Stable Path Tracking Control of a Mobile Robot

International Journal Control, and Systems, 3, no. 4, pp. 552-563, December 2005 JoonofSeop Oh,Automation, Jin Bae Park, and Yoon Hovol. Choi

552

Stable Path Tracking Control of a Mobile Robot Using a Wavelet Based Fuzzy Neural Network Joon Seop Oh, Jin Bae Park*, and Yoon Ho Choi Abstract: In this paper, we propose a wavelet based fuzzy neural network (WFNN) based direct adaptive control scheme for the solution of the tracking problem of mobile robots. To design a controller, we present a WFNN structure that merges the advantages of the neural network, fuzzy model and wavelet transform. The basic idea of our WFNN structure is to realize the process of fuzzy reasoning of the wavelet fuzzy system by the structure of a neural network and to make the parameters of fuzzy reasoning be expressed by the connection weights of a neural network. In our control system, the control signals are directly obtained to minimize the difference between the reference track and the pose of a mobile robot via the gradient descent (GD) method. In addition, an approach that uses adaptive learning rates for training of the WFNN controller is driven via a Lyapunov stability analysis to guarantee fast convergence, that is, learning rates are adaptively determined to rapidly minimize the state errors of a mobile robot. Finally, to evaluate the performance of the proposed direct adaptive control system using the WFNN controller, we compare the control results of the WFNN controller with those of the FNN, the WNN and the WFM controllers. Keywords: Fuzzy neural network, gradient descent method, Lyapunov stability, mobile robot, path tracking control, wavelet fuzzy model, wavelet neural network.

1. INTRODUCTION The localization and path tracking problems for mobile robots have been given great attention by automatic control researchers in recently published literature. The motion control of mobile robots is a typical nonlinear tracking control issue and has been discussed with different control schemes such as PID, GPC, sliding mode, predictive control, etc., [1-6]. Intelligent control techniques, based on neural networks and fuzzy logic, have also been developed for path tracking control of mobile robots [7,8]. Even though these intelligent control strategies have shown their effectiveness, especially for nonlinear systems, they have certain drawbacks due to their own characteristics. While conventional neural networks have good ability for self-learning, they also have some limitations such as slow convergence, the __________ Manuscript received February 28, 2005; revised August 22, 2005; accepted September 5, 2005. Recommended by Editor Jae-Bok Song under the direction of Editor-in-Chief Myung Jin Chung. Joon Seop Oh and Jin Bae Park are with the Department of Electrical and Electronic Engineering, Yonsei University, Sinchon-Dong, Seodaemun-Gu, Seoul 120-749, Korea (emails: {jsoh, jbpark}@control.yonsei.ac.kr). Yoon Ho Choi is with the School of Electronic Engineering, Kyonggi University, Kyonggi-Do, Suwon-Si, 443-760, Korea (e-mail: [email protected]). * Corresponding author.

difficulty in reaching the global minima in the parameter space, and sometimes even instability as well. In the case of fuzzy logic, it is a human-imitating logic, but lacks the ability for self-learning and selftuning. Therefore, in the research area of intelligent control, fuzzy neural networks (FNNs) are devised to overcome these limitations and to combine the advantages of both neural networks and fuzzy logic [9-11]. This provides a strong motivation for using FNNs in the modeling and control of nonlinear systems. The wavelet fuzzy model (WFM) has the advantage of wavelet transform by constituting the fuzzy basis function (FBF) and the conclusion part to equalize the linear combination of FBF with the linear combination of wavelet functions [12-15]. The conventional fuzzy model cannot provide a satisfactory result for the transient signal. On the contrary, in the case of the WFM, the accurate fuzzy model can be obtained because the energy compaction by the unconditional basis and the description of a transient signal by wavelet basis functions are distinguished [16,17]. Therefore, we have designed a FNN structure based on wavelet, which merges the advantages of neural network, fuzzy model and wavelet. The basic idea of the wavelet based fuzzy neural network (WFNN) is to realize the process of fuzzy reasoning of the WFM by the structure of a neural network and to make the parameters of fuzzy reasoning be expressed by the connection weights of a

