Modeling of a soft link continuum formulation

Transcription

1 Modeling of a soft link continuum formulation Stanislao Grazioso Thursday 12 th April, 2018 Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

2 Introduction Soft arm: 2D cross-sections moving upon a 3D reference curve. Kinematic assumption: the cross-sections behave as rigid bodies and, as a consequence, remain flat. The configuration of a soft robotic arm is the result of the infinite rigid body transformations of the cross-sections. α b n t α 1 : coordinate along the reference curve [if not indicated: α 1 = α] α 2, α 3 : coordinates along the cross-sections axes Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

3 Kinematics Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

4 Geometry of the reference curve initial configuration n 0 t 0 b 0 R 0 p current configuration t b n Rp u 0 p u 0 e 3 p u u p α e 1 e 2 α Local triad u0 (α) t 0 (α) = u 0 (α) n 0 (α) = 1 u 0 (α) κ 0 (α) t0 (α) = b 0 (α) = t 0 (α)n 0 (α) 1 u 0 (α) τ 0 (α) b0 (α) Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

5 Geometry of the reference curve (cont d) initial configuration n 0 t 0 b 0 R 0 p current configuration t b n Rp u 0 p u 0 e 3 p u u p α e 1 e 2 α Curvature and torsion of the curve κ 0 (α) = t0 (α) u 0 (α) τ 0 (α) = b0 (α) u 0 (α) Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

6 Geometry of the reference curve (cont d) initial configuration n 0 t 0 b 0 R 0 p current configuration t b n Rp u 0 p u 0 e 3 p u u p α e 1 e 2 α Frenet Serret formulas R 0 (α) = R 0 (α) f 0 ω(α) where f 0 ω(α) so(3) and the associated axial vector f 0 ω(α) reads f 0 ω(α) = u 0 (α) [τ 0 (α) 0 κ 0 (α)] T Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

7 Position field initial configuration n 0 t 0 b 0 R 0 p current configuration t b n Rp u 0 p u 0 e 3 p u u p α e 1 e 2 α Kinematic configuration of the soft link α R H(α) = H(R(α), u(α)) SE(3) [ ] [ ] up (α 1, α 2, α 3 ) p(α2, α = H(α 1 1 ) 3 ) 1 Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

8 Statics Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

9 Deformation field [ ] [ ] u p (α 1, α 2, α 3 ) = H p(α2, α (α 1 1 ) 3 ) 1 ɛ(α) = f 0 (α) = H (α 1 ) = H(α 1 ) f(α 1 ) f(α) se(3) = deformation twist [ ] [ ] f 0 u (α) e fω(α) 0 = u 0 (α) 1 τ 0 (α)e 1 + κ 0 (α)e 3 Since f(α) = f 0 (α) + ɛ(α), the strain vector is : [ ] [ ] fu (α) fu(α) 0 R f ω (α) fω(α) 0 = T (α)u (α) u 0 (α) e 1 R T (α)r(α) u 0 (α) (τ 0 (α)e 1 + κ 0 (α)e 3 ) Geometrically exact formulation = [ ] γ(α) κ(α) Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

10 Green Lagrange strain tensor [ ] u p (α 1, α 2, α 3 ) = 1 [ R (α) u (α) ] [ p(α2, α 3 ) 1 ] N(α 2, α 3 ) = [I 3 3 p(α 2, α 3 )] Deformation gradient u p (α 1, α 2, α 3 ) α 1 = R(α 1 )N(α 2, α 3 )f(α 1 ) u p (α 1, α 2, α 3 ) α 2 = R(α 1 )e 2 u p (α 1, α 2, α 3 ) α 3 = R(α 1 )e 3 G(α 1, α 2, α 3 ) = R(α 1 ) [N(α 2, α 3 )f(α 1 ) e 2 e 3 ] Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

11 Green Lagrange strain tensor (cont d) Green Lagrange strain tensor GL ij (α 1, α 2, α 3 ) = 1 2 The only non-vanishing terms are: ( u T p u p u0t p α i α j α i u 0 p α j GL 11 = f 0T N T Nɛ ɛt N T Nɛ GL 12 = GL 21 = 1 2 et 2 N T ɛ GL 13 = GL 31 = 1 2 et 3 N T ɛ ) Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

12 Green Lagrange strain tensor (cont d) gl(α 1, α 2, α 3 ) = [GL 11 2GL 12 2GL 13 ] T = D(α 1, α 2, α 3 )ɛ(α 1 ) f 0T N T N D(α 1, α 2, α 3 ) = e T 2 NT e T 3 NT Deformation measures from the left invariant vector field on SE(3)! Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

