Intelligent Control Techniques for Humanoid Robots

aroocarmineAI and Robotics

Oct 29, 2013 (4 years and 2 months ago)

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International Journal of Advanced Engineering Applications,
Vol.
2
, Iss.
2
, pp.
72
-
86

(2013)


72


Fragrance Journals

I
ntelligent
C
ontrol

T
echniques

for

H
umanoid
R
obots


Wu,
C. Y.
Tung, P. C.

Department of
Mechanical Engineerin
g
,
National Central University
,

Jongli
, Taiwan

93343016@
cc.ncu.edu.tw


Abstract
-

This paper focuses on the application of the intelligent control

techniques (neural networks, fuzzy logic and
genetic algorithms)

and their hybrid methods (
adaptive

fuzzy
and

slding mode

algorithms) in the area of humanoid

robotic
systems. This
paper represents an attempt to

give a report of the basic principles and concepts of intelligent

control in
humanoid robotics, with an outline of a number of recent

algorithms used in advanced control of humanoid robots.

Keywords
-

Humanoid Robots,

Fuzzy

Logic, Genetic Algorithms
,
Adaptive

Sliding Mode.


1

INTRODUCTION

Many aspects of modern life involve the use of intelligent machines

capable of operating under dynamic
interaction with its

environment. The field of biped locomotion is representative of

this interest concerning
human
-
like robots
(Hirai et al, 1998) and (
Sardain,

et al.1998
)
. The main

reasons for designing the humanoid
robots as service and maintenance

machines is to help us humans enjoy life and to relieve

us of many of the
mundane noncre
ative tasks which we all face

every day. Recently, significant progress has been made in the

design of a hardware platform of a humanoid robot and control

of humanoid robots, particularly in the
realization of dynamic

walking in several full
-
body humanoids

(
Yamazaki,

et al. 2000
)
. It is as

obvious as
interesting that anthropomorphic biped robots are

potentially capable to effectively move in all unstructured
environments

where humans do.

In order to accomplish high and complex demands, which they have as se
rvice machines, humanoid robots
must incorporate the intelligent capabilities. Intelligent humanoid robots are functionally oriented devices built
to perform sets of tasks instead of humans, as autonomous systems capable of extracting information from its
environment and using knowledge about its world and intelligence of their duties and proper governing
capabilities. Human operator can transfer to the robot his knowledge, experience and skill in advance, to make it
capable of solving complex tasks.

Natura
lly, the first approach to making humanoid robots more intelligent was the integration of
sophisticated sensor systems. However, today’s sensor products are still very limited in interactivity and
adaptability to changing environments. On the other hand, i
n to design robots and systems that best adapt to
their environment, research includes investigations in the field of mechanical robot design, environment
perception systems and embedded intelligent control. Also, in the case when the robot performs in an
unknown
environment, the previous knowledge may not be sufficient. Hence, the robot has to adapt to the environment
and to be capable of acquiring new knowledge through the process of learning.

Connectionist theory (NN
-

neural networks), fuzzy logic (FL),

and theory of evolutionary computation
(GA
-

genetic algorithms), are of great importance in the development of intelligent humanoid robot control
algorithms. Each of the proposed paradigms has their own merits and drawbacks. To overcome their drawbacks,
certain integration and synthesis of hybrid techniques (symbiotic intelligence) are important for efficient
application in humanoid robotics.


2

C
ONTROL PROBLEMS IN HUMANOID ROBOTICS


In spite of significant progress in the area of humanoid robots,

a lot
of work has still to be done in order to
improve actuators,

sensors, materials, energy accumulator, hardware and control

software that can be utilized to
realize user
-
friendly biped

robots. We are still in a initial stage in understanding the motor

control

principles and
the sensory integration subjacent to human

walking. The major problems associated with the analysis

and
control of bipedal systems are the high
-
order highly coupled

nonlinear dynamics and furthermore, the discrete
changes

in the dynamic phe
nomena due to the nature of the walking

gait. At the same time, the degree of
freedom formed between

the foot and the ground is unilateral and underactuated, so it is

necessary to synthesize
control method for underactuated degrees

of freedom. Also, the co
ntrol algorithm must accomplish

stable, fast
and reliable performance considering different characteristics

of walking and running.

