contact mechanics
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Contact Mechanics. B659: Principles of Intelligent Robot Motion Spring 2013 Kris Hauser. Agenda. Modeling contacts, friction Form closure, force closure Equilibrium, support polygons. Contact modeling. - PowerPoint PPT PresentationTRANSCRIPT
Contact MechanicsB659: Principles of Intelligent Robot MotionSpring 2013Kris Hauser
Agenda• Modeling contacts, friction• Form closure, force closure• Equilibrium, support polygons
Contact modeling• Contact is a complex phenomenon involving deformation and
molecular forces… simpler abstractions are used to make sense of it
• We will consider a rigid object against a static fixture in this class
• Common contact models:• Frictionless point contact• Point contact with Coulomb friction• Soft-finger contact
Point contact justification• Consider rigid objects A and B that make
contact over region R• Contact pressures (x) 0 for all x R• If R is a planar region, with uniform friction
and uniform normal, then all pressure distributions over R are equivalent to• A combination of forces on convex hull of R• If R is polygonal, a combination of forces on
the vertices of the convex hull of R• [“Equivalent”: one-to-one mapping between span of
forces/torques caused by pressure distribution over R and the span of forces/torques caused by forces at point contacts]
R
A
B
Frictionless contact points
• Contact point ci, normal ni for i=1,…,N• Non-penetration constraint on object’s motion: • Here is measured with respect to the motion of the object• Unilateral constraint
fixture
object
Frictionless dynamics• Assume body at rest• Consider pre-contact acceleration a, angular accel • Nonpenetration must be satisfied post-contact• Solve for nonnegative contact forces fi that alter acceleration
to satisfy constraints
fixture
object
a
Post impact velocity• Post impact velocities• Post-contact acceleration at contact: • Formulating nonpenetration constraints:
Forces at COM
Torques about COM
Matrix formulation• Note that the terms can be written
• With , , element-wise inequality• G is the grasp matrix (Jacobian of contact points w.r.t. rigid body
transform)
• Each of these linear inequalities in the fk’s must be satisfied for all i.
• Write out • (symmetric positive semi-definite)• (vector of initial contact accelerations in normal dir.)
Complementarity constraints• Nonpenetration constraints • Positivity constraints • Underconstrained system – how to prevent arbitrarily large
forces?
• Extra complementarity constraint: fi must be 0 whenever • Meaning: a contact force is allowed only if the contact remains
after the application of forces• Expressed as
• More compactly formulated as • Result: linear complementarity problem (LCP) that can be
solved as a convex quadratic program (QP) or using more specialized solvers (Lemke’s algorithm)
Note relationship to virtual work!
Frictional contact• Coulomb friction model• Normal force • Tangential force • Coefficient of friction μ• Constraint:
• Space of possible contact forces described by a friction cone
tan−1𝜇 nn
Quadratic constraint model•
• Cone specified exactly using following two constraints1. (quadratic nonconvex constraint)2. (linear)
• Constraint 1 is relatively hard to deal with numerically
Frictional contact approximations• In the plane, frictional contacts can be treated as two
frictionless contacts• The 3D analogue is the common pyramidal approximation to
the friction cone
• Caveats:• In formulation Af + b >= 0, A is no longer a symmetric matrix,
which means solution is nonunique and QP is no longer convex• Complementarity conditions require consideration of sticking,
slipping, and separating contact modes
High level issues• Zero, one, or multiple solutions? (Painlevé paradox)• Rest forces (acceleration variables) vs dynamic impacts
(velocity variables)• Active research in improved friction models• Most modern rigid body simulators use specialized algorithms
for speed and numerical stability• Often sacrificing some degree of physical accuracy• Suitable for games, CGI, most robot manipulation tasks where
microscopic precision is not needed
Other Tasks• Determine whether a fixture resists disturbances (form
closure)• Determine whether a disturbance can be nullified by active
forces applied by a robot (force closure)• Determine whether an object is stable against gravity (static
equilibrium)• Quality metrics for each of the above tasks
Form Closure• A fixture is in form closure if any possible movement of the
object is resisted by a non-penetration constraint• Useful for fixturing workpieces for manufacturing operations
(drilling, polishing, machining)• Depends only on contact geometry
Form closure Not form closure
Testing Form Closure• Normal matrix N and grasp matrix G• Condition 1: A grasp is not in form closure if there exists a
nonzero vector x such that NTGTx > 0• x represents a rigid body translation and rotation
• Definition: If the only x that satisfies NTGTx >= 0 is the zero vector, then the grasp is in first-order form closure• Linear programming formulation
• How many contact points needed?• In 2D, need 4 points• In 3D, need 7 points• Nondegeneracy of NTGT must be satisfied
Higher-order form closure• This doesn’t always work… sometimes there are nonzero
vectors x with NTGTx = 0 but are still form closure!• Need to look at second derivatives (or higher)
Form closure
Not form closure
Force Closure• Force closure: any disturbance force can be nullified by active
forces applied by the robot• This requires consideration of robot kinematics and actuation
properties• Form closure => force closure• Converse doesn’t hold in case of frictional contact
Force closure but not form closure
Not force closure
Static Equilibrium• Need forces at contacts to support object against gravity
mgf1
f2
021 mgff0)()( 2211 pcfpcf
Force balance
Torque balance
2211 , FCfFCf Friction constraint
Equilibrium vs form closure• Consider augmenting set of contacts with a “gravity contact”:
a frictionless contact at COM pointing straight downward• Form closure of augmented system => equilibrium
Support Polygon
Side
TopDoesn’t correspond
to convex hull of contacts projected
onto plane
Strong vs. weak stability• Weak stability: there exist a
set of equilibrium forces that satisfy friction constraints
• Strong stability: all forces that satisfy friction constraints and complementarity conditions yield equilibrium (multiple solutions)
• Notions are equivalent without friction
A situation that is weakly, but not strongly stable
Some robotics researchers that work in contact mechanics• Antonio Bicchi (Pisa)• Jeff Trinkle (RPI)• Matt Mason (CMU)• Elon Rimon (Technion)• Mark Cutkosky (Stanford)• Joel Burdick (Caltech)• (many others)
Recap• Contact mechanics: contact models, simulation• Form/force closure formulation and testing• Static equilibrium