Locomotion Concepts - ETH Z · Autonomous Systems Lab Autonomous Mobile Robots Roland Siegwart,...

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ASL Autonomous Systems Lab

Autonomous Mobile Robots

Roland Siegwart, Margarita Chli, Nick Lawrance

Slides courtesy of Marco Hutter


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Locomotion Concepts

Spring 2020

Locomotion Concepts - add ons 25/02/2020

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Week 2 overview

Locomotion Concepts - add ons

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On ground

Locomotion Concepts - add ons 3

Locomotion Concepts: Principles Found in Nature

25/02/2020

Reasons:

• Muscles don’t support actuation

• Nutrient transfer across rotational joints

• Traction

• Difficulties with obstacles (roll-over)

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Nature came up with a multitude of locomotion concepts

Adaptation to environmental characteristics

Adaptation to the perceived environment (e.g. size)

Concepts found in nature

Difficult to imitate technically

Do not employ wheels

Sometimes imitate wheels (bipedal walking)

The smaller living creatures are, the more likely they fly

Most technical systems today use wheels or caterpillars

Legged locomotion is still mostly a research topic

25/02/2020Locomotion Concepts - add ons 4

Locomotion Concepts

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Locomotion

physical interaction between the vehicle and its environment.

Locomotion is concerned with interaction forces, and the

mechanisms and actuators that generate them.

The most important issues in locomotion are:


Characterization of locomotion concept

stability

number of contact points

center of gravity

static/dynamic stabilization

inclination of terrain

characteristics of contact

contact point or contact area

angle of contact

friction

type of environment

structure

medium (water, air, soft or hard

ground)

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Legged systems can

overcome many

obstacles that are

not reachable by

wheeled systems!

But it is quite hard

to achieve this since

many DOFs must be controlled in a coordinated way

the robot must see detailed elements of the terrain

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Why legged robots?


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number of actuators

structural complexity

control expense

energy efficient

terrain (flat ground, soft

ground, climbing..)

movement of the involved

masses

walking / running includes up

and down movement of COG

some extra losses


Walking or rolling?

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The fewer legs the more complicated becomes locomotion

Stability with point contact – at least three legs are required for static stability

Stability with surface contact – at least one leg is required

During walking some (usually half) of the legs are lifted

thus losing stability?

For static walking at least 4 (or 6) legs are required

Animals usually move two legs at a time

Humans require more than a year to stand and then walk on two legs.


Mobile Robots with legs (walking machines)

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A minimum of two DOF is required to move a leg forward

a lift and a swing motion.

Sliding-free motion in more than one direction not possible

Three DOF for each leg in most cases (e.g. pictured below)

4th DOF for the ankle joint

might improve walking and stability

additional joint (DOF) increases the complexity of the design and

especially of the locomotion control.


Number of Joints of Each Leg (DOF: degrees of freedom)

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The gait is characterized as the distinct sequence of lift

and release events of the individual legs

it depends on the number of legs.

the number of possible events N for a walking machine with k legs

is:

For a biped walker (k=2) the number of possible events N

is:

For a robot with 6 legs (hexapod) N is already


The number of distinct event sequences (gaits)

! 12 kN

6123! 3! 12 kN

800'916'39! 11 N

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With two legs (biped) one can have 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

A distinct event sequence can be considered as a change from one state to

another and back.

So we have the following distinct event sequences (change

of states) for a biped:


The number of distinct event sequences for biped:

6! 12 kN

Leg down

Leg up

1 -> 2 -> 1

1 -> 3 -> 1

1 -> 4 -> 1

2 -> 3 -> 2

2 -> 4 -> 2

3 -> 4 -> 3

turning

on right leg

hopping

with two legs

hopping

left leg

walking

running

hopping

right legturning

on left leg

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Changeover

Walking (trot)

Bounding


Most Obvious Natural Gaits with 4 Legs are

Dynamic

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Roland Siegwart, Margarita Chli, Nick Lawrance 25/02/2020Locomotion Concepts - add ons 13

Muybridge 1878

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Most Obvious Gait with 6 Legs is Static

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Roland Siegwart, Margarita Chli, Nick Lawrance 15

Legged Robotics

Learning from Nature

Bio-inspired motor control 2012

[Iida]


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Fundamental principles behind locomotion