Walking biped robots can be classified in three different categories. First category represents static walkers,
whose motion is very slow so that system’s stability is completely described by the normal projection of the
Center of Gravity, which only depen
ds on joint’s position. Second category represents dynamic walkers, biped
robots with feet and actuated ankles. These walkers are potentially able to move in a static way, provided that
they have large enough feet and motion is slow. Third categpry represe
nts purely dynamicwalkers, robots
without feet. In this case the support polygon during the singlesupport phase is reduced to a point so that static
walking is not possible. With walking with dynamic balance, the projected center of mass is allowed outside

of
the area inscribed by the feet and the walker may essentially to fall during parts of the walking gait. The control
problems for dynamic walking are more complicated than for walking with static balance, but dynamic walking
International Journal of Advanced Engineering Applications,
Vol.
2
, Iss.
2
, pp.
72
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86

(2013)


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Fragrance Journals

patterns provide higherwalki
ng speed and great efficiency with more versatile walking structures. For all
mentioned categories of walking robots, issue of stable and reliable biped walk is the most fundamental and yet
unsolved with a high degree of reliability. The question has motiv
ated the definition of several dynamic
-
based
criteria for the evaluation and control of balance in biped locomotion. The most common criteria is a zero
moment point (ZMP) (
Vukobratovi´c,

1973
).

A humanoid robot is however, a kind of integrated machines: tw
o arm and two leg mechanism. Hence, we
must not only focus locomotion function but also arm’s function for this kind of machines. From this point of
the view, it is necessary to develop advanced control methods for mobile manipulation of humanoid robots.

B
iological investigations suggest that human’s rhythmic walking is the consequence of combined inherent
patterns and reflexive actions. The inherent dynamic pattern is rhythmic and periodic. It is considered as an
optimal feedforward motion pattern acquired

through development in the typical walk environments without
disturbances. The reflexive action is a rapid response due to the feedback control using sensory information. The
reflexive action determines stability against unexpected events such as external

disturbances or ground
irregularity. In this case, capabilities of adaptability and compensation of external disturbances must be included
in advanced control algorithms.


3

C
ONNECTIONIST
C
ONTROL

A
LGORITHMS

IN

H
UMANOID
R
OBOTICS

Recently, some researchers
have begun considering the use of

neural networks for control of humanoid
walking
(
Doerschuk,

et al, 1988
), (
Miller,

1994
), (
Kun
, et al, 1999
) and (
Wang,

et al, 1992
)
. The various type of
neural networks are used for gait synthesis

and control design of hu
manoid robot as it is multilayer

perceptrons,CMAC networks, fuzzy
-
neural network, RBF networks

or Hopfield networks, that are trained by
supervised or

unsupervised (reinforcement) learning methods. The neural

networks were used as efficient tool
for soluti
on of synthesis

and off
-
line and on
-
line adaptation of biped gait as well as for

solution the control
problem of static and dynamic balance during

process of walking and running on terrain with different

environment characteristics.

Kitamura et all. (
Kitamura,

et al, 1988
) proposed a walking controller based on Hopfield neural network in
combination with an inverted pendulum dynamic model.

Salatian et al. (
Salatan,

et al, 1997
) studied off
-
line and on
-
line reinforcement technique for adapting a gait
designed for horizontal surfaces in order to walk on sloping surfaces. They considered humanoid robot with 8
d.o.f and two force sensors on both foot. The control structure includes gait trajectory synthesizer and adaptive
neural unit that are tuning by re
inforcement signal from force sensors on the foot. The neuron unit includes more
neurons with inhibitory/excitatory inputs from sensor unit. These control algorithms without considering
kinematic and dynamic model of humanoid robot were evaluated only usin
g a biped dynamic simulation.

More recently Miller (
Miller,

1994
) and (
Kun
, et al, 1999
) has developed a hierarchical controller which
combines simple gait oscillators, classical feedback control techniques and neural network learning and does not
require
detailed equations of the dynamics of walking. The emphasis is on the real
-
time control studies using an
experimental ten axis biped robot with foot force sensors. There are 3 different CMAC neural networks for
humanoid posture control. The Front/Back Bala
nce CMAC neural network was used to provide for front/back
balance during standing, swaying and walking. The training of this network is realized using data from foot
sensors. The second CMAC neural network is used for Right/Left Balance in order to predic
t the correct knee
extension required achieving sufficient lateral momentum for lifting the corresponding foot for the desired
length of time. The training of this network is realized using temporal difference method based on error between
desired and real

time of foot rising. The third CMAC network is used to learn kinematically consistent robot
postures. In this case, training is also realized by data from foot sensors.

