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11//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
On Humanoid ControlOn Humanoid Control
HiroHiro HirukawaHirukawaHumanoid Robotics GroupHumanoid Robotics Group
Intelligent Systems InstituteIntelligent Systems InstituteAIST, JapanAIST, Japan
22//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Implies thatImplies thatWalking on a flat floor and rough terrainWalking on a flat floor and rough terrainGoing up and down stairs and laddersGoing up and down stairs and laddersLying down, crawling and getting upLying down, crawling and getting upFalling down safely and getting upFalling down safely and getting upOpening and closing doorsOpening and closing doors
Humanoids move on two, three or four feet.Humanoids move on two, three or four feet.
Humanoids can move the Humanoids can move the environment for humansenvironment for humans
33//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Humanoid as a Controlled Plant Humanoid as a Controlled Plant
A humanoid robot is a multiA humanoid robot is a multi--link structure that is not fixed link structure that is not fixed to the environment and moves to the environment and moves in the environment and/or in the environment and/or moves the environment by moves the environment by the the contact forcecontact force between the between the robot and the environment in robot and the environment in the gravity field.the gravity field.
44//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Humanoid Control ProblemHumanoid Control Problem
When the initial and final configurations of a When the initial and final configurations of a humanoid robot is given, find motions of humanoid robot is given, find motions of the robot that can transfer it from the initial the robot that can transfer it from the initial configuration to the final configuration configuration to the final configuration through through a sequence of the contact statesa sequence of the contact states..
55//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Control AlgorithmsControl Algorithms
Inverted Pendulum Scheme Inverted Pendulum Scheme 1.1. Plan motions of the robotPlan motions of the robot2.2. Change the position of the next footprint to Change the position of the next footprint to
keep the planned configuration of the robotkeep the planned configuration of the robot
ZMP (Zero Moment Point) based SchemeZMP (Zero Moment Point) based Scheme1.1. Plan a sequence of footprints.Plan a sequence of footprints.2.2. Change the configuration of the robot to Change the configuration of the robot to
keep the planned sequence of the footprintskeep the planned sequence of the footprints
66//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Motions vs. Contact ForceMotions vs. Contact Force
( )G CM − =g p f&&
( )G G CMp g p L τ× − − =
::::
G
C
C
pLfτ
Contact force
Contact torque
Position of the center of the gravity
Angular momentum about the COG
77//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
[Gubina, Hemami and McGee 1974]
Inverted Pendulum SchemeInverted Pendulum Scheme
88//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Footprints may have a constraint
99//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
[Vukobratovic and Stepanenko 1972]
And more….
ZMP based SchemeZMP based Scheme
1010//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
What is ZMP (Zero Moment Point)?What is ZMP (Zero Moment Point)?
Vukobratovic and Stepanenco, 1972
Center of Pressure
•ZMP NEVER leaves the support polygon!•ZMP can be measured by force sensors in feet.
1111//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
How ZMP is used?How ZMP is used?
When the ZMP is When the ZMP is insideinside the the support polygon, the contact support polygon, the contact between the feet and the between the feet and the floor should be kept.floor should be kept.
When the contact is kept, the When the contact is kept, the posture of the robot should posture of the robot should be kept without falling down.be kept without falling down.
