Control of the Chaotic Phenomenon in Robot Path using Differential

Flatness

Salah Nasr

, Amine Abadi

, Kais Bouallegue

and Hassen Mekki

Laboratory of Networked Objects Control and Communication Systems, National Engineering School of Sousse,

University of Sousse, Tunisia

Department of Electrical Engineering, Higher Institute of Applied Sciences and Technology of Sousse, Tunisia

Keywords:

Chaotic Phenomenon, Mobile Robot, Flatness Control, Chaos Theory.

Abstract:

This paper deals with the complex chaotic behavior that can appear in the dynamic trajectory of a mobile

robot, when the robot is broken or partly damaged. However, a ﬂatness-based controller is designed to ensure

the trajectory planning and tracking. Different mathematical tools have been used such as the ﬂatness control

technique and non linear chaotic systems. The simulation results for the kinematic controller are presented to

demonstrate the effectiveness of this approach.

1 INTRODUCTION

The control of mobile robots has been the subject

of much research in recent years, due to the increa-

singly frequent use in dangerous or inaccessible envi-

ronments where human beings can hardly intervene.

For autonomous mobile robotics, path generation and

execution are very important tasks. Path planning is

the process of generating a sequence of trajectory de-

riving from the assigned task to the mobile robot to be

able to perform it.

The general problem is reduced in most cases to

move the robot in a known or unknown environment

(Belaidi et al., 2017), while avoiding any ﬁxed or mo-

bile obstacles, to carry out a prescribed task. It should

deﬁne a strategy of movement (path planning)(Hargas

et al., 2015), and then execute the prescribed displa-

cement.

The robot controller, which is a major compo-

nent, has received a lot of attention from researchers.

This is why it has a direct impact on its robust-

ness and could prevent its deployment and applica-

bility in several domains (Kumar et al., 2014; Lai,

2014). Many control techniques have been propo-

sed for modern robots including the classical PID,

feedback linearization (Korayem et al., 2016; Tinh

et al., 2016), inverse dynamics, model predictive con-

trol (Klan

car and

Skrjanc, 2007), adaptive fuzzy-logic

control (Bakdi et al., 2017) etc.

Up to now, there has been no experimental work

that has treated the chaotic phenomenon in the ro-

bot trajectory. On the other hand, the interaction be-

tween the theory of chaos and mobile robotics has

been only recently studied, as it can be seen in (Nehm-

zow, 2003), for the generation of the unpredictable

trajectory for the robot. For example, the integration

between a chaotic system and the robot motion sy-

stem, dynamic systems, is used to impart the chaotic

behavior to a robot like the Arnold system in (Na-

kamura and Sekiguchi, 2001). An extension of this

strategy, applying various chaotic systems on the in-

tegration with the kinematics model of the robot, can

be found in (Jansri et al., 2004). In (Martins-Filho

et al., 2004), the author proposed an open loop cont-

rol approach to produce unpredictable trajectories so

as to control the velocities of the robot wheels, and the

state variables of the Lorenz chaotic system are used.

Neverthless, there has been no research work to solve

the chaotic phenomenon problem that can appear in

the robot trajectory.

In this context we propose to use a controller to

solve this problem and to facilitate the implementa-

tion of our work in a real mobile robot. One stra-

tegy of nonlinear control gaining popularity among

researchers is the differential-ﬂatness-based control

(Veslin Diaz et al., 2011; Levine, 2009). It has been

investigated to control ﬂexible robots (Markus et al.,

2012; Markus et al., 2017), mobile robots (Coulaud

and Campion, 2007), under-actuated planar robots

(Vivek et al., 2010), and so on. Differential ﬂatness is

known to be well suited for the problem of trajectory

generation and tracking (Markus et al., 2013). With

Nasr, S., Abadi, A., Bouallegue, K. and Mekki, H.

Control of the Chaotic Phenomenon in Robot Path using Differential Flatness.

DOI: 10.5220/0006828302370243

In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2018) - Volume 2, pages 237-243

ISBN: 978-989-758-321-6

237

this strategy, the trajectories (position, velocity and

acceleration ) of a nonlinear system can be easily in-

terpolated by deﬁning a smooth curve with initial and

ﬁnal conditions. The control variables and state can

be reconstructed without having to integrate the sy-

stem equations(Levine, 2009). Thus, we utilize the

ﬂatness control method to solve the problem of path

planning and chaotic phenomena , which can appear

in the robot trajectory; and we ensure that the mobile

robot tracks this trajectory.

