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JSAE Review, Vol.16, Issue 2, April 1995
270
Autonomous Parking of Vehicles with Intelligent
Fuzzy-Neural Networks
Antonio Moran, Masao Nagai
Tokyo University of Agriculture and Technology, Japan
amoran@ieee.org
ABSTRACT
This paper analyzes the performance and
practical implementation of fuzzy logic and neural
networks for the autonomous motion control of
mobile robots. One fuzzy controller and two
neural controllers were designed. The neural
controllers are one fuzzy-based neural controller
and one fuzzy-neural controller which are refined
versions of the fuzzy controller and are trained to
optimize a given cost function. It was found that
the mobile robot with fuzzy-neural controller
presents better tracking performance than the
robot with fuzzy controller and fuzzy-based
neural controller. The fuzzy-neural controller re-
quires less training time and has better
generalization properties than the fuzzy-based
neural controller.
1. INTRODUCTION
There is a significant interest in autonomous
mobile robots which can be defined as vehicles
that are capable of intelligent motion without
requiring neither a guide to follow nor
teleoperator (remote) control but autonomously
plan and control their own motion. Many are the
potential applications of autonomous mobile
robots including reconnaissance/exploratory
robots for land, air and undersea environments,
remote repair and maintenance in inaccessible
places, material handling systems, etc.
This paper analyzes the kinematical modeling of
mobile robots as well as the design of control
systems for the autonomous motion of the robot.
Several control systems have been designed on
the basis of fuzzy logic and neural networks
which are two widely used concepts in the fields
of artificial intelligence and nonlinear processing.
One fuzzy controller and two neural controllers
were designed. The fuzzy controller is designed
using the knowledge and experience of a human
operator. The neural controllers axe one
fuzzy-based neural controller and one fuzzy-
neural controller which are refined versions of the
fuzzy controller and are trained to optimize a
given cost function. Although both fuzzy-based
neural controller and fuzzy-neural controller
integrate the human knowledge processing
abilities of fuzzy logic with the adaptive
capabilities of neural networks, fuzzy-neural
networks implement the process of fuzzy
reasoning through a neural network structure so
that they behave as fuzzy systems with learning
capabilities.
A sensorless experimental mobile robot was
constructed to analyze the feasibility and
practical implementation of fuzzy and neural
controllers. It was found that the robot with fuzzy-
neural control has better performance than the
fuzzy controller and fuzzy-based neural controller
in terms of positioning accuracy and collision
avoidance.
2. PROBLEM DEFINITION
The autonomous mobile robot control problem
consists in designing a controller capable to
move the robot from an arbitrary initial position to
a goal position avoiding collision with objects that
could be present. The problem can be
decomposed in two sub-problems: path planning
and motion control. By solving the path planning
problem, the trajectory to be followed by the robot
is determined. By solving the motion control
problem, the control actions required to achieve
the desired trajectory axe determined.
The objective of this paper is to design an
autonomous control system for mobile robots
moving backwards toward the desired position or
to follow a desired trajectory. The problem
treated in this paper is shown in Fig.1. There is a
mobile robot inside a working area represented
by the plane X-Y of positive coordinates.
Autonomous fuzzy and neural controllers will be
designed to move the robot from arbitrary initial
positions to the goal position given by
coordinates . It is imposed the constraint
of no-collision with objects placed at both sides of
JSAE Review, Vol.16, Issue 2, April 1995
271
Fig.1. Mobile robot control problem
the goal position.
Figure 2 shows the robot model and the state
variables, and which exactly determine the
position of the mobile robot. The coordinate pair
specifies the position of the rear center of
the robot in the horizontal plane X-Y and the
angle specifies the inclination of the robot
respect to the X-axis. The steer angle at the front
wheels is the control input to be determined
given the present coordinate and inclination.
3. STATE EQUATION MODEL
3.1. Robot Kinematical Model
Considering only backward motion and assuming
the robot moves a fixed distance r at every stage,
the discrete-time equations describing the planar
motion of the robot are:
(1)
(2)
(3)
where
(4)
(5)
(6)
and
(7)
whereL represents the length of the robot.