The results indicate that experimental biped was able to learn the closed chain kinema
tics necessary to shift
body weight from side
-
to
-
side while maintaining good foot contact. Also it was able to learn the quasistatic
balance required to avoid falling forward or backward while shifting body weight from side
-
to
-
side at different
speeds. It
was able to learn the dynamic balance in order to lift a foot off the floor for a desired length of time
and different initial conditions. There are many limitations (limited step length, slow walking, no adaptation for
left
-
right balance, no possibility f
or walking at the sloping surface), hence in the paper (
Kun
, et al, 1999
), the
upgrade and improvement of the proposed approach was realized. The new dynamically balance scheme for
handling variable
-
speed gait was proposed that use preplanned but adaptive
motion sequences in combination
with closed
-
loop reactive control. There are new sensors( piezoresistive accelerometers and two solid
-
state rate
gyroscope) which are mounted on the new Humanoid Robot (Fig.1). The control structure on high
-
level control
lev
el include 7 components: gait generator, simple kinematics block and 5 CMAC controllers. The CMAC
neural network are used for compensation of right and left lift lean angle correction, reactive frontback offset,
right
-
left lean correction, right and left a
nkle


y correction and front
-
back lean correction. Training of neural
network is realized through process of temporal difference learning using information about ZMP from robot
foot sensors. The control structure on the lower control level include reactiv
e lean angle control together with
International Journal of Advanced Engineering Applications,
Vol.
2
, Iss.
2
, pp.
72
-
86

(2013)


74


Fragrance Journals

PID controller. The experimental results indicate that UNH biped robot can walk with forward velocities of the
range (21cm/min
-

72cm/min) with sideways leaning speed of the range (3.6 o/s
-

12.5 o/s).

The previously used

CMAC controller is particularly good option for robotic motor control. It has quality of
fast learning and simple computations in comparison with multilayer perceptrons and similar approximation
capabilities as radial basis function networks. But there ar
e problems with large memory requirements, function
approximation and stability of dynamic walking. These problems are addressed in the paper (
Pratt,

et al, 1999
)
where self
-
organizing CMAC neural network structure is proposed for biped control based on a
data clustering
technique together with adaptation of basic control algorithm. In this case, memory requirements are drastically
reduced and globally asymptotic stability is achieved in a Lyapunov sense. The structural adaptation of the
network centers is
realized to ensure adaptation to unexpected dynamics. Although robustness was enhanced in
terms of height and pitch tracking as well as external disturbance rejection, the adaptive controller does not
guarantee the long
-
term stability of the walking gait.





Fig.
1
Humanoid Robot


Wang et al. (Wang, et al, 1992) has developed a hierarchical controller for a three
-
link two
-
legged robot.
His approach uses the equations of motion, but only for the training of the neural networks, rather than to
directly
control the robot. Authors used very simplified model of biped with decoupled frontal and sagittal plane.
There are 3 neural network (multilayer perceptrons) for control of leg on the ground, control of leg in the air
and for body regulation. Training alg
orithm is standard back propagation algorithm based on error between
decoupled supervising control law and output of all three neural networks. There are no feedback in real time
control, hence it is great problem in the case when system uncertainties exis
t.

Beside considering the walking control problem, very little researc has been done on the intelligent control
of running control problem. Doerschuk et all (
Doerschuk,

et al, 1998
) presented an adaptive controller to control
the movement of simulated
jointed leg during a running stride (uniped control). The main idea of this approach
is using the modularity, i.e. using of separate controllers for each phase of the running stride ( takeoff, ballistic,
landing) thus allowing each to be optimized for the
specific objective of its phase. In takeoff phase, objective of
the controller to realize inverse feedforward control. The controller learns from experience to produce the
control signals that will produce the desired height, distance and angular momentum.