Stable Path Tracking Control of a Mobile Robot Using a Wavelet Based Fuzzy Neural Network

neural network. An approach that uses adaptive learning rates is driven via a Lyapunov stability analysis to guarantee fast convergence. In this paper, we design the direct adaptive control system using the WFNN structure. The control inputs are directly obtained by minimizing the difference between the reference track and the pose of a mobile robot that is controlled through a WFNN controller. The control process is a dynamic on-line one that uses the WFNN trained by the gradient descent (GD) method. Through computer simulations, we demonstrate the effectiveness and feasibility of the proposed control method and compare the control performance of the WFNN with those of the FNN, the WFM and the wavelet neural network (WNN). The remainder of this paper is composed as follows. Section 2 illustrates the network structure and learning algorithm of the WFNN. Section 3 then develops the direct adaptive control system and adaptive learning rates for the stable network. Sections 4 and 5 present the simulation results and conclusions, respectively.

2. STRUCTURE OF WAVELET BASED FUZZY NEURAL NETWORK While the WFM has the advantage of the wavelet transform, neural networks utilize their learning capability for automatic identification and tuning, but they have the following problems among others: (i) they need accurate input-output data, and (ii) their learning process is time-consuming. Therefore, we have designed a FNN structure based on wavelet that merges the advantages of the neural network, fuzzy modeling and wavelet. The basic idea of the WFNN is to realize the process of fuzzy reasoning of the wavelet fuzzy model by the structure of a neural network and to make the parameters of fuzzy reasoning be expressed by the connection weights of a neural network. WFNNs can automatically identify

Σ

-1

m11 x1

m21 Σ mK11

a1C D1C

a11

Σ

1 d11

A11 ( x1 )

f

-1

1 d 21

∏

A21 ( x1 )

f ∏

-1

1 d K11

-1

AK11 ( x1 )

∏

D11

ω 21

ˆ ∏ ˆ ∏

f ∏

ω11

ˆ ∏

y11

∏

y21

∏

ˆ ∏

ω j1

y j1

Σ

yˆ1

Σ

yˆ c

Σ

yˆ C

y K1 × K 2 "K N 1 D11

∏

ˆ ∏

DN 1

xn

∏

Σ

-1

m1N m2 N Σ

xN

mK N N a NC

Σ

1 d1N

∏

ˆ ∏

f

1 d2N

A2 N ( xN )

1 dKN N -1

DNC -1

DN 1

(A)

∏

ˆ ∏

f

ˆ ∏

f

ˆ ∏

ω jC

1

AK N N ( xN )

∏

y1C y 2C

∏

ω K × K "K

∏

aN1

∏

A1N ( xN )

2

N1

ω K ×K "K 1

2

NC

∏ ∏

y jC

D1C

DNC

y K1 × K 2 "K N C

, " , yˆ C ) , and K n membership functions in each input xn. The circles and the squares in the figure represent the units of the network. The denotations a , ω , m, d and the numbers (1,–1) between the units denote the connection weights of the network. WFNN can be divided into two parts according to the fuzzy reasoning process: the premise part and the consequence part. The premise part consists of nodes (A), (C) and (D), and the consequence part consists of nodes (D) through (F). The grades of the membership functions in the premise are calculated in nodes (A) and (C). Nodes (B) and (E) are used to equalize the linear combination of FBF with the linear combination of wavelet functions for the advantage of wavelet transform. Therefore, the output node (F) is equivalent to wavelet transform. Consequently, in our WFNN structure, the output yˆ c is calculated as follows: N

R

n =1

j =1

yˆ c = ∑ anc xn + ∑ B jc Φ j ,

where N

Φ j = ∏ φ kn n ( z k n n ) n =1

⎛ xn − mkn n = ∏−⎜ ⎜ dk n n =1 n ⎝ N

⎛ 1⎛ x −m ⎞ n kn n ⎟ exp ⎜ − ⎜ ⎟ ⎜ dk n ⎜ 2 n ⎠ ⎝ ⎝

⎞ ⎟ ⎟ ⎠

2

⎞ ⎟: ⎟ ⎠

mother wavelet function. The detailed descriptions of input and output nodes are as follows. Here, input and output nodes are denoted by I and O, respectively and the subscript denotes each node. Node A: xn − mknn . (2) OA = d kn n

(C)

(D)

(E)

(F)

OB =

N

∏O n =1

Fig. 1. WFNN structure.