If , then and Eqs.4 and 5 can be
simplified to:
(8)
(9)
Fig.2. Mobile robot model
Equations (4) to (7) have been derived
considering only the kinematics of the
longitudinal motion of the robot at two
consecutive steps. Assuming the robot moves at
low speed and without slipping or skidding, the
dynamics of the motion of the robot has been
neglected.
The state variables,, and the steer
angle are limited to the following ranges:
(10)
3.2. Control Strategy
The control objective is to place the mobile robot
at the position with coordinates ,
and with vertical inclination
. Since the robot must not collide with
objects fixed around the goal position, the
controller will be designed so that the robot firstly
moves to the middle of the working area
with vertical inclination. After
this the robot will move straightaway to
the goal position. This control strategy does not
require the measurement (or estimation) of
variable and can be accomplished if there is
enough space between the initial and goal
positions. In Section 5 it will be described how to
design fuzzy and neural controllers to accomplish
this control strategy.
4. EXPERIMENTAL MOBILE
ROBOT
The structure of the experimented mobile robot
and its control system is shown in Fig.3. The
robot is a four-wheel vehicle with front (steering)
and rear (traction) wheels connected to two
independently controlled stepping motors.
JSAE Review, Vol.16, Issue 2, April 1995
272
Fig.3. Experimental mobile robot
The front wheels axe properly turned to guide the
motion of the robot and the rear wheels are
driven to provide the tractive forces to move the
robot a fixed distance at every stage of control.
The steer angle at the front wheels is the
control input to be determined given the present
coordinate and inclination both of which
require to be measured or estimated. In order to
simplify the structure of the control system, the
coordinate and inclinationhave not been
measured but estimated using the mathematical
model of the robot kinematics given by Eqs.1 to
7.
From the robot mathematical model and the
algorithm of the fuzzy or neural controller, a
microcomputer computes the control signal (steer
angle) which is sent to a power amplifier
through an I/O board to control the angle of
rotation of the stepping motor connected to the
steering wheels. Since hard sensors are not used
to monitor the motion of the robot, it is clear that
the robot is controlled in an open-loop
configuration.
The stepping motor connected to the rear wheels
has been set to rotate a small angle at every
stage of control to move the robot a fixed
distance r. This motor rotates slowly enough to
avoid slipping of the traction wheels.
5. DESIGN OF FUZZY AND
NEURAL CONTROLLERS
Several autonomous control systems have been
designed on the basis of fuzzy logic and neural
networks which are two widely used concepts
inthe fields of artificial intelligence and nonlinear
processing. The fuzzy controller is designed
using the knowledge and experience of a human
operator. The neural controllers are a
fuzzy-based neural controller and a fuzzy-neural
controller which are refined versions of the fuzzy
controller and are trained to optimize a given cost
function. In the following the design method of
fuzzy and neural controllers will be described.
5.1. Fuzzy Controller
Fuzzy Reasoning is an efficient tool for
processing human knowledge and experience
expressed through linguistic IF-THEN rules. If a
human operator is capable to move the robot to
the desired position and to express linguistically
the steer actions he would take to perform this
task, then it is possible to construct a fuzzy
controller capable to emulate the control actions
of the human operator. In the following the design
process of fuzzy controllers is summarized.
(1) Divide the range of variation of controller input
and output variables in several parts each of one
representative of a particular position of the car.
(2) Define a membership function for every
division.
(3) Formulate a control rule base using the
knowledge and experience of a human operator.
(4) Calculate the output of the fuzzy controller
using the correlation inference method.
5.2. Neural Controller
As it is known, a human operator can properly
drive a robot from the initial to the goal position
but cannot perform the driving task in an optimum
way (in a minimum time, or with the minimum
path length, or a path with minimum overshoot,
etc.). In the same way, fuzzy controllers cannot
solve by itself optimization problems. Neural
networks, in the other hand, have learning and
self-tuning capabilities that can efficiently be used
to solve nonlinear optimization and control
problems. Moreover, the human knowledge
processing abilities of fuzzy systems and the
optimization capabilities of neural networks may
be integrated to construct fuzzy-based neural
controllers or fuzzy-neural controllers.
