Module 1.

Fundamentals of
Robot Technology
Ø Introduction
Ø Automation and Robotics
Ø Robotics in Science Fiction
Ø Progressive Advancement
Ø The Robotics trends and the future prospects
Ø Robot Anatomy
Ø Links, Joints and Joint Notation scheme
Mechanisms and Robotics Ø Degrees of Freedom (DOF)
Ø Required DOF in a Manipulator
Ø Arm Configuration
DE ES ZG561 Ø Wrist Configuration
Ø The End-Effector (EE)
BITS Pilani Lecture -1 Ø Human arm characteristics
Pilani Campus Ø Arm Configurations
Ø Robot drive systems

Introduction to Automation Introduction

v Ma ss p ro d uc t i on as s e mb l y lin e s we re
introduced in 1905 by Ford Motor Company.
v Specialized machines were designed to
manufacture for high volume production of
mechanical and electrical parts.
v When a production cycle ends and new
models of parts are introduced, production
machines have to be shut down and the
hardware retooled.
v Since periodic modification of the production
hardware is required, this type of automation
is called Hard Automation.
v When programmable mechanical systems are
used to perform manufacturing tasks, new
components can be made just by changing
the program in the machine, This type of
automation is called Soft Automation.

3 4
 Use of the word Robot Robotics in Science Fiction
v The word ''robot comes from the
Czech word ''robota'' which means
forced labor.
v It was first used in 1921 by the
novelist Karel Capek in his novel
''Rossum‘s Universal Robots".
Gort from the movie The
Robot in The Terminator
v The fantasy associated with robotics day the earth stood still.
offered by science fiction movies, Source: www.jeffbots.com

and printed and animated cartoons is

so far from reality that actual
industrial robots seem primitive.
C-3PO and R2-D2 Robots
A robot from Karel Capek's novel
Source: INDUSTRIAL ROBOTS PROGRAMMING, from Star wars
J. Norberto Pires ''Rossum 's Universal Robots Source: www.smithsonianmag.com
5 6

Industrial Robots Automation and Robotics

v Robotics Institute of America defines a robot as a
“reprogrammable multifunctional manipulator designed to move
mat erial, pa rts or specialized devices thro ugh varia ble
programmable motions for the performance of a variety of tasks”.

v A Robot is a software-controllable mechanical device that uses

sensors to guide or more end-effectors through programmed
motions in a workspace in order to manipulate physical objects.
vMaterial handling, v Contrary to popular notions about robots in science fiction
vplasma cutting, literature, today’s industrial robots are not Androids built to
vspot welding, and
v resistance welding The slim design of the Panasonic VR-5 impersonate humans but most are anthropomorphic in the sense
are all done with ease by the six axis AII robot makes it the perfect solution they are patterned after the human arm.
Panasoni c VR- 120H II. This high for compact work areas. This robot
performance robot is controlled by the model is typically outfitted for MIG
easy to use Panasonic robot controller. welding or material handling tasks.
7 8
Industrial Applications Industrial Applications

Robots Performing Spray Painting Robots Performing Arc Welding

9 10

Industrial Applications Medical Applications

Robotic system called Da Vinci, used for heart surgery, programmed to
follow the surgeon’s hand movements accurately without any tremors.

Spot Welding Robots Medical Robot Robot Performing Surgery

11 12
 Search and Rescue Applications Search and Rescue Applications

A robot picking a suitcase from a car

Image Source: https://newatlas.com/
Pacbot – A Robot for military applications
Source: www.army-technology.com

A robot used for disposing

explosive ordnance
https://www.qinetiq.com

13 14

Industrial Robot - History Industrial Robot - History

v The Unimate was the very first industrial robot.
v Conceived from a design for a mechanical arm patented
in 1954 (granted in 1961) by American inventor George
Devol
v The Unimate was developed as a result of the foresight
and business acumen of Joseph Engelberger - the
Father of Robotics.
v In 1961 Engelberger established Unimation, Inc., a
Condec Corp. company in Danbury, Connecticut, to
develop the business in the newly established robotics
industry he created.
Source: Industrial Robotics by Mikell P. Groover

15 16
 Industrial Robot - History Industrial Robot - History
v First industrial Robot produced by Unimation was Unimation also manufactured
installed in General Motors, USA in 1961. Programmable Universal
Machine for Assembly (PUMA
robot).

BITS Pilani, Pilani Campus BITS Pilani, Pilani Campus

Industrial Robot - History Industrial Robot

The Tomorrow Tool (T3 robot) manufactured by Cincinnati
A compact and computer- Milacron Inc. (shown in figure).
controlled robot for
Ø high speed, T3 robot is an articulated six axis robot.
Ø close-tolerance assembly,
Ø light materials handling, and
Ø inspection applications.

