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Control of the Motion Orientation of Autonomous Underwater Vehicle
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Procedia Computer Science 150 (2019) 69–77
13th International Symposium “Intelligent Systems” (INTELS’18) 13th International Symposium “Intelligent Systems” (INTELS’18)
Control of the Motion Orientation of Autonomous Underwater Vehicle Control of the Motion Orientation of Autonomous Underwater Vehicle Nguyen Quang Vinh*, Pham Van Phuc Nguyen Quang Vinh*, Pham Van Phuc
Academy of Military Science and Technology, 17 Hoang Sam, Cau giay, Ha Noi, Viet Nam Academy of Military Science and Technology, 17 Hoang Sam, Cau giay, Ha Noi, Viet Nam
Abstract Abstract In this paper we present an application of the hedge algebras controller and a direct adaptive fuzzy-neural output-feedback In this paper we present application the hedge algebras a direct adaptive fuzzy-neural output-feedback controller (DAFNOC) andan Hedge algebrasof(HA) in control of thecontroller orientationand of underwater vehicles. The experiments simulated controller (DAFNOC) Hedge (HA) in of the of algorithm underwaterofvehicles. The experiments on computers are done and to prove thealgebras effectiveness, thecontrol feasibility oforientation the proposed the neural controller undersimulated different on computers the effectiveness, the neural controller actions such asare thedone noisetoinprove the measuring devices,the thefeasibility influence of the proposed flow to thealgorithm motion ofofunderwater vehicles under different actions such as the noise in the measuring devices, the influence of the flow to the motion of underwater vehicles © 2019 The Author(s). Published by Elsevier B.V. © 2019 The Authors. Published by Elsevier B.V. © 2019 The Author(s). Published Elsevier B.V. This is an open access article underbythe CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of the scientific committee 13th Peer-review under responsibility of the scientific committee of of the the (https://creativecommons.org/licenses/by-nc-nd/4.0/) 13th International International Symposium Symposium“Intelligent “IntelligentSystems” Systems”(INTELS’18) (INTELS’18). Peer-review under responsibility of the scientific committee of the 13th International Symposium “Intelligent Systems” (INTELS’18) Keywords: neural controller; hedge algebras; underwater vehicles. Keywords: neural controller; hedge algebras; underwater vehicles.
1. Introduction 1. Introduction The inertial navigation algorithm generates navigation parameters with errors. Those errors increase over time The inertial navigation algorithm navigation parameters with errors. Those errors increase over time because the drifts are integral twice. generates To overcome those errors, the additional information from other navigation because the drifts are integral twice. To overcome those errors, the additional information from other navigation systems is often used to reduce the mentioned errors [1,2]. The use of a linear Kalman filter for solving the problem systems is often to reduce the by mentioned errors[3,4,9]. [1,2]. The use of athe linear filter mentioned aboveused has been studied many authors However, basicKalman drawback offor thissolving methodthe is problem to make mentioned above has been studied by many authors [3,4,9]. However, the basic drawback of this method to make two successive stages: the inertial navigation algorithm and the Kalman filter algorithm, which cause the is increment two successive stages: the inertial navigation algorithm and the Kalman filter algorithm, which cause the increment of computational time and the number of calculations affecting the response time of the system. In this paper, the of computational and the of number of calculations thecombined response with time measuring of the system. In this paper, the authors propose antime application the extended Kalman affecting filter (EKF) devices to remove authors propose an application of the extended Kalman filter (EKF) combined with measuring devices to remove the drifts and measuring errors. drifts and measuring errors.
* Corresponding author. Tel.: +7-903-226-3699 E-mail address:author. [email protected] * Corresponding Tel.: +7-903-226-3699 E-mail address: [email protected] 1877-0509 © 2019 The Author(s). Published by Elsevier B.V. This is an open access underPublished the CC BY-NC-ND 1877-0509 © 2019 The article Author(s). by Elsevier license B.V. (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the 13th International Symposium “Intelligent Systems” (INTELS’18) This is an open access article under CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the 13th International Symposium “Intelligent Systems” (INTELS’18)
1877-0509 © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the 13th International Symposium “Intelligent Systems” (INTELS’18). 10.1016/j.procs.2019.02.015
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Nguyen Quang Vinh et al. / Procedia Computer Science 150 (2019) 69–77 Nguyen Quang Vinh and Pham Van Phuc / Procedia Computer Science 00 (2019) 000–000
HA solve effectively problems in uncertain environments, therefore researchers often use them in control and automation areas. HA have been studied in some identification and prediction problems and have given significant successes in control areas (namely, in some approximation and control problems of with simple models [10]). However, the use of HA in control problems is not popular. The study for the successful application of an HA controller will confirm more effectiveness of HA theory, open application possibilities in practice. We started with this purpose and designed an HA controller applied in designing the orientation and depth controller under the influence of errors of measuring devices and invisible flows to the motion of the vehicle. The simulation results are the basis of studying and learning improvement of the control and experiment quality. DAFNOC is proposed for an torpedo shaped AUVs that is controlled by two horizontal rudders and two vertical rudders. Outputs control of AUVs are yaw, roll and pitch that are received from the navigation system on the AUVs. The DAFNOC uses the fuzzy-neural network Singleton to approximate the control law. The number of fuzzy-neural networks is the same as the number of outputs of the system. Input of each fuzzy-neural network is the estimated state error vector that is corresponding with each output system. The output weighting factors of fuzzy-neural networks are updated online according to adaptive control law so the DAFNOC can apply for highly nonlinear systems and the system has some none determinate parameters. From the fact the navigation parameters have been determined, the DAFNOC algorithm is proposed. DAFNOC algorithm for SISO system is presented in [9], development of DAFNOC algorithm in this paper is only using the status errors corresponding to the output of the system as input of fuzzy-neural networks, which allows to reduce the size of the fuzzy-neural networks, reduce the amount of computing systems and can apply DAFNOC for MIMO system by combining from SISO systems. 2. The dynamic model of autonomous underwater vehicles (AUVs) The motion of AUVs is described in the body frame Gb X bYb Z b that has origin coinciding with the center of gravity Gb (Fig. 1). The linear and angular velocity vector is denoted in the body frame by symbol [u , v, w, p, q, r ]T , where: p, q, r : elements of angular velocity on the coordinate.
