An introduction to rate-independent soft crawlers Paolo Gidoni - PowerPoint PPT Presentation

An introduction to rate-independent soft crawlers Paolo Gidoni CMAF-CIO, Universidade de Lisboa, Portugal Padova, 28 September 2017

An illustrated introduction to rate-independent soft crawlers Paolo Gidoni CMAF-CIO, Universidade de Lisboa, Portugal Padova, 28 September 2017

Crawlers in Nature

Elastic materials Large deformations Soft robotics Compliance and morphological computation Menciassi et al., 2006

Noselli& DeSimone, 2014 Seok et al., 2013 Umedachi et al., 2013 Jung et al., 2007

Application fields Interaction with fragile objects Activity in unknown/uncertain environment Medical intervention Bernth et al., 2017 Sanan et al., 2011

Soft robots, are also tough! Seok et al., 2013 Tolley et al., 2014

Why rate-independent systems (or SP)? → Dry friction → Elasticity → No inertial effects Why crawlers? → Simple enough for analytical approach → Complex enough to be meaningful → Simplexity

A classical system with friction x ( t ) ℓ ( t ) Dry friction on the contact point z ( t ) Force balance on the point ℓ ( t ) Neglect inertia

A classical system with friction x ( t ) ℓ ( t ) Energy E ( t , x ) = k 2 ( ℓ ( t ) − x − L rest ) 2 Dissipation potential R (˙ x ) = µ | ˙ x | Force balance: 0 ∈ ∂ ˙ z R (˙ z ) + D z E ( t , z ) Play operator Sweeping process on R with C ( t ) = [ − a , a ] + b ( t )

A minimal model of crawler L ( t ) k 2 ( x 2 − x 1 − L rest − L ( t )) 2 ≈ � A x , x � − � ℓ ( t ) , x � Energy E ( t , x ) = k Dissipation potential R (˙ x ) = µ | ˙ x 1 | + µ | ˙ x 2 | Energy is invariant for translation Our system has dimension 2, our control has dimension 1.

A minimal model of crawler? L ( t ) k Multiple solutions It is symmetric, so we do not expect it to go anywhere BAD EXAMPLE! What are we missing? (Don’t worry, it is a pathological example)

Three ways to asymmetry Anisotropic friction Complex shape change Friction manipulation

Anisotropic friction Noselli & DeSimone, 2014 It moves and the solution is unique! Bonus question: How do slanted bristles produce anisotropy? [G.& DeSimone, 2017]

Stasis domains shape shape e 2 e 2 C C ˆ ˆ C sh C sh position position e 1 e 1 Case 2 µ − = µ + Bad case µ − = µ + In general, for RIS, we have − D x E ( t , x ) ∈ C := ∂ R ( 0 ) In our case we get more: − D x E ( t , x ) ∈ ˆ C sh

shape � d � e � i � c � f � h � b � a � g net translation ˜ − ˙ u ( t ) ∈ N ˜ C ( t , u ) = C − ℓ ( t ) + ˆ C ( t , u ) ( u ) π ( u )

Complex shape change L 1 ( t ) L 2 ( t ) k k G. & DeSimone, 2016 Uniqueness fails only for µ + = 2 µ − and µ − = 2 µ + .

Three contact points, two scenarios µ − < µ + < 2 µ − µ + > 2 µ − (locomotion achievable in (only backwards locomotion) both directions)

Complex shape change Seok et al., 2013 Bernth et al., 2017 Onal et al., 2013 Jung et al., 2007

Friction manipulation Umedachi et al., 2013, Vikas et al. 2016 We control friction coefficients The dissipation potential R depends on time An extreme example is two-anchor crawling

L max Shape-change actuation strategy L ( t ) 0 t 1 . 5 µ µ 2 ( t ) Friction-manipulation µ strategy A µ 1 ( t ) 0 . 5 µ t 1 . 5 µ µ 2 ( t ) Friction-manipulation µ strategy B µ 1 ( t ) 0 . 5 µ t 2 µ µ 2 ( t ) Friction-manipulation strategy C µ 1 ( t ) µ t 0 . 5 1

t = 0 t = 0 . 2 t = 0 . 4 t = 0 . 5 t = 0 . 6 t = 0 . 8 Z ∗ ˆ Z ∗ ˆ Z ∗ ˆ Z ∗ ˆ Z ∗ ˆ Z ∗ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Y ∗ Y ∗ Y ∗ Y ∗ Y ∗ Y ∗ A Z ∗ ˆ Z ∗ ˆ Z ∗ ˆ Z ∗ ˆ Z ∗ ˆ Z ∗ ˆ Y ∗ ˆ Y ∗ ˆ Y ∗ ˆ Y ∗ ˆ Y ∗ ˆ Y ∗ ˆ B ˆ ˆ ˆ ˆ ˆ ˆ Z ∗ Z ∗ Z ∗ Z ∗ Z ∗ Z ∗ Y ∗ ˆ Y ∗ ˆ Y ∗ ˆ Y ∗ ˆ Y ∗ ˆ ˆ Y ∗ C

µ 1 ( t ) µ 2 ( t ) µ C sh ( t ) tension − µ − µ 1 ( t ) − µ 2 ( t ) t position 0 . 5 L max 0 t t 1 0 . 25 0 . 5 t 1 + 0 . 5 0 . 75

What we know Well-posedness of the approach (existence and uniqueness) → Coercivity and uniqueness of minimum in the “shape-sections” of R → Lipschitz continuity in time → Bounds and coercivity conditions uniform in time on R Abstract theorems on Hilbert spaces → Discrete and continuous crawlers → Planar crawlers (with modelling issues) Motility analysis → Does the crawler move? Can it move in both directions? → Common sense optimization (DeSimone, G. & Noselli, 2015; G. & DeSimone, 2016; G., prep. )

Some open problems Optimal control → Control on actuation or friction → We want to move (fast) → Constraints on control parameters Compliance → Everything Dynamical properties Problems with variational taste → State-dependent dissipation (Anisotropic friction for planar crawlers, obstacles) → Rate-dependent dissipation (mucus of the snail, biological fluids)

Thank you for your attention � pgidoni@fc.ul.pt