CINCINNATI MILACRON T3-776

BITS Pilani, Pilani Campus BITS Pilani, Pilani Campus

 The Robotics trends and Progressive Advancement
the future prospects
v Industry 4.0 is a name given to the current trend of v First Generation Robots
automation and data exchange in manufacturing v Repeating
v Non-servo, Pick and Place
technologies. It includes cyber-physical systems, the
Internet of things, cloud computing and cognitive computing. v Second Generation Robots
v Use of Sensing devices
Industry 4.0 is commonly referre d to as the f ourth
v Response to sensory feedback
industrial revolution
v Third Generation Robots
v Artificial intelligence
v Self learning
v Conclusion drawing from past experiences

v Fourth Generation Robots

v May be an android or a biological robot.

BITS Pilani, Pilani Campus BITS Pilani, Pilani Campus

The Robotics trends and

the future prospects The Robotics trends and the future prospects
v Public Security / Military v Complex mechanical hands with fingers attached to a
robot.
v Healthcare
v Coworkers
v Robots at Home
v Robots in Education
v Entertainment

BITS Pilani, Pilani Campus BITS Pilani, Pilani Campus

The Robotics trends and the future prospects
Coordinate Systems

• Cartesian

Humanoid robot ASIMO

from Honda
Atlas from Boston Dynamics

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Coordinate Systems Coordinate Systems

• Cylindrical • Spherical

27 28
 Robot Anatomy Mechanism Basics
Arm
v Study of the physical construction of Wrist v Mechanisms consist of connected parts with the objective
a Robot. of transferring motion and force from a power source to an
Shoulde output.
v The Manipulator consists of Rigid r
links and Joints.
v One part is designated the frame because it serves as the
v A Joint provides relative motion Elbow
frame of reference for the motion of all other parts. The
between two links.
Manipulator frame is typically a part that exhibits no motion.
Wrist
v Links are the individual parts of the mechanism. They are
considered rigid bodies and are connected with other links
Elbow
to transmit motion and forces
Shoulder
or
Hand End Effector Base

29 30

Degrees of Freedom
Joints
v Links in a mechanism are connected v Degrees of freedom is often used to describe the
using joints. number of directions that a robot can pivot or move a
joint.
v The joints can be of the following
types
v Pin Joint
v Sliding Joint (Prismatic Joint)

Source: Machines and Mechanisms, Applied Kinematic Analysis (Fourth Edition) – David H. Myszka

31 DE ZG/ES ZG561 Mechanisms and Robotics BITS Pilani

Degrees of Freedom Degrees of Freedom
v Consider an open kinematic chain as shown below.
v Hence link-1 can rotate about joint-1 w.r.t. ground link and
contributes one degree of freedom.
J2
L2 v Similarly link-2 can rotate about joint-2 w.r.t. link-1.
L1
v Or link-2 has one degree of freedom w.r.t. link-1.
J1
v Therefore this open loop chain has as many degrees of
freedom (DOF) as the number of joints in the chain, as each
joint is of one DOF.
v Link-1 is connected to the ground link-0 by a revolute
joint.

BITS Pilani, Pilani Campus DE ZG/ES ZG561 Mechanisms and Robotics BITS Pilani

Degrees of Freedom Links, Joints and Joint Notation

Scheme
v Thus in an n-DOF robotic manipulator n joints are used and n Types of Robot Joints
independent joint-link variables are required to completely Type Notation Symbol Description
specify the position and orientation of the each link.
Revolute R Rotary motion about an axis

v Hence for example in B

Linear L Linear motion along an axis
L2
case of two DOF (Prismatic)
L1 C
manipulator shown in
figure two joint-link
variables are required A
to define the position
and orientation of the
end point C.

BITS Pilani, Pilani Campus 36

 Types of Robot Joints Types of Robot Joints
Prismatic Joint

Pin Joint
(Revolute)

37 38

Further classification of
revolute Joints Human Arm Characteristics

prismatic joint L

rotational joint R

revolving joint V

twisting joint T

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 THANK YOU
The presenter is grateful to the authors of the following textbooks.

Machines and Mechanisms, Applied Kinematic Analysis (Fourth Edition) –

David H. Myszka

Industrial Robotics - Mikell P. Groover

Fundamentals of Robotics – Robert Schilling Mechanisms and Robotics

DE ES ZG561
BITS Pilani Lecture - 2
Pilani Campus

41

Fundamentals of Robot Technology Human Arm Characteristics

Robot Anatomy
Ø Arm Configuration
Ø Joint Notation scheme
Ø Degrees of Freedom (DOF)
Ø Required DOF in a Manipulator
Ø Wrist Configuration
Ø Design & Control issues
Ø Manipulation & Control
Ø Robot specifications

43 DE ZG/ES ZG561 Mechanisms and Robotics BITS Pilani

Joints and Joint Notation Scheme Required Degrees of Freedom
Types of Robot Joints • For moving objects placed on a line or a circle  1 dof
Type Notation Symbol Description • For moving objects anywhere on a flat surface  2 dof
Revolute R Rotary motion about an axis • Picking objects anywhere on a 3D space  3 dof
• Reaching in to complex profiles  > 3 dof
Prismatic P or L Linear motion along an axis

