Proceedings of the 2002 IEEE International Conference on Robotics & Automation Washington, DC • May 2002
Neural Net Based Nonlinear Adaptive Control for Autonomous Underwater Vehicles Ji-Hong Li*, Pan-Mook Lee** and Sang-Jeong Lee* * Department of Electronics Engineering, Chungnam National University 220 Goong-dong, Yusong-gu, Daejeon, 305-764, Korea
[email protected],
[email protected] ** Korea Research Institute of Ships & Ocean Engineering, KORDI
[email protected]
Abstract
networks were updated by using back propagation learning algorithms. Back propagation algorithms, one of the gradient descent methods, are widely used weights adjustment methods for neural networks and have been proved to be quite effective in practice. However, it is unable to obtain analytical results concerning the convergence and stability of the networks[10]. Recently, there have been a concentrated efforts toward the design and analysis of learning algorithms based on Lyapunov stability theory[10]-[15]. In these literatures, various learning algorithms were proposed to guarantee the boundedness of the network’s weights estimation errors and tracking errors. Kosmatopoulos et al.[10] developed gradient methods for recurrent high-order neural network models and analyzed their stability properties. Sannar and Slotine [12] introduced a radial basis Gaussian network, where parameter convergence of the network was satisfied only in the case that the signal matrix is persistently excited. In [11][13][14], a general LPNN scheme was used to design stable nonlinear adaptive controls, and network’s weights adaptation laws were analytically de-rived according to the Lyapunov theory. In order to guarantee the boundedness of the network’s weights estimation errors, Gong and Yao[15] proposed a projection algorithm accompanied with a general multi-layer neural network in [15].
Since the dynamics of autonomous underwater vehicles (AUVs) are highly nonlinear and their hydrodynamic coefficients vary with different operating conditions, a high performance control system of an AUV is needed to have the capacities of learning and adaptation to the variations of the AUV’s dynamics. In this paper, a linearly parameterized neural network (LPNN) is used to approximate the uncertainties of the vehicles’ dynamics, where the basis function vector of the network is constructed according to the vehicle’s physical properties. The proposed controller guarantees uniform boundedness of the vehicle’s trajectory tracking errors and network’s weights estimation errors based on Lyapunov stability theory, where the network’s reconstruction errors and the disturbances in the vehicle’s dynamics are bounded by unknown constant. Numerical simulation studies are performed to illustrate the effectiveness of the proposed control scheme.
1. Introduction The major difficulties in control problems for AUVs are that: the vehicle’s dynamics are nonlinear and timevarying and the hydrodynamic coefficients are hard to be estimated accurately because of their variations against different maneuvering conditions. Conventional controllers such as PID controllers are limited to handle these problems properly and may cause poor performance. Therefore, high performance control systems for AUVs are required to have the capacities of learning and adaptation to the changes of the vehicle’s dynamics to provide desired performance[1].
Another main issue, in the applications of neural networks, is the robustness of the designed control systems. When we approximate a poorly known system with networks, because of deficient information on the unknown dynamics, we could not exactly determined the number of weights between input layer and hidden layer[10][12][15] or the dimension of basis function vector[13][14]. Therefore, while we construct a neural network to approximate a given unknown dynamics, it always remains reconstruction error. Thus, the robustness of the designed control systems should be considered.
Based on the approximation capacity of neural networks for nonlinear mappings[2]-[5] and their learning characteristics, there have been considerable studies in exploring neural networks to the control systems for underwater robotics vehicles[6]-[9]. In these literatures, multi-layer neural networks were used to approximate the nonlinear dynamics of the vehicles and the weights of the neural 0-7803-7272-7/02/$17.00 © 2002 IEEE
This paper proposes a neural net based nonlinear adaptive controller for AUVs. Two-layer neural network, of which the architecture is similar to the one in [13], is applied to 1075
structed network, φ ( x) ∈ ℜ Nr is the constructed basis function vector, Nr is the dimension of φ (x) , and ε r (x) is network’s reconstruction error vector.
approximate the nonlinear uncertainties in the AUV’s dynamics and the basis function vector of the network is constructed according to the AUV’s physical properties. The uniform boundedness of the network’s weights estimation errors and tracking errors are guaranteed on the basis of Lyapunov stability theory, for the case where the network’s reconstruction errors are bounded by unknown constants. Numerical simulation studies on the motion control of an AUV are performed to illustrate the effectiveness of the proposed controller.
