BIRG (EPFL B) Crawling and Drumming - iCubwiki.icub.org/images/0/0f/Presentation_crawling.pdfThree Monthly Meeting - Lausanne - April 2008 BIRG (EPFL B) Crawling and Drumming Ludovic

Three Monthly Meeting - Lausanne - April 2008

BIRG (EPFL B)Crawling and Drumming

Ludovic Righetti, Sarah Dégallier and Auke J. [email protected], [email protected], [email protected]

Biologically Inspired Robotics Group (BIRG)Ecole Polytechnique Fédérale de Lausanne (EPFL)

– p.1

Outline of the presentation

General low-level control architecture based on MotorPrograms (MP) and Pattern Generators (PG)

Application to drumming iCub

Locomotion specific design of pattern generators

Sensory feedback for crawling

– p.2

Pattern Generators in robotics

Central PatternGenerator

Sensory feedbackM

ulti-

dim

ensi

onal

Osc

illat

ory

Out

puts

Sim

ple

Toni

cIn

puts

– p.3


Pattern Generators of movement primitives offer an interestingsolution for robotic control⇒ reduction of the dimensionality of the control problem

– p.4



Stable attractor dynamics - Fixed point or Limit cycle behavior(stability against perturbations)

– p.4




Smooth integration of sensory feedback⇒ online trajectory generation

– p.4





Smooth modulation of the movement (speed, amplitude,position)

– p.4






Low-level feedback integration (with other CPGs, with theenvironment and/or with the body dynamics)

– p.4






Low-level feedback integration (with other CPGs, with theenvironment and/or with the body dynamics)

They are modeled with dynamical systems

– p.4

Pattern Generator Based Architecture

– p.5

General control architecture

We propose a three layered control architectureinspired by a functional model of the motor system

At the Generator level there are several PatternGenerators (PGs) that generate complex coordinatedmovements

Activation and coordination of these PGs is made atthe Manager level, using Motor Programs (MPs)made of simple parameter sets

This architecture is compatible with the iCub cogni-

tive architecture, the planner corresponds to prospec-

tion by action simulation, the manager to the modula-

tion circuit and the generator to the phylogenetic self-

organizing perceptuo-motor skills

Perip

heral In

form

ation

Coordinator

PlannerHigher order cognitive functions

Manager

Generator

Referen

tials(IK

, task space...)

iCub

Discrete T

imin

g

MPMP

PGPGclock

– p.6

Discrete Pattern Generator

Critically damped differential equation to generate a discrete trajectory

Bell-shaped velocity profile

Speed of convergence is controlled by b, point of convergence is controlled by gi. These are theparameters controlled by the Motor Programs

Valid for any initial positions since the ODE has a fixed point attractor

ḣi = c(d − hi)

ẏi = h4

i vi

v̇i = d4−b2

4(yi − gi) − b vi.

0 0.2 0.4 0.6 0.8 1 1.2 1.40

1

2

3

4

5

6

TrajectoryVelocity Profile

– p.7

Rhythmic Pattern Generator

Rhythmic movement primitive (when mi > 0) that can be turned off via Hopf bifurcation (mi < 0)

Independent control of the ascending and descending phases of the oscillation

The parameters controlled by the MPs are the frequencies ωdown, ωup, and amplitude mi ofoscillations

Valid for any initial positions since the ODE has a stable limit cycle

ẋi = a`

mi − r2

i

´

xi − ωzi

żi = a`

mi − r2

i

´

zi + ωxi

ωi =ωdown

e−bzi + 1+

ωup

ebzi + 1

where ri =q

x2i + z2

i

0 5 10 15 20 25 30

−1

−0.5

0

0.5

1

Time

X

– p.8

Superimposition of movements

We superpose the previous PGs by injecting the discretemovement into the rhythmic one

ḣi = c(d − hi)

ẏi = h4i vi

v̇i = d4−b

2

4(yi − gi) − b vi.