Fig. 1. Reference frames for AUVs.
AUVs have a short working distance and move slowly. As a result, the local frame OX 0Y0 Z 0 can be considered as a navigation frame. The position and Euler angles of AUVs in the local frame are denoted here by symbol [ x, y, z , , , ]T ,
then J (),
(1) (2)
where J () is the direction cosine matrix that can be determined through four-parameters Rodring-Hamilton 0 , 1 , 2 , 3 as follows:
Nguyen Quang Vinh et al. / Procedia Computer Science 150 (2019) 69–77 Nguyen Quang Vinh and Pham Van Phuc / Procedia Computer Science 00 (2019) 000–000
202 212 -1 J ( ) 212 20 3 213 20 2
212 20 3 202 222 -1 22 3 20 1
213 20 2 22 3 20 1 . 202 232 -1
71 3
(3)
Motion equations of the AUVs are described in the body frame [6]: M RB CRB () RB ,
(4)
where M RB is the inertia matrix, CRB () is the Coriolis matrix, RB is a generalized vector of external forces and moments: RB M A C A () D () L() g () ,
(5)
where g ( ) is the buoyancy forces and moments, M A is the inertia matrix of added mass, C A () is the Coriolis matrix of added mass, D() is the potential damping matrix, L() is the forces and moments parameters matrix of rudder, is the forces and moments of rudder and propellers. From (2), (4), (5), the 6-DOF dynamic equation can be written as: M C ( ) D ( ) L( ) g ( ) , ( ) , y , J
(6)
where M M A M RB , C A ( ) CRB (), y is the output system. From (2), we have J () 1 ,
(7)
J () J () 1 ]. J () 1[
(8)
Substituting (7) and (8) into (6) yields. C (, ) g () , M ()
(9)
where M () J () T MJ ()1 , C (, ) J () T [ MJ () 1 J () C () D() L()], g () J () T g (), J () T .
From (2) and (9) the AUV dynamic equations in the body frame with multiple inputs and multiple outputs: ( ) , y , J M ( ) 1 C ( , ) M ( ) 1 g ( ) M ( ) 1 ,
An algorithm for detemining the navigation parameters of AUV vehicles is described in [4] of the author.
(10)
Nguyen Quang Vinh et al. / Procedia Computer Science 150 (2019) 69–77 Nguyen Quang Vinh and Pham Van Phuc / Procedia Computer Science 00 (2019) 000–000
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3. A direct adaptive fuzzy-neural output-feedback control for AUV AUVs are controlled to follow the yaw, roll and pitch angle. The pitch angle is controlled by two rudders drive in the horizontal plane s1 , s 2 with s1 s 2 . Two rudders h1 , h 2 control AUV’s both yaw and roll. Defining uh is yaw control signal, ul is roll control signal of the AUVs, the relationship between uh and ul is written ul
h1 h 2 h 2 h1 , uh , 2 2
(11)
rotation angles of rudder control yaw and roll is [4]:
h1 ul uh , h 2 ul uh .
(12)
The dynamic equation can be written as MIMO nonlinear equations (13). x A0 x B F ( x ) G ( x )u d ; y C T x,
(13)
T where inputs of the system are the rudder angles: u [u [ h , s , l ]T ; outputs of the system are the yaw, 1 , u2 , u3 ]
pitch B
y1 , y2 , y3 ]T [ , , ]T ; d and roll y [
is
the
ocean currents; A0 diag[ A01 , A02 , A03 ] R 66 ;
diag[ B1 , B2 , B3 ] R 63 ; C diag[C1 , C2 , C3 ] R 63 ;
0 1 0 1 A0 k , Bk 1 , Ck 0 , k 1 3 , use throughout this paper. 0 0 F ( x) M n ( ) 1 C ( , ) M n ( ) g ( ) M n ( ) 1 J ( ) T pl ; G ( x)u M ( ) 1 J ( ) T bl ; x [ y1 , y1 , y2 , y 2 , y3 , y3 ]T is a vector of states.