B. AUVs Dynamics Dynamic equations of AUVs in 6 degree of freedom can be expressed as [16] M (q)q&& + C D (q, q& )q& + g (q) + d = u ,
where q ∈ ℜ 6 is the position and orientation vector, M ∈ ℜ 6×6 is the inertia matrix (including added mass), CD ∈ ℜ6×6 is the matrix of the Coriolis, centripetal and damping term, g ∈ ℜ 6 is the gravitational forces and moments vector, and u ∈ ℜ 6 is the control input vector. d ∈ ℜ 6 is a disturbance vector.
2. Problem Statements A. Functional Approximation using Neural Networks A feedforward one-hidden layer neural network’s output can be expressed as y = W *T σ (V T x + θ ) , (1)
In equation (5), the inertia matrix M (q ) can be divided into the two arguments
where x ∈ ℜ n , y ∈ ℜ m , V ∈ ℜ n× N , W * ∈ ℜ N × m , θ ∈ ℜ N , N is the number of hidden neurons, and σ (⋅) is activation function (vector) defined as: with a given vector β = [ β 1 , L , β N ]T , then σ ( β ) = [σ ( β 1 ), L , σ ( β N )]T .
M (q) = M RB + M A ( q) ,
Given a desired trajectory q d (t ) ∈ ℜ 6 , vehicle’s tracking error vector is given by e(t ) = q (t ) − q d (t ) ,
(7)
and combined tracking error vector is defined as
(2)
s (t ) = e&(t ) + Λe(t ) ,
On the other hands, given a f ( x) ∈ C ( K ) , K ⊂ ℜ m , it can always be written in the parametric form [5] f ( x) = W *T φ * ( x) ,
(6)
where M RB = diag{m, L , m} > 0 is vehicle’s rigid-body inertia matrix and M A is added inertia matrix. In fact, M RB is constant and can be exactly known while M A is varying with vehicle’s different operating conditions.
Depending on a suitable activation function σ (⋅) , the form of the equation (1) is dense in the space of the continuous functions C (K ) defined on some compact set K ⊂ ℜm [2]-[5]. In other words, for a given f ( x) ∈ C ( K ) and ∀ε > 0 , there exists an integer N and real constants W * , V and θ such that || f ( x) − y ||< ε .
(5)
(8)
where Λ : Λ = ΛT > 0 is a design matrix. Using (6) ~ (8), the equation (5) can be rewritten as
(3)
where W * ∈ ℜ N ×m is a constant matrix and φ * ( x) ∈ ℜ N is a basis function vector of f (x) . Thus the problems of functional approximation using neural network can be expressed as the selection of a suitable activation function σ and parameters V and θ with σ (⋅) = φ * ( x) . If we exactly know the basis of a function, functional approximation problem can be referred to a parameter estimation problem. However, some basic questions, such as how to select the parameters V , θ and N in equation (1) for a given unknown function, still remain in open.
M RB s& = − M RB [q&&d − Λ (q& − q& d )] + f − d + u ,
(9)
f (q, q& , q&&) = − M A q&& − C D q& − g .
(10)
where In the applications of underwater vehicles, it is hard to priori determine the exact values of three terms of the right side of the equation (10) because of the highly uncertain nature of the vehicle’s dynamics. Using equation (4), AUV’s combined error dynamic equation (9) can be rewritten as
In practice, while we construct a neural network to approximate an unknown function, there always remain reconstruction errors because of deficient information on given unknown function. Thus, equation (3) can be expressed as (4)
M RB s& = −M RB [q&&d − Λ(q& − q& d )] + W T φ (q, q&, q&&) + ε r − d + u , (11) where W ∈ ℜ N ×6 is the idle constant weight matrix of constructed neural network, φ (q, q& , q&&) is the constructed basis function vector, and N is the dimension of φ (q, q& , q&&) .
where W ∈ ℜ Nr ×m is the idle weights matrix of the con-
In order to design a stable nonlinear adaptive control for
f ( x ) = W T φ ( x) + ε r ( x ) ,
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AUVs, we make the following practical assumptions on dynamics (9) and on the network’s reconstruction error vector.
M RB[(⋅) − Λ(⋅)]
qd q& d q&&d
Assumption 1: Nonlinear uncertainties f (q, q& , q&&) in AUV’s dynamics, expressed in (10), are continuous on some compact set K ⊂ ℜ 6 .
