1.4 Degree of Freedom of Mechanisms -2
In the previous examples, we have seen that the degree of freedom does not depend on the link lengths or the free parameter used. We must be able to determine an equation that relates the degree of freedom of a mechanism with the number of links, number of joints and the degree of freedom of the joints. To express these quantities in mathematical terms let us define:
=Degree
of freedom of space
=3
planar space
=
6 general (spatial) space
l= The number of links in a mechanism (including the fixed link)
j = The number of joints in a mechanism
fi =The degree of freedom of the i th joint in the mechanism
F = The degree of freedom of the mechanism
First consider l
links floating freely (no joints!!) in a space with
freedom. In such a case, apart from the fixed link (the fixed reference
frame is attached to this link. Therefore no parameter is required to
determine the position of this link), we will need
parameters for each link. Since there are l-1 floating links, when
there are no joints in the mechanism, the number of parameters required
to determine the position of every link is:
(l-1)
............(1.1 a)
Now, consider the
joints by utilizing a simple example. In the above figure there are four
floating links in planar space. If there are no joints, then the number
of parameters requires to determine the position of these links will be
3*4=12 . (A).If link 2 is connected to links 4 and 5 by revolute joints,
and if there is a cylinder in slot joint between links 2 and 3, the number
of parameters required to determine the position of these four links will
be less. We will still need 3 parameters to determine the position of
link 2. Once the position of link 2 is known, links 4 and 5 can only rotate
relative to link 2, and link 3 can rotate and translate along the slot
axis relative to link 2. To locate the position of link 4 we need 
angle (1 knew parameter), to locate the posion of link 5 we need 
(1 knew parameter) and to locate the position of link 3 we need 2 knew
parameters (s ve 
.
Thus the total number of parameters required is 3+1+1+2= 7 instead of
12, when there were no joints. In planar space we need 2 less parameters
for the revolute joint and one less parameter for the cylinder in slot
joint. This means, we don't need to define a parameter in the direction
for which the motion is constrained.
If the degree of
freedom of space is a kinematic
joint with fi degrees of freedoms constrains, ( -
fi ) degrees of freedom and we need not define this
many number of parameters. Since there are different joints with different
degrees of freedom, in a mechanism with j joints, the total number of
freedoms constrained by all the joints will be:
..........(1.1
b)
The degree of freedom of the mechanism will then be the degree of freedom of all the links without joints minus the degrees of freedom constrained by the joints.
F = Degrees of freedom without any joint (1.1a.) - constraints imposed by the joints(1.1b)
or
This equation is
known as the 
"General Degree-of-freedom Equation".
Although this equation is applicable to most of the mechanisms, there are exceptions, i.e. the degree-of-freedom of the mechanism may not obey the general degree-of-freedom equation. This is mainly due to the assumption that the constraints imposed by the joints are independent of each other, and the degree-of-freedom of the space is constant for the whole mechanism. Before discussing these exceptions let us see some mechanisms that obey the general degree-of-freedom equation.