The objective of this paper is to design a DAFNOC so that outputs of system y [ y1 , y2 , y3 ]T track reference signal y r [ yr1 , yr 2 , yr 3 ]T . Define y r [ yr1 , yr 2 , yr 3 ]T ; Y r [ yr1 , y r1 , yr 2 , y r 2 , yr 3 , y r 3 ]T ; e Y r x; eˆ Y r xˆ when T T [e1 , e12 , e21 , e22 , e31 e [e1 , e1 , e2 , e , e32 ]T E1 [e [e11 , e21 , e31 ]T , 2 , e3 , e3 ] 1 , e2 , e3 ] where eˆ and xˆ denote the estimates of e and x. Based on the certainty equivalence approach and Lyapunov, an optimal control law is: * T u G 1 ( x)[ F ( x) y r K c eˆ],
where K c
(14) T
diag [ K c1 , K c 2 , K c 3 ], K ckT [ K ck 1 , K ck12 ]. Choose K c such that Ak A0 k Bk K ck is Hurwitz matrix.
AUV’s dynamic equation (13) with output signal vector y is received from the navigation system, Since only the outputs of system are assumed to be measurable, the optimal control law cannot be implemented (14). Thus, control law of DAFNOC is proposed: u u f v,
(15)
Nguyen Quang Vinh et al. / Procedia Computer Science 150 (2019) 69–77 Nguyen Quang Vinh and Pham Van Phuc / Procedia Computer Science 00 (2019) 000–000
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T where u f [u [v1 , v2 , v3 ]T , here u f is designed to approximate the optimal control law (14) and f 1,u f 2 , u f 3 ] , v
the control term v is employed to compensate the external disturbance and the modeling error. Design the state observer [5]. eˆ A0 eˆ BK cT eˆ K 0 ( E1 Eˆ 1 ), T Eˆ C eˆ,
(16)
1
dial [ K 01 , K 02 , K 03 ], K 0 k dial[ K 0 k1 , K 0 k 2 ]T ; matrix A0 k K 0 k CkT is strictly Hurwitz matrix [5]. We define the observation errors as e e eˆ and E E Eˆ . From (13-16) we have
where K 0
1
1
1
* e ( A0 K 0 C T )e B[G ( x)u G ( x)u f G ( x)v d ],
E1 C T e.
(17)
To calculate u fk we use the configuration of a fuzzy-neural approximator (Fig. 2) [6]. The fuzzy inference engine uses the fuzzy IF-THEN rules to perform a mapping from an input estimated error vector eˆ k [eˆk1 , eˆk 2 ]T . The i -th fuzzy IF-THEN rule is written as (i 1, 2,3, ..., h). If eˆk1 is Aki 1 and eˆk 2 is Aki 2 than u fk is Bki , where Aki 1 , Aki 2 and Bki are fuzzy sets.
By using produce inference, center average and singleton fuzzifier, the output of the fuzzy-neural network can be expressed as [3]. (eˆkj ) i 1 j 1 T (eˆ ). u fk k k k h 2 Akji (eˆkj ) i 1 j 1 h
2
T
T
i k
Akji
(18)
T
We have u f u f (eˆ ) [1 1 (eˆ1 ), 2 2 (eˆ 2 ), 3 3 (eˆ3 )]T ; where k [ 1k , 2k , ..., hk ]T is an adjustable parameter 1, k (eˆ k ) [1k , k2 , ..., hk ]T is a fuzzy basis vector vector. ik is the point at which Bi (ik ) k
2
j 1
Akji
(eˆkj )
. (eˆ k ) h 2 Akji (eˆkj ) i 1 j 1 i k
(19)
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Nguyen Quang Vinh et al. / Procedia Computer Science 150 (2019) 69–77 Nguyen Quang Vinh and Pham Van Phuc / Procedia Computer Science 00 (2019) 000–000
Fig. 2. Configuration of a fuzzy-neural network.
In order to employ the SPR-Lyapunov design approach, (17) can be written as [6]. T
ek1 H k ( s ) Lk ( s )[ˆ k k (eˆ k ) v fk w fk ],
(20)
where H k ( s) CkT ( sI ( A0 k K 0 k CkT )) 1 Bk , k (eˆ k ) Lk1 ( s)[k (eˆ k )] and Lk ( s ) choosen so that Lk1 ( s ) is a proper stable transfer function and H k ( s ) Lk ( s ) is a proper SPR transfer function. Assumption w fk is w fk k , where k 0. Supposed that the update laws are chosen as k ek1k (eˆ k ) if || k || mk or T (|| k || mk and ek1 k k (eˆ k ) 0) k mk Pr( k ek 1k (eˆ k )) if || k || T and ek1 k k (eˆ k ) 0.
