Optimization of Closed-loop Multibody Systems print1 Jean-François Collard...

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Jean-François Collard([email protected])

March 21st 2007

Optimization of Closed-loopMultibody Systems

Computer-aided analysis ofmultibody dynamics (part 2)

Cost function

Begin

End

Begin

End

Begin

End

Mathematical Optimization

A wedding between optimization and MBS ?

IntroductionAssembling constraints

Mechanism synthesis

ContextMBS optimization at UCLOptimization prerequisites

• Geometry• Kinematics• Dynamics

Multibody System analysis

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MBS Optimization at UCL• Biomechanics (PhD thesis of M. Raison):

model identification using forward kinematics and inverse dynamics

• Multi-physics (IUAP demonstrator):comfort optimization of an Audi A6 equipped with a semi-active suspension

• Vehicle Dynamics (student thesis): optimization of a formula 3 on circuit, or a 2CV suspension

• Manipulator Kinematics (my PhD thesis): kinematical performance of 3D parallel robots

• Mechanisms Geometry (my PhD thesis): synthesis of steering mechanisms (Ackerman)

• …


Mechanism synthesis


MBS Optimization : “prerequisites”

Model formulation : assembling, equations of motion Assembling

Equations of motion

Model “fast” simulationCompact analytical formulation

Compact symbolical implementation (UCL)

Model portabilityAnalytical “ingredients”

Model exportation

Robustness

Efficiency

Flexibility


Mechanism synthesis


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MBS Optimization : “prerequisites”Model formulation

Assembling : nonlinear constraint equations : h(q, t) = 0

Equations of motion

« DAE »

« ODE »

Reduction technique (UCL)


Mechanism synthesis


Optimization : “prerequisites”Model “fast” simulation

Compact analytical formulation

Compact symbolical implementation (UCL)Formalism

parameters

operators

m z + k z + m g = 0

+, -, ...

m, k, z, ...

..SymbolicGenerator(Robotran)

Audi A6 dynamics : real time simulation !

# flops

# bodies

LagrangeRecursive

Newton-Euler


Mechanism synthesis


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MBS Optimization : “prerequisites”Model portability

Analytical “ingredients”

Model exportation

Reaction forces:Freact(q, q, q, m, …)

...

Inverse dynamics:Q(q, q, q, m, …)

...Direct dynamics:

q = f (q, m, I, F, L, …).. .

Direct kinematics:x = J(q) q. .

Inverse kinematics:q = (J-1)x. .

x.

. q

Q

Freact. q

SymbolicGenerator(Robotran)

MatlabSimulink

MultiphysicsPrograms (Amesim)

Optimizationalgorithms…


Mechanism synthesis


Optimization of Closed-loop MBS

Dealing with assembling constraintsArtificial penalty approach free-derivative search

Assembling penalty approach gradient-based algorithms

Comparison

Application to parallel manipulators (delta robot, Hexaslide robot)

Synthesis of mechanisms

Extensible-link approach + natural coordinates

Multiple local optima


Mechanism synthesis

Artificial penaltyAssembling penaltyApplications

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Dealing with assembling constraintsNumerical resolution of assembling constraints

Constraints involving joint variables q :h(q) = 0

Coordinate partitioning :q = [u v]

Newton-Raphson iterative algorithm:vi+1 = vi – [∂h/∂v]-1 h(q)

h(q)

Multiple closed loops

?h(q) = 0

u v ?

Types of problems encountered :

Singularity

∂h/∂v = 0

u v2

v1

Unclosable

h(q) ≠ 0 ∀v

u v2

v1


Mechanism synthesis


u

v1

v2

Dealing with assembling constraintsArtificial Penalization

-0.15 -0.1 -0.05 0 0.050.05

0.15

0.2

0.25

x [m]

y [m

]

Feasible domain of assembling constraints

0.1

G

xxx

X

The optimizer call f(X) return value ?

NR OK

xxF

det(Jc) = 0.004

NR KO

FG X

f(X)


Mechanism synthesis


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Dealing with assembling constraintsNumerical minimization of assembling constraints

Coordinate partitioning :q = [u v]

SQP iterative algorithm

h(q)

Multiple closed loops

?h(q) = 0

u v ?