This paper is organized as follows. In section 2,

we explain the basic description model of the robot.

The basic deﬁnition and control strategy of the diffe-

rential ﬂatness theory is presented in section 3. We

describe the kinematic system and its ﬂatness pro-

perty and we propose the control law to solve the tra-

jectory tracking problem. In section 4, we present the

chaotic phenomenon and the control law to solve the

trajectory problem. We give the concluding remarks

in section 5.

2 MODEL DESCRIPTION

The mobile robot considered in this work is a differen-

tial motion robot with two degrees of freedom, com-

posed by two independent active wheels, and a third

passive wheel (a kind of standard free-wheel). This

type of robots represents an important compromise

between the simplicity of control and the degrees of

freedom that allow the robot to accomplish the mobi-

lity requirements (Siegwart et al., 2011).

The robot structure is considered as a rigid body

operating on the horizontal plane (ﬁgure 1).Its kine-

matic model can be described as a differential system

composed of two control parameters, v and ω, which

respectively represent the values of linear and angu-

lar speeds. The state equation of the wheeled mobile

robot is written as follows:





˙x(t)

˙y(t)

θ(t)









cosθ(t) 0

sinθ(t) 0

0 1







v(t)

ω(t)



(1)

where x and y are the position of the robot and θ

is the orientation angle of the robot. The robot displa-

cement control can be performed by supplying the li-

near and angular velocities of the body, v(t) and ω(t),

called control variables or inputs.

Figure 1: Geometry of mobile robot on Cartesian plane.

3 FLATNESS CONTROL

METHOD

Flatness is a characteristic or property of a particular

system in which all solutions of the system can be pa-

rameterized by a ﬁnite number of functions and their

derivatives (Nicolau and Respondek, 2013). For the

analysis and design of controllers for nonlinear sys-

tems with this characteristic, this mathematical pro-

perty is extensively used.

3.1 Flatness Theory

Differential ﬂatness is a property of control systems

Dynamics, as presented by Fliess et al. (Fliess et al.,

1995). Differential ﬂatness, provides a uniﬁed ana-

lysis framework for trajectory planning and control

of nonlinear systems. This is particularly useful for

non-linear sub-actuated systems where it is difﬁcult

to plan and to analytically design possible trajecto-

ries. The necessary condition for a control system to

be differentially ﬂat is that it must be controlled.

From a control perspective, a good explanation of

differential ﬂatness for any nonlinear systems of the

form,

˙x = f (x, u);x ∈ R

, u ∈ R

(2)

The system can be stated to be differentially ﬂat if and

only if there exists a ﬁnite set of independent varia-

bles, equal to the number of inputs, called ﬂat outputs

y = [y

, ..., y

]

in such a way that :

y = y



x, u, ˙u, ...,

(p)



(3)

x = x(y, ˙y, ¨y, ...,

(r)

) (4)

u = u(y, ˙y, ¨y, ...,

(q)

) (5)

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

238

Moreover, for a ﬂat system, there is an inverti-

ble input and state transformations that can transform

non-linear systems into linear canonical forms (con-

trollable linear chain of integrators). An arbitrary

trajectory for ﬂat outputs corresponds to the original

state of the system of reference trajectories.This ma-

kes planning possible in the ﬂat output domain. In

addition, the linear feedback of the control can be de-

signed in the ﬁeld of linear ﬂat outputs by closing the

loop on errors in the ﬂat outputs and their derivatives.

3.2 Flatness Control Strategy

The Control design and trajectory planning for ﬂat sy-

stems are relatively easy because the trajectory can be

deﬁned in terms of ﬂat outputs while the required con-

trol input can be obtained using the ﬂatness property.

In order to prove how the kinematic model of the

mobile robot is differentially ﬂat, we choose the Car-

tesian position of the robot center (x, y) as ﬂat out-

puts. To design a diffeomorphism between ﬂat out-

puts and their derivatives and original states, the in-

put prolongation is utilized. Prolongation is a crucial

method used where the vector representing the state

is extended by some system parameters which is used

to describe a particular system as a differentially ﬂat

system. A very common prolongation way means is

the input prolongation where the input also becomes

a state. This property is utilized in optimal trajectory

generation and tracking control laws.

