Degrees of freedom of a Robot
Until now , we covered the configuration of a robot and DOF of a rigid body . In today’s lesson we will look at the DOF of a robot .
As we saw earlier ,the majority of robots consist of multiple mechanisms (joints + links). There are several types of joints :
Revolute joint (R) called also Hinge joint allowing rotational motion . So it have 1 DOF .While Prismatic joint (P) allows translational ( rectilinear ) motion (1 DOF) . There is also helical joint ( allows rotation and translation abouta screw axis ) , cylindrical ,universal , and spherical joint ( called also ball and socket ). Below , you can see the difference between the joints and their respective DOF.
Now that we have seen the most popular joints let’s see how to calculate the DOF of a mechanism as shown below :
The number of DOF of such a mechanism is calculated using Chebychev–Grübler–Kutzbach criterion or for short Grübler’s formula, which states that for a mechanism having N links (considering the ground as a link ) ,J joints ,let m be the number of DOF of a rigid body (m=3 for planar mechanism and m=6 for a spatial mechanism), and fi be the number of freedoms provided by joint i and Ci the number of constraints provided by joint i where m=fi+Ci. Then Grübler’s formula:
This formula holds only if all joint constraints are independent .
Let’s take an example :
This is the famous 4-bar mechanism. In this structure we have N=4 links ( including the ground ) ,J=4 joints ,m=3 because it is in the plane ,and fi=1 because there are only revolute joints . Now let’s apply Grübler’s formula:
dof = 3(4–1–4)+1*4=1
let’s see another example :
For this example , we can see it in 2 ways :
The first way is we can consider that this mechanism has 3 revolute joints and one prismatic joint so J=4 and each fi=1 and N=4 links (including the ground) then dof =3(4–1–4)+4*1=1 .
Or we can analyse it with a different approach which is : 2 revolute joints (fi=1) and one RP joint (fi=2) (concatenation of revolute joint and a prismatic joint ) .So J=3 and N=3 applying Grübler’s formula we obtain the same result which is DOF =1.
Grübler’s formula doesn’t work when the constraints provided by the joints are not independent . Let’s take the last example to illustrate this issue :
If we apply Grübler’s formula we will obtain 0 .So our body doesn’t move , but it is clearly that our mechanism has 1 DOF . The problem here is that the link in the middle is redundant and the joint does not provide any independent constraints .This means that our link in the middle is related to the motion of the other links and we can view it as there is no link in the middle .
Before we end up this lesson i want to distinguish between 2 types of mechanism :
1- open-chain mechanism (serial mechanism ) : is any mechanism with an open loop like :
2- closed chain mechanism which is the opposite of the serial mechanism :
And that’s it for today , see you in the next lesson when we will talk about the the configuration space