Robot Kinetics, Dynamics & Control

(Course Code:BRA403)

Department of Robotics & Automation

JSS Academy of Technical Education, Bangalore-560060
 TEXT BOOKS

• Theory of Machines, Sadhu Singh, 3rd ed., 2019.

• Theory of Machines, by R S Khurmi

REFERENCE BOOKS:

• Theory of Machines, Rattan S.S, Tata McGraw-Hill Publishing Company, 2014

Further Reference:
✓ National Programme on Technology Enhanced Learning (NPTEL)
https://nptel.ac.in/courses/112/104/112104121/
 Course Learning Objectives

• To understand the concept of machines, mechanisms and related terminologies.

• To identify and enumerate different link-based mechanisms with basic

understanding of motion
 Course Outcomes

At the end of the course, students will be able to,

CO1: Explain the fundamental principles of kinematic pair, kinematic chain, mechanisms and

their inversions.

CO2: Identify the mechanisms and predict their motions in mechanical devices.
 Question Paper Pattern

• The question paper will have 10 full questions.

• Each full question will be for 20 marks.

• There will be two full questions from each module.

• The students will have to answer 5 full questions, selecting 1 full question from

each module.
Kinematics of Machines

CHAPTER 1: Introduction to Kinematics of Machines

 Module 1
Introduction: Definitions Link or element, kinematic pairs, Kinematic chain, mechanism,

structure, Degrees of freedom, Grubler's criterion,, Mobility of Mechanism, Grashoff’s

criteria

Mechanism: Quick return motion mechanism: Whitworth Mechanism, Crank and slotted

lever mechanism, Oldham’s Coupling, Drag link mechanism,

Straight line motion mechanism: Peaucellier's mechanism, Robert's mechanism.

Intermittent Motion mechanisms: Geneva wheel mechanism, Ratchet and Pawl

mechanism, toggle mechanism, pantograph, condition for correct steering: Ackerman

steering gear mechanism

 Terms & Definitions
Machine

Device for transferring and/or transforming motion and force (power) from source (Input) to the load

(output).
 Classification of Motion

• Continuous rotation motion

• Linear motion / Rectilinear Motion / translatory motion

• Intermittent motion

• Angular Oscillations
 Basic Definitions Kinematic Chains and Inversions
• Kinematics

• Dynamics • Inversions of Four bar chain

• Kinetics • Single slider crank chain and

• Link or element • Double slider crank chain

• Kinematic pair

• Kinematic chain

• Mechanism, Machine & Structure

• Degrees of freedom (DOF)

• Grubler’s criterion

• Inversion
 Basic Definitions

Link / Kinematic Kinematic

Mechanism Machine
Element Pair Chain
 Kinematic Link or Element

Each part of a machine, which moves relative to some other part, is known

as a kinematic link (or link) or element.

 Types of Links

• Rigid link: Link which does not undergo any deformation while transmitting
motion.
e.g. Connecting rod

• Flexible link: Link which is partly deformed while transmitting the motion (without
affecting the motion).

e.g. belts, ropes, chains and wires

• Fluid link: A link formed by having a fluid in a container and the motion is
transmitted through the fluid by pressure.

e.g. hydraulic presses, jacks and brakes.

Types of Links
 Classification of Links

• Unary Link : Link with one Node (bucket of an excavator)

• Binary link : Link connected to other links at two points (nodes)

• Ternary link: Link connected to other links at three points (nodes)

• Quaternary link: Link connected to other links at four points (nodes)

 Structure

An assembly of a no. of resistant bodies (members) having no relative motion

between them and meant for carrying loads having straining action.

Examples: A railway bridge, a roof truss, machine frames. etc.

 Difference Between a Machine and a Mechanism

Sl.
No. Mechanism Machine

Mechanism transmits and modifies the

1 Machine changes the mechanical work.
motion.
A mechanism is an outline of the
Machine will have many mechanisms for
2 machine to produce define motion
transmitting the mechanical work or power.
between various links or joints

When kinematic chain is analysed as a In a machine, cross-section and proportions are to

3 mechanism, no consideration need to be be considered to give stiffness, strength, and
given to the cross-section of the links. clearance to the paths of the machine.