(1)

Node B:

∏

(B)

the fuzzy rules by modifying the connection weights of the networks using the GD scheme. Among various fuzzy inference methods, WFNNs use the sumproduct composition. The functions that are implemented by the networks must be differentiable in order to apply the GD scheme to their learning. Fig. 1 shows the configuration of the WFNN, which has N inputs ( x1 , x2 , " , x N ), C outputs ( yˆ1 , yˆ 2 , "

wavelet function, kn : k-th fuzzy variable of input n, kn : the number of fuzzy variables for the n-th input, N : the number of inputs, R : the number of fuzzy rules (the number of wavelet function), φ k n n ( z k n n ) :

∏

ω1C

ω2C

∏

553

Akn n

=

⎛ ⎛ x n −mk n ⎞ ⎞ n ⎟⎟ . ⎟⎟ d n =1 ⎝ ⎝ kn n ⎠⎠ N

∏ ⎜⎜ − ⎜⎜

(3)

554

Joon Seop Oh, Jin Bae Park, and Yoon Ho Choi

given by the product of the grades of the membership functions for the units in node (D). Here, μ j is the

Node C: ⎛ 1 ⎞ OC = Akn n ( xn ) = exp ⎜ − OA2 ⎟ ⎝ 2 ⎠ ⎛ 1⎛ x −m n kn n = exp ⎜ − ⎜ ⎜ 2 ⎜⎝ d kn n ⎝

⎞ ⎟ ⎟ ⎠

2

(4)

⎞ ⎟. ⎟ ⎠

The consequence part consists of nodes (D) through (F) and the fuzzy reasoning is realized as follows:

N

I D = μ j = ∏ OCk n =∏ Akn n ( xn ) n

n =1

n =1

⎛ 1⎛ x −m N n kn n = ∏ exp ⎜ − ⎜ ⎜ ⎜ 2 d n =1 kn n ⎝ ⎝

⎞ ⎟ ⎟ ⎠

2

(5)

⎞ ⎟. ⎟ ⎠

R

∑μ j =1

j

⎛ 1 ⎛ x − m ⎞2 ⎞ n kn n exp ⎜ − ⎜ ⎟ ⎟ ∏ ⎜ 2 ⎜⎝ d kn n ⎟⎠ ⎟ n =1 ⎝ ⎠ = , ⎛ 1 ⎛ x − m ⎞2 ⎞ ⎞ R ⎛ N n k n ⎜ exp ⎜ − ⎜ ∑ ∏ ⎜ 2 ⎜ d n ⎟⎟ ⎟⎟ ⎟⎟ j =1 ⎜ n =1 kn n ⎝ ⎠ ⎠⎠j ⎝ ⎝ N

(6)

where R = ∏ K k . k =1

y jc = ω jc

( j = 1, 2," , R and

xN is AkN N

c = 1, 2," , C ) ,

where Rj is the j-th fuzzy rule, Aknn is a fuzzy variable in the premise, and ω jc is a constant. Consequently, the output value of node (F) includes the inferred values. In our network structure, the network weight set, γ = {a, ω, d, m}, is tuned to minimize the model errors via the GD method. In order to apply the GD method, the squared error function is defined as follows: J=

N

1 (( y r1 − yˆ1 ) 2 + ( y r 2 − yˆ 2 ) 2 + " + ( y rC − yˆ C ) 2 ,(9) 2

ˆ = [ yˆ yˆ " yˆ ] are the output values of a where Y 1 2 C

Node E:

WFNN and Yr = [ y r1 y r 2 " y rC ] are the desired values. Using the GD method, the weight set, γ = {a, ω, d, m}, can be tuned as follows:

OE = y jc = ω jc OD OB ⎛ 1 ⎛ x − m ⎞2 ⎞ kn n ⎜− ⎜ n exp ⎟ ⎟ ∏ ⎜ 2 ⎜⎝ d kn n ⎟⎠ ⎟ n =1 ⎝ ⎠ = ω jc 2 ⎛ ⎛ N R ⎛ x − mkn n ⎞ ⎞ ⎞ ⎜ exp ⎜ − 1 ⎜ n ⎟ ⎟⎟ ∑ ⎜∏ ⎜ 2 ⎜⎝ d kn n ⎟⎠ ⎟ ⎟ j =1 n =1 ⎝ ⎠⎠ j ⎝ N

⎛ ⎛ x n − mk n ⎞ ⎞ n ⎟⎟ ⎟⎟ n =1 ⎝ ⎝ d kn n ⎠ ⎠ N

∏ ⎜⎜ − ⎜⎜

⎛ 1⎛ x −m ⎛ ⎛ x n − mk n ⎞ ⎞ n kn n n ⎜ −⎜ ⎟ exp ⎜ − ⎜ ⎟ ∏ ⎜ ⎜ dk n ⎟ ⎟ ⎜ 2 ⎜⎝ d kn n n =1 n ⎝ ⎠ ⎝ ⎠ ⎝ = ω jc ⎛ 1 ⎛ x − m ⎞2 ⎞ ⎞ R ⎛ N kn n ⎜ exp ⎜ − ⎜ n ⎟⎟ ⎟ ⎟ ∑ ∏ ⎜ ⎜ ⎟ ⎜ d 2 j =1 n =1 kn n ⎝ ⎠ ⎟⎠ ⎠ j ⎝ ⎝ N

γ p (k + 1) = γ p (k ) + Δγ p (k )

(7) ⎞ ⎟⎟ ⎠

2

⎞ ⎟ ⎟ ⎠

ω jc R

∑ IDj j =1

Node F: R

OF = yˆ c = ∑ anc xn + ∑ y jc n =1

j =1

N

R

n =1

j =1

= ∑ anc xn + ∑ B jc Φ j .

∂J ∂γ p (k )

= γ p (k ) − η

ˆ ∂J ∂Y ˆ ∂γ p (k ) ∂Y

(10)

where E = [( yr1 − yˆ1 ) ( yr 2 − yˆ 2 ) " ( yrC − yˆC )] , subscript p denotes each network weight and η is called the learning rate. ˆ with respect The gradient set of WFNN output Y to weight set is calculated as in (11), and each gradient of WFNN output yˆ with respect to each weight is presented as in (12) to (14):

.

N

= γ p (k ) − η

= γ p (k ) + η ⋅ E ⋅ υˆ p ,

= B jc Φ j ,

where B jc =

Rj: If x1 is Ak11 , " , xn is Akn n , " and Then

μj

OD = μˆ j =

normalized value of μ j . The fuzzy system realizes the center of gravity defuzzification formula using μˆ j in (6).

Node D: N

truth value of the j -th fuzzy rule and μˆ j is the

(8)

The input space is divided into R fuzzy subspaces. The truth value of the fuzzy rule in each subspace is

υˆ p =

ˆ ∂Y = [ υˆ a υˆ ω υˆ m υˆ d ] ∂γ p (k )

ˆ ˆ ˆ ˆ ⎤ ⎡ ∂Y ∂Y ∂Y ∂Y =⎢ ⎥, ⎣ ∂a(k ) ∂ω(k ) ∂m(k ) ∂d(k ) ⎦

(11)

Stable Path Tracking Control of a Mobile Robot Using a Wavelet Based Fuzzy Neural Network

υˆanc =

∂yˆ c = xn , ∂anc (k )

(12)

r

Ca

R

jc

∂ ∑ y jc ∂yˆ c j =1 = = = ∂ω jc (k ) ∂ω jc (k )

