Heading tracking control with an adaptive hybrid control for under actuated underwater glider

ISA Transactions xxx (xxxx) xxx–xxx

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Research article

Heading tracking control with an adaptive hybrid control for under actuated underwater glider Hongqiang Sanga, Ying Zhoua, Xiujun Sunb,c,∗, Shaoqiong Yangc a

School of Mechanical Engineering and Tianjin Key Laboratory of Advanced Mechatronic Equipment Technology, Tianjin Polytechnic University, Tianjin, 300387, China Physical Oceanography Laboratory, Ocean University of China, Qingdao, 266100, China c Qingdao National Laboratory for Marine Science and Technology, Qingdao, 266237, China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Underwater glider Heading tracking control Dynamic model Adaptive fuzzy incremental PID Anti-windup compensator

The underwater glider changes its direction to follow the preset path in the horizontal plane only by flapping its vertical rudder. Heading tracking control plays the core role in the navigation process. To deal with non-linear flow disturbance and saturation in actuator, a new hybrid heading tracking control algorithm was presented, which integrated an adaptive fuzzy incremental PID (AFIPID) and an anti-windup (AW) compensator to improve the adaptability and robustness of underwater glider's heading control. The dynamic model of an underwater glider named as Petrel-II 200 was modeled to serve as a controlled plant. The proposed heading tracking control algorithm was described in detail, where the rudder angle, a control quantum to the controlled plant were calculated to get forces and moments required for the desired glider heading. A closed loop motion control system with desired heading angle as input and actual heading angle output was put forward, which included the dynamic model of the Petrel-II 200 and the given heading tracking control algorithm. The simulations followed three typical mathematical signals and the experimental tests were carried out by taking in the dynamic parameters of the controlled plant. And the effectiveness of the proposed control algorithm was assessed and verified.

1. Introduction Autonomous underwater gliders (AUGs) are widely applied to perform various tedious and risky missions in military, scientific, civil as well as commercial areas such as oceanographic mapping, search for naval resources etc. [1]. However, the capabilities of long range time operation without supervision present a main challenge in the development of advanced AUGs, that the navigation control of the vehicle must be capable of safely and effectively guiding the AUG in dynamic and cluttered ocean environments [2]. Thus it is necessary to design a robust control strategy to perform precise navigation, which makes the AUG cruise on a planned path with pre-defined heading [3–5]. Heading tracking control plays the core role in the navigation process for under actuated underwater gliders. It reflects the possibility of a planned behavior during a mission using all present and future information about the area of operation [6]. Performing precise heading control of an AUG is a formidable task because of model nonlinearities, actuator saturations and time-varying disturbances [5]. Adaptive control has been much developed to deal with these nonlinearities in the past decade [7–15]. For model nonlinearities, various approaches have been proposed with adaptive controls, such as

∗

fuzzy control [7,8,15], neural network control [11] and adaptive backstepping control [10,13]. A hybrid control with PID and neural networks was used for an AUV to manage heading control [14]. This approach can find a robust solution when disturbances exist. An adaptive fuzzy PID control algorithm was presented to solve the uncertainties of PID parameters and the model of AUV [15]. Form the simulation results, it could be seen that the convergence time with a 20 amplitude step signal input using the proposed algorithm was 60 s, the overshoot was 7.05% and the undershoot was about 9.55%, which showed good performances in heading control. An adaptive sliding mode control based on a disturbance observer was designed for heading control of AUVs [5]. The nonsymmetrical dead-zone with unknown parameters and input saturation was considered in the adaptive heading control. Compared with traditional PID and sliding mode control, the adaptive sliding mode control has better performances in tracking step signal heading with much low errors. To dealing with input saturation problems, anti-windup design [16–20] has been developed. A hybrid control combined a model reference adaptive control and a modern anti-windup compensator (AW) was proposed to realize the heading control of an AUV in the presence of input saturations and uncertain dynamics [17]. A modern AW compensator was added to a model

Corresponding author. Physical Oceanography Laboratory, Ocean University of China, Qingdao, 266100, China. E-mail addresses: [email protected] (H. Sang), [email protected] (Y. Zhou), [email protected] (X. Sun), [email protected] (S. Yang).

https://doi.org/10.1016/j.isatra.2018.06.012 Received 25 May 2017; Received in revised form 28 March 2018; Accepted 29 June 2018 0019-0578/ © 2018 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Sang, H., ISA Transactions (2018), https://doi.org/10.1016/j.isatra.2018.06.012

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H. Sang et al.

Table 1 Relevant methods for underwater vehicle control. Method

Time domain performance

Main feature

Traditional PID [7,8]

Obvious overshoot and oscillation; long convergence time; introduced time delay Relatively small overshoot and declined oscillation

Simplicity; bad robustness

Conventional Fuzzy FNN [11]

Comparably short convergence time; improved resistance against disturbance No overshoot or oscillation No oscillation; still has overshoot phenomenon

AC [3]; DB [10] FASMC [9]; DCHM [12]; NNPID [14]; AFPID [15]; FPID [8]; VSPID + AW [19] VSPID [31]

Slight overshoot phenomenon

LQ + RBAW [20] APID + AW [17] ASMC [5] MRAC + AW [18]

Slight steady-state error with rectangle input Overshoot was reduced but was not eliminated Overcome the overshoot phenomenon Slight overshoot phenomenon

