Humanoid robots’ gait control strategy
based on the Lie logic technique and
LIPM model
Prof. Carlos Balaguer
University Carlos III of Madrid
IURS 2006 Robotics Summer School on
Humanoid Robots
Benicassim, 18-22/9/2006
http://roboticslab.uc3m.es
IURS 2006 Robotics Summer School on
Humanoid Robots
Benicassim, 18-22/9/2006
Content
1.
2.
3.
4.
5.
6.
7.
8.
9.
2
Humanoid robots configuration
Gait generation
Lie logic fundamentals
Application to Rh-1 humanoid robot
Linear Inverted Pendulum Model (LIPM)
3D Walking pattern generation
Simulations of Rh-1 humanoid robot
Experiments on Rh-1 humanoid robot
Conclusions
IURS 2006 Robotics Summer School on
Humanoid Robots
Benicassim, 18-22/9/2006
Humanoid robots configuration (I)
z
Rh-1 humanoid robot
developed by University
Carlos III of Madrid
–
–
–
Full-size: ~1.500 mm
~50 kg including batteries
21 DOF
z
z
z
–
–
–
3
6 DOF each leg
3 DOF each arm
1 DOF head
On board computers
On-board sensors
Wi-fi connection
IURS 2006 Robotics Summer School on
Humanoid Robots
Benicassim, 18-22/9/2006
Humanoid robots configuration (II)
z
Rh-1’s cantilever type
structure of hip joint:
–
–
–
–
4
YPR
RPY - Cantilever
Lower flexor torque of the
robot’s hip → smaller
actuator, less forces
Lower position of the
COM → more stability
Wide sphere of leg
motion
Distribute the flexion
wrench of the hip through
the body robot → more
difficult to control
IURS 2006 Robotics Summer School on
Humanoid Robots
Benicassim, 18-22/9/2006
Gait generation (I)
z
Gait types:
–
–
–
–
–
z
z
5
Forward motion
Backward motion
Lateral motion
Rotation motion
Climbing stairs motion
There are infinite
implementations of each
gait’s type
Combining gait types
most of the movements
must be possible
IURS 2006 Robotics Summer School on
Humanoid Robots
Benicassim, 18-22/9/2006
Gait generation (II)
z
Static formulation
–
6
During slow walking,
COM remains always
centered on the soles of
the feet.
z
Dynamic formulation
–
During fast and smooth
walking, COM is not
always centered on the
soles of the feet.
IURS 2006 Robotics Summer School on
Humanoid Robots
Benicassim, 18-22/9/2006
Gait generation (III)
z
For good & fast gait generation are necessary:
–
Foot position at every moment must be transformed
to joint position, i.e. space and time generation of
the joints’ paths.
z
–
Kinematics (direct and inverse) transformations at
least of 12 DOF with different reference systems
depends on the foot support.
z
7
Joints’ paths are defined by pre-selected patrons that
defines the form of walking (top model, sailor, etc.)
Traditional methods, like Denavit-Hartenberg, are difficult
to apply: no existence of close solutions and their
computation is very time consuming.
IURS 2006 Robotics Summer School on
Humanoid Robots
Benicassim, 18-22/9/2006
Lie logic fundamentals (I)
z
8
CHASLES’ theorem: Every rigid body motion
can be realized by a rotation about an axis
combined with a translation parallel to that
axis, this is a screw motion. The infinitesimal
version of a screw motion is the Lie algebra
se(3) – special Euclidian: TWIST ξ^.
IURS 2006 Robotics Summer School on
Humanoid Robots
Benicassim, 18-22/9/2006
Lie logic fundamentals (II)
z
Screw theory advantages:
–
–
–
9
It allows a global description without singularities
due to the use of local coordinates (as Euler
angles, Denavit-Hartenberg). It is possible to use
only two coordinate frames, the base “S” and the
tool “H” ones.
Truly geometric description of rigid motion to make
easer the kinematics analysis. A very natural and
explicit description of the “Jacobian” which has not
the drawbacks their local.
The same mathematical treatment for the different
robot joints: revolute and prismatic.
IURS 2006 Robotics Summer School on
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Benicassim, 18-22/9/2006
Lie logic fundamentals (III)
?
z
z
ξ ^θ
How to solve e . p = q ?
