DSpace at VNU: Trajectory Tracking Control of the Nonholonomic Mobile Robot using Torque Method and Neural Network

Trajectory Tracking Control of the Nonholonomic Mobile Robot using Torque Method and Neural Network T.. The inner loop generates control laws for the tangent and angular velocities to

Trajectory Tracking Control of the Nonholonomic Mobile Robot using Torque

Method and Neural Network

T T Hoang, D T Hiep, B G Duong and T Q Vinh

Department of Electronics and Computer Engineering University of Engineering and Technology Vietnam National University, Hanoi thuanhoang@donga.edu.vn

Abstract— This paper deals with the problem of tracking

control of the mobile robot with non-holonomic constraint A

controller with two control loops is designed The inner loop

generates control laws for the tangent and angular velocities to

control the robot to follow the target trajectory It is derived

based on the robot kinematics and the Lyapunov theory The

outer loop employs the torque method to control the robot

dynamics A neural network is implemented to compensate the

uncertainties caused by the dynamics model The asymptotic

stabilization of the whole system is proven by direct Lyapunov

stabilization theory Simulations in MATLAB confirmed the

validity of the proposed method

Keywords - Trajectory tracking, mobile robot, torque control,

neural networks, Lyapunov theory

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 However, we find it difficult to implement closely the dynamics model of the mobile robot due to the non-quantified parameters of dynamics as well as the usually-variable robot’s load These errors are mainly caused by the uncertainties of such a model and the non-parameters, which is composed of: 1-high frequency unmodeled dynamics, such as actuator dynamicsor structral vibrations; 2- measurement noise; 3-computation roundoff error and sampling delay Thus, the problem of kinematics and dynamics control of non-holonomic system is absolutely challenging The typical method is to solve this is

considered as “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 implemented neural network to compensate the uncertainties such as the combination between the backstepping method and the neural network in [9,11] In [21], a combination between the RBFNN controller and the sliding-mode techniques for the path following task of an omnidirectional wheeled mobile manipulators was applied The asymptotically stabilization were proven theoretically and experimentally 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 model If the dynamics model contains non-parameter uncertainties, the asymptotically stabilization is then not assured

In this paper, the tracking control algorithm based on the torque method is presented The controller is of feedforward being in combination with proportional type The non-parameter uncertainties and dynamics model errors are compensated by using the RBFNN The asymptotic

stabilization of the whole system is proven by the Lyapunov

theory

Our main job is that 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 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,

This controller is designed by the combination between the

feedforward, the scale techniques, and

compensated-nonparameter-RBFNN type

The paper is organized as follows Section 2 briefly is

introduced the kinematics and dynamics of the mobile robot

Section 3 is described the process to design the controller

Section 4 is presented the simulation results and finally is

section 5

II THEKINEMATICSANDDYNAMICSOF

NONHOLONOMICMOBILEROBOT

A typical example of a nonholonomic mobile robot is

shown in Fig 1 [9-11]

Figure 1 A noholonomic mobile robot platform

Prom [9-10], we have:

MȦ + CȦ = IJ (1)

where

2

2 2 2 2

2

0

0 4

c w

c w

I

r m d





and [ ]T

r l

= τ τ

IJ is the torque applied on the wheels;

r l

Ȧ is the angular velocity of the right and left

wheels in order; m=m c+2m w , in which m c is the mass of

the mobile robot platform, m w is the mass of one driving

wheel with the actuator; I=m d c 2+2m R w 2+I c, in which

,

c w

I I are 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.(1) into a linear form, [9]:

Y(Ȧ,Ȧ)p = IJ (2)

Y(Ȧ,Ȧ)p = MȦ + CȦ  (3)

where p is a 3x1 vector consisting of the known and

unknown robot dynamics, such as mass and moment of

inertia; Y( Ȧ,Ȧ) is a 2x3 coefficient matrix consisting of the known functions of the robot velocityȦ and acceleration

Ȧ which is referred as the robot regressor For the mobile robot shown in fig.1, we could easily compute:

T c w

I

ω ω −θω

Y(Ȧ,Ȧ)

p







A Outer control loop

Let the tangent and angular velocities of the robot be

v andω respectively We have:

1

r l

R

ω





(5)

The target of the control problem is to design an adaptive control law in which the position vecctor q and the

parameters of the mobile robot The desired position and velocity vectors are represented:

cos sin

T

θ θ θ

=

=

=

=







with v r > 0 for all t (6)

The position tracking error between the reference and the actual robot could be expressed in the mobile robot’s coordinator as follows [9-10]:

1 2 3

r

r

e

θ θ

−

¬ ¼

In this paper, we choose the control law for ν , ω like:

2r 2R

X

Y

O

P

3 1 1

3

3

cos

sin

r

v

e

e

+

=

3

3

sin

1

e

3

cos sin

r

r

e

ω ω

ω ω

− +

¬ ¼

e





(9)

With the control law in equation (8), it is easy to prove

asymptotical stability system due to e p →0 when t→ ∞

Choosing a positive definite function V as follows: p

p

p p

The 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

(11a)

Replacing (8) into (11a), we have

1 3 2 3 3

3

1 3 1 1 3 2 3 3 3 3 2

3

2 2

1 1 3 3

sin

V e v v e e v e e

e

e v e k e v e e v e e k e v e

e

k e k e



(11b)

It is straightforward to see that V p is continuous and

bounded according to the Barbalat theorem, which means

that Vp →0 when t→ ∞ , and e1→0,e3→0 when

t→ ∞ According to Barbalat theorem, we have:

e1→0,e3→0 (12) and the equation (8) becomes

v→v r (13)