When Pr( k ek1k (eˆ k )) k ek1k (eˆ k ) k
(21)
T ek1 k k (eˆ k ) k then || k || mk and || k || mk . Let vk be given as || k ||2
if ek 1 0 and | ek 1 | k , k vk k if ek1 0 and | ek1 | k , e if | e | , k1 k k k1 k
(22)
where k k 0 . Consider the Lyapunov 3
V Vk , where Vk k 1
1 T 1 T e k Pk e k k k with Pk PkT 0. 2 2 k
(23)
Differentiating Vk with respect to time we have: 1 T 1 T 1 Vk e k Pk e k e k Pk e k Tk k . k 2 2
(24)
Nguyen Quang Vinh et al. / Procedia Computer Science 150 (2019) 69–77 Nguyen Quang Vinh and Pham Van Phuc / Procedia Computer Science 00 (2019) 000–000
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QkT 0 such that Because H k ( s ) Lk ( s ) is SPR, there exists Q k 1 T 1 T Vk e k Qk e k e k 1[ k k (eˆ k ) v fk w fk ] k k . k 2
(25)
By using assumptions we have: 1 Vk min (Qk ) | ek 1 |2 . 2
(26)
As a result 3
V
V
3 1 min (Qk ) | ek 1 |2 . 2
k k 1 k 1
(27)
The stability of the overall control system is analyzed based on the Lyapunov method. Using Barbalat’s lemma, we have lim | ek1 (t ) | 0. Because A0 k Bk K ckT is Hurwitz matrix, e k (t ) and eˆ k (t ) are t
bounded, e e k eˆ k and ek1 , e k L we conclude that ek1 0 as t . The update law (21) with all signals of k the closed-loop are bounded, E1 (t ) 0 as t . The hedge algebra controller looks at the [4]. 4. The orientation control of underwater vehicles
Following Zhang et al one can use the target function for the controller in the following form [6]: E k1
2 1 2 1 kd k 1 RK 1rk2 , 2
(28)
where RK and rk , respectively are the steering angle and the changing orientation rate of the underwater vehicle at the moment k , respectively, constants 1 , 1 and 1 , respectively are the proportion coefficient, the feedback coefficient and the orientation differential coefficient, respectively. The HAC or DAFNOC is described as in Fig. 3.
Fig. 3. The simulation scheme of the orientation control of the underwater vehicle using HA or DAFNOC.
5. Simulations, computations, discussions
To test the algorithms above, we use the input data of underwater vehicle [3]. The wanted trajectory is the line connecting point AUVs touching water (the starting position) and the pickup shot poin(the destination) determined before throwing AUV. Suppose that the starting position of AUV has coordinates ( x0 , y0 , z0 ) (100,100, 20) with beginning state angles ( 0 , 0 , 0 (00 , 180 ,30 ), the finishing point
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Nguyen Quang Vinh et al. / Procedia Computer Science 150 (2019) 69–77 Nguyen Quang Vinh and Pham Van Phuc / Procedia Computer Science 00 (2019) 000–000
has coordinates ( x2 , y2 , z2 ) (1100,1100,120). External noise is the influence of ocean flows considered at the moment 20 s: [uc , vc , wc ]T [4,4,0.5]T (m/s). The actual test model is shown in Fig. 4.
Fig.4. AUV experiment on the sea.
Consider the vertical plane with assumptions w p q 0, the orientation angle is small [4]. We use the HA orientation controller with the above designing parameters responsible to the orientation control of underwater vehicles as in Fig. 5. The simulation results in Fig. 5 show that under the influence of ocean flows the parameters of the controller are updated online, hence the turning angle of the helm will change to make the system less affected and catch the wanted trajectory quickly. The trajectory of the underwater vehicle at the starting moment of simulation has a deviation since the starting orientation angle does not coincide with the line of sight of the starting position and the destination (LOS). With the DAFNOC approach the yaw angle quickly achieves the required set value and low mobility, the HA method is highly mobile, with a slow but fast calculation. The simulation results show that with the online updating law of the controller parameters, outputs of the system quickly adapt to the effects of the ocean currents and the interaction between the control channels which suggests possible application DAFNOC to control AUVs in real time.
Fig. 5. The simulation results of the AUV orientation control.
6. Conclusions
In this paper we have presented the system of the motion equations of an underwater vehicle controlled by two orientation helms, two depth helms of the 6th degree of freedom under the influences of the flow and measurement noise. We have applied HA and DAFNOC to simulate the single channel orientation and compare them using the direct adaptive fuzzy-neural output-feedback control and analyzed the adaptive ability of the system under the influences of the flow and measurement noise.
Nguyen Quang Vinh et al. / Procedia Computer Science 150 (2019) 69–77 Nguyen Quang Vinh and Pham Van Phuc / Procedia Computer Science 00 (2019) 000–000
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In the future we will do research on more various conditions, in more difficult tasks and analyze the adaptive ability of the controller and also prove the stability of the entire system in order to construct a controller for the entire motion function of underwater vehicles. References [1] Wang R, Wang S, Wang Y. A Hybrid Heading Control Scheme for a Biomimetic Underwater Vehicle. 26th International Ocean and Polar Engineering Conference. Rhodes, Greece: ISOPE, 2016; p. 619-625. [2] Fossen ThI. Maritime Control Systems – Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles. Marine Cybernetics. Trondheim, Norway, 2002. [3] Nguyen Quang Vinh, Truong Duy Trung. Guidance, navigation and control of Autonomous underwater vehicle. International Symposium on Electrical-Electronics, 2013, 36: p. 44-49. [4] Nguyen Quang Vinh, Control of the motion orientation and the depth of underwater vehicles by hedge algebras. The XVIth International symposium “Intelligent Systems” , INTELS’16,2016, Procedia Computer Science. 103: p. 331–338. [5] Nguyen Quang Vinh. An algorithm for determining the navigation parameters of UAVs based on the combination of measuring devices. IEEE Trans. on Automatic Control, 1996; 41-3: p. 447-451. [6] Zhang Y, Hearn GE, Sen P. Neural network approaches to a class of ship control problems. Part I, II. Eleventh Ship Control Systems Symposium 1997; 1. [7] Zhang T, Ge SS, Hang CC. Adaptive neural network control for strict-feedback nonlinear systems using backstepping design. Automatica, 2000; 36: p. 1835-1846. [8] Brandt RD, Lin F. Adaptive interaction and its application to neural networks, 1999. Elsevier, Information Science. 121: p. 201-215. [9] Fossen TI, Pettersen KY, Nijmeijer H. Sensing and control for autonomous vehicles. Springer, 2017; 474. [10] Nguyen Quang Vinh, Nguyen Duc Anh, Phan Tuong Lai. The control of the motion of airplanes using hecke algebras. Journal of military scientific and technological researches 2015; 37: p. 28-102.
