If a rigid body is fully immobilized by a set of rigid stationary fixtures, we say it
is in form closure.
In particular, first-order form closure means that only the zero twist satisfies the impenetrability
constraints for all the contacts.
This condition is equivalent to the condition that the positive linear span of the contact
normal wrenches is the entire wrench space, which is 6-dimensional for spatial bodies
and 3-dimensional for planar bodies.
Remembering that at least n+1 vectors is needed to positively span an n-dimensional space,
this means that first-order form closure requires at least 4 point contacts for a planar body
and at least 7 point contacts for a spatial body.
These are minimum requirements.
Some objects, like a sphere, cannot be form-closure grasped for any number of contacts, as there
is no way to kinematically prevent rotation of the sphere.
This figure shows a planar body with three point contacts, as indicated by the contact
normals.
The body is not in first-order form closure, as it has a non-empty cone of feasible twists,
drawn as rotation centers.
In any case, because there are only 3 contacts, the body cannot be in first-order form closure.
If we add a fourth contact at the top left, the set of feasible rotation centers is reduced
to a small region with a minus label.
The body can still rotate clockwise about any point inside the gray region.
If we change the angle of the fourth contact constraint, however, the feasible rotation
centers vanish and the body is in form closure.
This figure shows a bowtie-shaped planar body in a form-closure grasp by 2 fingers creating
4 contact normals.
Our graphical methods are convenient for visualizing form closure in the plane, but we can also
define a computational test for form closure.
Let F be the matrix of wrenches due to the j contact normals, where each wrench is a
column of the matrix.
Then the contacts create first-order form closure if and only if the rank of the F matrix
is n, where n is 3 for planar bodies and 6 for spatial bodies, and F times k is equal
to zero, where k is a j-vector of positive coefficients multiplying the wrenches.
These two conditions taken together ensure that any wrench can be generated as a positive
linear combination of the individual wrenches.
This test can be implemented as a linear program in any scientific computing environment.
The planar triangle shown here is not in form closure, because the full rank condition is
not satisfied.
The three contact normals cannot prevent pure rotation about the center of the triangle.
Similarly, a first-order analysis tells us that this large triangle is also not in form
closure, because our graphical contact analysis does not rule out rotation about the center
of the triangle.
Finally, a first-order analysis indicates that this concave body can rotate about any
rotation center on the vertical line.
In fact, however, both the large triangle and the concave body are in form closure by
a more detailed analysis of the contact geometry, while the small triangle still is not in form
closure.
By a higher-order analysis, form closure can sometimes be achieved with as few as 2 contacts.
To summarize, if an object is in form closure by a first-order analysis, then it is also
in form closure by a higher-order analysis.
But if a first-order analysis concludes that only sliding and rolling is possible, then
a higher-order analysis may conclude form closure.
You can think of the first-order test as a conservative test for form closure.
This ends our purely kinematic analysis of contact.
In the next video, we will begin to study the forces that can be transmitted through
contacts.