Robotics done

Introduction to Robotics
Ref: S.K.Saha, McGraw Hill Publisher
By Avinash Juriani
M.Tech IIT ISM DHN
B.Tech SRM Chennai
M.Tech Notes Advanced Robotics

Laws of Robotics
• A robot must not harm a human being, nor
through inaction allow one to come to harm.
• A robot must always obey human beings,
unless that is in conflict with the 1st law.
• A robot must protect from harm, unless that is
in conflict with the 1st two laws.
• A robot may take a human being’s job but it
may not leave that person jobless. [Fuller(1999)]

Robot: Definition
• Reprogrammable, multifunctional
manipulator designed to move material
through variable programmed motions for the
performance of a variety of tasks. (ISO)
• Robotics Institute of America (RIA)
• Japan Industrial Robot Association (JIRA)
• British Robot Association (BRA)

The Unimate Robot
UNIMATION:
UNIversal+autoMATION
Robot is a universal tool that can
be used for many kind of tasks

Special-purpose Robots
• A special-purpose robot is the one that is used
in other than a typical factory environment.

Special-purpose: Automatic Guided Vehicles (AGVs)

Can move sideways also (3 DOF), used
in hospitals, security etc.

Walking Robots: used in military, undersea exploration,
and places where rough terrains exist

Parallel robots: a parallel structure with 6 legs to
control the moving platform used as a flight
simulator for imparting training to …

Thumb rules on the decision of a robot usage
• Four Ds of Robotics: i.e. is the task dirty, dull,
dangerous, or difficult?
• Robot may not leave a human jobless.
• Whether you can find people who are willing
to do the job.
• Robots and automation must make short-term
and long-term economic sense.

Books recommended
• John J. Craig, Introduction to Robotics: Mechanics and
Control, Prentice Hall
• Mark W. Spong, Robot Modeling and Control, Wiley
• S. K. Saha, Introduction to Robotics, McGraw Hill
• K. S. Fu, R. C. Gonzalez, C. S. G. Lee, Robotics: Control,
Sensing, Vision and Intelligence McGraw-Hill
• S.R. Deb, Robotics Technology and Flexible Automation,
TMH

@ McGraw-Hill Education
1
Serial Robots

2
Outline
• Robot Subsystems (Focus: Serial-type)
– Motion
– Recognition
– Control
• Robot Classifications By
– Application
– Coordinate system
– Actuation system
– Control method
– Programming method

3
Robot Subsystems

4
Robot Subsystems (contd.)
• Motion: Manipulator (Arm + Wrist),
End-effector, Actuators (Set in motion),
and Transmissions
• Recognition: Sensors (Measure
status), and ADC
• Control (Supervision): DAC, and
Digital Controller

5
Motion Subsystem
i) Manipulator: Mechanical arm + wrist
ii) End-effector
- Welding torch, painting brush, etc.
- Robot hand
- Simple grippers

6
(iii) Actuator
- Pneumatic, Hydraulic, Electric
(iv) Transmission
- Belt and chain drives
- Gears
- Link mechanisms

7
i) Manipulator: Mechanical arm + wrist
PUMA: Programmable Universal Manipulator for Assembly

8
ii) End-effector: Robot hand

9
More end-effectors: Simple grippers

10
iv) Transmission: Belt and Chain drives

11
Other transmission system: Gears

12
Another transmission: Link mechanisms

13
Recognition Subsystem
(ii) Analog-to-Digital
Converter (ADC)
- Electronic device
(i) Sensors (Essentially transducers)
- Converts a signal
to another
Fig. 2.8 An analog-to-digital converter
[Courtesy: http://www.eeci.com/adc-16p.htm]

14
Control Subsystem
(i) Digital Controller
- CPU, Memory, Hard disk (to store programs)
Controller
Robot
Sensor
Desired end-effector
trajectory
Driving
input
Actual end-effector
configuration
Joint displacement
and velocity
Fig. 2.9 Control subsystem
[Courtesy: http://www.abb.com/Product/seitp327/f0cec80774b0b3c9c1256fda00409c2c.aspx]
(a) Control scheme of a robot (b) ABB Controller

15
Control Subsystem (contd.)
(ii) Digital-to-Analog Converter (DAC)
(iii) Amplifier
- Amplify weak commands from DAC
Fig. 2.10 A digital-to-analogue converter
[Courtesy: http://www.eeci.com]