Now, on performing one prolongation of v as an

additional state, we describe the prolonged systems

by:











˙x = v cos θ

˙y = v sin θ

˙v =

θ =

(6)

Here,

are the new inputs for the prolonged

system that satisfy:

= ˙v

= ω

(7)

By choosing the ﬂat outputs

Fo = [Fo

, Fo

]

= [x, y]

(8)

All the inputs and the state variables can be ex-

pressed in terms of ﬂat outputs and their derivatives.

With (x, y) = (Fo

, Fo

)

v =

, θ = arctan





, (9)

The inputs

can be deﬁned as follows:

= ˙v =

(10)

θ =

(11)

By differentiating the ﬂat outputs up to an input

appears, an invertible relation between inputs and hig-

her derivatives of the ﬂat outputs can be equivalently

build from equation 10 and equation 11 as described

follows:





= D





(12)

With

D =



cosθ −vsinθ

sinθ vcosθ



(13)

the inputs are choosing as





= D

−1

V =



vcosθ vsinθ

− sinθ cosθ



V (14)

Then equation 12 can be written as

Fo = V. (15)

The reference trajectory must allow the robot

to move, from an initial position with coordinates

(x, y) at time t=0 to a ﬁnal position with coordinates

(x f , y f ) at time t = 10 s, with minimum of energy and

also avoid some static circular obstacles. These ob-

stacles are deﬁned by the following equation:

= (x − x

)

+ (y − y

)

− R (16)

Where x

andy

are the coordinates of the center of

the circle and r denotes the radius, i is the number of

obstacles.

The constraint which means that the mobile robot

avoids the obstacle is deﬁned as follows:

(x, y) = (x − 2)

+ (y − 2)

− 1 ≥

0 (27)

(x, y) = (x − 6)

+ (y − 3)

− 1 ≥

0 (28)

(x, y) = (x − 8)

+ (y − 5)

− 1 ≥

0 (29)

(x, y) = (x − 6)

+ (y − 6)

− 1 ≥

0 (29)

(x, y) = (x − 2)

+ (y − 6)

− 1 ≥

0 (29)

To meet these objectives, the problem of reference

trajectory generation is formulated as an optimization

problem in the following way:

Control of the Chaotic Phenomenon in Robot Path using Differential Flatness

239

min

˙x

+ ˙y

(17)

≥ 0 (18)

This problem of optimization is solved by the

most efﬁcient method based on the ﬂatness and the

B-spline function (Bahrami et al., 2009).

x(0) = 0 x(10) = 9

y(0) = 0 x(10) = 9

θ(0) = 0 θ(10) = 0

v(0) = 0 v(10) = 0

(19)

Figure 2: Simulation results of reference and real trajecto-

ries of x position.

Figure 3: Simulation results of reference and the real trajec-

tories of y position.

In Figure 2and 3, we show that the ﬂatness con-

trol input deﬁned by equation 10 and equation 11

permits a good tracking of the desired trajectory for

the mobile Robot. Therefore, the ﬂatness property is

considered as a powerful tool for path planning and

tracking trajectory. As depicted in Figure6, the mo-

bile robot can easily avoid the deﬁned static obstacle.

Figure 4: Simulation results of the control input U1.

Figure 5: Simulation results of the control input U2.

Figure 6: Simulation results of optimal trajectory with ob-

stacle avoidance.

4 CHAOTIC PHENOMENA

Deterministic chaos has been employed for develo-

ping consumer electronic products and intelligent in-

dustrial systems.

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

240

4.1 Chaos Theory

During the 20th century, three great revolutions

occurred: quantum mechanics, relativity and chaos.

The theory of chaos, also called the dynamical system

theory, is the study of unstable aperiodic behavior in

deterministic dynamical systems, which show a sen-

sitive dependence on initial conditions (Vaidyanathan,

2013).

The chaos theory has drawn a great deal of at-

tention in the scientiﬁc community for almost two

decades. Chaos is a very interesting phenomenon

in nonlinear dynamical systems, which has been in-

tensively studied during the last decades and used

in several possible commercial applications (Trejo-

Guerra, 2008).

The Lorenz system has become one of paradigms

in the research of chaotic systems. The Lorenz chao-

tic system is utilized for investigation. The dynamical

equations of the Lorenz system is given as follows:







= −10X

+ 10.X

= 28X

− X

= −

+ X

(20)

The implementation of this dynamic system is

presented in ﬁgure 7.

