A Hierarchical Design Methodology for Multibody Systems with - PowerPoint PPT Presentation

A Hierarchical Design Methodology for Multibody Systems with Frictional Contacts Vijay Kumar GRASP Lab Mechanical Engineering and Applied Mechanics Computer and Information Science University of Pennsylvania Joint work with Bharath Mukundakrishnan Jonathan Fink Peng Song GRASP Laboratory GRASP Lab, Penn Mechanical Engineering University of Pennsylvania ECS, RPI Rutgers University Jeffrey Trinkle Steve Berard Jong-Shi Pang Computer Science Computer Science Mathematical Sciences Rennselaer Polytechnic Institute Rennselaer Polytechnic Inst. Rennselaer Polytechnic Institute University of Pennsylvania GRASP 1

1. Decentralized Multirobot Manipulation Motion plans derived from geometric models  Can we generalize to dynamic models?  Pereira, Campos and Kumar, WAFR 02 University of Pennsylvania GRASP 2

2. Part Feeding, Assembly Dynamic models are necessary. [Kraus, 2001] [Boothroyd, 1968] Design with geometric and kinematic models is possible. University of Pennsylvania GRASP 3

3. Micro Manipulation  100 µ dia probe attached to10g load cell  0.4mm x 0.8mm part assembly Test Fixture Configuration B Configuration A 2 mm 1.5 mm University of Pennsylvania GRASP 4

Design Process or Plan Task Dynamical System  Intermittent contacts  Contact state transitions  Multiple contacts Fundamental difficulties  Static indeterminacy with traditional models (jamming, wedging)  Whitney, Dupont  No consistent models for frictional impacts  Goldsmith, Pfeiffer, Keller, Brach, Wang and Mason, Stronge, Chatterjee and Ruina  No unified treatment for design and planning University of Pennsylvania GRASP 5

Outline 1. Background  Contact models  Normal and tangential compliance  Frictional contacts  Time-stepping methods 2. Hierarchical Approach  Models at different levels of fidelity  Abstraction and model reduction  Example 3. Algorithms for design optimization  Randomized algorithms  Time-stepping algorithms 4. Case Study: Part Feeder  Modeling  Iterative design process University of Pennsylvania GRASP 6

Systems with Frictional Contacts O c c O µ λ N T λ F c T O Friction T λ F University of Pennsylvania GRASP 7

Compliant Contact Models Rigid Rigid N Rigid Rigid N Core Core Core Core Deformed Shell Undeformed Viscoelastic Viscoelastic Shell Layer Layer T ϕ δ N N Gross motion slip δ T ϕ T Fine deformation & = + f ( ) g ( , ) λ δ δ δ N , i N , i N , i N , i N , i N , i & = + = + f ( ) g ( , ) i 1 , , n n L λ δ δ δ T , i T , i T , i T , i T , i T , i R S University of Pennsylvania GRASP 8

More Generally… Elastic Bodies Linear Elastic, Counterformal Contacts Ø i n A δ i n undeformed deformed B University of Pennsylvania GRASP 9

Advantages of Compliant Contact Model λ Τ λ Τ rigid actual linear elastic Coulomb u u  Proof of uniqueness and existence  Contact forces can always be determined  More realistic friction model  Tangential compliance  Gross slip is preceded by small local deformations  Hysteresis Disadvantages  Identification of parameters  Computational time University of Pennsylvania GRASP 10

Time Stepping Model Equations of Motion [Anitescu, Pang, Potra, Stewart, and Trinkle] University of Pennsylvania GRASP 11

Extension 1: Compliant Models Constitutive law Contact compliance relative gross separation/slip deformations motion University of Pennsylvania GRASP 12

Extension 2: Frictional Contact (cf. Peng Song’s talk tomorrow) University of Pennsylvania GRASP 13

Compliant Frictional Contacts: Technical Results Single frictional contact [Song and Kumar, 2003]  For a single contact with a lumped compliance model, a unique trajectory always exists Multiple frictional contacts [Song, Pang and Kumar, 2003]  A discrete-time solution trajectory always exists  There exists a µ *>0, such that if µ *> µ i >0, a unique trajectory exists  Under “certain conditions” the discrete time trajectory converges to converges to that obtained by using the rigid body model time-stepping algorithm University of Pennsylvania GRASP 14

