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 n

n

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 mgff

0)()( 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

Top

Doesn’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