ẋi = a(

mi − r2i

)

(xi − yi) − ωzi

żi = a(

mi − r2i

)

zi + ω (xi − yi)

ωi =ωdown

e−bzi + 1+

ωup

ebzi + 1

Dis

cret

eP

Gs

Rhy

thm

icP

Gs

– p.9

Example of trajectories (drumming)

20 40 60 80 100−100

−50

0

Time [s]

Pos

ition

[Deg

]

Left Shoulder Flexion/Extension

20 40 60 80 100−1

0

1

2

Time [s]

Mot

or P

rogr

am

Rhythmic On/OffAnchor positionFrequency

– p.10

Applications to crawling and drumming

[Dégallier, Santos, Righetti and Ijspeert, Humanoids 2006]

[Dégallier, Righetti and Ijspeert, IROS 2007]

– p.11

– p.12

Pattern Generators in the iCub

Simple Motor Programs can generate complex and coordinatedmovements

Implemented in Yarp modules that can easily be instantiated foreach part of the body

Could be used easily from any higher-level module

Currently running on the PC104 of the iCub but could be movedto the DSPs (low computational cost)

– p.13

Design of locomotion specific CPGs

– p.14

Context of locomotion

We need locomotion specific pattern generators and feedbackintegration

These CPGs should be general enough to be applied to verydifferent legged robots

Generation of different coordinated gaits

Constant swing duration and stance duration related to speedof locomotion (as for all quadruped mammals including infants)

Integration of sensory feedback compatible with observationsfrom mammals

Compatibility with the previous architecture

– p.15

Architecture of CPGs

We present a generic way to construct networks of coupleddynamical systems able to generate any desired pattern ofoscillation (i.e. gait)

The design of the network is independent of the internaldynamics of each cell

We use the theory of symmetric coupled cells networkdeveloped by Golubitsky et al.

The analytic problem of finding ODEs such that the desiredpattern of oscillations exists is transformed into an algebraicone (use of group theory)

[Righetti and Ijspeert, Robotics Science and Systems 2006]

– p.16

Symmetric coupled cells network

Principle: design a network of dynamical systems that has thesame symmetry group as the desired gait

A symmetry in a coupled cells network is a permutation of thecells that preserves the architecture of the network(independent of the underlying ODEs)

We distinguish 2 kinds of symmetries in sets of ODEsSpatial symmetries γ are such that for any solution x(t) wehave γx(t) = x(t)Spatio-temporal symmetries ϕ are such that the orbits of x(t)and ϕx(t) are the same (if x(t) is a periodic solution thenϕx(t) = x(t+ ψ))

– p.17

Example of symmetries

1 2

34

Right Arm

Right LegLeft Leg

Left Arm

Trot-like gaitDiagonal limbs are in phase

Left and right limbs are half a period out of phase

– p.18


1 2

34

Right Arm

Right LegLeft Leg

Left Arm

The trot gait has the following symmetries{(13)(24), 0}

{(12)(34), 12}

{(14)(23), 12}

– p.18


1 2

34

Right Arm

Right LegLeft Leg

Left Arm

This coupling architecture has the same symmetry groupThere exists periodic solutions with the trot symmetry

– p.18


1 2

34

Right Arm

Right LegLeft Leg

Left Arm

We find the other solutions of the network by calculating thesubgroups of the symmetry group

– p.18

Generic CPGs

Trot/Bound/Pace Network Walk Network

1Left

forelimb

2Right

forelimb

3Left

hindlimb

4Right

hindlimb

1Left

forelimb

2Right

forelimb

3Left

hindlimb

4Right

hindlimb

Generic design of CPGs able to support the most common symmetrical

gaits (walk, trot, bound and pace)

The design is independent of the internal dynamics of each cell

– p.19

Generic CPGs

Trot/Bound/Pace Network Walk Network

1Left

forelimb

2Right

forelimb

3Left

hindlimb

4Right

hindlimb

1Left

forelimb

2Right

forelimb

3Left

hindlimb

4Right

hindlimb

Generic design of CPGs able to support the most common symmetrical

gaits (walk, trot, bound and pace)

The design is independent of the internal dynamics of each cell

⇒ We can now design the dynamics of the cells

– p.19

Dynamics of the cells

During locomotion, swing duration is constant, stance durationcontrols speed of locomotion

However there is no oscillator in which we can independentlycontrol the duration of the ascending and descending phases ofthe oscillations (swing and stance)

