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Optimization of Multibody Systems

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.

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Optimization of Multibody Systems

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

  2. 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) = JT (q) l (International benchmark 1991) Mechanisms (KULeuven, 2002) Automotive suspension (UCL 1995) (Automatic System 2002) (Tenneco-Monroe 2000, 2004, 2006) 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” Formalism parameters Symbolic Generator (Robotran) .. m z + k z + m g = 0 m, k, z, ... +, -, ... Audi A6 dynamics : real time simulation ! operators 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” Direct kinematics: x = J(q) q . . Inverse dynamics: Q(q, q, q, m, …) . .. Direct dynamics: q = f (q, m, I, F, L, …) .. . Reaction forces: Freact(q, q, q, m, …) . .. Inverse kinematics: q = (J-1)x . . . . q q Symbolic Generator (Robotran) 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 3 dof 3 dof Rb la z lb Rp 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 h(q) v1 q2 u q1 v2 q3 h(q)  0 v h/v = 0 h(q) = 0 ? v1 v1 u u u v ? v2 v2 Singularity Unclosable Multiple closed loops 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) 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 det(Jc) = 0.004 G G F X F X Penalization of assembling constraints Cost function penalty 0.25 assembling constraints f(X) 0.2 0.15 NR OK y [m] 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 Initial design Using free-derivative algorithm: Simplex method (Nelder-Mead) Optimum design 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 Optimal mechanism Initial mechanism Target Multibody Dynamics Optimization prerequisites Applications Isotropy of manipulators Comfort of vehicles Biomechanics of motion Synthesis of mechanisms

  19. Synthesis of mechanisms Objective: Path-following OR Function-generation 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 Non-Linear Least-Squares Optimization Objective: 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 Refine possibly the grid 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

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