1212//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
0 1 2 3 4 5 6 7
-0.1-0.05
00.05
0.1
y [m
]
time [s]
CoM
Trajectory of the center of mass
From the ZMP to the COGFrom the ZMP to the COG
0 1 2 3 4 5 6 7
-0.1-0.05
00.050.1
zmp
[m]
time [s]
zmp ref
Target ZMP pattern
1313//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Motions vs. Contact ForceMotions vs. Contact Force
( )G CM − =g p f&&
( )G G CMp g p L τ× − − =
::::
G
C
C
pLfτ
Contact force
Contact torque
Position of the center of the gravity
Angular momentum about the COG
1414//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
CartCart--Table ModelTable Model
• A running cart on a mass-less table• The table has a small support area
M
cz
x0
x
O0ZMP =
1515//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
The ZMP of the CartThe ZMP of the Cart--Table ModelTable Model
M
hz
0xx
O
x&&
0ZMPτ =
0( )0
ZMP hMg x x Mxzτ = − −=
&&
0hzx x x
g= − &&
ZMP equation
Balance of moment
1616//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
ZMP
x
M
0 1 2 3 4 5 6 7
-0.1-0.05
00.050.1
zmp
[m]
time [s]
zmp ref
0 1 2 3 4 5 6 7
-0.1-0.05
00.05
0.1
y [m
]
time [s]
CoM
0hzx x x
g= − &&
Input and OutputInput and Output
0xCart trajectory
Walking pattern generation↓
Find the cart trajectory to realize the given ZMP pattern
1717//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
0x
ZMP
Servo tracking control of the ZMPServo tracking control of the ZMP
0x
+ −
Target ZMP Controller
0refx
u
0
0 1 0 00 0 1 00 0 0 1
1 0 h
x xd x x udt
x x
xzx xg
x
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎡ ⎤⎡ ⎤ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎢ ⎥⎣ ⎦
& &
&& &&
&
&&
Proper dynamics ofCart-Table model
M
1818//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
From ZMP reference to Cart motion
ZMP referenceController
x
0refx + − x
ZMP
The cart must move beforebefore ZMP changes !→Servo controller must use FUTURE information
0 0.5 1 1.5 2 2.5
0
0.1
0.2
0.3
x [m
]
time [s]
referencex
Ordinal servo control
0 1 2
0
0.1
0.2
0.3
x [m
]
time [s]
zmp refx
Walking pattern
1919//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
The Preview ControlThe Preview Control
Concept and naming Concept and naming [Sheridan 1966][Sheridan 1966]LQ optimal controllerLQ optimal controller[Tomizuka and Rosenthal 1979][Tomizuka and Rosenthal 1979][Katayama et.al 1985][Katayama et.al 1985]
0
0 00
( 1)
1 2
0( )
[ , , ]( )
refk
Nref
k
kref
k I k k x ki
N
u Kx
f fx fx
x K+
=+
⎡ ⎤⎢ ⎥− ⎢= ⎥⎢⎣ ⎦
−⎥
− −∑ L Mx
State feedback Preview gainTarget ZMP ofN-step future
AccumulateServo error
Control law
On a winding road, we steer a car by watching ahead, by previewing the future reference.
2020//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Preview gainPreview gain1 ( 1)
1
( ) ( )
( )
T T T i Ti
T T
f R B PB B A BK C Q
K R B PB B PA
− ∗ −
−
= + −
≡ +
+0 +0.5 +1 +1.5 +2 0
500
1000
1500
future time [s]
prev
iew
gai
n
if
・The preview gain is approximately zero after 1.6s future.・Controller does not need the future ZMP farther than 1.6s.
LQ optimal feedback gain
2121//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
ZMP Tracking ExampleZMP Tracking Example
0 1 2 3 4 5 6 7
0
0.2
0.4
0.6
0.8
1zm
px [m
]xzmpzmp_ref
0 1 2 3 4 5 6 7
-0.1
-0.05
0
0.05
0.1
0.15
zmpy
[m]
time [s]
yzmpzmp_ref
2222//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Walking Pattern GeneratorWalking Pattern Generator
HRP-2L HRP-2P[Kajita et al.]
