02_leggedlocomotion

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ETH Master Course: 151-0854-00L

Autonomous Mobile Robots

Legged Locomotion

Roland Siegwart

Margarita Chli

Martin RufliDavide Scaramuzza


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Lecture OverviewMobile Robot Control Scheme

2

raw data

positionglobal map

Sensing Acting

InformationExtraction

PathExecution

CognitionPath Planning

knowledge,data base

missioncommands

Real World

Environment

LocalizationMap Building

MotionControl

Perce

ption

actuator

commands

environment modellocal map

path

2 - Legged Locomotion


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Locomotion PrinciplesPrinciples Found in Nature

3



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Locomotion PrinciplesBiped Walking and Pure Rolling

Biped walking mechanism approximates purerolling via polygonal motion

The smaller the step size, the more

the polygon tends to a disk (wheel)

Work against gravity is required

Allows to overcome larger ob-

stacles (compared to rolling)

4



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Locomotion PrinciplesSelection of Locomotion Concept

Selection depends on terrain properties

robot weight and complexity

desired operating speed

maximal energy expenditure

required energy efficiency etc.

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Legged LocomotionNumber of Legs vs. Control Complexity

The number of legs influences Mechanical complexity

Control complexity

Insects can walk directly upon birth

Most mammals require several minutes to stand

Humans require more than a year to walk on two legs

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Legged LocomotionNumber of Distinct Gait Sequences

The gait is characterized as a distinct sequence oflift and release events of the individual legs

The number of possible events Nfor a walking machine with klegs is

For a biped walker, the number of possible events becomes

For a robot with 6 legs (hexapod)

7

!12

kN

6!3!12 kN

800'916'39!11 N



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CoG

Legged LocomotionStatic Locomotion

Static Locomotion

Characteristics

Body weight supported by atleast three legs

CoG withing support triangle

Safe, slow and inefficient

8



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Legged LocomotionStatic Locomotion

9

Most widespread static sequence with 6 legs



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Legged LocomotionStatic Locomotion on ALoF

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Legged LocomotionDynamic Locomotion

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CoG

3


Dynamic Locomotion

Characteristics

The robot falls unless it moves

legs can be in ground contact

Fast, efficient, but demanding for

actuation and control


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Legged LocomotionDynamic Locomotion with 4 Legs

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Changeover walking

Galloping



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Legged LocomotionDynamic Locomotion with 2 Legs

In principle only dynamic walking feasible Large feet allow for static walking, however

Two legs (biped) allow for four different states

1. Both legs down

2. Right leg down, left leg up

3. Right leg up, left leg down

4. Both leg up

13

1 2 1

1 3 1

1

4

1

2 3 2

2 4 2

3

4

3

turning

on right leg, orlimping

hopping

with two legs

hopping on

left leg

walking,

running

hopping onright leg

turningon left leg, orlimping



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Legged Locomotion(Dynamic) Locomotion on ASIMO

14

Courtesy K. Moriyama



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Legged LocomotionEnergy Optimization of Gaits

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Legged LocomotionEnergy Opt. via Series Elastic Actuation

The optimal actuator should be back-drivable

be able to perform negative work

have a low inertia and gear ratio

be highly efficient

Series Elastic Actuators can emulate someof these properties

16

Series Elastic Actuator

x

Fground

u



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Legged LocomotionDynamic Locomotion on StarlETH

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Legged LocomotionFrom Legs to Links and Joints

A minimum of two DOF required to move leg a lift and a swing motion.

Sliding-free motion in more than one direction not possible

Three DOF for each leg required in most cases

Additional joints (and thus DOF) increase the complexity of the designand especially of the locomotion control

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Forward KinematicsDefinition and Introduction

Forward KinematicsGiven is a set of joint angles

Determine resulting end-effector position

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0y

0

x

),(gg yx

g

321

321321211

321321211

)sin()sin(sin

)cos()cos(cos

g

g

g

aaay

aaax

1a

2a

3a



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0y

0x

Forward KinematicsChain of Coordinate Frames

20


21


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Forward KinematicsHomogeneous Transformation Matrix

21

0y

0x

),( gg yx

1

10

0

11

g

g

g

g

y

x

Ty

x

: any two frames can be connected viaat most 1 rotation and 2 translation parameter

1y

1

x

2


a

22


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Forward KinematicsHomogeneous Transformation Matrix

22

: any two frames can be connected viaat most 3 rotation and 3 translation parameter

i

g

g

g

i

i

i

i

i

i

i

ii

g

g

g

z

y

x

zy

x

z

y

x

110001

1

1

111

ii

R1

cossin0

sincos0001

1 ii

xR

cos0sin

010sin0cos

1 ii

yR

100

0cossin0sincos

1

ii

zR

3


23


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Forward KinematicsDenavit-Hartenberg Convention

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Denavit-Hartenberg (DH) reference frame layout Adds structure into kinematic chains

Involves 4 parameter only (instead of 6 for the general case)

Procedure

C

ourtesyE.

Tira-Th

ompson


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Forward KinematicsDenavit-Hartenberg Convention

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Resulting Denavit-Hartenberg transform. matrix

(joint angle): rotation about . Angle of w.r.t.

(twist angle): rotation about . Angle of w.r.t.

(link length): distance between axis and axis (i.e., and )(link offset ): offset along axis

1000

cossin0

sinsincoscoscossin

cossinsincossincos

11

111

111

1

iii

iiiiiii

iiiiiii

ii

d

a

a

T

i 1iz 1ix

1i ix

1ia 1i i

id 1iz

1iz iz

ix

iz 1iz


25


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Forward KinematicsDH Coordinate Frames on a PUMA Arm

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26


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Inverse KinematicsDefinition and Introduction

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Inverse KinematicsGiven is a desired end-effector position

Determine corresponding joint angles

Problem is non-trivial and generally not well-posed

No Solution One Solution >1 Solution


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For 2-linkmanipulators employ e.g. cosine law(and solve for )

Inverse KinematicsSolution via Closed-form Approach

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0x

0y

),( gg yx

221

2

2

2

1

22cos2 aaaayx gg ),( 21

2a

1a


28


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Inverse KinematicsSolution via Iterative Search

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From forward kinematics we know

his often not easily invertible in closed form

Approach: iteratively perform the following steps

1. Start from a known forward-kinematic solution (e.g., viasampling).

2. Linearize around , resulting in the Jacobian

....................................................................................................................................................................................................

..................................................................................................

........

3. Invert the Jacobian to obtain

4 Move by in direction

),,( 1 nT

ggg hzyx

h ),,( 1 n

TiiiT

zyxJ 121

),,( 1 nT

iii hzyx

T

igigig zzyyxx

i

n

mm

n

hh

hh

J

1

1

1

1

02_leggedlocomotion

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