IEEE Humanoids 2013, Workshop: Benchmarking of Human-Like Robotic Locomotion Atlanta GA USA

Balance Stability and Energy Efficiency: Towards the Analogy

Atlanta, GA, USA

Balance Stability and Energy Efficiency: Towards the Analogy between Robotic and Human Gait through Unified Models

Joo H. Kim, Ph.D.Assistant ProfessorAssistant Professor

Department of Mechanical and Aerospace EngineeringPolytechnic Institute of New York University (NYU-Poly)

Brooklyn, New York, USABrooklyn, New York, USA

Outlines

Two Human-like FeaturesPart I: Balance Stability

Part II: Energy Consumption Models

Modeling, Design, and Control

RoboticsandM h i

Biomechanics andBi di l i iMechanisms

Principles of Motions and Structures

Biomedical engineering

Principles of Motions and Structures

Part I: Balance Stability

Background (Balance Stability Model)

Phase space approaches for balance stability of legged mechanisms • Balance criteria using inverted pendulum

Motion flexibility Controllability

Collision avoidance Balance criteria using inverted pendulum • Stability boundary in phase plane (position and velocity of CoM)• Ankle, hip, and stepping strategies• Single-support capturability in a given state to come to a stop without falling Instability

Risk of falling

Biped Mechanism

by taking step(s)

Li it l P i é

g

Control

VelocitiesCMP

ZMP/COP

FRI

Sufficient conditions

Balancing

Nonperiodic motionLimit cycle Poincaré map

Pai & Pattron 97’ A. Goswami 99’ A. Bottaro et al. 05’

TorquesFRI

Necessary conditions

Stephens (2007) etc. Mark W. Spong (1999) etc. S.M. Bruijn (2010) etc.

Based on specific control law and parametersPratt et al. 06’ B. Stephens 07’ Lee & Goswami 07’

Balanced, Falling, and Fallen States

Fallen: part(s) of the mechanism other than its feet are in contact with the ground

Static equilibrium

contact with the groundFalling: a trajectory passing through the state necessarily

terminates in fallen state under a balance controllerBalanced: when applying a balance controller will not lead

to a fallen configuration

Balanced

Falling

Fallen

: Initial state

: Intermediate state

: Final state

to a fallen configuration Specific controller domain

Feasible Balanced State Domain (D) Explicit Conditions on Balanced StateFeasible Balanced State Domain (D)iC D i

: controller-specific balanced state domains( i = 1, 2, … )

iC

Explicit Conditions on Balanced State Initial state exists within balanced state domain At least one trajectory exists leading to a final

balanced state of static equilibrium subject to given q j gactuation limits

ZMP within FSR under no-slip condition and actuation limits

Dynamic Models of Actuation and Ground Reaction

Single-segmental Legged Mechanism

( ) ( ) ( ) ( ) ( ) ( )[ ]fict p p p r r r Tq q q q q qΦ 0

Constraints assigned to fictitious joints :

Y

23 footL

2l

2m

g

footL

1mX

Y

Inertial frame

O

( ) ( ) ( ) ( ) ( ) ( )1 2 3 4 5 6[ ]f p p pf f f f f fq q q q q q Φ 0

( ) ( ) ( ) ( ) ( ) ( )1 2 3 4 5 6[ ]p p p r r r T fict T

constr f f f f f f qf Φ λRequired actuations (Generalized constraint force)

global forces global moments

Multi-segmental Legged

XAnkle joint

1q

1

OFictitious joints

Z global forces global moments

1xZMPN

( )3p

y fF N ( )2p

x fF Normal Tangential

Differential-algebraic equations for multibody dynamic modelg gg

Mechanism

2m

3m

3l2 Case1: fixedCase2: free to move

YY

fict T

fict fictfictt

q

qq

τ v g hm Φ qΦ q ΦΦ 0 λ

fixed

0q Fictitious jointsAnkle joint0x

0y

23 footL 2l

1q

gfootL

1m X

Y

Z

Inertial frame

1

XZ

XZ

1 0q Fictitious joints

Inertial frame

Numerical Construction of Balanced State Manifold

For a given initial position, two problems:• Minimize and Maximize C t i t t

( )initialt Constraints set:

C1. Joint variable range of motion

C2 Maximum actuator torque

( ) [ , ]L Uj j jt 1,2; [ , ]initial finalj t t t

C5. Positive normal GRF

C6 No slip condition( ) 0N t [ , ]initial finalt t t

C2. Maximum actuator torque

C3. Initial joint variable

( ) [ , ]L Uj j ju t u u 1,2; [ , ]initial finalj t t t

0( ) andt

C6. No-slip condition

C7. Final static equilibrium( ) ( ) 0t t

( )( )x

s

F tN t

[ , ]initial finalt t t

2 ( ) 0i i i lt 1 2j

velocity

C4. ZMP constraint1( ) andinitialt

( ) (0, )x footZMP t L [ , ]initial finalt t t

( ) ( ) 0j final j finalt t 2 ( ) 0initialt 1,2j

Balanced State

MaximaMaxima

Balanced State ManifoldFalling / Fallen State

Minima

Minima

maximum actuator torqueangle

Iterative Algorithm

START

For(                   ;                   ;           )m ax m axLu u m ax m ax

Uu u m axu

For(             ;            ;       )L U MinimumFeasible Velocity

maxu

SQP Solver

Maximum  &  Minimum    

MaximumFeasible Velocity

NoU

No

Yes

Y

m ax m axUu u

maxu

Yes

END

Numerical Results: Single-segmental mechanism

Numerical Results: Multi-segmental mechanism

Human Walking Motion

0.6

0.8

1

1.2

( )y m

Human Walking Motion

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

( )x m

( )y m

0.8

1

Robotic Walking Motion

0.8 1 1.2 1.4 1.6 1.8 2

( )x m

( )y m

0

0.2

0.4

0.6

( )x m

Part II: Energy Consumption Models

Background (Energy Consumption Models)

