Degrees of freedom -DOF

Hello again . This is the second lesson of the robotic course. In today’s lesson we will look at an important concept which Degrees Of Freedom.

To recapitulate what we have seen in the last lesson i will give a short definition of the configuration of the robot.

The configuration of a robot is a specification of the positions of all point of the robot. For instance , a point laying on 2D plane . We can describe the position of the point by 2 coordinates (x,y) .Thus the configuration of the robot is q=(x,y) .

Now ,after finishing this short introduction ,let’s move to our main concept which is Degrees Of Freedom or for short DOF .

The number of degrees of freedom of a robot is the minimum number of real-valued coordinates needed to represent its configuration or it can be seen as the number of ways a robot can move. In the previous example the minimum number to represent the configuration is 2 because we have 2 coordinates x and y . Let’s see the example of a coin laying on a plane :

As we saw in the last lesson , the configuration of this coin is q =(x,y,theta) which is the minimum number of real-valued coordinates needed to represent its position .To see why you can check this link. Thus , the number of DOF is 3 .

But if we don’t have the configuration of this coin , how can we determine the number of DOF of this coin ?

Actually there is a formula to define the number of DOF. But i want you to understand how we got it . To do that , let’s take the same example of our coin laying on 2D plane .

First , choose randomly a point on the coin :

Here ,our point is defined by 2 coordinates A(Xa,Ya) ===> 2 DOF. Next we choose another point B(Xb,Yb):

If you say that B has 2 DOF YOU ARE WRONG ! why ?

According to the definition of a rigid body the distance between two point is always constant wherever the coin is . So our point B is restricted to the constraint :

A rule of thumb if you have n coordinates and k constraints then DOF is n-k.

In our example , we have 4 coordinates and 1 constraint .So DOF is 3.

To get the intuition of this result , maintain the position of A constant and move the coin . The only possible positions that B take with the constraint of maintaining A where it is ,is the circle of center A and radius d(A,B) ,look at the image below :

The minimum number of coordinates to describe the position ( or configuration ) of the coin is 3 : (Xa,Ya) and an angle theta that represent how much we turn the coin. Thus , DOF is 3 . Are we done yet ? or when do we stop adding points ? to see how let’s add another point C(Xc,Yc):

In the image above , the possible positions of C with maintaining A and B constants is the intersections of the blue circle with the rainbow circle ,that is 2 positions .These two solutions correspond to the head or tail position .Here we choose head .

Now , we have 3 constraints , d(A,B) must be constant , also d(A,C) and d(B,C) and 5 coordinates (Xa,Ya) ,theta ,(Xc,Yc). But we see here that d(A,C) and d(B,C) eliminate 2 coordinates (Xc,Yc) because these two constraints are not independent . Intuitively, if we maintain A and B in their positions and try to move C there is no other location that C can take so C is fixed (we choose the coin to be in the head position ). So we finish with 3 DOF (Xa,Ya,theta) .

As you can see here , when we add a point and the DOF don’t change we stop adding points ,so we can have only 2 points A and B ,and the DOF that we found is the actual DOF .

If A and B don’t belong to the coin then there won’t be any constraints , and that implies that we will have 6 degrees of freedom . For N points in 2D plane , we have 3N degrees of freedom.

We end up with the formula that i promised to give you :

Degrees of freedom= number of variables(coordinates)-number of independent constraint

This formula can be extended to the 3D space (physical space ) .

To demonstrate that a rigid body in 3D space has 6 DOF let’s do the analogue demonstration by picking a random point A(Xa,Ya,Za) from the the body ; So 3 DOF . We add another point B(Xb,Yb,Zb) this point is subject to the constraint d(A,B) must be constant by maintaining the position of A constant and finding all the position where d(A,B) is constant ,this means that B must lie on the sphere of radius d(A,B) centered at A =====> 3–1=2 DOF , then B can be expressed by 2 coordinates like longitude and latitude , finally we choose another point C(Xc,Yc,Zc) . C is subject to 2 constraints (it must lay on the intersection of the 2 other spheres (we maintained A and B constants)======> 3–2=1 DOF for C . So a rigid body has 6 degrees of freedom (3+2+1=6)

Another important formula is the DOF of a rigid body in the n-dimensional space :

For example the DOF of a rigid body in the 4-dimensional space is 10 :

That’s it for today, next we will see the number of DOF for a robot composed of multiple mechanisms.

Reference :

MODERN ROBOTICS MECHANICS, PLANNING, AND CONTROL by Kevin M. Lynch and Frank C. Park May 3, 2017