45 46

Arm Configurations Arm Configurations

S. No. Configuration Joints

1. Cartesian (3 Linear joints)
2. Cylindrical (2 Linear joints, 1 Rotary Joint)
3. Polar (Spherical) (1 Linear joint, 2 Rotary joints)
4. Articulated (jointed arm) (3 Rotary Joints)

47 48
 Robot Configurations Robot Configurations
• Cartesian • Cylindrical

49 50

Robot Configurations Robot Configurations

• Spherical
• SCARA (Selective compliance assembly robotic arm)
• A variation of the cylindrical configuration involving,
1 linear joint and 2 rotary joints

51 52
 Arm Configurations Arm Configurations

S. No. Configuration Joints

1. Cartesian (3 Linear joints)
2. Cylindrical (2 Linear joints, 1 Rotary Joint)
3. Polar (Spherical) (1 Linear joint, 2 Rotary joints)
4. Articulated (jointed arm) (3 Rotary Joints)

53 54

Robot Configurations and Work Envelope Robot Configurations and Work Envelope
• Cartesian • Cylindrical

Wo r k Volu m e or Wor k
Envelope is a term that
refers to the space within
which a robot can
manipulate its wrist’s end

55 56
 Robot Configurations and Work Envelope Robot Configurations and Work Envelope

Jointed Arm Configuration

• Polar

57 58

Robot Configurations Wrist Movements

Selective compliance assembly robotic arm
• SCARA ()
• A variation of the cylindrical configuration, involving
• 1 linear joint and 2 rotary joints

59 60
Degrees of Freedom
Further classification of Joints
vDegrees of freedom is often used to describe the prismatic joint L
number of directions that a robot can pivot or
move a joint.
rotational joint R

revolving joint V

twisting joint T

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Joint Notation Scheme Joint Notation Scheme

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Joint Notation Scheme Joint Notation Scheme

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Sketch the following arm configurations Required Degrees of Freedom

For the following arm and body designations describe the • For moving objects placed on a line or a circle  1 dof
robots using sketches • For moving objects anywhere on a flat surface  2 dof
• Picking objects anywhere on a 3D space  3 dof
• LLR • Reaching into complex profiles  > 3 dof
• RLR
• LRR
• LVR
• LL - TRL

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Required Degrees of Freedom Required Degrees of Freedom

69 70

Required Degrees of Freedom Required Degrees of Freedom

71 72
 Required Degrees of Freedom Required Degrees of Freedom

73 74

Wrist Configurations Wrist Configurations

v Arm configurations carry and position the wrist, the v For an arbitrary orientation of wrist in 3-D space, the wrist

second major part of a Manipulator, Wrist is attached to the must possess at least 3-DOF.

end point of the arm

v This 3-DOF gives three rotations about the three principal

v Wrist subassembly enables the manipulator to orient the axes.

end-effector to perform the desired task properly. v Less than 3-DOF wrists are also used depending on the
requirement.
v The end effector, if gripper, must be oriented at an
appropriate angle to pick and grasp a work piece. v Wrist must be compact and at the same time must not
affect the performance of the arm.
BITS Pilani, Pilani Campus BITS Pilani, Pilani Campus
 Wrist Configurations Wrist Configurations
Two Axis Wrist (RT) Two Axis Wrist (TR) Three Axis Wrist (TRT)

DE ZG/ES ZG561 Mechanisms and Robotics BITS Pilani DE ZG/ES ZG561 Mechanisms and Robotics BITS Pilani

Wrist Configurations Design and Control Issues

v Wrist with Roll, Pitch and Yaw motions is shown below. • Robots require higher mobility and
dexterity than conventional
machine tools.
• The mechanical structure of a
robot consists of rigid cantilever
beams connected by hinged joints.
• This is inherently poor in stiffness,
accuracy and load carrying
capacity.

BITS Pilani, Pilani Campus 80

 Design and Control Issues Manipulation and Control
• The errors accumulate
because the joints are in
series.
• The position and motion of
each joint is affected by the
position and motion of other
joints.
• The weight and inertial load • In the analysis of spatial mechanisms (manipulators), the location of links,
of each link is carried by the joints and end effector in 3-D space is continuously required.
previous link and the links • These need to be calculated using mathematical methods.
undergo rotary and linear • To describe the position and orientation of a body in space a frame or a
motion, making centrifugal coordinate system is attached to the body
and Coriolis accelerations • The position and orientation of this frame with respect to a reference
significant. frame, mathematically describes the location of the body.

