quang c. yoshih:ura · —g(x + y) f z + — + ixfy — pty , ai (c, y)v(c, y)dcd v(c, y)dcdy cc,y...

Stéphane Caron University of Tokyo Yoshihiko Nakamura University of Tokyo Quang‐Cuong Pham Nanyang Technological University Motivation Reduction to a discrete vector field Back to Questions Simulation Experiment Wrench Cone for Rectangular Surfaces Coulomb Friction G2D example) Positional constraint: Complementarity between accelerations and forces: contact modes Fixed: Sliding: Poster design by Felix 9reuer — License) Creative Commons 8ttributionDShare8like qGw Unported f W f E f M f q f W f E f M f q f M f W f E f f q 6 6 6 τ f τ Limbed robots need to use contacts to locomote. Questions Will a contact hold when executing a motion? How to make the best use of one contact? How to change stance Gset of contacts)? Combining Contacts at the Gravito‐Inertial Wrench Definition of the GI wrench) We can show from the Equation of Motion that) Cone Duality ww)ww ww)w( ww)WE ww)Ew ww)Ew ww)EW ww)qq ww)qW ww)Ez ww)Eq ww)EE ww)EW Stepping on a box using a E5 o tilted plane and a W m ledge No solution using staticallyDstable configurations Found a dynamic solution with TimeDOptimal Path ParamG contribution Local: a contact holds by CWC Whole‐body: all contacts hold by GIWC How to change stance Gset of contacts)? Autonomous discovery of locomotion trajectories? contribution a link in surface contact Continuous stressDpressure field Resultant wrench) Equivalent generator using barycentring coordinates) Input constraints) Reduction of the polyedral convex cone pFourierDMotzkin1 yields three conditions) WG Coulomb friction on the resultant force EG CoP inside the support area qG 8 new condition on the resultant yaw torque [to appear at ICR8 EwW5] a continuum of stress and pressure fields an equivalent and discrete force field minimum dimension) one wrench wholeDbody multiDcontact planning [Hirukawa et alG? ICR8 EwwO] [Ramos et alG? Humanoids EwWE] [Saad et alG? TRO EwWq] [Escande et alG? R8S EwWq] [Hauser? IJRR EwWM] [8udren et alG? IROS EwWM]

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Page 1: Quang C. Yoshih:ura · —g(X + Y) f z + — + IXfY — PTY , ai (c, y)v(c, y)dcd v(c, y)dcdy cc,y A v(c, y)dcdy Multi-Contact Motion Planning for Humanoids with Cone Duality

Stéphane.CaronUniversity.of.Tokyo

Yoshihiko:NakamuraUniversity:of:Tokyo

Quang‐Cuong.PhamNanyang.Technological.University

Motivation

Reduction(to(a(discrete(vector(field

Back(to(Questions

Simulation(Experiment

Wrench(Cone((((for(Rectangular(Surfaces

Coulomb(Friction(G2D(exampleI

Positional(constraint:

Complementarity(between(accelerationsand(forces:(contact(modes

Fixed:

Sliding:

Poster:design:by:Felix:9reuer:—:License):Creative:Commons:8ttributionDShare8like:qGw:Unported

fWfE

fWfE fM

fMfW

6τf

Limbed.robots.need.to.use.contacts.to.locomote.

Questions

Will.a.contact.hold.when.executing.a.motion?

How.to.make.the.best.use.of.one.contact?

How.to.change.stance.Gset.of.contactsI?

Combining(Contacts(atthe(Gravito‐Inertial(Wrench

Definition:of:the:GI:wrench)

We:can:show:from:the:Equation:of:Motion:that)

Cone:Duality

ww)ww ww)w( ww)WE ww)Ew ww)Ew ww)EW

ww)qqww)qW:ww)Ezww)Eqww)EEww)EW

Stepping:on:a:box:using:a:E5o:tilted:plane:and:a:W:m:ledge

No:solution:using:staticallyDstable:configurations

Found:a:dynamic:solution:with:TimeDOptimal:Path:ParamG

contrib

ution

Local:.a.contact.holds.by.CWC

Whole‐body:.all.contacts.hold.by.GIWC

How.to.change.stance.Gset.of.contactsI?

Autonomous.discovery.of.locomotion.trajectories?

contribution

a:linkin:surface

contact

Continuous:stressDpressure:field:

Resultant:wrench)

Equivalent:generator:using:barycentring:coordinates)

Input:constraints)

Reduction:of:the:polyedral:convex:cone:pFourierDMotzkin1:yields:three:conditions)

WG:Coulomb:frictionon:the:resultant:force

EG:CoP:inside:the:support:area

qG:8:new:conditionon:the:resultant:yaw:torque

[to:appear:at:ICR8:EwW5]

a:continuum:of:stressand:pressure:fields

an:equivalentand:discrete

force:field

minimumdimension):one

wrench

wholeDbodymultiDcontact

planning

[Hirukawa:et:alG?:ICR8:EwwO][Ramos:et:alG?:Humanoids:EwWE][Saad:et:alG?:TRO:EwWq][Escande:et:alG?:R8S:EwWq][Hauser?:IJRR:EwWM][8udren:et:alG?:IROS:EwWM]

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Y v C RAO-Engineering Thermodynamics Through Examples Incomplete

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arXiv:2008.04346v1 [cond-mat.str-el] 10 Aug 2020 · =v;c L ;kˆ sL y ;k 1 2 fLy ;k L ;k;ˆ sg : (6) where the dissipator operators are L v;k = L y c;k = c y c;k c v;k. Associated

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