2323//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Experiment of HRPExperiment of HRP--22
2424//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Configuration of the Feedback ControllerConfiguration of the Feedback Controller
Gyro
Accelerometer
Force/torquesensor
Joint servo
Kalmanfilter
Bodyinclination
Stabilizer
References of Joint angles, body inclination and ZMP
Balancer Module
encoder
ZMPZMPcalculator
2525//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Feedback Controller is essentialFeedback Controller is essential
Without stabilizer With stabilizer
2626//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Feedback Control of the Table OrientationFeedback Control of the Table Orientation
Cart on a table
2 2( ) ( sin cos ) 2 0c c cx z xz g z x xx+ θ+ − θ+ θ + θ =&& &&& &
Equation of motion:
2 2( )c c cx z gx gz z x+ θ = − θ−&& &&
Linearize at 0θ ≈
Table inclination Cart acceleration
0= zc
xM
fµ
2727//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Feedback Control of the Cart PositionFeedback Control of the Cart Position
M
cz
px
O0ZMP =
01 ( )
1hzx x x
sT g= −
+&&
ZMP equation with sensor delay T
0 01/ 1/ 0 /( )0 0 1 00 0 0 1
cx T T x z gTd x x xdt
x x
− −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
&&
&
System representation
1 0 2 3x k x k x k x= − − −&& &Stabilization by state feedback
ZMP errorCart position modification
[Nagasaka, Inaba and Inoue, 1999]
Used in the walking controller of H7.Good for robots with hard feet.
2828//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Experiment on a SlopeExperiment on a Slope
不整地歩行予備実験
-不整地移動技術(1/2)-
2929//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
HRPHRP--2 walks on a Rough Terrain2 walks on a Rough Terrain
Gap < ±20 mm Slope < 5%
3030//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Japanese Traditional DanceJapanese Traditional Dance
[Nakaoka et al. 2005]
3131//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
The First Running HumanoidThe First Running HumanoidSony Sony QrioQrio [Dec.,2003][Dec.,2003]
Qrio runs at 0.84 km/h.
3232//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Running Biped [AIST Apr.,2004]Running Biped [AIST Apr.,2004]
Slow motionsSpeed 0.58km/hour
3333//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Walking on a floor with low frictionWalking on a floor with low friction
μ: 0.5 → 0.1 μ: 0.5 → 0.05
μ=0.15 between a tire of a car and a wet snow surface
3434//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Reduction of the Slip by a Tuning of Walking PatternReduction of the Slip by a Tuning of Walking Pattern
•Rotation about Yaw-axis may occur at μ=0.3 due to the change of the acceleration when the supporting leg is exchanged.
•The pattern generator is tuned to reduce the peak of the jerk.
μ=0.3 Conventional pattern μ=0.3 Improved pattern0 0.5 1 1.5
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
showzmp0.m (sliptest3b_mu03\sim-astate.log) /ZMP trajectory
x [m]
y [m
]
Walk direction
0 0.5 1 1.5
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8showzmp0.m (sliptest3_mu03\sim-astate.log) /ZMP trajectory
x [m]
y [m
]
Walk direction
3535//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Walk on a Floor with a Low FrictionWalk on a Floor with a Low Friction
3636//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Implies thatImplies thatWalking on a flat floor and rough terrainWalking on a flat floor and rough terrainGoing up and down stairs and laddersGoing up and down stairs and laddersLying down, crawling and getting upLying down, crawling and getting upFalling down safely and getting upFalling down safely and getting upOpening and closing doorsOpening and closing doors
Humanoids move on two, three or four feet.Humanoids move on two, three or four feet.
Humanoids can move the Humanoids can move the environment for humansenvironment for humans
3737//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Contact States GraphContact States Graph
3838//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Balance Control for the TransitionBalance Control for the Transition
The position of the The position of the torso link is under a torso link is under a compliance control.compliance control.
Landing
ZMP[
m] toe
heel
3939//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Dynamic SimulationDynamic Simulation
4040//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Lying down and Getting upLying down and Getting up
4141//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Implies thatImplies thatWalking on a flat floor and rough terrainWalking on a flat floor and rough terrainGoing up and down stairs and laddersGoing up and down stairs and laddersLying down, crawling and getting upLying down, crawling and getting upFalling down safely and getting upFalling down safely and getting upOpening and closing doorsOpening and closing doors
Humanoids move on two, three or four feet.Humanoids move on two, three or four feet.