81 82

Robot Specifications Drive System

• Drive System 1. Hydraulic drive: gives a robot great speed and strength. These
– Electrical systems can be designed to actuate linear or rotational joints. The
– Hydraulic main disadvantage of a hydraulic system is that it occupies floor
space in addition to that required by the robot.
– Pneumatic
• Speed of Motion 2. Electric drive: compared with a hydraulic system, an electric system
• Load-Carrying Capacity (Pay – Load) provides a robot with less speed and strength. Accordingly, electric
drive systems are adopted for smaller robots. However, robots
• Control Systems
supported by electric drive systems are more accurate, exhibit better
• Precision of Movement repeatability, and are cleaner to use.
– Spatial Resolution
– Accuracy 3. Pneumatic drive: are generally used for smaller robots. These robots,
with fewer degrees of freedom, carry out simple pick-and-place
– Repeatability
material handling operations.
• Work Volume

83 84
 Speed of Motion Spatial Resolution, Accuracy, Precision
• Maximum speed of 500 degrees/second can be
achieved.
• Highest speed can be achieved by large robots
with arm extended fully.
• Hydraulic robots tend to be faster than electric drive
robots.
• Robots with electric drives have better control.
• Selection of the most desirable speed also
depends on Spatial Resolution is the smallest increment of movement into which the
robot can divide its work volume.
– Accuracy of positioning of the object
– Weight of the object to be handled Accuracy refers to a robot’s ability to position its wrist at a target point
– Distances to be moved. within its work volume.

85 86

Spatial Resolution, Accuracy,

Repeatability Spatial Resolution, Accuracy, Precision

Repeatability is the ability of the robot to position its wrist at a point in

space that had previously been taught to the robot.

87 88
 Load Carrying Capacity THANK YOU
1. The load the robot can carry at its weakest position
The presenter is grateful to the authors of the following textbooks.
without including the weight of the end effector, while
maintaining its specified accuracy. Machines and Mechanisms, Applied Kinematic Analysis (Fourth Edition) –
David H. Myszka

Industrial Robotics - Mikell P. Groover

2. Effective/Net load carrying Capacity =
Fundamentals of Robotics – Robert Schilling
(Load carrying capacity – weight of the end effector)@ the
weakest position

89 90

Fundamentals of Robot Technology

Analysis & modelling of mechanisms

• Mechanisms
• Spatial Mechanisms
• Four bar planar and Spatial Mechanisms
Mechanisms and Robotics

DE ES ZG561
BITS Pilani Lecture-3
Pilani Campus

92
 Mechanisms Mechanisms
• Kinematic Link: Each part of a machine, which moves • Kinematic Chain: When the kinematic pairs are
relative to some other part is known as a kinematic link coupled in such a way that the last link is joined with the
or link or element. first link to transmit definite motion, it is called kinematic
• Types of Links: chain (Closed).
– Rigid Link • Mechanism: When one of the links in a kinematic chain
– Flexible Link is fixed, the chain is known as a mechanism.
– Fluid Link • Machine: When a mechanism is required to transmit
power or do some particular work, it is called a machine.
• Structure: Assemblage of resistant bodies having no
relative motion between them and meant for carrying
loads.
• Kinematic Pair: The two links or elements of a machine,
when in contact with each other if the relative motion
between them is completely constrained, the pair is
known as kinematic pair.
93 94

Links and Kinematic symbols Links and Kinematic symbols

Simple Link Complex Link

Simple Link
with Point of Pin Joint
interest

95 96
 Links and Kinematic symbols Links and Kinematic symbols

Slide Joint

97 98

Links and Kinematic symbols Kutzbach criterion

• m = 3(l - 1) - 2j – h
When h = 0, Kutzbach criterion
Gear pair becomes Grubler’s Equation
• j = number of binary joints
• l = no. of links
• h – no of higher pairs (Cams and Gears).
• m – mobility, degree of freedom

• Binary joint. When two links are joined at the same

connection, the joint is known as binary joint.
• Ternary joint. When three links are joined at the
Cam and Follower same connection, the joint is known as ternary joint. It
is equivalent to two binary joints as one of the three
links joined carry the pin for the other two links

99 100
 Robot Construction Joint types
v A robot is mechani cally
constructed by connecting a
set of bodies, called links, to
each other using va rious
types of joints.

v Actuators, such as electric

mot ors, de liver forces or
torques to t he joints th at
cause the robot’s links to
move.

101 102

Joint dof Find the degrees of freedom

103 104
 Find the degrees of freedom Find the degrees of freedom

Dof = 4 Dof = 2

Dof = 1
Dof = 1

105 106

Find the degrees of freedom Find the degrees of freedom

N=8
J=9
Dof = 3

107 108
 Gr¨ubler’s formula for degrees of
freedom
• l = 12 v Consider a mechanism consisting of
N links, where ground is also
• j = 15 regarded as a link. Let J be the
• Dof = 3 number of joints, m be the number of
degrees of freedom of a rigid body.
Ø (m = 3 for planar mechanisms and
Ø m = 6 for spatial mechanisms)
v f i b e t h e n u mb e r o f d eg r ee s of
freedom provided by joint i, and ci be
the number of constraints provided
by joint i (it follows that fi + ci = m for
all i).
v Then Gr¨ubler ’s formula for the
degrees of freedom (dof) of the robot
is
109 110

Find the dof Find the dof

The Stewart-Gough platform consists of The Stewart-Gough platform consists of
two platforms—the lower one stationary two platforms—the lower one stationary
and regarded as ground, the upper one and regarded as ground, the upper one
m obi l e— c on n ec t ed by s i x un i v er sa l- m o bi le — co nn e ct ed by si x u n i v e r s al -
prismatic-spherical (UPS) serial chains. prismatic-spherical (UPS) serial chains.