⊕-
Linear Feedback
⊕
q & q q&&
AUV
Adaptive Neural Network
Assumption 2: Network’s reconstruction error and disturbance are bounded although their bounds may not be known. Thus, without loss of generality, assume that || ε r − d ||2 ≤ || ε r ||2 + || d ||2 ≤ ε max , where || ⋅ || 2 denotes Euclidean norm, and ε max is an unknown positive constant.
Fig. 1 Structure of neural net based adaptive controller
control law expressed by (15), and the neural network’s weights’ adaptation laws are chosen as & Wˆ = φ (q, q& , q&&) s T − αWˆ ,
3. Neural Net Based Adaptive Controller
(17)
where α is a design parameter. Then the combined tracking error s and the network’s weights estimation error ~ W are all uniformly bounded.
The difficulty to handle the AUV’s control problem is mainly induced by nonlinear uncertainties in the AUV’s dynamics. In this paper, we attempt to approximate these uncertainties using a neural network.
Proof: In order to illustrate the boundedness of the combined tracking errors and network’s weights estimation errors, we define the Lyapunov candidate as ~ ~ V = 1 / 2 s T M RB s + 1 / 2tr (W T W ) (18)
According to equation (4), the output of the constructed neural network can be written as fˆ (q, q& , q&&) = Wˆ T φ (q, q& , q&&) ,
+
(12)
where Wˆ is the estimation of W in (4). The purpose for the approximation of f (q, q& , q&&) expressed in (10) is to remove this nonlinear uncertainty from the combined error dynamics (9). Let we select the control law as
Differentiating equation (18) yields
(13)
~ ~& (19) V& = s T M RB s& + tr (W T W ) ~& & & where W = W& − Wˆ = −Wˆ , and trace tr denotes the sum of the diagonal elements of the matrix.
where the selection of u a will be discussed later. Substituting equation (13) into (11), we can get ~ M RB s& = − M RB [q&&d − Λ (q& − q& d )] + W T φ (q, q& , q&&) (14) + ε r − d + ua ,
Substituting (16) and (17) into (19), we can get ~ ~ V& = − s T K PD s + s T (ε r − d ) + s T W T φ − tr (W T (φs T − αWˆ )) ~ ~ = − s T K PD s + s T (ε r − d ) + s T W T φ − tr (W T φs T ) (20) ~ + αtr (W T Wˆ )
u = u a − fˆ (q, q& , q&&) ,
~
where W = W − Wˆ is the weights’ estimation error matrix. Since first term of the right side of (14) is an undesirable dynamics, we attempt to remove this term using u a . Consequently, the selected control law (Fig. 1) is u = M RB [q&&d − Λ (q& − q& d )] − u PD − Wˆ T φ (q, q& , q&&) ,
For arbitrary given two vectors x, y ∈ ℜ n , it is obvious that tr ( xy T ) = x T y = y T x , from which we can conclude ~ ~ that s T W T φ = tr (W T φs T ) . So the equation (20) becomes ~ V& = − s T K PD s + s T (ε r − d ) + αtr (W T Wˆ ) ~ ~ ~ = − s T K PD s + s T (ε r − d ) + αtr (W T W ) − αtr (W T W ) ~ 2 ~ ≤ − s T K PD s + s T (ε r − d ) + α || W || F || W || F −α || W || F ~ 2 ≤ − a min || s || 2 + || s || 2 || ε r − d || 2 +α || W || F || W || F ~ 2 ~ 2 2 − α || W || F ≤ − a min || s || 2 +ε max || s || 2 − α || W || F ~ + α || W || F || W || F , (21)
(15)
where linear feedback control term u PD is selected as u PD = K PD s , where K PD = diag{a1 , L , a 6 } and a i is a positive design parameter. Substituting equation (15) into (11), AUV’s combined error dynamics can be rewritten as ~ M RB s& = − K PD s + W T φ (q, q& , q&&) + ε r − d . (16)
where || ⋅ || F denotes Frobenius matrix norm, and a min is defined as a min = min{a1 , L , a 6 } with a i defined in (15).
Theorem 1: Consider the nonlinear dynamical system described by (5) with Assumptions 1-2. If we suggest a
Boundedness of tracking error
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Rewrite the equation (21) as 2 V& ≤ − a min || s || 2 +ε max
+
α
|| W || F ≤ − a min || s || 2 +ε max || s || 2 + 2
4
the functional approximation capacity of neural networks completely, the constructed basis function vector needs to satisfy exciting conditions in practice.