Examples: clocks, typewriter, steering

4 Examples: slotting, lathe, shaper etc.
mechanism in a car, etc.
 Difference between a Machine and a Structure
Sl.
No. Structure Machine

In structure, no relative motion In machine, relative motion exists

1
exists between its members. between its parts.
Members meant for carrying Machine links are meant to transmit
2 loads are subjected to straining motion and forces which are dynamic
action. (both static and kinetic).

Structure serves to modify and Machine serves to modify and

3
transmit forces. transmit mechanical work.

Examples: roof trusses, bridges, Examples: slotting, shaper, lathe, and

4
building and machine frames. screw jack.
 Kinematic Pair

The two links or elements in contact with each other, said to form a pair, if the
relative motion between them is completely or successfully constrained (i.e. in a
definite direction)

Completely constrained Successfully constrained

 Constrained Motions
1. Completely constrained motion

If motion between a pair is limited to a definite direction.

Example:
✓The motion of a square bar in a square hole

✓The motion of a shaft with collars at each end in a circular hole

2. Incompletely constrained motion

If the motion between a pair take place in more than one direction.

E.g.: A circular bar or shaft in a circular hole

• It may either rotate or slide in a hole.

• Both motions are independent.
3. Successfully constrained motion

If the motion between the elements, forming a pair, is such that the constrained
motion is not completed by itself, but by some other means

E.g.: Shaft in a foot-step bearing, I C engine valve, Piston inside an cylinder.

 Classification of Kinematic Pairs

According to the type of relative motion between the elements

1. Sliding pair / Prismatic pair (P)

When the two elements of a pair are connected in

such a way that one can only slide relative to the
other, has a completely constrained motion.

DOF = 1 E.g.: The piston and cylinder, ram and its guides in shaper, tail
stock on the lathe bed
 2. Turning Pair / Revolute Pair (R)

When the two elements of a pair are connected in such a way that one can only
turn or revolve about a fixed axis of another link.

Examples Lathe spindle supported in head stock.

DOF = 1
3. Spherical pair (S)

When the two elements of a pair are connected in such a way that one element
turns or swivels about the other fixed element.

E.g.: ball and socket joint., Vehicle mirror, Joystick

DOF = 3
 4. Screw pair (H)

When the two elements of a pair are connected in such a way that one element can
turn about the other by screw threads.

E.g.:

• Bolt with a nut

DOF = 1
5. Cylindrical pair (C)

If the relative motion between the pairing elements is the combination of turning

and sliding, then it is called as cylindrical pair.

DOF = 2
6. Rolling Pair

When the two elements of a pair are connected in such a way that one rolls
over another fixed link.

E.g.: Ball and roller bearings

Belt and pulley

DOF = 1
 Classification of Kinematic Pairs

According to the type / nature of contact between the elements / links.

1. Lower pair

When the two elements of a pair have a surface contact, and the surface of one
element slides or rolls over the surface of the other.

E.g. sliding pairs, turning pairs and screw pairs form lower pairs.
 Classification of Kinematic Pairs

According to the type / nature of contact between the elements / links.

2. Higher pair

When the two elements of a pair have a line or point contact, and the motion
between the two elements is partly turning and partly sliding.

E.g.: toothed gearing, belt and rope drives, ball and roller bearings and cam and
follower.
 Classification of Kinematic Pairs
According to the Mechanical arrangement.
1. Self closed pair

Two elements of a pair are connected mechanically in such a way that the relative
motion occurs.
E.g. Lower pairs are self closed pair.

