Degrees of Freedom

Degrees of Freedom(dof)

“No. of parameters needed to specify completely the spatial pose of a rigid body”

A rigid body in ‘d’ dimensions has  d(d+1)/2 no. of degrees of freedom.

d(d+1)/2        =         d          +           d(d-1)/2

           (dof)                 (translations)          (rotations)

• A non-rigid body has infinite no. of dof.

In 3 dimensional space for a rigid body there are 6 dof.

 

• 6 dof:

1.  Moving up and down(heaving)

2.  Moving left and right( swaying)

3.  Moving forward and backward (pitching)

4.  Tilting forward and backward (pitching)

5.  Turning left and right( yawing)

6.  Tilting side to side (rolling)

Dof for a system: A system is defined to be a set of rigid bodies.Its dof can be calculated by

Dof of system = (Dof of bodies) –(internal constraints they may have on relative motion)

Here the dof represents the no. of parameters needed to specify the spatial pose of the system.  

Eg:1

A human arm has 7 dof.

    I.        Shoulder----3----pitch,yaw,& roll

   II.        Elbow--------1----pitch

  III.        Wrist----------3---pitch,yaw,roll 

·         Only 3 of these movements would be necessary to move the hand to any point in space. But people would lack the ability to group things form different angels or directions. 

·         Total no. of dof is 6 

·         A robot/object that has mechanisms to control all 6 physical dof is said to be holonomic 

·         An object with few dof is said to be non-holonomic. 

·         An object with more controllable Dof( than total dof) such as human arm is said to be redundant.

Eg:2

An air plane is a non-holomonic. Since it has only 4 dof

                    I.        Forward motion

                  II.        Rolling

                III.        Pitch (only some extent)

                IV.        Yaw

Eg:3

A car has max. of 15 dof

15 = 6 DOF for the car body+ 1 for each wheel's vertical motion+ 1 for each wheel spinning + 1 for the engine

Eg :4

A human wrist has 27 dof .

• It is the most complicated design that ever made on the earth and it is the system which has the max. no of dof.

•  4 in each finger ( 3 for extension and flexion and one for abduction and adduction)                                     =                                4 x 4

•  the thumb is more complicated and has 5 Dof     = 5

• 6 DOF for the rotation and translation of the wrist =  6

                                                                                           -------------

                                                                                                 27        

                                                                                         -----------              

• Human Hand Not only design but also animating the wrist movements is so difficult such that they animate with only 12 dof neglecting the 15 other dof.

• It clearly unclouded that He is the best designer than any one who born on the earth