Multifingered Hand Kinematics
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In this chapter, we study the kinematics of a multifingered robot hand grasping an object. Given a description of the fingers and the object, we derive the relationships between finger and object velocities and forces, and study conditions under which a grasp can be used to manipulate an object. In addition to the usual fixed contact case, we also include a complete derivation of the kinematics of grasp when the fingers are allowed to roll or slide along the object.
Chapter Summary
The following are the key concepts covered in this chapter:
- A contact is described by a mapping between forces exerted by
a finger at a point on the object and the resultant wrenches in some
object reference frame. The contact basis
describes the set
of wrenches that can be exerted by the finger, written in the contact
coordinate frame. For contacts with friction, the friction cone
models the range of allowable contact forces that can be applied. The
friction cone satisfies the following properties:
-
is a closed subset of 
with non-empty interior.
-
for 
.
-
- A grasp is a collection of fingers which exert forces on
an object. The net object wrench is determined from the individual
contact forces by the relationship
, where 
is the grasp map:
is a the wrench transformation
between the object and contact coordinate frames. The contact forces
must all lie within the friction cone 
.
- A grasp is force-closure when finger forces lying in the
friction cone span the space of object wrenches
A grasp is force-closure if and only if the grasp map is surjective and there exists an internal force
which satisfies 
and 
.
- The fundamental grasp constraint describes the relationship
between finger velocity and object velocity:
where
is the vector of finger joint angles and
is the configuration of the object frame relative to
the palm frame.
The hand Jacobian 
is defined as
where
is the spatial Jacobian for the \th{i} finger
and 
is the twist transformation between the base
and contact frames. For contacts in which rolling does not occur, 
is a constant matrix.
- The relationships between the forces and velocities in a multifingered grasp are summarized in the following diagram: \begin{center} \input \figdir/graspCD.pst \end{center}
- A grasp is manipulable when arbitrary motions can be
generated by the fingers
A force-closure grasp is manipulable if and only if
is surjective.
- The contact kinematics describe how the contact points move
along the surface of the fingers and object.
For an individual rolling contact, the contact kinematics are
where
are the geometric parameters for a given
coordinate chart on the surface.
The contact kinematics allow and
to be computed using 
rather than solving for 
in terms of 
.