AFIPID + AW

No overshoot and desirable heading tracking performance

reference adaptive PID controller for underwater vehicles, various simulations have been carried out in a nonlinear six degrees of freedom model [18]. A comparison of multivariable saturating control and AW control was also addressed in Ref. [16]. Table 1 summarizes underwater vehicle controls. Although many adaptive controls dealing with nonlinearities and approaches using AW compensators have been successfully applied to underwater vehicles, heading tracking control performance with an AW compensator and the different types of desired heading tracking capabilities of AUGs are rare in existing literature. Besides, heading tracking capabilities including tracking a fixed heading and tracking a time-varying heading largely reflect the robustness and the adaptability of heading control algorithms. However, it is a well-known fact that disturbances caused by wind, wave and current are immeasurable and may depart the vehicle from a preset heading [14]. In some non-uniform flow situations, the influence of actuator saturation worked on underwater gliders is great. Thus, it is important and practical issue for heading tracking control of underwater gliders to deal with actuator saturation along with uncertain disturbance. Combining an adaptive control with an AW compensator can be a convenient and practical way to deal with nonlinearities of AUGs. This paper puts forward a new model reference adaptive fuzzy incremental PID (AFIPID) control algorithm with an AW compensator. The proposed control algorithm is employed in heading tracking control of the PetrelII 200.

Good robustness; declined control precision with simple fuzzy processing Approximation accuracy for systematic nonlinearity was improved without considering environment disturbance Good adaptability and improved robustness Prevention of integrator windup; Big energy consumption due to frequent regulation of the actuator Restricted effect of input saturation AW was used to settle actuator saturation Big heading error was not considered Actuator saturation was considered; only be validated by simulation No limitations (at current research)

Fig. 1. The coordinate frame assignment of the Petrel-II 200.

The body coordinate frame O−xyz and the geodetic coordinate frame E−ηξς (i.e. the inertial frame) of the Petrel-II 200 are assigned, which is shown in Fig. 1. Let O = (ξO, ηO , ςO )Τ be a position vector of the glider from the origin of the body coordinate frame to the origin of the geodetic coordinate frame. And assume Ω = (φ , θ , ψ)Τ is an attitude vector of the glider in the geodetic coordinate frame, which is used for transformation from the body coordinate frame to the inertial frame. Besides, O and Ω can also be used to describe the position and the attitude of the glider, which responds the six degrees of freedom (DOFs) in the following dynamic model of the Petrel-II 200. υ = (u, v, w )Τ and ω = (p , q, r )Τ are a transitional velocity vector and an angular velocity vector in the body coordinate frame, respectively. F = (X , Y , Z )Τ and T = (K , M , N )Τ are a transitional external force vector of the vehiclefluid system and a total moment vector of the glider in the body coordinate frame, respectively. The attitude transformation matrix R OE of the glider can be defined as:

2. Dynamics of the Petrel-II 200 To better appreciate the effect of the heading tracking control method and imitate the real glider in an unsteady, non-uniform flow field, the full dynamic model of the Petrel-II 200 was developed by combining rigid body dynamic equation with Hydrodynamic equation [21–25]. The Petrel-II 200, a kind of under actuated AUGs, was modeled as a rigid body with fixed wings and a tail immersed in a fluid with buoyancy control and controlled internal moving mass.

x y z ⎛ Oξ Oξ Oξ ⎞ R OE = (x OEyOE zOE) = ⎜ x Oη yOη z Oη ⎟= ⎜ x Oς y ⎟ Oς z Oς ⎠ ⎝ cos ψ cos θ sin φ cos θ sin φ sin ψ + cos ψ sin θ cos φ ⎞ ⎛ ⎜−sin ψ cos φ + cos ψ sin φ sin θ cos ψ cos φ + sin ψ sin θ sin φ −sin ψ sin θ cos φ + cos ψ sin φ ⎟ ⎜ sin ψ sin φ + cos ψ sin θ cos φ cos ψ sin φ + sin ψ sin θ cos φ ⎟ cos θ cos φ ⎝ ⎠

2

(1)

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The dynamics of the Petrel-II 200 can be expressed as: . b = R OEυ

(2)

. R OE = R OE ω

(3)

ˆ

and the velocity acted on the center of mass of the glider, respectively. Assumed that the position of the glider's center of mass is constant related to itself, thus the velocity vG and its derivative can be expressed as:

⎛ 0 −r q ⎞ 0 − p ⎟ represents a 3 × 3 skew-symmetric matrix. where ω = ⎜ r ⎜− q p 0 ⎟⎠ ⎝ According to the principle of coordinate transformation, the angular velocity ω in the body coordinate frame can be expressed as: . ω = ROΩ Ω (4)

⎧ v. G ≈ v. + ω × rG . v = v + ω × v + ω × rG + ω × (ω × rG) ⎨ ⎩ G

ˆ

where rG = (x G , yG , z G)Τ is the position vector of center of mass in the body coordinate frame. The sum of moments of external forces acted on the center of mass TG can be described as:

0 − sin θ ⎞ ⎛1 whereR OΩ = ⎜ 0 cos φ sin φ cos θ ⎟ is a rotation transformation ma⎜ 0 − sin φ cos θ cos φ ⎟ ⎠ ⎝ trix. The wind coordinate frame W − w1 w2 w3 is introduced to express hydrodynamic force and hydrodynamic moment of the glider. When the flow is static, the velocity of the glider coincides with the w1 axis of the wind coordinate frame. Therefore, the velocity υ of the glider can be expressed as:

υ = R OW υ w1

. . TG = Π = IG ωG = T + rG × F

(10)

where Π = IG ωG is the expression of π in the body coordinate frame, ⎛ IG xx − IG xy − IG xz ⎞ and IG = ⎜− IGyx IG yy − IG yz ⎟ is the moment of inertia matrix. ⎜ − IG − IG IGzz ⎟ zx zy ⎠ ⎝ According to the authors' previous research [21], transforming Equation (7) from the inertial coordinate frame to the body coordinate frame, and combined with Equations (8) and (10), the six DOFs motion of the Petrel-II 200 can be expressed as:

(5)

⎛ cos α sin β sin β sin α cos β ⎞ where R OW = ⎜− cos α sin β cos β − sin α cos β ⎟ is a transformation ⎜ ⎟ 0 cos α ⎝ − sin α ⎠ matrix from the wind coordinate frame to the body coordinate frame. The total stationary mass or total body mass m can be defined as:

m= ms + m m = mW + mh + mb + m p + mr

(9)

. . . 2 2 ⎧ X = m(u + z G q − yG r − vr + wq − x G (q + r ) + yG pq + z G pr ) . . . ⎪Y = m(v − z G p + x G r − wp + ur − y (p2 + r 2) + z G rq + x G pq) G ⎪ . . . ⎪ Z = m(w + yG p − x G q − uq + vp − z G (q2 + p2 ) + x G pr + yG qr ) ⎪ K = I p. + (I − I ) qr + m[y (w. + vp − uq) − z (v. + ur − wp)] x z y G G ⎪ . . + myG z G (r 2 − q2) + myG x G (−r + pr ) + mx G z G (−r − pq) ⎨ . . . ⎪ M = Iy q + (Ix − Iz) pr + m[z G (u + wq − vr ) − x G (w + vp − uq)] ⎪+ mx z (p2 − r 2) + my z (−r. + pq) + my x (−p. − rq) G G G G G G ⎪ ⎪ N = Iz r. + (Iy − Ix ) pq + m[x G (v. + ur − wp) − yG (u. + wq − vr )] ⎪ . . 2 2 ⎩+ myG x G (q − p ) + mx G z G (−p + qr ) + myG z G (−q − pr )

(6)

where ms is the mass of relatively static rigid body including the mass of rigid ballast body mW , the mass of floating rigid body mh , and the mass of variable-mass rigid body mb . m m is the mass of relatively moving rigid body consisting of the mass of pitching rigid body m p and the mass of rolling rigid body mr . To simplify dynamic model, the glider is regarded as a 6 DOFs floating rigid body. Let p be linear momentum of the glider, π be angular momentum with respect to the origin of the inertial frame. The following equation can be derived by linear momentum theorem and angular momentum theorem as: . ⎧ p. = mgi3 + REOFwater ⎪ π = bG × mgi3 + b × REOFwater + REOT ⎪. ⎪ pP = m p gi3 + REOfp−h ⎪ π. p = bp × m p gi3 + bp × REOfp−h + REOτ p−h ⎪. ⎪ pr = mr gi3 + REOfr−h . ⎨ πr = br × mr gi3 + br × REOfr−h + REOτr−h ⎪. ⎪ pw = m w gi3 + REOfw−h ⎪ π. w = b w × m w gi3 + b w × REOfw−h + REOτ w−h ⎪. ⎪ p b = mb gi3 + REOfb−h ⎪ π. b = b b × mb gi3 + b b × REOfb−h + REOτ b−h (7) ⎩

(11) which models a submerged vehicle with general body, fixed wings, vertical rudder and thruster. And the first three equations in Equation (11) represent the translational motion, while the last three equations represent the rotational motion of the Petrel-II 200. To obtain more accurate dynamic model, viscous hydrodynamic force Fvisc including lift force FL , drag force FD and lateral force FS , and viscous hydrodynamic moment Tvisc including rotational moments Mroll , Mptich and Myaw related to three coordinate axes in the wind coordinate frame are considered in this paper, which can be expressed as:

⎛ Mroll ⎞ ⎛− FD ⎞ RWO·Fvisc = ⎜ FS ⎟, RWO·Tvisc = ⎜ Mptich ⎟ ⎟ ⎜ ⎜M ⎟ ⎝ − FL ⎠ ⎝ yaw ⎠

(12)

Combined with Equations (11) and (12) and references [21–27], the dynamic model with hydrodynamics of the Petrel-II 200 can be simplified as Equations (13) and (14). And its validation has been verified in our previous research [21].

where Fwater means the water action forces including hydrodynamic force and buoyancy exerted on the glider. The force f*−h and the moment τ *−h (∗=p,r,w,b) are a force and a moment exerted on the centers of mass of other bodies including m w , mb , m p , mr through the floating rigid body mh in the body coordinate frame, respectively. Let b∗ (∗=p,r,w,b) be the position vector about the center of mass of the internal rigid body m∗ (∗=p,r,w,b) , bG be the position vector about the center of gravity of the glider in the inertial frame. The external force FG acted on the center of mass of the glider is equal to the external force F acted on buoyant center, which can be described as: . . FG = F = P = mvG (8)