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Static walking can be represented by inverted pendulum

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Static walking principles

Inverted Pendulum


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Biped walking mechanism

not too far from real rolling

rolling of a polygon with side length equal

to the length of the step

the smaller the step gets, the more the

polygon tends to a circle (wheel)

But…

rotating joint was not invented by nature

work against gravity is required

more detailed analysis follows later in this

presentation


Biped Walking

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Forward falling combined with passive leg swing

Storage of energy: potential kinetic in combination with low

friction


Case Study: Passive Dynamic Walker

C y

outu

be

mate

rial

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Exploit this in so-called passive dynamic walkers

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Inverted Pendulum

Energetically very efficient

usedE m g h hCOT

m g d m g d d


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Efficiency = cmt = |mech. energy| / (weight x dist. traveled)

Locomotion Concepts - add ons 21

Efficiency Comparison

C J. Braun, University of Edinburgh, UK

25/02/2020

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Exploit this in so-called passive dynamic walkers

Add small actuation to walk on flat ground

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Inverted Pendulum

Cornell Ranger

Total distance: 65.24 km

Total time: 30:49:02

Power: 16.0 W

COT: 0.28


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Towards Efficient Dynamic Walking: Optimizing

Gaits

C Structure and motion laboratory University of London

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Biomechanical studies suggest SLIP models to describe

complex running behaviors

Simple step-length rule to adjust the velocity


Dynamic Locomotion

Spring-Loaded Inverted Pendulum (SLIP) model

[Dickinson, Farley, Full 2000] [Geyer 2006]

, ,

1

2

FB

F HC des st R HC des HC HCT k h r r r r

Constant speed Accelerate Decelerate[Raibert 1986]

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McGuigan & Wilson, 2003 – J. Exp. Bio.

Spring loaded inverted pendulum (SLIP)

are robust against collisions

can better handle uncertainties

can temporarily store energy

reduce peak power

[Alexander 1988,1990,2002,2003]

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Dynamic locomotion

Leg Structure


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Inverted pendulum does not fully explain GRF

Use SLIP model to better explain what happens in nature

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From static to dynamic locomotion


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Early Raibert hoppers (MIT leg lab) [1983]

Pneumatic piston

Hydraulic leg “angle” orientation

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Dynamic Locomotion

SLIP principles in robotics


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Static locomotion (IP) Dynamic Locomotion (SLIP)

How to determine stability?

How to control (actively stabilize)?

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Understanding Locomotion

Summary


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Consider a simple one-dimensional dynamic system,

described by an ODE:

ሶ𝑥 = 𝑓 𝑡, 𝑥

𝑥 𝑡′ = 𝑥0 +න0

𝑡′

𝑓 𝑥, 𝑡 𝑑𝑡

Consider the set of periodic systems, where:

𝑓 𝑡 + 𝑇, 𝑥 = 𝑓 𝑡, 𝑥 ∀(𝑡, 𝑥)

A simple example:

ሶ𝑥 = 𝑓 𝑡, 𝑥

= −𝑥 + 2 cos 𝑡 (𝑇 = 2𝜋)


Periodic dynamic systems

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For our simple system:

if 𝑥0 = 1

𝑥 2𝜋 = 1 = 𝑥0

This is a stable limit cycle with a period of 2𝜋

What happens for other values of 𝑥0?



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Cyclic stability and Poincaré maps

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if 𝑥0 = 1.1

𝑥 2𝜋 = 1.000186

Not quite a stable cycle… but appears to be approaching

one

For this simple system, we can solve the ODE, and

calculate the expression for 𝑥 𝑡, 𝑥0 :

𝑥 𝑡, 𝑥0 = 𝑒−𝑡 𝑥0 − 1 + cos 𝑡 + sin(𝑡)

Then, as 𝑡 → ∞, the first term disappears, and:

𝑥 𝑡 ≈ cos 𝑡 + sin(𝑡)


Cyclic stability and Poincaré maps

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The Poincaré map is defined as the function that

maps the endpoints of successive cycles:

𝑃 𝑥0 = 𝑥 𝑇 = 𝑥1

A fixed point 𝑥∗ describes a stable cycle:

𝑃 𝑥∗ = 𝑥∗

For our example, with period T = 2𝜋:

