1/13

Nonlinear parametric equationsHigher-order continuation methods

Applications

A full higher-order AD continuation and bifurcationframework

I. Charpentiera and B. Cochelinb

aLaboratoire des sciences de l’ingénieur, de l’informatique et de l’imagerieUMR 7357, CNRS&University of Strasbourg, France

bLaboratoire de Mécanique et AcoustiqueUPR 7051 CNRS, Marseille, France

15/09/2016

I. Charpentier and B. Cochelin A full higher-order AD continuation and bifurcation framework

2/13

Nonlinear parametric equationsHigher-order continuation methods

Applications

Goals of the talk

Recall the higher-order continuation framework

Discuss bifurcation detection and branch switching

Present robotic issues

Outline

1 Nonlinear parametric equationsAn example in roboticsResidual and workspace equationsWorkspace boundary equations

2 Higher-order continuation methodsContinuationBifurcation analysisBifurcation and geometric series

3 ApplicationsElasticaRobotic planar armsClassification of singularities

I. Charpentier and B. Cochelin A full higher-order AD continuation and bifurcation framework

3/13

Nonlinear parametric equationsHigher-order continuation methods

Applications

An example in roboticsResidual and workspace equationsWorkspace boundary equations

An example in robotics

Case study in robotics RR’s Forward kinematics{x = l1 cos(θ1) + l2 cos(θ1 + θ2)y = l1 sin(θ1) + l2 sin(θ1 + θ2)

inputs: v = (θ1, θ2) ∈ [−π/3, π/3]outputs: u = (x, y) ∈ W

Workspace

W ={

u | ∃ v s.t. RW(u, v) = 0}

Goal: Determine workspace boundaries

I. Charpentier and B. Cochelin A full higher-order AD continuation and bifurcation framework

4/13

Nonlinear parametric equationsHigher-order continuation methods

Applications

An example in roboticsResidual and workspace equationsWorkspace boundary equations

Residual and workspace equations

Mechanics

General residual eq. for continuation

R(u, λ) = 0

for a varying scalar input λ

Robotics

General residual eq. for mechanisms

RW(u, v) = 0

for varying inputs v(no favored parameter)

I. Charpentier and B. Cochelin A full higher-order AD continuation and bifurcation framework

5/13

Nonlinear parametric equationsHigher-order continuation methods

Applications

An example in roboticsResidual and workspace equationsWorkspace boundary equations

Workspace boundary equations

Boundaries are singularities

Augmented equations (stability domain, [Seydel 1979]; robot workspace, [Litvin 1980])

∂W = {u ∈ W | ∃ (ξ, v) s.t.(RWv

)ᵀ(u, v).ξ = 0 and ξᵀ.ξ = 1}

Implementations"Math version": MATCONT, for ODE and stab. domains [Govaerts&Kuznetzov 2007] uses AD

"Robotic version"

Handcoded Jacobian in [Haug et al. 1996] + First order continuation

Source transformation [Hentz et al. 2016] + Higher order continuation

RW (u, v)Augment−→

system

RW (u, v)(RW

)ᵀ(u, v).ξ

ξᵀ.ξ − 1

Diff.−→

w.r.t. v

RW (u, v)RWv (u, v)(RW

)ᵀ(u, v).ξ(

RWv)ᵀ

(u, v).ξξᵀ.ξ − 1

Pick up−→

informationR∂W (u, v, ξ)

Solution for R∂W(u, v, ξ) by higher order continuation

I. Charpentier and B. Cochelin A full higher-order AD continuation and bifurcation framework

6/13

Nonlinear parametric equationsHigher-order continuation methods

Applications

ContinuationBifurcation analysisBifurcation and geometric series

Higher-order continuation (see [Charpentier 2012] for a review)

General under-determined nonlinear residual problem (unknown: U)

R(U(a)) = 0

Pseudo-arclength equation to close the system

a =

⟨U(a)− U(0),

∂U∂a

(0)⟩

Approximations using Taylor coefficients (under analyticity assumptions)

U(a) =K∑

k=0

ak Uk where Uk =1k!∂k U∂ak

(0)

Iterative sequence of K linear systems {R1}U1 = −{R1|U1=0

}, 〈U1,U1〉 = 1

{R1}Uk = −{Rk|Uk =0

}, 〈Uk ,U1〉 = 0, for k = 2, ..,K

Applies to classicalR(u, λ) (pseudo-arc-length),RW (u, v) (prescribed path)

andR∂W (u, v, ξ) (2D or 2D projections)

I. Charpentier and B. Cochelin A full higher-order AD continuation and bifurcation framework

7/13

Nonlinear parametric equationsHigher-order continuation methods

Applications

ContinuationBifurcation analysisBifurcation and geometric series

Higher-order bifurcation analysis

Rank deficiency of the Jacobian at bifurcation points

Bif. detection(1st order equation)Rank deficiency of the Jacobian

Sign of det(R1)

