Degrees of Freedom of Mechanisms: Mobility Analysis of Planar Linkages

Degrees of Freedom of
Mechanisms
Mobility Analysis of
Planar linkages
Mobility of Mechanisms
Mobility = Degrees of Freedom#
 The number of independent coordinates needed to

describe the position and orientation of all rigid
elements of a mechanism.
 Mobility of a mechanism is the number of input parameters

(pair variables) that must be independently controlled to
bring the device into a particular position.
#Do not confuse with DOF of joints

Understanding Mobility
The motion of a mechanism is fully determined,

if the motion of one or more of its links is
prescribed. The number of links, driving its
motion, corresponds with the number of
degrees of freedom (DOF) of the mechanism.
The number of degrees of freedom determines
the number of required actuators as well as the
boundaries within which a body (linked to the
output link) can be manipulated.
Animated Robotic Manipulator
No of Degrees of Freedom=?
Constraints generated by joints
y
Dy
Dq Dx Dq
hinge
x
Unconstrained rigid body constrained rigid body
in a plane in a plane
 A rigid body in plane has maximum 3 degrees-of-freedom
 A simple hinge imposes 2 constraints
Degrees of Freedom removed by Joints
Gruebler criterion
DOF of a mechanism
F=3(n-1)-2p
n= number of links
p= number of 1-dof kinematic pairs
Kutzbach Criterion
for planar mechanisms with n-1 movable links and 1 fixed link
F  3(n 1)  2 p1  p2
F: Degrees of freedom (or mobility)
n: Number of links
p1: Number of joints of single degree-of-freedom
p2: Number of joints of two degrees-of-freedom
F 1 F 0 F 0
Structure Statically
Mechanism
indeterminate
structure
Proof of Kutzbach Criterion
1. An unconstrained rigid body has 3 degrees-of-freedom

2. (n-1) movable unconstrained links have 3(n-1) degrees
of freedom
3. A 1-dof joint imposes 2 constraints and hence removes
2-dof
4. A 2-dof joint imposes 1 constraint and hence removes 1
dof
Notes on using Kutzbach
criterion
 Fixed link has to be counted
 Higher-order 1-dof joints are to be
replaced by simple hinges according to
the following formula:
p1   iji
i
where ji is the number
of hinges connecting
i+1 links.
Examples of Linkage Mobility
Delta-Triplet
A structural
building block
Example #1
Dump truck
n=4, p1=4,p2=0 => F=3*(4-1)-2*4=1

Transom mechanism (window over door)
Example #2 n=4, p1=4, p2=0 => F=1
Example #3
n=8
p1=8+2=10
2nd Order joint
p2=0
F=3*(8-1)-2*10-0=1
Example #4
n=6
p1=5+2=7
p2=1
F=3*(6-1)-2*7-1=0
Example #6
DOF=?
A well designed spring should

not constrain the DOF. Thus, Note: The spring is replaced
may be omitted in kinematic Dyad as an
by a
diagram
instant equivalent
Paradoxes:
Is Kutzbach Criterion Always Right ?
From Kutzbach Criterion:

Actually for equal link
n=5, p1=6 lengths of 2, 5 and 4
F=3*(5-1)-2*6=0 DOF=1
E-quintet- another structural building block
Friction Wheel
For pure rolling:
n=3, p1=3 =>F=0
For Rolling and sliding:
n=3, p1=2, p2=1=>F=2
What happens when the distance between the pivots are not
equal to the sum of the radii of the wheels?
redundant DOF
1. Let us consider the roller is not fixed to the link 2
n=5
p1=4
p2=2
F=3(5-1)-2*4-2=2
2. Let us consider the roller is fixed to the link 2

n=4
p1=3
p2=2
F=3(4-1)-2*3-2=1
Note: The rotation of the roller does not influence the relationship of the input and
output motion of the mechanism. Hence, the degree of freedom of the roller will not
be considered; It is called a passive or redundant degree of freedom.
Example 5 revisited
n=5
P1=5
P2=1
F=3*(n-1)-2*p1-p2=1
Redundant DOF : examples
DOF=? DOF=?
Redundant links
Redundant link
Redundant joints
Redundant joint
Modified Kutzbach criterion
F  3(n  nr 1)  2( p1  p1r )  p2  Fr
Subscript r denotes redundant component

Closure
 Revolute joints in any If all revolute joints in a
fourbar mechanism is
loop can be replaced by replaced by prismatic
prismatic joints with no joints, it will be a 2-DOF
mechanism
change in DOF of the
mechanism, provided
that at least two If three revolute joints are
replaced by prismatic joints,
revolute joints remain in the revolute joint will not turn
the loop and the DOF of the
mechanism will be zero.
Closure
 Any full joint can be replaced by a half joint,
but this will increase the DOF by one
 Removal of a link will reduce the DOF by one

Degrees of Freedom of Mechanisms: Mobility Analysis of Planar Linkages

Uploaded by

Copyright:

Available Formats

Degrees of Freedom of Mechanisms: Mobility Analysis of Planar Linkages

Uploaded by

Document Information

Original Description:

Original Title

Copyright

Available Formats

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

Available Formats

Degrees of Freedom of Mechanisms: Mobility Analysis of Planar Linkages

Uploaded by

Copyright:

Available Formats

Degrees of Freedom of

Mobility = Degrees of Freedom#

 The number of independent coordinates needed to

 Mobility of a mechanism is the number of input parameters

#Do not confuse with DOF of joints

The motion of a mechanism is fully determined,

1. An unconstrained rigid body has 3 degrees-of-freedom

n=4, p1=4,p2=0 => F=3(4-1)-24=1

A well designed spring should

From Kutzbach Criterion:

2. Let us consider the roller is fixed to the link 2

F  3(n  nr 1)  2( p1  p1r )  p2  Fr

Subscript r denotes redundant component

You might also like