Manipulation and grasping are branches of robotics and involve notions from kinematics, mechanics, and CONTROL THEORY. Grasping is concerned with characterizing and achieving the conditions that will ensure that a robot gripper holds an object securely, preventing, for example, any motion due to external forces. Manipulation, on the other hand, is concerned with characterizing and achieving the conditions under which a robot or a part held by a robot will perform a certain motion. Research in both areas has led to practical systems for picking up parts from a conveyor belt or a pallet, reorienting them, and inserting them into an assembly (e.g., Tournassoud, Lozano-Perez, and Mazer 1987; Peshkin and Sanderson 1988; Goldberg 1993), with promising applications in flexible manufacturing.

This entry focuses on a quasi-static model of mechanics that neglects inertial forces and dynamic effects. This is valid in typical grasping and manipulation tasks when all velocities are small enough, and allows for a geometric analysis of object motion under kinematic constraints. Our discussion of manipulation is restricted to the problem of characterizing the motion of an object pushed by one or several fingers, and it excludes some fundamental problems such as general robot motion planning in the presence of obstacles.

Grasping emerged as a field of its
own in the early eighties with the introduction of dextrous multifinger
grippers such as the Salisbury Hand (Salisbury 1982) and the Utah-MIT
Dextrous Hand (Jacobsen et al. 1984). Much of the early work was
conducted in Roth's research group at Stanford (e.g., Salisbury
1982; Kerr and Roth 1986) drawing on notions of form and force closure
from *
screw theory* (Ball 1900), which provides a unified representation
for displacements and velocities as well as forces and torques using
a line-based geometry. Namely, when a hand holds an object at rest,
the forces and moments exerted by the fingers should balance each
other so as not to disturb the position of this object. Such a grasp
is said to achieve *equilibrium.* An equilibrium grasp achieves *force closure* when
it is capable of balancing any external force and torque, thus holding
the object securely. A *form closure* grasp achieves the
same result by preventing any small object motion through the geometric
constraints imposed by the finger contacts. Intuition suggests that
the two conditions are equivalent, and it can indeed be shown that
force closure implies form closure and vice versa (Mishra and Silver
1989). A secure grasp should also be *stable;* in particular,
a compliant grasp submitted to a small external disturbance should return
to its equilibrium state. Nguyen (1989) has shown that force or
form closure grasps are indeed stable.

Screw theory can be used to show that, in the frictionless case, four or seven fingers are both necessary and, under very general conditions, sufficient (Lakshminarayana 1978; Markenscoff, Ni, and Papadimitriou 1990) to construct frictionless form or force closure grasps of two- or three-dimensional objects, respectively. As could be expected, friction "helps" and it can also be shown that only three or four fingers are sufficient in the presence of Coulomb friction (Markenscoff, Ni, and Papadimitriou 1990). In fact, it can also be shown that any grasp achieving equilibrium for some friction coefficient µ will also achieve form or force closure for any friction coefficient µ' > µ (Nguyen 1988; Ponce et al. 1997).

Screw theory can also be used to characterize the geometric arrangement of contact forces that achieve equilibrium (and thus form or force closure under friction). In particular, two forces are in equilibrium when they oppose each other and share the same line of action, and three forces are in equilibrium when they add to zero and their lines of action intersect at a point. The four-finger case is more involved, but a classical result from line geometry is that the lines of action of four noncoplanar forces achieving equilibrium lie on the surface of a (possibly degenerated) hyperboloid (Ball 1990). In turn, these geometric conditions have been used in algorithms for computing optimal grasp forces given fixed finger positions (e.g., Kerr and Roth 1986), constructing at least one (maybe optimal) configuration of the fingers that will achieve force closure (e.g., Mishra, Schwartz, and Sharir 1987; Markenscoff and Papadimitriou 1989), and computing entire ranges of finger positions that yield force closure (e.g., Nguyen 1988; Ponce et al. 1997). The latter techniques provide some degree of robustness in the presence of the unavoidable positioning uncertainties of real robotic systems.

As shown in Rimon and Burdick (1993),
for example, certain grasps that are not form closure nevertheless
immobilize the grasped object. For example, three frictionless fingers
positioned at the centers of the edges of an equilateral triangle
cannot prevent an infinitesimal rotation of the triangle about its
center of mass, although they can prevent any finite motion. Rimon and Burdick
(1993) have shown how to characterize these grasps by mapping the
constraints imposed by the fingers on the motion of an object onto
its *configuration space,* that is, the set of object positions
and orientations. In this setting, screw theory becomes a *first-order* theory
of mobility, where the curved obstacle surfaces are approximated
by their tangent planes, and where immobilized object configurations
correspond to isolated points of the free configuration space. Rimon
and Burdick have shown that second-order (curvature) effects can
effectively prevent any finite object motion, and they have given operational
conditions for immobilization and proven the dynamic stability of
immobilizing grasps under various deformation models (Rimon and Burdick
1994). An additional advantage of this theory is that second-order
immobility can be achieved with fewer fingers than form closure (e.g.,
four fingers instead of seven are in general sufficient to guarantee
immobility in the frictionless case). Techniques for computing second-order
immobilizing grasps have been proposed in Sudsang, Ponce, and Srinivasa
(1997) for example .