~ || s || 2 −α (|| W || F −1 / 2 || W || F ) 2 2
α 4
2
|| W || F ,
4. Simulation Studies A. Construction of two-layer neural network
Since a min > 0 , if || s || 2 ≥
ε max + ε 2 max + a min α || W || F 2 2a min
,
The constructed neural network depicted in Fig. 2 has a two-layer structure, and the basis function vector is constructed according to the physical properties of AUVs.
(22)
In the AUV’s 6 DOF dynamical equation (5), position and orientation vector q can be expressed as
then V& ≤ 0 whatever values the network’s weights estimation errors take.
q = [ x y z φ θ ψ ]T ,
Boundedness of network’s weights estimation errors Rewrite the equation (21) as
where x, y, z denote the position along the three axis, φ , θ ,ψ are Euler angles of the vehicle (in earth frame).
ε ~ 2 ~ V& ≤ −α || W || F +α || W || F || W || F −a min (|| s || 2 − max ) 2 2a min +
ε
2
max
4a min
(24)
In general, underwater vehicle’s dynamics on each axis are coupled with each other by q& , q&& and Euler angles. Here we consider the nonlinear uncertainties expressed in (10) again. According to the physical properties of AUVs[16], gravitational term g in the right side of (10), is concerned with signoidal function of φ and θ , therefore the corresponding basis function vector component φ 3 is selected as φ 3 = [G T (G ⊗ G ) T ]T , (25)
ε max ~ 2 ~ ≤ −α || W || F +α || W || F || W || F + . 4a min 2
Since α > 0 , if || W || F + || W || F +ε 2 max /(αa min ) ~ || W || F ≥ , (23) 2 then V& ≤ 0 whatever value the tracking error takes. 2
where G is defined as G = [sin φ sin θ cos φ cos θ ]T and ⊗ denotes Kronecker product. The first argument of the right side of (10) is concerned with q&& and the second is concerned with q& . The basis function vector components φ1 and φ 2 are selected as
This demonstrates that the tracking errors and the network’s weights estimation errors are all uniformly bounded on the basis of the Lyapunov stability theory.
Remark 1: AUV’s dynamics are subject to structured and/or unstructured uncertainties. If the basis function vector φ * ( q, q& , q&&) of f (q, q& , q&&) in (10) is exactly known, then AUV’s dynamics are subject only to structured uncertainties (unknown idle weights values). However, in practice we could not exactly know the basis function vector, so there always remain unstructured uncertainties (neural network’s reconstruction errors). In this paper we assume that these unstructured uncertainties are bounded even though their bounds may not be known, as well as the disturbances in AUV’s dynamics. Remark 2: According to equation (22) and (23), it is obvious that increasing design parameter a min can result in decrease of boundaries of tracking error and network’s weights estimation errors, and parameter α offers practical trade-off between tracking error bounds and weights estimation error bounds as Lewis et al.[13] described as well. Remark 3: In this paper, the main focus is taken on to guarantee the stability of the proposed control scheme when the control system contains unknown but bounded unstructured uncertainties. However, in order to exploit
φ1 = q&&, φ 2 = [q& T (q& ⊗ q& ) T ]T .
(26)
According to above discussions, we construct the basis function vector as
φ (q, q& , q&&) = [φ1T φ 2 T φ 3 T ]T = [q&&T q& T (q& ⊗ q& ) T G T (G ⊗ G ) T ]T .
(27)
In most of the presented literatures on applications of underwater vehicles, only up to second order velocity of vehicles are considered because the parameters of higher orders are negligible and hard to be determined exactly. The structure of the basis function vector expressed in (27) needs to be modified in practice to satisfy suitable performance for other applications.
B. Simulation Results The proposed neural net based adaptive control scheme is applied to the motion control of an AUV, where the nonlinear dynamics expressed in (10) are assumed to be unknown. SAUV(semi-autonomous underwater vehicle)
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q& q&&
Input Signal Pre-Processing
q
wˆ ij
5. Conclusions fˆ1
This paper presents a neural net based nonlinear adaptive controller for an AUV, where AUV’s dynamics contain nonlinear uncertainties. Applied network has a two-layer structure and network’s basis function vector is constructed according to the physical properties of AUVs. Neural network’s weights adaptation laws are derived according to Lyapunov stability theory, and practical boundary of the tracking errors and the network’s weights estimation errors are given. Simulation studies show the suitability of proposed control scheme for high performance motion control of an AUV.
fˆ2 fˆ6
Fig. 2 Two-layer neural network structure developed in KRISO[17] is used in this simulation study. SAUV has four propellers mounted at the stern of the vehicle, and the control input torque is taken as
τ = [τ x 0 0 0 τ θ τ ψ ]T ,
6. Acknowledgment This work is supported in part by the Ministry of Maritime Affairs and Fisheries and the Ministry of Science and Technology, Korea.