⎧ ξ˙O = u cos θ cos ψ + v (sin ψ sin φ + sin θ cos ψ cos φ) ⎪ + w (sin ψ cos φ + sin θ cos ψ sin φ) ⎪ ⎪ η˙ = u sin θ + v cos θ cos φ − w cos θ sin φ O ⎪. ⎪ ζO = −u cos θ sin ψ + v (cos ψ sin φ + sin θ sin φ cos φ) ⎨ + w (cos ψ cos φ − sin θ sin ψ sin φ) ⎪. ⎪ φ = p − (q cos φ − r sin φ)tan θ ⎪. q cos φ − r sin φ ⎪ψ = cos θ . ⎪ θ q φ + r cos φ sin = ⎩

where P and vG are the expression of p in the body coordinate frame 3

(13)

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. . . ⎧ (m+λ11) u + mz G q − myG r − mvr + mwq − mx G (q2 + r 2) + myG pq ⎪ ⎪ + mz G pr ⎪= − ΔB sin θ + G sin θ + B sin θ + X ′ ρV 2 SX ′ n B O ⎪ ⎪ (m+λ22) v. + mz G p. + (mx G + λ26) r. − mwp + mur − my (p2 + r 2) G ⎪ ⎪ + mz G rq + mx G pq ⎪ ⎪ = ΔB cos θ cos φ + G cos θ cos φ + B cos θ cos φ ⎪ 1 ′ ′ ′ 2 ⎪ + 2 ρVB S (Yα α + Y γ γ + Yδh δh ) ⎪ . . . ⎪ (m+λ33) w + myG p − (mx G + λ35) q − muq + mvp − mz G (p2 + q2) ⎪ ⎪ + mx G rp + myG rq ⎪ ⎪ = − ΔB cos θ sin φ−G cos θ cos φ−B cos θ cos φ ⎪ + 1 ρV 2 S (Z ′ β + Z ′ p + Z ′ q + Z ′ δ ) B β p q δv v 2 ⎪ ⎪ (Jx + λ 44) p. − Jxz r. − Jxy q. − my w. + Jzy (r 2 − q2) + (Jz − Jy) qr G ⎪ + myG (vp − uq) − mz G (ur − wp) + Jxy pr − Jxz pq ⎨ ⎪ = − Gz G cos θ cos φ − GyG cos θ sin φ + Kn′ ρn2D5 ⎪ 1 ′ ′ ′ ′ 2 ⎪ + 2 ρVB SL (Kβ β + K p p + K q q + K δv δ v ) ⎪ . . . . . ⎪ (Jy + λ 55) q − Jxy p − Jyz r − mz G u − (mx G + λ35) w + Jzx (p2 − r 2) ⎪ ⎪ + (Jx − Jz) pr ⎪+ mz G (wq − vr ) − mx G (vp − uq) + Jxy qr − Jyz pq ⎪ ⎪ = ΔB Ib cos θ sin φ + Gz G cos θ cos φ + Gx G cos θ sin φ ⎪ ′ ⎪ + 1 ρVB2 SL (Mβ′ β + Mp′ p + Mq′ q + Mδv δv) 2 ⎪ . . . ⎪ (Jz + λ 66) r − myG u + (mx G + λ26) v − Jzx p. − Jzy q. + Jxy (q2 − p2 ) ⎪ ⎪ + (Jy − Jx) pq ⎪ ⎪+ mx G (ur − wp) − myG (wq − vr ) + Jzx qr − Jyz rp ⎪ ⎪ = ΔB Ib cos θ cos φ + Gx G cos θ cos φ−GyG sin θ ⎪ + 1 ρV 2 SL (N ′ α + N ′ γ + N ′ δ ) B α γ δh h 2 ⎩ (14)

Table 3 Dynamic coefficients of the Petrel-II 200.

As previously mentioned, the proposed control method is implemented on a validated six DOFs model of the Petrel-II 200. Dynamic equations have nonlinear characteristic. Thus, the control system should be sufficiently adaptive to enable it to handle variations in the dynamics of the vehicle under different disturbances arising from

Description

λ11 λ22 λ33 λ 44 λ55 λ 66 λ26 λ35 Jpx

5.1 90.27 65.04 5.1 90.27 65.04 19.53 −22.06 0.0412

kg kg kg kg·m·s2 kg·m·s2 kg·m·s2 kg·m kg·m kg·m2

Additional mass

Jpy

0.0503

kg·m2

Jpz Jmx Jmy

0.0510

kg·m2

0.8993 65.4539

kg·m2 kg·m2

Jmz

65.0187

kg·m2

Parameter

Value

Description

Kp′

−0.2

10

0

Rolling resistance moment coefficient

Zα′

Kβ′

Zq′

2

Vertical resistance coefficient

′ K δv

0

Kr′

0

Mα′ ′ Mδh

−0.61

Mq′

−0.13

CF

0.18

CT

0.03

XO′

0.31

0.21

′ Zδh

−0.6

Y β′

3

Pitching resistance moment coefficient Propeller coefficient

′ Yδv Y p′

0.21

Nβ′

0.38

′ Nδv Np′

0.21

Forward resistance coefficient

Nr′

−0.14

Lateral resistance coefficient

0

0.089

Deflection resistance moment coefficient

Parameter

Value

Unit

Description

m mp Δm g ρ L D n Ib Ip

65 9.9 ± 0.41 9.8 1024 1.98 0.16 ± 15 0.74 0.37

kg kg

Total mass of the vehicle Pitching rigid body mass

kg m/s2 kg/m3 m m rad/s m m

δv

± 40

deg

Net buoyancy regulation mass Gravity constant Seawater density Total length of the vehicle Propeller diameter Propeller speed Buoyancy center in the body coordinate frame Pitching rigid body center in the body coordinate frame Vertical rudder angle range