13 Strain energy, stress vector and stiffness matrix Strain energy S(α 1, α 2, α 3 ) G(α 1, α 2, α 3 ) V int = 1 S ij GL ij dv 2 second Piola Kirchhoff stress tensor Green Lagrange strain tensor V Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

14 Strain energy, stress vector and stiffness matrix (cont d) Strain energy V int = 1 s T gl dv 2 V s = [S 11 S 12 S 13 ] T gl = [GL 11 2GL 12 2GL 13 ] T = Dɛ Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

15 Strain energy, stress vector and stiffness matrix (cont d) Strain energy V int = 1 ɛ T (α 1 )σ(α 1 ) dα 1 = 1 (γ T (α 1 )n(α 1 ) + κ T (α 1 )m(α 1 )) dα 1 2 L 2 L [ ] n(α1 ) σ(α 1 ) = = m(α 1 ) A DT s da stress resultants over the cross-sections Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

16 Strain energy, stress vector and stiffness matrix (cont d) Constitutive equations S ij = C ijkl GL kl C ijkl = λg 0,ij G 0,kl + µ(g 0,ik G 0,jl + G 0,il G 0,jk ) Elasticity tensor λ, µ: Lame s coefficients 1 α 3 u 0 τ 0 α 2 u 0 τ 0 G 0 (α 1, α 2, α 3 ) = 1 d. d + (α 3 u 0 τ 0 ) 2 α 2 α 3 ( u 0 τ 0 ) 2 SYM. d + (α 2 u 0 τ 0 ) 2 controvariant metric of the initial configuration d = u 0 2 (1 α 2 κ 0 ) 2 Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

17 Strain energy, stress vector and stiffness matrix (cont d) Constitutive equations s(α 1, α 2, α 3 ) = C(α 1, α 2, α 3 ) gl(α 1, α 2, α 3 ) C µα 3 u 0 τ 0 2µα 2 u 0 τ 0 C(α 1, α 2, α 3 ) = u 0 2. C dd 22 µα 2 α 3 ( u 0 τ 0 ) 2 SYM. C 33 C 11 = 1 D ((λ + 2µ)(1 α 2κ 0 ) 2 + 4µ(α α 2 3)(τ 0 ) 2 ) C 22 = µ u 0 2 ((1 α 2 κ 0 ) 2 + (α 3 τ 0 ) 2 ) C 33 = µ u 0 2 ((1 α 2 κ 0 ) 2 + (α 2 τ 0 ) 2 ) D = (1 α 2 κ 0 ) 2 + (α2 2 + α3)(τ 2 0 ) 2 λ = λµ/(λ + µ) Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

18 Strain energy, stress vector and stiffness matrix (cont d) Constitutive equations σ(α 1 ) = K(α 1 ) ɛ(α 1 ) Stiffness matrix K(α 1 ) = A D T (α 2, α 3 )C(α 1, α 2, α 3 )D(α 2, α 3 ) da K(α 1 ) = [ ] Kuu K uω SYM K ωω Initially straight beam + reference curve neutral axis of the beam (i.e. n 0 and b 0 are chosen to be the principal axes of the cross-sections) K(α 1 ) is diagonal: K uu = diag(ea, GA 2, GA 3 ) K ωω = diag(gj, EI 2, EI 3 ) Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

19 Strain energy, stress vector and stiffness matrix (cont d) Strain energy V int = 1 ɛ T Kɛ dα 1 2 L Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

20 Example: stiffness matrix of a curved beam e 2 u 0 (α 1) α α 1 e 1 e 3 r Beam with a cantilever arc as reference curve (of length L = αr) Squared cross section, side b Homogeneous elastic material (properties λ, µ) Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

21 Example: stiffness matrix of a curved beam (cont d) e 2 u 0 (α 1) α α 1 e 1 e 3 r Position vector and derivative u 0 (α 1 ) = r cos ( α 1 ) r sin ( α 1 ) r u 0 (α 1 ) = sin ( α 1 ) r cos ( α 1 ) r 0 0 Frenet triad sin ( α 1 ) t 0 r (α 1 ) = cos ( α 1 ) cos ( α 1 ) r n 0 r (α 1 ) = sin ( α 1 ) 0 r b 0 = κ 0 = 1/r; τ 0 = 0 Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

22 Example: stiffness matrix of a curved beam (cont d) e 2 u 0 (α 1) α α 1 e 1 e 3 r Initial deformation vector f 0 = [ e1 1 r e 3 ] Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