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Procedia Computer Science 150 (2019) 69–77
13th International Symposium “Intelligent Systems” (INTELS’18) 13th International Symposium “Intelligent Systems” (INTELS’18)
Control of the Motion Orientation of Autonomous Underwater Vehicle Control of the Motion Orientation of Autonomous Underwater Vehicle Nguyen Quang Vinh*, Pham Van Phuc Nguyen Quang Vinh*, Pham Van Phuc
Academy of Military Science and Technology, 17 Hoang Sam, Cau giay, Ha Noi, Viet Nam Academy of Military Science and Technology, 17 Hoang Sam, Cau giay, Ha Noi, Viet Nam
Abstract Abstract In this paper we present an application of the hedge algebras controller and a direct adaptive fuzzy-neural output-feedback In this paper we present application the hedge algebras a direct adaptive fuzzy-neural output-feedback controller (DAFNOC) andan Hedge algebrasof(HA) in control of thecontroller orientationand of underwater vehicles. The experiments simulated controller (DAFNOC) Hedge (HA) in of the of algorithm underwaterofvehicles. The experiments on computers are done and to prove thealgebras effectiveness, thecontrol feasibility oforientation the proposed the neural controller undersimulated different on computers the effectiveness, the neural controller actions such asare thedone noisetoinprove the measuring devices,the thefeasibility influence of the proposed flow to thealgorithm motion ofofunderwater vehicles under different actions such as the noise in the measuring devices, the influence of the flow to the motion of underwater vehicles © 2019 The Author(s). Published by Elsevier B.V. © 2019 The Authors. Published by Elsevier B.V. © 2019 The Author(s). Published Elsevier B.V. This is an open access article underbythe CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of the scientific committee 13th Peer-review under responsibility of the scientific committee of of the the (https://creativecommons.org/licenses/by-nc-nd/4.0/) 13th International International Symposium Symposium“Intelligent “IntelligentSystems” Systems”(INTELS’18) (INTELS’18). Peer-review under responsibility of the scientific committee of the 13th International Symposium “Intelligent Systems” (INTELS’18) Keywords: neural controller; hedge algebras; underwater vehicles. Keywords: neural controller; hedge algebras; underwater vehicles.
1. Introduction 1. Introduction The inertial navigation algorithm generates navigation parameters with errors. Those errors increase over time The inertial navigation algorithm navigation parameters with errors. Those errors increase over time because the drifts are integral twice. generates To overcome those errors, the additional information from other navigation because the drifts are integral twice. To overcome those errors, the additional information from other navigation systems is often used to reduce the mentioned errors [1,2]. The use of a linear Kalman filter for solving the problem systems is often to reduce the by mentioned errors[3,4,9]. [1,2]. The use of athe linear filter mentioned aboveused has been studied many authors However, basicKalman drawback offor thissolving methodthe is problem to make mentioned above has been studied by many authors [3,4,9]. However, the basic drawback of this method to make two successive stages: the inertial navigation algorithm and the Kalman filter algorithm, which cause the is increment two successive stages: the inertial navigation algorithm and the Kalman filter algorithm, which cause the increment of computational time and the number of calculations affecting the response time of the system. In this paper, the of computational and the of number of calculations thecombined response with time measuring of the system. In this paper, the authors propose antime application the extended Kalman affecting filter (EKF) devices to remove authors propose an application of the extended Kalman filter (EKF) combined with measuring devices to remove the drifts and measuring errors. drifts and measuring errors.