Figure 7: Lorenz chaotic system.

4.2 Chaos Analysis in Mobile Robot

The most applications of chaos in robotics are clas-

siﬁed into two types: chaos synthesis and chaos ana-

lysis ; chaos synthesis in robotics is deﬁned as the

application of chaotic systems for motion planning of

autonomous robots and entails the generation of artiﬁ-

cial chaos to make different mobile robots accomplish

speciﬁc tasks (Aihara and Katayama, 1995). Whe-

reas, chaos analysis implies the observation of chaotic

behavior in autonomous robots. Therefore, control-

ling the chaotic behavior of the mobile robot becomes

a worthwhile endeavor.

In this subsection, based on the Lorenz chaotic sy-

stem, we give a chaotic behavior to the mobile robot.

Subsequently, we use the control technique based on

differential ﬂatness to control this chaotic behavior

in order to allow the robot to complete its trajectory,

despite its behavior, and to achieve its objective.

By using the dynamic equation of the Lorenz sy-

stem, introduced in equation 20, we will ﬁnd the robot

equation of motion as follows:











= −10X

+ 10.X

= 28X

− X

= −

+ X

˙x = v cos(X

)

˙y = v sin(X

)

(21)

The proposed system described in equation 21

generates an unpredictable path by giving a chaotic

behavior of the mobile robot with two independent

active wheels.

Figure 8: Chaotic phenomena in mobile robot.

As depicted in ﬁgure 8, the sensitivity to initial

conditions makes the robot trajectory extremely un-

predictable. Then with this behavior, the robot can

not reach its objective. Thus, moving from an initial

position to a ﬁnal one is almost impossible with this

behavior.

In this context, control over ﬂatness may be a good

solution to solve this problem. We adopt the techni-

que used in section 3 to restore the control of the new

kinematic system combined with the Lorenz chaotic

system. In this case, we choose θ = X

5 DISCUSSION

Figures 2, 3, 4 and 5 illustrate the effectiveness of the

closed-loop ﬂatness control which allows the mobile

robot to follow the desired reference trajectory pro-

perly. By ensuring a good tracking of the trajectory,

Control of the Chaotic Phenomenon in Robot Path using Differential Flatness

241

Figure 9: Flatness control of x chaotic trajectory of mobile

robot.

Figure 10: Flatness control of y chaotic trajectory of mobile

robot.

Figure 11: Flatness control of x-y chaotic trajectory of mo-

bile robot with obstacle avoidance.

the mobile robot can move by avoiding the static ob-

stacles with the minimal energy and by choosing the

optimal trajectory.

Time [s]

0 10 20 30 40 50 60 70 80 90 100

-4

-2

Figure 12: Flatness control of second chaotic trajectory of

mobile robot.

Figure 7 shows the behavior of the Lorenz chaotic

system in the Cartesian plan.

In ﬁgures 9, 10 and 12 we present the good ef-

fectiveness of chaotic trajectory tracking , and we en-

sure that the robot better reaches its desired trajectory.

Even more, as depicted in ﬁgure 11, we can show the

robustness of the control strategy with chaotic pheno-

menon and in the presence of obstacles.

Next, some endeavors for uncovering the chaotic

behavior of robots are presented. Chaos can be em-

ployed for analyzing robotic arms, and chaos quan-

tiﬁers can be used for analyzing chaotic dynamics in

robot-environment interaction.

6 CONCLUSIONS

This article has described the path planning and the

ﬂatness based tracking control of a wheeled mobile

robot. The ﬂatness-based approach to trajectory con-

trol and optimal trajectory tracking offers a fast al-

ternative to the classical control for such robots. Ha-

ving determined the ﬂat output of the mobile robot,

the trajectory control has been determined with rea-

sonable accuracy. Secondly, we have presented a cha-

otic phenomenon tuned to the behavior of the auto-

nomous mobile robot, so we have solved the problem

related to this phenomenon using the differential ﬂat-

ness method.

In recent years, the discovery of chaos has attrac-

ted much interest among investigators. Deterministic

chaos leads to a quantitative analysis, which is the es-

sence of science. In spite of several efforts to ﬁnd evi-

dence of chaotic dynamics in robotics, useful appli-

cations of deterministic chaos in robotics have rarely

been studied.

ICINCO 2018 - 15th International Conference on Informatics in Control, Automation and Robotics

242

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