Energy consumption of humanoid (battery) and human (metabolism)• Actual energy expended during a task (and not just work, KE, or )• As functions of kinematic and kinetic variables

2

1

( )n

ii

• As functions of kinematic and kinetic variables• As functions of time (and not just as a time average) • Numerical simulations and optimization • Realistic cost of transportp

Joint Space as Generalized Coordinates

1[ ... ]T nnq q q R 1 2( , ,..., , )i i mq q M M M t (i = 1, 2, …, n)

activation contraction1Ma

1M

M

l

v

muscle space (j=1…m)

excitation dynamics dynamics

musculoskeletalmodel

1

1

MT

MT

l

v1

1

qq

1

1M

v

F

1 1q ,

joint space (i=1…n)

2 2q ,

excitation

activationdynamics

contractiondynamics

2Ma 2

2

2

M

M

M

l

v

F

excitationmapH

uman

oid

Hum

an

( )t

( )t

activationdynamics

contractiondynamics

MTl

Mma

Mm

Mm

Mm

l

v

F

n nq ,

excitation

musculoskeletalmodel

2

2

MT

MT

l

v2

2

qq

ping

( )( ) ( ) ( )app befdi tV t V t Ri t Ldt

musculoskeletal

model

MTm

MTm

l

vm

m

qq

contactforces

muscle

( ) ( )bef EV t k t

( ) ( )Mt k i t muscleforces

m

I

bT

lT

lMT=lT+lMcosθ

θ

( ) ( , ) ( ) ( )M max LV M M M PE MF t F F l v a t F l m

mImb

12

( , )( ) | ( ) | ( )max

c q qa t t c t

First Law of Thermodynamics

totW

InletExit

System of interest, ,i i iV e

, ,e e eV e

,e

System boundary

Q

extE Q W k p p t extE E U U E Q W k p p int extE E U W W

First Law:

Work and Energy Principle: E E U W W

intE W Q

gy p

E W Q

Energy Consumption Models

Food Water O2

Biochemical EnergyHum

anoi

dH

uman

Total Robot Energy Consumption

Electrical Energy

Total MEE

Muscle MEE BMR

Muscle Work Muscle Heat

Mass loss (sweat, Air, etc.)

Stored EnergyTotal Robot Energy Consumption

Mechanical Work Heat  Noise SensorsMicrocontroller, motor drivers etc

Actuators Electronics 

Activation Heat Maintenance Heat Shortening Heat Lengthening Heat

2( ) ( ) ( ) ( )n n

motor iRE t t q t t

drivers, etc.

( , )( ) ( ) ( ) ( )amn n

met ih q qE 21 1

( ) ( ) ( ) ( )i i ii i i

E t t q t tk

1 1

1 1

( , )( ) ( ) ( ) ( )

( , ) ( ) ( ) ( )

met ii i imax

i i isln n

ccii i imax

i i

h q qE t t q t t

h q q t q t Q t BMR

1 1i ii

( )basal basalBMR Q W

Measurements and Parameter Identification

Hum

anoi

dH

uman

x

z

y

Results

00

450

500

550

150

200

250

300

350

400

ME

E R

ate

(Wat

ts)

Hum

anoi

dH

uman

0 10 20 30 40 50 60 70 80 90 1000

50

100

% Gait Cycle

70

Hip

XY

80

Knee

XY 160

180Ankle

XY

10

20

30

40

50

60

ME

E S

agitt

al P

lane

(Wat

ts)

YZ

10

20

30

40

50

60

70YZ

20

40

60

80

100

120

140

60 YZ

20 40 60 80 100% Gait Cycle

20 40 60 80 100

% Gait Cycle

20 40 60 80 100

% Gait Cycle

Subject Trial Reference Emet (J) Predicted Emet (J) % ErrorMale 1 265 259 -2.1

2 257 267 3.834

262270

273282

4.14 54

5270271

282269

4.5-0.5

Female 1 281 283 1.02 294 304 3.53 288 300 4.24 287 298 4.15 284 296 4.1

Balance Stability Model

Conclusions

Balance Stability Model Balanced state manifold (viability kernel) - reachable superset of all possible controller-

specific domains Balanced state manifold was constructed from explicit forms of necessary and sufficient

conditionsconditions Showed valid features and consistent with experimental data Analysis of robotic and human gait balance within unified framework

Energy Consumption Models Energy Consumption ModelsActual energy expended for general functional task and realistic cost of transportAs functions of kinematic and kinetic variablesAs functions of time (and not just as a time average) ( j g )Numerical simulations and optimization Analysis of robotic and human gait balance within unified framework

Future Work