16
Classification
• By Applications
• By Coordinate System
• By Actuation System
• By Control Method
• By Programming Method

17
By Application
• Welding robot
• Assembly robot
• Heavy-duty robot
- Special features like maximum speed,
accuracy, etc. are incorporated keeping
the application in mind
- See videos in
http://www.directindustry.com/video/industrial-
robots-robotic-cells-AM.html

18
By Coordinate System
(a) Cartesian
(b) Cylindrical
(c) Spherical
(d) Anthropomorphic
(e) Gantry  (a)
(f) SCARA (Selective Compliance
Assembly Robot Arm)

19
(a) Cartesian robot

20
(b) Cylindrical robot

21
(c) Spherical robot

22
(d) Articulated robot

23
(e) Gantry robot

24
(f) SCARA arm

25
Fundamental Configurations
Type Joints
1 (base): Motion 2 (elevation):
Motion
3 (reach): Motion
Cartesian
Cylindrical
Spherical
Revolute
P: travel, x
-P+R+900@Z
R: rotation θ
R: -do-
R: -do-
P: height y
P: -do-
-P+R+900@Z
R: angle φ
R: -do-
P: reach z
P: -do-
P: -do-
-P+R+900@Z
R: angle ψ

26
Comparison (for selection)
Configuration Advantages Disadvantages
Cartesian (3 linear axes)
x: base travel
y: height
z: reach
- Easy to visualize
- Rigid structure
- Easy offline programming
- Easy mechanical stops
- Reach only front and back
- Requires large floor space
- Axes are hard to seal
- Expensive
Cylindrical (1 rotation
and 2 linear axes)
θ: base rotation
y: height
z : reach
- Can reach all around
- Rigid y, z-axes
- θ-axes easy to seal
- Cannot reach above itself
- Less rigid θ-axis
- y, z-axes hard to seal
- Won’t reach around obstacles
- Horizontal motion is circular
Spherical (2 rotating and
1 linear axes)
θ: base rotation
φ: elevation angle
z: reach
- Can reach all around
- Can reach above or
below obstacles
- Large work volume
- Cannot reach above itself
- Short vertical reach
Articulated (3 rotating
axes)
θ: base rotation
φ: elevation angle
ψ: reach angle
- Can reach above or
below objects
- Largest work area for
least floor space
- Difficult to program off-line
- Two or more ways to reach
a point
- Most complex robot

27
By Actuation System
• Pneumatic (in factory floors)
• Hydraulic (for heavy applications)
• Electric (more common these days)

28
By Control Method
• Servo/Non-servo control
– Servo  closed-loop (Hydraulic & Electric)
– Non-servo  open-loop (Pneumatic)
• Path control
– Continuous path  trajectory (welding etc)

29
By Programming Method
• Online programming
– Direct use of the robot
– Teach pendant
• Offline programming (saves time)
– Using a computer on a new task
– Download when ready

30
Summary of the Chapter
• Focus on serial-type robots (not parallel
or mobile, etc.)
• Different subsystems are explained
• Five ways are explained to classify a
robot

1
Actuators

2
Outline
• An actuation system
• Pneumatic actuators
– Advantages and Disadvantages
• Hydraulic actuators
• Electric actuators
– Stepper motors
– DC motors
– AC motors
• Selection of motors

3
An Actuation System
• A power supply
• A power amplifier
• A motor
• A transmission system
Actuator vs. Motor?
(Interchangeably used)

4
Schematic of Actuation System

5
• One of fluid devices
• Uses compressed air [1-7 bar; ~.1 MPa/bar]
• Components
1) Compressor; 2) After-cooler; 3) Storage tank;
4) Desiccant driers; 5) Filters; 6) Pressure
regulators; 7) Lubricants; 8) Directional control
valves; 9) Actuators
Pneumatic Actuators

6
Fig. 3.2(a) Pneumatic actuator components

8
Advantages vs. Disadvantages
• Advantages
– Cheapest form of actuators.
– Components are readily available.
– Compressed air is available in factories.
– Compressed air can be stored, and
conveyed easily over long distances.
– Compressed air is clean, explosion-proof
& insensitive to temp. var. Many
applns.