Once an object has been grasped,
it can of course be manipulated by moving the gripper while keeping
its fingers locked, but the range of achievable motions is limited
by physical constraints. For example, the rotational freedom of
a gripper about its axis is usually bounded by the mechanics of
the attached robot wrist. A simple approach to fine manipulation
in the plane is to construct *finger gaits* (Hong et al.
1990). Assume that a disk is held by a four-finger hand in a three-finger
force closure grasp. A certain amount of, say, counterclockwise
rotation can be achieved by rotating the wrist. To achieve a larger
rotation, first position the fourth finger so that the disk will
be held in force closure by the second, third, and fourth fingers,
then release the first finger and reposition it. By repositioning
the four fingers in turn so that their overall displacement is clockwise,
we can then apply a new counterclockwise rotation of the wrist,
and repeat the process as many times as necessary. (See Li, Canny,
and Sastry 1989 for a related approach to dextrous manipulation,
which includes coordinated manipulation, as well as rolling and
sliding motions.)

Thus far, our discussion has assumed
implicitly that a workpiece starts *and remains* at rest
while it is grasped. This will be true when the part is very heavy
or bolted to a table, but in a realistic situation, it is likely
to move when the first contact is established and contact may be
immediately broken. Moreover, the actual position and orientation
of the object with respect to the hand are usually (at best) close
to the nominal ones. Mason (1986) proposed that an appropriate characterization
of the mechanics of pushing would at the same time provide the means
of (1) predicting (at least partially) the motion of the manipulated
object once contact is established, and (2) reducing the uncertainty
in object position without sensory feedback. Assuming Coulomb friction,
he constructed a program that predicts the motion of an object with
a known distribution of support forces being pushed at a single
contact point. He also devised a simple rule for determining the
rotation sense of the pushed object when the distribution is unknown.

Extensions of this approach have
been applied to a number of other manipulation problems. Fearing
(1986) has shown how to exploit local tactile information to determine
how a polygonal object rotates while it is grasped, and demonstrated
the capture of an object by the three-finger Salisbury hand as well
as the execution of other complex manipulation tasks such as part "twirling." In
the manufacturing domain, Peshkin and Sanderson (1988) have shown
how to use static fences to reorient parts carried by a conveyor
belt, and Goldberg (1993) has used a modified two-jaw gripper to
plan a sequence of grasping operations that will reorient a part with
unknown original orientation. A variant of this approach has also
been used to plan a set of tray-tilting operations that will reorient
a part lying in a tray (Erdmann and Mason 1988). More recently, Lynch
and Mason (1995) have derived sufficient conditions for *stable
pushing,* namely, for finding a set of pushing directions that
will guarantee that the pushed object remains rigidly attached to
the pusher during the manipulation task. They have also proven conditions
for local and global controllability, and given an algorithm for
planning pushing tasks in the presence of obstacles.

The kinematics of pushing are important
as well, because they determine the relative positions and orientations
of the gripper-object pair during the execution of a manipulation
task. Brost (1991) has shown how to construct plans for pushing
and compliant motion tasks through a detailed geometric analysis
of the obstacle formed by a rigid polygon in the configuration space
of a second polygon. More recently, Sudsang, Ponce, and Srinivasa
(1997) have introduced the notion of *inescapable configuration
space* (ICS) region for a grasp. As noted earlier, an object
is immobilized when it rests at an isolated point of its free configuration
space. A small motion of a finger away from the object will transform
this isolated point into a compact region of free space (the ICS)
that cannot be escaped by the object. For simple pushing mechanisms,
it is possible to compute the maximum ICS regions and the corresponding
range of finger motions, and to show that moving the finger from
the far end of this range to its immobilizing position will cause the
ICS to continuously shrink, ensuring that the object ends up in
the planned immobilizing configuration. Thus a grasp can be executed
in a robust manner, without requiring a model of the part motion
at contact. More complex manipulation tasks can also be planned
by constructing a graph of overlapping maximum ICS regions. This
approach has been applied to grasping and in-hand manipulation with
a multifingered reconfigurable gripper (Sudsang, Ponce, and Srinivasa
1997), and more recently, to manipulation tasks using disk-shaped
mobile platforms in the plane.

- BEHAVIOR-BASED ROBOTICS
- HAPTIC PERCEPTION
- MOBILE ROBOTS
- ROBOTICS AND LEARNING
- WALKING AND RUNNING MACHINES

- Coordinated Manipulation of Objects in a Plane - Rus (ResearchIndex)
- Nat'l Academy Press, Motion, Control, and Geometry: (1997), 3: GEOMETRIC PHASES, CONTROL THEORY, AND ROBOTICS
- Vision and Touch Guided Manipulation

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