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 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 fixture object

Frictionless dynamics fixture object a

Post impact velocity Forces at COM Torques about COM

Matrix formulation

Complementarity constraints Note relationship to virtual work!

Frictional contact n n

Quadratic constraint model

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 closureNot 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 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

Higher-order form closure This doesn’t always work… sometimes there are nonzero vectors x with N T G T x = 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 mgmg f1f1 f2f2 Force balance Torque balance 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