2. Force - closed pair

Two elements of a pair are kept in contact by the action of

external forces.
E.g.: Cam and follower
 Kinematic Pairs

Lower pair Higher pair Self / Form closed pair Force - closed pair

• Sliding pairs Rolling pair Lower pairs

• Turning pairs

• Screw pairs
Based on the possible motions (Few Important Types only)

Name of Pair Letter Symbol D.O.F

1. Revolute / Turning Pair R 1

2. Prismatic / Sliding Pair P 1

3. Helical / Screw Pair H 1

4. Cylindrical Pair C 2

5. Spherical / Globular Pair S (or) G 3

6. Flat / Planar Pair E 3

 Kinematic Chain

When the kinematic pairs are coupled in such a way that the last link is joined to
the first link to transmit (definite / specific) motion (i.e. completely or successfully
constrained motion), it is called a kinematic chain.
Or
Assembly of links (Kinematic link / element) and Kinematic pairs to transmit
required / specified output motion(s) for given input motion(s)
 Mechanism
• When one of the links of a kinematic chain is fixed, the chain is known as
mechanism.
• Used for transmitting or transforming motion.

e.g. printing machine, windshield wiper, robot arms

• A mechanism may be regarded as a machine in which each part is reduced to

the simplest form to transmit the required motion.
 Degrees of Freedom (DOF) / Mobility of a Mechanism

Number of independent coordinates, required to describe / specify the

configuration or position of all the links of the mechanism, with respect to the

fixed link at any given instant.

or

No. of relative motions / Total no. of motions a link has.

 Degrees of freedom (DOF)

In a kinematic pair, depending on the constraints imposed on the motion, the

links may loose some of the six degrees of freedom.
 Grubler’s Criterion / Equation

Grubler’s mobility equation

M = Mobility or Total no. of DOF
M = 3 (n-1) - 2 j1 - j2
n = Total no. of links in a mechanism
J1 = No. of joints having 1 DOF
J2 = No. of joints having 2 DOF

If, M>0, It gives mechanism with M DOF

M=0, it gives a statically determinate structure
M<0, it gives statically indeterminate structure
Find the Degrees of Freedom / Determine the mobility of the following structure.
Find the Degrees of Freedom / Determine the mobility of the following structure.
 Inversion of Mechanism

• A mechanism is one in which one of the links of a kinematic chain is fixed.

• Different mechanisms can be obtained by fixing different links of the same

kinematic chain.
 Types of Kinematic Chains

1. Four bar chain or quadric cyclic chain

2. Single slider crank chain

3. Double slider crank chain

1. Four bar chain or quadric chain (Grashof’ s criteria / Law)

• Four bar chain (mechanism) is the basic kinematic chain

and consists of four rigid links
• Each of them forms a turning pair at A, B, C and D.

The link that makes a complete revolution is called a crank

• The four links may be of different lengths.

• According to Grashof ’s law for a four bar mechanism, “the sum of the
shortest and longest link lengths should not be greater than the sum of the
remaining two link lengths” if there is to be continuous relative motion between
the two links.
 1. Four bar chain or quadric chain

• The shortest link, will make a complete revolution relative to the other three links. In Fig., AD (link 4 )
is a crank / driver.

• link BC (link 2) which makes a partial rotation or oscillates is known as lever/ rocker/ follower

• link CD (link 3) which connects the crank and lever is called connecting rod or coupler.

• The fixed link AB (link 1) is known as frame of the mechanism.

The mechanism transforms rotary motion into oscillating motion.

Single Slider Crank Chain
1. Single Slider Crank Chain
 A mechanism used in shaping and slotting
machines, where the metal is cut intermittently.

• Link AC (i.e. link 3) is fixed.

• Crank CB revolves with uniform angular speed
about the fixed center C.

• Sliding block attached to the crank pin at B slides

along the slotted bar AP, causing link AP to oscillate
about point A.
• Short link PR transmits the motion from AP to the
ram which reciprocates along the line of stroke
R 1R 2
• The forward or cutting stroke: when the

crank rotates from the position CB1 to CB2

(or through an angle β) in the clockwise

direction.