– p.20




ẋi = α(µ − r2i )xi − ωyi

ẏi = β(µ − r2i )yi + ωxi

– p.20





ẏi = β(µ − r2i )yi + ωxi

ω =ωstance

e−by + 1+

ωswing

eby + 1

– p.20





ẏi = β(µ − r2i )yi + ωxi +

∑

kijyj

ω =ωstance

e−by + 1+

ωswing

eby + 1

– p.20


0 5 10 15 20 25 30

−1

−0.5

0

0.5

1

Time

X

By changing ωswing and ωstance we can independently control the duration of the ascending and

descending phases of the oscillation (i.e. we control the duration of swing and stance phases)

– p.21

Gait generation

0 0.5 1 1.5

X4

X3

X2

X1

Time

Trot

0 0.5 1 1.5

X4

X3

X2

X1

Time

Bound

0 0.5 1 1.5

X4

X3

X2

X1

Time

Pace

0 0.5 1 1.5

X4

X3

X2

X1

Time

Walk

– p.22

Sensory feedback

The integration of sensory information at the CPG level isphase dependent during animal locomotion

Transitions between swing and stance phases are critical

Sensory feedback strongly couples the neural controller withthe system it controls

We design phase dependent sensory feedback in such a waythat it explicitly shapes the dynamics of the oscillators“the CPG is controlled by the mechanics”

[Righetti and Ijspeert, ICRA 2008]

– p.23

Sensory feedback integration

ui = 0 ui = ±F ui = −ωxi −P

kijyj

ẋi = α(µ − r2

i )xi − ωyi

ẏi = β(µ − r2

i )yi + ωxi +X

kijyj + ui

ω =ωstance

e−by + 1+

ωswing

eby + 1

– p.24

Sensory feedback integration

kstance

k sw

ing

0.1 0.2 0.3 0.4 0.5 0.6 0.70.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

kstance

k sw

ing

0.1 0.2 0.3 0.4 0.5 0.6 0.70.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 10 20 30 40 50 600

20

40

Time

Tou

ch fo

rce

[N]

0 10 20 30 40 50 60−1

0

1

Time

X

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

X

Y

Tests where performed with 3 differentsimulated robots, from rigid ones to oneswith passive dynamics

In all simulations, speed is independent ofswing duration and correlated to stanceduration as in mammals!

With the feedback, locomotion is morerobust to parameter uncertainty

The robots can locomote on uneven terrainsand slopes

The controller is tightly coupled to the robotand its environment (mutual entrainment)

The same controller works for differentkinds of robots and gaits (e.g. walk and trotwith the Aibo, walk and bound with a robotwith passive dynamics)

Speed no feedback Speed with feedback

No Feedback Feedback Enabled

– p.25

Integration in the general architecture

Generation of different behaviors using the same PatternGenerators with different Motor Programs

Crawling

Reaching while crawling

Crawling then reaching

– p.26

Conclusion

We developed a modular architecture based on PatternGenerators and Motor Programs

Any higher-level cognitive function could use this framework bysimple parameter specifications to the Pattern Generators(goal, speed, frequency...)

Complete integration as modules in Yarp, successfully testedon the iCub for drumming

Locomotion specific sensory feedback was successfullyintegrated

– p.27

large Outline of the presentationlarge Pattern Generators in roboticslarge Pattern Generators in roboticslarge General control architecturelarge Discrete Pattern Generatorlarge Rhythmic Pattern Generatorlarge Superimposition of movementslarge Example of trajectories (drumming)large Applications to crawling and drumminglarge Pattern Generators in the iCublarge Context of locomotionlarge Architecture of CPGslarge Symmetric coupled cells networklarge Example of symmetrieslarge Generic CPGslarge Dynamics of the cellslarge Dynamics of the cellslarge Gait generationlarge Sensory feedbacklarge Sensory feedback integrationlarge Sensory feedback integrationlarge Integration in the general architecturelarge Conclusion

Documents

BIRG (EPFL B) Crawling and Drumming - iCubwiki.icub.org/images/0/0f/Presentation_crawling.pdfThree Monthly Meeting - Lausanne - April 2008 BIRG (EPFL B) Crawling and Drumming Ludovic