Using canonicals sub-problems
–
–
–
–
z
10
Paden-Kahan one
Paden-Kahan two
Paden-Kahan three
Pardos one
All of them use exponentials instead matrices
IURS 2006 Robotics Summer School on
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Lie logic fundamentals (IV)
z
Paden-Kahan one: rotation about a single axis
e
ξ ^θ
.p = q
⎡ v ⎤ ⎡− w × r ⎤
ξ =⎢ ⎥=⎢
⎥
w
w
⎣ ⎦ ⎣
⎦
v´= v − wwT v
θ = a tan 2[wT (u´×v´), u´T .v´]
11
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Lie logic fundamentals (V)
z
Paden-Kahan two: rotation about two
ξ1^θ1 ξ 2 ^θ 2
ξ1^θ1
subsequent axis.
e .e . p = e .c = q
⎡− w1 × r ⎤
⎡− w2 × r ⎤
=
^
ξ
⎥ 2 ⎢ w ⎥
2
⎣ w1 ⎦
⎣
⎦
ξ1 = ⎢
(w w )w u − w v (w w )w v − w u
α=
(w w ) − 1;β = (w w ) − 1
T
1
2
T
1
T
2
T
1
T
1
2
2
2
T
1
T
1
2
T
2
2
u − α 2 − β 2 − 2αβ w1T w2
2
γ2 =
w1 ×w2
2
c = r + αw1 + βw2 ± γ (w1 × w2 )
12
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Lie logic fundamentals (VI)
z
Paden-Kahan three: rotation to a given
distance.
ξ ^θ
e .p − q = δ
⎡v⎤
⎡− w × r ⎤
ξ =⎢ ⎥=⎢
⎥
⎣ w⎦ ⎣ w ⎦
u´= u − wwT u
v´= v − wwT v
2
2
2
T
δ ´ = δ − w ( p − q)
θ 0 = a tan 2[wT (u´×v´), u´T .v´]
2
2
⎛
+
− δ ´2 ⎞⎟
u
´
v
´
−1 ⎜
θ = θ 0 ± cos
⎜
⎟
2 u´ v´
⎝
⎠
13
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Lie logic fundamentals (VII)
z
14
Pardos one: traslation to a given distance.
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Lie logic fundamentals (VIII)
POE (Product of exponentials)
e
e
e
15
ξ1^θ1
.e
ξ 3 ^θ 3
ξ 2 ^θ 2
ξ 2 ^θ 2
.e
.e
.p
ξ 3 ^θ 3
ξ 3 ^θ 3
.p
.p = q
IURS 2006 Robotics Summer School on
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Lie logic fundamentals (IX)
n
g st (θ ) = ∏ e
ξ i ^θ i
⋅ g (0 ) = e
ξ1^θ1
⋅e
ξ 2 ^θ 2
⋅e
ξ 3 ^θ 3
⋅e
ξ 4 ^θ 4
⋅e
ξ 5 ^θ 5
⋅e
ξ 6 ^θ 6
⋅ g st (0 )
i =1
z
z
z
z
16
The above concepts (screw, twis, POE) could
be used to solve direct and inverse kinematics
of manipulator robot.
Remember that Denavit-Hartemberg
approach is based on products of matrices.
Each robot DOF is an axis (ωi).
A Matlab toolbox was implemented.
