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Mitigation of the bullwhip effect via novel supply chain forecasting and stock control techniques based on big data obtained from organizations collaborative mechanisms
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Alberto Luviano-Juárez
Trajectory Tracking Control of a Mobile Robot Through a Flatness-Based Exact
Feedforward Linearization Scheme
DOI: 10.1115 / 1.4028872
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Hebertt Sira-Ramirez
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Discontinuos control of nonlinear systems
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John Cortes-Romero
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Article in the Journal
of Dynamic Systems Measurement and Control · May 2015
Instituto Politécnico Nacional
Center for Research and Advanced Studies of the National Polytechnic Institute
National University of Colombia
Hebertt Sira-Ramirez
Machine Translated by Google

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An important property of linear and nonlinear systems, which is
strongly related to the structural property controllability, is the
differential flatness. Differential flatness allows a complete
parameterization of all system variables in terms of a limited set of
special, differentially independent, output variables, called the flat
outputs, and a finite number of their time derivatives
[17–19].
differentially flat when there is no slippage condition as shown in
Ref.
[20],
or by appropriate inertia distribution of the links for a class
of underactuated mobile manipulators
[21].
Several methods have been proposed, and applied, to solve the
regulation and trajectory tracking tasks in mobile robots. These
methods range from sliding mode techniques
[6–8],
backstepping
[9], neural networks approaches
[10],
linearization techniques
[11],
and classical control approaches (see Ref.
[12])
among many other
possibilities. Some classical comprehensive contributions to this
area are given in Refs.
[1]
and
[13].
A useful approach to control
nonholonomic mechanical systems is based on linear time-varying
control schemes (see Refs.
[14]
and [15]). In the pioneering work of
Samson
[16],
smooth feedback controls (depending on an exoge
nous time variable) are proposed to stabilize a wheeled cart.
The main characteristic of a number of mobile robotic systems is
that they present nonintegrable (ie, nonholonomic) kinematic
constraints (see Ref. [4]). As a consequence, these constraints can
not be expressed purely in terms of the underlying generalized
coordinates. Second, in many of these cases, the lack of compli
ance with Brockett's necessary conditions for smooth stabilization
[5]
makes the output trajectory tracking problems particularly
challenging. These features restrict the possible control strategies
and, in addition, some ad hoc procedures are necessary to decou
ple the system in order to accomplish the control task.
MAY 2015, Vol. 137 / 051001-1
[DOI: 10.1115 / 1.4028872]
Controlling nonholonomic mobile robots has been an active topic
of research during the past three decades due to the wide variety of
applications such as: mine excavation, process monitor ing, material
inspection, planetary exploration, military tasks, materials
transportation, man – machine interfaces, etc.
[1–3].
Journal of Dynamic Systems, Measurement, and Control
Copyright VC 2015 by ASME
In this article, a multivariable control design scheme is proposed for the reference trajec
tory tracking task in a kinematic model of a mobile robot. The control scheme leads to time-
varying linear controllers accomplishing the reference trajectory tracking task. The proposed
controller design is crucially based on the flatness property of the system lead ing to
controlling an asymptotically decoupled set of chains of integrators by means of a linear
output feedback control scheme. The feedforward linearizing control scheme is invoked and
complemented with the, so called, generalized proportional integral (GPI) control scheme.
Numerical simulations, as well as laboratory experimental tests, are presented for the
assessment of the proposed design methodology.
The feedforward linearization scheme that a feedforward, open
loop, control scheme, computed on the basis of flatness, is capable
of locally maintaining the actual states of the system close to a
vicinity of a desired trajectory. The open loop controller is com
plemented with a linear control scheme intimately associated with
the linearization of the system. In our developments, we consider
the feedforward linearization scheme in connection with a flatness
based control scheme. Since the obtained linearized system is a set
of two second-order integrator chains, they can be controlled by
means of regular proportional integral derivative (PID)
With the help of a nominal feedforward input, deduced from the
flatness of the system, it is possible to reduce the control task to
that of a linearized multivariable system. This general control ler
design approach was proposed by Hagenmeyer and Delaleau, and
it is known as “exact feedforward linearization” [22]. The method is
valid for a large class of differentially flat systems and its application
requires that initial conditions are found within a vicinity of the
desired nominal trajectory. One of the advantages of the method is
the fact that an open-loop linearizing feedforward input can,
practically, yield a linear system in Brunovsk y form, thus, reducing
the problem of controlling a nonlinear system (possibly with
nonholonomic constraints), to that of controlling a simple linear
system in a canonical form. The feedforward lineari zation technique
provides a good local solution for the regulation and trajectory
tracking of a large class of nonlinear systems. The devised
controllers perform quite well, even in the case of com plex
trajectories to follow.
The differential parameterization leads to a flatness based control
ler whose performance is closely related to the exact knowledge of
the gain matrix, which is highly nonlinear and whose variations may
lead to instabilities or singularities. Besides, the perfect knowledge
of this matrix needs the precise measurement of the first time
derivative of the set of flat outputs, which is not given in practice,
even the positions provided by vision systems are likely to present
additive noises for instance.
For the case of the differential mobile robot, the center of the axis
connecting the wheels can be considered as the flat output.
It has been shown that some mobile robotic systems are
1
Alberto Luviano-Ju arez1
John Cort es-Romero Department
of Electrical and Electronics
Engineering,
Universidad Nacional de
Colombia, Carrera
30 No. 45-03, Bogot a, CP
111321, Colombia e-mail: [e-mail]
Exact Feedforward
Through a Flatness-Based
Linearization Scheme
Trajectory Tracking
Control of a Mobile Robot