Previous types of problems encountered :

Singularity

∂h/∂v = 0

u v2

v1

Unclosable

h(q) ≠ 0 ∀v

u v2

v1

u

v1

v2

Constraints involving joint variables q :min hT(q)h(q)

s.t. c(q)>=0v

Unique if c is well chosen MinimizedAvoidable if c is well chosen


Mechanism synthesis


Dealing with assembling constraintsAssembling Penalization

-0.15 -0.1 -0.05 0 0.050.05

0.15

0.2

0.25

x [m]

y [m

]

0.1

G

X • f* = cost function (best assembly)• h* = MIN. of assembling constraint• g = penalized cost function

G X

g(X)

B

B h*2

f*g

g = f* + w h*2

weighted factor

Zero-value domain of assembling constraints


Mechanism synthesis


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ComparisonArtificial Penalization

Advantages:• g is differentiable since f and h* are• efficient use of gradient-based

algorithms• one parameter w to tune

Disadvantages:• computation of sensitivity analysis may

be expensive• local optimal solution

Assembling Penalization

h*2

f*g

g = f* + w h*2

BFG

fext

s

B

Δ

Advantages:• fext is easy to compute• possible use of simple free-derivative

search algorithms (e.g. Nelder-Mead Simplex, G.A.,…)

Disadvantages:• fext is not differentiable in F• three parameters to tune (G,s,Δ)• problems if domain is not convex


Mechanism synthesis


Application to Delta robot isotropyProblem statement

qx J=& &

1x&2x&

3x&

1q& 2q&

3q& J

3 dof

1q2q

3q

( )1 2 3, ,x x x

Rb

z

Rp

la

lb

3 dof

Objective : Maximize isotropy index over a 2cm sided cubeParameters : la, lb, z, Rb, Rp

( )1

1N

i icond JN

Isotropy index ==∑


Mechanism synthesis


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Application to Delta robot isotropyOptimization results

Optimum design

Initial designOptimum values

Average isotropy = 95%la = 13.6 cmlb = 20 cmz = 13.5 cmRb = 13.1 cmRp = 10.4 cm

Using free-derivativealgorithm: Simplex method (Nelder-Mead)


Mechanism synthesis


Application to Hexaslide robot isotropyProblem statement

Parameters :Gz, l, RB, α, β, H, ψ

Objective :Maximize average dexterity over a 6-dimensional hypercube(3 positioning and 3 orienting coordinates)

Problem :Forward kinematic Jacobian dimensionaly inhomogeneous !

6 dof


Mechanism synthesis


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Application to Hexaslide robot isotropyNormalizing characteristic length* Lc

Goal :Making the Jacobian matrix dimensionally homogenous

xx q

q

q

δδ

ϑ δϑδ

⎡ ⎤⎢ ⎥⎡ ⎤ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦

&&

&

No units

[m-1]

*Angeles J., Fundamentals of Robotic Mechanical Systems. Springer-Verlag, 1997.

[m-1]

[m-1]

cx

q

q

xq

Lϑ δϑ

δ

δδ⎡ ⎤⎢ ⎥

⎡ ⎤ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎢ ⎥

⎢ ⎥⎣ ⎦

&&

& =

Lc value ?It can also be obtained by optimization as an additional optimization variable.


Mechanism synthesis


Application to Hexaslide robot isotropy

Parameters :Gz, l, RB, α, β, H, ψ, Lc

Inspired from :J. Ryu & J. Cha, « Volumetric error analysis and architecture optimizationfor accuracy of HexaSlide type parallel manipulators », Mechanism andMachine Theory 38 (2003) 227-240.

Initial design

Optimal design

Optimum valuesAverage isotropy = 50%

Gz = 0.456 ml = 1.676 m

RB = 0.521 mα = 48°β = 47°

H = 0.276 m= 104.4°

Lc = 0.002 mm


Mechanism synthesis


Optimization results

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Mechanism synthesisInitial mechanism Optimal mechanism

Target


Mechanism synthesis

IntroductionExtensible-link modelMultiple local optima

Mechanism synthesisProblem statement

Requirements

Variables: point coordinates & design parameters

Constraint: assembling the mechanism

Function-generationPath-following ORObjective:

δi δo


Mechanism synthesis


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Mechanism synthesisExtensible-link model


Mechanism synthesis


Mechanism synthesisExtensible-link model

Advantage: no assembling constraints

( )( ) ( )( )1, 1

1min , ,2

N T

i=

− −∑N

i i i il f fd t f l K d t f l

KObjective:

Non-LinearLeast-SquaresOptimization


Mechanism synthesis


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Mechanism synthesisMultiple solution with Genetic Algorithms

Different local optima !


Mechanism synthesis


Mechanism synthesisOptimization strategy

Find equilibrium ofeach configuration

Group grid points w.r.t.total equilibrium energy

Perform global synthesisstarting from best candidates

Create grid overthe design space

Refine possibly the grid

7x7 grid = 49 points


Mechanism synthesis


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Optimization parameters:ONLY point coordinates


Mechanism synthesis









Mechanism synthesis


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4 groups = 4 candidates

Global synthesis

2 local optima:

Optimization parameters:point coordinates

AND design parameters


Mechanism synthesis








4 groups = 4 candidates

2 local optima:

Global synthesisOptimization parameters:

point coordinatesAND design parameters


Mechanism synthesis


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Mechanism synthesisApplication to six-bar linkage: multiple local optima

83521 gridpoints

284 groups

14 local optima

1 « global »optimum

Additionaldesign criteria


Mechanism synthesis


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Optimization of Closed-loop Multibody Systems print1 Jean-François Collard...

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