1. IPE-409 CAD/CAM
Dr. Nafis Ahmad
Professor
Department of IPE, BUET
Email:nafis@ipe.buet.ac.bd

2. 7.
Industrial Robots:
Manipulator Kinematics
2

3. 3
Introduction
 Manipulator
Kinematics
 Accuracy and
repeatability with
which the robot can
position its end
effector

4. 4
Introduction

5. 5
Manipulator Kinematics
 Manipulator kinematics is concerned with the position and
orientation of the robot's end of-arm or the end effector
attached to it as a function of time but without regard for
the effects of force or mass.
 Kinematic analysis neglects the effect of mass of the
manipulator's links and joints, the end effector and load
being carried on position and orientation.
 Here discussion on manipulator kinematics will be limited to
the mathematical representation of the position and
orientation of the robot's end-of-arm

6. 6
Manipulator Kinematics
• Figure 1: Two manipulators with two degrees-of-freedom: (a) an OO
robot and (b) an RR Robot. The robot manipulator consists of a sequence
of joints (J1, J2 ) and links (L1, L2)

7. 7
Joint space method vs. world
space method
• The values of the positions of the joints relative to their
respective inputs links as shown in Figure 1[a] are λ1 and λ2
. In Figure 1[b] the values are θ1 and θ2.
• Figure 1[a]: Pj = ( λ1 , λ2 ) …...(5.1)
• Figure 1[b]: Pj = ( θ1 , θ2 ) …...(5.2)
• This is known as joint space method of representation,
because it defines position and orientation (symbolized as
Pj) in term, of the joint values.
• Position can also be represented by Cartesian or World
Coordinate known as world space method of representation.

8. 8
World space method
• Position by Cartesian or World Coordinate known as world
space method of representation. The end-of-arm position
Pw is defined in Cartesian or World Coordinate as :
• Pw = ( x , z )
• For a robot with six joints operating in 3-D space, the end-
of-arm position and orientation Pw, can be defined as
• Pw = (x, y, z, α, β, γ)
• where x, y, and z specify the Cartesian coordinates in world
space α, β and γ specify the angles of rotation of the three
wrist joints (orientation).

9. 9
World space method
• Orientation cannot be independently established for our
two robots in Figure 1.
• For the OO manipulator, the end-of-arm orientation is
always vertical; and
• For the RR manipulator, the orientation is determined
by the joint angles θ1 and θ2.
• RR robot has two possible ways of reaching a given set
of x and z coordinates, and so there are two alternative
orientations of the end-of-arm that are possible for all
x-z values within the manipulator's reach except for
those coordinate positions making up the outer circle of
the work volume when θ2 is zero. Figure 2.

10. 10
Manipulator Kinematics
Figure 2: The two alternative pairs of joint values for RR
robot

11. 11
Forward and Backward
Transformation
• Mapping from joint space to world space is called forward
transformation and converting from world space to joint
space is called backward transformation.
• The forward and backward transformations are readily
accomplished for the Cartesian coordinate robot.
• Forward transformation : x = λ2 and z = λ1
• Backward transformation : λ1= z and λ2 = x
• Where x and z are the coordinate values in world space and
λ1 and λ2 are the values in joint space.

12. 12
Forward and Backward
Transformation
• For the RR robot of the forward transformation is calculated by
noting that the lengths and directions of the two links might be
viewed as vectors in space :
• r1 = {L1 cos θ1 , L1 sin θ2} ; r2 = {L2 cos (θ1 + θ2), L2 sin (θ1 + θ2)}
• Vector addition of r1 and r2 (and taking account of link L0 ) yields
the coordinate values of x and y at the end-of-arm:
• x = L1 cos θ1 + L2 cos (θ1 + θ2 ) ; z = L0 + L1 sin θ1 + L2 sin (θ1 + θ2
)
• Given the link values L1 and L2 , the following equations can be
derived for the two angles θ1 and θ2 :

13. 13
Forward and Backward
Transformation
• Forward and Backward Transformation for a Robot with
Three Joints:
• For the forward transformation, we can compute the x
and z coordinates in a way similar to that used for the
previous RR robot. The values of x and z can be
computed as follows:
• x = L1 cos θ1 + L2 cos (θ1 + θ2 ) + L3 cos (θ1 + θ2 + θ3 )
• z = L1 sin θ1 + L2 sin (θ1 + θ2 ) + L3 sin (θ1 + θ2 + θ3 )
• (0,0)--?
• The angle made by the wrist with the horizontal:
• α = θ1 + θ2 + θ3

14. 14
Forward and Backward
Transformation
Forward and Backward Transformation for a Robot with Three
Joints. Coordinates of joint 3
X3 = x – L3 cos α
Z3 = z – L3 sin α
θ3 = α – (θ1 + θ2 )

15. 15
Examples
Example 7.1: Given the world coordinates for a RR:R
robot (similar to earlier one) as x = 300 mm, Z = 400
mm, and α = 30°;and given that the links have values L1
= 350 mm, L2 = 250 mm and L3 = 50 mm, determine
the joint angles θ1, θ2 and θ3

16. 16
Manipulator Kinematics
The first step is to find X3 and Z3 using given coordinates
x = 300 and z = 400.
X3 ~ 300 - 50 cos30 = 256.7, Z3 = 400 - 50 sin30 = 375
θ3 can be determined as follows:

17. 17
Robot with 4 DOF

18. 18
Robot with 4 DOF

19. 19
Accuracy and repeatability
The capacity of the robot to position and orient the
end of its wrist with accuracy and repeatability is an
Important control attribute in nearly all industrial
applications. There are several terms that must be
defined in the context of this discussion:
(1) control resolution,
(2) accuracy, and
(3) repeatability.

20. 20
Accuracy and repeatability
Control resolution refers to the capability of the robot's
controller and positioning system to divide the range of
the joint into closely spaced points that can be identified
by the controller.
These are called addressable points because they
represent locations to which the robot can be commanded
to move. The capability to divide the range into
addressable points depends on two factors:
(1) limitations of the electromechanical components that
make up each joint-link combination and
(2) the controller's bit storage capacity for that joint.

21. 21
Accuracy and repeatability
(2) Accuracy: Accuracy is a measure of the robot's ability to
position the end of its wrist at a desired location in the
work volume.
(3) Repeatability: Repeatability is a measure of the robot's
ability to position its end-of-wrist at a previously taught
point in the work volume.

22. Thanks!
Any questions?
You can find me at:
@ahmadn
nafis@ipe.buet.ac.bd
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