Δδ v (k ) = Kp (Δψ (k ) − Δψ (k − 1)) + Ki Δψ (k ) + K d (Δψ (k ) − 2Δψ (k − 1) + Δψ (k − 2))

Table 2 Dynamic parameters of the Petrel-II 200. Unit

Description

internal and external sources [28]. The main task of the proposed control is to track the desired heading with high accuracy and rapid convergence. PID control is of interest to engineers due to its simplicity and its effective performance [29]. In particular, reliability is significant in autonomous control systems [30]. However, traditional PID cannot satisfy accurate control requirements of the Petrel-II 200 nonlinear control system. This study is considered as an extension of adaptive PID with an AW compensator, which will focus on the heading tracking control of the Petrel-II 200. The motion of the Petrel-II 200 is created by the forward driving system and heading control system. An incremental PID is employed to implement control command as follows:

3. Heading tracking control based on the AFIPID with an AW

Value

Value

Table 4 Non-dynamic parameters of the Petrel-II 200.

Tables 2–4 give the dynamic parameters, coefficients and non-dynamic parameters of the Petrel-II 200, respectively. The dynamic parameters and coefficients have been obtained by CFD simulation, empirical formulas and previous experience [21]. And the non-dynamic parameters have been obtained by physical measurement.

Parameter

Parameter

(15)

where Δδ v (k ) = δ v (k ) − δ v (k − 1) , Δψ (k ) = ψe (k ) − ψc (k ) . For nonlinearities, a real-time AFIPID algorithm based heading tracking control was proposed to dynamically adjust the vertical rudder angle of the Petrel-II 200, which is shown in Fig. 2. Actual heading angleψc and desired heading angleψe are used as inputs of the proposed control, the IPID control as forward path is used for the heading tracking control outputs. The adaptive fuzzy system as a feedback control adaptively adjusts the parameters of the IPID control by a fuzzy inference system (FIS) in real time. To deal with actuator saturation problem, an AW compensator was introduced into the heading tracking control. The control rules of the FIS can be obtained by experiments, which adaptively regulate parameters of the IPID algorithm. Based on traditional IPID control, adaptive fuzzy IPID control can be expressed as:

Additional moment of inertia

Additional static moment Pitching moment of inertia

The whole vehicle moment of inertia except pitching module

4

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Fig. 2. The heading tracking control system of the Petrel-II 200.

Δδ v (k ) = Kp (k )(Δψ (k ) − Δψ (k − 1)) + Ki (k )Δψ (k ) + K d (k )(Δψ (k ) − 2Δψ (k − 1) + Δψ (k − 2))

Table 6 The fuzzy rules of. ΔKi .

(16)

Combining Δψ (k ) with Δψ (k ) − Δψ (k − 1) , the adaptive adjustment rules of the IPID control can be established as:

⎧ Kp (k ) = Kp (k − 1) + w1 (k ){Δψk , Δψk − Δψk − 1} p ⎪ Ki (k ) = Ki (k − 1) + w2 (k ){Δψk , Δψk − Δψk − 1}i ⎨ ⎪ K d (k ) = K d (k − 1) + w3 (k ){Δψk , Δψk − Δψk − 1}d ⎩

NB NS ZO PS PB

wi (k ) = ηi Δψ (k )(Δψ (k ) + Δψ (k ) − Δψ (k − 1)) δ (k − 1) i=1,2,3

ZO

PS

PB

NB NB NB NS ZO

NB NB NS ZO ZO

NS NS ZO PS PS

NS ZO PS PS PB

ZO PS PS PB PB

NB

NS

ZO

PS

PB

PS NB NB NB PS

NB NS NS NS ZO

ZO NS NS NS ZO

ZO ZO ZO ZO ZO

PB PS PS PS PB

Table 7 The fuzzy rules of. ΔK d .

(18)

where ηi (i = 1,2,3) is the coefficient of the adaptive control. The fuzzy input space was defined considering the control response rate and the maneuverability of the Petrel-II 200, thus the fuzzy subsets can be defined as: π 2

π

Δψ (k ) = {NB NS ZO PS PB } = − π − 2 0

{

Δψ (k ) − Δψ (k − 1) = {NB NS ZO PS PB } = −

π π 6

NB NS ZO PS PB

}

Remark (Tables 5–7): the horizontal axis is Δψ (k ) ; the vertical axis is Δψ (k ) − Δψ (k − 1) . π

− 12 0

π 12

π 6

}

fuzzy control were on-line trained with variable learning rates using the difference between the desired heading and the actual heading. When the actuator is in saturation, the control system of the Petrel-II 200 will be uncontrollable. Therefore, an AW compensator is introduced into the heading tracking control to solve above problem. The saturation nonlinearity is given by:

ΔKP , ΔKi, ΔK d = {NB NS ZO PS PB } where NB: negative big; NS: negative small; ZO: zero; PS: positive small; PB: positive big. Tables 5–7 give the fuzzy rules of ΔKp , ΔKi and ΔK d , respectively. The membership functions determined by simulations and experiments are shown in Fig. 3. Each group of the inputs (Δψ (k ) and Δψ (k ) − Δψ (k − 1) ) corresponds to one fuzzy rule of each IPID control parameter variation. To improve the reliability of the control system, the adaptive adjustment was given to change the execution weights of the various rules. The connective weights and biases of the adaptive

