Transcript The Linear Biped Model and Application to Humanoid Estimation

The Linear Biped Model and Application to
Humanoid Estimation and Control
Benjamin Stephens
Carnegie Mellon University
Monday June 29, 2009
Introduction
2
Motivation
 Robotics
 Find simple models for complex systems
 Develop algorithms that use simple models to make
humanoid control simpler
 Better way to understand and explain dynamic balance
and locomotion
 Human Physiology
 Evaluating biomechanical models
 Understand and prevent falls, which can lead to
hip/wrist fractures.
3
Take-Home Message
“The Linear Biped Model is a simple model of
balance that can describe a wide range of
behaviors and be directly applied to humanoid
robot estimation and control”
4
Outline
 Modeling
 Balance Overview
 Linear Biped Model
 Orbital Energy Control
 Lateral Foot Placement Control
 Humanoid Robot
 Center of Mass Estimation
 Feed-forward Control
 Future Work
 Conclusion
5
Modeling Balance
6
Intro to Modeling Balance
 Sum of forces
 Center of pressure
Fy
 Base of support
Fy
FL
Fg
Feq Feq
FR
7
Fg
Feq
FR
Feq
FL
center of pressure
center of pressure
Linear Inverted Pendulum Model
 Features:
y
 All mass concentrated at CoM
 Massless legs
 Does not move vertically
 Linear
I  mgL sin   
(Linearize)
mLy  mgy  
L
F

g
  g
y   y 
   y  ycop 
L
mg  L
ycop
8
y
Kajita, S.; Tani, K., "Study of dynamic biped locomotion on rugged terrain-derivation and
application of the linear inverted pendulum mode," IEEE International Conference on Robotics
and Automation, vol.2, pp.1405-1411, 1991.
0
Stability of Linear Inverted Pendulum
 What’s the best we can do?
 Apply maximum allowable force to the ground
 Move center of pressure to edge of base of support


g
g
y  max  y  y  max
L
mL
L
mL
y
d   y  d
k
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Benjamin Stephens, "Humanoid Push Recovery," The IEEE-RAS 2007 International Conference on
Humanoid Robots, Pittsburgh, PA, November 2007
The Linear Biped Model
 Weighted sum of the dynamics due to two linear
inverted pendulum models (rooted at the feet) y
my  FLY  FRY
mg  FLZ  FRZ
FLY 
L
L

FLZ
 y  yL 
L
FRY 
u
R
L

FRZ
 y  yR 
L
 u mg
 y  yL   wR  u  mg  y  yR 
my  wL  
L L

L L

mg  wL mg  wR mg
wL  wR  1
10
 R  wR u
 L  wLu
L
FR
FL
R
FRZ  wR mg
L
FLZ  wL mg
Benjamin Stephens, " Energy and Stepping Control of Linear Biped Model in the Coronal Plane,"
Submitted to The IEEE-RAS 2009 International Conference on Humanoid Robots.
yR
y
yL
The Double Support Region
 We define the “Double Support Region” as a fixed
fraction of the stance width.
y
D  W
 1
yD
wL  
 2D
 0
wR  1  wL
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,
yD
,
y D
,
yD
L
FR
R
FL
y 0
2D
2W
L
2d
Dynamics of Double Support
 The dynamics during double support simplify to a
simple harmonic oscillator
 u mg
 y  yL   wR  u  mg  y  yR 
my  wL  
L L

L L

my 
 R  L
L

mg
D  y  y  W   D  y  y  W 
2 DL
y 
y 
g

y
L
mL
LIPM Dynamics
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 R  L
g
D  W  y

mL
DL
g
u
y   y 
L
mL

1 

Stability of the Linear Biped Model
 What’s the best we can do?
 Apply maximum allowable force to the ground
 Move center of pressure to edge of base of support
u
u
gg
mgd
g
mgd
 yy max  y   y  max
LL
mL
mL
L
mL

FZ
d
 R  wR mgd
 L  wL mgd
 R   L  u  mgd
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Phase Space of LiBM
y
Double Support Region
y
FFRR
yR
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Location of feet
yL
FLFL
Controlling Balance
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Static Balance Control
 Goal: Return to a state of static balance (zero velocity)
 Strategies:
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Periodic Balance
 Goal: Balance while moving in a cyclic motion,
returning to the cycle if perturbed.
y
y
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Slow
Fast
Swaying
Swaying
Marching
in Place
or Walking
Orbital Energy Control
 Orbital Energy:
1 2 g 2
E  y 
y
2
2L
 Solution is a simple harmonic oscillator:
 g 
2 LEd
1 2 g 2
y 
y  Ed  0  y 
sin 
t 
2
2L
g
 L 
 We control the energy:
e  Ed  E
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g 