* Corresponding author. Tel.: +7-903-226-3699 E-mail address:author. [email protected] * Corresponding Tel.: +7-903-226-3699 E-mail address: [email protected] 1877-0509 © 2019 The Author(s). Published by Elsevier B.V. This is an open access underPublished the CC BY-NC-ND 1877-0509 © 2019 The article Author(s). by Elsevier license B.V. (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the 13th International Symposium “Intelligent Systems” (INTELS’18) This is an open access article under CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the 13th International Symposium “Intelligent Systems” (INTELS’18)
1877-0509 © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the 13th International Symposium “Intelligent Systems” (INTELS’18). 10.1016/j.procs.2019.02.015
70 2
Nguyen Quang Vinh et al. / Procedia Computer Science 150 (2019) 69–77 Nguyen Quang Vinh and Pham Van Phuc / Procedia Computer Science 00 (2019) 000–000
HA solve effectively problems in uncertain environments, therefore researchers often use them in control and automation areas. HA have been studied in some identification and prediction problems and have given significant successes in control areas (namely, in some approximation and control problems of with simple models [10]). However, the use of HA in control problems is not popular. The study for the successful application of an HA controller will confirm more effectiveness of HA theory, open application possibilities in practice. We started with this purpose and designed an HA controller applied in designing the orientation and depth controller under the influence of errors of measuring devices and invisible flows to the motion of the vehicle. The simulation results are the basis of studying and learning improvement of the control and experiment quality. DAFNOC is proposed for an torpedo shaped AUVs that is controlled by two horizontal rudders and two vertical rudders. Outputs control of AUVs are yaw, roll and pitch that are received from the navigation system on the AUVs. The DAFNOC uses the fuzzy-neural network Singleton to approximate the control law. The number of fuzzy-neural networks is the same as the number of outputs of the system. Input of each fuzzy-neural network is the estimated state error vector that is corresponding with each output system. The output weighting factors of fuzzy-neural networks are updated online according to adaptive control law so the DAFNOC can apply for highly nonlinear systems and the system has some none determinate parameters. From the fact the navigation parameters have been determined, the DAFNOC algorithm is proposed. DAFNOC algorithm for SISO system is presented in [9], development of DAFNOC algorithm in this paper is only using the status errors corresponding to the output of the system as input of fuzzy-neural networks, which allows to reduce the size of the fuzzy-neural networks, reduce the amount of computing systems and can apply DAFNOC for MIMO system by combining from SISO systems. 2. The dynamic model of autonomous underwater vehicles (AUVs) The motion of AUVs is described in the body frame Gb X bYb Z b that has origin coinciding with the center of gravity Gb (Fig. 1). The linear and angular velocity vector is denoted in the body frame by symbol [u , v, w, p, q, r ]T , where: p, q, r : elements of angular velocity on the coordinate.
Fig. 1. Reference frames for AUVs.
AUVs have a short working distance and move slowly. As a result, the local frame OX 0Y0 Z 0 can be considered as a navigation frame. The position and Euler angles of AUVs in the local frame are denoted here by symbol [ x, y, z , , , ]T ,
then J (),
(1) (2)
where J () is the direction cosine matrix that can be determined through four-parameters Rodring-Hamilton 0 , 1 , 2 , 3 as follows:
Nguyen Quang Vinh et al. / Procedia Computer Science 150 (2019) 69–77 Nguyen Quang Vinh and Pham Van Phuc / Procedia Computer Science 00 (2019) 000–000
202 212 -1 J ( ) 212 20 3 213 20 2
212 20 3 202 222 -1 22 3 20 1
213 20 2 22 3 20 1 . 202 232 -1
71 3
(3)
Motion equations of the AUVs are described in the body frame [6]: M RB CRB () RB ,
(4)
where M RB is the inertia matrix, CRB () is the Coriolis matrix, RB is a generalized vector of external forces and moments: RB M A C A () D () L() g () ,
(5)
where g ( ) is the buoyancy forces and moments, M A is the inertia matrix of added mass, C A () is the Coriolis matrix of added mass, D() is the potential damping matrix, L() is the forces and moments parameters matrix of rudder, is the forces and moments of rudder and propellers. From (2), (4), (5), the 6-DOF dynamic equation can be written as: M C ( ) D ( ) L( ) g ( ) , ( ) , y , J
(6)
where M M A M RB , C A ( ) CRB (), y is the output system. From (2), we have J () 1 ,
(7)
J () J () 1 ]. J () 1[
(8)
Substituting (7) and (8) into (6) yields. C (, ) g () , M ()
(9)
where M () J () T MJ ()1 , C (, ) J () T [ MJ () 1 J () C () D() L()], g () J () T g (), J () T .
From (2) and (9) the AUV dynamic equations in the body frame with multiple inputs and multiple outputs: ( ) , y , J M ( ) 1 C ( , ) M ( ) 1 g ( ) M ( ) 1 ,
An algorithm for detemining the navigation parameters of AUV vehicles is described in [4] of the author.
(10)
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3. A direct adaptive fuzzy-neural output-feedback control for AUV AUVs are controlled to follow the yaw, roll and pitch angle. The pitch angle is controlled by two rudders drive in the horizontal plane s1 , s 2 with s1 s 2 . Two rudders h1 , h 2 control AUV’s both yaw and roll. Defining uh is yaw control signal, ul is roll control signal of the AUVs, the relationship between uh and ul is written ul
h1 h 2 h 2 h1 , uh , 2 2
(11)
rotation angles of rudder control yaw and roll is [4]:
h1 ul uh , h 2 ul uh .
(12)
The dynamic equation can be written as MIMO nonlinear equations (13). x A0 x B F ( x ) G ( x )u d ; y C T x,
(13)
T where inputs of the system are the rudder angles: u [u [ h , s , l ]T ; outputs of the system are the yaw, 1 , u2 , u3 ]
pitch B
y1 , y2 , y3 ]T [ , , ]T ; d and roll y [
is
the
ocean currents; A0 diag[ A01 , A02 , A03 ] R 66 ;
diag[ B1 , B2 , B3 ] R 63 ; C diag[C1 , C2 , C3 ] R 63 ;
0 1 0 1 A0 k , Bk 1 , Ck 0 , k 1 3 , use throughout this paper. 0 0 F ( x) M n ( ) 1 C ( , ) M n ( ) g ( ) M n ( ) 1 J ( ) T pl ; G ( x)u M ( ) 1 J ( ) T bl ; x [ y1 , y1 , y2 , y 2 , y3 , y3 ]T is a vector of states.
