2. By The end of this Lecture,
Questions we should be able to answer:
- Types of robot configuration, and robot notations.
- Robot degree of freedom.
- Types of robot frame of reference.
- Robot work space.
4. Number of Degrees of Freedom (DOF)
is equal to
the minimum number
of
parameters specifying the configuration.
So, we can say that:
5. Robots degrees of freedom
is
equal to the dimensions of the configuration
space (The number of independent position
variables) which would have to be specified to
locate all parts of a mechanism.
In most manipulators,
DOF is usually the number of joints (nodes).
6.
7. 1- Mobility or DOF of a spatial manipulator
We’re just going to discuss , specifically, how to
calculate, mathematically, the mobility or the
degree of freedom of a spatial manipulator.
The spatial manipulator is, as we mentioned before,
“A robot has 3 coordinates (x, y, z), and 3
orientations (around x, around y, around z)”.
So, we’re just going to start with the spatial
manipulator which is operating in a 3-D space.
9. Connectivity / Degree of freedom of a joint
In generally, it represents
The number of rigid bodies that can be
connected to a fixed rigid body through a joint
It means:
How many rigid link
can be connected
to one fixed link through a particular joint.
10. So,
Let’s consider a manipulator with (n) rigid moving links and
(m) joints. As we discussed that each rigid body in 3-D space
has got 6 DOF. So, we’ve got n such rigid links. So, we’ve got
6n total DOF for total of n rigid links. As we know the
definition of the connectivity (ci) is:
“ How many link can be connected to one fixed link through
that particular ith- joint”
As mentioned before,
A revolute or prismatic joint has a DOF = 1.
As a rule,
For serial manipulator the connectivity will be ci = 1.
A note that should be memorized: (i….. is the joint digit and
will varies for 1 up to m, where m is the number of joints).
11. For a particular joint, say i-th joint,
If it has the connectivity ci, it is going to put constraint (ξi)
equals:
ξi = 6 – ci
So, the total number of constraints (ξ) will be summation
on i from 1 to m (number of joints). This means:
ξ = σ𝒊=𝟏
𝒊=𝒎
ξi = σ𝒊=𝟏
𝒊=𝒎
(6 – 𝒄𝒊)
Note that ci … for spatial manipulator is equal 1.
We have to know that for any spatial manipulator :
The number of availability is (6n).
12. So,
The mobility of the manipulator (M) is defined as:
the difference between the availability minus the
total number of constraints. Mathematically,
M = 6n – ξ
= 6n – σ𝒊=𝟏
𝒊=𝒎
(6 – 𝒄𝒊)
This particular formula is very well-known as:
Grubler’s Criterion.
By using this particular criterion, very easily we can
find out what should be the DOF of a particular
robotic system.
13.
14. The same technique can be applied to the planar
system same to determine mobility or DOF of a
planar manipulator, which is working on 2-D
plane.
Mobility (DOF) of a planar manipulator
15. In 2-D planar manipulator, we have
For (n) number of moving links and (m) number of joints
that have been considered. The connectivity (ci) for the ith
joint as mentioned before, is:
“ How many link can be connected to one fixed link
through that particular ith- joint”
So, for any serial manipulator, each joint has a
connectivity equals one, (ci = 1).
Now, the number of constraints put by ith joint (ξi) is:
ξi = 3 – ci
Each of the joints is having one connectivity, (ci = 1), and
each of the joints is going to put two constraints (ξi = 2) .
16. Now,
The mobility of the manipulator (M), as mentioned
before, is defined as: the difference between the
availability minus the total number of constraints.
Mathematically,
M = 3n – ξ
= 3n – σ𝑖=1
𝑖=𝑚
(3 – 𝑐𝑖)
So, this is nothing but the mobility of a planar
manipulator. This particular formula, as mentioned
before, is once again the well-known as:
Grubler’s Criterion.
17. Now, a very important remark,
We just want to mention one issue, particularly in
the previous slides (slides of spatial manipulator).
We are using a particular term, that is the mobility.
So,
in place of the term DOF
we’re using the term mobility.
Now, we’ve an issue regrading
. On principle, as we told by
definition,
18. Supposing that
one redundant manipulator is having certain
DOF, truly speaking, we should not call, it is
having 10 DOF. Instead we should say that, it
has got the mobility levels of 10.
Why?
Because, by definition, for spatial manipulator,
the maximum DOF should be equal to 6, and that
is why if it is more than 6, we generally use the
term mobility.