Humanoids can move the Humanoids can move the environment for humansenvironment for humans
4242//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Preliminary Experiment for FallingPreliminary Experiment for Falling
With the knee extended With the knee bended
4343//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Falling Motion of a Leg RobotFalling Motion of a Leg Robot
4444//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Impact TestImpact Test
4545//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Falling Motion of Humanoid RobotFalling Motion of Humanoid Robot
4646//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Implies thatImplies thatWalking on a flat floor and rough terrainWalking on a flat floor and rough terrainGoing up and down stairs and laddersGoing up and down stairs and laddersLying down, crawling and getting upLying down, crawling and getting upFalling down safely and getting upFalling down safely and getting upOpening and closing doorsOpening and closing doorsArms and legs coordinationArms and legs coordination
Humanoids move on two, three or four feet.Humanoids move on two, three or four feet.
Humanoids can move the Humanoids can move the environment for humansenvironment for humans
4747//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Walking and TouchingWalking and Touching Leaning on Leaning on BalancingBalancing
PullingPulling PushingPushing
Several Configurations of Arm/Leg CoordinationSeveral Configurations of Arm/Leg Coordination
4848//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
A Generalized ZMP A Generalized ZMP [Harada et al. 2004][Harada et al. 2004]
2D Convex Hull 3D Convex Hull
4949//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
A Generalized ZMP A Generalized ZMP [Harada et al. 2004][Harada et al. 2004]
5050//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
ZMP
A Generalized ZMP A Generalized ZMP [Harada et al. 2004][Harada et al. 2004]
Small Acceleration
5151//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
ZMP
Large Acceleration
Moment around the edge
A Generalized ZMP A Generalized ZMP [Harada et al. 2004][Harada et al. 2004]
5252//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Y
X Z
pG ( )=~ pG~
Projection of the ZMPProjection of the ZMP
5353//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Numerical ExampleNumerical Example
5454//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Area of the Generalized ZMPArea of the Generalized ZMPFront Front
5555//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Push a heavy objectPush a heavy object
25.9 kg
[Harada et al. 2004]
5656//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Pushing a button withPushing a button withthe support of a handthe support of a hand
[Harada et al. 2004]
5757//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
An Open QuestionAn Open Question
Can we plan motions of humanoid robots Can we plan motions of humanoid robots based on the unified criterion?based on the unified criterion?
What the ZMP criterion can judge?What the ZMP criterion can judge?The ZMP can judge if the contact should be The ZMP can judge if the contact should be kept without solving the equations of motions kept without solving the equations of motions when the robot moves when the robot moves on a flat planeon a flat plane under under the sufficient friction assumption.the sufficient friction assumption.
5858//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Our Goal areOur Goal are
to create a new criterion that can judge the to create a new criterion that can judge the contact stability of humanoids which may contact stability of humanoids which may touch an arbitrary terrain with two, three or touch an arbitrary terrain with two, three or four feet, andfour feet, and
to prove that the criterion is equivalent to to prove that the criterion is equivalent to ZMP in a specific case and more universal, ZMP in a specific case and more universal, and to claim to say and to claim to say ““Adios ZMPAdios ZMP””..