𝑥 𝑡, 𝑥0 = 𝑒−𝑡 𝑥0 − 1 + cos 𝑡𝑃 𝑥0 = 𝑥 2𝜋, 𝑥0 = 𝑒−2𝜋 𝑥0 − 1 + 1

= 𝑒−2𝜋𝑥0 + 1 − 𝑒−2𝜋


Poincaré maps

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Poincaré maps

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We can consider the Poincaré map iteratively:

𝑥 (𝑘 + 1)𝑇 = 𝑃 𝑥 𝑘𝑇

𝑥 𝑘𝑇 = 𝑃 𝑃 …𝑃 𝑥0

Something about the evolution of the Poincaré map

determines stability…

Formally, the derivative of 𝑃(𝑥) determines stability:

𝑖𝑓 𝑃′ 𝑥0 < 1 then 𝑥 𝑡 is asymptotically stable

𝑃′ 𝑥0 > 1 then 𝑥 𝑡 is unstable


Poincaré maps

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For our example, with period 2𝜋:

𝑥 𝑡, 𝑥0 = 𝑒−𝑡 𝑥0 − 1 + cos 𝑡 + sin 𝑡

𝑃 𝑥0 = 𝑥 2𝜋, 𝑥0 = 𝑒−2𝜋 𝑥0 − 1 + 1

= (𝑒−2𝜋)𝑥0 + 1 − 𝑒−2𝜋

Thus, the Poincare map is a line with gradient:

𝑒−2𝜋 ≈ 1.87 × 10−3


Poincaré maps

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Poincaré maps

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Poincaré maps

Sta

ble

(𝑃′𝑥0

<1)

Unsta

ble

(𝑃′𝑥0

>1)

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Determining the derivatives in higher dimensions often

gets a bit tricky…

Essentially linearize 𝜕𝑃 𝑎

𝜕𝑥0around a fixed point 𝑎, and

calculate the eigenvalues of the linearized solution. If

𝜆 > 1 for any of the eigenvalues, the point 𝑎 is unstable.

Often requires numerical integration (no analytic solution)


Poincaré maps in higher dimensions

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Dynamic Locomotion

Stability Analysis

x

yq Point mass:

Equation of Motion:

flight

stance

0x

y gq

, ,spring f k lF

xspring

o

yspring

o

F

x m

y Fg

m

q

Discuss the edX

limit-cycle movie


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Stability of Locomotion

Limit Cycle Analysis

x

yq Point mass:

Equation of Motion:

flight

stance

0x

y gq

, ,spring f k lF

xspring

o

yspring

o

F

x m

y Fg

m

q


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x

yq Point mass:

System state

How does the state

propagate from one

apex to the next?

x

y

x

y

z

y

xz

analysis at

apex

kz 1kz


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Given the Poincaré mapping from one apex to the next

P includes differential equations that cannot be analytically solved!

Analyze periodic stability of a fixed point

How can we do this? Linearization…

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1k kP z z

* *Pz z

1k k k

P

z z Φ z

z

?

* *

*

2* * * 2

1 1 2...k k k k k

P PP P

x x x xz

z z z z z z z zx x

0


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Find the linearization of the Poincaré map around fix-point

Choose , with a small h

Simulate until the next apex, starting from

Calculate

Do the same thing for

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Numerical Limit Cycle Analysis

*

1

1

k k

k k

y yP

x x

x xx

1

0

k

k

k

yh

x

z

1 *

1

1

0

k

k

yP h

x

z

*

*

1

*

1

*

*

k

k

y y

P h

x x

h

x xx

0

1

k

k

k

yh

x

z


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The previous evaluation results in a 2x2 matrix

Eigenvalue analysis:

System is energy conservative:

System can be

stable

unstable:

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Numerical Limit Cycle Analysis

*

P

x xx

1 1

2 1

2 1


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Analyzing stability through limit cycles

static gaits

"System is statically stable"

Does not fall if stopped

Poincaré Map

Fix-Point

Linearization of mapping

The system is stable iff:

1k kP x x

* *Px x

1i Φ1k k k

P

x x Φ x

x

[C. David Remy, 2011]

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Week 2 overview

Locomotion Concepts - add ons

2

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Locomotion Concepts - ETH Z · Autonomous Systems Lab Autonomous Mobile Robots Roland Siegwart,...

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