N =14
J = 18
Dof = 6

111 112
Find the dof Find the dof

N= 8
J=9
Fi = 15
Dof = 3
113 114

Find the dof Find the dof

Each parallelogram linkage has one
degree of freedom of motion, so that
each leg of the Delta robot is
kinematically equivalent to an RUU
chain.

L= 8
J=9
Fi = 15
Dof = 3

115 116
Find the dof Find the dof

117 118

Home Work THANK YOU

The presenter is grateful to the authors of the following text books for their
wonderful work.

v Machines and Mechanisms, Applied Kinematic Analysis (Fourth Edition) –

David H. Myszka

v Industrial Robotics - Mikell P. Groover

v Fundamentals of Robotics – Robert Schilling

v INTRODUCTION TO ROBOTICS MECHANICS, PLANNING, AND

CONTROL- F. C. Park and K. M. Lynch

v Modern Robotics: Mechanics, Planning, and Control," Kevin M. Lynch and

Frank C. Park, Cambridge University Press, 2017.

v Theory of Machines, R.S. Khurmi.

119 120
 Fundamentals of Robot Technology

vKinematic Synthesis of Mechanisms

Ø The form or type synthesis
Ø The number synthesis
Ø The dimensional synthesis
vSynthesis of a 4 bar mechanism
Mechanisms and Robotics
vNumerical Example

DE ES ZG561
BITS Pilani Lecture-4
Pilani Campus

122

Kinematic Synthesis Kinematic Synthesis

v Kinematics is the study of the motion of mechanisms and Form or Type Synthesis
methods of creating them. v This refers to the kind of mechanism selected, like a belt drive, gear
drive, linkage or cam mechanism.
v Kinematic synthesis may be treated as a reverse
problem of kinematic analysis and aims at designing a Number Synthesis
mechanism which can satisfy prescribed motion v Number syn the sis is b ase d on th e most o bviou s exte rn al
characteristics of a kinematic chain, namely the number of links
characteristics, such as displacement, velocity and together with the number of joints along with the joint type.
acceleration.
Dimensional synthesis
v Synthesis is accomplished in three interrelated phases v Dimensional synthesis aims at determining significant dimensions of
the links and starting position of links in a mechanism, to accomplish
Ø The form or type synthesis specified task and prescribed motion characteristics.
Ø The number synthesis
Ø The dimensional synthesis
123 124
Algebraic Method of Kinematic Synthesis

B Y

r2
r3 A

r1

ϕ
C X
r0 O
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Algebraic Method of Kinematic Synthesis Algebraic Method of Kinematic Synthesis

B Y

r2
r3 A

r1

ϕ
C X
r0 O
BITS Pilani, Pilani Campus DE ZG/ES ZG561 Mechanisms and Robotics BITS Pilani
Algebraic Method of Kinematic Synthesis Approximate and Exact Synthesis
v If the required link follows the
exact curve specified, the
synt h e si s is ca l le d e xac t
synthesis.

v If the link meets the exact

path at only a few points it is
approximate synthesis.

v The difference between the

t wo fu n ct i o n s ( e x a ct a n d
Equation (1) is called Freudenstein’s equation and can approximate) is called
be used for synthesis of a 4 bar mechanism..
structural error.

DE ZG/ES ZG561 Mechanisms and Robotics BITS Pilani DE ZG/ES ZG561 Mechanisms and Robotics BITS Pilani

Chebyshev’s Spacing of
accuracy points Function Generator

Function generator is a device

v When the mechanism is designed to generate a for indicating corresponding
specified function or trace a given curve over a given values of x and y.
range, the function is exactly generated at a finite
number of points.

v These are known as accuracy points obtained by

using the above formula.

v It is also the same as drawing a circle with a

diameter equal to the range of the function

v The next step is to inscribe in it a regular polygon of

sides 2n

v The polygon must be inscribed such that at least two

sides are perpendicular to the x axis.
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Numerical Example Solution

DE ZG/ES ZG561 Mechanisms and Robotics BITS Pilani DE ZG/ES ZG561 Mechanisms and Robotics BITS Pilani

Solution Solution

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Solution Solution

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Robot Kinematics
Direct Kinematics

• Forward kinematics v A robotic manipulator can be modelled as a chain

of rigid bodies called links.
• Rotation
v The links are interconnected to one other by joints.

v The objective is to control both the position and the

orientation of the tool in three dimensional space.

v In order to program the tool we must formulate the

relationship between the joint variables and the
position and the orientation of the tool.

v This is called the Direct Kinematics problem.

