DSpace at VNU: Trajectory Tracking Control of a Mobile Robot by computed torque method with on-line learning neural network

 Abstract—This paper proposes a novel control algorithm for the mobile robot with nonholonomic constraint.. The algorithm consists of two control loops: one is based on the kinematics



Abstract—This paper proposes a novel control algorithm for

the mobile robot with nonholonomic constraint The algorithm

consists of two control loops: one is based on the kinematics

and Lyapunov theory to derive the control laws for the tangent

and angular velocities to control the robot to follow a target

trajectory, the other controls the robot dynamic based on the

moment method in which a neural network namely RBFNN is

introduced to compensate the uncertainty of dynamic

parameters The convergence of the estimators based on

RBFNN of Stone-Weierstrass is proven The asymptotically

stabilization of the whole system is confirmed by direct

Lyapunov stabilization theory The effectiveness of the method

is verified by simulations in Matlab

Keywords — Trajectory tracking, mobile robot, torque

control, neural networks, Lyapunov theory

I INTRODUCTION

Controlling a mobile robot to follow a predefined

trajectory is a challenging task due to the nonlinear

chacteristic and nonholonomic constraint of the robot

According to Brocket theory, a nonholonomic system is not

able to be asymptotically stable using the smooth and time

invariant control laws Some methods to stablize the

nonholonomic system through feedback control have been

proposed They however often assume ideal conditions

Others focus on determining uncertainties in measurements

and model parameters and try to fix them by using hybrid

feedback control or velocity chart control These methods

are usually complex and difficult to implement

Our approach is the use of Lyapunov function technique

to design a stable controller for nonholonomic systems

[1],[2] The goal is the optimization in motion of the robot

during the path following process From the robot

kinematics, the uncertainties in system parameters are

determined and compenstated by the implementation of an

extended Kalman filter But this stage only focuses on the

kinematics while the dynamics parameters such as the

robot’s load which plays an important role in the stable of

the robot are not concerned In addition, non-parameter

uncertainties such as high-frequency unmodeled dynamics,

actuator dynamics, structural vibrations, measurement

noises, computstion roundoff error, and sampling delay also

need be considered Thus, the problem of kinematics and

dynamics control of nonholonomic system is challenging

A number of approaches to control the system with nonholonomic constraint have been introduced [3],[4],[6] In [9-16], authors were combined the dynamics model of the mobile robot to the kinematics controller with nonholonomic constraint

One of the most popular methods to solve this class of control problem is adaptive control For example, the backstepping method of Wang et al [17] and R.Fierro et al [18], the sliding-mode techniques in [19-20], were applied to

reduce sway for an offshore container crane These methods

also employed neural network to compensate the uncertainties such as the combination between the backstepping method and the neural network reported in [9, 11] In [21], Dong Xu et al were applied a combination between the RBFNN controller and the sliding-mode techniques for the path following task of an omnidirectional wheeled mobile manipulators

In [9], the author presented a control method using neural network in which, online learning law of weight factors is used to compensate the uncertainties caused by error in dynamics modelling The asymptotically stabilization were theorically proven and confirmed by experiments Nevertheless, if the dynamics model contains non-parameter uncertainties, the asymptotically stabilization

is then not assured

The new point in this paper is the splitting of the path following tasks into two independent control loops The outer loop is employed to control the kinematics such as the determination of tangent and angular velocities so that the errors in position and direction go toward zero (globally asymptotically stabilization according to the Lyapunov theory); output of this controller is sent to the inner control loop The inner control loop is used to control the dynamics

In this control loop, we control by computed torque method This controller is designed by the combination between the Feedforward and the scale techniques The RBFFN is used to compensate the non-parameter uncertainties and dynamic modeling errors The globally asymptotic stabilization of the system is proven by the Lyapunov theory

The paper is organized as follows Section 2 briefly introduces the kinematics and dynamics of the mobile robot,

as presented in [9-11] Section 3 describes the process to design the controller Section 4 presents the simulation results and section 5 is the conclusion

Trajectory Tracking Control of a Mobile Robot by computed

torque method with on-line learning neural network

Thuan Hoang Tran1, Van Tinh Nguyen2 , Minh Tuan Pham2, Thuong Cat Pham2

1University of Engineering and Technology, Vietnam National University, Hanoi

2The Institute of Information Technology, Vietnamese Institute of Science and Technology, Hanoi

1thuanhoang@donga.edu.vn, 2nvtinh@ioit.ac.vn

II NONHOLONOMICMOBILEROBOTKINEMATICS

ANDDYNAMIC

A typical example of a nonholonomic mobile robot is

shown in Fig 1 [9-11]

