Optimization of Multibody Systems

Jean-François Collard

Paul Fisette

24 May 2006

Optimization of Multibody Systems

Multibody DynamicsMotion analysis of complex mechanical systems

(UCL 1995)

Mobile Robot

(McGill 1997)

Parallel manipulator

(Tenneco-Monroe 2000, 2004, 2006)

Automotive suspension

(International benchmark 1991)

Off-road vehicle

(Automatic System 2002)

Mechanisms

(Bombardier 1993, 2003, 2006)

Railway vehicle

.M(q) q + c (q, q) = JT (q) ..ROBOTRAN

(KULeuven, 2002)

Serial manipulator

« Computer simulation »

Multibody DynamicsOptimization prerequisites

Applications

Motion analysisHistorical aspects

Multibody DynamicsHistorical aspects 1970 …

Satellites : “first” multibody applications Analytical linear model – Modal analyses

1980 … Vehicle dynamics, Robotics (serial robots) “Small” nonlinear models, Time simulation of “small systems”

1990 … Vehicle, machines, helicopters, mechanisms, human body, etc. Flexible elements, Non-linear simulations, Sensitivity analysis, …

2000 … Idem + Multiphysics models (hydraulic circuits, electrical actuator, …) Idem + Optimization of performances


Applications

Motion analysisHistorical aspects

Optimization : “prerequisites”

Model formulation : assembling, equations of motion

Assembling

Equations of motion

Model “fast” simulation

Compact analytical formulation

Compact symbolical implementation (UCL)

Model portability

Analytical “ingredients”

Model exportation


Applications

Model formulationModel « fast » simulationModel portability

Optimization : “prerequisites”Model formulation Assembling : nonlinear constraint equations : h(q, t) = 0

Equations of motion

« DAE »

« ODE »

Reduction technique (UCL)


Applications


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

Lagrange

RecursiveNewton-Euler


Applications


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…


Applications


Optimization: applications

Isotropy of parallel manipulators

Assembling constraints and penalty method

Comfort of road vehicles

Multi-physics model

Biomechanics of motion

Identification of kinematic and dynamical models

Synthesis of mechanisms

Extensible-link approach

Multiple local optima


Applications

Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms


Problem statement

qx J

1x2x

3x

1q 2q

3q J

3 dof

1q

2q

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

N

iicond J

NIsotropy index


Applications



Dealing with 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

q1

q2

q3u

v1

v2

Unclosable

h(q) 0 v

u v2

v1


Applications



Penalization of assembling constraints

-0.15 -0.1 -0.05 0 0.050.05

0.15

0.2

0.25

Cost function penalty

x [m]

y [m

]

assembling constraints

0.1

G

xx

x

X

The optimizer call f(X) return value ?

NR OK

xxF

det(Jc) = 0.004

FG X

f(X)

NR KO


Applications



Results for the Delta robot

Optimum design

Initial designOptimum values

Average isotropy = 95%

la = 13.6 cm

lb = 20 cm

z = 13.5 cm

Rb = 13.1 cm

Rp = 10.4 cm

Using free-derivative algorithm: Simplex method (Nelder-Mead)


Applications



Model: Audi A6 with a semi-active suspension

+MBS

Hydraulic

+MBS

Hydraulic

Multibody Model(UCL – ULg)

Front left Front right

Rear left Rear right

+

Front leftFront left Front rightFront right

Rear leftRear left Rear rightRear right

+

ROBOTRAN (UCL) : symbolic

OOFELIE (ULg) : FEM - numerical

Hydraulic Model(TENNECO Automotive)

Control Model(KULeuven)


Applications



Optimization using Genetic Algorithms

Objective : Minimize the average of the 4 RMS vertical

accelerations of the car body corners

Parameters : 6 controller parameters

Input : 4 Stochastic road profiles

0 2 4 6 8 10 12-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Time [s]

Ver

tical

pos

ition

[m]

rear-left wheelrear-right wheelfront-left wheelfront-right wheel

0.710.52Initial uncontrolled

RMS accelerations [m/s2]

0.580.41Optimum

0.660.46Initial controlled

Ride & handlingComfort

0.710.52Initial uncontrolled

RMS accelerations [m/s2]

0.580.41Optimum

0.660.46Initial controlled

Ride & handlingComfort


Applications



Objective : Quantification of joint and muscle efforts

+ ElectroMyoGraphy (EMG) :

Fully equipped subject :


Applications



Kinematics optimization

• MAX relative error = 2.05 % • MEAN relative error = 0.05 % MEAN absolute error = 3.1 mm

xmod and xexp superimposed :


Applications



Muscle overactuation: optimization Forearm flexion/extension

From : triceps brachii EMG biceps brachii EMG

find : triceps brachii force biceps brachii force

and the corresponding elbow torque QEMG

that best fit the elbow torque QINV

obtained from inverse dynamics.

In progress…


Applications



Initial mechanism

Optimal mechanism

Target


Applications



Problem statement

Requirements

Variables: point coordinates & design parameters

Constraint: assembling the mechanism

Function-generationPath-following ORObjective:


Applications



Extensible-link model


Applications



Extensible-link model

Advantage: no assembling constraints

1,

1

1min , ,

2

NT

i

N

i i i il f f

d t f l K d t f l

Objective:Non-Linear Least-Squares Optimization


Applications



Multiple solution with Genetic Algorithms

Different local optima !


Applications



Optimization strategy

Find equilibrium of each configuration

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

Perform global synthesis starting from best candidates

Create grid over the design space

Refine possibly the grid

7x7 grid = 49 points


Applications









Optimization parameters:

ONLY point coordinates


Applications










Applications









4 groups = 4 candidates

Global synthesis

2 local optima:

Optimization parameters:

point coordinates

AND design parameters


Applications









4 groups = 4 candidates

2 local optima:

Global synthesisOptimization parameters:

point coordinates

AND design parameters


Applications



Application to six-bar linkage: multiple local optima

83521 grid points

284 groups

14 local optima

1 « global » optimum

Additional design criteria


Applications


Thank you for your attention

Optimization of Multibody Systems

Documents

Transcript of Optimization of Multibody Systems