Geometric series[van Dyke 1974][Cochelin&Médale 2013]

Branch switching(2nd order equation)

Tangents from ABE[Doedel et al. 1991]

Series[Cochelin&Médale 2013, AD2016]

Other options

PertubationDeflation [Farrell et al. 2016]

Curvature(3rd order equation)

Direct solution

(Many critical cases inrobotics)

Other options

Approximate curvature

Perturbation

I. Charpentier and B. Cochelin A full higher-order AD continuation and bifurcation framework

8/13

Nonlinear parametric equationsHigher-order continuation methods

Applications

ContinuationBifurcation analysisBifurcation and geometric series

Bifurcation and geometric series

Singular Jacobian at point Ub

{Rb1}φ

i = 0, i = 1, 2, and ψ∗.{Rb1} = 0

Algebraic Bifurcation Equation (ABE) for (α, β) s.t. {Rb1}(αφ

1 + βφ2) = 0

c11α2 + 2c12αβ + c22β

2 = 0, with cij = ψ∗.{Rb2}φ

iφj for i = 1, 2

1st order representation of the two branches U(α, β) = Ub + αUB1

1 + βUB2

1

Perfect case: αβ = 0, U(α, β) lies on one or the other branch

Numerical case: αβ = µ 6= 0 and a geometric series emerges

U(a, s) = Ub − sUB11 −

(µ

s

)UB2

1︸ ︷︷ ︸U(0, s)

+aUB11 +

(µ

s

)(a

a − s

)UB2

1

Four steps: (1) Detect a geometric series, (2) Locate bif. and clean the series(3) Compute the tangent, (4) Compute the ranges of validity

I. Charpentier and B. Cochelin A full higher-order AD continuation and bifurcation framework

9/13

Nonlinear parametric equationsHigher-order continuation methods

Applications

ElasticaRobotic planar armsClassification of singularities

Elastica

A classical case study in continuation and mechanics (CC case)[Doedel et al. 1991; Cochelin&Médale 2013; Farrell et al. 2016]

Diamanlab tool

Higher-order continuation (in Matlab)

Higher-order bifurcation (simple bif)

GUI

Interesting points

Large adaptive steps

Accurate continuation and detection

Interactive branch switching

Download http://icube-web.unistra.fr/cs/index.php/Software_download#Diamanlab

I. Charpentier and B. Cochelin A full higher-order AD continuation and bifurcation framework

10/13

Nonlinear parametric equationsHigher-order continuation methods

Applications

ElasticaRobotic planar armsClassification of singularities

Workspace boundaries of the RR planar arm

RR Implementation

RW : R2 → [−π3 ,π3 ]

2 → Wv 7→ (θ1, θ2) 7→ u = (x , y)

Joint constraintsθi = sin(π3 vi), for vi ∈ R

Forward kinematics{x = l1 cos(θ1) + l2 cos(θ1 + θ2)y = l1 sin(θ1) + l2 sin(θ1 + θ2)

R∂W(v,u,ξ) equations

Continuation with no favoredparameter

Critical cases

2 angular sets for u(elbow up, elbow down)

Infinite number of v for u

Bifurcation with identical tangents

Different curvatures

I. Charpentier and B. Cochelin A full higher-order AD continuation and bifurcation framework

11/13

Nonlinear parametric equationsHigher-order continuation methods

Applications

ElasticaRobotic planar armsClassification of singularities

RRR planar mechanism (3 links, 3 revolute joints)

[Haug et al. 1996; Hentz et al. 2016b]

RW : R3 → [−π3 ,π3 ]

3 → Wv 7→ (θ1, θ2, θ3) 7→ u = (x , y)

I. Charpentier and B. Cochelin A full higher-order AD continuation and bifurcation framework

12/13

Nonlinear parametric equationsHigher-order continuation methods

Applications

ElasticaRobotic planar armsClassification of singularities

Classification of singularities

Classification of singularities from rank deficiency of Jacobians[Zlatanov et al. 1998], operated with AD in [Hentz et al. 2016b]

x

-4 -3 -2 -1 0 1

y

-3

-2

-1

0

1

2

3

AA

23

1

5

4

Workspace boundaries

Bifurcation diagram

Selected solution for planar robots (for now)

Higher order series for continuation and bifurcation detection

Jacobian calculation for classification

Perturbation method for branch switching

I. Charpentier and B. Cochelin A full higher-order AD continuation and bifurcation framework

13/13

Nonlinear parametric equationsHigher-order continuation methods

Applications

ElasticaRobotic planar armsClassification of singularities

Conclusions

Accurate and reliable HO continuation and bifurcation framework for classicalapplications

Very specific issues in robotics

HO continuation + HO bif. detection + Perturbation method for branch switching

Planar mechanism workspace or 2D projections

I. Charpentier and B. Cochelin A full higher-order AD continuation and bifurcation framework