Example  Linear, visco-elastic contacts  Initial value problem  Five springs at each contact  m = 0.05 kg.  ε =10 -10 N/m 2  Δ t~10 -4 seconds University of Pennsylvania GRASP 15

Example (continued) University of Pennsylvania GRASP 16

Design Optimization Design Optimization External inputs or disturbances Difficulties: (1) high dimensionality; (2) non smoothness University of Pennsylvania GRASP 17

Abstractions and hierarchy S 1 System S 1 System S 2 S 2 Transformation S 2 is an abstraction of S 1 if for any δ > 0 and all inputs u ( t ), there exists v ( t ) such that for all x* is reachable for a given design implies z*=h ( x *) is reachable for the same design University of Pennsylvania GRASP 18

Example of Abstraction Kinematic (first order model) More generally…  Dynamics with compliance  Rigid body dynamic Geometric (zeroth order model)  Kinematic (quasi-static)  Geometric University of Pennsylvania GRASP 19

Case Study: Design of a Part Feeder University of Pennsylvania GRASP 20

Definitions  State space Original state space augmented by all parameters  Inputs/disturbances Geometric model - virtual input Dynamic model - gravitational force  Design space Initial conditions (original state space) + parameter choices  Search space x 2 Focus on “search” and “satisfaction” rather than optimization x 1 University of Pennsylvania GRASP 21

Explorating the Design Space: The RRT method random state  Explore motions from the chosen Find the state, Xnear, “closest” to the vertex by trying all possible inputs. random state among all explored state  Choose the state closest to the s. random state random state, Xnew Choose the state “closest" to the random state, Xnew. initial state  Explore motions from the Key: A vertex with a larger Voronoi chosen vertex by trying all Grow the tree until a solution possible inputs region has higher probability of is found or the no. of vertices being chosen as X near reaches a certain value [Lavalle and co-workers, 1999-2003] University of Pennsylvania GRASP 22

Rapidly Exploring Random Tree x init , q init Target set (e.g., successful assembly) University of Pennsylvania GRASP 23

Rapidly Exploring Random Tree x init , q init x rand , q rand University of Pennsylvania GRASP 24

Rapidly Exploring Random Tree x near , q near x init , q init x rand , q rand University of Pennsylvania GRASP 25

Rapidly Exploring Random Tree x init , q init x rand , q rand University of Pennsylvania GRASP 26

Rapidly Exploring Random Tree x new , q new x init , q init x rand , q rand University of Pennsylvania GRASP 27

Coverage and Growth New trees are started when the growth rate slows below a specified threshold. Plots show 8 designs being explored. University of Pennsylvania GRASP 28

RRT Generated from the Geometric Model with a Given Design Red (thick) geometrically feasible successful path. Green (thin) geometrically feasible trajectories. University of Pennsylvania GRASP 29

Sampling the 12-Dimensional Design Space: Geometric Model Example: different chute angles University of Pennsylvania GRASP 30

Exploring the Design Space: Geometric Model University of Pennsylvania GRASP 31

Pruning the Design Space: Kinematic Model First order model further restricts the choice of design parameters! University of Pennsylvania GRASP 32

Initial Design for Dynamic Analysis Geometric Kinematic University of Pennsylvania GRASP 33

Dynamic Analysis: Inelastic Impacts Song et al , ICRA 2004 Heavy end last Heavy end first 1. LCP solver, time-stepping algorithm [Stewart & Trinkle] 2. No external input/disturbance University of Pennsylvania GRASP 34

Dynamic Analyis: Visco-Elastic Contacts Visco-elastic contacts  LCP solver, time-stepping [Song, Pang, & Kumar]  Exact detection of collisions [Esposito & Kumar] University of Pennsylvania GRASP 35

Experimental Prototype Experimental data digitized at 500 Hz., played back at 1/10 normal speed University of Pennsylvania GRASP 36

Summary • Explore design space using a family of models • Simpler models are used as abstractions for more complex models initially • Can incorporate uncertainty in parameters Related •Enhancement: Optimization [ICRA 04] •Alternative: Use “unified” (implicit, NCP) model to solve boundary-value problem [RSS 05] (cf. Peng Song’s talk tomorrow) University of Pennsylvania GRASP 37