Computer Science A family of rigid body models: connections between quasistatic and dynamic multibody systems Jeff Trinkle Computer Science Department Rensselaer Polytechnic Institute Troy, NY 12180 Jong-Shi Pang, Steve Berard, Guanfeng Liu
Computer Science Motivation Dexterous Manipulation Parts Feeder Design Planning Parts feeder design goals: 1) Exit orientation independent of entering orientation 2) High throughput Valid quasistatic plan exists Part enters cg down Design geometry of feeder to guarantee 1) and maximize 2). Feeder geometry has 12 design parameters Evaluate feeder design via simulation No quasistatic plan found, Part enters cg up but dynamic plan exists
Computer Science LIGA Tribology Test “Vehicle” LIGA – German acronym for process for making micro- scale parts from metals, ceramics, and plastics. Typical dimensions are on the order of 1 . 0 0.001 mm ± Sandia wants to understand function, efficiency, robustness before building. Optimal design.
Computer Science Micro-Machine Assembly Tweezers Pawl (2.3 mm) and washer (1.0 mm) subassembly. Pins (0.169 mm) in holes (0.165 mm). Need fixture to hold and align washer and pawl. Fixture should guarantee unique positions and orientations of parts.
Computer Science Pawl in Fixture
Computer Science Simulation of Pawl Insertion
Computer Science Past Work in Quasistatic Multibody Systems Grasping and Walking Machines – late 1970s. Used quasistatic models with assumed contact states. Whtney, “Quasistatic Assembly of Compliantly Supported Rigid Parts,” ASME DSMC, 1982 Caine, Quasistatic Assembly, 1982 Peshkin, Sanderson, Quasistatic Planar Sliding, 1986 Cutkosky, Kao, “Computing and Controlling Compliance in Robot Hands,” IEEE TRA, 1989 Kao, Cutkosky, “Quasistatic Manipulation with Compliance and Sliding,” IJRR, 1992 Peshkin, Schimmels, Force-Guided Assembly, 1992
Computer Science Past Work in Quasistatic Multibody Systems Mason, Quasistatic Pushing, 1982 - 1996 Brost, Goldberg, Erdmann, Zumel, Lynch, Wang Trinkle, Hunter, Ram , Farahat, Stiller, Ang, Pang, Lo, Yeap, Han, Berard, 1991 – present Trinkle Zeng, “Prediction of Quasistatic Planar Motion of a Contacted Rigid Body,” IEEE TRA, 1995 Pang, Trinkle, Lo, “A Complementarity Approach to a Quasistatic Rigid Body Motion Problem,” COAP 1996
Computer Science Hierarchical Family of Models • Models range from pure geometric to Model Space dynamic with contact compliance Dynamic • Required model “resolution” is dependent on design or planning task • Approach: Quasistatic – Plan with low resolution model first – Use low resolution results to speed planning with high resolution model Kinematic – Repeat until plan/design with required accuracy is achieved Geometric Compliant Rigid
Computer Science Components of a Dynamic Model Newton-Euler Equation Defines motion dynamics Kinematic Constraints Quasistatic model: time-scale the Describe unilateral and bilateral constraints Newton-Euler Normal Complementarity equation. Prevents penetration and allows contact separation Friction Law Defines friction force behavior: Bounded magnitude Maximum Dissipation Leads to tangential complementarity Maintains rolling or allows transition from rolling to sliding
Computer Science Complementarity Problems n z Let be an element of and ℜ n n n w ( z ) w ( z ) : ℜ → ℜ let be a given function in . ℜ z Find such that: 0 w z 0 ≤ ≥ ⊥ Linear Complementarity Problem of size 1. R z b Given constants and , find such that: w w Rz b = + 0 w z 0 ≥ ≤ ⊥ z
Computer Science Newton-Euler Equation Non-contact q - configuration forces v - generalized velocity M - symmetric, positive definite inertia matrix f - non-contact generalized forces G - Jacobian relating generalized velocity and time rate of change of configuration & M ( q ( t )) v ( t ) f ( t , q ( t ), v ( t )) = & dx q ( t ) G ( q ( t )) v ( t ) = & x = where dt