Fig 1 A noholonomic mobile robot platform

The position of the robot in an inertial Cartesian frame

{O,X,Y} can be described by a vector ON, and orientation

between mobile robot base frame {P,X p ,Y p} and Cartesian

frame.The configuration of the robot can be described by five

generalized coordinates

r l

x y

where x, y are the coordinates of P, θ is the orientation angle

of mobile robot, and   are the angles of the right and left r, l

driving wheels Therefore the nonholonomic constraint of the

mobile robot can be expressed as,

r l





(2)

All kinematic equality constraints are considered

independent of time and can be expressed as,

A z z  0 (3)

Where   sincos sincos 0 0 00

Let S z  be a matrix formed by a set of smooth and

linearly independent vector fields spanning the null space of

 

A z , i.e.,

A z S z =    0 (5)

It is easy to verify that S z  is given by

 

cos cos

sin sin

S z (6)

Now, we easily have the following pair of equations [9-10]:

z = S z v    t (7)

Mv + Cv = τ (8) where

2

2

0

0 4

c w

c w

I

r m d





r l

τ is the torque applied on the right and left wheels,  T

r l



v represents the angular velocity of the right and left wheel, m m c2m w, Im d c 22m R w 2I c

Here, m c is the mass of the mobile robot platform, m w is the mass of one driving wheel with the actuator, I I c, ware moments of inertia of platform about the vertical axis through

P, the wheel with the actuator about the wheel axis,

respectively

We can rewrite the system dynamic Eq (8) into a linear form,

 

Y v, v p = τ

Y v, v p = Mv + Cv



  (9) where p is a 3x1 vector consisting of the known and

unknown robot dynamics, such as mass and moment of inertia; Y v, v is a 2x3 coefficient matrix consisting of the

known functions of the robot velocity v and acceleration vwhich is referred as the robot regressor For the mobile robot shown in fig.1, we can get:

 

r l l

l r r

T c w

I

Y v, v

p







 (10)

2r 2R

X

Y

O

P

III CONTROLLERDESIGN

A Outer control loop

Let the tangent and angular velocities of the robot be

v and respectively We have:

1

r

l

y

         



The objective of the control problem is to design an

adaptive control law so that the position vector q and the

velocity q to hold the position vector qr t and the desired

velocity qr t , in case the robot dynamics parameters are

not known exactly Desired position and velocity vectors are

represented by:

cos

sin

T

r r r r

r r r

r r r

r r















q







with v r > 0 for all t (12)

The tracking error position between the reference robot

and the actual robot can be expressed in the mobile basis

frame as below [9-10]:

1 2 3

r

r

e

 



 

The derivation of the position tracking error can be

expressed as:

3

cos sin

r

r

e





 

 

 

e





(14)

There are some methods in the literature to select the

smooth velocity input In this research, we choose a new

control law for v, as:

3 3 2

3

cos

sin

r

v

e

k e v e

e



(15)

where k k1, 3> 0 In this control law, when e30 then

3

3

sin

1

e

e  , and  always be bounded

With the control law in equation (15), it is easy to prove

asymptotical stability system due to ep0 when t 

Choosing a positive definite function V p as follows:

T

p p p

Derivation of V p with respect to time V p is:

1 1 2 2 3 3

T

p p p

V

e e e e e e



e e

(17)

Replacing (15) into (17), we have

3

3

1 1 3 3

cos sin

sin cos cos sin

e

e

k e k e

      

 



(18)

It is easy to see that V p is continuous and bounded according to the Barbalat theorem It means that Vp0 when t , consequently e10,e30 when t  Appling the Barbalat theorem again for the derviation, we get:

e10,e30 (19) and the equation (15) becomes

r

vv (20)

r

 (21) Combining (14), (19), (20), (21) infers: e20,e20 Thus the control law (15) assures the proximity control system ep0 when t 

B Inner control loop

The deviation of stick angular velocity of driven wheels is:

r cr

l cl

v v



e v v (22)

where v is the desired angular velocity of the robot

control wheel torque We must find the control law of the angular velocity using computed-torque method in order to

0

c 

Derivating and multiplying both sides of equation (22) with the matrix M obtain:

Mec M v vc τ CecY pc (23-a)

In case of non-parametric uncertainty component d in

the mobile robot dynamics model, equation (23-a) is rewritten as below

Mec M v vc τ CecY p dc  (23-b) Where Y p Mvc  cCvc (24)