IURS 2006 Robotics Summer School on
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Lie logic: Puma robot example (I)
Definitions:
⎡0 ⎤
⎡0 ⎤
⎡1⎤
⎡1⎤
⎡0 ⎤
⎡1⎤
w1 = ⎢⎢1⎥⎥; w2 = ⎢⎢0⎥⎥; w3 = ⎢⎢0⎥⎥; w4 = ⎢⎢1⎥⎥; w5 = ⎢⎢0⎥⎥; w6 = ⎢⎢0⎥⎥
⎢⎣0⎥⎦
⎢⎣0⎥⎦
⎢⎣1⎥⎦
⎢⎣0⎥⎦
⎢⎣0⎥⎦
⎢⎣0⎥⎦
⎡v ⎤
⎡ v3 ⎤ ⎡− w3 × r ⎤
⎡ v2 ⎤ ⎡− w2 × k ⎤
⎡− w × k ⎤
;
ξ
;
ξ
=
=
=
⎥ 2 ⎢w ⎥ ⎢ w ⎥ 3 ⎢w ⎥ = ⎢ w ⎥
3
2
⎦
⎣ 2⎦ ⎣
⎦
⎦
⎣ 3⎦ ⎣
⎡v ⎤
⎡ v5 ⎤ ⎡− w5 × p ⎤
⎡ v6 ⎤ ⎡− w6 × p ⎤
⎡− w4 × p ⎤
=
=
=
;
ξ
;
ξ
6
5
⎢w ⎥ ⎢ w ⎥
⎢w ⎥ = ⎢ w ⎥
w4 ⎥⎦
6
5
⎣ 5⎦ ⎣
⎦
⎣ 6⎦ ⎣
⎦
ξ1 = ⎢ 1 ⎥ = ⎢ 1
⎣ w1 ⎦ ⎣ w1
ξ4 = ⎢ 4 ⎥ = ⎢
⎣ w4 ⎦ ⎣
17
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Lie logic: Puma robot example (II)
g sh (θ ) = eξ1 ^θ1 .eξ 2 ^θ 2 .eξ3 ^θ 3 .eξ 4 ^θ 4 .eξ5 ^θ5 .eξ 6 ^θ 6 .g sh (0 )
Initial conditions:
⎡1
⎢0
g sh (0 ) = ⎢
⎢0
⎢
⎣0
18
⎤
1 0 p y − S y ⎥⎥
0 1 H z − pz ⎥
⎥
0 0
1 ⎦
0 0
0
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Lie logic: Puma robot example (III)
z
Solving θ3 by
Padan-Kahan
three
g sh (θ ) ⋅ g sh (0 ) ⋅ p − k = e
−1
ξ1^θ1
⋅⋅⋅ e
ξ 6 ^θ 6
⋅ p−k
Twist properties :
1. Rotation over their own axis :
e
ξ ^θ
⋅ r = r → eliminates angles 1 and 2
2. Rotational screw conservs the norm :
e
ξ ^θ
⋅ p − r = p − r → eliminates angle 4, 5 and 6
⎯
⎯→ δ = e
19
ξ 3 ^θ 3
⋅ p−k
−K −3
⎯P⎯
⎯→ θ 3
Etc.
IURS 2006 Robotics Summer School on
Humanoid Robots
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Humanoid kinematics by Lie logic (I)
z
20
Human body locomotion control
IURS 2006 Robotics Summer School on
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Humanoid kinematics by Lie logic (II)
z
SKD (Sagital kinematics division)
Boundary
conditions:
same position
and orientation
for the common
parts (pelvis,
thoracic,
cervical) of the
left and right
humanoid.
21
Rh-1
IURS 2006 Robotics Summer School on
Humanoid Robots
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Humanoid kinematics by Lie logic (III)
Each manipulator is
treated as an open
kinematics chain
separating leg and arm
Right 12 DOF
manipulator with
the base in the
right foot
22
Left 13 (12+1)
DOF manipulator
with the base in
the left foot
IURS 2006 Robotics Summer School on
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Humanoid kinematics by Lie logic (IV)
z
Direct kinematics
g st (θ ) = e
ξ1^θ1
⋅e
ξ 2 ^θ 2
Λe
ξ12 ^θ12
⎡υ ⎤ ⎡− ωi × qi ⎤
ξ =⎢ ⎥=⎢
⎥
ω
ω
i
⎣ ⎦ ⎣
⎦
∧
i
23
⋅ g st (0 )
IURS 2006 Robotics Summer School on
Humanoid Robots
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Humanoid kinematics by Lie logic (V)
z
Inverse kinematics: crucial points
t: is a point on
the axis of the
last DOF
s: is a point not
on the axis of
the last physical
DOF (hip)
24
p: is a common
point for axes of
the last three
DOF (femur)
q: is a common
point for axes of
the two first DOF
(foot)
IURS 2006 Robotics Summer School on
Humanoid Robots
Benicassim, 18-22/9/2006
Humanoid kinematics by Lie logic (VI)
z
e
Inverse kinematics (I)
−ξ 6 ^θ 6
⋅⋅⋅ e
−ξ1^θ1
⋅ g st (θ ) ⋅ g st (0 ) ⋅ p − q = e
−1
⎯
⎯→ δ = eξ9 θ 9 ⋅ p − q
^
e
−ξ 6 ^θ 6
⋅⋅⋅ e
−ξ1^θ1
ξ 7 ^θ 7
⋅⋅⋅ e
ξ12 ^θ12
− K −3
⎯P⎯
⎯→ θ 9
⋅ g st (θ ) ⋅ g st (0 ) ⋅ p = e
−1
ξ 7 ^θ 7
⋅⋅⋅ e
ξ12 ^θ12
− K −2
⎯
⎯→ q ' = eξ 7 θ 7 ⋅ eξ8 θ8 ⋅ p ' ⎯P⎯
