Fast and Robust Legged Locomotion

Fast and Robust Legged Locomotion

Sean BaileyMechanical Engineering Design Division

Advisor: Dr. Mark CutkoskyMay 12, 2000

Intro Biomimesis Design Analysis Conclusions

Overview• Introduction

• Functional Biomimesis

• Robot Design

• Model Analysis

• Conclusions


Fast, Robust Rough Terrain Traversal• Why?

– Mine clearing– Urban Reconnaissance

• Why legs?

• Basic Design Goals– 1.5 body lengths per second– Hip-height obstacles– Simple


Traditional Approaches to Legged Systems• Statically stable

– Tripod of support– – Slow– Rough terrain

• Dynamically stable– No support

requirements– – Fast– Smooth terrain

0 F

0 F


Biological Example• Death-head cockroach Blaberus discoidalis

• Fast– Speeds of up to 10 body/s

• Rough terrain– Can easily traverse fractal terrain of

obstacles 3X hip height

• Stability– Static and dynamic

FunctionalBiomimesis

“Biomimetic” configuration

Extract fast rough terrain locomotion capabilities

Too complex!


Biomimesis Options


Biological Inspiration• Control heirarchy

– Passive component– Active component


MechanicalSystem

Environment

MechanicalFeedback(Preflexes)

SensoryFeedback(Reflexes)

Neural System

FeedforwardMotor Pattern

Passive DynamicSelf-Stabilization

Locomotion

Is Passive Enough?• Passive Dynamic Stabilization

– No active stabilization– Geometry– Mechanical system properties


Cockroach Geometry

•Passive Compliant Hip Joint•Effective Thrusting Force

Functional Biomimesis

•Damped, Compliant Hip Flexure•Embedded Air Piston

Robot Implementation

Geometry

•Rotary Joint•Prismatic Joint


Sprawlita• Mass - .27 kg• Dimensions - 16x10x9 cm• Leg length - 4.5 cm• Max. Speed - 39cm/s

2.5 body/sec

• Hip height obstacle traversal


Movie

• Compliant hip• Alternating tripod• Stable running• Obstacle traversal


Mechanical System Properties• Prototype: Empirically tuned properties• Design for behavior

?Mechanical

SystemProperties

Modeling


“Simple” Model

• Body has 3 planar degrees of freedom– x, z, theta– mass, inertia

• 3 massless legs (per tripod)– rotating hip joint - damped torsional spring– prismatic leg joint - damped linear spring– 6 parameters per leg

18 parameters to tune - TOO MANY!

Full 3D model Planar model Symmetry assumption

K, B, nom

k, b, nom


Simplest Locomotion Model

• Body has 2 planar degrees of freedom– x, z– mass

• 4 massless legs– freely rotating hip joint – prismatic leg joint - damped linear spring– 3 parameters per leg

6 parameters to tune, assuming symmetry

g

g

k, b, nom

g

Biped QuadrupedBiped


• Time-Based Mode Transitions– Clock-driven motor pattern– “Groucho running”1

• One “reset” mode– Two sets of legs - Two modes– Symmetric - treat as one mode

• Mode initial conditions– Nominal leg angles– Instant passive component compression

Modeling assumptions

1 McMahon, et al 1987

Leg Set2

Leg Set1

Leg Set2

Leg Set 1

Sta

teTime

x0

= state trajectoryStride Period

T T T T

g







Leg Set2

Leg Set1

Leg Set2

Leg Set 1

Sta

teTime

x0


g

t = 2T-

T T T T







Leg Set2

Leg Set1

Leg Set2

Leg Set 1

Sta

teTime

x0


t = 2T+

T T T T

g







Leg Set2

Leg Set1

Leg Set2

Leg Set 1

Sta

teTime

x0


t = 2T + 1/3T

T T T T

g







Leg Set2

Leg Set1

Leg Set2

Leg Set 1

Sta

teTime

x0


t = 2T + 2/3T

T T T T

g







Leg Set2

Leg Set1

Leg Set2

Leg Set 1

Sta

teTime

x0


t = 3T-

T T T T

g







Leg Set2

Leg Set1

Leg Set2

Leg Set 1

Sta

teTime

x0


t = 3T+

T T T T

g







Leg Set2

Leg Set1

Leg Set2

Leg Set 1

Sta

teTime

x0


t = 3T + 1/3T

T T T T

g


Non-linear analysis tools• Discrete non-linear system

• Fixed points– numerically integrate to find– exclude horizontal position information

)(1 kk xfx = state trajectory= fixed points

)( ** xfx

xk+1 = xk = x*

Leg Set2

Leg Set1

Leg Set2

Leg Set 1

Sta

teTime

x0

= state trajectory

Stride Period

T T T T


Non-linear analysis tools• Floquet technique

– Analyze perturbation response

– Digital eigenvalues via linearization - examine stability

– Use selective perturbations to construct M matrix

][,...,1 Meign ll

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4

3

2

1

4

3

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1 d

dddddddd

dddd

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û

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000d

kx

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ë

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4

3

2

1

1

dddd

kx

pertxperttotal xxx *

pertk

pertk

pertk Mxx

xxxfx

*1 |)(

*x = nominal trajectory

NumericallyIntegrate


Non-linear analysis tools• Floquet technique

][,...,1 Meign ll

)(1 kk xfx )f(xx **

)()()( **1

* pertk

pertk

pertk xfxfxxfxx

...)()( ** | tohx

xxxfxf pert

k

pertk

pertk

pertk Mxx

xxxfx

*1 |)(

pertk

pertk x

xxxfxxx *

*1

* |)(

Perturbation Response

0.83 0.84 0.85 0.86 0.87 0.880

0.005

0.01

0.015

0.02

0.025Perturbation Response over 3 Mode Transitions

X (meters)

Z (m

eter

s)

Nominal OrbitPerturbed Trajectory



• Relationships– damping vs. speed and

“robustness”– stiffness, leg angles, leg

lengths, stride period, etc

• Use for design– select mechanical properties– select other parameters

• Insight into the mechanism of locomotion

6.5 7 7.5 8 8.5 9 9.5 101.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

Damping (N-s/m)

Recovery RateHorizontalVelocity

X_dot (m/s)

1/m

ax[e

ig(M

)]

Analysis trends


Design Example RobustnessSpeed

Stiffness

Damping

Stiffness

Damping

Stiffness

Damping

Speed = 0 Speed = 13 cm/s Speed = 23.5 cm/s


Locomotion Insight

StaticallyUnstable

RegionInitialcondition

ModeEquilibrium

Trajectory

LegExtension

Limit

Leg Pre-Compressions

• Body tends towardsequilibrium point

• Parameters andmechanical propertiesdetermine how


• Current leg systems are either fast or can handle rough terrain• Biology suggests emphasis on good mechanical design

– enhances capability– simplifies control

• Purely clock-driven systems can be fast and robust

• Floquet technique can be used to indicate locomotion robustness• Trends can be established to improve design and provide insight

Summary and Conclusions


Future Work• Extend findings and insights to more complex models• Develop easily modeled 4th generation robot

• Utilize sensor feedback in high level control• Examine other behaviors

Thanks!• Center for Design Research• Dexterous Manipulation Lab• Rapid Prototyping Lab

• Mark Cutkosky• Jorge Cham, Jonathan Clark


Fast and Robust Legged Locomotion

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