Jean-Fran

1
Optimization of Multibody Systems

• Jean-François Collard
• Paul Fisette
• 24 May 2006

2
Multibody Dynamics
Motion analysis of complex mechanical systems
ROBOTRAN
 Computer simulation 
Multibody Dynamics Optimization
prerequisites Applications
Motion analysis Historical aspects
3
Multibody Dynamics
Historical 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

Multibody Dynamics Optimization
prerequisites Applications
Motion analysis Historical aspects
4
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

Multibody Dynamics Optimization
prerequisites Applications
Model formulation Model  fast  simulation Model
portability
5
Optimization prerequisites
Model formulation

• Assembling nonlinear constraint equations
  h(q, t) 0
• Equations of motion

 DAE 
Reduction technique (UCL)
 ODE 
Multibody Dynamics Optimization
prerequisites Applications
Model formulation Model  fast  simulation Model
portability
6
Optimization prerequisites
Model fast simulation

• Compact analytical formulation
• Compact symbolical implementation (UCL)

flops
Lagrange
RecursiveNewton-Euler
bodies
Multibody Dynamics Optimization
prerequisites Applications
Model formulation Model  fast  simulation Model
portability
7
Optimization prerequisites
Model portability

• Analytical ingredients
• Model exportation

Q
.
x
Freact
Matlab Simulink
MultiphysicsPrograms (Amesim)
Optimization algorithms

Multibody Dynamics Optimization
prerequisites Applications
Model formulation Model  fast  simulation Model
portability
8
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

Multibody Dynamics Optimization
prerequisites Applications
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
9
Isotropy of parallel manipulators
Problem statement

Objective Maximize isotropy index over a 2cm
sided cube
Parameters la, lb, z, Rb, Rp
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
Multibody Dynamics Optimization
prerequisites Applications
10
Isotropy of parallel manipulators
Dealing with assembling constraints

Constraints involving joint variables q h(q) 0
Coordinate partitioning q u v
Newton-Raphson iterative algorithm vi1 vi
?h/?v-1 h(q)
Types of problems encountered
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
Multibody Dynamics Optimization
prerequisites Applications
11
Isotropy of parallel manipulators
Penalization of assembling constraints

Cost function penalty
0.25
assembling constraints
f(X)
0.2
0.15
y m
NR OK
x
x
x
x
x
NR KO
0.1
The optimizer call f(X) ? return value ?
0.05
-0.15
-0.1
-0.05
0
0.05
x m
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
Multibody Dynamics Optimization
prerequisites Applications
12
Isotropy of parallel manipulators
Results for the Delta robot

Optimum values Average isotropy 95 la
13.6 cm lb 20 cm z 13.5 cm Rb 13.1
cm Rp 10.4 cm
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
Multibody Dynamics Optimization
prerequisites Applications
13
Comfort of road vehicles
Model Audi A6 with a semi-active suspension

OOFELIE (ULg) FEM - numerical
Multibody Dynamics Optimization
prerequisites Applications
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
14
Comfort of road vehicles
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
Multibody Dynamics Optimization
prerequisites Applications
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
15
Biomechanics of motion
Objective Quantification of joint and muscle
efforts

ElectroMyoGraphy (EMG)
Fully equipped subject
Multibody Dynamics Optimization
prerequisites Applications
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
16
Biomechanics of motion
Kinematics optimization

? xmod and xexp superimposed

• MAX relative error 2.05
• MEAN relative error 0.05
• ? MEAN absolute error 3.1 mm

Multibody Dynamics Optimization
prerequisites Applications
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
17
Biomechanics of motion
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
Multibody Dynamics Optimization
prerequisites Applications
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
18
Synthesis of mechanisms
Initial mechanism

Target
Multibody Dynamics Optimization
prerequisites Applications
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
19
Synthesis of mechanisms
Problem statement

Requirements
Variables point coordinates design parameters
Constraint assembling the mechanism
Multibody Dynamics Optimization
prerequisites Applications
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
20
Synthesis of mechanisms
Extensible-link model

Multibody Dynamics Optimization
prerequisites Applications
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
21
Synthesis of mechanisms
Extensible-link model

Advantage no assembling constraints
Multibody Dynamics Optimization
prerequisites Applications
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
22
Synthesis of mechanisms
Multiple solution with Genetic Algorithms

Different local optima !
Multibody Dynamics Optimization
prerequisites Applications
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
23
Synthesis of mechanisms
Optimization strategy

Create grid over the design space
7x7 grid 49 points
Find equilibrium of each configuration
Group grid points w.r.t. total equilibrium energy
Perform global synthesis starting from best
candidates
Refine possibly the grid
Multibody Dynamics Optimization
prerequisites Applications
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
24
Synthesis of mechanisms
Optimization strategy

Optimization parameters ONLY point coordinates
Create grid over the design space
Find equilibrium of each configuration
Group grid points w.r.t. total equilibrium energy
Perform global synthesis starting from best
candidates
Refine possibly the grid
Multibody Dynamics Optimization
prerequisites Applications
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
25
Synthesis of mechanisms
Optimization strategy

Create grid over the design space
Find equilibrium of each configuration
Group grid points w.r.t. total equilibrium energy
Perform global synthesis starting from best
candidates
Refine possibly the grid
Multibody Dynamics Optimization
prerequisites Applications
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
26
Synthesis of mechanisms
Optimization strategy

4 groups 4 candidates
Create grid over the design space
Optimization parameters point coordinates AND
design parameters
Global synthesis
Find equilibrium of each configuration
2 local optima
Group grid points w.r.t. total equilibrium energy
Perform global synthesis starting from best
candidates
Refine possibly the grid
Multibody Dynamics Optimization
prerequisites Applications
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
27
Synthesis of mechanisms
Optimization strategy

4 groups 4 candidates
Create grid over the design space
Optimization parameters point coordinates AND
design parameters
Global synthesis
Find equilibrium of each configuration
2 local optima
Group grid points w.r.t. total equilibrium energy
Perform global synthesis starting from best
candidates
Multibody Dynamics Optimization
prerequisites Applications
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
28
Synthesis of mechanisms
Application to six-bar linkage multiple local
optima

83521 grid points
284 groups
14 local optima
Additional design criteria
1  global  optimum
Multibody Dynamics Optimization
prerequisites Applications
Isotropy of manipulators Comfort of
vehicles Biomechanics of motion Synthesis of
mechanisms
29
Thank you for your attention