19. So, for the particular redundant manipulator
By definition,
if this serial manipulator has
10 DOF,
that is greater than 6 DOF,
instead of saying that, we will say it is having
mobility level of 10.
22. Example 1
For serial planar manipulator shown in figure,
calculate its DOF or mobility.
23. Solution
joint 1 , 2, and 4 are revolute
joint 3 is a linear (prismatic)
Connectivity is equal 1 for each joint, so ci = 1
So,
= 4
DOF
Number of links
24. Discussion
Now, this serial manipulator has firstly a fixed base. and
finally an end effector. So, let’s try to calculate its DOF or
mobility.
Now, here, (n) is nothing but the
number of moving links for this
example, n = 4 links. So, there are
four number of moving links.
The joints are, firstly revolute joint
then secondly revolute joint, then the
linear (prismatic) joint, the revolute
joint. The number of joints (m) is
equal to m = 4 joints.
4
3
R
2
1
R
R
P
25. Connectivity for each particular joint
We have joint 1 , 2, and 4 are revolute joints, while
joint 3 is a linear (prismatic) one. So, for the
revolute and linear joints, we know that the joints
connectivity is equal 1 for each, so
c1 = c2 = c3 = c4 = 1
So, supposing that this particular serial planar
manipulator, which is working on 2-D plane,
Each joint is having one connectivity,
26. Then, for this serial planar manipulator
The number of constraint (ξi) for each joint is equals:
ξi = 3 – ci
So, the total number of constraints (ξ) which is the
summation on i from 1 to m (number of joints) is going to
be nothing but:
ξ = σ𝒊=𝟏
𝒊=𝒎
(3 – 𝒄𝒊)
= σ𝑖=1
𝑖=4
(3 –1)
= 8
27. So, we have to know
We have to know that for a serial planar
manipulator the number of availability is (3n).
So, the DOF or mobility is calculated using
Grubler’s Criterion as:
M = 3n – ξ
= 3× 4 – 8
Hence, DOF or M is nothing but 4
28. Although,
This is a planar serial manipulator, it is having 4 DOF,
that means, this is:
“One redundant serial planar manipulator”
Another observation we should take. For this serial
manipulator, the DOF or mobility which is equal 4 is
nothing but the sum of all ci values. Since ci’s are
equal and equal 1, and if we sum them up we will be
getting 4. So,
DOF/M = σ𝒊=𝟏
𝒊=𝟒
𝒄𝒊 = 4 (redundant manipulator)
This particular condition is true only for the serial
manipulator, but not for the parallel manipulator.
29.
30. This is very simple parallel manipulator.
Parallel planar manipulator
31. It consists of fixed base, a revolute joint, 3 legs
and a top plate which is nothing but the end-
effector. At each leg we’ve got 2 revolute joints,
and prismatic joint.
Now, we have to count
how many constraint, how
many joints. That we’ll
have to count.
Parallel planar manipulator
32. Now, on each leg
we’ve got 2 links, so for the three legs we’ve got 6
links and the end-effector will be considered as one
link.
So, we ‘ve got a total of 6
links plus the end-effector,
hence that is 7 links.
So, we’ve got (n)
which is the total number
of links is equal 7.
1
2
7
4
3
6
5
33. on each leg we have 3 joints multiplied by 3 the
number of legs, so we’ve got 9 joints. These joints
are, on each leg, we have 2 revolute joints, and one
prismatic joint.
As mentioned before, each
of these joints has connectivity
of 1, (ci = 1). That means each
joint is going to put:
ξi = 3 – ci
= 2 (constraints for
each joint)
Now, we have
R
R
R
R
R
R
P
P
P
34. So, each leg is going to give
two constraints for each joint, multiplied by 3 joints.
Hence, each leg is going to give 6 constraints. For the 3
legs we’ve got 18 constraints.
So, the total number of constraints (ξ) which is the
summation on i from 1 to m
(number of joints) for each leg
is going to be nothing but:
ξ = σ𝒊=𝟏
𝒊=𝒎
(3 – 𝒄𝒊)
= σ𝑖=1
𝑖=3
(3 –1)
= 6 for each leg
For the three legs, ξ = 18
35. Substituting in Grubler’s Criterion
we’ve got:
M = 3n – ξ
= 3n – 3 [σ𝑖=1
𝑖=𝑚
(3 – 𝑐𝑖)]
= 3n – 3 [σ𝑖=1
𝑖=3
(3 –1)]
= 3n – 18
For n = 7 link, we get:
M = 3×7 – 18 = 3 DOF
36. So,
The mobility or DOF
is
coming to be equal to 3
Hence, we can conclude that this manipulator which
is (a parallel planar manipulator)
is
an ideal manipulator
This is the way by which actually we can find out
the degree of freedom or the mobility of different
types of manipulators.