⎯→ θ 7 , θ 8
^
25
^
⋅ p−q
⋅p
IURS 2006 Robotics Summer School on
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Humanoid kinematics by Lie logic (VII)
z
e
Inverse kinematics (II)
−ξ 9 ^θ 9
⋅⋅⋅ e
−ξ1^θ1
⎯
⎯→ q ' = e
e
−ξ11^θ11
⋅⋅⋅ e
ξ10 ^θ10
−ξ1^θ1
⎯
⎯→ q ' = e
26
⋅ g st (θ ) ⋅ g st (0 ) ⋅ t = e
−1
⋅e
ξ11^θ11
ξ10 ^θ10
⋅e
− K −2
⋅ p ' ⎯P⎯
⎯→ θ10 , θ11
⋅ g st (θ ) ⋅ g st (0 ) ⋅ s = e
ξ12 ^θ12
⋅e
ξ11^θ11
−1
ξ12 ^θ12
− K −1
⋅ p ' ⎯P⎯
⎯→ θ12
⋅s
ξ12 ^θ12
⋅t
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Lie logic vs Denavit-Hartenberg
Y0
k
H
θ3
θ2
X0
Z0
θ1
Y
X
Z
27
S
0,4
ms
2,5
ms
0,2
ms
25
ms
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Kinematic (static) gait generation
v2
g2
w
f2
Foot in fly
28
w
h2
g1
+
1
2
r2
v v
1
h1
u
w f1
r1
v
Marius Sophus Lie
(1842-1899)
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Dynamic gate generation
z
Advantages of the dynamic gate generation:
–
–
–
29
It is not necessary to pass exactly through the ZMP
shadow during the walking
The robot can mover faster and with more natural
and smooth gait
The inertial forces can be taken in account for
other different applications than walking: seating,
stairs climbing, etc.
IURS 2006 Robotics Summer School on
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Distributed mass model
z
z
30
To generate the humanoid gait is necessary
to take in account its dynamical model
Classical Newton-Euler mass distributed
model is extremely difficult to compute
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Distributed vs concentrated models
z
Advantages of the concentrated mass model:
–
–
–
z
Advantages of the distributed mass model:
–
–
31
Less computation time with easer algorithm
Easer control architecture and strategy
Easy analytical model of the robot
Exact model of the robot
Easer prediction of the robots walking
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Concentrated mass model
• Pendulum ball moves like a free ball in a
plane following the inverted pendulum laws in
the gravity field
32
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Inverted Pendulum Model (I)
z
2D model
τ
Fx = M .&
x&= f . sin θ + . cos θ
r
τ
Fz = M .&
z&= f . cosθ − M .g − . sin θ
r
33
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Inverted Pendulum Model (II)
z
34
3D model
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Inverted Pendulum Model (III)
z
Following the right hand rule, p = (x,y,z) is uniquely
specified by a set of state variables q = (qr,qp,r)
⎛
⎜ 0
∂p ⎜
= ⎜ − rCr
J=
∂q ⎜
− rCr S r
⎜
⎝ D
rC p
0
− rC p S p
D
S p = sin(θ p )
C p = cos(θ p )
S r = sin(θ r )
Cr = cos(θ r )
D = 1 − sin(θ r ) 2 − sin(θ p ) 2
35
⎞
Sp ⎟
⎟
− Sr ⎟
⎟
D ⎟
⎠
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Inverted Pendulum Model (IV)
x&⎞
⎛&
⎜ ⎟
m⎜ &
y&⎟ = J T
⎟
⎜&
&
z
⎝ ⎠
⎛τ r ⎞ ⎛ 0 ⎞
⎟
⎟ ⎜
−1 ⎜
.⎜τ p ⎟ + ⎜ 0 ⎟
⎜ f ⎟ ⎜ − mg ⎟
⎠
⎝ ⎠ ⎝
( )
⎛
⎜ 0
⎜
m⎜ rC p
⎜
⎜ Sp
⎜
⎝
36
− rCr
0
− Sr
− rCr S r ⎞
⎛ − rCr S r ⎞
⎟
⎜
⎟
D ⎟⎛ &
D
x&⎞ ⎛ τ ⎞
⎜
⎟
− rC p S p ⎟⎜ ⎟ ⎜ r ⎟
rC
S
−
p p ⎟
y&⎟ = ⎜τ p ⎟ − mg ⎜
⎜&
⎟
⎜
D
D ⎟
⎜
⎟
⎜
⎟
⎜ D ⎟
z&⎠ ⎝ f ⎠
D ⎟⎝ &
⎟
⎜
⎟
⎠
⎝
⎠
⎛ D⎞
⎟τ p + mgx
m( z &
x&− x&
z&) = ⎜
⎜C ⎟
⎝ p⎠
⎛D
m(− z &
y&+ y &