5959//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Related WorksRelated Works
Legged RobotsLegged RobotsZMP [ZMP [VukobratovicVukobratovic 1972]1972]Locomotion with hand contact [Locomotion with hand contact [YonedaYoneda 1996]1996]FRI [FRI [GoswaniGoswani 1999]1999]FSW [FSW [SaidaSaida 2003]2003]Generalized ZMP [Harada 2004]Generalized ZMP [Harada 2004]
Mechanical AssemblyMechanical AssemblyStrong and Weak Stability [Strong and Weak Stability [TrinkleTrinkle 1997]1997]
6060//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
FormulationFormulation
1 1
( )K L
l lC k k k k
k l
ε µ= =
= +∑∑f n t
1 1
( )K L
l lC k k k k k
k l
ε µ= =
= × +∑∑τ p n t
L
( )G GM= −f g p&&( )G G GM= × − −τ p g p L
Gravity and Inertia Wrench
Gp
⇒ Polyhedral Convex Cone
Gp : Center of the mass
: Angular momentum around COG
Contact Wrench
6161//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Strong Stability CriterionStrong Stability Criterion
The contact state The contact state mustmust be stable if (be stable if (--ffGG,,--ττGG)) is is an internal element of the contact wrench cone an internal element of the contact wrench cone under the sufficient friction assumption.under the sufficient friction assumption.
(proof)(proof)
1 1
( )K L
l l lC k k k k
k l
f n tε ε= =
= +∑∑1 1
( )K L
l l lC k k k k k k
k l
τ p n p tε ε= =
= × + ×∑∑
, ,( , ) 0, ( ) int( ); ( , ) ( ) 0,G G G G G G G Gx f CWC x fδ τ δ τ∀ Ω ≠ − − ∈ Ω <
where the CWC is given by
The work done by ((ffGG,,ττGG)) is negative for any motion;is negative for any motion;
6262//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
1 2
1
( )K
G k kk
Mx ε ε=
= −∑3 4
1
( )K
G k kk
My ε ε=
= −∑
3 4 1 2
1
( ) ( ) )K
G G G G z k k k k k kk
Mx y My x x yε ε ε ε=
− + = − − −∑L
0
1
( )K
G G G G x k kk
M z g y My z yε=
+ − + =∑L
0
1
( )K
G G G G y k kk
M z g x Mx z xε=
− + + + =−∑L
0
1
( )K
G kk
M z g ε=
+ =∑Strong stability holdsif the moment alongthe horizontal axesis inside the CWC.
Horizontal forceshould balance the friction from the assumption
Example 1. Walking on a horizontal plane Example 1. Walking on a horizontal plane with sufficient frictionwith sufficient friction
6363//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Strong Stability Determination by ZMPStrong Stability Determination by ZMP
1
1, 0K
k kk
λ λ=
= ≥∑
1
( )( )
KG G G G x
k kkG
M z g y My z yM z g
λ=
+ − + =+ ∑L
1
( )( )
KG G G G y
k kkG
M z g x Mx zx
M z gλ
=
+ − −=
+ ∑L
6464//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
1
1, 0K
k kk
λ λ=
= ≥∑
0
1
( )K
G G G G x k kk
M z g y My z yε=
+ − + =∑L
0
1
( )K
G G G G y k kk
M z g x Mx z xε=
− + + + =−∑L
1
( )( )
KG G G G x
k kkG
M z g y My z yM z g
λ=
+ − + =+ ∑L
1
( )( )
KG G G G y
k kkG
M z g x Mx zx
M z gλ
=
+ − −=
+ ∑L
0
1
( )K
G kk
M z g ε=
+ =∑
ZMP
CWC
Equivalence between the ZMP and the CWCEquivalence between the ZMP and the CWC
Dividing the equations by
6565//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Equivalence between the ZMP and the CWCEquivalence between the ZMP and the CWC
1
1, 0K
k kk
λ λ=
= ≥∑
0
1
( )( )
KG G G G x k
kkG
M z g y My z yM z g
εε=
+ − + =+ ∑L
0
1
( )( )
KG G G G y k
kkG
M z g x Mx zx
M z gεε=
− + + +=−
+ ∑L
1
( )( )
KG G G G x
k kkG
M z g y My z yM z g
λ=
+ − + =+ ∑L
1
( )( )
KG G G G y
k kkG
M z g x Mx zx
M z gλ
=
+ − −=
+ ∑LZMP
CWC
0 0
1
1, 0K
k k
k
ε εε ε=
= ≥∑
6666//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
The CWC for a 2DThe CWC for a 2D--Robot on a LineRobot on a Line
z
xfx
fz
τy
6767//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
A Desired Trajectory in the CWCA Desired Trajectory in the CWC
fx
fz
τy
The CWC is the direct product of a 2D polyhedral cone and 1D linear subspace,which is identical for the single and double support phases.
