optimization of multibody systems
DESCRIPTION
Optimization of Multibody Systems. Jean-François Collard Paul Fisette 24 May 2006. Multibody Dynamics. Mobile Robot. Railway vehicle. Parallel manipulator. ( Bombardier 1993 , 2003, 2006). Off-road vehicle. Serial manipulator. ( McGill 1997 ). M(q) q + c (q, q) = J T (q) l. - PowerPoint PPT PresentationTRANSCRIPT
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
Multibody DynamicsOptimization prerequisites
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
Multibody DynamicsOptimization prerequisites
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)
Multibody DynamicsOptimization prerequisites
Applications
Model formulationModel « fast » simulationModel portability
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
Multibody DynamicsOptimization prerequisites
Applications
Model formulationModel « fast » simulationModel portability
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…
Multibody DynamicsOptimization prerequisites
Applications
Model formulationModel « fast » simulationModel portability
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 DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
Isotropy of parallel manipulators
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
Multibody DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
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: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
Multibody DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
Isotropy of parallel manipulators
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
Multibody DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
Isotropy of parallel manipulators
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)
Multibody DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
Comfort of road vehicles
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)
Multibody DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
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
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
Multibody DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
Biomechanics of motion
Objective : Quantification of joint and muscle efforts
+ ElectroMyoGraphy (EMG) :
Fully equipped subject :
Multibody DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
Biomechanics of motion
Kinematics optimization
• MAX relative error = 2.05 % • MEAN relative error = 0.05 % MEAN absolute error = 3.1 mm
xmod and xexp superimposed :
Multibody DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
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 DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
Synthesis of mechanisms
Initial mechanism
Optimal mechanism
Target
Multibody DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
Synthesis of mechanisms
Problem statement
Requirements
Variables: point coordinates & design parameters
Constraint: assembling the mechanism
Function-generationPath-following ORObjective:
Multibody DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
Synthesis of mechanisms
Extensible-link model
Multibody DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
Synthesis of mechanisms
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
Multibody DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
Synthesis of mechanisms
Multiple solution with Genetic Algorithms
Different local optima !
Multibody DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
Synthesis of mechanisms
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
Multibody DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
Synthesis of mechanisms
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
Optimization parameters:
ONLY point coordinates
Multibody DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
Synthesis of mechanisms
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
Multibody DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
Synthesis of mechanisms
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
4 groups = 4 candidates
Global synthesis
2 local optima:
Optimization parameters:
point coordinates
AND design parameters
Multibody DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
Synthesis of mechanisms
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
4 groups = 4 candidates
2 local optima:
Global synthesisOptimization parameters:
point coordinates
AND design parameters
Multibody DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
Synthesis of mechanisms
Application to six-bar linkage: multiple local optima
83521 grid points
284 groups
14 local optima
1 « global » optimum
Additional design criteria
Multibody DynamicsOptimization prerequisites
Applications
Isotropy of manipulatorsComfort of vehiclesBiomechanics of motionSynthesis of mechanisms
Thank you for your attention