# Contact Mechanics

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Contact Mechanics

B659: Principles of Intelligent Robot Motion

Spring

2013

Kris

Hauser

Slide2Agenda

Modeling contacts, friction

Form closure, force closure

Equilibrium, support polygons

Slide3Contact 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

Slide4Point contact justification

Consider rigid objects A and B that make contact over region RContact pressures (x) 0 for all x RIf R is a planar region, with uniform friction and uniform normal, then all pressure distributions over R are equivalent toA combination of forces on convex hull of RIf 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

Slide5Frictionless contact points

Contact point ci, normal ni for i=1,…,NNon-penetration constraint on object’s motion: Here is measured with respect to the motion of the objectUnilateral constraint

fixture

object

Slide6Frictionless dynamics

Assume body at restConsider pre-contact acceleration a, angular accel Nonpenetration must be satisfied post-contactSolve for nonnegative contact forces fi that alter acceleration to satisfy constraints

fixture

object

a

Slide7Post impact velocity

Post impact velocities

Post-contact acceleration at contact:

Formulating nonpenetration constraints:

Forces at COM

Torques about COM

Slide8Matrix formulation

Note that the terms can be writtenWith , , element-wise inequalityG 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.)

Slide9

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 forcesExpressed 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!

Slide10Frictional contact

Coulomb friction modelNormal force Tangential force Coefficient of friction μConstraint: Space of possible contact forces described by a friction cone

n

n

Slide11Quadratic constraint model

Cone specified exactly using following two constraints (quadratic nonconvex constraint) (linear)Constraint 1 is relatively hard to deal with numerically

Slide12

Frictional contact approximations

In the plane, frictional contacts can be treated as two frictionless contactsThe 3D analogue is the common pyramidal approximation to the friction coneCaveats:In formulation Af + b >= 0, A is no longer a symmetric matrix, which means solution is nonunique and QP is no longer convexComplementarity conditions require consideration of sticking, slipping, and separating contact modes

Slide13High level issues

Zero, one, or multiple solutions? (Painlevé paradox)Rest forces (acceleration variables) vs dynamic impacts (velocity variables)Active research in improved friction modelsMost modern rigid body simulators use specialized algorithms for speed and numerical stabilityOften sacrificing some degree of physical accuracySuitable for games, CGI, most robot manipulation tasks where microscopic precision is not needed

Slide14Other 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

Slide15Form Closure

A fixture is in form closure if any possible movement of the object is resisted by a non-penetration constraintUseful for fixturing workpieces for manufacturing operations (drilling, polishing, machining)Depends only on contact geometry

Form closure

Not form closure

Slide16Testing 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

N

T

G

T

x

> 0

x represents a rigid body translation and rotation

Definition

: If the only x that satisfies

N

T

G

T

x

>= 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 N

T

G

T

must be satisfied

Slide17Higher-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

Slide18Force Closure

Force closure: any disturbance force can be nullified by active forces applied by the robotThis requires consideration of robot kinematics and actuation propertiesForm closure => force closureConverse doesn’t hold in case of frictional contact

Force closure but not form closure

Not force closure

Slide19Static Equilibrium

Need forces at contacts to support

object against gravity

m

g

f

1

f

2

Force balance

Torque balance

Friction constraint

Slide20Equilibrium vs form closure

Consider augmenting set of contacts with a “gravity contact”: a frictionless contact at COM pointing straight downwardForm closure of augmented system => equilibrium

Slide21Support Polygon

Side

Top

Doesn’t correspond to convex

hull of contacts projected onto plane

Slide22Strong vs. weak stability

Weak stability: there exist a set of equilibrium forces that satisfy friction constraintsStrong 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

Slide23Some 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)

Slide24Recap

Contact mechanics: contact models, simulation

Form/force closure formulation and testing

Static equilibrium

## Contact Mechanics

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