so that, the engine shut down command is

Fig. 33. Subsurface trajectory with initial condition of 68 degrees to the desired waypoint.

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Fig. 34. Yaw angle with respect to the North and Brng angle corresponding to Fig. 33.

Fig. 35. Rudder deflections during the tracking maneuver.

its path. To fix this, as the subsurface closes to the target (about 2 m), the engines will shut down by the controller. The first test tracking behavior is illustrated in Fig. 31. It shows the subsurface trajectory in the XY plane from point (4, 1.5) to (0, 14.1). Table 5 shows the initial conditions of tests. In the first test, the subsurface departed at the angle of −63 degrees to the North. After adjusting the path towards the desired waypoint and getting close to the error range, the subsurface motor is turned off and the controller stops commanding. Fig. 31 shows that in the presence of the designed controller, the system tracks the planned path toward the desire point and shows a reasonable difference with the nonlinear simulation. Since the controller is designed based on the equivalent linear model derived from frequency response identification, as the rudder deflections or the yaw angle increases, the validity of the linear model could be questioned due to existence of the nonlinearities. In the second test, the initial angle between the subsurface and the desired waypoint is 47 degrees, which is fairly large and causes a slight difference between the simulation model and the experimental results. Besides, the GPS error may cause some differences between the physical and the simulation models. Fig. 32 shows the trajectory of the subsurface departing from point (6.5, 1) toward point (0, 14.1). Here, the subsurface is released with the yaw angle of −42 degrees to the North and 47 degrees relative to the desired waypoint. This initial condition causes a moderate trajectory deviation between the physical model and the simulation model. In the third test, the subsurface is released with a yaw angle of −3 degrees to the North and 68 degrees relative to the desired waypoint that causes a large trajectory deviation, as shown in Fig. 33. Fig. 34 shows the yaw angle of the subsurface to the North, the angle between the subsurface and the desired waypoint that is named Brng (the angle of an imaginary line that is connected from the centerline of the subsurface to the desired waypoint with respect to the North, as shown in Fig. 32) as well as the difference between these two angles. These angles vary as the AUV is tracking the waypoint. The time history of these angles for the subsurface motion is shown in Fig. 34. At first, the yaw angle of the subsurface to the North and the Brng angle are around −3 and −71 degrees, respectively. Then, these angles gradually get closer and coincide at the angle of −153 degrees. This angle is where the defined error range is achieved.

added to the simulation. Fig. 23 shows the 3D motion path of the AUV from the origin to the desired point at (300, 50, −50) m. The control command (rudder deflections) for the horizontal and vertical planes motion are shown in Fig. 24. As seen in this figure, the effectiveness of the pitch control at the beginning of the maneuver is limited by the rudder deflection saturation. As the subsurface reaches the desired position, the rudder returns to its initial value. Moreover, the performance of the controller has been examined in the presence of uncorrelated noise source in the measurement with zero mean, the power ratio of 10% of the maximum signal amplitude and the cut of frequency of 12 rad/s. Fig. 25 shows the noise on the yaw angle during the movement to the desired position. The simulation results of the controller, which are shown in Figs. 26– 28. Fig. 26, show that the rudder deflections in the presence of noise approaches to their expected values in less than 20 s Figs. 27 and 28 show the trajectory of the subsurface and the corresponding yaw angle in the horizontal plane, with and without the existence of noise. It is evident that despite the existence of noise, the new trajectory is almost identical to the initial trajectory. Further investigations reveal that the controller is robust until the noise power ratio reaches 60%. 8. Validation of controller As shown in Fig. 29, the subsurface position, the distance to the target point and the deviation angle, are received from the GPS. They are then compared with the subsurface attitude in the horizontal plane of motion that is measured by the compass. The difference of these angles enters to the microcontroller that computes the control commands according to the designed controllers. This computational cycle, allows the subsurface to track the correct path and reach the target waypoint. It should be noted that as the GPS is not available under the water, it is mounted at the top of a one-meter antenna, which is installed over the propeller at the bottom of the subsurface. The device is tested in a swimming pool with the length of 14.5 and width of 10 m, as shown in Fig. 30. The main goal of this test is to evaluate the performance of the designed controllers tracking performance towards a desire point. However, the GPS provides the longitudes and latitudes with the accuracy of about 5 m, which causes the subsurface to deviate from 167