Φj R

(13)

,

Dr W ivin , he g el

∑ IDj

j =1

υˆmk

,d n n kn n

⎛H ⎞ ∂⎜⎜ ∑ B jc Φ j ⎟⎟ ∂yˆ c j =1 ⎠ = = ⎝ ∂mk n n , d k n n (k ) ∂mk n n , d k n n (k )

b

k =1

KN

, NUM = Φ j , DEN = ∑ I D j , j =1

⎛ N φ kn n ( z kn n ) 1 ⎜∏ ⎛ 2 ⎛ 1 2 ⎜ n =1 =− ⎜ OAknn − 1 exp ⎜ − OAknn d kn n ⎜ φ kn n ( z kn n ) ⎝ ⎝ 2 ⎜ ⎝

(

DEN (mkn n ) =

)

⎞ ⎞ ⎞ ⎟ ⎟⎟⎟ , ⎠⎠⎟ ⎟ ⎠

∂zkn n ∂DEN ∂mkn n ∂zkn n

⎛ N ⎜ ∏ OCknn 1 =− ∑ ⎜ n =1 d kn n h =1 ⎜ OCk n n ⎜ ⎝ H

NUM (dkn n ) =

⎞ ⎛ ⎛ 1 2 ⎞⎞⎟ ⎜ −OAknn exp ⎜ − OAknn ⎟ ⎟ ⎟ , ⎝ 2 ⎠⎠⎟ ⎝ ⎟ ⎠h

∂zkn n ∂NUM ∂dkn n ∂zkn n

⎛ N ⎞ ⎟ OA2k n ⎜ ∏φkn n ( zkn n ) ⎛ 2 1 ⎞ ⎛ ⎞ 2 = − n ⎜ n =1 ⎜ OAknn − 1 exp ⎜ − OAknn ⎟ ⎟ ⎟ , dkn n ⎜ φkn n ( zkn n ) ⎝ ⎝ 2 ⎠⎠⎟ ⎜ ⎟ ⎝ ⎠ ∂zkn n ∂DEN DEN (d kn n ) = ∂d kn n ∂zkn n

(

=−

O

2 Aknn

d kn n

World Coordinate

x

Fig. 2. Mobile robot model and world coordinate. R

∂zkn n ∂NUM ∂mkn n ∂zkn n

NUM (mkn n ) =

y

⎞⎞ ⎟⎟ , ⎟⎟ ⎠⎠h

N

where H =

( x y)

(14)

⎛ NUM (mk n , d k n ) DEN (mk n , d k n ) NUM H ⎛ n n n n = ∑ ⎜ ω jc ⎜ − ⎜ DEN DEN 2 h =1⎜ ⎝ ⎝

∏ Kk

r

θ

υˆω

s te

555

⎛ N O H ⎜ ∏ C kn n ∑ ⎜ n =O1 h =1 ⎜ Ckn n ⎜ ⎝

)

⎛ ⎛ 1 2 ⎜ −OAknn exp ⎜ − OAknn ⎝ 2 ⎝

⎞ ⎞ ⎞ ⎟ ⎟⎟⎟ . ⎠⎠⎟ ⎟ ⎠h

3. PATH TRACKING CONTROL FOR MOBILE ROBOT USING THE WFNN 3.1. Dynamic model of mobile robot The mobile robot used in this paper is composed of two driving wheels and four casters. It is fully described by a three dimensional vector of generalized coordinates constituted by the coordinates of the

midpoint between the two driving wheels, and by the orientation angle with respect to a fixed frame as shown in Fig. 2. The equation for motion dynamics is as follows:

δθ k ⎤ ⎡ ⎢δ d k cos(θ k + 2 ) ⎥ ⎡ X k +1 ⎤ ⎡ X k ⎤ ⎢ ⎥ δθ k ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢Yk +1 ⎥ = ⎢Yk ⎥ + ⎢δ d k sin(θ k + 2 ) ⎥ , ⎢⎣θ k +1 ⎥⎦ ⎢⎣θ k ⎥⎦ ⎢ ⎥ ⎢ ⎥ δθ k ⎣⎢ ⎦⎥

(15)

d − dl d r − dl and dθ = r are linear 2 b velocity and angular velocity, respectively, and d r ,

where δd =

d l and b are two incremental distances of two driving wheels and distance between these two wheels, respectively. In this model, the control input vector is represented by U = [u d uθ ]T = [δd δθ ]T .

