Grasps

Contacts

Hard Contact

Force exerted only normally to the surface

Frictional Contact

Force can be exerted normally and tangentially

Soft Contact

Finger can exert a pure torsional moment

Closure Properties

Equilibrium

Defintion

• The contact forces and moments satisfy the contact constraints

• The object is in static equilibrium : Wc + g = 0 with g external wrench, W wrench matrix (what is really the wrench matrix?), c vector of wremch intensities

• sum of normal forces \ge 0

• Frictional Coulomb inequality if frictional contact

Force closure

For any arbitrary wrench it is in equilibrium

The forces applied at the contact points ensure the object immobility

Force closure implies that for any arbitrary wrench \w there exists an intensity vector \lambda such that W\lambda = \w

Force closure =/=> stability (what happens when perturbation from equilibrium occurs)

But some ways of filtering good force closures is to look for regions where neighbor candidates also yield force closure , another mentionned approach is to test for stability by perturbing the system according to Data-Driven Grasp Synthesis - A Survey.

Form closure

Force closed with frictionless contact:

\exist \lambda > 0 such that W\lambda = 0

Form closure ==> Force closure

Force Analysis

internal forces : contact forces lying in the nullspace of the grasp matrix W

Exists a unique decomposition of grasp forces into internal and equilibrating forces.

force distribution problem : find actuator or contact forces required by the task while preventing slip

Stability

Grasp is stable if the stiffness matrix is positive definite (stiffness matrix?)

Independent contact regions (ICRs)

Set of regions where each finger can be independently placed anywhere without the grasp loosing force-closure property

Interesting because provides robustness to the grasp in front of positioning errors.

Equilibrium

Equilibrium grasp : given a set of point fingers p_1 .. p_n in contact with an object with contact normals n_1 ...n_n, there is a frictionless equilibrium grasp if there is a linear combination of the normals the sums to 0 with all coeffs of the linear combination positive and at least one of them not zero.

<==> to form closure ?

Grasp quality

Several exist, for instance

• maximal wrench that can be resisted, regardless of direction

• task-specific quality

• associated with hand configuration

Defintions

Configuration space

Definition : refers to the set of reachable positions by a robot's end-effector.

For a multi-finger robot, with n fingers in the workspaces $W_n \simeq |R^d, i \in [1, n]$ , the configurations space is $W_1 x ... x W_n \simeq (|R^d)^n$

The free workspace for each finger is $W_i \ int[O]$ where O is the object to manipulate

The configuration space obstacle is O^M = ${p_1 ... p_n} \in M | \exists p_i \in O$

Caging

Definition: configuration that insures the object will move with the cage (follow), although the object is not immobilized in the cage. Alternative def : configuration in which the mobility of the object is bound.

Polytope

Definition : bounded polyhedron where polyhedron is composed of a finite number of intersections of half-spaces : spaces that can be defined as {Ax <= b}

projection of polytope

{x1 such that there exist x1 such that [x1, x2] belongs to polytope}

MIT slides for more about polytopes

• includes explanation of Fourrier Moltzkin elimination

Physics

Wrench ou torseur (d'actions mécaniques)

At point Pi : {force, torque}_Pi

Twist

At point Pi : {linear velocity, angular velocity}