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Lambert, J.D., 1973. Computational Methods in Ordinary Differential Equations. Wiley, (ISBN: 978-0-471-51194-6). Mahfouz, A.B., Haddara, M.R., 2003. Effect of the damping and excitation on the identification of the hydrodynamic parameters for an underwater robotic vehicle. Ocean Eng. 30 (8), 1005–1025. Marchand, M., Fu, K.H., 1985. Frequency domain parameter estimation of aeronautical systems without and with time delay. In: Proceedings of the 7th IFAC Symposium on Identification and System Parameter Estimation, York, UK, pp. 669–674. Myring, D.F., 1976. A theoretical study of body drag in subcritical axisymmetric flow. Aeronaut. Q. 27 (3), 186–194. Naeem, W., Sutton, R., Chudley, J., 2003. System Identification, Modelling and control of an autonomous underwater vehicle. In: Maneuvering and Control of Marine Craft 2003 (MCMC 2003): A Proceedings Volume from the 6th IFAC Conference, Girona, Spain, pp. 19–25. Nikusokhan, M., Banazadeh, A., 2014. Frequency domain identification of friction and Backlash for Servomechanisms. Sharif Mech. Eng. J. 1 (3), 31–40. Perez, T., Fossen, T.I., 2011. Practical aspects of frequency-domain identification of dynamic models of marine structures from hydrodynamic data. Ocean Eng. 38 (2), 426–435. Prestero, T.T.J., 2001. Verification of a Six-Degree of Freedom Simulation Model for the REMUS Autonomous Underwater Vehicle (M.Sc. thesis). Ocean Engineering, Massachusetts Institute of Technology Massachusetts. Rahimian, M., Seif, M.S., Sayadi, H., 2009. Dynamic Simulation of Underwater Vehicles (M.Sc. thesis). Sharif University of Technology, Tehran. Rhee, K.P., Yoon, H.K., Sung, Y.J., Kim, S.H. and Kang, J.N., 2000. An experimental study on hydrodynamic coefficients of submerged body using planar motion mechanism and coning motion device. In: International Workshop on Ship Maneuverability, pp. 1-20. Sadeghzadeh, B., Mehdigholi, H., Seif, M.S., 2009. Numerical Simulation of Flow around Autonomous Underwater Vehicle Body and Hydrodynamic Forces Analysis (M.Sc. thesis). Sharif University of Technology, Tehran. Sarhadi, P., Ranjbar Noei, A., Khosravi, A., 2016. Model reference adaptive PID control with anti-windup compensator for an autonomous underwater vehicle. Robot. Auton. Syst. 83, 87–93. Sarkar, T., Sayer, P.G., Fraser, S.M., 1977. A study of autonomous underwater vehicle hull forms using computational fluid dynamics. Int. J. Numer. Methods Fluids 25 (11), 1301–1313. Selvam, R.P., Bhattacharyya, S.K., Haddara, M., 2005. A frequency domain system identification method for linear ship maneuvering. Int. Shipbuild. Progress. 52 (1), 5–27. Shafiei, M.H., Binazadeh, T., 2015. Application of neural network and genetic algorithm in identification of a model of a variable mass underwater vehicle. Ocean Eng. 96, 173–180. Shi, C., Zhao, D., Peng, J., Shen, C., 2009. Identification of ship maneuvering model using extended Kalman filters. Int. J. Mar. Navig. Saf. Sea Transp. 3 (1), 105–110. Silva, J., Terra, B., Martins, R., Sousa, J., 2007. Modeling and simulation of the LAUV autonomous underwater vehicle. In: Proceedings of the 13th IFAC Conference on Methods and Models in Automation and Robotics, 713–718. Tiano, A., Sutton, R., Lozowicki, A., Naeem, W., 2007. Observer Kalman filter identification of an autonomous underwater vehicle. Control Eng. Pract. 15 (6), 727–739. Tischler, M.B., Remple, R.K., 2006. Aircraft and rotorcraft system identification: engineering methods with flight test example. Am. Inst. Aeronaut. Astronaut., (Chapters 1, 2, 4 and 6.). Toxopeus, S.L., 2009. Deriving mathematical manoeuvring models for bare ship hulls using viscous flow calculations. J. Mar. Sci. Technol. 14 (1), 30–38. Xianbo, X., Dong, C., Caoyang, Yu., Lei, M., 2013. Coordinated 3D path following for autonomous underwater vehicles via classic PID controller. In: IFAC Proceedings Volumes 46(20), 327–332. Xu, F., Zou, Z.J., Yin, J.C., Cao, J., 2013. Identification modeling of underwater vehicles' nonlinear dynamics based on support vector machines. Ocean Eng. 67, 68–76. Young, P., Patton, R.J., 1990. Comparison of test signals for aircraft frequency domain identification. J. Guid. Control Dyn. 13, 430–438. Yuh, J., 1990. Modeling and control of underwater robotic vehicles. IEEE Trans. Syst. Man Cybern. 20 (6), 1475–1483. Zhang, X.G., Zou, Z.J., 2011. Identification of Abkowitz model for ship manoeuvring motion using ɛ-support vector regression. J. Hydrodyn. 23 (3), 353–360.