5.2.1. Fuzzy-Based Neural Controller
A fuzzy-based neural controller is a conventional
multilayer neural network with sigmoid neurons
which at the first step of its learning process is
trained to emulate a fuzzy controller and
afterwards is trained to optimize a given cost
function. The internal structure of fuzzy-based
neural networks does not have any particular
representation.
The steps to design a fuzzy-based neural
controller are:
(1) Train the neural network to emulate a roughly
designed fuzzy controller. To do that, the network
is trained with input and output training signals
selected from representative points of the fuzzy
controller.
JSAE Review, Vol.16, Issue 2, April 1995
273
Fig.4. Structure of fuzzy-neural network
(2) Define a cost function to be optimized.
(3) Optimize the cost function by updating the
connection weights of the neural network using
any gradient-based optimization algorithm.
5.2.2. Fuzzy-Neural Controller
Differing from conventional neural networks
whose internal structure is unclear, fuzzy-neural
networks have a particular structure which
represents the process of fuzzy reasoning
through a multilayer neural network. In this
sense, fuzzy-neural networks have premise and
consequence parts as in fuzzy reasoning.
Figure 4 shows the structure of a fuzzy-neural
network with inputs and and output.
Although these variables do not correspond to
the mobile robot control problem analyzed in this
study, they will be used to describe in a simple
way the basic structure of fuzzy-neural networks.
Similarly as the IF-THEN rules of fuzzy logic,
fuzzy-neural networks have premise and
consequence parts connected by a relation
function which relates normalized membership
functions of input and output variables. The first
layer of a fuzzy-neural network is composed of
neurons each of one represents a particular
division of the input variables ( with 3 divisions
and with 2 divisions). The nonlinear function of
these neurons is selected to be the membership
function of the respective division. Neurons in the
premise and consequence parts are properly
connected to represent the process of fuzzy
reasoning, and the nonlinear function of neurons
in intermediate layers are selected to
mathematically represent the AND and OR
relations of fuzzy logic.
Fuzzy-neural networks can be easily configured
to have a symmetrical structure. A detailed
description of the process to structure fuzzy-
neural networks as well as the several types of
fuzzy-neural networks can be found in Ref.1.
The process to design a fuzzy-neural network is
as follows:
(1) Represent the fuzzy controller through a
fuzzy-neural network. A roughly designed fuzzy
controller is enough in the most of cases.
(2) Determine approximate initial values for the
connection weights of the relation part of the
fuzzy-neural network. For this, linear equations
may be formulated by equating the outputs of the
fuzzy controller and fuzzy- neural controller for a
given set of inputs.
(3) Define a cost function to be optimized.
(4) Optimize the cost function by updating the
connection weights of the relation part of the
fuzzy-neural network. Any gradient-based
algorithm may be used to update the connection
weights.
Given the symmetry of the robot kinematics, the
inputs of the neural controllers are not and
but and.
Since the control strategy is firstly to move the
robot to the closest place with coordinate
and with inclination, the following cost
function J has been defined:
(11)
and are the coordinate and inclination of the
robot when it reaches the limits of the working
area. is a positive weighting coefficient
determined in order to achieve the performance
specifications.The parameters to be adjusted
during the network training process are the
neurons connection weights. The training of
the neuro-controller can be performed using
any gradient based optimization algorithm such
as the back propagation algorithm. The
parameters of the neural controller are
iteratively updated according to:
(12)
JSAE Review, Vol.16, Issue 2, April 1995
274
where is the iteration counter and is
calculated from the following equation:
(13)
where is the learning rate and is the moment
coefficient. The total partial derivative
is
calculated considering the kinematics of the robot
motion as given by Eqs.1 to 7. Deriving Eq.12
can be expanded as:
(14)
Defining the state vector:
(15)
the total partial derivatives
and
inEq.14 can be calculated while the robot is
running using the following recurrent equation:
(16)
The Jacobianand the partial derivative
are calculated using the mathematical
model of the robot, and the partial derivatives
and are calculated using the
neural controller and with a proper modification of
the back propagation algorithm.
Compared with fuzzy-based neural controllers,
fuzzy-neural controllers present the following
characteristics:
(1) The fuzzy structure of fuzzy-neural controllers
allows an easy and direct integration of the
human processing abilities of fuzzy logic with the
nonlinear and self-tuning capabilities of neural
networks.