23 Example: stiffness matrix of a curved beam (cont d) e 2 u 0 (α 1) α α 1 e 1 e 3 r Stiffness matrix K = b/2 b/2 b/2 b/2 D T CD dα 2 dα 3 = [ ] Kuu K uω SYM K ωω Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

24 Example: stiffness matrix of a curved beam (cont d) e 2 u 0 (α 1) α α 1 e 1 e 3 r K uu = c 1 diag ( (λ + 2µ)b 2, µb 2, µb 2) ( ) c1 + c 2 K ωω = diag µ b4 2 6, c 1(λ + 2µ) b4 12, c 2(λ + 2µ) b4 12 K uω = 0 0 c 3(λ + 2µ)b c 3 µb Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

25 Example: stiffness matrix of a curved beam (cont d) e 2 u 0 (α 1) α α 1 e 1 e 3 r c 1 = 4 4 (κ 0 b) 2 c 2 = 12 ( (κ 0 b) 3 2ln( 2 ) κ0 b 2 + κ 0 b ) + (1 + c 1)κ 0 b c 3 = 1 ( (κ 0 b) 2 ln( 2 ) κ0 b 2 + κ 0 b ) + c 1κ 0 b Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

26 Static equilibrium equations Principle of Virtual Work δ(v int ) = δ(v ext ) δ(ɛ) = δ(f) = (δh) + fδh δ(v int ) = δ(ɛ) T σ dα = L = [ δh T σ ] L 0 δh T (σ ˆf T σ ) dα L δ(v ext ) = +δh(0) T g ext (0) δh(l) T g ext (L) δh T g ext (α) dα g ext (α) = [ g T ext,u gext,ω T ] L Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

27 Static equilibrium equations (cont d) Static equilibrium equations [ weak form δh T (σ g ext ) ] L 0 L δht (σ ˆf T σ g ext ) dα = 0 strong form σ ˆf T σ = g ext Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

28 Equations of motion (static case) kinematic equations constitutive equation H = H(f 0 + ɛ) σ = Kɛ boundary conditions δh(l) (K(L)ɛ(L) g ext(l)) δh(0) ( K(0)ɛ(0) g ext(0) ) = 0 static equilibrium equations σ ˆf T σ = g ext Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

29 Exercise 1: planar bending of a cantilever beam (integrable analytic solution) Derive the strain field and SE(3) field of a cantilever beam subject to a torque τ at its free end. τ y x z Solution: Exercise1.pdf Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

30 Dynamics Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

31 Velocity field Ḣ(α) = H(α) η(α) [ ] v(α) η(α) = ω(α) Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

32 Kinetic energy ρ density A cross-section area J I J II K = 1 η T Mη dα 2 L [ ] ρai3 3 J M = T I J I J II first moment of inertia of the cross section (computed in the local axes of the arm) second moment of inertia of the cross section (computed in the local axes of the arm) Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

33 Dynamic equilibrium equations Hamilton s principle t1 t 0 (δ(k) δ(v int ) + δ(v ext )) dt = 0. δ(v int ) = δ(ɛ) T σ dα = L = [ δh T σ ] L 0 δh T (σ ˆf T σ ) dα L δ(v ext ) = +δh(0) T g ext (0) δh(l) T g ext (L) δh T g ext (α) dα L Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

34 Dynamic equilibrium equations (cont d) Hamilton s principle t1 t1 t 0 (δ(k) δ(v int ) + δ(v ext )) dt = 0. δ(η) = (δh) + ηδh t1 t 0 L [ Mη ] t1 dαdt = t1 δ(k) dt = δ(η) T = δh T Mηdα δh T (M η ˆη T Mη) dα dt L L t 0 t 0 Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

35 Dynamic equilibrium equations (cont d) Dynamic equilibrium equations weak form strong form [ δh T (σ g ext) ] L 0 L δht ( M η + ˆη T Mη + σ ˆf T σ + g ext) dα = 0 M η ˆη T Mη σ + f T σ = g ext Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

36 Equations of motion (dynamic case) Kinematic equations Material constitutive law Compatibility equations Ḣ = H η H = H(f 0 + ɛ) σ = Kɛ η ɛ = ηf Boundary conditions δh(l) (K(L)ɛ(L) g ext (L)) = δh(0) (K(0)ɛ(0) g ext (0)) Dynamic equilibrium equations M η η T Mη σ + f T σ = g ext Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

37 Exercise 2: planar rotation of a beam (integrable analytic solution) Derive the strain field, velocity field and SE(3) field of a beam rotating at a constant velocity ω in its own plane. A x ω B z Solution: Exercise2.pdf Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38

38 Modeling of a soft link continuum formulation Stanislao Grazioso Geometric Theory of Soft Robots Thursday 12 th April, / 38