Fig. 35 shows the rudder deflections that reaches the maximum value of −14.3 degrees, at the beginning of the test. It is notable that, as the difference between the Brng and yaw angles decreases and the rudder deflection reduces, as well. After three different test scenarios, it is clear that by increasing the initial reference angle of the subsurface with respect to the desired waypoint causes more deviation between the outputs of the physical and simulation models. As mentioned before, the main reason is that the controller is designed based on the equivalent linear model, which is optimal for small angles. It should be mentioned that the difference between the physics and simulation does not lead to the target waypoint miss distance. However, the large initial angles make the subsurface to follow a non-optimal trajectory, slower tracking and more penalty on the control effort and travelled distance. 9. Conclusion and closing remarks The main goal of this paper was to proof the applicability of frequency domain identification to extract the transfer functions of an AUV in horizontal and vertical planes motion with the purpose of tracking control design. The frequency range of applicability is found to be 0.4–24 rad/s according to the coherence values. The extracted linear models are verified and showed less than 12% error in comparison with the nonlinear models. Experimental results proved that the identified transfer functions are more accurate than those obtained by the conventional linearization methods. Dynamic models and physical characteristics of the actuators and saturation limits are also extracted and considered in the controller design process and simulation. In addition, the controller performance is validated by the experimental data, for different initial conditions. The results show that the proposed controller is satisfactory for tracking purposes. References Avila, J.P., Donha, D.C., Adamowski, J.C., 2013. Experimental model identification of open-frame underwater vehicles. Ocean Eng. 60, 81–94. Banazadeh, A., Ghorbani, M.T., 2013. Frequency domain identification of the Nomoto model to facilitate Kalman filter estimation and PID heading control of a patrol vessel. Ocean Eng. 72, 344–355. Beverley, C., 2009. Assigning Closely Spaced Targets to Multiple Autonomous Underwater Vehicles (Ph.D. thesis). Mechanical Engineering, University of Waterloo, Waterloo, Ontario, Canada. Bhattacharyya, S.K., Haddara, M.R., 2006. A frequency domain parametric identification for nonlinear ship maneuvering. J. Ship Res. 50 (3), 197–207. Craven, P.J., Sutton, R., Burns, R.S., 2000. Control strategies for unmanned underwater vehicles. J. Navig. 51 (1), 79–105. De Barros, E.A., Pascoal, A., De SA, E., 2008. Investigation of a method for predicting AUV derivatives. Ocean Eng. 35 (16), 1627–1636. Fossen, T.I., 1994. Guidance and Control of Ocean Vehicles. John Wiley & Sons Ltd., England, (Chapters 2 and 4, ISBN: 978-0-471-94113-2). Herman, P., 2009. Decoupled PD set-point controller for underwater vehicles. Ocean Eng. 36 (6), 529–534. Jun, B.H., Park, J.Y., Lee, F.Y., Lee, P.M., Lee, C.M., Kim, K., Lim, Y.K., Oh, J.H., 2009. Development of the AUV ‘ISiMI’ and a free running test in an Ocean Engineering Basin. Ocean Eng. 36 (1), 2–14. Källström, C.G., Åström, K.J., 1981. Experiences of system identification applied to ship steering. Automatica 17 (1), 187–198.

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