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. 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

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.

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

978-1-4673-6322-8/13/$31.00 c 2013 IEEE

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 nonparameters, which is composed of: 1-high frequency
unmodeled dynamics, such as actuator dynamics or 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 nonparameter uncertainties and dynamics model errors are

compensated by using the RBFNN. The asymptotic

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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 compensatednonparameter-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.

THE KINEMATICS AND DYNAMICS OF
NONHOLONOMIC MOBILE ROBOT

Yp

Xp

2R

P
2r

O

X

Figure 1. A noholonomic mobile robot platform

Prom [9-10], we have:
 + CȦ = IJ
MȦ

(1)

where

(

)

ª r 2 mR2 + I
«
+ Iw
«

4 R2
M=«
2
2
« r mR − I
«
¬
4R 2

(

)

and IJ = [ τr

Ȧ = [ ωr

(

r 2 mR2 − I

)

º
ª
r 2 mc d θ º
»
0
«
»

»
4R
4 R2 »
;C = «
»
2
« r mc d θ
»
r 2 mR2 + I
»
0 »
«−
+ Iw »
2
2
¬
¼
R
4
¼
4R
2

(

l
ω
r
ω

(


º
θω
l
 »
−θω
r¼

)

ª§ r 2 mR2 + I
·
+ Iw ¸
p = Ǭ
2
Ǭ
¸
4R
¹
©

(

§ r 2 mR2 − I
¨
¨
4R 2
©

) ¸·

§ r 2 mc d θ ·º
¨
¸»
¸ ©¨ 4R2 ¹¸»
¹
¼

T

(4)

THE CONTROLLER DESIGN

A. Outer control loop
Let the tangent and angular velocities of the robot be
v and ω respectively. We have:

ª1 R º
ª x º ªcos θ 0º
ªωr º « r r » ª v º
« » «
» ªvº
Ȧ=« »=«
(5)
» « » ; q = « y » = «sin θ 0» « »
¬ω¼
¬ ωl ¼ « 1 −R » ¬ω¼

«¬θ »¼ «¬0
»
1¼
¬« r r ¼»
The target of the control problem is to design an
adaptive control law in which the position vecctor q and the
velocity q to hold the position vector qr (t) , and the
demanded velocity q r (t) in terms of unknown dynamics
parameters of the mobile robot. The desired position and
velocity vectors are represented:

θ

d

ªω

 =« r
Y(Ȧ,Ȧ)
l
¬ω

III.

A typical example of a nonholonomic mobile robot is
shown in Fig. 1 [9-11].

Y

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]:
 =IJ
(2)
Y(Ȧ,Ȧ)p


(3)
Y(Ȧ,Ȧ)p = MȦ + CȦ
where p is a 3x1 vector consisting of the known and
unknown robot dynamics, such as mass and moment of
 is a 2x3 coefficient matrix consisting of the
inertia; Y(Ȧ,Ȧ)
known functions of the robot velocity Ȧ and acceleration
 which is referred as the robot regressor. For the mobile
Ȧ
robot shown in fig.1, we could easily compute:

)

τl ] is the torque applied on the wheels;
T

ωl ] is the angular velocity of the right and left
T

wheels in order; m = mc + 2mw , in which mc is the mass of
the mobile robot platform, mw is the mass of one driving
wheel with the actuator; I = mc d 2 + 2mw R 2 + I c , in which

I c , I w are moments of inertia of platform about the vertical

qr = [ xr

yr

θr ]

T

xr = vr cos θ r
(6)
with vr > 0 for all t
y r = vr sin θ r
θr = ωr
The position tracking error between the reference and
the actual robot could be expressed in the mobile robot’s
coordinator as follows [9-10]:
ª e1 º ª cos θ sin θ 0 º ª xr − x º
e p = ««e2 »» = «« − sin θ cos θ 0 »» «« yr − y »»
(7)
«¬ e3 »¼ «¬ 0
0
1 ¼» ¬«θ r − θ ¼»