9
– Few moving parts Reliable + low maint.
costs
– Relevant personnel are familiar with the tech.
– Very quick Fast work cycles
– No mech. transmission is required.
– Safe in explosive areas as no elect. contact
– Systems are compact.
– Control is simple. Mechanical stops.
– Components are easy to connect.

10
• Disadvantages
– Air is compressible.
– Precise control of speed/position is not
easy.
– If no mechanical stops resetting is slow.
– Not suitable for heavy loads
– If moisture penetrates rusts occur.
Compressibility of the air can be
advantageous.
Prevents damage due to overload.

11
Major Components
• Compressor: Compresses air
• After-cooler: Cools air after
compression as hot air contains vapor
• Storage tank: Provides const. high
press.
• Desiccant Drier: Air passes through
chemicals to remove moisture
• Filters: Removes water droplet
• Pressure Regulator: Poppet valve

12
Fig. 3.3(a) Hydraulic actuator components

14
Advantages vs. Disadvantages
• Advantages
– High  + power-to-size ratio.
– Accurate control of speed/pos./dirn.
–Few backlash prob. Stiffness +
incompressibility of fluid
–Large forces can be applied at
locations.

15
Backlash  Unwanted play in
transmission components
- Greater load carrying cap.
- No mech. linkage  Mech. simplicity.
- Self lubricating  Low wear + non-corrosive
- Due to 'storage' sudden demands can be met.
- Capable of withstanding shock.

16
• Disadvantages
– Leakages occur  Loss in performance
– Higher fire risk.
– Power pack is (70 dBA)
– Temp. change alters viscosity.
– Viscosity at temp. causes sluggishness.
– Servo-control is complex
70 dbA  Noise of heavy traffic

17
Electric Actuators
• Electric motors
+
• Mechanical transmissions
• First commercial electric motor: 1974 by
ABB

18
Advantages vs.
Disadvantages
• Advantages
– Widespread availability of power supply.
– Basic drive element is lighter than fluid
power.
– High power conversion efficiency.
– No pollution
– High accuracy + hight repeatability
compared to cost.
– Quiet and clean

19
– Easily maintained and repaired.
– Components are lightweight.
– Drive system is suitable to electronic
control.
• Disadvantages
– Requires mechanical transmission
system.
– Adds mass and unwanted movement.
– Requires additional power + cost.
– Not safe in explosive atmospheres.

20
Electric Motors
• Stepper motors
– Variable Reluctance
– Permanent Magnet
– Hybrid
• Small/Medium end of industrial range
• Digitally controlled  No feedback
• Incremental shaft rotation for each
pulse

21
• Steps range from 1.8 – 90 deg.
• To know final position, count # of
pulses
• Velocity = No. of pulse per unit time
• 500 pulses/sec  150 rpm (1.8o/pulse)
• Pulses cease, motor stops. No brake,
etc.
• Max. torque at low pulse rate
• Many steppers from same source.
Exact synchronization

23
Features: Variable Reluctance
• Patented: 1919; Commercial: 1950
• Magnetic reluctance  Elec. Resistance
• Magnetic flux only around closed path
• Rotor is soft steel, and 4 poles
• Rotor + stator teeth aligned with the
minimum reluctance  rotor is at rest
• To rotate, AA’ is off BB’ is on

25
Features: Permanent Magnet
• Two sets of coils: A and B
• Rotor is permanent magnet
• Each pole is wound with field winding
• Coil A is reversed  A’. Rotates 45o
CCW
• Coil B is reversed  B’. Another 45o

26
Hybrid Stepper
• Combines the features of Variable
Reluctance and Permanent Motor
• Permanent magnet with iron caps that
have teeth
• The rotor sets itself in minimum reluctance
position

27
• Direct Current: Used in toys etc.
• Electrically driven robots us DC
– Introduced in 1974 by ABB
– Powerful versions available
– Control is simple
– Batteries are rarely used
– AC supply is rectified to DC
DC Motors

28
lBif
fr
a
  sin2

29
Principle of a DC Motor
• Magnetic Field  Stator
– Field coils wound on the stators
– Permanent magnet
• Conductor (Armature)  Rotor
– Current via brushes + commutators
• Maximum torque for  = 90o

31
Features of a DC Motor
• High voltage in stator coils  Fast
speed (simple speed control)
• Varying current in armature 
Controls torque
• Reversing polarity  Turns opposite
• Larger robots: Field control DC motor
– Current in field coils  Controls torque
– High power at high speed + High
power/wt.