39. Wrist Configurations
Wrist assembly is attached to end-of-arm. End effector is
attached to wrist assembly.
Function of wrist assembly is to orient end effector, while,
Body-and-arm determines global position of end effector.
End effector has three degrees of freedom:
(Roll, Pitch, Yaw).
Notation : RRT
40. Recently,
Many robots today can be designed to move with 7 DOF
+
3DOF in
END EFFECTOR
in
ARM
6DOF
MANIPULATOR
41. Types of Robotic Chains
a) Opened chains b) Closed chains
Open chain serial robot arm Stewart platform (Parallel chain).
42. Types of Robotic Chains
a) Opened chains
In open chains consists of series of
links has an unattached link at the end
of the serial chain.
• As mentioned before:
DOF = σ𝒊=𝟏
𝒊=𝟒
𝒄𝒊 = 4
Assignment:
Prove the relation Example of 4 DOF in open chain
For a robot just by calculating the rotations axes and prismatic axes. In the
example up we have four revolute joints that means we have four degrees of
freedom
43. b) Closed chains
To calculate degree of freedom for closed chains, we need to define
how many links, revolute joints and prismatic joints (linear links) .
prismatic joints
Links
revolute joints
Degree of freedom = 3(n-1)-2 𝐽𝑅 -2 𝐽𝑝 (*)
Where, = 3n - σ𝑖=1
𝑖=𝑚
(3 – 𝑐𝑖)
𝐽𝑅 : Number of revolute joints
𝐽𝑝 : Number of prismatic joints
n: Number of links
44. Example 1
In the example: Degree of freedom can be calculated in this way:
We have 5 links, and 5 revolute joints and zero linear links.
Substituting in equation (*) we get:
3(5-1)-2(5)-2(0)= 2 dof
45. To calculate Stewart platform DOF,
we need to define how many links,
revolute joints and prismatic joints.
Degree of freedom = 6 (n-1) - 3𝐽𝑠 - 4 𝐽ℎ - 5 𝐽𝑅 - 5 𝐽𝑃
Where,
𝐽𝑠 ….. number of spherical joints
𝐽ℎ ….. number of hooks joints
𝐽𝑅 ….. number of revolute joints
𝐽𝑝 … . . number of prismatic joints
n …… number of linkages
Stewart platform (Parallel chains)
49. Robot Workspace (Envelope)
The volume of space
that
a robot operates within
or,
The space around a robot
that is
accessible for
the End Effector or Gripper.
50. As a robot moves around the limits
of its reaches it traces out a specific
shape. The Cylindrical robot in the
middle image has a visible work area
of a cylinder. A Cartesian robot
sweeps out a rectangular volume. A
Polar robot has a partial sphere
workspace.
Robot Workspace (Envelope)
Cylindrical robot
Cartesian robot Spherical robot
SCARA robot
51. Types of Robot Frames
A robot’s World, Joint, and Tool reference
frames.
Most robots may be programmed to
move relative to either of these reference
frames.
𝑥𝑗
𝑦𝑗
𝑧𝑗
𝒙′
𝒚′
𝒛′
𝒚
𝒛
𝒙
52. Advantages of Robots
Robots increase productivity, safety, efficiency,
quality, and consistency of products.
Robots can work in hazardous environments without
the need.
Robots need no environmental comfort.
Robots work continuously without experiencing
fatigue of problem.
Robots have repeatable precision at all times.
Robots can be much more accurate than human.
Robots can process multiple tasks simultaneously.
53. Industrial Robots versus humans
1- Competitive Advantage
Robots can do some things more efficiently and quicker than humans.
2- Mechanical Advantage
• Robots never get sick or need to rest, so they can work 24
hours a day, 7 days a week.
• Greater output per hour with consistent quality
• Continuous precision in repetitive operation.
54. Disadvantages of old version Robots
Robots lack capability to respond in emergencies.
Robots, although superior in certain senses, have limited
capabilities in Degree of freedom, Dexterity, Sensors, Vision
system, real time response.
Robots are costly, due to Initial cost of equipment, Installation
costs, Need for Peripherals, Need for training, Need for
programming.
Robots replace human workers creating economic problems.
55. Today's robots:
• Are creative or innovative
• Can think independently
• Can make complicated decisions
• Can learn from mistakes
• Can adapt quickly to changes in their surroundings
Every successful business must depend on real
people for these abilities.