Computer Science Kinematic Quantities at Contacts ˆ t Normal and tangential displacement ˆ n i i functions: ( t , q ( t )) ψ in ψ ( t , q ( t )) i 1 ,..., N it ψ = ψ it in ( t , q ( t )) ψ io q Locally, C-space is represented as: 1 L ( t , q ( t )) 0 ; i , , N ψ ≥ = in
Computer Science Normal Complementarity λ i λ Define the contact force it T i t ( ) [ ] λ λ = λ λ λ in in it io Normal Complementarity ˆ t 0 ( t , q ) ( t ) 0 ≤ ψ ⊥ λ ≥ i n n ˆ n i L ψ L T [ ] ψ = where n in L L T [ ] λ = λ n in
Computer Science Dry Friction Assume a maximum dissipation law & & ( , ) arg min( ( t , q , v ) ( t , q , v ) ) λ λ = ψ λ + ψ λ it io it it io io ( , ) ( ); where λ λ ∈ ℑ µ λ i 1 ,..., N ∀ = it io i in & & ( , ) ψ it ψ is the contact slip rate io Linearized Coulomb Coulomb Friction Friction Friction Slip Slip Slip
Computer Science Instantaneous-Time Dynamic Model & Non-contact q = Gv forces & M v f ( t , q , v ) W W W = + λ + λ + λ n n t t o o 0 ( t , q ) 0 ≤ ψ ⊥ λ ≥ n n & T ( , ) arg min( ( t , q , v ) λ λ = ψ λ t o t t & T ( t , q , v ) ) + ψ λ o o ( , ) ( ) λ λ ∈ ℑ µ λ t o n
Computer Science Scale the Times of the Input Functions t t Scale the driving inputs. Replace with in the driving input functions. ε & q ( t ) G ( q ) v ( t ) = & M ( q ) v ( t ) f ( t , q , v ) W ( t , q ) ( t ) W ( t , q ) ( t ) W ( t , q ) ( t ) = ε + λ + λ + λ n n t t o o 0 ( t , q ) ( t ) 0 ≤ ψ ε ⊥ λ ≥ n n & & T T ( ( t ), ( t )) arg min( ( t , q , v ) ( t ) ( t , q , v ) ( t )) λ λ = ψ ε λ + ψ ε λ t o t t o o ( , ) ( ) λ λ ∈ ℑ µ λ t o n
Computer Science Time-Scaled Dynamic Model Change variables ~ 1 t ~ ~ ( ) ( t ) t q ( ) q ( t ) v ( ) v ( ) − λ τ = λ τ = ε τ = τ = ε Application of chain rule and algebra yields: ~ d v ~ ~ ~ ~ ~ 2 M f ( , q , v ) W W W ε = τ ε + λ + λ + λ n n t t o o d τ ~ ~ 0 ( , q ) 0 ≤ ψ τ ⊥ λ ≥ n n d d ~ ~ ~ ~ ψ ψ ~ ~ ~ ~ T T t o ( , ) arg min( ( , q , v ) ( , q , v )) λ λ = λ τ ε + λ τ ε t o t o d d τ τ ~ ~ ~ ( , ) ( ) λ λ ∈ ℑ µ λ t o n
Computer Science Time Stepping Methods 1 − l l dx / d ( x x ) / h + Approximate derivatives by: τ ≈ th l l h x x ( l ) where is the time step, , and is the = τ τ l scaled time at which the state of the system was obtained. ~ ~ ~ ~ ~ 2 l 1 l l 1 M ( v v ) f ( , q , v ) W + + ε − = τ ε + λ ~ ∂ ψ ~ l T l 1 l 1 n 0 ( W v ) h 0 + + ≤ ψ + + ⊥ λ ≥ n n n ∂ τ ~ ~ ~ ~ ∂ ψ ∂ ψ ~ ~ l 1 l 1 l 1 T T l 1 l 1 T T l 1 t o ( + , + ) arg min(( + ) ( W v + ) ( + ) ( W v + )) λ λ = λ + + λ + t o t t o o ∂ τ ∂ τ ~ ~ ~ l 1 l 1 ( , ) ( ) + + λ λ ∈ ℑ µ λ t o n ~ ~ ~ l 1 l l 1 q q G v h + + = +
Computer Science LCP Time-Stepping Problem 6 B F N N ~ 0 2 l 1 2 M W W 0 v Mv hf + − ε − ε − n f ~ l 1 + T l 1 W 0 0 0 + / h / ρ λ ψ + ∂ ψ ∂ τ n n n n n = ~ + l 1 T + l 1 W 0 0 E / + ρ λ ∂ ψ ∂ τ f f f n l 1 T l 1 + 0 U E 0 0 + ζ − σ ~ l 1 l 1 Constraint + + ρ λ n n ~ Stabilization Kinematic l 1 l 1 + 0 0 + ≤ ρ ⊥ λ ≥ Control f f l 1 l 1 + + ζ σ Size 6 B 2 N F = + + ~ ~ ~ l 1 l l 1 q q G v h + + = +
Computer Science Example: Fence and Particle Assume: Particle is constrained from below T f [ 0 0 mg ] Non-contact force: = − Fence is position-controlled Wall is fixed in place Expected motion: Quasistatic: no motion when not in contact with fence. Dynamic: if deceleration of paddle is large, then particle can continue sliding without fence contact
Computer Science Time-Scaled Fence and Particle System Dynamic Quasistatic Boundary
Computer Science Time-Scaled Fence and Particle System Dynamic Quasistatic
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