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

2013 IEEE 8th Conference on Industrial Electronics and Applications (ICIEA)

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ª vr cos e3 + k1e1
º
ªvº «
»
(8)
«ω» = « ω + k e + v e sin e3 »
3 3
r 2
¬ ¼ « r
»
e
3 ¼
¬
where k1 , k3 > 0. In this law, when e3 → 0 then
sin e3
→ 1 , and ω always be bounded.
e3
ª e1 º ªω e2 − v + vr cos e3 º
e p = «e2 » = « −ω e1 + vr sin e3 »
(9)
« » «
»
«¬ e3 ¼» «¬
»¼
ωr − ω
With the control law in equation (8), it is easy to prove
asymptotical stability system due to ep → 0 when t → ∞ .
Choosing a positive definite function V p as follows:

(

1
1
V p = epT ep = e12 + e22 + e32
2
2
The derivation of Vp with respect to time Vp is:

)

(10)

Vp = eTpe p

= e1e1 + e2e2 + e3e3
= e1 ( ωe2 −v + vr cos e3 ) + e2 ( −ωe1 + vr sin e3 ) + e3 ( ωr −ω)

(11a)

= e1 ( −v + vr cos e3 ) + e2vr sin e3 + e3 ( ωr −ω)
Replacing (8) into (11a), we have
Vp = e1 ( −v +vr cose3 ) +e2vr sine3 +e3 ( ωr −ω)

§
sine ·
= e1 ( −vr cose3 −k1e1 +vr cose3 ) +e2vr sine3 +e3 ¨ωr −ωr −k3e3 −vre2 3 ¸
e3 ¹
©

(11b)

=−k1e12 −k3e32

It is straightforward to see that Vp 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 → vr
(13)

ω → ωr

(14)

Combining (7), (12), (13), (14) infers: e2 → 0, e2 → 0 .
Thus the control law (8) assures the proximity control
system e p → 0 when t → ∞ .

B. Inner control loop
The deviation of stick angular velocity of driven wheels
is:

angular velocity using computed-torque method which is
satisfied ec → 0 and ep → 0 when t → ∞ .
Derivating and multiplying both sides of equation (15)

with the matrix M, we obtain:
 -Ȧ
 c ) = IJ - Cec - Yc p
Me c = M(Ȧ
(16-a)
In case of non-parametric uncertainty component d in
the mobile robot of dynamics model, equation (16-a) is
rewritten as below:
 -Ȧ
 c ) = IJ - Cec - Yc p + d (16-b)
Me c = M(Ȧ
 c + CȦ
c
Yc p = MȦ
(17)
Where
Torque output of the controller is:
IJ = IJ NN - K Dec + Yc pˆ
(18)
where K D is the positive definite matrix and pˆ is the
matrix esimated by the matrix p.
Substituting (17) into (16), we have:
Me c = IJ NN - (K D + C)ec + d + Yc p
(19)
with p = pˆ - p .
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:
d = Wı + İ = dˆ + İ

(20)
where, where W is the weight matrix of an online
updated network; İ is the approximate error and is bounded

İ ≤ ε0 .

by

The neural network Wı 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:

σj

(s
= exp−

j

− cj )

λ j2

2

; j = 1, 2,...n

where, c j , λ j are the expectation and variance of the
Gaussian function chosen in [17]. And we must determine
the parameters c j , λ j differently , which cover the
uncertainty d in terms of amplitude and frequecy. The
output dˆ is an approximation of d.
e
Choosing IJ NN = (η + 1) Wı − δ c to satisfy our control
ec
law.

ªω − ωcr º
ec = Ȧ - Ȧ c = « r
(15)
»
¬ ωl − ωcl ¼
where Ȧ is the desired angular velocity of the robot
calculated by (5), Ȧc is the output of the angular velocity
control wheel torque. We must find the control law of the

1800

(21)

2013 IEEE 8th Conference on Industrial Electronics and Applications (ICIEA)


Because of IJ NN = (Ș + 1)Wı - į

ec

and the matrix C is
ec

symetric, ecT Cec = 0. So that:
ª e
º
V = −eTc K D ec + ecT «-δ c - İ »
»¼
¬« ec
≤ −ecT K D ec - δ ec + ec . İ
≤ −ecT K D ec - δ ec + ec .ε 0

If we choose δ = ε 0 + γ with γ > 0 then
V ≤ −ecT K D ec - Ȗ ec

Figure 2. The approximate neural network of W σ function

Theorem: The dynamics of the mobile robot (1) using
neural network (20) will be tracked the desired trajectory qd
with the error ep → 0 , if we choose the control algorithm

 i as follows:
IJ and learning neural network w
IJ = -K Dec + (η +1) Wı - δ

ec
+ Yc pˆ
ec

 i = −η ecσ i

w

σ j = exp−

(22)

( ecj − c j )

2

IV.