z&) = ⎜⎜
⎝ Cr
⎞
⎟⎟τ r − mgy
⎠
IURS 2006 Robotics Summer School on
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Inverted Pendulum Model (V)
z
Motion constraints:
–
–
–
37
Limits the motion in a plane with given normal
vector (kx, ky, -1) and z intersection zc,
The normal vector should match the slope of the
ground and the z intersection should be the
expected average distance of the center of the
robot’s mass from the ground,
The motion is constraints to the sagital plane
g
1 D
g
&
&
x= x
&
τp
x&= x +
zc
zc
mzc C p
1 D
g
&
τr
y&= y −
zc
mzc Cr
&
y&=
g
y
zc
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Linear Inverted Pendulum Model (LIMP)
With initial conditions “i” at time ti, the mass
concentrated trajectory is:
t − ti
t − ti
) + Tc x&i sinh(
)
Tc
Tc
x
t − ti
t − ti
x&(t ) = i sinh(
) + x&i cosh(
)
Tc
Tc
Tc
x(t ) = xi cosh(
y (t ) = yi cosh(
y&(t ) =
38
t − ti
t − ti
) + Tc y&i sinh(
)
Tc
Tc
yi
t − ti
t − ti
sinh(
) + y&i cosh(
)
Tc
Tc
Tc
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Gait generation by LIPM (I)
x
Stopping phase
x
y
x
Starting phase
y
39
y
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Gait generation by LIPM (II)
Support foot
(left)
Support foot
(left)
40
Support foot
(right)
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Gait generation by LIPM (III)
41
x temporal trajectories
(position-blue line, velocity-red line)
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Gait generation by LIPM (IV)
y temporal trajectory
(position-blue line, velocity-red line)
42
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Gait generation by LIPM (V)
Foot
trajectories
are
computed
by
single
splines taking in account
some constraints as,
• step length,
• max height of foot,
• lateral foot motion,
• foot orientation and
• speed
43
in order to avoid fall down
and reduce the impact
force on landing foot
IURS 2006 Robotics Summer School on
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Gait generation by LIPM (VI)
Any direction walking patterns could been
developed using rotation matrix around z-axis of
local frame
Pattern
Connection
44
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Simulations of Rh-1 humanoid robot
• MATLAB based simulator
• VRML toolbox
• Lie logic implementation
• LIPM model
45
IURS 2006 Robotics Summer School on
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Gait generation architecture
46
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Gait correction procedure (I)
z
47
Gait compensation (correction) due the
mechanical flexion (flexor torques)
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Gait correction procedure (II)
without correction
48
with correction
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Experiments on Rh-1 humanoid robot (I)
49
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Experiments on Rh-1 humanoid robot (II)
50
IURS 2006 Robotics Summer School on
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Experiments on Rh-1 humanoid robot (III)
2005 (Static gait)
Lp=130 mm, Tp=20 seg (0.02Km/h)
51
2006 (Dynamic gait)
Lp=180 mm, Tp=1.25 seg (0.52Km/h)
About 20 times faster!
