Abstract: This paper proposes a simple walking control method for a 10 degree of freedom (DOF) biped robot with stable and humanlike walking using simple hardware configuration. The biped robot is modeled as a 3D inverted pendulum. From dynamic model of the 3D inverted pendulum and under the assumption that center of mass (COM) of the biped robot moves on a horizontal constraint plane, zero moment point (ZMP) equations of the biped robot depending on the coordinate of the center of the pelvis link obtained from the dynamic model of the biped robot are given based on the D’Alembert’s principle. A walking pattern is generated based on ZMP tracking control systems that are constructed to track the ZMP of the biped robot to zigzag ZMP reference trajectory decided by the footprint of the biped robot. An optimal tracking controller is designed to control the ZMP tracking control system. When the ZMP of the biped robot is controlled to track the x and y ZMP reference trajectories that always locates the ZMP of the biped robot inside stable region known as area of the footprint, a trajectory of the COM is generated as a stable walking pattern of the biped robot. Based on the stable walking pattern of the biped robot, a stable walking control method of the biped robot is proposed by using the inverse kinematics. From the trajectory of the COM of the biped robot and an arc referenceinput of the swinging leg, the inverse kinematics solved by the solid geometry method is used to compute the angles of each joint of the biped robot. These angles are used as references angles. Because the reference angles of the biped robot are computed from the stable walking pattern of the biped robot, the walking of the biped robot is stable if the angles of each joint of the biped robot are controlled to track those reference angles. The stable walking control method of the biped robot is implemented by simple hardware using PIC18F4431 and dsPIC30F6014. The simulation and experimental results show the effectiveness of this proposed control method Keywords: Optimal Tracking Controller; ZMP Tracking Control System; Biped Robot.
Trang 1A Simple Walking Control Method for Biped Robot
with Stable Gait
Tran Dinh Huy, Nguyen Thanh Phuong, Ho Dac Loc, *Tran Quang Thuan
Ho Chi Minh City University of Technology, Vietnam
* Posts and Telecommunications Institute of Technology branch Hochiminh city
e-Mail: phuongkorea2005@yahoo.com
Abstract:
This paper proposes a simple walking control method for a 10 degree of freedom (DOF) biped robot with stable and human-like walking using simple hardware configuration The biped robot is modeled as a 3D inverted pendulum From dynamic model of the 3D inverted pendulum and under the assumption that center of mass (COM) of the biped robot moves on a horizontal constraint plane, zero moment point (ZMP) equations of the biped robot depending on the coordinate of the center of the pelvis link obtained from the dynamic model
of the biped robot are given based on the D’Alembert’s principle A walking pattern is generated based on ZMP tracking control systems that are constructed to track the ZMP of the biped robot to zigzag ZMP reference trajectory decided by the footprint of the biped robot An optimal tracking controller is designed to control the ZMP tracking control system When the ZMP of the biped robot is controlled to track the x and y ZMP reference trajectories that always locates the ZMP of the biped robot inside stable region known as area
of the footprint, a trajectory of the COM is generated as a stable walking pattern of the biped robot Based on the stable walking pattern of the biped robot, a stable walking control method of the biped robot is proposed by using the inverse kinematics From the trajectory of the COM of the biped robot and an arc reference input of the swinging leg, the inverse kinematics solved by the solid geometry method is used to compute the angles of each joint of the biped robot These angles are used as references angles Because the reference angles of the biped robot are computed from the stable walking pattern of the biped robot, the walking of the biped robot is stable if the angles of each joint of the biped robot are controlled to track those reference angles The stable walking control method of the biped robot is implemented by simple hardware using PIC18F4431 and dsPIC30F6014 The simulation and experimental results show the effectiveness of this proposed control
method
Keywords: Optimal Tracking Controller; ZMP Tracking Control System; Biped Robot
1 Introduction
Research on humanoid robots and biped robots
locomotion is currently one of the most exciting
topics in the field of robotics and there exist many
ongoing projects Although some of those works
have already demonstrated very reliable dynamic
biped walking [11], it is still important to
understand the theoretical background of the biped
robot The biped robot performs its locomotion
relatively to the ground while it is keeping its
balance and not falling down Since there is no
base link fixed on the ground or the base, the gait
planning and control of the biped robot is very
important but difficult Up so far, numerous
approaches have been proposed The common
method of these numerous approaches is to restrict
zero moment point (ZMP) within a stable region to
protect the biped robot from falling down [2]
In the recent years, a great amount of scientific
and engineering research has been devoted to the
development of legged robots able to attain gait patterns more or less similar to human beings Towards this objective, many scientific papers have been published, focusing on different aspects
of the problem Sunil, Agrawal and Abbas [3] proposed motion control of a novel planar biped with nearly linear dynamics They introduced a biped robot that the model was nearly linear The motion control for trajectory following used nonlinear control method Park [4] proposed impedance control for biped robot locomotion so that both legs of the biped robot were controlled
by the impedance control, where the desired impedance at the hip and the swing foot was specified Huang and Yoshihiko [5] introduced sensory reflex control for humanoid walking so that the walking control consisted of a feedforward dynamic pattern and a feedback sensory reflex In these papers, the moving of the body of the robot was assumed to be only on the sagittal plane The
Trang 2biped robot was controlled based on the dynamic
model The ZMP of the biped robot was measured
by sensors so that the structure of the biped robot
was complex and the biped robot required a high
speed controller hardware system
This paper presents a stable walking control of a
biped robot by using the inverse kinematics with
simple hardware configuration based on the
walking pattern which is generated by ZMP
tracking control systems The robot’s body can
move on the sagittal and the lateral planes
Furthermore, the walking pattern is generated
based on the ZMP of the biped robot so that the
stability of the biped robot during walking or
running is guaranteed without the sensor system to
measure the ZMP of the biped robot In addition,
the simple inverse kinematics using the solid
geometry is used to obtain angles of each joints of
the biped robot based on the stable walking
pattern The biped robot is modeled as a 3D
inverted pendulum [1] The ZMP tracking control
system is constructed based on the ZMP equations
to generate a trajectory of COM A continuous
time optimal tracking controller is also designed to
control the ZMP tracking control system From the
trajectory of the COM, the inverse kinematics of
the biped robot is solved by the solid geometry
method to obtain angles of each joint of the biped
robot It is used to control walking of the biped
robot
2 Mathematic Model Of The Biped Robot
A new biped robot developed in this paper has 10
DOF as shown in Fig 1
Fig 1 Configuration of 10 DOF biped robot
The biped robot consists of five links that are one
torso, two links in each leg those are upper link
and lower link, and two feet The two legs of the
biped robot are connected with torso via two DOF
rotating joints which are called hip joints Hip
joints can rotate the legs in the angles 5 for right leg and 7 for left leg on sagittal plane, and in the angles 4 for right leg and 6 for left leg on in frontal plane The upper links are connected with lower links via one DOF rotating joints those are called knee joints which can rotate on sagittal plane The lower links of legs are connected with feet via two DOF of ankle joints The ankle joints can rotate the feet in angle 1 (for right leg) and 10
(for left leg) on the sigattal plane, and in angle 2
for left leg and 9 for right leg on the in frontal plane The rotating joints are considered to be friction-free and each one is driven by one DC motor
2.1 Kinematics model of biped robot
It is assumed that the soles of robot do not slip In the world coordinate system w which the origin is set on the ground, the coordinate of the center of the pelvis link and the ankle of swing leg can be expressed as follows:
3 1
2 1 1 b
c x l sin l sin
2 4
3
2 1 3 2 2 1 b c cos 2 l
sin cos
l sin l y y
(2)
2 1 3 2
2 1 1 b c
sin 2
l cos cos
l
cos cos l z z
(3)
In choosing Cartesian coordinate a which the origin is taken on the ankle, position of the center
of the pelvis link is expressed as follows:
3 1
2 1 1
ca l sin l sin
3
2 1 3 2 2 1 ca
cos 2 l
sin cos
l sin l y
(5)
3 2 1 3 2
2 1 1 ca
sin 2
l cos cos
l
cos cos l z
(6)
where, x ca , y ca , z ca are position of the center of the
pelvis link in a Similarly, position of the ankle joint of swing leg
is expressed in the coordinate system h which the origin is taken on the center of pelvis link as:
8 7
1 7 2
8 7 6
1 6 2 3
eh l sin l cos sin 2
l
8 7 6 1
7 6 2
eh l cos cos l cos cos
It is assumed that the center of mass of each link is concentrated on the tip of the link and the initial
z
x
y
Knee
a
b
3
8
1
5
10
6
7
Ankle
Pelvis Torso
z h x h
y h
z a
x a
y a
l2
l 1
0
B 2(x b ,y b ,z b )
K1
B2
B1
C
B
K
E
C(x c ,y c ,z c )
Foot
Hip
Trang 3position is located at the origin of the w This
means that x b = 0 and y b = 0
The COM of the robot can be obtained as follows:
e 4 3 c 2 1 b
e e 4 4 3 3 c c 2 2 1
1
b
b
com
m m m m m m m
x m x m x m x m x m x
m
x
m
x
e 4 3 c 2 1 b
e e 4 4 3 3 c c 2 2 1
1
b
b
com
m m m m m m m
y m y m y m y m y m y
m
y
m
y
e 4 3 c 2 1 b
e e 4 4 3 3 c c 2 2 1
1
b
b
com
m m m m m m m
z m z m z m z m z m z
m
z
m
z
where (x b , y b , z b ) and (x e , y e , z e) are coordinates of
the ankle joints B2 and E, (x 1 , y 1 , z 1 ) and (x 4 , y 4 , z 4)
are coordinates of the knee joints B1 and K1, (x 2,
y 2 , z 2 ) and (x 3 , y 3 , z 3) are coordinate of the hip
joints B and K, (x c , y c , z c) is coordinate of the
center of pelvis link C, m b and m e are the mass of
ankle joints B2 and E, m 1 and m 4 are the mass of
knee joints B1 and K1, m 2 and m 3 are the mass of
hip joints B and K, and m c is the mass of the center
of pelvis link C
If the mass of links of legs is negligible compared
with mass of the trunk, Eqs (1)~(3) can be
rewritten as follows:
c
com x
c
c
It means that the COM is concentrated on the
center of the pelvis link
In this paper, Eqs (13)~(15) are used
2.2 Dynamic model of biped robot
When the biped robot is supported by one leg, the
dynamics of the robot can be approximated by a
simple 3D inverted pendulum whose leg is the foot
of biped robot and head is COM of biped robot as
shown in Fig 2
Fig 2 Three dimension inverted pendulum
The length of inverted pendulum r is able to be
expanded or contracted The position of the mass
point p = [x ca , y ca , z ca]T can be uniquely specified
by a set of state variable q = [ r , p , r]T as follows
[1]:
p rS p sin r
r r
rD sin
sin 1 r
[r , p , f]T is defined as actuator torques and force associated with the variables [r , p , r]T The Lagrangian of the 3D inverted pendulum is
ca 2
ca 2 ca 2
ca y z ) mgz x
( m 2
1
where m is the total mass of the biped robot, g is
the gravity acceleration
Based on the Largange’s equation, the dynamics
of 3D inverted pendulum can be obtained in the Cartesian coordinate as follows:
D D
S
rC D
S rC
mg f z y x
D S
S rC 0 rC
D S rC rC 0
r r
p r
ca ca ca
r p
p p p
r r r
Multiplying the first row of the Eq (20) by D/C r
yields
r ca r
C
D z
rS y rD
Substituting Eqs (16) and (17) into Eq (21), the
dynamics equation of inverted pendulum along y ca
axis can be obtained as
where
r C
D
is the torque around x axis
Using similar procedure, the dynamics equation of
inverted pendulum along x ca axis can be derived from the second row of the Eq (20) as
z ca x ca x ca z ca y mgx ca
p y C
D
The motions of the point mass of inverted pendulum are assumed to be constrained on the
plane whose normal vector is [k x , k y , -1]T and z intersection is z c The equation of the plane can be expressed as
c ca y ca x
ca k x k y z
where k x , k y , z c are constant
Second order derivative of Eq (31) are
ca y ca x
x a
P r
r
P
r
f P
f r
0
p
f
Trang 4Substituting Eqs (24) and (25) into Eqs (22) and
(23), the equation of motion of 3D inverted
pendulum under constraint can be expressed as
c ca ca ca ca c
x ca
c
ca
mz
1 y
x y x z
k y
z
g
c ca ca ca ca c
y ca
c
ca
mz
1 y
x y x z
k
x
z
g
It is assumed that the biped robot walks on the flat
floor and horizontal plane In this case, k x and k y
are set to zero It means that the mass point of
inverted pendulum moves on a horizontal plane
with the height z ca = z c Eqs (26) and (27) can be
rewritten as
x c ca
c
ca
mz
1 y
z
g
y c ca
c
ca
mz
1 x
z
g
When inverted pendulum moves on the horizontal
plane, the dynamic equation along the x ca axis and
y ca axis are independent and linear differential
equations[1]
(x zmp , y zmp) is defined as location of ZMP on the
floor as shown in Fig 3
ZMP is such a point where the net support torque
from floor about x ca axis and y ca axis is zero From
pendulum under constraint can be expressed as
ca c
ca
g
z
x
ca c
ca
g
z
y
Fig 3 ZMP of inverted pendulum
Eq (30) shows that position of ZMP along x ca axis
is linear differential equation and it depends only
on the position of mass point along x ca axis
Similarly, position of ZMP along y ca axis do not
depend on x ca but only on y ca axis
3 WALKING PATTERN GENERATION
The objective of controlling the biped robot is to realize a stable walking or running The stable walking or running of the biped robot depends on
a walking pattern The walking pattern generation
is used to generate a trajectory for the COM of the biped robot For the stable walking or running of the biped robot, the walking pattern should satisfy the condition that the ZMP of the biped robot always exists inside the stable region Since position of the COM of the biped robot has the close relationship with the position of the ZMP as shown in Eqs (25)~(26), a trajectory of the COM can be obtained from the trajectory of the ZMP Based on a sequence of the desired footprint and the period time of each step of the biped robot, a reference trajectory of the ZMP can be specified Fig 3 illustrates the footprint and the zigzag reference trajectory of the ZMP to guarantee a stable gait
Left foot
Right foot
ZMP reference trajectory
y [m]
x [m]
0.1
0.1
-0.1
t 1
t 8
Fig 3 Footprint and reference trajectory of the
ZMP
The x and y ZMP trajectories versus times
corresponding to the zigzag reference trajectory of the ZMP in Fig 3 can be obtained as shown in Figs 4 and 5
0 10 20 30 40 50 60 70 80 -0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Time (sec)
t 2
t 3
t 4
t 5
t 6
t 7
Fig 4 x ZMP reference trajectory versus time
Foo
0
z c
x a
z a Mass
point
x ca
x zmp
Foo
0
z c
y a
z a
Mass point
y ca y zmp
Trang 50 10 20 30 40 50 60 70 80
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Time (sec)
t 1
t 2 t 3 t 4 t 5 t 6 t 7
t 8
Fig 5 y ZMP reference trajectory versus time
3.1 Walking pattern generation based on
optimal tracking control of the ZMP
When the biped robot is modeled as the 3D
inverted pendulum which is moved on the
horizontal plane, the ZMP’s position of the biped
robot is expressed by linear independent equations
along x a and y a directions which are shown as Eqs
(30)~(31)
ca ca
dt
d
dt
d
defined as the time derivatives of the horizontal
acceleration along x a and y a directions of the
COM, uxand u y are introduced as inputs Eqs
(30)~(31) can be rewritten in strictly proper form
as follows:
, x x x g
z 0
1
x
, u 1 0 0 x x x 0 0
0
1 0
0
0 1
0
x
x
x
t ca ca
ca cd zmp
x
t ca ca ca
t
ca
ca
ca
x
x x
x C
B x
A
x
y y y g
z 0
1
y
, u 1 0 0
y y y
0 0
0
1 0
0
0 1
0
y
y
y
t ca ca
ca cd zmp
y
t ca ca ca
t
ca
ca
ca
y
y y
x C
B x
A
x
where x zmp is position of the ZMP along x a axis as output of the system (32), y zmp is position of the
ZMP along y a axis as output of the system (33),
ca
x and ycaare positions of the COM with respect
to x a and y a axes, andxca, xca, yca, yca are horizontal velocities and accelerations with respect
to xa and ya directions, respectively
The systems (32) and (33) can be generalized
as a linear time invariant system as follows:
Cx
B Ax x
y
u
(34)
where x n1 is state vector of system, u c is
input signal, y is output, A nn , B n1 and C 1n
Instead of solving differential Eqs (30)~(31), the position of the COM can be obtained by constructing a controller to track the ZMP as the outputs of Eqs (32)~(33) When x zmp and y zmp are controlled to track the x and y ZMP reference
trajectories, the COM trajectories can be obtained from state variablesxcaandyca According to this pattern, the walking or running of the biped robot are stable
3.2 Continuous Time Controller Design for ZMP Tracking Control
The system (34) is assumed to be controllable and observable The objective designing this controller
is to stabilize the closed loop system and to track the output of the system to the reference input
An error signal between the reference input r(t)
and the output of the system is defined as follows:
e (35) The objective of the control system is to regulate
the error signal e(t) equal to zero when time goes
to infinity
As shown in Figs 4 and 5, the x and y ZMP
reference trajectories include segments as a ramp function and segments as a step function and have singular points To control the output of the ZMP tracking control systems to track the ramp
segments of the x and y ZMP reference
trajectories, the designed controller should satisfy the internal model principle This means that the reference input should be assumed to be a ramp signal input However, when the outputs of the ZMP tracking control systems track the ZMP
(32)
(33)
Trang 6reference trajectories, an overshoot occurs at
singular points of the ZMP reference input because
at these points the time derivative of the ZMP
reference input does not exist Moreover, the
singular points of the ZMP reference trajectories
are very important points The overshoot at these
points makes the ZMP of the biped robot to move
outside the stable region if the maximum value of
the overshoot is larger than the chosen value of
stability margin In this case, the biped robot
becomes unstable When the outputs of the ZMP
tracking control systems are controlled to track the
step reference inputs, the errors between the
outputs of the ZMP systems and the ramp
segments of the ZMP reference inputs are
constant Because the ramp segments of the ZMP
reference trajectories are segments that the biped
robot changes its ZMP in two leg supported phase,
the errors mean that the outputs of the ZMP
systems are delayed by time compared with the
ZMP reference trajectories However, the walking
pattern generation is generated in offline process
so that the errors at the ramp segments of the ZMP
reference trajectory are not important The
reference signal is assumed to be a step function in
this paper
The first order and second order derivatives of the
error signal are expressed as follows:
t r t y t
From the time derivative of the first row of Eq
(34) and Eq (36) the augmented system is
obtained as follows:
w 0 e 0 e
dt
a a
X
B x C
0 A
x
(37)
where wu is defined as a new input signal
A scalar cost function of the quadratic form is
chosen as
0
2 c
ec n 1
1 n n n
Q
0
0 0
symmetric semi-positive definite matrix, R c
and Q ec are positive scalar
The control signal w that minimizes the cost
function (38) of the system (37) can be obtained as
e K u
w KcXa K1cx c (39)
1c
n+1n+1 is solution of the following Ricatti equation with symmetric positive definite matrix
0
a a c a c c T
a P P A P B B P Q
When the initial conditions are u c (0) = 0 and
x(0) = 0, Eq (39) yields
t
0
c e t dt K
t t
Block diagram of the closed loop optimal tracking control system is shown as follows:
Fig 6 Block diagram of the closed loop optimal
tracking control system
4 Walking Control
Based on the stable walking pattern generation discussed in previous section, a continuous time trajectory of the COM of the biped robot is generated by the ZMP tracking control system The continuous time trajectory of the COM of the
biped robot is sampled with sampling time T c and
is stored into micro-controller The ZMP reference trajectory of the ZMP system is chosen to satisfy the stable condition of the biped robot The control objective for the stable walking of the biped robot
is to track the center of the pelvis link to the COM trajectory The inverse kinematics of the biped robot is solved to obtain the angle of each joint of the biped robot The walking control of the biped robot is performed based on the solutions of the inverse kinematics which is solved by the solid
geometry method
Solving the inverse kinematics problems directly from kinematics models is complex An inverse kinematics based on the solid geometry method is presented in this section During the walking of the biped robot, the following assumptions are supposed
- Trunk of the biped robot is always kept on the sagittal plane: 2 4 and 9 6
- The feet of the biped robot are always parallel with floor: 3 1 5
- The walking of the biped robot is divided into 3 phase: Two legs supported, right leg supported and
Trang 7left leg supported
- The origin of the 3D inverted pendulum is
located at the ankle of supported leg
4.2 Inverse kinematics of biped robot in one leg
supported
When the biped robot is supported by right leg,
left leg swings A coordinate system a that takes
the origin at the ankle of supported leg is defined
Since the trunk of robot is always kept on the
sagittal plane, the pelvis link is always on the
horizontal plane as shown in Fig 7
The knee joint angle of the biped robot is gotten as
follows:
2 1
2 2 2 2 1 1 3
l l 2 k h l l cos
The ankle joint angle 2 k can be obtained from
Eq (43) The angle 1 k can be obtained from Eq
(44)
1
2 2 2 1 2 1 ca
1 1
1
l k h
l l k h cos k h
k x sin
DOB
k
where r k l y k
4
l
k
2
3
Fig 7 Inverted pendulum and supported leg
4.1 Inverse kinematics of swing leg
When the biped robot is supported by right leg,
left leg is swung as shown in Fig 8
Fig 8 Swing leg of biped robot
A coordinate system hwith the origin that is taken at the middle of pelvis link is defined During the swing of this leg, the coordinate y of eh
the foot of swing leg is constant r ' k is defined
as the distance between foot and hip joint of swing
leg at k th sample time It is expressed in the coordinate system has follows
z k
2
l k y k x k '
2 3 eh 2
eh 2
where (x eh (k),y eh (k),z eh (k)) is coordinate of the
ankle of swing leg in the coordinateh at k th
sample time
The hip angle 6 k of the swing leg is obtained
based on the right triangle KEF as
k ' r 2 / l k y sin EKF
6
The minus sign in (46) means counterclockwise The hip angle 7 k is equal to the angle between
link l 2 and KG line It is can be expressed as
2
2 2 2 1 eh
1
1 7
l k ' r 2
l l k ' r cos k
' r k x sin
EKK GKE
k
(47)
Using the cosin’s law, the angle of knee of swing leg can be obtained as
2
2 2 2 1 1
8
l 2 k ' r l l cos E
KK
When robot is supported by two legs, the inverse kinematics is calculated by similar proceduce of one leg supported
5 Simulation And Experimental Results
The walking control method proposed in previous section is implemented in the CIMEC-1 biped robot developed for this paper as shown in Fig 9
l3/2 COM
r
r
P
x a
y a
z a
z c
3
0
C
B
l1
l2
h
D
2
A
1
B 1
F
E
G
H
x h
z h
C
l 1
l 2 r'
(x ca ,y ca ,z ca)
7
l 3 /2
K 1
Trang 8Fig 9 HUTECH-1 biped robot
A simple hardware configuration using three
PIC18F4431 and one dsPIC30F6014 for the
CIMEC-1 biped robot is shown in Fig 10
Master unit dsPIC 30F6014
Motor
Potentiomate
r
Potentiomate
r Potentiomate
r
r Potentiomate
r
r Potentiomate
r
r Potentiomate
r
r
Slave unit 1 PIC 18F4431
Slave unit 2 PIC 18F4431
Slave unit 3 PIC 18F4431
Hip joint
Ankle joint
Hip joint
Ankle joint
Knee joint
Knee joint
Fig 10 Hardware configuration of the
CIMEC-1 biped robot
dsPIC30F6014 is used as a master unit, and
PIC18F4431 is used as slave units The master unit
and the slave units communicate each other via
I2C communication The master unit is used to
solve the inverse kinematics problem based on the
trajectory of the center of the pelvis of the biped
robot and the trajectory of the ankle of the
swinging leg which are contained in its memory It
can also communicate personal computer via
RS-232 communication The angles at the k th sample
time obtained from the inverse kinematics are sent
to the slave units as reference signals
The block diagram of proposed controller for biped robot is shown in Fig 11
Fig 11 Block diagram of proposed controller
To demonstrate the walking performance of the biped robot based on the ZMP walking pattern generation combined with the inverse kinematics, the simulation results for walking on the flat floor
of the biped robot using Matlab are shown Fig
10 shows one step walking pattern of the biped robot on the flat floor
The period of step is 10 sec That is, changing time
of supported leg is 5 sec and moving time of swing leg is 5 sec The length of step is 20 cm During the moving of the biped robot, the height of the center of pelvis link is constant In the swing phase, the ZMP is located at the center of the supported foot When two legs of the biped robot are contacted to the ground, the ZMP moves from current supported leg to geometry center of the new supported foot The parameter values of the biped robot used in the simulation are given in Table 5.1
Table 5.1 Numerical values used in simulation
1
2
3
c
The footprint and ZMP desired trajectory are shown in Fig 13
Desired ZMP Trajectory
x ZMP Trajectory
y ZMP Trajectory
y ZMP servo system
x ZMP servo system x COM
y COM
Desired trajectory of swing leg
Biped robot
angle joints
i
Inverse kinematics
of the biped robot
Swing phase Changing supported leg
Fig 12 One step walking pattern
ZMP servo system
Trang 9ZMP desired trajectory Left foot
Right foot
Fig 13 Footprint and desired trajectory of ZMP
The simulation results are shown in Figs 14~20
Fig 14 presents x, y ZMP reference, output and
coordinate of COM with respect to time Figs
15~16 show control signals and tracking errors
Figs 17 ~19 show joints’ angle of one leg of the
robot, the joints’ angle of opposite side leg are
similar Fig 20 presents movement of the center of
pelvis link in the world coordinate system
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time [sec]
ZMP reference input Position of COM ZMP output
a) x ZMP reference, output and COM
-0.1
-0.05
0
0.05
0.1
0.15
Time [sec]
ZMP reference input Position of COM ZMP output
a) y ZMP reference, output and COM
Fig 14 x, y ZMP reference, output and COM
0 10 20 30 40 50 60 70 80
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2x 10
-4
Time (sec)
a) Control signal u of y ZMP
-5
-4
-3
-2
-1
0
1
2
3
4
-4
Time (sec)
b) Control signal u of x ZMP Fig 15 Control signal input
-2 0 2 4 6 8
-3
Time [sec]
a) x tracking error
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025
Time [sec]
b) y tracking error
Fig 16 Tracking error
10 15 20 25 30 35 40 45 50 55
Time (sec)
1
Experiment Result Simulation result
Fig 17 The ankle joint angle 1
-15 -10 -5 0 5 10 15 20
Time (sec)
2
Experiment result Simulation result
Fig 18 The ankle joint angle 2
40 45 50 55 60 65 70 75 80 85 90 95
Time (sec)
Experiment result Simulation result
Fig 19 The knee joint angle 3
Trang 10-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-0.1
-0.05
0
0.05
0.1
x (m)
Fig 20 Coordinate of center of pelvis link
5 Conclusion
In this paper, a 10 DOF biped robot is developed
The kinematic and dynamic models of the biped
robot are proposed An continuous time optimal
tracking controller is designed to generate the
trajectory of the COM for its stable walking The
walking control of the biped robot is performed
based on the solutions of the inverse kinematics
which is solved by the solid geometry method A
simple hardware configuration is constructed to
control the biped robot The simulation and
experimental results are shown to prove
effectiveness of the proposed controller
REFERENCES
[1] S Kajita, F Kanehiro, K Kaneko, K, Yokoi
and H Hirukawa, “The 3D Linear Inverted
Pendulum Mode: A simple modeling for a
biped walking pattern generation”, Proc of
IEEE/RSJ International conference on
Intelligent Robots and Systems, pp 239~246,
2001
[2] C Zhu and A Kawamara, “Walking Principle
Analysis for Biped Robot with ZMP Concept,
Friction Constraint, and Inverted Pendulum
Model”, Proc of IEEE/RSJ International
conference on Intelligent Robots and Systems,
pp 364~369, 2003
[3] S K Agrawal, and A Fattah, “Motion
Control of a Novel Planar Biped with Nearly
Linear Dynamics”, IEEE/ASME Transaction
on Mechatronics, Vol 11, No 2, pp
162~168, 2006
[4] J H Part, “Impedance Control for Biped
Robot Locomotion”, IEEE Transaction on
Robotics and Automation, Vol 17, No 3, pp
870~882, 2001
[5] Q Huang and Y Nakamura, “Sensor Reflex
Control for Humanoid Walking”, IEEE
Transaction on Robotics, Vol 21, No 5, pp
977~984, 2005
[6] B C Kou, “Digital Control Systems”,
International Edition, 1992
[7] D Li, D Zhou, Z Hu, and H Hu, “Optimal
Priview Control Applied to Terrain Following
Flight”, Proc of IEEE Conference on
Decision and Control, pp 211~216, 2001
[8] D Plestan, J W Grizzle, E R Westervelt and
G Abba, “Stable Walking of A 7-DOF Biped
Robot”, IEEE Transaction on Robotics and Automation, Vol 19, No 4, pp 653-668,
2003
[9] F L Lewis, C T Abdallah and D.M Dawson, “Control of Robot Manipulator”,
Prentice Hall International Edition, 1993
[10] G F Franklin, J D Powell and A E Naeini,
“Feedback Control of Dynamic System”,
Prentice Hall Upper Saddle River, New Jersey
07458
[11] G A Bekey, “Autonomous Robots From Biological Inspiration to Implementation and
Control”, The MIT Press 2005
[12] H K Lum, M Zribi and Y C Soh, “Planning
and Control of a Biped Robot”, International Journal of Engineering Science ELSEVIER,
Vol 37, pp 1319~1349, 1999
[13] H Hirukawa, S Kajita, F Kanehiro, K Kaneko and T Isozumi, “The Human-size Humanoid Robot That Can Walk, Lie Down
and Get Up”, International Journal of Robotics Research Vol 24, No 9, pp
755~769, 2005
[14] K Mitobe, G Capi and Y Nasu, “A New Control Method for Walking Robots Based on Angular Momentum”, Journal of Mechatronics ELSEVIER, Vol 14, pp
164~165, 2004
[15] K Harada, S Kajita, K Kaneko and H Hirukawa, “Walking Motion for Pushing
Manipulation by a Humanoid Robot”, Journal
of the Robotics Society of Japan, Vol 22, No
3, pp 392–399, 2004
Tran Dinh Huy received the B.E and
engineering from HoChiMinh City University of Technology in 1995 and
1998, respectively He is currently a PhD student of Open University Malaysia His research interests include robotics and motion control
Nguyen Thanh Phuong received
the B.E., M.E degrees in electrical engineering from HoChiMinh City University of Technology, in 1998,
mechatronics in 2008 from Pukyong National University, Korea respectively He is currently a Lecturer in the Department of