ω→ωr (14)

Combining (7), (12), (13), (14) infers: e2→0,e2→0

Thus the control law (8) assures the proximity control

system 0e p → when t→ ∞

B Inner control loop

The deviation of stick angular velocity of driven wheels

is:

r cr

l cl

ω − ω

= «¬ω − ω »¼

where Ȧ is the desired angular velocity of the robot

calculated by (5), Ȧ is the output of the angular velocity c

control wheel torque We must find the control law of the

angular velocity using computed-torque method which is satisfied 0e c → and e p→0 whent→ ∞

Derivating and multiplying both sides of equation (15)

with the matrix M, we obtain:

Me = M(Ȧ - Ȧ ) = IJ - Ce - Y p   (16-a)

In case of non-parametric uncertainty component d in

the mobile robot of dynamics model, equation (16-a) is rewritten as below:

Me = M(Ȧ - Ȧ ) = IJ - Ce - Y p + d   (16-b) Where Y p = MȦ + CȦ c c c (17) Torque output of the controller is:

ˆ

IJ = IJ - K e + Y p (18)

where KDis the positive definite matrix and ˆp is the matrix esimated by the matrix p

Substituting (17) into (16), we have:

Me = IJ - (K + C)e + d + Y p (19) with p = p - p ˆ

Because d is unknown parameter, it then could be

considerd as uncertainty component We are able to approximate this by a finite neural network [3,4,6] as follows:

ˆ

d = Wı + İ = d + İ (20)

where, where W is the weight matrix of an online

updated network; İ is the approximate error and is bounded

by İ ≤ ε0

The neural network W ı is approximated by Gaussian

RBF network consisting of three layers: input layer, hidden layer with n nodes that contains the Gaussian function, and the output layer with linear function of n neurons (Fig 2) 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

σ

λ

−

where, cj, λj are the expectation and variance of the Gaussian function chosen in [17] And we must determine the parameters cj, λj differently , which cover the

uncertainty d in terms of amplitude and frequecy The output ˆd is an approximation of d

Choosing =(η+1) −δ c

NN

c

e

e to satisfy our control law

Figure 2 The approximate neural network of Wσ function

Theorem: The dynamics of the mobile robot (1) using

neural network (20) will be tracked the desired trajectory q d

with the error e p →0, if we choose the control algorithm

IJ and learning neural network wias follows:

2

ˆ

exp

cj j j

j

η σ σ

λ

+1

= −

−

c

c

c

e

e

where the optional parameter K D is a symmetric

positive definite matrix, ,η δ > 0

Figure 3 Mobile robot control based on torque method with online

learning neural network

Proof:

Selecting a positive function V as follows:

1

2

i =1

e Me + p ī p + w w (23) Derivating V with respect to time, we obtain:

ˆ

V

ι σ

=

¦

¦

2

i =1 2 T

i=1





(24)

From (19), we achieve:

= −

T



NN

c

e

IJ = (Ș + 1)Wı - į

e and the matrix C is symetric, e Ce T c c= 0 So that:

δ

≤ −

≤ −

c T

T

e

e



If we choose δ ε= 0+ with γ γ >0 then

V ≤ − T

 (25) From equation (24), (25), we can see that V → so that 0 0

c →

e , 0p→ and 0e p → , p →p The control system in Fig 3 is the asymptotic stability

→ d

q q Otherwise, the error between the tracking trajectory and the desired one is closely to zero Additionally, the proposed controller exactly estimate the required parameters of mobile robot dynamics ( ˆp→p) Both theorem and the global asymptotic stability of the system with torque control using neural network depicted in Fig 3 have proved

IV SIMULATIONRESULTS

In our scenario, we simulates the mobile robot model

with the following paramters: 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 = diag(5,5); k D 1=k3=2; δ = 10 Suppose that we only estimate ˆp=0.6p and non-parametric uncertainty components are: d = ª¬sin(0.25 )t cos 0.25( t)º¼ T

5

t

5≤ <t 25 : v r =0.5, ωr = 0

2

r

v t

2

r

v t

5

t

35≤ <t 40 : v r =0.5, ωr = 0

The simulation results are presented in below figures:

A Using the neural network with the parameter estimation

algorithm of dynamics model:

0

0.5

1

1.5

2

2.5

3

3.5

X (m)

Actually Trajectory

Desired Trajectory

Figure 4 The desired trajectory and the actual trajectory with τMN

-1.5

-1

-0.5

0

0.5

Time (s)

X direct Error

Y direct Error Orient Error

Figure 5 The position error with τMN

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

w11 w12 w21 w22

Figure 6 The neural network weights with τMN

B Without the neural network with the paramter estimation

algorithm of mobile robot:

0

0.5

1

1.5

2

2.5

3

3.5

X (m)

Actually Trajectory

Desired Trajectory

Figure 7 The desired trajectory and the actual trajectory without τMN

-1.5 -1 -0.5 0 0.5

Time (s)

X direct Error

Y direct Error Orient Error

Figure 8 The position error without τMN

Comparing Fig.5 and Fig.8, we can verify the efficiency

of components IJNN( created by RBFNN) in compensating

uncertainties in the model dynamics

V CONCLUSION This paper proposes a control method for trajectory tracking of nonholonomic mobile robot The main contribution is the proposal of a controller with two control loops One is for the kinematics The other is for the dynamics In addition, a neural network is introduced to deal with uncertainties of the dynamics model The global stability of the system is proven The simulation results confirmed the effectiveness of the method

ACKNOWLEDGMENT This work was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED)

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