The objective of this paper is to design a DAFNOC so that outputs of system y [ y1 , y2 , y3 ]T track reference signal y r [ yr1 , yr 2 , yr 3 ]T . Define y r [ yr1 , yr 2 , yr 3 ]T ; Y r [ yr1 , y r1 , yr 2 , y r 2 , yr 3 , y r 3 ]T ; e Y r x; eˆ Y r xˆ when T T [e1 , e12 , e21 , e22 , e31 e [e1 , e1 , e2 , e , e32 ]T E1 [e [e11 , e21 , e31 ]T , 2 , e3 , e3 ] 1 , e2 , e3 ] where eˆ and xˆ denote the estimates of e and x. Based on the certainty equivalence approach and Lyapunov, an optimal control law is: * T u G 1 ( x)[ F ( x) y r K c eˆ],
where K c
(14) T
diag [ K c1 , K c 2 , K c 3 ], K ckT [ K ck 1 , K ck12 ]. Choose K c such that Ak A0 k Bk K ck is Hurwitz matrix.
AUV’s dynamic equation (13) with output signal vector y is received from the navigation system, Since only the outputs of system are assumed to be measurable, the optimal control law cannot be implemented (14). Thus, control law of DAFNOC is proposed: u u f v,
(15)
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T where u f [u [v1 , v2 , v3 ]T , here u f is designed to approximate the optimal control law (14) and f 1,u f 2 , u f 3 ] , v
the control term v is employed to compensate the external disturbance and the modeling error. Design the state observer [5]. eˆ A0 eˆ BK cT eˆ K 0 ( E1 Eˆ 1 ), T Eˆ C eˆ,
(16)
1
dial [ K 01 , K 02 , K 03 ], K 0 k dial[ K 0 k1 , K 0 k 2 ]T ; matrix A0 k K 0 k CkT is strictly Hurwitz matrix [5]. We define the observation errors as e e eˆ and E E Eˆ . From (13-16) we have
where K 0
1
1
1
* e ( A0 K 0 C T )e B[G ( x)u G ( x)u f G ( x)v d ],
E1 C T e.
(17)
To calculate u fk we use the configuration of a fuzzy-neural approximator (Fig. 2) [6]. The fuzzy inference engine uses the fuzzy IF-THEN rules to perform a mapping from an input estimated error vector eˆ k [eˆk1 , eˆk 2 ]T . The i -th fuzzy IF-THEN rule is written as (i 1, 2,3, ..., h). If eˆk1 is Aki 1 and eˆk 2 is Aki 2 than u fk is Bki , where Aki 1 , Aki 2 and Bki are fuzzy sets.
By using produce inference, center average and singleton fuzzifier, the output of the fuzzy-neural network can be expressed as [3]. (eˆkj ) i 1 j 1 T (eˆ ). u fk k k k h 2 Akji (eˆkj ) i 1 j 1 h
2
T
T
i k
Akji
(18)
T
We have u f u f (eˆ ) [1 1 (eˆ1 ), 2 2 (eˆ 2 ), 3 3 (eˆ3 )]T ; where k [ 1k , 2k , ..., hk ]T is an adjustable parameter 1, k (eˆ k ) [1k , k2 , ..., hk ]T is a fuzzy basis vector vector. ik is the point at which Bi (ik ) k
2
j 1
Akji
(eˆkj )
. (eˆ k ) h 2 Akji (eˆkj ) i 1 j 1 i k
(19)
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Fig. 2. Configuration of a fuzzy-neural network.
In order to employ the SPR-Lyapunov design approach, (17) can be written as [6]. T
ek1 H k ( s ) Lk ( s )[ˆ k k (eˆ k ) v fk w fk ],
(20)
where H k ( s) CkT ( sI ( A0 k K 0 k CkT )) 1 Bk , k (eˆ k ) Lk1 ( s)[k (eˆ k )] and Lk ( s ) choosen so that Lk1 ( s ) is a proper stable transfer function and H k ( s ) Lk ( s ) is a proper SPR transfer function. Assumption w fk is w fk k , where k 0. Supposed that the update laws are chosen as k ek1k (eˆ k ) if || k || mk or T (|| k || mk and ek1 k k (eˆ k ) 0) k mk Pr( k ek 1k (eˆ k )) if || k || T and ek1 k k (eˆ k ) 0.
