An Alternative for Trajectory Tracking in Mobile Robots Applying Differential Flatness

Abstract

One solution for trajectory tracking in a non-holonomic vehicle, like a mobile robot, is proposed in this chapter. Using the boundary values, a desired route is converted into a polynomial using a point-to-point algorithm. With the properties of Differential Flatness, the system is driven along this route, finding the necessary input values so that the system can perform the desired movement.

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Chapter 8
Elkin Yesid Veslin Díaz
Universidad de Boyacá, Colombia
Jules G. Slama
Universidade Federal do Rio de Janeiro, Brazil
Max Suell Dutra
Universidade Federal do Rio de Janeiro, Brazil
Omar Lengerke Pérez
Universidad Autónoma de Bucaramanga, Colombia
Hernán Gonzalez Acuña
Universidad Autónoma de Bucaramanga, Colombia
An Alternative for Trajectory
Tracking in Mobile Robots
Applying Differential Flatness
ABSTRACT
One solution for trajectory tracking in a non-holonomic vehicle, like a mobile robot, is proposed in this
chapter. Using the boundary values, a desired route is converted into a polynomial using a point-to-point
algorithm. With the properties of Differential Flatness, the system is driven along this route, nding the
necessary input values so that the system can perform the desired movement.
DOI: 10.4018/978-1-4666-2658-4.ch008

149
An Alternative for Trajectory Tracking in Mobile Robots Applying Differential Flatness
INTRODUCTION
Throughout time, mobile robots have acquired
great importance because of a wide variety of
applications arising from the potential for au-
tonomy that they present; some examples include
autonomous robots and transportation systems
(AGV—Automated Guided Vehicles). The func-
tion of an autonomous robot is to carry out different
tasks without any human intervention in unknown
environments, in which transportation systems
move objects from place to place without needing
a driver. In these operations, the main task is to
control the displacement of a robot through a given
route. However, the main problem that exists in
the control systems is precisely the performance
of the system from one space to another, and the
mobile robot is not an exception.
Nowadays, researchers in this field have de-
veloped several inquiries such as the application
of chaotic routes for the exploration of uncertain
spaces and the control of AGV systems in order
to be applied in industry; the former highlights the
application of flexible systems of manufacturing
theories or FSF for the generation of routes in
vehicles with trailers (Tavera, 2009; Lengerke,
2008).
Due to the existence of friction during displace-
ment, this kind of system presents non-holonomic
restrictions in its kinematic structural and, there-
fore, the mobility is reduced (Siciliano, 2009).
It has also been shown that these systems are
differentially flat; meaning that the system has a
set of outputs called flat outputs, which according
to the properties of flat systems, the outputs and
their derivatives allow one to describe the whole
system (Fliess, 1994). This paper highlights this
property in the case of trajectory tracking, with
a point-to-point steering algorithm (van Nieuw-
stadt, 1997; De Doná, 2009). The desired route is
reformulated through a function in time and space.
The parameterization, which is combined with the
differential flatness systems concepts, determines
a set of inputs that allows for the control of robot
movement by means of such routes.
The methodology designed is conceived from
a brief description of differential flatness systems
concepts; afterwards the concept of parameter-
ization of routes is introduced, with the tracking
method from one point to another in order to be
implemented in the system. This analysis provides
different graphical simulations that show the
outcomes. Finally, a discussion is opened about
the advantages of the implementation and future
possibilities for studies.
BACKGROUND
Flatness Systems
The differential flatness systems’ concept was
introduced by Michel Fliess, and his teamwork
through the concepts of differential algebra (Fliess,
1994). They conceive a system as a differential
field, which is generated by a set of variables
(states and inputs). Later, Martin (1997) redefined
this concept in a more geometric context, in which
flatness systems could be described in terms of
absolute equivalence.
According to Fliess (1994), a system is defined
as differentially flat if there is a set of outputs (with
the same number of inputs) called flat outputs.
These outputs can determine the state of the system
and its inputs without the need of integrations; this
means that a system is flat when a set of outputs
Y ∈ can be found in the way:
y h x u u u r
=
( )
( )
, , , ,
… (1)
Such that the states (x) and the inputs (u) could
be defined by a set of outputs and its derivatives as:
x f y y u q
=
( )
( )
, , ,
… (2)
u g y y u q
=
( )
( )
, , ,
… (3)

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