δe (k ) δeac (k ) = ⎧ ⎨ ⎩ δmax sign (δe (k ))

NB

NS

ZO

PS

PB

PB PB PS PS ZO

PB PB PS ZO ZO

PS PS ZO NS NS

PS ZO NS NS NB

ZO NS NS NB NB

δe (k ) ≤ δmax δe (k ) ≥ δmax

(19)

where δeac is the output signal of the actuator, δmax is the amplitude limit of the actuator. To solve the problem of actuator control subjected to saturation, a variable-structure PID control was presented to adaptively correct the integral parameter [31]. Though this control can successfully deal with the saturation problem, the control signal with high frequency oscillation will be a great damage on the actuator and increase the power consumption. Thus, based on the existing AW control, an improved AW compensator is put forward, which can be described as:

Table 5 The fuzzy rules of.ΔKp .

NB NS ZO PS PB

NS

(17)

where wi (k ) (i = 1,2,3) is the corrected coefficient, and the range of wi (k ) iswi (k ) ∈ (0,1) . The adaptive control rule is built as:

{

NB

5

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Fig. 5. The closed loop control system used for heading tracking control simulations.

heading error. 4. Closed loop motion control system Based on the accurate dynamic model of the Petrel-II 200, the heading tracking control algorithm is embedded into a closed loop motion control system and the heading angle of the glider used as a feedback is sampled by compass sensor. When the actual position and its next preset target path are updated by the satellite navigation system on the sea, the desired heading of the vehicle can be confirmed. The heading tracking control is mainly used to maintain actual heading of the glider being consistent with the desired heading in the horizontal plane during navigation motion. Then vertical rudder flapping adjusts the heading of the glider to follow the preset path. Fig. 5 gives the closed loop control system for heading tracking control simulations. The dynamics of the Petrel-II 200 is served as the plant to be controlled in the simulation platform. This control system gives a corresponding rudder angle based on the desired heading concluded by the control system and the actual heading acquired by sensors in real time.

Fig. 3. The membership functions.

δeac (k ) = δeac (k − 1) + Kp (Δψ (k ) − Δψ (k − 1) + ηp κp) + Ki Δψ (k )

5. Simulation and experiment validation

+ K d (Δψ (k ) − 2Δψ (k − 1) + Δψ (k − 2) − ηd κd ) To evaluate and validate performance capabilities of the proposed heading tracking control, simulations and experiments of the AFIPID + AW control, conventional IPID control and the AFIPID control were performed. Considering the actual operations of the Petrel-II 200, all the simulations were performed with a time step of 1 s. The initial control parameters selected by simulations and experiments were set as Kp _ init = 2.5, Ki _ init = 0.5 and K d _ init = 2 . When η1 = 0.1, η2 = 0.085, η3 = 0.1, the adaptive control could be slowly evolved to prevent diverging effect. The parameters of the AW compensator were selected as ηp = 2, ηd = 1 and ℓ = 0.01, which could be suitable to all the saturation conditions. The velocity of the Petrel-II 200 was 0.4 m/s on the horizontal plane and the range of the corresponding rudder flapping angle was from −40° to 40°. The control variables {ΔB, ΔIp , δr, δ v , δh, n} were initially set as {0,0,0,0,0,15rad/s} and the

(20)

−ℓ(δe − δeac )/ Kp, d δe ≠ δeac , Δψ (k ) δe > 0 is the rule of whereκp, d = ⎧ ⎨ δe = δeac ⎩0 the AW control, ηp , ηd andℓ with all positive constants are the AW compensator parameters which can be selected by simulations. The structure of the proposed hybrid control algorithm for heading tracking control of the Petrel-II 200 is shown in Fig. 4. The fuzzy control hybrid with the adaptive control dynamically adjust the control parameters based on not only the error between the model reference input and the process output but also the control sensitivity which can be described with the combinations of δeac (k − 1) and the changing rate of Δψ (k ) . The AW compensator is considered as a modified optimization control that operates on the saturation problem caused by a high

Fig. 4. The fuzzy adaptive IPID control algorithm with AW compensator. 6

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Fig. 7. Heading tracking control with different AW compensators in the presence of actuator saturation.