IURS 2006 Robotics Summer School on
Humanoid Robots
Benicassim, 18-22/9/2006
Experiments on Rh-1 humanoid robot (IV)
Hip and Foot spatial trajectories
COM
trajectory
0
-1
-2
-3
-4
Right foot
trajectory
-5
-6
-7
-8
1
1.5
0
1
-1
0.5
-2
0
-0.5
-3
-1
-4
52
-1.5
Spatial hip and feet trajectories without correction (black line),
with correction (red line)
IURS 2006 Robotics Summer School on
Humanoid Robots
Benicassim, 18-22/9/2006
Conclusions
z
z
z
z
z
53
Locomotion of the humanoid robots is a complex and,
in general, not solved problem
The small amount of computation for LIPM method,
allow us dynamic walking control in real time on
actual humanoid robots.
Dynamic walking was successfully implemented in
Rh-1 humanoid robot using the Lie logic and LIPM
model
The implemented dynamic gait is smooth and natural,
and about twenty times faster than static one
It have been demonstrated that any direction patterns
could been generated
IURS 2006 Robotics Summer School on
Humanoid Robots
Benicassim, 18-22/9/2006
References (I)
[1]
M. Arbulú, J.M. Pardos, L.M. Cabas, P. Staroverov, D. Kaynov, C. Pérez, M.A.
Rodríguez; C. Balaguer, “Rh-0 humanoid full size robot`s control strategy based on the Lie logic
technique”, IEEE-RAS International Conference on Humanoid Robots (Humanoids'2005).
Tsukuba. Japan. Dec, 2005
[2]
M. Arbulú, I. Prieto, D. Gutiérrez, L. Cabas, P. Staroverov, C. Balaguer, “User friendly
graphical environment for gait optimization of the humanoid robot Rh-0”, 7tn International
Conference on Climbing and Walking Robots (Clawar'2004). Madrid. Spain. Sep, 2004
[3]
M. Arbulú, F. Prieto, L. Cabas, P. Staroverov, D. Kaynov, C. Balaguer, “ZMP Human
Measure System”, 8th International Conference on Climbing and Walking Robots (Clawar'2005).
London. United Kingdom. Sep, 2005.
[4]
K. Hirai, M. Hirose, Y. Hikawa and T. Takanaka, “The development of Honda
humanoid robot”, IEEE International Conference on Robotics and Automation (ICRA 1998) Leuven
(Belgium)
[5]
J. Yamaguchi, E. Soga, S. Inoue A. and Takanishi, “Development of a bipedal
humanoid robot control method of whole body cooperative dynamic bipedal walking”, IEEE
International Conference on Robotics and Automation (ICRA’ 1999), Detroit, (USA).
[6]
S. Kajita, F. Kaneiro, K. Kaneko, K. Fujiwara, K. Yokoi and H. Hirukawa, “Biped
walking pattern generation by a simple 3D inverted pendulum model”, Autonomous Robots, vol 17,
nª2, 2003
[7]
K. Löeffler, M. Gienger, F. Pfeiffer and H. Ulbrich, “Sensors and Control Concept of a
Biped Robot”, IEEE Industrial Transactions on Industrial Electronics, Vol. 51, Nº 5, October 2004
54
IURS 2006 Robotics Summer School on
Humanoid Robots
Benicassim, 18-22/9/2006
References (II)
[8] Y. Choi, B. You and S. Oh, “On the Stability of Indirect ZMP Controller for Biped
Robot Systems”, Proceedings of 2004 IEEE/RSJ International Conference on
Intelligent Robots and Systems, Sept. 2004, Sendai, Japan
[9] M.H. Raibert, Legged robots that balance, MIT Press:Cambridge, 1986
[10]A.-J. Baerveldt, R. Klang. “A low cost and Low-weight Attitude Estimatin System
for an Autonomous Helicopter”. Proc. of IEEE International Conference on
Intelligent Engineering Systems, Budapest, Hungary, 391-391, 1997.
[11]Q.Huang; K.Kaneko; K.Yokoi; S.Kajita; T.Kotoku; N.Koyachi; H.Arai; N.Imamura;
K.Komoriya; K.Tanie. “Balance Control of a Biped Robot Combining Off-line
Pattern with Real-time Modification”. Proc. Of IEEE International conference on
Robotics and Automation. San Francisco, USA. April, 2000.
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