• The return stroke: when the crank rotates

from the position CB2 to CB1 (or through

angle α) in the clockwise direction.

Whitworth Quick return Motion mechanism
Whitworth Quick return Motion mechanism
• Link CD (link 2) is fixed.
• The crank CA (link 3) rotates at a uniform angular speed.
• The slider (link 4) attached to the crank pin at A slides along the slotted bar PA (link 1) which
oscillates with respect to pivot point D.
• The connecting rod PR carries the ram at R.
• The motion of the tool is constrained along the line RD produced.
• When the crank CA turns from the position CA1 to CA2 through an angle α in the clockwise direction,
the tool moves from the left hand end to the right hand end.
• When the crank moves from the position CA2 to CA1 through an angle β in the clockwise direction,
the tool moves back from right hand end to the left hand end.
• The time taken during the cutting stroke (or forward stroke) is more than the time
taken during the return stroke.
• The ratio between the time taken during the cutting and return strokes is given by;
 Content

• Quick return motion Mechanisms

• Drag link Mechanism

• Whitworth Mechanism

• Crank and slotted lever Mechanism

• Straight line motion Mechanisms (Approx. & Exact St. Line)

• Peaucellier's Mechanism

• Robert's Mechanism

• Watt’s Mechanism
 Content

• Intermittent Motion mechanisms

• Geneva wheel Mechanism

• Ratchet and Pawl Mechanism

• Toggle mechanism

• Pantograph

• Ackerman steering gear mechanism

Straight line motion Mechanisms

1. Peaucellier's mechanism

2. Robert's mechanism

3. Watt’s mechanism
Peaucellier's mechanism
• Used for tracing a straight line.

• OO1 - fixed link and straight links - O1A, OC, OD, AD, DB, BC & CA

• The pin at A is constrained to move along the circumference of a circle with the
fixed diameter OP, by means of the link O1A.
Peaucellier's mechanism

It may be proved that the product OA × OB remains

constant, when the link O1A rotates
From right angled triangles ORC and BRC,

• Since OC and BC are of constant length, therefore the product OB × OA remains constant.
• The links AB & DE act as levers.
• End A & E of the levers are fixed.
• The AB & DE are parallel in the mean position.
• Coupling rod BD is perpendicular to the levers AB & DE.

On displacement of the mechanism, the point ‘C’ traces 8 shaped path, a portion of which will be
approximately straight line.
Robert's mechanism

• It is a four bar mechanism, used for tracing a straight line.

• Link OA and O1B are of equal length and OO1 is fixed.
• A bar PQ is attached to the link AB at its Centre P.
• On displacement of the mechanism (dotted lines), the point Q will trace out an
approximately straight line.
Intermittent Motion Mechanisms

1.Geneva wheel mechanism

2.Ratchet and Pawl mechanism

3.Toggle Mechanism
Geneva wheel
Geneva wheel

• It consists of a driving wheel (D) carrying a pin (P), engages in a slot on the follower (F)
• During one quarter revolution of the driving plate, the Pin and follower remain in contact.
• For one complete rotation of the driving wheel, the follower is turned by one fourth of a
rotation
Ratchet and Pawl mechanism

• Ratchet and Pawl mechanism consists of a Ratchet wheel and a pawl .

• When the lever carrying pawl is raised, the ratchet wheel rotates in the counter clock wise direction.

• As the pawl is lowered the pawl slides over the ratchet teeth.

• Secondary pawl is used to prevent the ratchet from reversing.

• Applications: Feed mechanisms, lifting jacks, clocks, watches and counting devices.
Pantograph

• A device / instrument, used to enlarge or reduce the size/scale of the drawing

(reproduces the motion).

 Pantograph

• It consists of a parallelogram ABCD as shown in Fig.

• The links BA and BC are extended to O and E respectively, such that

• For all relative positions of links, the triangles OAD and OBE are similar and the
points O, D and E are in straight line.