initial conditions {u 0 , v0, w0, p0 , q0 , r0, ψ0 , φ0 , θ0 , ξo0, ηo0 , ζ o0} were set as {0.001m/s, 0,0,0,0,0,0,0,0,0,0,0} . The initial values were concluded by physical measurement and motion conditions. To compare the three heading tracking control algorithms, the step, rectangular and sinusoidal heading tracking signals were adopted, respectively. The desired heading tracking, the actual heading and the control input rudder angle in three different heading tracking responses without disturbance were illustrated in Fig. 6, where the glider tracked desired heading by using traditional IPID, the fuzzy IPID, the AFIPID and the AFIPID with the AW, respectively. Compared with traditional IPID control in all the three responses, the fuzzy IPID control showed a better heading tracking performance with smaller heading error and much restrained oscillation, which suggested that the fuzzy control has better robustness in such nonlinear control system. As seen in Fig. 6 (a), π for a large step desired heading command with the amplitude of 2 rad , the heading errors in both the fuzzy IPID control and the AFIPID control had smaller deviations while the big jump occurred on the expected heading, however, obvious inhibitory effect on the overshoot and the oscillation was still not obtained. The AFIPID control with the AW provided a better performance compared with the other control algorithms, which worked at a higher convergence speed, with overshoot close to zero and without steady-state error. More specifically, the proposed heading tracking control showed strong anti-saturation capability with the AW compensator, which helped the AFIPID control to recover the heading tracking properly. Fig. 6 (b) showed the control effect of the rectangular response. It could be noted that the heading tracking performance of the AFIPID control with or without the AW compensator was similar when the heading error was small and the saturation problem was not caused. However, when the heading error was large and the actuator saturation would be happened, the performance of the proposed AFIPID control with the AW was more accurate and more effective with the smoothest transition and the shortest recovery time. In contrast, oscillations in the control signal and overshoot in the heading tracking phenomena could be found in the other three controls. In the response of tracking sinusoidal shown in Fig. 6 (c), the IPID control had a notable difference with a longer convergence time, about 15 s oscillation and about 3 s time-delay in the heading tracking control performance. Compared with traditional IPID control, the proposed control system showed a better performance with relative low heading error, fast convergence time and low power consumption. Since the error of the heading was small in the sinusoidal heading tracking process, the AFIPID with or without the AW compensator was

Fig. 6. Heading tracking control under different input signal without external disturbance.

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Fig. 9. The AFIPID control with the AW compensator applied to the LOS guidance law.

Fig. 8. Heading tracking control under different input signal with external disturbance.

similar in this condition. To evaluate and validate performance of the improved AW compensator, comparisons between traditional AW compensator and the improved AW compensator were presented in Fig. 7. Generally, it could be clearly seen that the performance of the proposed AW compensator was more accurate and more effective. Especially the improved AW compensator avoided high frequency oscillation of the actuator output, which greatly helped to decrease power consumption. In the following simulation, the heading tracking performance was assessed in rigorous perturbed conditions. For this purpose, Gaussian white noise as an environmental disturbance was added into the heading tracking control of the Petrel-II 200, ranging from −0.2 rad to 0.2 rad. As shown in Fig. 8, sharps and fluctuations occurred in all three control signals caused by the disturbance. Fig. 8 (a) showed that the AFIPID control with the AW compensator performed better with a lower heading error for its stronger resistance to the disturbance, especially when the disturbance together with the actuator saturation occurred. Fig. 8 (b) demonstrated the robustness against the disturbance of the proposed control with the sinusoidal heading tracking signal input. It could be seen that though the AFIPID control with or

Fig. 10. Experimental test for heading tracking control of the Petrel-II 200.

without the AW showed a similar performance when heading error was small and the saturation was not occurred, relative higher overshoot and the jump in control signal caused by the saturation would ruin the control performance when the AW compensator was not introduced. 8

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the difference among the disturbance worked on the glider, the experimental results demonstrated the efficiency of the proposed heading tracking control in practice. To better compare the proposed heading tracking control with other algorithms, quantitative comparisons using Integral Absolute Error N (IAE, IAE = ∫0 ψe (k ) − ψc (k ) dk ) and Total Variation (TV, N

TV = ∑k = 2 δ (k ) − δ (k − 1) ) were finished, which was shown in Table 8. IAE represents the comprehensive performance of the control system and TV reflects the smoothness of the control signal. It could be seen from Table 8 that the proposed heading tracking control had the smallest IAE and TV values among all the control algorithms, which showed the proposed heading tracking control had a better performance in both heading control output and smoothness of the control input. 6. Conclusion Heading tracking control plays the core role in the navigation process. The underwater glider is an under actuated dynamic system. The underwater glider has three DOFs including yaw, lateral and forward in the horizontal plane, however there is only one control variable, the angle of flapping rudder. A new heading tracking control with an adaptive fuzzy incremental PID combined with an anti-windup compensator was put forward in this paper. Performance of the proposed heading tracking control was investigated. To validate the effectiveness and correction of the proposed heading tracking control, simulations and experiments of the closed loop control system were carried out to follow three typical mathematical signals. Furthermore, the ability to cope with the saturation problem was provided by simulation contrast with the proposed control and the conventional AFIPID control. Meanwhile, the simulations and experimental tests were also performed under the consideration of unsteady, non-uniform flow environment. The simulation and experiment results show that the heading tracking control with the AFIPID algorithm and the AW compensator has an excellent performance, which can maintain the desired heading under large disturbance and saturation. The self-adaptation of the parameters in the heading tracking control has been realized by the adaptive fuzzy inference system, which greatly reduces the cost of manual power and material resources caused by the sea test of underwater glider, and greatly improves research efficiency of underwater glider. The AW compensator in the heading tracking control optimized the control performance. In future, new heading control algorithms will be developed on the basis of current research and intelligent algorithms will be adopted for obtaining the adaptive adjustment of the parameters. In addition, the navigation control strategy will be researched and the navigation control simulation platform will be built under the closed loop heading tracking controller with the controller and the controlled object.

Fig. 11. Heading tracking control performance of the Petrel-II 200 during the experiments.

