Jean-François Collard Paul Fisette 24 May 2006 Optimization of Multibody Systems

Multibody Dynamics Motion 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) = J T (q).. ROBOTRAN (KULeuven, 2002) Serial manipulator « Computer simulation » Multibody Dynamics Optimization prerequisites Applications Motion analysis Historical aspects

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

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

Optimization : “prerequisites” Model formulation Assembling : nonlinear constraint equations : h(q, t) = 0 Equations of motion « DAE » « ODE » Reduction technique (UCL) Multibody Dynamics Optimization prerequisites Applications Model formulation Model « fast » simulation Model 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,..... Symbolic Generator (Robotran) Audi A6 dynamics : real time simulation ! # flops # bodies Lagrange Recursive Newton-Euler Multibody Dynamics Optimization prerequisites Applications Model formulation Model « fast » simulation Model portability

Optimization : “prerequisites” Model portability Analytical “ingredients” Model exportation Reaction forces: F react (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 F react. q Symbolic Generator (Robotran) Matlab Simulink Multiphysics Programs (Amesim) Optimization algorithms … Multibody Dynamics Optimization prerequisites Applications Model formulation Model « fast » simulation Model 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 Dynamics Optimization prerequisites Applications Isotropy of manipulators Comfort of vehicles Biomechanics of motion Synthesis of mechanisms

Isotropy of parallel manipulators Problem statement 3 dof Rb z Rp la lb 3 dof Objective : Maximize isotropy index over a 2cm sided cube Parameters : la, lb, z, Rb, Rp Multibody Dynamics Optimization prerequisites Applications Isotropy of manipulators Comfort of vehicles Biomechanics of motion Synthesis 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: v i+1 = v i – [  h/  v] -1 h(q) h(q) Multiple closed loops ? h(q) = 0 u v ? Types of problems encountered : Singularity  h/  v = 0 u v2v2 v1v1 q1q1 q2q2 q3q3 u v1v1 v2v2 Unclosable h(q)  0  v u v2v2 v1v1 Multibody Dynamics Optimization prerequisites Applications Isotropy of manipulators Comfort of vehicles Biomechanics of motion Synthesis of mechanisms

Isotropy of parallel manipulators Penalization of assembling constraints Cost function penalty x [m] y [m] assembling constraints 0.1 G x x x X The optimizer call f(X)  return value ? NR OK x x F det(Jc) = FG X f(X) NR KO Multibody Dynamics Optimization prerequisites Applications Isotropy of manipulators Comfort of vehicles Biomechanics of motion Synthesis of mechanisms

Isotropy of parallel manipulators Results for the Delta robot Optimum design Initial design Optimum 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 Dynamics Optimization prerequisites Applications Isotropy of manipulators Comfort of vehicles Biomechanics of motion Synthesis of mechanisms

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

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

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

Biomechanics of motion Kinematics optimization MAX relative error = 2.05 % MEAN relative error = 0.05 %  MEAN absolute error = 3.1 mm  x mod and x exp superimposed : Multibody Dynamics Optimization prerequisites Applications Isotropy of manipulators Comfort of vehicles Biomechanics of motion Synthesis 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 Q EMG that best fit the elbow torque Q INV obtained from inverse dynamics. In progress… Multibody Dynamics Optimization prerequisites Applications Isotropy of manipulators Comfort of vehicles Biomechanics of motion Synthesis of mechanisms

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

Problem statement Requirements Variables: point coordinates & design parameters Constraint: assembling the mechanism Function-generationPath-followingORObjective: Multibody Dynamics Optimization prerequisites Applications Isotropy of manipulators Comfort of vehicles Biomechanics of motion Synthesis of mechanisms

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

Extensible-link model Advantage: no assembling constraints Objective: Non-Linear Least-Squares Optimization Multibody Dynamics Optimization prerequisites Applications Isotropy of manipulators Comfort of vehicles Biomechanics of motion 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

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 Dynamics Optimization prerequisites Applications Isotropy of manipulators Comfort of vehicles Biomechanics of motion 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 Dynamics Optimization prerequisites Applications Isotropy of manipulators Comfort of vehicles Biomechanics of motion 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 Dynamics Optimization prerequisites Applications Isotropy of manipulators Comfort of vehicles Biomechanics of motion 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 Dynamics Optimization prerequisites Applications Isotropy of manipulators Comfort of vehicles Biomechanics of motion 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 synthesis Optimization parameters: point coordinates AND design parameters Multibody Dynamics Optimization prerequisites Applications Isotropy of manipulators Comfort of vehicles Biomechanics of motion Synthesis of mechanisms

Application to six-bar linkage: multiple local optima grid points 284 groups 14 local optima 1 « global » optimum Additional design criteria Multibody Dynamics Optimization prerequisites Applications Isotropy of manipulators Comfort of vehicles Biomechanics of motion Synthesis of mechanisms

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