Collision Response and Contact Handling for Articulated Body

47 Pages·2007·0.57 MB·English

Collision Response and Contact Handling for
Articulated Body Dynamics
Sachin Patil
October 29, 2007
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Collision Response in Rigid Bodies
CollisionResponseand
(cid:4)
Collision detection and generating appropriate collision responses is
ContactHandlingfor
RigidBodies
an integral part of any rigid body simulator
Overviewof
Featherstone’sABM
(cid:4)
Contacts are primarily of two kinds:
Impulse-basedcollision
responseforABD
Constraint-based (cid:3) Colliding contacts : vrel < −(cid:15)
collisionresponsefor
ABD
(cid:4) Bodies “bounce” off each other, coefficient of restitution
Collisionresponsefor
governs bounciness
AdaptiveABD
(cid:4) Instantaneous change in velocity
(cid:4) Requires restart of the solver
(cid:3) Resting contacts : −(cid:15) < v < (cid:15)
rel
(cid:4) Gradual contact forces prevent interpenetration
(cid:4) No instantaneous changes in velocity
(cid:4) Examples include sliding, rolling, stacking etc.
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Colliding contacts (Impacts)
CollisionResponseand
(cid:4)
These occur when two rigid bodies interpenetrate in a given time
ContactHandlingfor
RigidBodies
step of the simulation
Overviewof
Featherstone’sABM
(cid:4)
Two approaches to resolve this:
Impulse-basedcollision
responseforABD
(cid:3)
Force driven methods
Constraint-based
collisionresponsefor
ABD
(cid:4) Penalty (based on penetration distance) restoring force is
Collisionresponsefor
applied to the bodies
AdaptiveABD
(cid:4) Easier, but slow objects react in a ’slow’ fashion to collision
(cid:3)
Impulse driven methods
(cid:4) Apply impulses to colliding objects at point of collision
causing instantaneous change in velocities
(cid:4) For frictionless bodies, direction is the same as the normal
direction at point of contact
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Continuous contacts
CollisionResponseand
(cid:4)
Any continuous contact is a constraint
ContactHandlingfor
RigidBodies
Overviewof (cid:4) Two approaches to resolve such contacts
Featherstone’sABM
Impulse-basedcollision
(cid:3)
Impulse based methods
responseforABD
Constraint-based
(cid:4) Inherently local in nature
collisionresponsefor
ABD
(cid:4) Faster and simpler
Collisionresponsefor
AdaptiveABD (cid:4) No explicit contact constraints
(cid:4) Cannot guarantee stability after resolution
(cid:3)
Constraint based methods
(cid:4) Inherently global in nature
(cid:4) Much more stable than local, impulse based methods
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Resting contact response
CollisionResponseand
ContactHandlingfor
RigidBodies
Overviewof
Featherstone’sABM
Impulse-basedcollision
responseforABD
Constraint-based
collisionresponsefor
ABD
Collisionresponsefor
AdaptiveABD
Figure 1: Resting Contact
(cid:4) Apply normal forces fi nˆi at each contact point
cp cp
(cid:4)
Solve all contact forces simultaneously (since they influence each
other) which yields a linear system
(cid:4)
All forces subject to three conditions (outlined in the next slide)
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Resting contact response
CollisionResponseand
(cid:4) Interpenetration must be prevented : a ≥ 0
ContactHandlingfor cp
RigidBodies
Overviewof (cid:4) Forces can only be repulsive (to prevent further interpenetration) :
Featherstone’sABM
f ≥ 0
cp
Impulse-basedcollision
responseforABD
(cid:4)
Constraint-based Forces should equal zero when the bodies in contact start to
collisionresponsefor
T
separate : a f = 0
ABD cp cp
Collisionresponsefor
AdaptiveABD (cid:4)
Normal accelerations depend linearly on the normal forces :
a = Af + b
cp cp
(cid:4)
This is a Linear Complementarity Problem (LCP)
(cid:3)
This formulation can also be used with impulse based methods
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Linear Complementarity Problem (LCP)
CollisionResponseand
(cid:4) Need to solve a quadratic program to solve for the forces f
ContactHandlingfor cp
RigidBodies
Overviewof (cid:3) Generally solving a LCP is an NP-complete problem
Featherstone’sABM
Impulse-basedcollision
responseforABD (cid:4)
There are two kinds of methods to solve such problems:
Constraint-based
collisionresponsefor
(cid:3)
ABD Pivoting algorithms
Collisionresponsefor
AdaptiveABD (cid:4) Use a finite number of steps and require the recursive
solution of systems of linear equations
(cid:4) Do not provide useful intermediate result
(cid:4) Theoretically faster than iterative methods
(cid:3)
Iterative methods
(cid:4) Converge to the solution
(cid:4) Can be interrupted in the middle to yield a valid result
(cid:4) Gracefully deal with singular systems
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Linear Complementarity Problem (LCP)
CollisionResponseand
(cid:4) More details on how to compute the A and b matrices and the LCP
ContactHandlingfor
RigidBodies
formulation for resolving resting contacts available in:
Overviewof
Featherstone’sABM
(cid:3)
Analytical Methods for Dynamic Simulation of Non-penetrating
Impulse-basedcollision
responseforABD
Rigid Bodies, D.Baraff, Computer Graphics Proceedings, 1989
Constraint-based
collisionresponsefor
ABD
(cid:4)
More details on how to solve LCP problems available in:
Collisionresponsefor
AdaptiveABD
(cid:3)
Dantzig’s algorithm : Fast contact force computation for
non-penetrating rigid bodies, D.Baraff, Computer Graphics
Proceedings, 1994
(cid:3)
Lemke’s algorithm : Lemke’s algorithm, the hammer in your
toolbox, C.Hecker, 2004,
http://www.d6.com/users/checker/dynamics.htm
(cid:4)
Incorporate friction in previous formulation for realistic looking results
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Overview of forward dynamics methods
CollisionResponseand
(cid:4)
Previously we came across the following methods to compute
ContactHandlingfor
RigidBodies
forward dynamics for an articulated body
Overviewof
Featherstone’sABM
(cid:3)
Composite Rigid Body algorithm
Impulse-basedcollision
responseforABD
(cid:3)
Lagrange Multipliers method
Constraint-based
collisionresponsefor O O
(cid:3) Featherstone’s (n) Articulated Body Method and (lg(n))
ABD
Collisionresponsefor Divide-and-Conquer method
AdaptiveABD
(cid:4)
This lecture will use Featherstone’s ABM as the primary forward
dynamics algorithm for illustrating concepts in collision response
and contact handling
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Forward dynamics methods
CollisionResponseand
(cid:4)
Traditionally two kinds of approaches adopted towards solving this
ContactHandlingfor
RigidBodies
problem:
Overviewof
Featherstone’sABM
(cid:3)
Maximal coordinates
Impulse-basedcollision
responseforABD
(cid:4) For m links, there are 6m state variables where each link
Constraint-based
collisionresponsefor is a rigid object
ABD
Collisionresponsefor (cid:4) Constraints used to keep the links together
AdaptiveABD
(cid:4) Straightforward extension of simple rigid body dynamics
(cid:3)
Reduced (generalized) coordinates
(cid:4) The number of state variables is the same as the number
of joints in the articulation hierarchy
(cid:4) In spite of an intuitive representation, the equations of
motion turn out to be much more complicated
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