• It may be proved that point E traces out the same path as defined by point D.
 Pantograph

From similar triangles OAD and OBE, we find that,

Let O be fixed and the points D and E move to some new positions D′ and E′, then

• The line DD′ is parallel to the line EE′.

Pantograph
 Pantograph

Pantographs are a special

devices mounted on electric

trains to collect current from

one or several contact wires.

They consist of a

pantograph head, frame,

base, and drive system, and

their geometrical shape is

variable.
Toggle mechanism

Links are arranged in such a way that a small force

applied at one point can create a much larger force at
another point.

A toggle mechanism is used when large forces to be applied through a short distance.

E.g.: Stone crushers, presses, riveting machines, clamps etc.,

Toggle mechanism
For equilibrium cond. of the slider 6
FΤ2
tan 𝛼 =
P
F
P=
2 tan α

tan 0 = 0; 𝑃 = ∞

Two extreme positions: 0 to ꝏ

• For small angle of α, F (effort) is much smaller than P (resistance)

• Toggle mechanism are used to obtain large force amplifications in applications like
sheet metal punching and forming machines.
Toggle clamp
Steering Gear Mechanism
• In vehicle, the front wheels are mounted over the front axles.

• The front wheels are pivoted at the points A and B.

• Rear wheels are placed over the back axle.

• When the vehicle takes a turn, the front wheels along with the respective axles turn

about the respective pivoted points.

• The rear wheels remain straight and do not turn. Therefore, steering is done by front

wheels only.

• In order to avoid skidding (i.e. slipping), the two front wheels must turn about the

same instantaneous center “I” which lies on the axis of the back wheels.
Fundamental equation for correct steering:

If this condition is satisfied, there will be no skidding of the wheels, when the
vehicle takes a turn.
Ackermann Steering Gear
Ackermann Steering Gear
• In Ackerman steering gear, the links ABCD is a four bar chain.

• Shorter links BC and AD are of equal length, connected with front wheel axles.

• Longer links AB and CD are of unequal length.

 Ackermann Steering Gear

The following are the only three positions for correct steering.

• When the vehicle moves in a straight path, the

longer links AB and CD are parallel and shorter
links BC and AD are equally inclined to the
longitudinal axis as shown by continuous lines

• When the vehicle is steering to the left, the

axes of the front wheel axle intersect on the
axes of the back wheel axle at I, for correct
steering.

• When the vehicle is steering to the right, the

similar position may be obtained.
 Drag link mechanism

• A drag link converts rotary motion of a crank to a

second link in a different plane or axis.

• Converts rotation of a steering arm to a center link

(Linear motion) and to tie rod links which carries the

wheels to be steered.

• A drag link is used when the steering arm operates in

a plane above the other links.

Drag link mechanism

Coupler

Frame
Drag link mechanism
• Used for connecting two parallel shafts whose axes are at a small distance apart.
• The shafts are coupled in such a way that if one shaft rotates, the other shaft also
rotates at the same speed.
• This inversion is obtained by fixing the link 2.
• The link 1 and link 3 form turning pairs with link 2.
• The flanges have diametrical slots cut on their inner faces.
• The intermediate piece have two tongues on each face at right angles to each other.
• The tongues on the link 4 closely fit into the slots in the two flanges (link 1 and link 3).
• The link 4 can slide or reciprocate in the slots in the flanges.
• When the shaft A is rotated, the flange C causes the intermediate piece to rotate
at the same angle through which the flange has rotated, and further rotates the
flange D at the same angle and thus the shaft B rotates.
• Hence links 1, 3 and 4 have the same angular velocity at every instant.
 Mechanisms

Finger Engine Exercise bikers

 Mechanisms

Linkage for 180 deg. interrupted rotation Cam-operated Ratchet Pawl

ABB IRB 460 with Gripper and Pallet Drawbridge - Small assembly
 Mechanisms

embossing tool
Arbor Press

Geneva transport mechanism - Linear intermittent motion

 Mechanisms

6 Bar mechanism

Wobbling disk mechanism

Wiper mechanism
End of Module