fx
fz
τy
6868//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
10
21
1 2
( )G G G
y y
G xK
k kk
F F
M z g y M
z
y
y z
z L
ε λ λ=
+ − +
−= −∑
Example 2. Robot on a Stair (1/2)Example 2. Robot on a Stair (1/2)
1yλ 2Fz
0
11 1 2 2
( )G G G G y
K
k kx
k
xF Fz
M z g x Mx z
x z
L
λ λε=
− + + +
=− + +∑
2yλ
1xλ
1xλ
1Fz
6969//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Example 2. Robot on a Stair (2/2)Example 2. Robot on a Stair (2/2)
01 1 2 2
1
0 1 21 2
1 1 2 1 2
1 2
( )G G G G xK
y yk k F F
k
y yK
k k G F Fy y y yk
y yG
M z g y My z
y z z
y My z z
whereMy
L
ε λ λ
λ λελ λ λ λ
λ λ
=
=
+ − +
= − −
⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎟⎜ ⎟ ⎟⎜ ⎜ ⎟⎟ ⎟= − +⎜⎜ ⎜ ⎟⎟ ⎟⎜⎜ ⎜ ⎟⎟ ⎟⎜ ⎜ ⎟⎜ + +⎝ ⎠ ⎝ ⎠⎝ ⎠
= +
∑
∑
0
1
1 21 2
1 2 1 2
( ) ( )K
G G G G x k kk
y y
F F Fy y y y
FM z g y My z y
z z
z
z
L ε
λ λλ λ λ λ
=
+ − − + =
⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟= +⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜+ +⎝ ⎠ ⎝ ⎠
∑1yλ
2yλ
1xλ
1xλ
2Fz1Fz
7070//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Pattern Generation of the COGPattern Generation of the COG
( )
0 1 0 00 0 1 00 0 0 1
0 ( )
G G
G G y
G G
G
tx G G
G
y G
ref refy
F
tx y
y yd y y udt
y y
yMg M z y
yu y
L
zξ
ξ τ
⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟= +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠
⎛ ⎞⎜ ⎟= − − ⎜ ⎟⎜ ⎟⎝ ⎠
=
= −
& &
&& &&
&
&&
&&
&
7171//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
The CWC for a 2DThe CWC for a 2D--Robot on a StairRobot on a Stair
z
xfx
fz
τy
7272//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
A Desired Trajectory in the CWCA Desired Trajectory in the CWC
fx
fz
τy
fx
fz
τy
The CWC is the direct product of a 2D polyhedral cone and 1D linear subspace,which is not identical for the single support phases of the lower and the higher feet,and is the product of the 2D cone and 2D linear subspace for a double support phase.
7373//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Vertical Trajectory of the ZMPVertical Trajectory of the ZMP
Pseudo Plane on which the ZMPtrajectory is defined [Honda]
Equivalent trajectory of the ZMPbased on the proposed criterion
Not admissiblein the singlesupport phase
7474//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Horizontal Contact Force while Climbing StairsHorizontal Contact Force while Climbing Stairs
fx
-100
-80
-60
-40
-20
0
20
40
60
80
100
0 0.1 0.2 0.3
fy
-80
-60
-40
-20
0
20
40
60
80
0 0.1 0.2 0.3
Black curve is generated from a continuous trajectory in the CWCRed curve is generated from a discontinuous one in the CWC
single double single single double single
7575//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
ZMPZMP vs.vs.CWCCWC
ZMPZMP CWCCWC
Flat planeFlat planeFoot contactFoot contactSufficient frictionSufficient friction
Strong StabilityStrong Stability Strong StabilityStrong Stability
Strong StabilityStrong StabilityArbitrary terrainArbitrary terrainHand/Foot contactHand/Foot contactSufficient frictionSufficient friction
N/AN/A
7676//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Summary of the CWCSummary of the CWC
The proposed criterion is equivalent to ZMP The proposed criterion is equivalent to ZMP in the specific case and can judge the strong in the specific case and can judge the strong stability in generic cases.stability in generic cases.