(2) The number of connection weights is
generally lesser for fuzzy-neural networks. The
number of connection weights to be determined
by training can be reduced even more if some
desired properties of the neural network (such as
symmetry) are taken into account.
(3) Pre-training is not required for fuzzy-neural
networks. Initial connection weights are
determined by solving a set of linear equations.
(4) The training time is shorter for fuzzy-neural
networks.
(5) The structure of fuzzy-neural networks allows
a simple tractability of the training process and
training objectives. Some desired properties such
as symmetry, rotational invariance, etc., can be
easily achieved with fuzzy-neural networks.
6. RESULTS AND DISCUSSION
After the fuzzy controller, fuzzy-based neural
controller and fuzzy-neural controller were
designed, their positioning and tracking
performance were evaluated by placing the robot
in several initial positions. Figure 5 shows the
trajectory of the mobile robot with (a) fuzzy
controller, (b) fuzzy-based neural controller and
(c) fuzzy-neural controller.
(a) Fuzzy controller
(b) Fuzzy-based neural controller
(c) Fuzzy-neural controller
Fig.5. Trajectories of mobile robot with (a) fuzzy
controller, (b) fuzzy-based neural controller
and (c) fuzzy-neural controller
JSAE Review, Vol.16, Issue 2, April 1995
275
The initial position is a difficult position in the
sense that there is not much clearance between
the initial and goal positions so that collision with
obstacles around the goal position can easily
happen for inefficient controllers.
In Fig.5(a) it can be noted that the robot with
fuzzy controller collides with obstacles before
reaching the goal position. In Figs.5(b) and (c), it
can be noted that the robot with fuzzy-based
neural controller and fuzzy-neural controller
achieve the desired position with vertical
inclination and without colliding with the obstacles
around the goal position. It can also be noted that
the trajectory of the robot with fuzzy-neural
controller does not present overshoot and
reaches the coordinate with inclination
faster than the robot with fuzzy-based
neural controller.
Figure 6 shows the trajectory of the mobile robot
for a curved reference trajectory using
(a) fuzzy-based neural controller, (b) fuzzy-neural
controller and (c) a controller designed by LQ
theory using exactly linearized motion equations
of the robot. It can be noted that although the
robot with the three controllers is able to track the
desired reference and without colliding with the
fixed obstacles, the trajectory of the robot with
fuzzy-based neural controller and LQ controller
present steady-state error. That is not the case of
the robot with fuzzy-neural controller which
presents a trajectory which fastly readies the de-
sired trajectory with zero steady-state error. The
robot with fuzzy controller failed to follow the
desired trajectory and it is not shown in Fig.6.
From these results the better positioning and
tracking performance of the mobile robot with
fuzzy-neural controller are clearly manifest.
7. CONCLUSIONS
This paper has analyzed the autonomous motion
of mobile robots with fuzzy controller, fuzzy-
based neural controller and fuzzy-neural
controller. It was found that the robot with fuzzy-
neural controller has better positioning accuracy
and tracking performance than the robot with
fuzzy controller and fuzzy-based neural
controller.
Fuzzy-neural networks, whose internal structure
represents the process of fuzzy reasoning,
require less training time than fuzzy-based neural
networks, and its fuzzy structure allows a simple
tractability of the training process and a simple
way to formulate a symmetrical network.
REFERENCES
[1] A. Moran, and M. Nagai, ’Autonomous
Parking of Vehicles with Intelligent Fuzzy-Neural
Networks’, Proc. of AVEC’94, pp.270-275,
Tsukuba, Japan.
[2] J. Alexander, and J. Maddocks, ’On the
Kinematics of Mobile Wheeled Robots’,
Autonomous Robot Vehicles,Springer Verlag,
New York, 1990.
[3] A. Moran, and M. Nagai, ‘Efficient On-Line
Training of Recurrent Networks for the
Identification and Optimal Control of Dynamical
Systems’, Proc. of IJCNN’93, Vol.2, pp.1789-
1792, Nagoya, Japan.
Fig.6. Trajectories of mobile robot with
(a) fuzzy-basedneural controller,
(b) fuzzy-neural controllerand (c) LQ controller