139

140
 Kinematic Parameters Coordinate Systems

v Directions of Positive X Y Z axes

Joint Link
Parameters Parameters

141 142

Rotation Matrices
Coordinate Systems

Directions of Positive X Y Z rotations

vAnticlockwise rotation is positive

when seen from the opposite
direction of the arrow head

143 BITS Pilani, Pilani Campus

 Rotation Matrices THANK YOU
The presenter is grateful to the authors of the following text books for their
Ø Fundamental rotation matrices for rotation about x axis y axis and z wonderful work.
axis are given as follows.
v Machines and Mechanisms, Applied Kinematic Analysis (Fourth Edition) –
David H. Myszka

v Industrial Robotics - Mikell P. Groover

v Fundamentals of Robotics – Robert Schilling

v Mechanism and Machine Theory, Ashok G. Ambekar

BITS Pilani, Pilani Campus 146

Robot Kinematics

• Forward kinematics
• Rotation
• Translation
• Post-multiplication Vs. Pre-multiplication
• Homogeneous Transformation
Mechanisms and Robotics • Introduction to Denavit – Hartenberg
Parameters
• θ – r manipulator
DE ES ZG561
Lecture - 5
BITS Pilani
Pilani Campus

148
Direct Kinematics Kinematic Parameters

v A robotic manipulator can be modelled as a chain

of rigid bodies called links.

v The links are interconnected to one other by joints.

v The objective is to control both the position and the

orientation of the tool in three dimensional space.

v In order to program the tool we must formulate the

relationship between the joint variables and the
position and the orientation of the tool.

v This is called the Direct Kinematics problem.

Joint Link
Parameters Parameters
149 150

Coordinate Systems Coordinate Systems

Directions of Positive X Y Z rotations

v Directions of Positive X Y Z axes

vAnticlockwise rotation is positive

when seen from the opposite
direction of the arrow head

151 152
 Rotation Matrices Rotation Matrices

Ø Fundamental rotation matrices for rotation about x axis y axis and z

axis are given as follows.

BITS Pilani, Pilani Campus BITS Pilani, Pilani Campus

Problem 1 Problem 1 - Solution

v Find the result of rotating the point [7,2,5]T through 90° in

the positive direction about the z-axis.

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 Homogeneous Transformation
Homogeneous Transformation Matrices Matrices

v When we want to describe a generalized transformation by a single matrix

that combines the effects of translation and rotation, we define a vector by
adding 1.

BITS Pilani, Pilani Campus BITS Pilani, Pilani Campus

Homogeneous Transformation Matrices Post-Multiplication Vs. Pre-

Multiplication
Composite Rotations

vIf the mobile coordinate frame M is

to be rotated by an amount ϕ about
the K t h unit vector of the fixed
coo rdi nat e f rame F, t h e n p re -
multiply R by Rk(ϕ)
vIf the mobile coordinate frame M is
to be rotated by an amount ϕ about
its own K th unit vector, then post-
multiply R by Rk(ϕ)

BITS Pilani, Pilani Campus BITS Pilani, Pilani Campus

Problem 2 Problem 2 - Solution

v Find the result of rotating the point [7,2,5]T through 90° in the positive direction about the
z-axis followed by a rotation of 90° about the x-axis, followed by a translation of (4,5,6).

BITS Pilani, Pilani Campus BITS Pilani, Pilani Campus

Denavit - Hartenberg
Representation

ØDenavit and Hartenberg (1955) proposed a systematic

notation scheme for assigning right handed orthonormal
coordinate frames, one to each link in an open kinematic
chain of links.
ØO n c e t h e s e c o o r d i n a t e d f r a m e s a r e a s s i g n e d ,
transformations between adjacent coordinate frames can
be represented by a single standard 4X4 homogeneous
coordinate transformation matrix.

BITS Pilani, Pilani Campus BITS Pilani, Pilani Campus

 2 dof manipulator – Forward
Kinematics

BITS Pilani, Pilani Campus 166

2 dof manipulator – Link

2 dof manipulator – D-H Parameters
Coordinate Diagram

Link i θ d  a Home
Solution: 1 θ1 90 L1 90
2 d2
[x’,y’,z’] = Rotz (q1)*Transx(L1)*Rotz(90)* Rotx(90)* Transz(d2)* [0,0,0]’

167 168
 2 dof manipulator – Forward
2 dof manipulator – Forward Kinematics
Kinematics

169 170

Cylindrical arm – Forward Kinematics THANK YOU

The presenter is grateful to the authors of the following text books.

vIndustrial Robotics - Mikell P. Groover

vIndustrial Robotics Computer Interfacing and Control – Wesley E.

Snyder

vFundamentals of Robotics – Robert Schilling

172

171
 Forward kinematics

Ø 3 dof Cylindrical Arm

Ø 3 dof Articulated Arm
Ø 5 dof Articulated Arm

Mechanisms and Robotics

DE / ES ZG561

BITS Pilani Lecture - 6

Pilani Campus

174

Coordinate Systems Coordinate Systems

v Directions of Positive X Y Z axes Directions of Positive X Y Z rotations

vAnticlockwise rotation is positive

when seen from the opposite
direction of the arrow head

175 176
 Homogeneous Transformation
Homogeneous Transformation Matrices Matrices

v When we want to describe a generalized transformation by a single matrix

that combines the effects of translation and rotation, we define a vector by
adding 1.