Therefore we claim to say Therefore we claim to say ““Adios ZMPAdios ZMP””, and , and the voice can be louder when we can plan the voice can be louder when we can plan motions in a variety of cases based on the motions in a variety of cases based on the proposed criterion.proposed criterion.
7777//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Open Problems in the ControlOpen Problems in the Control
Robust walkingRobust walkingBiped walking is still not robust enough for a Biped walking is still not robust enough for a large disturbance.large disturbance.
Walking on a natural rough terrainWalking on a natural rough terrainWalking must be more generalized with the Walking must be more generalized with the recognition of the working environment.recognition of the working environment.
Falling motion controlFalling motion controlThe humanThe human--size humanoid just crashes when size humanoid just crashes when it falls down without a proper control.it falls down without a proper control.
7878//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Humanoids in Real EnvironmentHumanoids in Real Environment
7979//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Research PlatformsResearch Platforms
HRP-2 (Kawada)154cm, 0.5M EuroAvailable at LAAS, France
HOAP (Fujitsu)60cm, 50K Euro
HRP-2m Choromet(General Robotix)35cm, 5K Euro
8080//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Free Research PlatformFree Research Platform
OpenHRP: Open Architecture Humanoid Robotics Platformhttp://www.aist.go.jp/is/humanoid/openhrp/
8181//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Implementation Features of Implementation Features of OpenHRPOpenHRP
Concurrent development using an arbitrary Concurrent development using an arbitrary operating system and language operating system and language
OpenHRPOpenHRP is written in Java, C++ and runs on Linux is written in Java, C++ and runs on Linux and Windowsand Windows
Distributed Object System based on CORBA
8282//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
CORBA Objects of CORBA Objects of OpenHRPOpenHRP
8383//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
ISEISE : I: Integrated ntegrated SSimulation imulation EEnvironmentnvironment
VRMLfile
ISE
VRMLfile
Modelfile
Projectfile
8484//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Our Current ChallengeOur Current Challenge
A Famous Project of Takeo KanadeA Famous Project of Takeo KanadeEyeVisionEyeVision at the at the SuperballSuperballLetLet’’s watch NBA in the court.s watch NBA in the court.
Our ChallengeOur ChallengeLetLet’’s go to the cafeteria with a humanoid.s go to the cafeteria with a humanoid.
Robust biped walkingRobust biped walkingGoing up and down stairsGoing up and down stairsOpening and closing doorsOpening and closing doors3D SLAM3D SLAM
8585//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Powered SuitsPowered Suits
HAL [U of Tsukuba] Bleex [UC Berkeley]
8686//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
Autonomous WalkingAutonomous Walking--AidAid
Range Sensor
Stairs
ObstaclesRough Terrain
8787//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
A Measure of the Ability of a RobotA Measure of the Ability of a Robot
Artificial IntelligenceArtificial IntelligenceThis robot has the intelligence that is This robot has the intelligence that is compatible to three years old child.compatible to three years old child.
Mobility of a HumanoidMobility of a HumanoidThis robot has the mobility that is compatible This robot has the mobility that is compatible to eighty years old person.to eighty years old person.
8888//8888Hiro Hirukawa The Onassis Foundation Lecture Series 2006
ΕνχαριστωΕνχαριστω
May, my dog on a summer vacation