Benjamin StephensCarnegie Mellon University

9th IEEE-RAS International Conference on Humanoid Robots

December 8, 2009

Modeling and Control of Periodic Humanoid Balance Using the Linear Biped Model

Introduction

2

Motivation

3

Simple models for complex systems

Make complex robot control easier

Models for human balance control

Achieve stable balance on force-controlled robot

Force Controlled Balance

4

How to handle perturbations when using low-impedance control on a torque-controlled humanoid robot

Force Controlled Balance

5

How to handle perturbations when using low-impedance control on a torque-controlled humanoid robot

Sarcos Humanoid Robot

6

Hydraulic ActuatorsForce Feedback Joint Controllers33 major DOFs (Lower body = 14)Total mass 94kgOff-board pump (3000 psi)

Sarcos Hydraulic Humanoid Robot

Contributions

7

Linear biped model for force control of balance

Simple description of periodic balance control

Application of model to estimation and control of Humanoid robot

Outline

Modeling BalanceControlling BalanceApplications to Humanoid Robot ControlConclusion

8

Modeling Balance

9

General Biped BalanceAssumptions:

Zero vertical accelerationNo torque about COM

Constraints:COP within the base

of support

10

gF

PF

PCP

CF

RP LP

REFERENCE:Kajita, S.; Tani, K., "Study of dynamic biped locomotion on rugged terrain-derivation and application of the linear inverted pendulum mode," ICRA 1991

CP PPL

mgF

General Biped Balance Stability

11

COM Position

COM

Vel

ocity

minmaxCC PP

L

gPPP

L

g

max2

minCC PP

PP

Linear constraints on the COP define a linear stability region for which the ankle strategy is stable

REFERENCE:Stephens, “Humanoid Push Recovery,” Humanoids 2007

The Linear Biped ModelContact force is distributed linearly to the two feet.

12

LR FFFF LR ww

1 LR wwRF LF

y

LyRy

L

y

RXM LXM

gF

PF

CF

The Linear Biped ModelBiped dynamics resemble two superimposed linear

inverted pendulums.

13

mgwF

mgwF

LLZ

RRZ

L

Myy

L

FF

L

Myy

L

FF

LXL

LZLY

RXR

RZRY

RF LF

y

LyRy

L

y

RXM LXM

gF

PF

CFXLLX

XRRX

MwM

MwM

The Double Support Region

14

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

WD

LR

L

ww

Dy

DyD

DyDy

w

1

,0

,2

,1

RF LF

y

LyRy

L

y

RXM LXM

gF

PF

CF

D2W2

d2

Dynamics of Double Support

15

The dynamics during double support simplify to a simple harmonic oscillator

L

Myy

L

mg

D

yD

L

Myy

L

mg

D

yDym X

RX

L 22

mL

My

L

gy X 0

1

mLy

L

gy

LIPM Dynamics

RYLYY FFF

RwLw

Controlling Balance

16

Phase Space of LiBM

y

y

RF LFRF LFRF LF

Ry LyLocation of feet

LFRF

Double Support Region

17

Periodic BalanceGoal: Balance while moving in a cyclic motion,

returning to the cycle if perturbed.

18

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 ddes

19

20

-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-ve

l

Energy Control Trajectories

21

22

Application to Humanoid Balance

24

Humanoid Applications

25

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.

Center of Mass Filtering

26

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

Joint Kinematics

HipAccelerometer

FeetForce Sensors

Kalman FilterPeriodic

Humanoid Balance

CoM State

27

Feed-Forward Force Control

28

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

Full-body inverse dynamics can be reducedto force control of 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

29

30 -0.03 -0.02 -0.01 0 0.01 0.02 0.03-0.1

-0.05

0

0.05

0.1

Position

Vel

ocity

Simulation

desiredactual

31

32 -0.03 -0.02 -0.01 0 0.01 0.02 0.03

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

Position

Vel

ocity

Simulation

desiredactual

Impulsive Push

Limit Cycle

Robot Experiments

33

Future Work

34

3D Linear Biped ModelRobot Behaviors

Foot PlacementPush RecoveryWalking

Robust Control/EstimationPush Force EstimationRobust control of LiBM

RxF

x

Lx

y

RyF

LzF

RzF

LyFLxF

xy

z

LyRx

Ry

Conclusion

35

Linear biped model for force control of balanceSimple description of periodic behaviors and balance controlApplied to estimation and control of humanoid robot

y

Slow Swaying Fast Swaying

Marching in Place or Walking

y

Joint Kinematics

HipAccel

Force Sensors

Kalman Filter

Periodic Humanoid

BalanceCoM State

RF LF

)(qJ L)(qJR

RF LF

y

LyRy

L

y

RXM LXM

gF

PF

CF

Thank you. Questions?