Final simulation was devoted to assess performance of the proposed heading tracking control applied into the practical path planning navigation as shown in Fig. 9. A Line-of-Sight (LOS) guidance law [32] was adopted for straight line path following of the Petrel-II 200. The proposed heading tracking control loaded with the aforementioned disturbance was embedded into the LOS guidance law, which was applied to the Petrel-II 200 dynamic model. The AFIPID control with the AW compensator performed well when it tracked the desired heading obtained from the LOS guidance law. Furthermore, the path error was used to describe the accuracy and efficiency of the AFIPID with the AW, which was maintained below in 5 m. The heading tacking capability, the robustness to the disturbance and the abilities to cope with saturation problem could be demonstrated through the following simulation results. To prove the validity of the proposed heading tracking control in real conditions with unknown disturbances and the associated difficulties with real time implementation, several real-time tests were conducted in the South Ocean of China. As shown in Fig. 10, the goal of the experimental tests was to track a fixed target heading in uncontrolled environment due to the wind conditions and a high current flow. To better validate the proposed heading tracking control, traditional IPID control and the AFIPID control without the AW compensator were also embedded into the Petrel-II 200 control system. The experimental results were shown in Fig. 11. Due to the real-time varying disturbance influence, three heading tracking performances in the experimental tests were different to some extent. Ignoring the impacts of Table 8 Quantitative comparisons using IAE and TV. Figure

Algorithm

IAE (rad)

TV (rad)

Figure

Algorithm

IAE (rad)

TV (rad)

Fig. 6(a)

IPID FIPID AFIPID AFIPID IPID FIPID AFIPID AFIPID IPID FIPID AFIPID AFIPID AFIPID

30.4649 27.1271 26.9753 26.3195 22.6140 20.4147 20.2406 19.6705 26.8439 15.4353 13.1307 6.6225 76.0126

1.9157 0.9232 0.9500 0.8730 8.6458 5.1475 5.3214 4.6392 3.8931 3.3321 3.3192 3.2627 74.6780

Fig. 7 Fig. 8(a)

AFIPID IPID AFIPID AFIPID IPID AFIPID AFIPID IPID AFIPID AFIPID IPID AFIPID AFIPID

35.7687 60.5686 50.5850 43.2515 68.6407 54.9017 54.0758 33.9812 26.9050 24.6099 17.8809 13.9497 10.7734

7.4059 226.6831 235.1727 184.5482 418.1284 225.0025 216.5040 10.1095 6.5121 5.0151 11.9834 10.8829 10.8437

Fig. 6(b)

Fig. 6(c)

Fig. 7

+ AW

+ AW

+ AW + TAW

Fig. 8(b)

Fig. 9

Fig. 11

9

+ IAW

+ AW

+ AW

+ AW

+ AW

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Acknowledgment

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This research is supported by Wenhai Program of Qingdao National Laboratory for Marine Science and Technology (No.2017WHZZB0101), Tianjin Municipal Natural Science Foundation (No.18JCZDJC40100), the Program for Innovative Research Team in University of Tianjin (No. TD13-5037), the National Key Research and Development Plan of China (No. 2017YFC0305900) and Key R&D Program of Shandong (No. 2016CYJS02A02). The authors appreciate all staff members in research and development team of Petrel-II 200. References [1] Fossen TI. Marine control systems: guidance, navigation and control of ships, rigs and underwater vehicles. Marine Cybernetics 2002. [2] Zhuang YF, Sharma S, Subudhi B, Huang HB, Wan J. Efficient collision-free path planning for autonomous underwater vehicles in dynamic environments with a hybrid optimization algorithm. Ocean Eng 2016;127:190–9. [3] Rezazadegan F, Shojaei K, Sheikholeslam F, Chatraei A. A novel approach to 6-DOF adaptive trajectory tracking control of an AUV in the presence of parameter uncertainties. Ocean Eng 2015;107:246–58. [4] Geranmehr B, Nekoo SR. Nonlinear suboptimal control of fully coupled non-affine six-DOF autonomous underwater vehicle using the state-dependent Riccati equation. Ocean Eng 2015;96:248–57. [5] Cui RX, Zhang X, Cui D. Adaptive sliding-mode attitude control for autonomous underwater vehicles with input nonlinearities. Ocean Eng 2016;123:45–54. [6] Zeng Z, Sammut K, He F, Lammas A. Efficient path evaluation for AUVs using adaptive B-Spline approximation. MTS/IEEE oceans conference. Virginia beach, VA. 2012. [7] Xiang XB, Yu CY, Zhang Q. Robust fuzzy 3D path following for autonomous underwater vehicle subject to uncertainties. Comput Oper Res 2017;84:165–77. [8] Hammad MM, Elshenawy AK, EI. Singaby MI. Trajectory following and stabilization control of fully actuated AUV using inverse kinematics and self-tuning fuzzy PID. PLoS One 2017;12:1–35. [9] Zhang W, Liang ZC, Guo Y, Meng DT, Zhou JJ, Han YF. Fuzzy adaptive sliding mode controller for path following of an autonomous underwater vehicle. OCEANS 2015MTS/IEEE. Washington, DC, USA. 2015. [10] Liang X, Qu XR, Hou YH, Zhang JD. Three-dimensional path following control of underactuated autonomous underwater vehicle based on damping backstepping. Int J Adv Rob Syst 2017:1–9. [11] Huang H, Zhang GC, Qing HD, Zhou ZX. Autonomous underwater vehicle precise motion control for target following with model uncertainty. Int J Adv Rob Syst 2017:1–11.

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