When Pr( k ek1k (eˆ k )) k ek1k (eˆ k ) k
(21)
T ek1 k k (eˆ k ) k then || k || mk and || k || mk . Let vk be given as || k ||2
if ek 1 0 and | ek 1 | k , k vk k if ek1 0 and | ek1 | k , e if | e | , k1 k k k1 k
(22)
where k k 0 . Consider the Lyapunov 3
V Vk , where Vk k 1
1 T 1 T e k Pk e k k k with Pk PkT 0. 2 2 k
(23)
Differentiating Vk with respect to time we have: 1 T 1 T 1 Vk e k Pk e k e k Pk e k Tk k . k 2 2
(24)
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QkT 0 such that Because H k ( s ) Lk ( s ) is SPR, there exists Q k 1 T 1 T Vk e k Qk e k e k 1[ k k (eˆ k ) v fk w fk ] k k . k 2
(25)
By using assumptions we have: 1 Vk min (Qk ) | ek 1 |2 . 2
(26)
As a result 3
V
V
3 1 min (Qk ) | ek 1 |2 . 2
k k 1 k 1
(27)
The stability of the overall control system is analyzed based on the Lyapunov method. Using Barbalat’s lemma, we have lim | ek1 (t ) | 0. Because A0 k Bk K ckT is Hurwitz matrix, e k (t ) and eˆ k (t ) are t
bounded, e e k eˆ k and ek1 , e k L we conclude that ek1 0 as t . The update law (21) with all signals of k the closed-loop are bounded, E1 (t ) 0 as t . The hedge algebra controller looks at the [4]. 4. The orientation control of underwater vehicles
Following Zhang et al one can use the target function for the controller in the following form [6]: E k1
2 1 2 1 kd k 1 RK 1rk2 , 2
(28)
where RK and rk , respectively are the steering angle and the changing orientation rate of the underwater vehicle at the moment k , respectively, constants 1 , 1 and 1 , respectively are the proportion coefficient, the feedback coefficient and the orientation differential coefficient, respectively. The HAC or DAFNOC is described as in Fig. 3.
Fig. 3. The simulation scheme of the orientation control of the underwater vehicle using HA or DAFNOC.
5. Simulations, computations, discussions
To test the algorithms above, we use the input data of underwater vehicle [3]. The wanted trajectory is the line connecting point AUVs touching water (the starting position) and the pickup shot poin(the destination) determined before throwing AUV. Suppose that the starting position of AUV has coordinates ( x0 , y0 , z0 ) (100,100, 20) with beginning state angles ( 0 , 0 , 0 (00 , 180 ,30 ), the finishing point
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has coordinates ( x2 , y2 , z2 ) (1100,1100,120). External noise is the influence of ocean flows considered at the moment 20 s: [uc , vc , wc ]T [4,4,0.5]T (m/s). The actual test model is shown in Fig. 4.
Fig.4. AUV experiment on the sea.
Consider the vertical plane with assumptions w p q 0, the orientation angle is small [4]. We use the HA orientation controller with the above designing parameters responsible to the orientation control of underwater vehicles as in Fig. 5. The simulation results in Fig. 5 show that under the influence of ocean flows the parameters of the controller are updated online, hence the turning angle of the helm will change to make the system less affected and catch the wanted trajectory quickly. The trajectory of the underwater vehicle at the starting moment of simulation has a deviation since the starting orientation angle does not coincide with the line of sight of the starting position and the destination (LOS). With the DAFNOC approach the yaw angle quickly achieves the required set value and low mobility, the HA method is highly mobile, with a slow but fast calculation. The simulation results show that with the online updating law of the controller parameters, outputs of the system quickly adapt to the effects of the ocean currents and the interaction between the control channels which suggests possible application DAFNOC to control AUVs in real time.
Fig. 5. The simulation results of the AUV orientation control.
6. Conclusions
In this paper we have presented the system of the motion equations of an underwater vehicle controlled by two orientation helms, two depth helms of the 6th degree of freedom under the influences of the flow and measurement noise. We have applied HA and DAFNOC to simulate the single channel orientation and compare them using the direct adaptive fuzzy-neural output-feedback control and analyzed the adaptive ability of the system under the influences of the flow and measurement noise.
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In the future we will do research on more various conditions, in more difficult tasks and analyze the adaptive ability of the controller and also prove the stability of the entire system in order to construct a controller for the entire motion function of underwater vehicles. References [1] Wang R, Wang S, Wang Y. A Hybrid Heading Control Scheme for a Biomimetic Underwater Vehicle. 26th International Ocean and Polar Engineering Conference. Rhodes, Greece: ISOPE, 2016; p. 619-625. [2] Fossen ThI. Maritime Control Systems – Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles. Marine Cybernetics. Trondheim, Norway, 2002. [3] Nguyen Quang Vinh, Truong Duy Trung. Guidance, navigation and control of Autonomous underwater vehicle. International Symposium on Electrical-Electronics, 2013, 36: p. 44-49. [4] Nguyen Quang Vinh, Control of the motion orientation and the depth of underwater vehicles by hedge algebras. The XVIth International symposium “Intelligent Systems” , INTELS’16,2016, Procedia Computer Science. 103: p. 331–338. [5] Nguyen Quang Vinh. An algorithm for determining the navigation parameters of UAVs based on the combination of measuring devices. IEEE Trans. on Automatic Control, 1996; 41-3: p. 447-451. [6] Zhang Y, Hearn GE, Sen P. Neural network approaches to a class of ship control problems. Part I, II. Eleventh Ship Control Systems Symposium 1997; 1. [7] Zhang T, Ge SS, Hang CC. Adaptive neural network control for strict-feedback nonlinear systems using backstepping design. Automatica, 2000; 36: p. 1835-1846. [8] Brandt RD, Lin F. Adaptive interaction and its application to neural networks, 1999. Elsevier, Information Science. 121: p. 201-215. [9] Fossen TI, Pettersen KY, Nijmeijer H. Sensing and control for autonomous vehicles. Springer, 2017; 474. [10] Nguyen Quang Vinh, Nguyen Duc Anh, Phan Tuong Lai. The control of the motion of airplanes using hecke algebras. Journal of military scientific and technological researches 2015; 37: p. 28-102.