BITS Pilani, Pilani Campus BITS Pilani, Pilani Campus

Cylindrical arm – Forward Kinematics Link Coordinate Diagram

179 180
 D-H Parameters D-H Parameters
Solution: • Solution:
[x’,y’,z’] = [Rotz(q1).Transz(L1)].[Transz(d2).Rotx(-90).transz(L2)].Transz(d3).[0,0,0]’ Solution: [x’,y’,z’] = [Rotz(q1).Transz(L1)].[Transz(d2).Rotx(-90).transz(L2)].
Transz(d3).[0,0,0]’

D-H Parameters
Link i θ d  a Home
1 θ1 L1
2 d2, L2 -90
3 d3

181 182

3 dof Articulated Robot – Forward 3 dof Articulated Robot – Link Coordinate

Kinematics Diagram

183 184
3 dof Articulated Robot – Forward 3 dof Articulated Robot – Forward
Kinematics Kinematics
• 3 dof Articulated manipulator

Link i θ d  a Home
1 θ1 L1 90
2 θ2 L2
3 θ3 90 L3 90

185 186

5 dof manipulator 5 dof manipulator

Solution:

P=
[rotz(theta1)*transz(L1)*rotx(90)]*[rotz(theta2)*transx(L2)]*[rotz(theta3)*transx(L
3)]*[rotz(theta4)*rotz(90)*rotx(90)]*[rotz(theta5)*transz(L5)]*origin

L1 = 200mm, L2 = 250mm,
L3 = 300mm, L4 = 150mm,

187 188
 5 dof manipulator 5 dof manipulator
P=
[rotz(theta1)*transz(L1)*rotx(90)]*[rotz(theta2)*transx(L2)]*[rotz(theta3)*tra
nsx(L3)]*[rotz(theta4)*rotz(90)*rotx(90)]*[rotz(theta5)*transz(L4)]*origin

Axis θ d a α Home
1 θ1 L1 90
2 θ2 L2
3 θ3 L3
4 θ4 90 90
5 θ5 L5

189 190

5 dof manipulator
5 dof manipulator
P=
[rotz(theta1)*transz(L1)*rotx(90)]*[rotz(theta2)*tran
sx(L2)]*[rotz(theta3)*transx(L3)]*[rotz(theta4)*roty
(90)*transz(L4)]*[rotz(theta5)]*origin

191 192
 THANK YOU

The presenter is grateful to the authors of the following text

books for their wonderful work.

vIndustrial Robotics Computer Interfacing and Control –

Wesley E. Snyder

vFundamentals of Robotics – Robert Schilling Mechanisms and Robotics

DE ZG561
Lecture - 7
BITS Pilani
Pilani Campus

193

2 dof manipulator – Inverse

Inverse kinematics
Kinematics
Ø Inverse Kinematics of a 2dof Planar Arm
Ø Inverse Kinematics of a 3 dof Polar Arm
P (700,700,0)
Ø Inverse Kinematics of a 3 dof Cylindrical Arm
Ø Inverse Kinematics of a 3 dof Articulated Arm

L1 = 500 mm

195

196
2 dof manipulator – Link Coordinate 2 dof manipulator – Inverse
Diagram Kinematics

Link i θ d  a Home
1 θ1 90 L1 90
2 d2
Solution:

[x’,y’,z’] = Rotz (q1)*Transx(L1)*Rotz(90)* Rotx(90)* Transz(d1)* [0,0,0]’

197 198

Inverse Kinematics of a 3 dof Inverse Kinematics of a 3 dof

Polar Arm Polar Arm

Find the joint angles and extension d3 for the

tool point P(650,150,350)
P = [Rotz(q1).Transz(L1).Rotx(90)]*[Rotz(q2).Transx(L2)*Rotz(90).Rotx(90)]*[Transz(d3)]*Origin
L1 = 250 mm
L2= 500 mm

BITS Pilani, Pilani Campus BITS Pilani, Pilani Campus

Inverse Kinematics of a 3 dof Equations
Polar Arm
D-H Parameters

Axis d a Home
1 L1 90
2 L2 90 90
3 d3

BITS Pilani, Pilani Campus BITS Pilani, Pilani Campus

Answers Inverse Kinematics of a 3 dof Cylindrical Arm

q1 = 12.9946 °
q2 = 8.5255 °
d3 = 174.5368 mm
L1 = 700 mm
L2 = 600 mm

P (750,750,300)

BITS Pilani, Pilani Campus BITS Pilani, Pilani Campus

Inverse Kinematics of a 3 dof Cylindrical Arm Inverse Kinematics of a 3 dof Cylindrical Arm

Axis d a Home
1 L1 L2 90 90
2 d1 90
3 d2

BITS Pilani, Pilani Campus BITS Pilani, Pilani Campus

BITS Pilani, Pilani Campus BITS Pilani, Pilani Campus

 3 dof Articulated Articulated

Axis d a Home
1 L1 90
P (650, 200, 250)
2 L2
L1 = 150 mm 3 L3 90 90
L2 = 400 mm
L3 = 300 mm