3.2. The direct adaptive control system using the WFNN In our control system, the direct adaptive control system is designed using the WFNN structure. The purpose of our control system is to minimize the state error E(e x , e y , eθ ) between the reference trajectory

Yr ( xr , yr ,θ r ) and the controlled trajectory Y( x, y,θ ) of a mobile robot. For this purpose, the parameters of the WFNN are trained via the GD method. The overall control system is shown in Fig. 3. The WFNN controller calculates the control input

U = [u d uθ ]T by training the inverse dynamics of the plant iteratively. However, the updating of parameters of the WFNN through the variation rate J ( γ, Y) in the GD method cannot be calculated directly. So, we train the parameters of a WFNN through the transformation of the output error of the plant. In our

556

Joon Seop Oh, Jin Bae Park, and Yoon Ho Choi

- Calculation of the partial derivative of the cost function with respect to the parameter set of a WFNN controller:

Updating the parameters of WFNN

u c (δd , δθ )

The Direct Adaptive Controller Based on WFNN

controlled trajectory Y ( x , y ,θ )

Mobile Robot

−

∑

+

∂C ∂x ∂y ∂θ = − ex − ey − eθ ∂γ p ∂γ p ∂γ p ∂γ p

reference trajectory Yr ( xr , y r ,θ r )

Feedforward Jacobian of Mobile Robot

= − ex

Gradient Descent Method and Lyapunov Stability Analysis

state error E(e x ,e y ,eθ )

= −E J (u )

Fig. 3. Direct adaptive control system. where E (e x , e y , eθ )

ex Fuzzy Premise Part

#

Fuzzy Consequence Part

#

#

Equalizer for Wavelet Transform

ω R1 # ω R2

eθ

uθ

#

E (e x , e y , eθ )

an 2

Fig. 4. WFNN structure for mobile robot.

WFNN structure, inputs, multidimensional wavelets, and two outputs are considered as shown in Fig. 4. In this structure, inputs are composed of errors between the reference trajectory and the controlled trajectory, and outputs are control variables. Each control variable is as follows: 3

R

3

R

n =1

j =1

n =1

j =1

3

R

3

R

n =1

j =1

n =1

j =1

u d = ∑ and en + ∑ y jd = ∑ and en + ∑ B jd Φ j , (16)

uθ = ∑ anθ en + ∑ y jθ = ∑ anθ en + ∑ B jθ Φ j , where 3

B jc Φ j = ω jc

2 ⎞⎞ ⎛ ⎛ R ⎜ 3 ⎜ 1 ⎛⎜ en − mknn ⎞⎟ ⎟ ⎟ ∑ ⎜ ∏ exp⎜ − ⎜ ⎟⎟ j =1 ⎜ n =1 ⎜ 2 ⎝ d knn ⎟⎠ ⎟ ⎟ ⎝ ⎠⎠ j ⎝

,

The partial derivative of the control input U with respect to the parameters of a WFNN controller can be calculated by using (20) and (21). - Updating of the parameters of the WFNN via the following iterative GD method:

γ p (k + 1) = γ p (k ) + Δγ p (k ) ∂C ∂U , = γ p (k ) − η E J (u ) ∂γ p ∂γ p

(20)

where η is the learning rate of a WFNN. From (18) and (19), each gradient of the controller output u c with respect to each weight is presented as follows: (21)

R

∂ ∑ y jc

Training Procedure The purpose of training the parameters of the WFNN is to minimize the state errors E (ex , ey , eθ ) . To do this, we present the following training procedure: - Definition of the following cost function so as to train a WFNN controller based on direct adaptive control technique: 1 (( xr − x) 2 + ( y r − y ) 2 + (θ r − θ ) 2 ) . 2

and

∂θ k δd k δθ k ⎤ ⎡ ⎢cos(θ k + 2 ) − 2 sin(θ k + 2 ) ⎥ ⎢ ⎥ δθ δd k δθ ⎢ .(19) cos (θ k + k ) ⎥⎥ J (u ) = ⎢ sin(θ k + k ) 2 2 2 ⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎥⎦ 0 1 θ k =θ k −1

∂u c = en , ∂anc

and c = {d ,θ }.

