AUTONOMOUS GAIT PATTERN FOR A DYNAMIC BIPED

WALKING

Christophe Sabourin, Kurosh Madani

Laboratoire Images, Signaux, et Syst

emes Intelligents (LISSI EA / 3956)

Universit

e Paris-XII, IUT de S

enart, Avenue Pierre Point, 77127 Lieusaint, France.

Olivier Bruneau

Laboratoire Vision et Robotique - Ecole Nationale Sup

erieure d’Ing

enieurs de Bourges

10 Boulevard Lahitolle 18020 BOURGES, France.

Keywords:

Autonomous gait pattern, biped dynamic walking, Fuzzy-CMAC neural networks.

Abstract:

In this paper, we propose an autonomous gait pattern for a dynamic biped walking. Our approach takes

simultaneously advantage from a Fuzzy-CMAC based computation of robot’s swing leg’s desired trajectory

and a high level control strategy allowing regulating the robot’s average velocity. The main interest of this

approach is to proffer to the walking robot autonomy and adaptability involving only one parameter: the

average velocity. Furthermore, this approach allows increasing the robustness of the walking robot regarding

the forwards pushed force.

1 INTRODUCTION

The design and the control of biped robots are one

of the more challenging topics in the ﬁeld of robotics

and were treated by a large number of research works

over past decades. The potential applications of this

research area are very foremost in the middle and long

term. Indeed this can lead ﬁrstly to a better compre-

hension of the human locomotion mechanisms, what

can be very helpful for the design of more efﬁcient

orthosis. Secondly, the humanoid robots are intended

to replace the human for interventions in hostile en-

vironments or to help him in the daily tasks. How-

ever, in addition to the problems related to autonomy

and decision of such humanoid robots, their basic lo-

comotion task is still today a big challenge. If it is

true that a number of already constructed prototypes,

among which the most remarkable are undoubtedly

the robots Asimo (1) and HRP-2P (2), have proved the

feasibility of such robots, it is also factual that the per-

formances of these walking machines are still far from

equalizing the human’s dynamic locomotion process.

The design of new control laws allowing real time

control for real dynamic walking in unknown envi-

ronments is thus today fundamental. Moreover, such

robots must be able to adapt themselves automatically

to indoor and outdoor human environments. Conse-

quently, it is necessary to develop more autonomous

biped robots with robust control strategies in order to

allow them, on the one hand to adapt their gait to the

real environment and on the other hand, to counteract

external perturbations.

In the ﬁeld of biped locomotion, the control strate-

gies can be classiﬁed in two main categories. The

ﬁrst is based on a kinematics and dynamic model-

ing of the mechanical structure. This implies to iden-

tify perfectly the intrinsic parameters of biped robot’s

mechanical structure, requires a high precision mea-

surement of the joints’ angles, velocities and accel-

erations and needs a precise evaluation of interaction

forces between feet and ground. Moreover, the con-

trol strategies based on a precise kinematics and dy-

namic modeling require a lot of computation. For all

these reasons, the computing of the on-line trajecto-

ries generally are given by using a simpliﬁed model-

ing and the stability is ensured by the control of the

Zero Moment Point (ZMP) (3) (4) (5) (6). The sec-

ond solution consists to use the soft-computing tech-

niques (fuzzy logic, neural networks, genetic algo-

rithm, etc..) and/or pragmatic rules resulting from

the expertise of the walking human. Two main ad-

vantages distinguish this second class of approaches.

Firstly, it is not necessary to know perfectly the char-

acteristics of the mechanical structure. Secondly, this

category of techniques takes advantage from learning

(off-line and/or on-line learning) capabilities. This

last point is very important because generally the

learning ability allows increasing the autonomy of the

biped robot.

In this paper, we present a control strategy for

Sabourin C., Madani K. and Bruneau O. (2006).

AUTONOMOUS GAIT PATTERN FOR A DYNAMIC BIPED WALKING.

In Proceedings of the Third International Conference on Informatics in Control, Automation and Robotics, pages 26-33

DOI: 10.5220/0001203000260033

 SciTePress

Figure 1: Prototype RABBIT.

an under-actuated robot: RABBIT (ﬁgure 1) (7) (8).

This robot constitutes the central point of a project,

within the framework of CNRS ROBEA program (9),

concerning the control of walking and running biped

robots, involving several French laboratories.

This robot is composed of two legs and a trunk and

has no foot as shown on ﬁgure 1. If it is true, from de-

sign point of view, that RABBIT is simpler compared

to a robot with feet, from the control theory point of

view, the control of this robot is a more challenging

task, particularly because, in phase of single support,

the robot is under-actuated. In fact, this kind of robots

allows studying real dynamical walking leading to the

design of new control laws in order to improve biped

robots’ current performances. It is pertinent to note

that the ZMP approach, generally used for humanoid

robots, is not appropriated for the case of a biped

without feet, because the contact surface between the

foot and the ground is limited to a point.

This project has been the subject of many publica-

tions concerning the ﬁeld of control strategies emerg-

ing on the one hand from rigorous mathematical mod-

eling, and on the other hand issued from the use of

CMAC neural networks. Developed approaches have

been subject of experimental validations (10) (11).

In this paper, we present an extension of the control

strategy using the CMAC neural network. In our pre-

vious work (10), the CMAC was used to generate the

joint trajectories of the swing leg but these trajecto-

ries were ﬁxed. Consequently, the step length could

not be changed during the walking. Today, our aim is

to develop a control strategy able to generate a fully

autonomous biped walking based on a soft-computing

approach. In this paper, we show how it is possible to

change the walking gait by using the fusion of differ-

ent trajectories learned by several CMAC neural net-

works. In fact, our control strategy is based on two

stages :

• The ﬁrst one uses a set of pragmatic rules allowing

to stabilize the pitch angle of the trunk and to gen-

erate the leg motions (12). This control strategy al-

lows generating a stable dynamic walking with step

length and velocity transitions. During this ﬁrst

stage, the robot is supposed to move in an ideal en-

vironment (without disturbance). We also assume

that frictions are negligible. However, in the case of

our intuitive control, it is not possible to counteract

external (pushed force) and internal (friction) dis-

turbances. Consequently, we propose to use a neu-

ral network allowing to increase the robustness of

our control strategy. In fact, in the ﬁrst stage, the

pragmatic rules are used as a reference control to

learn, by a set of CMAC neural networks, a set of

joint trajectories.

• In the second stage, we use these neural networks

to generate and to modulate the trajectory of the

swing leg. This trajectory is obtained by fusing

outputs of several neural networks. In fact, the

data contained in each CMAC represent a reference

walking carried out during the ﬁrst stage. The fu-

sion is realized by using fuzzy logic. Consequently,

it is possible to modulate, for example, step length

according to average velocity. Furthermore, the fu-

sion allows us to generate an inﬁnity of trajectories

only from a limited number of walking references.

This paper is organized as follows. Section 2

presents the characteristics of our virtual under-

actuated robot. In Section 3, we explain the method

used to train each CMAC neural network. Section 4

presents the control strategy using the Fuzzy-CMAC

neural networks. In section 5, we give the main re-

sults obtained in simulation. Conclusions and further

developments are ﬁnally given.

2 VIRTUAL MODELING OF THE

ROBOT

The robot RABBIT has only four articulations: one

for each knee, one for each hip. Motions are included

in the sagittal plane by using a radial bar link ﬁxed at

a central column that allows to guide the direction of

progression of the robot around a circle. Each artic-

ulation is actuated by one servo-motor RS420J. Four

encoders make it possible to measure the relative an-

gles between the trunk and the thigh for the hip, and

between the thigh and the shin for the knee. Another

encoder, installed on the bar link, allows to give the

pitch angle of the trunk. Two binary contact sensors

detect whether or not the leg is in contact with the

ground. Based on the informations given by encoder,

it is possible to calculate the step length L

step

when

the two legs are in contact with the ground. The dura-

AUTONOMOUS GAIT PATTERN FOR A DYNAMIC BIPED WALKING

tion of the step t

step

is computed by using the contact

sensor informations (duration from takeoff to landing

of the same leg). Furthermore, it is possible to esti-

mate the average velocity V

by using (1).

step

(1)

The characteristics (masses and lengths of the

limbs) are summarized in table 1. Since the contact

between the robot and the ground is just one point

(passive DOF), the robot is under-actuated during the

single support phase: there are only two actuators (at

the knee and at the hip of the contacting leg) to control

three parameters (vertical and horizontal position of

the platform and pitch angle). In fact, this robot rep-

resents the minimal system able to generate a biped

walking and running gaits.

Table 1: Masses and lengths of the limbs of the robot.

Limb Weight (Kg) Length (m)

Trunk 12 0.2

Thigh

6.8 0.4

Shin

3.2 0.4

The numerical model of the robot previously de-

scribed was designed with the software ADAMS

(Fig. 2).

Figure 2: Modeling of the robot with ADAMS.

This software, from the modeling of the mechani-

cal system (masses and geometry of the segments) is

able to simulate the dynamic behavior of this system

and namely to calculate the absolute motions of the

platform and the relative motions of the limbs when

torques are applied on the joints by the virtual actua-

tors . Figure 3 shows references for the angles and the

torques required for the development of our control

strategy. q

and q

are respectively the measured an-

gles at the hip and the knee of the leg i. q

corresponds

ADAMS is a product of MSC software

to the pitch angle. T

knee

and T

hip

are the torques ap-

plied respectively at the knee and at the hip during the

swing phase, T

knee

and T

hip

are the torques applied

during the stance phase.

Figure 3: Parameters for angles and torques.

The model used to simulate the interaction between

feet and ground is exposed in (13). The normal con-

tact force is given by equation (2):



0 if y > 0

−λ

|y| ˙y + k

|y| if y ≤ 0

(2)

Where y and ˙y are respectively the position and the

velocity of the foot (limited to a point) with regard

to the ground. k

and λ

are respectively the gen-

eralized stiffness and damping of the normal forces.

They are chosen in order to avoid the rebound and to

limit the penetration of the foot in the ground. The

tangential contact forces are computed with the equa-

tion (3) in the case of a contact without sliding or with

the equation (4) if sliding occurs.



0 if y > 0

−λ

˙x + k

(x − x

) if y ≤ 0

(3)



0 if y > 0

−(sgn( ˙x))λ

− µ

˙x if y ≤ 0

(4)

Where x and ˙x are respectively the position and the

velocity of the foot with regard to the position of the

contact point x

at the instant of impact with ground.

and λ

are respectively the generalized stiffness

and damping of the tangential forces. λ

is the co-

efﬁcient of dynamic friction depending on the nature

of surfaces in contact and µ

a viscous damping coef-

ﬁcient during sliding. After each iteration, the normal

and tangential forces are computed from the equations

(2) and (3). But, if F

is located outside the cone

of friction (kF

k > µ

k with µ

the static fric-

tion coefﬁcient), then the tangential force of contact

is computed with equation (4). The interest of this

ICINCO 2006 - ROBOTICS AND AUTOMATION

model is that it is possible to simulate walking with

or without phases of sliding allowing us to evaluate

the robustness of the control.

Within the framework of a real robot’s control, the

morphological description of this one is insufﬁcient.

It is thus necessary to take into account the techno-

logical limits of the actuators in order to implement

the control laws used in simulation on the experimen-

tal prototype. From the characteristics of servo-motor

RS420J used for RABBIT, we thus choose to apply

the following limitations :

• when velocity is included in [0; 2000]rpm, the

torque applied to each actuator is limited to 1.5Nm

what corresponds to a torque of 75Nm at the out-

put of the reducer (ration gear equal to 50),

• when velocity is included in [2000; 4000]rpm the

power of each actuator is limited to 315W ,

• when the velocity is bigger than 4000rpm, the

torque is imposed to be equal to zero.

3 TRAINING OF CMAC NEURAL

NETWORKS

In this section, we present ﬁrstly the CMAC neural

network and secondly, the principle which we use to

train the CMAC.

3.1 CMAC Neural Networks

The CMAC is a neural network imagined by Albus

from the studies on the human cerebellum (14) (15).

Despite its biological relevance, its main interest is

the reduction of the training and computing times in

comparison to other neural networks (16). This is of

course a considerable advantage for real time control.

Because of these characteristics, the CMAC is thus a

neural network relatively well adapted for the control

of complex systems with a lot of inputs and outputs

and has already been the subject of some researches

in the ﬁeld of the control of biped robots (17) (18).

The CMAC is an associative memory type neural

network which is a set of N detectors regularly dis-

tributed on several C layers. The receptive ﬁelds of

these detectors are distributed on the totality of the

limited range of the input signal. On each layer, the

receptive ﬁelds are shifted of a quantiﬁcation step q.

Consequently, the widths of the receptive ﬁeld are not

always equal. The number of detectors N depends on

the one hand of the width of the receptive ﬁelds and

on the other hand of the quantiﬁcation step q. When

the value of the input signal is included in the recep-

tive ﬁelds of a detector, this one is activated. For each

value of the input signal, the number of activated de-

tector is equal to the number of layers C (parameter

of generalization). Figure 4 shows a simpliﬁed or-

ganization of the receptive ﬁelds having 14 detectors

distributed on 3 layers. Being given that there is an

overlapping of the receptive ﬁelds, neighboring in-

puts will activate common detectors. Consequently,

this neural network is able to carry out a generaliza-

tion of the output calculation for inputs close to those

presented during learning.

Figure 4: Description of the simpliﬁed CMAC with 14 de-

tectors distributed on 3 layers. For each value of the in-

put signal, the number of activated detector is equal to 3.

A = [0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0], Y = A(u)

W =

+ w

The output Y of the CMAC is computed by us-

ing two mappings. The ﬁrst mapping projects an

input space point u into a binary associative vector

A = [a

..a

]. Each element of A is associated with

one detector and N is the number of detector. When

one detector is activated, the corresponding element

A of this detector is 1 otherwise it is equal to 0. The

second mapping computes the output Y of the net-

work as a scalar product of the association vector A

and the weight vector W = [w

..w

] (equation (5)).

Y = A(u)

W (5)

3.2 Learning Phase

During this learning phase, we use an intuitive control

allowing us to perform dynamic walking of our virtual

under-actuated robot without references trajectories.

This intuitive control strategy is based on three points:

• the observation of the relations between joint mo-

tions and the evolution of the parameters describing

the trajectory of the robot platform,

• an interpretation of the muscular behavior,

• the analysis of the intrinsic dynamics of a biped.

Based on these considerations, it is possible to deter-

mine a set of pragmatic rules. The objective of this

strategy is to generate the movements of the legs by

using a succession of passive and active phase. Also,

it is possible to modify step length, average velocity

AUTONOMOUS GAIT PATTERN FOR A DYNAMIC BIPED WALKING

by an adjustment of several parameters (12). Con-

sequently, this approach allow us to generate several

reference trajectories which are learned by several

CMAC neural networks.

Figure 5 shows the method used during this training

phase. The trajectories of the swing leg (in terms of

joint positions and velocities) are learned with four

”single-input/single-output” CM AC

(k = 1, .., 4)

neural networks. Indeed, two CMAC are necessary to

memorize the joint angles q

and q

and two other

CMAC for angular velocities ˙q

and ˙q

. q

and q

are respectively the measured angles at the hip and

the knee of the leg i; ˙q

and ˙q

are respectively the

measured angular velocities at the hip and the knee of

the leg i.

Figure 5: Principle of the learning phase of CMAC neural

network (u = q

When leg 1 is in support (q

= 0), the angle q

is applied to the input of each CM AC

(u = q

)

and when leg 2 is in support (q

= 0), this is the an-

gle q

which is applied to the input of each CMAC

(u = q

). Consequently, the trajectories learned by

the neural networks are not function of time but func-

tion of the geometrical pattern of the robot. Further-

more, we consider that the trajectories of each leg in

swing phase are identical. This allow on the one hand

to divide by two the number of CMAC and on the

other hand to reduce the training time. The weights

of each CMAC

are updated by using the error be-

tween the desired output Y

of each CM AC

and the

computed output Y

of each CMAC

4 CONTROL STRATEGY USING

FUZZY-CMAC

Figure (6) shows the global strategy which is used to

control the walking robot. It should be noted that the

architecture of this control can be decomposed into

three parts:

• The ﬁrst is used to compute the trajectory of the

swing leg from several output of the CMAC

neu-

ral networks and a Fuzzy Inference System.

• The second allows regulating the average velocity

from a modiﬁcation of the desired pitch angle.

• The third is composed by four PD control in order

to ensure the tracking of the reference trajectories

at the level of each articulation.

Figure 6: Principle of the control strategy used Fuzzy-

CMAC trajectories.

4.1 Reference Trajectories

During the training stage, ﬁve trajectories with an

average velocity V

included in [0.4..0.8]m/s have

been learned by ﬁve CM AC

. Each CM AC

composed of four single input/single output CMAC

(two CMAC

for the angular positions q

and q

and two other CMAC

for the angular velocities ˙q

and ˙q

). Table 2 gives the main parameters which

are used during the learning phase according to the

desired average velocity V

, where V

is calculated

by using (1). q

and q

are respectively the desired

relative angle between the two thighs and the desired

pitch of the trunk. q

corresponds to the desired an-

gle of the knee at the end of the knee extension of the

swing leg just before the double contact phase.

Table 2: Parameters used during the learning stage.

(m/s) q

(

◦

) q

(

◦

) q

(

◦

)

CMAC

0.4 20 −7 3.5

CMAC

0.5 25 −10 3

CMAC

0.6 30 −15 2.5

CMAC

0.7 35 −20 8

CMAC

0.8 40 −25 8

ICINCO 2006 - ROBOTICS AND AUTOMATION

It must be pointed out that the step length L

step

increases when V

increases. Table (3) gives L

step

according to V

Table 3: V

and L

step

for the ﬁve references trajectories.

(m/s) L

step

(m)

CMAC

0.4 0.23

CMAC

0.5 0.28

CMAC

0.6 0.31

CMAC

0.7 0.36

CMAC

0.8 0.4

4.2 Fuzzy-CMAC Trajectories

During the walking, the measured angle q

, if leg 1

is in support, or q

, if leg 2 is in support, is applied

at each input of each CM AC

. The desired angular

position q

and q

, and the desired angular velocity

˙q

and ˙q

are carried out by using a fusion of the ﬁve

learned trajectories. This fusion is realized by using a

Fuzzy Inference System (FIS). This FIS is composed

of ﬁve rules:

• IF V

IS V erySmall THEN Y = Y

• IF V

IS Small THEN Y = Y

• IF V

IS Medium THEN Y = Y

• IF V

IS Big THEN Y = Y

• IF V

IS V eryBig THEN Y = Y

Where Y

corresponds at the output of the CMAC

Figure 7 shows the membership functions. The aver-

age velocity is modeled by ﬁve fuzzy sets (VerySmall,

Small, Medium, Big, VeryBig). Each desired trajec-

tory Y

is computed by using equation (6).

Figure 7: Membership functions used to compute the

FuzzyCMAC trajectories.

l=1

(6)

Consequently, the trajectory depends on the one

hand of the geometrical position of the stance leg and

on the other hand of the measured average velocity.

4.3 High Level Control

The high level control allows us to regulate the aver-

age velocity by adjusting the pitch angle of the trunk

at each step by using the error between the average

velocity V

and the desired average velocity V

and

of its derivative as described in ﬁgure 6.

At each step, ∆q

, which is computed by using the

error between V

and V

and of its derivative (equa-

tion 7), is then added to the pitch angle of the previous

step q

(n) in order to carry out the new desired pitch

angle of the following step q

(n + 1) as shown in

equation (8).

∆q

= K

− V

) + K

− V

) (7)

(n + 1) = q

(n) + ∆q

(8)

4.4 PD Control

In order to ensure the tracking of the desired trajecto-

ries, the torques T

knee

and T

hip

applied respectively

at the knee and at the hip are computed by using PD

control. During the swing stage, the torques are car-

ried out by using equations (9) and (10). q

and

˙q

are respectively the reference trajectories (position

and velocity) of the swing leg from the output of the

Fuzzy-CMAC (j = 1 for the hip, j = 2 for the knee).

hip

= K

hip

− q

) + K

hip

( ˙q

− ˙q

) (9)

knee

= K

knee

− q

) + K

knee

( ˙q

− ˙q

) (10)

Secondly, the knee of the stance leg is locked, with

= 0 and ˙q

= 0 (equation 11), and the torque

applied to the hip allows to control the pitch angle of

the trunk (equation 12). q

and ˙q

are respectively the

measured absolute angle and angular velocity of the

trunk. q

is the desired pitch angle.

knee

= −K

knee

− K

knee

˙q

(11)

hip

= K

trunk

− q

) − K

trunk

˙q

(12)

5 RESULTS

The control strategy presented in section 4 allows:

• To generate the joint trajectories of the swing leg

from the geometrical conﬁguration of the robot and

the real average velocity.

• To regulate, at each step, the average velocity

thanks to an adjustment of the pitch angle.

AUTONOMOUS GAIT PATTERN FOR A DYNAMIC BIPED WALKING

The main interest of this approach is that the walk-

ing robot is autonomous and is able to adapt, for ex-

ample, the step length from only one parameter: the

average velocity. Furthermore, if the robot is pushed

forwards, the average velocity increases and conse-

quently the step length increases too. In this manner,

it is easier for the robot to compensate this kind of

perturbation.

5.1 Autonomous Walking Gait

Figure 8 shows the stick diagram of the walking of

the robot when the desired average velocity increases.

It must be noticed that the control strategy allows to

adapt automatically the pitch angle and the step length

as the human being.

Figure 8: Stick diagram of the walking robot when the av-

erage velocity increases.

Figure 9 shows the desired average velocity V

the measured velocity V

(equation 1) and the step

length L

step

. When V

increases from 0.3m/s

to 0.9m/s, V

increases gradually and converges

fowards the new value of V

. L

step

increases au-

tomatically from 0.25m to 0.4m from the measured

average velocity at each step.

Figure 9: Average velocity and step length when the desired

average velocity increase from 0.3m/s to 0.9m/s.

It must be pointed out that the walking gait can de-

pends to the desired average velocity V

. But in this

case, the step length transition is abrupt. Furthermore,

the fact that the walking gait depends of the measured

average velocity allows increasing the robustness of

the walking of the robot.

5.2 Evaluation of the Robustness

According to a Pushed Force

During walking, a robot moving in real environment

can be subjected to external forces involving an im-

balance. Consequently, the control strategy must react

quickly in order to compensate this perturbation and

to avoid the fall of the robot. Generally, when human

being is pushed forwards, he increases the step length.

In the case of the proposed control strategy, the step

length depends of the real average velocity V

. If

the robot is pushed forwards, the duration of the step

step

decreases and consequently, V

increases. In

this manner, at the next step after this perturbation,

step

is adjusted.

The ﬁgure 10 shows the evolution of the average

velocity before and after the force perturbation. At

t = 15s, we applied on the trunk of the robot an im-

pulsive pushed force. The duration and the amplitude

of this force are respectively 0.2s and 50Nm. Before

this perturbation, the robot walks with an average ve-

locity to 0.7m/s which corresponds at the desired av-

erage velocity. After this perturbation, V

increases

considerably but our control strategy allow to com-

pensate slowly this perturbation.

Figure 10: Average velocity V

when the robot is pushed

forwards at t = 15s.

The ﬁgures 11 and 12 show respectively the evolu-

tion of the step length and the pitch angle of the trunk.

It must be pointed out that:

• L

step

is adjusted automatically from the measured

average velocity.

• The pitch angle of the trunk q

decreases just after

the application of the pushed force in order to slow

down the velocity of the progression of the robot.

6 CONCLUSION

In this paper, we have proposed an autonomous gait

pattern for a dynamic biped walking. Our approach

is based ﬁrstly on Fuzzy-CMAC issued computation

of robot’s swing leg’s desired trajectory and secondly

on a high level control strategy allowing regulation

ICINCO 2006 - ROBOTICS AND AUTOMATION

Figure 11: Step length L

step

when the robot is pushed for-

wards at t = 15s.

Figure 12: Pitch angle of the trunk when the robot is pushed

forwards at t = 15s.

of the robot’s average velocity. The main interest of

this approach is to proffer to the walking robot au-

tonomy and adaptability involving only one parame-

ter: the average velocity. The obtained results show

the adaptability of the walking step length and the

issued additional robustness of the proposed control

strategy. In fact, when the robot is pushed forwards,

the average velocity increases and consequently the

step length increases too. In this manner, it is easier

to compensate this kind of perturbation.

Future works will focus ﬁrstly on the extension of

the Fuzzy-CMAC approach in order to increase the

autonomy of the walking robot according to the na-

ture of the environment (get up and down stairs for

instance), avoidance and dynamic crossing obstacles

and secondly on the experimental validation of our

approach.

REFERENCES

Y. Sakagami, R. Watanabe, C. Aoyama, S. Matsunaga, N.

Higaki, K. Fujimura. The intelligent ASIMO: system

overview and integration. Proc. IEEE Conf. on Intelli-

gent Robots and Systems, 2002, 2478–2483.

K. Kaneko, F. Kanehiro, S. Kajita, H. Hirukawa, T.

Kawasaki, M. Hirata, K. Akachi, T. Isozumi. Hu-

manoid robot HRP-2. Proc. IEEE Conf. on Robotics

and Automation, 2004, 1083–1090.

M. Vukobratovic, B. Borovac. Zero moment point - thirty

ﬁve years of its live. International Journal of Hu-

manoid Robotics, 2004, Vol.1 N

◦

1, 157–173.

S. Kajita, F. Kanehiro, K. Kaneko, K. Fujiwara, K. Harada,

K. Yokoi and H. Hirukawa. Biped walking pattern

generation by using preview control of Zero-Moment

Point Proc. IEEE Conf. on Robotics and Automation,

2003, 1620–1626.

Q. Huang, K. Yokoi, S. Kajita, K. Kaneko, H. Arai, N. Koy-

achi, K. Tanie. Planning walking patterns for a biped

robot. IEEE Transactions on Robotics and Automa-

tion, 2001, Vol.17, N

◦

3, 280–289.

K. Hirai, M. Hirose, Y. Haikawa, T. Takenaka. The devel-

opment of honda humanoid robot. Proc. IEEE Conf.

on Robotics and Automation, 1998, 1321–1326.

C. Chevallereau, G. Abba, Y. Aoustin, F. Plestan, E.R.

Westervelt, C. Canudas-de-Wit, J.W. Grizzle. RAB-

BIT: A testbed for advanced control theory. IEEE

Control Systems Magazine, 2003, Vol.23, N

◦

5, 57–79.

http://robot-rabbit.lag.ensieg.inpg.fr/

http://www.laas.fr/robea/

C. Sabourin, O. Bruneau. Robustness of the dynamic walk

of a biped robot subjected to disturbing external forces

by using CMAC neural networks. Robotics and Au-

tonomous Systems, 2005, Vol.23, 81–99.

E.R. Westervelt, Gabriel Buche, J.W. Grizzle. Experimental

validation of a framework for the design of controllers

that induce stable walking in planar bipeds. The Inter-

national Journal of Robotics Research, 2004, Vol.23

◦

6, 559–582.

C. Sabourin, O. Bruneau, J-G.Fontaine. Start, stop and tran-

sition of velocities on an underactuated bipedal robot

without reference trajectories. Internationnal Journal

of Humanoid Robotics, 2004, Vol.1, N

◦

2, 349–374.

O. Bruneau, F.B. Ouezdou. Distributed ground/walking

robot interactions. Robotica, Cambridge University

Press, 1999, Vol.17, N

◦

3, 313–323.

J. S. Albus. A new approach to manipulator control:

the cerebellar model articulation controller (CMAC).

Journal of Dynamic Systems, Measurement and Con-

trol, (1975), 220–227.

J. S. Albus. Data storage in the cerebellar model articula-

tion controller (CMAC). Journal of Dynamic Systems,

Measurement and Control, 1975, 228–233.

W. T. Miller, F. H. Glanz, L. G. Kraft. CMAC: An asso-

ciative neural network alternative to backpropagation.

Proceedings of the IEEE, Special Issue on Neural Net-

works, vol.78, N

◦

10, 1990, 1561-1567.

A. L. Kun, T. Miller. The design process of the uniﬁed walk-

ing controller for the UNH biped. Proc. IEEE Conf. on

Humanoid Robots, 2000.

A. Brenbrahim, J. Franklin. Biped dynamic walking us-

ing reinforcement learning. Robotics and Autonomous

Systems, 1997, Vol.22 , 283–302.

AUTONOMOUS GAIT PATTERN FOR A DYNAMIC BIPED WALKING