Torque output of the controller is:

τ = τNNK eD cY pcˆ (25)

where KD is the positive definite matrix pˆ is the matrix

estimated by matrix p

Substituting (24) into (23), we have

Mec τNNKDC e cf (26)

Because fY p d Y p Y p dc  c  cˆ is unknown so it

can be seen as an uncertain component We can approximate

this uncertain component by a finite neural network [3, 4, 6]

which has the following structure:

f Y p d Wσ ε f εc    ˆ (27)

where W is the weight matrix of an online updated

network; ε is the approximate error and is bounded by

0





The neural network Wσ is approximated by Gaussian

RBF network consisting of three layers: input layer, hidden

layer with 2 nodes that contains the Gaussian function, and

the output layer with linear function of 2 neurons (Fig 2)

[7]

Fig 2 Neural network approximationWσ funcion.

The RBF network structure satisfies the conditions of the

Stone-Weierstrass theorem Hidden layer neuron is the

Gaussian function with the form:

2

exp j j ; 1, 2

j

j

s c

j







where cj, j are the expectation and variance of the

Gaussian function chosen as follows [17]

NN

c

e to satisfyour

control law

Following the Stone-Weierstrass theorem, we have the

diagram Mobile robot control by torque method with online

learning neural network as shown in the figure 3 with

2

ˆ 1

exp

c

c

i c i

cj j j

j

 





 



e

e

where the optional parameter K D is a symmetric positive definite matrix, the coefficients  , 0

Fig 3 Mobile robot control by torque method with online learning neural network

Proof:

Selecting a positive function V as follows:

1

1 2

T

c c i i i

V



e Me w w T  (30) Derivating V with respect to time gets

2 1 2 1

T

c c i i i T

c c i i

i

V









T



(31)

According to (23), we have:

T

c NN D c

c D c c c c NN





(32)

NN

c

e and the matrix C

is symmetric, T 0

c c 

e Ce So

c D c c

c

V      

e

e



T

c D c  c c

 e K e  e  e ε

0

T

c D c  c c 

 e K e  e  e

If we choose   0 with  0 then

T

c D c c

V e K e  e

We see V0 when ec 0 and V0 if and only if 0

c 

e According to the Lyapunov stability principle [1], [2] ec 0 meansep 0

1



2



2

1

ˆ

j j

f w 

2

1

ˆ

j j

f w 

1

c

e

2

c

e

The control system in Fig 3 is the asymptotic stability

d

q

q  or in other words, the trajectory of the robot

follows the desired trajectory with error 0 Theorem as well

as the global asymptotic stability of the system with torque

control using neural network depicted in Fig 3 has been

proved

IV SIMULATIONRESULTS

The control algorithm developed in section III was

implemented in Matlab-Simulink In the course of the

simulation test, we use a mobile robot model having the

following dynamic parameters r = 0.15m; R= 0.75m;

d=0.2m; m c = 30kg; m w = 30kg; I c = 15.625 kgm 2 ; I w =

0.0005kgm 2 ; I m = 0.0025 kgm2 The parameters of the

controller are: K D = diag(5,5); k1=k3=2;  = 10 Suppose that

we only estimate pˆ0.6p and non-parametric uncertainty

components are:

d

Process variability of tangent and angular velocity with

time is:

5

t

5 t 25 : v r 0.5, r 0

2

r

v t

2

r

v t

5

t

35 t 40 : v r 0.5, r 0

Fig 4 shows the choosen reference velocity.

Below is the graph reflecting the simulation results

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Time (s)

Actual Desired

Fig 5 The tracking errors without τNN

-0.1 0 0.1 0.2 0.3 0.4

Time (s)

X Direct

Y Direct Orient

Fig 6 Position error without τNN

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5

X Axis (m)

Actual Desired

Fig 7 The tracking errors with τNN

-0.1 0 0.1 0.2 0.3 0.4

Time (s)

X direct

Y direct Orient

Fig 8 Position error with τNN

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Time (s)

w11 w12 w21 w22

Fig 9 Neural network weights with τNN

Comparing Fig.6 and Fig.8, we can verify the efficiency

of components τNN( created by RBFNN) in compensating uncertainties in the model dynamics

V CONCLUSIONS This paper proposes a control model using neural networks to compensate the uncertainties of the robot and

X Axis (m)

assure the global stability of the system Simulation in

Matlab-Simulink is consistent with the principles of the

proposed control law

ACKNOWLEDGMENT This work was supported by Vietnam National Foundation

for Science and Technology Development (NAFOSTED)

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