C=

ex = xr − x, e y = yr − y, eθ = θ r − θ ,

= γ p (k ) − η

2⎞ ⎛ ⎛ ⎛ e n − mk n ⎞ ⎞ ⎜ 1 ⎛⎜ en − mknn ⎞⎟ ⎟ n ⎟⎟ ⎜ ⎜ − − exp ∏ ⎜ 2⎜ d ⎟ ⎟⎟ d knn ⎟⎠ ⎟ n =1⎜ ⎜ ⎜ kn n ⎝ ⎠ ⎠ ⎝ ⎝ ⎠ ⎝

∂U , ∂γ p

∂Y is the feedforward Jacobian of a mobile ∂U robot and is as follows:

ud

#

(18)

J (u ) =

a n1

ω 11 ω12

ey

∂x ∂U ∂y ∂U ∂θ ∂U − ey − eθ ∂U ∂γ p ∂U ∂γ p ∂U ∂γ p

(17)

∂u c j =1 = = ∂ω jc ∂ω jc (k )

Φj R

,

(22)

∑ IDj

j =1

⎛H ⎞ ∂⎜⎜ ∑ B jc Φ j ⎟⎟ ∂uc j =1 ⎠ = ⎝ ∂mk n n , d k n n (k ) ∂mk n n , d k n n (k ) ⎛ NUM (mk n , d k n ) DEN (mk n , d k n ) NUM H ⎛ n n n n = ∑ ⎜ ω jc ⎜ − 2 ⎜ ⎜ DEN DEN h =1 ⎝ ⎝

and the detailed description is shown in (14).

(23) ⎞⎞ ⎟⎟ , ⎟⎟ ⎠⎠h

Stable Path Tracking Control of a Mobile Robot Using a Wavelet Based Fuzzy Neural Network

3.3. Convergence and stability of the WFNN controller In the update rule of (20), selection of the values for the learning rate η has a significant effect on the control performance. Generally, if η is too big, the system is unstable. For the small η , although the convergence is guaranteed, the control speed is very slow. Therefore, in order to train the WFNN effectively, adaptive learning rates, which guarantee both fast convergence and stability, must be derived. In this subsection, the specific learning rates for the type of network weights are derived based on the convergence analysis of a discrete type Lyapunov function. Theorem 1: Let η p, c be the learning rate for the

T

⎡ ∂e (k ) ⎤ Δeθ (k ) ≈ ⎢ θ ⎥ Δγ p (k ) . ⎢⎣ ∂γ p (k ) ⎥⎦ From (18), (19) and (20), Δγ p (k ) is defined as Δγ p (k )= −η p,c

∂C ∂γ p (k )

⎡ ∂u (k ) ⎤ .⎢ c ⎥, ⎢⎣ ∂γ p (k ) ⎥⎦

and the error difference can be represented by T

WFNN. G p , c (k ) and G p ,c,max (k ) are defined as ∂u (k ) and G p,c,max (k ) ≡ max k G p,c (k ) , G p, c (k ) = c ∂γ p ( k )

⎡ ∂u ( k ) ⎤ ∂x(k ) η p ,c " = −⎢ c ⎥ ⎢⎣ ∂γ p (k ) ⎥⎦ ∂uc (k )

respectively, and

is the Euclidean norm in ℜ n .

⋅

Here, subscripts p and c denote each weight and output, respectively. Then the convergence is guaranteed if η p, c is chosen as follows:

0 < η p ,c