In the previous video, we learned that the
number of degrees of freedom of a robot is
equal to the total number of freedoms of the
rigid bodies minus the number of constraints
on their motion.
The constraints on motion often come from
joints.
The most common type of joint is the revolute
joint.
It places 5 constraints on the motion of the
second spatial rigid body relative to the
first, and therefore the second body has only
one degree of freedom relative to the first
body, given by the angle of the revolute joint.
Another common joint with one degree of freedom
is the prismatic joint, also called a linear
joint.
We can also have joints with more than one
degree of freedom, like this universal joint,
which has two degrees of freedom.
The spherical joint, also called a ball-and-socket
joint, has three degrees of freedom: the two
degrees of freedom of the universal joint
plus spinning about the axis.
This table summarizes the previous four joints,
plus two other types of joints, the one-degree-of-freedom
helical joint and the two-degree-of-freedom
cylindrical joint.
This table shows the number of degrees of
freedom of each joint, or equivalently the
number of constraints between planar and spatial
bodies.
Using this table of freedoms and constraints
provided by joints, we can come up with a
simple expression to count the degrees of
freedom of most robots, using our formula
from Chapter 2.1.
Let's say the robot has N links.
By historical convention, N includes ground
as a link.
The robot has J joints.
And we define m to be the degrees of freedom
of a single body, so m equals 3 for a rigid
body moving in the plane and m equals 6 for
a rigid body moving in 3-dimensional space.
We can write our equation in terms of these
variables: N-1 is the number of links other
than ground, and m times N-1 is the total
number of freedoms of the bodies if they are
not constrained by joints.
Then we subtract off the constraints provided
by the J joints.
Since the number of constraints provided by
joint i is equal to m minus the number of
freedoms allowed by joint i, we can replace
ci by m minus fi and rewrite the equation
like this.
Rearranging once more, we get this.
This is called Grubler's formula, and it assumes
that the constraints provided by the joints
are independent.
Let's apply Grubler's formula to a few mechanisms.
The first mechanism is called a serial, or
open-chain, robot, because there is a single
path from ground to the end of the robot.
It's called a 3R robot, meaning it has three
revolute joints.
This planar robot has, m=3, N=4, J=3, and
one freedom at each joint.
Grubler's formula tells us, 3(4-1-3)+3=3.
The robot has 3 degrees of freedom, as we
expect.
The next mechanism is called a four-bar linkage,
obtained by pinning the endpoint of the 3R
robot to a particular location in the plane.
This is called a closed-chain mechanism, because
there's a closed loop.
As before, we have, m=3 and N=4, but now we
have J=4 joints.
Grubler's formula tells us that this mechanism
has, 3(4-1-4)+4, is equal to one degree of
freedom.
We would also predict this by the fact that
pinning the endpoint of the 3R robot to a
particular x-y location creates two constraints,
so we can subtract 2 from the 3 freedoms of
the 3R robot to see that there is one degree
of freedom.
The next mechanism is like the four-bar, except
now it adds one more link and two more joints.
Grubler's formula would tell us that this
mechanism has zero degrees of freedom, but
that's wrong; it still has one degree of freedom,
just like the four-bar.
The reason that Grubler's formula does not
apply is that the joint constraints are not
independent.
Testing whether joint constraints are independent
is not an easy task, and we won't pursue it
further.
Finally, we have a spatial closed-chain mechanism
called a Stewart platform.
It has 6 legs connecting the bottom platform
to the top platform, and each leg consists
of two links and a universal joint, a prismatic
joint, and a spherical joint.
The prismatic joints are actuated, creating
motion of the top platform as you see in the
video.
Since each leg has 2 links, there is a total
of 12 links in the legs, and adding ground
and the top platform makes 14 links total.
Each leg has 3 joints with 6 degrees of freedom
total, for a total of 18 joints with 36 total
freedoms.
The mechanism moves in 3-dimensional space,
making m equal to 6.
Grubler's formula tells us the Stewart platform
has, 6(14-1-18)+36, is equal to 6 degrees
of freedom.
The top platform can be moved with all 6 degrees
of freedom of a rigid body.
There are limits to the range of motion, of
course, but these limits do not reduce the
number of degrees of freedom.
In the next video we will explore another
important property of a configuration space:
its topology.