209 210

Articulated Solution - Articulated

P=
[rotz(theta1)*transz(L1)*rotx(90)]*[rotz(theta2)*transx(L2)]*[rotz(theta3)*transx(L3)*rotz(90)rotx(90)]*ori
gin

Ø T 0  1 = [rotz(theta1)*transz(L1)*rotx(90)]
Ø T 1  2 = [rotz(theta2)*transx(L2)]
Ø T 2  3 = [rotz(theta3)*transx(L3)*rotz(90)rotx(90)]

Ø P = [T 0  1] * [T 1  2] * [T 2  3] *origin

211 212
 • Theta 1 = 17.10 deg
• Theta 2 = 17.77 deg
• Theta 3 = - 22.153 deg

213 214

THANK YOU

The presenter is grateful to the authors of the following text books for their
wonderful work.

vIndustrial Robotics Computer Interfacing and Control – Wesley E.

Snyder
Mechanisms and Robotics
vFundamentals of Robotics – Robert Schilling

DE ZG561
Lecture - 8
BITS Pilani
215 Pilani Campus
Inverse Kinematics Homogeneous Transformation Matrices

• Inverse Kinematics
– Inverse Kinematics of a 4 dof SCARA Arm
– Inverse Kinematics of a 2 dof RR Planar Arm using Matrix inverse
– Inverse Kinematics of a 3 dof Articulated Arm using Matrix inverse

8 October 2020 DE ZG561, MECHANISMS AND ROBOTICS 217

BITS Pilani, Pilani Campus

• 4 dof SCARA

Find the joint angles and the extension of the 4 dof

SCARA Robot for the target point (650,250,200)
• 4 dof SCARA
L1= 500mm L2 = 450, L23 = 50, L3 = 350, L4 = 200.

P=
Transz(L1)*rotz(theta1)*transx(L2)*transz(L23)*rotz(theta2)*transx(L3)*rotx(180)*transz(
d3)*rotz(theta4)*transz(L4)*origin
 • 4 dof SCARA

• d3 = 150 mm
• 4 dof SCARA
• Theta1 = 46.69 deg (21.0375+25.6525)
• Theta2 = 59.4735 deg

2 dof Planar Arm – Inverse

2 dof Planar Arm
Kinematics using matrix inverse

8 October 2020 DE ZG561, MECHANISMS AND ROBOTICS 223 8 October 2020 DE ZG561, MECHANISMS AND ROBOTICS 224
2 dof Planar Arm 2 dof Planar Arm

Coupled
Equations

Decoupled
Equations

8 October 2020 DE ZG561, MECHANISMS AND ROBOTICS 225 8 October 2020 DE ZG561, MECHANISMS AND ROBOTICS 226

2 dof Planar Arm 2 dof Planar Arm

solving

8 October 2020 DE ZG561, MECHANISMS AND ROBOTICS 227 8 October 2020 DE ZG561, MECHANISMS AND ROBOTICS 228
Home Work – Inverse Kinematics of a 3 dof Home Work – Inverse Kinematics of
arm using matrix inverse a 3 dof arm using matrix inverse
Solution from Forward
Kinematics
• Find the angles for the given tool point

P (650, 200, 250)

L1 = 150 mm
L2 = 400 mm
L3 = 300 mm

8 October 2020 DE ZG561, MECHANISMS AND ROBOTICS 229 8 October 2020 DE ZG561, MECHANISMS AND ROBOTICS 230

Home Work – Inverse Kinematics of

5 dof manipulator
a 3 dof arm using matrix inverse
Solution from Forward Kinematics

P = (400,350,250)

L1 = 200mm,
L2 = 400mm,
L3 = 300mm,
L5 = 150mm,

8 October 2020 DE ZG561, MECHANISMS AND ROBOTICS 231 8 October 2020 DE ZG561, MECHANISMS AND ROBOTICS 232
 5 dof manipulator 5 dof manipulator
P=
Solution: [rotz(theta1)*transz(L1)*rotx(90)]*[rotz(theta2)*transx(L2)]*[rotz(theta3)*transx(L3)]*
[rotz(theta4)*rotz(90)*rotx(90)]*[rotz(theta5)*transz(L4)]*origin
P=
rotz(theta1)*transz(L1)*rotx(90)]*[rotz(theta2)*transx(L2)]*[rotz(theta3)*transx(L3)]*
[rotz(theta4)*rotz(90)*rotx(90)]*[rotz(theta5)*transz(L4)]*origin Axis θ d a α Home
1 θ1 L1 90
2 θ2 L2
3 θ3 L3
4 θ4 90 90
5 θ5 L4

8 October 2020 DE ZG561, MECHANISMS AND ROBOTICS 233 8 October 2020 DE ZG561, MECHANISMS AND ROBOTICS 234

THANK YOU
The presenter is grateful to the authors of the following text books for their wonderful
work.

vIndustrial Robotics Computer Interfacing and Control – Wesley E. Snyder

vFundamentals of Robotics – Robert Schilling

8 October 2020 DE ZG561, MECHANISMS AND ROBOTICS 236

8 October 2020 DE ZG561, MECHANISMS AND ROBOTICS 235