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.

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