Three different types of
neural networks are investigated (multi layer perceptrons, Cerebellar Model Articulation Controller (CMAC)
and neuro
-
fuzzy nets). It was concluded that neuro
-
fuzzy nets achieve more accurate results that both others
methods. The n
euro
-
fuzzy takeoff controller very accurately controls the angular momentum of the stride after
only two learning iterations. The ballistic controller controls the movement of the leg while the foot is in the air.
The ballistic controller combines neural n
etwork learning with classic PD control. It is typical feedback error
learnig schema. The controller learn the dynamic model of leg from experience generated by PD controller and
improved upon its performance. CMAC controller is used for neural network lea
rning part with possibility to
very accurate control the movement of the leg along a target trajectory even during the first attempt. Ballistic
learning is on
-
line method without the need for precomputed examples. This enables the great humanoid robot
adap
tability to various changes and new conditions.

The neural networks can be efficiently used for generation of trajectories (gait) of humanoid robots (
Juang,

et al, 1996
) and (
Kurematsu,

et al, 1991
). For example, Juang and Lin (
Juang,

et al, 1996
) used bac
k propagation
through time algorithm for gait synthesis of a biped robot. The complex inverse dynamic computations were
eliminated by using linearized inverse biped model.

International Journal of Advanced Engineering Applications,
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, pp.
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4
SYSTEM IDENTIFICATION AND CONTROL


A. Description of the measured signals from
the servos

A set of output signals can be retrieved from the Humanoid Robot servos. These signals provide
information regarding the actual servos angular position, angular velocity, D/C current, temperature and voltage.
The angular position, temperature
and voltage signals are sampled at 100 Hz, while the angular velocity and
load are sampled at a rate 10times slower.


B. Close loop position control

By default the servos are configured for position control. In fact, all servos have an internal feedback
po
sition control loop. This characteristic can be easily confirmed by the simple experiment shown in Figure 2.
The servo tries to follow the desired time varying sinusoidal reference position by changing its actual D/C
current (load) charge through time, eve
n in the presence of an external torque applied at time instant t=6 sec.


C. Open loop velocity control

In contrast, experiments suggest that the servos do not have internally any angular velocity feedback control.
This can be experimentally confirmed by a
pplying an external torque to the servo while in constant rotating
velocity. From Figure 3 it can be concluded that the current consumption is not able to respond accordingly after
the fifth second, when a torque is applied, so the reference angular positi
on cannot be followed.



Fig.
2
Servo following a position (close loop control)



Fig.
3
Servo following a reference velocity with open

loop velocity control


D. Stiction

Stiction is a physical phenomenon that is present in almost any system with moving
components. Therefore,
its characterization is essential for obtaining an accurate dynamic model of the servos. A simple way to quantify
stiction can be made through the following experiment: starting with the servo rotating at a constant speed in
one dire
ction, progressively slowing it down until it stops, and then slowly increase its rotating speed in the
opposite direction. With this experiment it should be possible to identify the typical dead
-
zone effect due to
stiction. In our case this was clearly qu
antified to be around 7
-
10% of the full range in case no load is applied to
the servo, as can be seen from Figure 4.


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Fig.
4
Stiction dead
-
zone


E. Voltage supply

Another parameter with relevance to the behavior of the system is the voltage supplied to
the servos.
Experiments show that the output estimated velocity error is proportional to the voltage supplied to the servo. In
fact, good output velocity estimation is achieved only if the battery is charged around 10 V, as can be seen from
Figure 5.



Fi
g.
5
Effects of the supplied voltage to the servos in theoutputs velocity response


F. Humanoid identification

In order to capture the static and the dynamic properties of the Humanoid Robot, both mechanical
properties of all its components, such as mass
and inertia, as well as its servos dynamic responses, must be
known to a certain degree of accuracy. These dynamic properties will be used in Simulink and simMechanics in
order to get an accurate simulator for the real Humanoid Robot aiming at a good contr
ol strategy.


(1) Mechanical properties identification

An accurate static model of the Humanoid Robot can be obtained based on the physical properties of their
components. Typically, by knowing the mass, center of mass and the inertia tensor of each elemen
t of the HR it
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is possible to get a quite reliable model that can be further used in simulation and control. For quantifying the
masses of each element, a precision scale with a resolution of 0.05 grams was used. The centroid of each mass
was then found by

using the SolidWorks® software package, after the detailed elements of all the pieces
involved were drawn in this 3D CAD software. It was assumed here that, except for the servos, all the pieces are
of isotropic nature. A simple experiment has shown that
the maximum error obtained for the geometric position
of the centroid is of 0.5 mm on each Cartesian direction. Finally, the inertia tensor of each element was
determined through the SolidWorks® software.


Fig.
6
3D models of a servo and a component of
the

Humanoide robot


(
2
)

Dynamic properties identification

For the identification of the dynamic behavior of the servos

it was considered the relation between the
reference input

velocity and the correspondent estimated velocity obtained

through the
following equation:



(1)


where

is the estimated velocity at time instant t
.

is the angular position at time instant t
.

is the angular position in the previous time instant
.

is the sampling period.


The classical prediction error method was used for the

identification of the servo dynamic model
(
Ljung,

1987
)
,
using the

identification data shown in Figure
7.



Fig.
7

Servo Identification Data


After testing several tentative models with different orders,

a BJ(2,1,2,1) was found to best approximate the
desired

dynamical behavior of the servo. The BJ model that results

in the best data fit is the following:





(2)



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Figure
8

compares the real output of the servo with the one

estimated by the BJ model for the validation data. It
can be

concluded that the dynamic characteristics of the servo are

well captured by the BJ model.



Fig.
8

Servo
v
alidation
d
ata


5

F
UZZY

CONTROL ALGORITHMS IN HUMANOID ROBOTICS

As one of methodologies applied for biped gait synthesis and control of bipedwalking, some researchers
used the fuzzy logic (
Vukobratovi´c,

1995
) and (
Zhou,

et al, 2000
). Fuzzy logic were used dominantly as part of
c
ontrol systems on the executive control level, for generation and tuning PID gains, fuzzy control supervising,
direct fuzzy control by supervised and reinforcement error signals. In paper (
Vukobratovi´c,

1995
), fuzzy logic
is applied at the level of local
control for tuning of gains of local PID controller, while the complete control
structure includes nominal feedforward control (based on dynamic model of biped), also. It have showed that
aggregationdecomposition method for stability analysis of complete b
iped system is applicable in the cases
when local subsystems are stabilized with fuzzy regulators.

The problem of biped gait synthesis using the reinforcement learning with fuzzy evaluative feedback is
considered in (
Zhou,

et al, 2000
). As first, initial g
ait from fuzzy rules is generated using human intuitive
balancing scheme. Simulation studies showed that the fuzzy gait synthesizer can only roughly track the desired
trajectory. A disadvantage of the proposed method is the lack of practical training data.

In this case there are no
numerical feedback teaching signal, only evaluative feedback signal exists (failure or success), exactly when the
biped robot falls (or almost falls) down. Hence, it is a typical reinforcement learning problem. The dynamic
balanc
e knowledge is accumulated through reinforcement learning constantly improving the gait during walking.
Exactly, it is fuzzy reinforcement learning that uses fuzzy critical signal. For human biped walk, it is typical to
use linguistic critical signals such

as ”near
-
falldown”, almost
-
success”, ”slower”, ”faster”, etc. In this case, the
gait synthesizer with reinforcement learning is based on a modified GARIC (Generalized Approximate
Reasoning for Intelligent Control) method. This architecture of gait synthes
izer consists of three components:
action selection network (ASN), action evaluation network (AEN), and stochastic action modifie (SAM) (Figure
9
). The ASM maps a state vector into a recommended action using fuzzy inference. The training of ASN is
achieved

as with standard neural networks using error signal of external reinforcement. The AEN maps a state
vector and a failure signal into a scalar score which indicates the state goodness. It is also used to produce
internal reinforcement. The SAM uses both re
commended action and internal reinforcement to produce a
desired gait for the biped. The reinforcement signal is generated based on the difference between desired ZMP
and real ZMP in the x
-
y plane. In all cases, this control structure includes on
-
line adap
tation of gait synthesizer
and local PID regulators. The approach is verified using simulation experiments. In the simulation studies, only
even terrain for biped walking is considered, hence the approach should be verified for irregular and sloped
terrain
. In the Figure
9

,

are the ZMP coordinates
;

,
are the desired joint angles of the
biped gait.


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Fig.
9

The architecture of the reinforcement learning based

gat
synthesizer


6

G
ENETIC

A
LGORITHM

IN

H
UMANOID ROBOTICS

With locomotion robots, GA can be efficiently applied for hierarchical trajectory generation of natural
motion of biped using energy optimization (
Arakawa,

et al, 1997
). The hierarchical trajectory
generation method
consists of two layers, one is the GA level which minimizes the total energy of all actuators and the other is the
evolutionary programming (EP) layer which optimizes the interpolated configuration of biped locomotion
robots. The chromoso
me in the EP level represents the interpolated configuration expressed by 12 state
variables (angles) of the biped. Also, a chromosome in a GA level consists of two parts, the first of them
representing the set of interpolated configurations, while the sec
ond part includes a bit which represents the
effectiveness of configuration (0 or 1). The process runs in cyclic procedure through the application of mutation
and selection at the EP level, transfer of generated interpolated configuration into the GA level
, and complete
evolution process through crossover, mutation, evaluation and selection at the GA level. The fitness function at
the GA level is connected to the optimization of total robot energy in order to ensure the natural movement of
biped. The fitnes
s function also contains some constraints related to the robot motion. The final result represents
the optimized trajectory similar to natural human walking that was demonstrated by experiment.

Another example is the application of GA to PD local gain tuni
ng and determination of nominal trajectory
for dynamic biped walking (
Cheng,

et al, 1997
). The biped with 5 links is considered. In the proposed GA, 19
controller gains and 24 final points for determination of nominal trajectory are taken into account. Des
igns to
attain different goals, such as the capability of walking on an inclined surface, walking at high speed, or walking
with specified step size, have been evolved with the use of GA. The fitness functions are connected to total time
of effective walki
ng, average speed of biped body and the size of the walking step. Total number of generation
for problem solving was between 10 and 80 generations. The research showed excellent results in the evaluation
of control parameters as well as in optimization of
mechanical design of biped.


7

A
DAPTIVE FUZZY SIGNED DISTANCE SLIDING MODE CONTROL

IN

H
UMANOID
ROBOTICS

Basic Signed Distance Fuzzy Logic Control Concept

Consider a class of nth
-
order dynamic systems


,







(
3
)


,

with


,

where

is the state vector;

and

are the control input and the system output;

is a nonlinear continuous function whose upper bound is known as

; and

is not exactly
known, but is of a known sign and is a bounded continuou
s function with upper bounds of

,

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.

The goa
l is to find a suitable control

to force the system output

to follow a given reference


input signal

in the presence of model uncertainties on
and

.

Using the two fuzzy input variables, the error and the change
-
of
-
error, as the reasoning rule, the
conventional FLC has the following form:



: If

is

and

is

then
is
,

(
4
)




,
,

where
,
and
are the linguistic values taken by the system state variables,


,
, and
is
the fuzzy output variable.
The required fuzzy reasoning rule numbers are

.
For a complex higher order
control system, fuzzy input variables require all system states. Hence, the number of required reasoning rules
becomes huge and the generation of the
rule table is quite difficult.
Reducing the fuzzy reasoning rules is crucial
f
or developing the FLC control.

In this section, a modification, the idea of [Choi 1999], the signed distance

, is introduced briefly; this
is used as the antecedent part of the fuzzy logic control to simplify the fuzzy rules from tw
o
-
dimensions to one
-
dimension.
Figure
10

shows a simplified 2
-
dimensional space hyperplane of

and

.

It is called the
switching line and is expressed as follows:


,



(
5
)



Fig.
10

Signed distance in state space.

for SMC. Let

be an end point of a line perpendicular to the switching line in the state space; while
is the other end point
located on the switching line.
The distance

between

and
can be expressed as














(
6
)


.

The signed distance

for the actual state point

in state space is defined as follows:








,











(
7
)

where



Figure
11

shows a skew
-
symmetry rule table, for infinitesimal quantization leve
l having the UNLP FLC
pattern.
There exists some similarity between the signed distance

in Fig. 1 and the infinitesimal quantization
level in Fig.
11
. Moreover, the absolute magnitude of the control input is proportional to the distance from its
main diagonal hyperplane. Thus, the signed distance can be used to simplify the reasoning rules by reducing
the two
-
dimensional skewed
-
symmetric UNLP patter
n FLC rule tables to one
-
dimensional single
-
input fuzzy
variables.

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Fig.
11

Skew
-
symmetry fuzzy rule table with UNPL pattern.


The concept of signed distance can also be extended to a higher order dynamic system for a general
n
-
input
FLC case as follows:

IF


is

and …and


is
,

THEN


is
,




(
8
)



,

where

is the number of fuzzy sets for fuzzy input variables and

(
) is the linguistic value
taken by the system state variable

in the
k
th rule. In this case, the rule table is established in the
n
-
dimensional space of
,
,

and
.
The number of rules

is usually huge, so that it is very difficult
to obtain the control rules. If


the hyperplane

is defined as follows:

,








(
9
)

the signed distance

can also be defined with a general form as


.




(
10
)

This represents the signed distance from the operating point to the switching hyperplane

in Eq. (
9
).
Equation (
10
) contains the required inform
ation about the system states.
Therefore, we obtain the same
conclusion, that the general signed distance can reduce the number of reasoning rules for a FLC control process
from

to
.


Basic Concept of Fuzzy Sliding Mode Control

There
are two parts in this section.
First, we discuss some fu
zzy logic control definitions.
Second, the
conventional sliding mode control concept is briefly reviewed, and the main ideas behind the adaptive signed
distance fuzzy sliding mode control are then pre
sented.


Some Fuzzy Logic Control Definitions

In this paper, the adopted membership functions are Gaussian functions in the form of
.
Moreover, the fuzzy set

can be expressed as
.
m is center of the
Gaussian function, while

is the width.

Definition 1:

A fuzzy rule base,
, that is, a union of fuzzy rules in which each rule

can be
expressed as

IF


is

and …and

is
,



THEN


is
,
,
.






(
11
)

Definition 2:
The firing strength of the

rule is

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.


(1
2
)

Definition 3:

The fuzzy premise function,
, is determined by the premise part of
, and is defined as

,
.








(1
3
)

Definition 4:

The weighted averaged defuzzification [1
6
-
1
8
], is defined as

,









(1
4
)

where
,
.

For complete rule sets, all the fuzzy premise functions,

should be well defined, and
.


Sliding Mode Control

If we define the tracking error
, where
, and define the error vector

as follows:

=
,









(1
5
)

then, equation (1) can be described by a closed
-
loop error dynamic equation of the following form


.











(1
6
)

In a conventional SMC design, the designer must first choose a sliding f
unction.

It can be viewed as a set of
states in which the mapping

is equal to zero, i.e.



,











(1
7
)




,

where

is the coefficient vector of the sliding surface

predetermined by the designer.

There are two steps in
the SMC design procedure. First, a feedback control law capable of achieving the hitting condition, such that
the states can stay on the sliding surfa
c
e is found.
This is called hitting phase; the control effort is the hitting
control
. In order to overcome the presence of modeling imprecision and of disturbances, the control law has
to be discontinuous across the sliding surface
.

Second, if the dynamics are exactly known, we solve the equation formally,

, and we can obtain the
equivalent control,
, which can be interpreted as the continuous control law that maintains

(Shih et al,
1985) and (Nguyen et al,

1995
).
Therefore, the sliding mode controller can be expressed as

.












(1
8
)

The equivalent control

in Eq. (1
8
) can be obtained by letting
, i.e.,

,









(1
9
)

,



(
20
)

where
. In the sliding mode, when the trajectory of the states slides on the surface
and
, the system dynamics can be
characterized by the following (n
-
1)
th
-
order polynomial

. (
21
)

The coefficients

can be properly chosen such that all the roots in Eq. (
21

) will be in the open left
-
half of
the complex plane. Therefore, if the state trajectory can be forced to slide on the sliding surface, the desired
stability of control system performance has been

achieved. However, if the functions

and

are not
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exactly known, there is no way to obtain the equivalent control
. In this paper, a set of fuzzy rules is applied
to approximate the estimated equivalent control
.


Adaptive Fuzzy Signed Distance sliding mode Control

In this section, we first construct a fuzzy sliding mode control and show how to develop an adaptive fuzzy
signed distance controller for obtaining the equivalent control

through rule adaptation.
Then, we construct
a hitting control to guarantee the system’s stability. The main task in this section is to derive an adaptive law to
adjust the parameter vector

such that the estimated equivalent control

ca
n be optimally approximated to
the actual equivalent control

of the SMC, without knowing the functions of

and
.

To design the adaptive law

for the sliding mode controller, let the sliding mode coefficients

be
predefined as in Eq. (
21
), such that a Hurwits polynomial

has all its roots
in the open left half of the complex plane. The variable

denotes the Laplace transformation variable.
Moreover, the fuzzy sliding mode controller can be expressed as

,











(2
2
)

where

is the estimated equivalent control obtained by the adaptive fuzzy sliding mode control. Substitute
Eq. (1
0
) and (2
2
) into Eq. (1
6
); the closed
-
loop error dynamic equation in Eq. (1
6
) can be rewritten as

,









(
23
)

where

,
.

We derive the adaptive law

that is the controller’s parameter vector, such that the estimated equivalent
control

can well approximate the actual equivalent control. Suppose that there exists a set of parameters

in the universal discourse
, such that the estimated equivalent control

has the minimum
approximation error

.












(2
4
)

We further define the approximated equivalent control and actual equivalent control (Wang, 1993), (Lin and
Chen, 1994) and (Li and Tong, 2003) as


and
.










(2
5
)

Then, we define a Lyapunov function

as

,










(2
6
)

where

is the parameter error and

is a positive constant. By using Eqs. (1
9
), (2
3
), (2
4
), and
(2
5
), we
obtain the derivative of the Lyapunov function with respect to time as follows:







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.










(2
7
)

The first term on the right
-
hand side of Eq. (2
7
) is zero
1
. If we choose the adaptive law,




(2
8
)


then Eq. (25) becomes

.












(2
9
)

The hitting control

in Eq. (2
2
) is used to force the states to reach the sliding surface
, to zero, in a
finite time, and then to maintain this condition

for all future timed; that is, we make

a positively
inva
riant set of the closed
-
loop system. If the following hitting control is selected:

,










(
30
)

then equation (
30
) makes Eq. (
29
) less than or equal to zero, and the hitting condition which guarantees the stability of the
sliding mode control system can be satisfied. Therefore, the dynamics of the system expressed in
Eq. (
3
) are stable.

The application of the fuzzification operati
on to fuzzify the crisp signed distance
, in Eq. (
7
) or Eq. (
10
),
is the main concept of the signed distance sliding mode control. Figure
12

illustrates this concept.


Fig.
12

Fuzzification of the crisp
signed distance



8

CONCLUSION

In spite of the intensive development and experimental verification

of various humanoid robots, it is
important to further

improve their capabilities using advanced hardware and control

software solutions to make
humanoid
robots more autonomous,

intelligent and adaptable to the environment and humans. The

presented
survey indicates that the intelligent techniques, if applied

in an appropriate manner, can be very powerful tools
for

attaining these goals.

The
adaptive fuzzy s
lding mode

were used for the synthesis and on
-
line

adaptation of biped gait, as well as
for the control of humanoid

robots to ensure static and dynamic balance during the process

of walking and
running on the terrain with different environment

characterist
ics. The main advantages are the compensation

of
system’s uncertainties and the inclusion of learning

capabilities. The majority of the proposed control algorithms

were verified by simulation, while there were few experimental

verification on real biped an
d humanoid robots.
Besides,

the inclusion of complex nonlinear models in real
-
time control,

limited realized steps and slow walking
are the problems in implementation

of connectionist control algorithms. Fuzzy logic

was used mainly as part of
control syste
ms on the executive

control level, for generation and efficient tuning of PID gains

and direct fuzzy
control by supervised and reinforcement error

signals. The main problem in using fuzzy control algorithms

for
biped robots remains the inclusion of a compl
ex dynamic

model and learning capabilities. The GA represents an
efficient

tool for searching the optimised solutions of gait synthesis and

biped control, the main problem being
how to cope with the

reduction of GA optimisation process in real time and pre
serve

stability of the motion. The
hybrid methods using complementary

characteristics of intelligent techniques have a great potential

in the field
International Journal of Advanced Engineering Applications,
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, pp.
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-
86

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of intelligent humanoid robots. An important

idea from the area of artificial life is the use of simultaneou
s

evolution of the robot design and control.


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