λ j2

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:
2
·
1§
V = ¨ ecT Mec + p T ī-1p + ¦ w Ti w i ¸
2©
i=1
¹

(23)

Derivating V with respect to time, we obtain:
2

V = ecT Me c + pˆ T ī-1p + ¦ w Ti w i
i=1

2

§
·
= ¨ Mec - Yc p - Ș¦ w iσ ι ¸
i=1
©
¹
From (19), we achieve:
V = ecT [ IJ NN - (K D + C)ec - d - ȘWı ]

(25)

From equation (24), (25), we can see that V → 0 so that
ec → 0 , p → 0 and ep → 0 , p → p .
The control system in Fig. 3 is the asymptotic stability
q → qd . 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 ( pˆ → p ).
Both theorem and the global asymptotic stability of the
system with torque control using neural network depicted in

Fig. 3 have proved.
SIMULATION RESULTS

In our scenario, we simulates the mobile robot model
with the following paramters: r = 0.15m; R= 0.75m;
d=0.2m; mc = 30kg; mw = 30kg; Ic = 15.625 kgm2; Iw =
0.0005kgm2; Im = 0.0025 kgm2. The parameters of the
controller are: K D = diag(5,5); k1=k3=2; δ = 10. Suppose
that we only estimate pˆ = 0.6 p and non-parametric
uncertainty components are: d = ª¬sin(0.25t ) cos ( 0.25t ) º¼
πt ·
§
0 ≤ t < 5 : vr = 0.25 ¨ 1 − cos ¸ , ωr = 0
5 ¹
©
5 ≤ t < 25 : vr = 0.5, ωr = 0

T

vr
2π t ·
§
25 ≤ t < 30 : vr = 0.15π ¨ 1 − cos
¸ , ωr = −
5
1.5
©
¹
vr
2π t ·

§
30 ≤ t < 35 : vr = 0.15π ¨ 1 − cos
¸ , ωr =
5 ¹
1.5
©
π
t
§
·
35 ≤ t < 40 : vr = 0.25 ¨1 + cos ¸ , ωr = 0
5 ¹
©
35 ≤ t < 40 : vr = 0.5, ωr = 0

(24)

eTc

= −eTc K D ec - ecT Cec + ecT [ IJ NN - Wı - İ - ȘWı ]

2013 IEEE 8th Conference on Industrial Electronics and Applications (ICIEA)

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The simulation results are presented in below figures:

(m, rad)

A. Using the neural network with the parameter estimation
algorithm of dynamics model:

0.5

0

-0.5

3.5
Actually Trajectory
Desired Trajectory

3

-1

2.5

-1.5
0

5

10

15

Y (m )

2

20

25
30
Time (s)

35

X direct Error
Y direct Error
Orient Error
40
45
50

Figure 8. The position error without τ MN

1.5
1
0.5
0
0

2

4

6

X (m)

8

10

12

Comparing Fig.5 and Fig.8, we can verify the efficiency
of components IJ NN ( created by RBFNN) in compensating
uncertainties in the model dynamics.

Figure 4. The desired trajectory and the actual trajectory with τ MN

0

-0.5

X direct Error
Y direct Error
Orient Error

-1.5
0

5

10

15

20

25
Time (s)

30

35

40

45

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.

0.5

-1

V.

ACKNOWLEDGMENT

50

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

Figure 5. The position error with τ MN

Neural Network Weights

1
w11
w12
w21
w22

0.8
0.6

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0.4
0.2
0

[2]

-0.2
-0.4
0

5

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Time (s)

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40

45

50

Figure 6. The neural network weights with τ MN

[3]

B. Without the neural network with the paramter estimation

algorithm of mobile robot:

[4]

3.5
3

Actually Trajectory
Desired Trajectory

[5]

Y (m)

2.5

[6]

2

1.5

[7]

1
0.5
0
0

[8]

2

4

6

X (m)

8

10

12

Figure 7. The desired trajectory and the actual trajectory without τ MN

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2013 IEEE 8th Conference on Industrial Electronics and Applications (ICIEA)

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