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An introduction to rate-independent soft crawlers Paolo Gidoni - - PowerPoint PPT Presentation
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,
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Crawlers in Nature
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Soft robotics
Elastic materials Large deformations Compliance and morphological computation
Menciassi et al., 2006
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Noselli& DeSimone, 2014 Seok et al., 2013 Jung et al., 2007 Umedachi et al., 2013
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Application fields
Interaction with fragile
- bjects
Activity in unknown/uncertain environment Medical intervention
Bernth et al., 2017 Sanan et al., 2011
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Tolley et al., 2014
Soft robots, are also tough!
Seok et al., 2013
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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
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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
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A classical system with friction
x(t) ℓ(t)
Energy E(t, x) = k
2(ℓ(t) − x − Lrest)2
Dissipation potential R(˙ x) = µ |˙ x| Force balance: 0 ∈ ∂˙
zR(˙
z) + DzE(t, z) Play operator Sweeping process on R with C(t) = [−a, a] + b(t)
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A minimal model of crawler
L(t) k
Energy E(t, x) = k
2(x2 − x1 − Lrest − L(t))2 ≈ Ax, x − ℓ(t), x
Dissipation potential R(˙ x) = µ |˙ x1| + µ |˙ x2| Energy is invariant for translation Our system has dimension 2, our control has dimension 1.
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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)
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Three ways to asymmetry
Anisotropic friction Complex shape change Friction manipulation
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Anisotropic friction
Noselli & DeSimone, 2014
It moves and the solution is unique! Bonus question: How do slanted bristles produce anisotropy?
[G.& DeSimone, 2017]
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Stasis domains
position shape e2 e1
C ˆ Csh
position shape e2 e1
C ˆ Csh Case 2µ− = µ+ Bad case µ− = µ+
In general, for RIS, we have −DxE(t, x) ∈ C := ∂R(0) In our case we get more: −DxE(t, x) ∈ ˆ Csh
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a
- b
- c
- d
- e
- f
- g
- h
- i
- shape
net translation
−˙ u(t) ∈ N˜
C(t,u)(u)
˜ C(t, u) = C − ℓ(t) + ˆ π(u)
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Complex shape change
L1(t) L2(t) k k
- G. & DeSimone, 2016
Uniqueness fails only for µ+ = 2µ− and µ− = 2µ+.
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Three contact points, two scenarios
µ+ > 2µ− (only backwards locomotion) µ− < µ+ < 2µ− (locomotion achievable in both directions)
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Complex shape change
Bernth et al., 2017 Seok et al., 2013 Jung et al., 2007 Onal et al., 2013
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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
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t L(t) Lmax t t 1.5µ µ 0.5µ 1.5µ µ 0.5µ t µ 2µ 1 0.5 µ1(t) µ2(t) µ2(t) µ1(t) µ2(t) µ1(t)
Shape-change actuation strategy Friction-manipulation strategy A Friction-manipulation strategy B Friction-manipulation strategy C
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t = 0 t = 0.2 t = 0.4 t = 0.5 t = 0.6 t = 0.8 A
ˆ Y ∗ ˆ Z∗ ˆ Y ∗ ˆ Z∗ ˆ Y ∗ ˆ Z∗ ˆ Y ∗ ˆ Z∗ ˆ Y ∗ ˆ Z∗ ˆ Y ∗ ˆ Z∗
B
ˆ Y ∗ ˆ Z∗ ˆ Y ∗ ˆ Z∗ ˆ Y ∗ ˆ Z∗ ˆ Y ∗ ˆ Z∗ ˆ Y ∗ ˆ Z∗ ˆ Y ∗ ˆ Z∗
C
ˆ Y ∗ ˆ Z∗ ˆ Y ∗ ˆ Z∗ ˆ Y ∗ ˆ Z∗ ˆ Y ∗ ˆ Z∗ ˆ Y ∗ ˆ Z∗ ˆ Y ∗ ˆ Z∗
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µ2(t) −µ2(t) µ1(t) −µ1(t) t µ −µ 0.5Lmax t1 t1 + 0.5 t position 0.25 0.5 0.75 tension
Csh(t)
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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.)
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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)
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