e  Ke  0   y  y  y  Ke  0
L 

g
 y   y  Ky Ed  E 
L
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Energy Control Trajectories
0.4
0.3
0.2
y-vel
0.1
0
-0.1
-0.2
-0.3
-0.4
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-0.2
-0.15
-0.1
-0.05
0
y-pos
0.05
0.1
0.15
0.2
Stepping Control
 Because we define double support region, when to
step is pre-determined, we only have to decide how
far to step
x2
u1
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yR
y
x0
u0
x1 DSP region moves yL
y
N-Step Controller
 Because DSP region is fixed, we know when to take a
step, only need to decide where
 N-Step lookahead over a set foot step distances
cost  K1  stance _ width   K 2  y 
2
 Benefits:




22
Very fast
Works for any desired energy
Recovers from Pushes
Stabilizes position
2
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24
Application to Humanoid Balance
25
Humanoid Applications
Linear Biped Model predicts gross body motion and
determines a set of forces that can produce that motion
State Estimation
 Combine sensors to predict important features, like center
of mass motion.
Feed-Forward Control
 Perform force control to generate the desired ground
contact forces.
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Robot Sensing Overview
High Level
Controller
PROCESS
NOISE
Joint Level
Controller
State Estimate
Robot
MEASUREMENT
NOISE
Estimate
Fusion & Filter
Position Measurement
Kinematics
Model
Potentiometers
MEASUREMENT
NOISE
Flatness
Calculation
Force/Torque
Sensors
Force Measurement
Robot
Model
Acceleration Measurement
Acceleration
Estimate
Joint Torques
IMU
MEASUREMENT
NOISE
Center of Mass Filtering
 A (linear) Kalman Filter can combine multiple
measurements to give improved position and velocity
center of mass estimates.
Joint
Kinematics
Hip
Accelerometer
Kalman Filter
Periodic
Humanoid
CoM State
Balance
Feet
Force Sensors
 NOTE: Because we measure force, we should also be
able to estimate push/disturbance magnitudes
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29
Feed-Forward Force Control
 LiBM can be used for feedforward control of a
complex biped system.
 Torques can be generated by force control
of the CoM with respect to each foot
 L  J LT FL
 R  J RT FR
J R (q)
J L (q)
 Additional controls are applied to bias
towards a home pose and to keep the
torso vertical.
FR FL
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0.1
velocity
0.05
0
-0.05
-0.1
-0.02
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-0.01
0
0.01
position
0.02
0.03
Movie Summary
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Conclusion
“The Linear Biped Model is a simple model of balance that
can describe a wide range of behaviors and be directly
applied to humanoid robot estimation and control”
Joint
Kinematics
y
Slow Swaying
Fast Swaying
y
Periodic
CoM State
Humanoid
Balance
Kalman
Filter
Hip
Accel
Force
Sensors
L
y
FR
R
FL
J R (q)
L
J L (q)
Marching in Place or Walking
FR FL
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Future Work
x
y
 3D Linear Biped Model
FLy
FLx
 Refine Robot Behaviors
z
 Foot Placement
 Push Recovery
x
y
 Sliding Mode Control of LiBM
FLz
FRy FRz
 Walking
 Robust Control/Estimation
FRx
 Rx
 Ry
 Lx
 Push Force Estimation
 Online LiBM Parameter Estimation/Adaptation
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 Ly
The End
 Thanks to Research Committee Members:
 Chris Atkeson
 Jessica Hodgins
 Martial Herbert
 Stuart Anderson
 Questions?
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Dynamic Constraints


FZ
d
RSP
DSP
LSP
mgd
 R  wR mgd
L
 L  wL mgd
u  mgd
u   R  L
y
R
 mgd
2D
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Friction Constraints on LiBM
FLY

FLZ
L
mg
 y  yL 
 L
L
  L

L mg
 
L mg L  y  yL    L  L mgL  y  yL 
 
L
u
mg
 y  yL 
 L
L
L

L mg
mg  L  y  y L   u  mg L  y  y L 
mg  L  y  y R   u  mg L  y  y R 
 L mgd   L  L mgd
Double Support
Right
Support
 R  L
Left
Support
mgd
L

R
2W
 mgd
y
L mg L  y  yL    L  L mgL  y  yL 
Hybrid Orbital Energy
In DSP region, we use
the same energy
equation as before, x is
relative to half way
between feet
EDSP 
1 2 g 2
x 
x
2
2L
In SSP region, we use
the orbital energy, x is
relative to stance foot
ESSP 
1 2 g 2
x 
x
2
2L
Energy at middle of SSP
determines curve
DSP region!