32
Specification & Characteristic
Technical Specifications of DC Motors
Brand Parvalux
Manufacturer Part No. PM2 160W511109
Type Industrial DC Electric Motors
Shaft Size (S,M,L) M
Speed (rpm) 4000 rpm
Power Rating (W) 160 W
Voltage Rating (Vdc) 50 V(dc)
Input Current 3.8 A
Height × Width × Length 78 mm ×140 mm × 165 mm

34
Permanent Magnet (PM) Motor
• Two configurations
– Cylindrical [Common in industrial robots]
– Disk

35
Permanent Magnet (PM) Motor (cont.)
• No field coils
• Field is by permanent magnets (PM)
• Some PM has coils for recharge
• Torque  Armature current [Const. flux]

36
Advantages of PM DC Motors
• No power supplies for field coils
• Reliability is high
• No power loss due to field supply
• Improved Efficiency + Cooling

37
Brushless PM DC Motor
• Problem with DC motors
– Commuter and brushes  Periodical
reversal of current through each armature
coil
– Brushes + Commutators  Sliding
contact  Sparks  Wear  Change
brushes + Resurface commuators
• Solution: Brushless motors
– Sequence of stator coils
– PM rotor

39
Principles of Brushless PM
• Reverse principle than convention DC
• Current carrying conductor (stator)
experience a force
• Magnet (rotor) will experience a reaction
(Newton’s 3rd law)
• Current to stator coils is electronically
switched by transistors (Expensive)
• Switching is controlled by rotor position
 Magnet (rotor) rotates same direction

40
Advantages of Brushless PM
• Better heat dissipation
• Reduced rotor inertia
• Weigh less  Less expensive +
Durable
• Smaller for comparable power
• Absence of brushes  Reduced
maintenance cost
• Electric robots  Hazardous areas with
flammable atmospheres (Spray
painting)

41
AC Motors
• Alternating Current: Domestic supply
• 50 Hz; 220 V (India)
• 60 Hz; 110 V (USA)
• Difficult to control speed  Not suitable for
robots

42
Principle of an AC Motor
• External electromagnets (EM) around a
central rotor
• AC supply to EM  Polarity change
performs the task of mech. Switching
• Magnetic field of coils will appear to rotate
 Induces current in rotor (induction) or
makes rotor to rotate (synchronous)

43
Specification & Characteristic
Technical Specifications of AC Motor
Brand ABB
Manufacturer Part No. 1676687
Type Industrial 1-, 3-Phase Electric Motors
Supply Voltage 220 – 240 Vac 50 Hz
Output Power 180 W
Input Current 0.783 A
Shaft Diameter 14 mm
Shaft Length 30 mm
Speed 1370 rpm
Rated Torque 1.3 Nm
Torque Starting 1.3 Nm
Height × Length × Width 150 mm × 213 mm × 120 mm

45
Features of an AC Motor
• Higher the frequency  Fast speed
• Varying frequency to a number of robot
axes has been impractical till recently
• Electromagnetism is used for regenerative
braking (also for DC)  Reduces
deceleration time and overrun
• Motor speed cannot be predicted (same
for DC)  Extra arrangements required

46
Classification of an AC Motor
• Single-phase [Low-power requirements]
– Induction
– Synchronous
• Poly-phase (typically 3-phase) [High-
power requirements]
– Induction
– Synchronous
• Induction motors are cheaper  Widely
used

48
Single-phase AC Induction Motor
• Squirrel cage rotor (Cu or Al bars into slot in the
end)  Circuit is complete
• Stator has windings
 Alternating current
 Alternative magnetic field
• EM forces induces current in the rotor
conductors
• When rotor is stationary no resultant torque (not
self-starting)

49
Single-phase AC Induction Motor
• Auxiliary starting winding
• Motor speed  Frequency
• 50 rev/sec  50 Hz
• No exact match
• Slip: 1 to 3%

50
Three-phase AC Induction Motor
• Three windings in stator at 120o apart
• Each winding is connected to one of the three
lines of the supply
• Direction reversal  Interchange any of two line
connections
• Rotation of field is much smoother
• Self-starting

52
AC Synchronous Motor
• