Humanoid Robots - New Developments Part 4 ppsx

Dynamic Simulation of Single and Combined Trajectory Path Generation and
Control of A Seven Link Biped Robot 97
(27)
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))2/cos()cos(
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Accordingly, the linear acceleration of the links can be calculated easily. After generation
of the robot trajectory paths with the aid of interpolation process and with utilization of
MATLAB commands, the simulation of the biped robot can be performed. Based on the
all above expressed relations and the resulted parameters and subsequently with
inserting the parameters into the program, the simulation of the robot are presented in
simulation results.
3. Dynamic of the robot
In similarity of human and the biped robots, the most important parameter of stability of
the robot refers to ZMP. The ZMP (Zero moment point) is a point on the ground whose sum
of all moments around this point is equal to zero. Totally, the ZMP mathematical
formulation can be presented as below:
(30)
)cos(
)sin()cos(
1
111
i
n
i
i
n
i
ii
i
ii
n
i

iii
n
i
i
zmp
zgm
Izxgmxzgm
x




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

¦
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O
TOO
Where,
i
x

and
i
z

are horizontal and vertical acceleration of the link's mass center with

respect to F.C.S where
i
T

is the angular acceleration of the links calculated from the
interpolation process. On the other hand, the stability of the robot is determined according
to attitude of ZMP. This means that if the ZMP be within the convex hull of the robot, the
stable movement of the robot will be obtained and there are no interruptions in kinematic
parameters (Velocity of the links). The convex hull can be imagined as a projection of a
Humanoid Robots, New Developments98
pyramid with its heads on support and swing foots and also on the hip joint. Generally, the
ZMP can be classified as the following cases:
1) Moving ZMP
2) Fixed ZMP
The moving type of the robot walking is similar to human gait. In the fixed type, the
ZMP position is restricted through the support feet or the user's selected areas.
Consequently, the significant torso's modified motion is required for stable walking of
the robot. For the explained process, the program has been designed to find target angle
of the torso for providing the fixed ZMP position automatically. In the designed
program,
torso
q shows the deflection angle of the torso determined by the user or
calculated by auto detector mood of the program. Note, in the mood of auto detector,
the torso needed motion for obtaining the mentioned fixed ZMP will be extracted with
respect to the desired ranges. The desired ranges include the defined support feet area
by the users or automatically by the designed program. Note, the most affecting
parameters for obtaining the robot's stable walking are the hip's height and position. By
varying the parameters with iterative method for
sded
xx , [Huang and et. Al, 2001] and

choosing the optimum hip height, the robot control process with respect to the torso's
modified angles and the mentioned parameters can be performed. To obtain the joint’s
actuator torques, the Lagrangian relation [Kraige, 1989] has been used at the single
support phase as below:
(31)
)(),()(
ii
qGqqqCqqH 

W
where,
6,2,0  i
and
GCH ,,
are mass inertia, coriolis and gravitational matrices of the
system which can be written as following:
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tor
G
G
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G
G
qG
5
4
3
2
1
)(
Obviously, the above expressed matrices show the double support phase of the movement
of the robot where they are used for the single support phase of the movement. On the other
hand, the relation (31) is used for the single support phase of the robot. Within the double
support phase of the robot, due to the occurrence impact between the swing leg and the
ground, the modified shape of relation (31) is used with respect to effects of the reaction
forces of the ground [Lum and et. Al. 1999 and Westervelt, 2003, and Hon and et. Al., 1978].
For the explained process and in order to obtain the single support phase equations of the
robot, the value of
0
q
(as can be seen in figure (1.4)) must be put equal to zero. The
calculation process of the above mentioned matrices components contain bulk mathematical
relations. Here, for avoiding the aforesaid relations, just the simplified relations are
presented:
Dynamic Simulation of Single and Combined Trajectory Path Generation and
Control of A Seven Link Biped Robot 99

torso
torsoctorsotorsoctorso
torsoctorsoeeectorsotor
fswingfswingcfswingfswingfswingcfswing
fswingfswingcfswing
fswingfswingcfswing
fswingfswingecfswinge
eeecfswing
ccc
eceee
ccc
eceecc
ecececc
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IIqqllqqllqqll
qllqllqlllllm
qqllqqll
qqllqqllqqll
qqllqqllqqll
qqllqqllqllqll
qllqllqlllllllm
qqllqqllq
qllqqllqqll
qqllqllqllqllqll
llllmqqllqqllqqll
qllqllqlllllmqqll
qllqllllmqlllm
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)cos()cos()cos(([
))]cos(2)cos(2)cos(2)cos(2)
cos(2
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torsotorsoctorso
torsoctorsoctorsotor
fswingfswingcfswingfswingfswingcfswing
fswingfswingcfswing
fswingfswingcfswing
cfswing
ccc
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cccc
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Humanoid Robots, New Developments100
torso
torsoctorsotorsoctorsoctorsotor
fswingfswingcfswingfswingfswingcfswing
fswingfswingcfswing
fswingfswingcfswing
cfswing
ccc
cc
cccc
IIIII
qqllqqllqqllllm
qqllqqll
qqllqqll
qqllqqllqqll
qqllqqllqqllllllm
qqllqqllqqllqqll
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))]2/(cos(2))2/(cos()cos(([
))])2/(cos(2))2/(cos(2
)cos(2))2/(cos(2
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))]cos(2)cos(2)cos(2)cos(
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qqllqqllqqllqqll
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ll
mqqllqqllqqllqqll
qqllllmqqllqqlllm
fswingfswingcfswingfswingfswingcfswing
fswingfswingcfswing
fswingfswingcfswingcfswing
ccc
cccc
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IIqqllqqll
qqllqqll
qqllqqllqqlll
lmqqllqqllqqlllm
fswingfswingcfswingfswingfswingcfswing
fswingfswingcfswing
fswingfswingcfswingcfswing
cccc

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qqllqqlllm
fswingfswingcfswingfswingfswingcfswing
fswingfswingcfswingfswingfswingcfswingcfswing
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17
SS
Dynamic Simulation of Single and Combined Trajectory Path Generation and
Control of A Seven Link Biped Robot 101
torsotorsoctorsotorsoctorso
torsoctorsoeectorsotor
fswingfswingcfswingfswingfswingcfswing
fswingfswingcfswing
fswingfswingcfswing
fswingfswingecfswing

eeecfswing
ccc
ecee
cccec
eccecc
IIIIIqqllqqll
qqllqllqllllm
qqllqqll
qqllqqllqqll
qqllqqllqqll
qqllqqllqll
qllqllqllllllm
qqllqqllqqllqqll
qq
llqqllqllqllqll
lllmqqllqqllqqllqll
qllllmqqllqlllm
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))]2/(cos(2))2/(cos(
)cos())2/(cos()cos(([
))])2/(cos(2))2/(cos(2
)cos(2))2/(cos(2)cos(2
)cos(2))2/(cos()cos(
)cos()cos())2/(cos(
)cos()cos()cos(([
))]cos(2)cos(2)cos(2)cos(
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([
))]cos(2)cos()cos()cos(
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torso
torsoctorsotorsoctorsoctorsotor
fswingfswingcfswingfswingfswingcfswing
fswingfswingcfswing
fswingfswingcfswing
cfswingcc
ccc
cccc
IIIII
qqllqqllqqllllm
qqllqqll
qqllqqllqqll
qqllqqllqqllqqllqqll
llllmqqllqqllqqll
qqllqqllqqlllllmqqll
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22111221
22

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3344
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233211144113311221
22
4
2
3
2
25433442422332
414113311221
2
4
2
3
2
243232
31311221
2
3
2
231221
2
2222
))]2/(cos(2))2/(cos()cos(([
))])2/(cos(2))2/(cos(2
)cos(2))2/(cos(2)cos(2
)cos(2))2/(cos()cos()cos()cos(
([))]cos(2)cos(2)cos(2
)cos()cos()cos(([))]cos(2

)cos()cos(([))]cos(([h
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SS
ESES
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torso
torsoctorsoctorsotorfswingfswingcfswing
fswingfswingcfswingfswingfswingcfswing
cfswingcc
cccc
IIIII
qqllllmqqll
qqllqqllqqll
qqllqqllllllmqqllqqll
qqlllllmqqllllmlm

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))]2/(cos(2([))])2/(cos(2
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Humanoid Robots, New Developments102
543
33443443
2224422332
22
4
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35
433442422332
2
4
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))])2/(cos(2))2/(cos(2)cos(2
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))]cos(2)cos()cos(([))]cos(([h
III
qqllqqllqqll
qqllqqllqqlllllm
qqllqqllqqllllmqqlllm
fswingfswingcfswingfswingfswingcfswing
fswingfswingcfswingcfswing
ccccc
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IIqqll
qqllqqllqqll
qqllllmqqllqqlllm
fswingfswingcfswing
fswingfswingcfswingfswingfswingcfswing
cfswingccc
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llmqqllqqllqlllm
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fswingfswingcfswing
fswingfswingcfswing
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ecfswingcc
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cccecc
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Dynamic Simulation of Single and Combined Trajectory Path Generation and
Control of A Seven Link Biped Robot 103
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1114411331
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354334424223324141
1331

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fswingfswingcfswingfswingfswingcfswing
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37

Humanoid Robots, New Developments104
54
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qqllqqllqqll
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47

Dynamic Simulation of Single and Combined Trajectory Path Generation and
Control of A Seven Link Biped Robot 105
533
4422
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22
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61
S
SMS
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Humanoid Robots, New Developments106
torsotorsotortorsotortor
Iqqllqqlllmasstorsoh  ))])2/(cos())2/(cos(([
2211
2
62
SS
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22
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63
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torsotor
Ilmasstorsoh  )]([
2
67
, 0
66,65,64
h
In the all of above mentioned components, the following parameters have been used and
they can be seen in figure (1.6) :

(32)
,.,
.00. cmcm
qq
E
O
M
E
I
 
(a)
(b)
Fig. (1.6). (a) The foot’s and (b) The support link’s geometrical configurations.
Dynamic Simulation of Single and Combined Trajectory Path Generation and
Control of A Seven Link Biped Robot 107
Where,
fswing
l
and
fswing
E
refer to indexes of the swing leg with respect to geometrical
configurations of the mass center of the swing leg as can be deducted from figure (1.6).
The coriolis and gravitational components of the relation (30) can be calculated easily.
After calculation of kinematic and dynamic parameters of the robot, the control process
of the system will be emerged. In summery, the adaptive control procedure has been
used for a known seven link biped robot. The more details and the related definitions
such as the known and unknown system with respect to control aspect can be found in
reference [Musavi and Bagheri, 2007 and Musavi, 2006, and Bagheri and Felezi, and et.
Al., 2006]. For the simulation of the robot, the obtained parameters and relations are

inserted into the designed program in Matlab environment. As can be seen in
simulation results section, the most concerns refer to stability of the robot with respect
to the important affecting parameters of the robot movements which indicate the ankle
and hip joints parameters [Bagheri and Najafi and et. Al. 2006]. As can be seen from the
simulations figures, the hip height and horizontal positions have considerable effects
over the position of the ZMP and subsequently over the stability of the robot. To
minimize the driver actuator torques of the joints, especially for the knee joint of the
robot, the hip height which measured from the F.C.S has drastic role for diminution of
the torques.
4. Simulation Results
In the designed program, the mentioned simulation processes for the two types of
ZMP have been used for both of the nominal and un-nominal gait. For the un-nominal
walking of the robot, the hip parameters (hip height) have been changed to consider
the effects of the un-nominal motion upon the joint's actuator torques. The results are
presented in figures (1.8) to (1.15) while the robot walks over declined surfaces for the
single phase of the walking. Figure (1.15) shows combined path of the robot. The used
specifications of the simulation of the robot are listed in table No. 1. Figures (1.8),
(1.10) and (1.12) display the moving type of ZMP with the nominal walking of the
robot. Figures (1.9), (1.11) and (1.13) show the same type of ZMP and also the un-
nominal walking of the robot (with the changed hip height form the fixed coordinate
system). Figure (1.14) shows the fixed ZMP upon descending surface. As can been seen
from the table, the swing and support legs have the same geometrical and inertial
values whereas in the designed program the users can choose different specifications.
Note, the swing leg impact and the ground has been regarded in the designed program
as given in references [Lum and et. Al. 1999 and Westervelt, 2003, and Hon and et. Al.,
1978]. Below, the saggital movement and stability analysis of the seven link biped
robot has been considered whereas the frontal considerations are neglected. For
convenience, 3D simulations of the biped robot are presented. In table No. 1,
aban
ll ,

and
af
l present the foot profile which are displayed in figure (1.7). The program
enables the user to compare the results as presented in figures where the paths for the
single phase walking of the robot have been concerned. In the program with the aid of
the given break points, either third-order spline or Vandermonde Matrix has been
used for providing the different trajectory paths. With the aid of the designed
program, the kinematic, dynamic and control parameters have been evaluated. Also,
Humanoid Robots, New Developments108
the two types of ZMP have been investigated. The presented simulations indicate the
hip height effects over joint’s actuator torques for minimizing energy consumption and
especially obtaining fine stability margin. As can be seen in figures (9.h), (11.h) and
(13.h), for the un-nominal walking of the robot with the lower hip height, the knee's
actuator torque values is more than the obtained values as shown in figures (8.h),
(10.h) and (12.h) (for the nominal gait with the higher hip height). This is due to the
robot's need to bend its knee joint more at a low hip position. Therefore, the large knee
joint torque is required to support the robot. Therefore, for reducing the load on the
knee joint and consequently with respect to minimum energy consumption, it is
essential to keep the hip at a high position. Finally, the trajectory path generation
needs more precision with respect to the obtained kinematic relations to avoid the
link’s velocity discontinuities. The presented results have an acceptable consistency
with the typical robot.
.Sh
l
Ti
l
.To
l
an
l

ab
l
af
l
m3.0 m3.0 m3.0 m1.0 m1.0 m13.0
.Sh
m
.Th
m
.To
m
.Fo
m
s
D
c
T
kg7.5 kg10 kg43 kg3.3
m5.0 s9.0
d
T
m
T
ao
H
ao
L
ed
x
sd

x
s18.0 s4.0 m16.0 m4.0 m23.0 m23.0
gs
g
gf
g
min
H
max
H
s
h
s
H
00 m60.0 m62.0 m1.0 m15.0
shank
I
tight
I
torso
I
foot
I
2
02.0 kgm
2
08.0 kgm
2
4.1 kgm
2

01.0 kgm
Table 1. The simulated robot specifications.
Fig. 1.7. The foot configuration.
an
l
af
l
ab
l
Dynamic Simulation of Single and Combined Trajectory Path Generation and
Control of A Seven Link Biped Robot 109
(a) Stick Diagram (b) ZMP
(c) Velocity (d) Acceleration (e) Angular Vel. (f) Angular Acc.
(j) Inertial Forces (h) Driver Torques
Fig. 1.8. (a) The robot’s stick diagram on
$
0
O
, Moving ZMP,
mHmH 62.0,60.0
maxmin

(b) The moving ZMP diagram in nominal gait which satisfies stability criteria (c) __: Shank M.C
velocity, : Tight M.C velocity (d)__: Shank M.C acceleration, :Tight M.C acceleration (e) __:
Shank angular velocity, : Tight angular velocity (f) __: Shank angular acceleration, : Tight
angular acceleration (j) __: Shank M.C inertial force, : Tight M.C inertial force (h) __: Ankle joint
torque, : Hip joint torque, …: Shank joint torque
Humanoid Robots, New Developments110
(a) Stick Daigram (b) ZMP
(c) Velocity (d) Acceleration (e) Angular Vel. (f) Angular Acc.

(j) Inertial Forces (h) Driver Torques
Fig. 1.9. (a) The robot’s stick diagram on
$
0
O
, Moving ZMP,
mHmH 52.0,50.0
maxmin

(b) The moving ZMP diagram in nominal gait which satisfies stability criteria (c) __: Shank M.C
velocity, : Tight M.C velocity (d)__: Shank M.C acceleration, :Tight M.C acceleration (e) __:
Shank angular velocity, : Tight angular velocity (f) __: Shank angular acceleration, : Tight
angular acceleration (j) __: Shank M.C inertial force, : Tight M.C inertial force (h) __: Ankle joint
torque, : Hip joint torque, …: Shank joint torque
Dynamic Simulation of Single and Combined Trajectory Path Generation and
Control of A Seven Link Biped Robot 111
(a) Stick Diagram (b) ZMP
(c) Velocity (d) Acceleration (e) Angular Vel. (f) Angular Acc.
(j) Inertial Forces (h) Driver Torques
Fig. 1.10 (a) The robot’s stick diagram on
$
10
O
, Moving ZMP,
mHmH 62.0,60.0
maxmin

(b) The moving ZMP diagram in nominal gait which satisfies stability criteria (c) __: Shank M.C
velocity, : Tight M.C velocity (d)__: Shank M.C acceleration, :Tight M.C acceleration (e) __:
Shank angular velocity, : Tight angular velocity (f) __: Shank angular acceleration, : Tight

angular acceleration (j) __: Shank M.C inertial force, : Tight M.C inertial force (h) __: Ankle joint
torque, : Hip joint torque, …: Shank joint torque
Humanoid Robots, New Developments112
(a) Stick Diagram
(b) ZMP
(c) Velocity (d) Acceleration
Dynamic Simulation of Single and Combined Trajectory Path Generation and
Control of A Seven Link Biped Robot 113
(e) Angular Vel. (f) Angular Acc.
(j) Inertial Forces (h) Driver Torques
Fig. 1.11.
(a) The robot’s stick diagram on
$
10
O
, Moving ZMP,
mHmH 52.0,50.0
maxmin
(b) The moving ZMP diagram in nominal gait which satisfies stability criteria
(c) __: Shank M.C velocity, : Tight M.C velocity
(d)__: Shank M.C acceleration, :Tight M.C acceleration
(e) __: Shank angular velocity, : Tight angular velocity
(f) __: Shank angular acceleration, : Tight angular acceleration
(j) __: Shank M.C inertial force, : Tight M.C inertial force
(h) __: Ankle joint torque, : Hip joint torque, …: Shank joint torque
Humanoid Robots, New Developments114
(a) Stick Diagram
(b) ZMP
(c) Velocity (d) Acceleration
Dynamic Simulation of Single and Combined Trajectory Path Generation and

Control of A Seven Link Biped Robot 115
(e) Angular Vel. (f) Angular Acc.
(j) Inertial Forces (h) Driver Torques
Fig. 1.12.
(a) The robot’s stick diagram on
$
8
O
, Moving ZMP,
mHmH 62.0,60.0
maxmin

(b) The moving ZMP diagram in nominal gait which satisfies stability criteria
(c) __: Shank M.C velocity, : Tight M.C velocity
(d)__: Shank M.C acceleration, :Tight M.C acceleration
(e) __: Shank angular velocity, : Tight angular velocity
(f) __: Shank angular acceleration, : Tight angular acceleration
(j) __: Shank M.C inertial force, : Tight M.C inertial force
(h) __: Ankle joint torque, : Hip joint torque, …: Shank joint torque
Humanoid Robots, New Developments116
(a) Stick Diagram
(b) ZMP
(c) Velocity (d) Acceleration
Dynamic Simulation of Single and Combined Trajectory Path Generation and
Control of A Seven Link Biped Robot 117
(e) Angular Vel. (f) Angular Acc.
(j) Inertial Forces (h) Driver Torques
Fig. 1.13
(a) The robot’s stick diagram on
$

8
O
, Moving ZMP,
mHmH 52.0,50.0
maxmin

(b) The moving ZMP diagram in nominal gait which satisfies stability criteria
(c) __: Shank M.C velocity, : Tight M.C velocity
(d)__: Shank M.C acceleration, :Tight M.C acceleration
(e) __: Shank angular velocity, : Tight angular velocity
(f) __: Shank angular acceleration, : Tight angular acceleration
(j) __: Shank M.C inertial force, : Tight M.C inertial force
(h) __: Ankle joint torque, : Hip joint torque, …: Shank joint torque
Humanoid Robots, New Developments118
(a) Stick Diagram
(b) ZMP
(c) Velocity (d) Acceleration
Dynamic Simulation of Single and Combined Trajectory Path Generation and
Control of A Seven Link Biped Robot 119
e) Angular Vel. (f) Angular Acc.
(j) Inertial Forces (h) Driver Torques
Fig. 1.14
(a) The robot’s stick diagram on
$
8
O
, Fixed ZMP,
mHmH 62.0,60.0
maxmin

(b) The fixed ZMP diagram in nominal gait which satisfies stability criteria
(c) __: Shank M.C velocity, : Tight M.C velocity
(d)__: Shank M.C acceleration, :Tight M.C acceleration
(e) __: Shank angular velocity, : Tight angular velocity
(f) __: Shank angular acceleration, : Tight angular acceleration
(j) __: Shank M.C inertial force, : Tight M.C inertial force
(h) __: Ankle joint torque, : Hip joint torque, …: Shank joint torque
Humanoid Robots, New Developments120
(a) Stick Diagram (b) ZMP (c) Inertial Forces
(d) Driver Torques (e) Driver Torques
Fig. 1.15 (a) The robot’s stick diagram on combined surface, nominal motion, Moving ZMP,
8
O
$
(b) The moving ZMP diagram in nominal gait which satisfies stability criteria (c) Inertial
forces: __: Supp. tight, : Supp. shank, …: Swing tight, : Swing shank (d) Joint’s torques: __: Swing
shank joint, : Swing ankle joint, …: Supp. hip joint, : Swing hip joint (e) Joint’s torques: __: Supp.
ankle joint, : Supp. shank joint
5. References
Peiman Naseradin Mousavi, Ahmad Bagheri, “Mathematical Simulation of a Seven Link Biped Robot
and ZMP Considerations”, Applied Mathematical Modelling, Elsevier, 2007, Vol. 31/1.
Q. Huang, K. Yokoi, S. Kajita, K. Kaneko, H. Arai, N. Koyachi, K. Tanie, “Planning Walking
Patterns For A Biped Robot”, IEEE Transactions on Robotics and Automation, VOL
17, No. 3, June 2001.
John J. G, “Introduction to Robotics: Mechanics and Control”, Addison-Wesley, 1989.
H. K. Lum, M. Zribi, Y. C. Soh, “ Planning and Contact of a Biped Robot”, International
Journal of Engineering Science 37(1999), pp. 1319-1349
Eric R. Westervelt, “ Toward a Coherent Framework for the Control of Plannar Biped
Locomotion”, A dissertation submitted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy, (Electrical Engineering Systems), In the

University of Michigan, 2003.
H. Hon, T. Kim, and T.Park, “Tolerance Analysis of a Spur Gear Train,” in Proc. 3
rd
DADS
Korean user’s Conf. 1978, pp. 61-81
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7
Analytical Criterions for the Generation of
Highly Dynamic Gaits for Humanoid Robots:
Dynamic Propulsion Criterion and
Dynamic Propulsion Potential
Olivier Bruneau
Université de Versailles Saint-Quentin-en-Yvelines,
Laboratoire d’Ingénierie des Systèmes de Versailles (LISV, EA 4048)
France
Anthony David
Université d’Orléans, Ecole Nationale Supérieure d’Ingénieurs de Bourges,
Laboratoire Vision et Robotique (LVR, EA 2078)
France
1. Introduction
Many studies have been made to develop walking anthropomorphic robots able to move in
environments well-adapted to human beings and able to cooperate with them. Among the
more advanced projects of humanoid robots, one can mention : the Honda robots P2, P3

(Hirai, 1997) (Hirai et al., 1998) and Asimo (Sakagami et al., 2002), the HRP series developed
by AIST (Kaneko et al., 1998) (Kajita et al., 2005) (Kaneko et al., 2004) (Morisawa et al., 2005),
the small robot QRIO proposed by Sony (Nagasaka et al., 2004), the KHR series developed
by KAIST (Kim et al., 2004) (Kim et al., 2005), the last robot of Waseda University having
seven degrees of freedom per leg (Ogura et al., 2004), Johnnie (Lohmeier et al., 2004) and H7
(Kagami et al., 2005). These robots are namely able to climb stairs and to carry their own
power supply during stable walking. The problem of dynamic locomotion and gait
generation for biped robot has been studied theoretically and experimentally with quite
different approaches. However, when searchers study the behavior or the design of
dynamic walking robots, they inevitably meet a number of intrinsic difficulties related to
these kinds of systems : a large number of parameters have to be optimized during the
design process or have to be controlled during the locomotion task; the intrinsic stability of
walking machines with dynamic behaviors is not robust; the coordination of the legs is a
complex task. When human walks, it actively uses its own dynamic effects to ensure its
propulsion. Today, some studies exploit the dynamic effects to generate walking gaits of
robots. In this research field, four kinds of approaches are used. The first one uses pragmatic
rules based on qualitative studies of human walking gaits (Pratt et al., 2001) (Sabourin et al.,
2004). The second one focuses on the mechanical design of the robot in order to obtain
natural passive dynamic gaits (Collins et al., 2005). The third one deals with theoretical