(28)
where τ x , τ θ and τ ψ are coupled thrust and moments depending on the propellers’ RPM.
References
The desired trajectory is selected as q& d = [2.5 + sin ωt , 0, 0, 0, 0, 0]T ,
[1] S. K. Choi and J. Yuh, “Experimental Study on a Learning Control System with Bound Estimation for Underwater Robots,” Int. Journal of Autonomous Robots, Vol. 3, pp. 187-194, 1996. [2] G. Cybenko, “Approximation by Superpositions of a Sigmoidal Function,” Mathematics of Control, Signals and Systems, vol. 2, pp. 303-314, 1989. [3] K. Hornik, M. Stinchcombe, and H. White, “Multilayer Feedforward Networks are Universal Approximators,” Neural Networks, Vol. 2, pp. 359-366, 1989. [4] K. I. Funahashi, “On the Approximate Realization of Continous Mappings by Neural Networks,” Neural Networks, Vol. 2, pp. 183-192, 1989. [5] F. Girosi and T. Poggio, “Networks and the Best Approximation Property,” Biological Cybernetics, 63, pp. 169-176, 1990. [6] K. P. Venugopal, R. Sudhakar, and A. S. Pandya, “On-Line Learning Control of Autonomous Underwater Vehicle Using Feedforward Neural Networks,” IEEE Journal of Oceanic Engineering, Vol. 17, No. 4, pp. 308-319, October 1992. [7] Yuh, J., “Learning Control for Underwater Robotic Vehicles,” IEEE Control Systems Magazine, pp. 39-46, 1994. [8] K. Ishii, T. Fujii, and T. Ura, “An On-line Adaptation Method in a Neural Network Based Control System for AUV’s,” IEEE Journal of Oceanic Engineering, Vol. 20, No. 3, pp. 221-228, July 1995. [9] N. E. Leonard, “Control Synthesis and Adaptation for an Underactuated Autonomous Underwater Vehicle,” IEEE Journal of Oceanic Engineering, Vol. 20, No. 3, pp. 211220, July 1995. [10] E. B. Kosmatopoulos, M. M. Polycarpou, M. A.
(29)
and the parameters used in the simulations are as follows: Λ = diag{0.5, 0.5, 0.5, 1, 1, 1}, K PD = diag{200, 0, 0, 0, 300, 300}, K rb = diag{600, 0, 0, 0, 50, 50},
(30)
Φ 1 = 0.01, Φ 5 = Φ 6 = 0.005, Φ 2 = Φ 3 = Φ 4 = 0, where K rb is the sliding mode switching discontinuity gain matrix, and Φ i is a boundary layer thickness near the sliding surface. Simulation results are depicted in Fig. 3 and 4. Fig. 3 shows the comparison between the true values of nonlinear uncertainties f x and f θ express in (10) and their approximations by neural network. The true values of the uncertainties are calculated with the full nonlinear equations of motion of SAUV. In order to illustrate the effecttiveness of the proposed control scheme, we performed two simulation tests, one is the proposed control scheme, and the other is one in which linear feedback control is applied combined with a sliding mode scheme. Tests results are shown in Fig. 4. From Fig. 4(b), we can see that the tracking error on z axis is diverging in the case where only the sliding mode and linear feedback schemes are used. On the other hand, when the proposed controller is applied, the tracking error on z axis is bounded within a small value. From these simulation results, we can see that the constructed neural network has a satisfactory approximation capacity under the above simulation conditions and weights update laws (17) and clearly result in superior tracking performance.
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[16] T. I. Fossen, Guidance and Control of Ocean Vehicles, John Wiley & Sons, 1994. [17] S. W. Hong et al., “Development of Technologies for Navigation and Manipulator System of a SemiAutonomous Underwater Vehicle”, Tech. Rep. 99-M-DU21-C-01, KORDI, Daejeon, Korea, 2000(in Korean).
(a)
(b)
(a)
(b)
(c)
Fig. 3 Nonlinear uncertainties and their approximations by neural network. (a) Uncertainty in the dynamics of x axis. (b) Uncertainty in pitch dynamics.
Fig. 4 Tracking performance comparison between linear feedback + sliding mode and linear feedback + neural network. (a) Along x axis. (b) Along z axis. (c) Along pitch axis.
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