Benjamin StephensCarnegie Mellon University

Monday June 29, 2009

The Linear Biped Model and Application to Humanoid Estimation and Control

Introduction

2

Motivation

3

RoboticsFind simple models for complex systemsDevelop algorithms that use simple models to make

humanoid control simplerBetter way to understand and explain dynamic balance

and locomotionHuman Physiology

Evaluating biomechanical modelsUnderstand and prevent falls, which can lead to

hip/wrist fractures.

Take-Home Message

4

“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”

OutlineModeling

Balance OverviewLinear Biped ModelOrbital Energy ControlLateral Foot Placement Control

Humanoid RobotCenter of Mass EstimationFeed-forward Control

Future WorkConclusion

5

Modeling Balance

6

Sum of forcesCenter of pressureBase of support

Intro to Modeling Balance

7

gF

RF

LF

yF

eqF

center of pressure

gF

RF

LFyF

eqF eqF

center of pressure

eqF

Linear Inverted Pendulum ModelFeatures:

All mass concentrated at CoMMassless legsDoes not move verticallyLinear

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.

F

0

L

y

y

sinmgLI mgyymL

copyyL

g

mgy

L

gy

(Linearize)

copy

8

Stability of Linear Inverted PendulumWhat’s the best we can do?

Apply maximum allowable force to the groundMove center of pressure to edge of base of support

mLy

L

gy

mLy

L

g maxmax

9 Benjamin Stephens, "Humanoid Push Recovery," The IEEE-RAS 2007 International Conference on Humanoid Robots, Pittsburgh, PA, November 2007

dyk

yd

The Linear Biped ModelWeighted sum of the dynamics due to two linear

inverted pendulum models (rooted at the feet)

RF

LF

y

LyRy

L

y

R L

10 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.

RYLY FFym

LLZL

LY yyL

F

LF

RRZR

RY yyL

F

LF

1 RL ww

RZLZ FFmg

RRLL yy

L

mg

L

uwyy

L

mg

L

uwym

uw

uw

LL

RR

u

mgwmgwmg RL

mgwF

mgwF

LLZ

RRZ

The Double Support Region

11

We define the “Double Support Region” as a fixed fraction of the stance width.

RF

LF

y

L

y

R L

D20

W2d2

WD

LR

L

ww

Dy

DyD

DyDy

w

1

,0

,2

,1

Dynamics of Double Support

12

The dynamics during double support simplify to a simple harmonic oscillator

RRLL yy

L

mg

L

uwyy

L

mg

L

uwym

WyyDWyyDDL

mg

Lym LR

2

yWDDL

g

mLy LR

mL

uy

L

gy

1

mLy

L

gy

LIPM Dynamics

Stability of the Linear Biped Model

13

What’s the best we can do?Apply maximum allowable force to the groundMove center of pressure to edge of base of support

mL

uy

L

gy

mL

uy

L

g maxmax

mgdw

mgdw

LL

RR

mgduLR

dFZ

mL

mgdy

L

gy

mL

mgdy

L

g

Phase Space of LiBM

y

y

RF LFRF LFRF LF

Ry LyLocation of feet

LFRF

Double Support Region

14

Controlling Balance

15

Static Balance Control

16

Goal: Return to a state of static balance (zero velocity)

Strategies:

Periodic BalanceGoal: Balance while moving in a cyclic motion,

returning to the cycle if perturbed.

17

y

y

Slow SwayingFast SwayingMarching in Place or Walking

Orbital Energy ControlOrbital Energy:Solution is a simple harmonic oscillator:

We control the energy:

22

22

1y

L

gyE

t

L

g

g

LEyEy

L

gy d

d sin2

022

1 22

EEe d

00

Keyy

L

gyKee

EEyKyL

gy d

18

19

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

y-pos

y-vel

Energy Control Trajectories

20

Stepping ControlBecause we define double support region, when to

step is pre-determined, we only have to decide how far to step

y

y

Ry Ly

0x2x

1x

0u1u

DSP region moves21

N-Step ControllerBecause DSP region is fixed, we know when to take a

step, only need to decide whereN-Step lookahead over a set foot step distances

Benefits:Very fastWorks for any desired energyRecovers from PushesStabilizes position

222

1 _ yKwidthstanceKcost

22

23

24

Application to Humanoid Balance

25

Humanoid Applications

26

Linear Biped Model predicts gross body motion and determines a set of forces that can produce that motion

State EstimationCombine sensors to predict important features, like center of mass motion.

Feed-Forward ControlPerform force control to generate the desired ground contact forces.

RobotJoint Level Controller

Potentiometers

Force/Torque Sensors

IMU

Joint Torques

KinematicsModel

Flatness Calculation

Position Measurement

High Level Controller

Acceleration Estimate

Estimate Fusion & Filter

RobotModel

Acceleration Measurement

Robot Sensing Overview

State Estimate

PROCESS NOISE

MEASUREMENT NOISE

MEASUREMENT NOISE

MEASUREMENT NOISE

Force Measurement

Center of Mass Filtering

28

A (linear) Kalman Filter can combine multiple measurements to give improved position and velocity center of mass estimates.

NOTE: Because we measure force, we should also be able to estimate push/disturbance magnitudes

Joint Kinematics

HipAccelerometer

FeetForce Sensors

Kalman FilterPeriodic

Humanoid Balance

Periodic Humanoid

BalanceCoM State

29

Feed-Forward Force Control

30

LiBM can be used for feedforward control of a complex biped system.

Torques can be generated by force controlof the CoM with respect to each foot

Additional controls are applied to biastowards a home pose and to keep the torso vertical.

LTLL FJ R

TRR FJ

RF LF

)(qJ L)(qJR

31

-0.02 -0.01 0 0.01 0.02 0.03

-0.1

-0.05

0

0.05

0.1

position

velocity

Movie Summary

32

Conclusion

33

“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”

RF

LF

y

L

R L

y

Slow Swaying Fast Swaying

Marching in Place or Walking

y

Joint Kinematics

HipAccel

Force Sensors

Kalman Filter

Periodic Humanoid

Balance

Periodic Humanoid

BalanceCoM State

RF LF

)(qJ L)(qJR

Future Work

34

3D Linear Biped ModelRefine Robot Behaviors

Foot PlacementPush RecoveryWalking

Robust Control/EstimationSliding Mode Control of LiBMPush Force EstimationOnline LiBM Parameter Estimation/Adaptation

RxF

x

Lx

y

RyF

LzF

RzF

LyF LxF

xy

z

LyRx

Ry

The End

35

Thanks to Research Committee Members:Chris AtkesonJessica HodginsMartial HerbertStuart Anderson

Questions?

37

Dynamic Constraints

mgdw

mgdw

LL

RR

y

R

L

D2

DSP RSP LSP

mgd

mgd

LRu

mgdu

38

dFZ

Friction Constraints on LiBM

LZ

LY

F

F

mg

yyLmg

L

L

LLL

LLLLL yyLmgyyLmg

mg

yyLmg

Lu

L

LLL

RR

LL

yyLmguyyLmg

yyLmguyyLmg

y

R

L

W2

Double Support

Right Support

LeftSupport

mgd

mgd

LR

mgdmgd LLL

LLLLL yyLmgyyLmg

Hybrid Orbital Energy

DSP region!

22

22

1x

L

gxEDSP

In DSP region, we use the same energy equation as before, x is relative to half way between feet

22

22

1x

L

gxESSP

In SSP region, we use the orbital energy, x is relative to stance foot

Energy at middle of SSP determines curve