Transcript The Linear Biped Model and Application to Humanoid

Modeling and Control of Periodic Humanoid
Balance Using the Linear Biped Model
Benjamin Stephens
Carnegie Mellon University
9th IEEE-RAS International Conference on
Humanoid Robots
December 8, 2009
Introduction
2
Motivation
 Simple models for complex systems
 Make complex robot control easier
 Models for human balance control
 Achieve stable balance on force-controlled robot
3
Force Controlled Balance
 How to handle perturbations when using low-
impedance control on a torque-controlled humanoid
robot
4
Force Controlled Balance
 How to handle perturbations when using low-
impedance control on a torque-controlled humanoid
robot
5
Sarcos Humanoid Robot
 Hydraulic Actuators
 Force Feedback Joint Controllers
 33 major DOFs (Lower body = 14)
 Total mass 94kg
 Off-board pump (3000 psi)
Sarcos Hydraulic Humanoid Robot
6
Contributions
 Linear biped model for force control of balance
 Simple description of periodic balance control
 Application of model to estimation and control of
Humanoid robot
7
Outline
 Modeling Balance
 Controlling Balance
 Applications to Humanoid Robot Control
 Conclusion
8
Modeling Balance
9
General Biped Balance
Assumptions:
FP
 Zero vertical acceleration
 No torque about COM
mg
FP  P  PC 
L
FC
Fg
Constraints:
 COP within the base
of support
PC PR
10
P
REFERENCE:
Kajita, S.; Tani, K., "Study of dynamic biped locomotion on rugged terrain-derivation and
application of the linear inverted pendulum mode," ICRA 1991
PL
General Biped Balance Stability
Linear constraints on the COP define
a linear stability region for which the
ankle strategy is stable




COM Velocity

g
g
max


P  PC  P  P  PCmin
L
L
P
min
PC  2  P  PCmax
COM Position
11
REFERENCE:
Stephens, “Humanoid Push Recovery,” Humanoids 2007
The Linear Biped Model
 Contact force is distributed linearly to the two feet.
FP
y
F  wR  wL F  FR  FL
wR  wL  1
L
Fg
FC
FL
FR
M RX
12
yR
M LX
y
yL
The Linear Biped Model
 Biped dynamics resemble two superimposed linear
inverted pendulums.
FRZ  wR mg
FP
y
FLZ  wL mg
M RX  wR M X
M LX  wL M X
FRY
FLY
13
L
Fg
FC
FRZ
M RX
 y  yR  

L
L
FLZ
M LX
 y  yL  

L
L
FL
FR
M RX
yR
M LX
y
yL
The Double Support Region
 We define the “Double Support Region” as a fixed
fraction of the stance width.
FP
y
D  W
 1
yD
wL  
 2D
 0
wR  1  wL
,
yD
,
y D
,
yD
L
FL
FR
M RX
yR
14
Fg
FC
M LX
y
2D
2W
yL
2d
Dynamics of Double Support
 The dynamics during double support simplify to a
simple harmonic oscillator
FY  FLY  FRY
MX
 D  y  mg
 y  yL  
my  

L
 2 D  L
MX 
  D  y  mg
 y  yR  



L 
  2 D  L
wL
y 
g

y
L
mL
LIPM Dynamics
15
wR
g
M
y   y  X
L
mL

1 

0
Controlling Balance
16
Phase Space of LiBM
y
Double Support Region
y
FFRR
yR
17
Location of feet
yL
FLFL
Periodic Balance
 Goal: Balance while moving in a cyclic motion,
returning to the cycle if perturbed.
y
y
18
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
19
g 

e  Ke  0   y  y  y  Ke  0
L 

g
des
 y   y  Ky Ed  E 
L
20
Energy Control Trajectories
0.4
0.3
0.2
y-vel
0.1
0
-0.1
-0.2
-0.3
-0.4
21
-0.2
-0.15
-0.1
-0.05
0
y-pos
0.05
0.1
0.15
0.2
22
Application to Humanoid Balance
24
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.
25
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
Feet
Force Sensors
26
Kalman Filter
Periodic
Humanoid
CoM
State
Balance
27
Feed-Forward Force Control
 LiBM can be used for feedforward control of a
complex biped system.
 Full-body inverse dynamics can be reduced
to 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.
28
FR FL
29
Simulation
0.1
desired
actual
Velocity
0.05
0
-0.05
-0.1
30
-0.03 -0.02 -0.01
0
0.01
Position
0.02
0.03
31
Simulation
0.1
desired
actual
0.05
0
Velocity
Limit Cycle
-0.05
-0.1
-0.15
Impulsive Push
-0.2
-0.25
32
-0.03 -0.02 -0.01
0
0.01
Position
0.02
0.03
Robot Experiments
33
Future Work
 3D Linear Biped Model
 Robot Behaviors
 Foot Placement
 Push Recovery
 Walking
 Robust Control/Estimation
 Push Force Estimation
 Robust control of LiBM
34
x
y
FLy
FLx
z
x
y
FRx
FLz
FRy FRz
 Rx
 Ry
 Lx
 Ly
Conclusion
 Linear biped model for force control of balance
 Simple description of periodic behaviors and balance control
 Applied to estimation and control of humanoid robot
Joint
Kinematics
FP
Slow Swaying
y
Fast Swaying
y
Periodic
CoM State
Humanoid
Balance
Kalman
Filter
Hip
Accel
Force
Sensors
L
Fg
FC
M RX
yR
35
y
FL
FR
M LX
y
yL
J R (q)
J L (q)
Marching in Place or Walking
Thank you. Questions?
FR FL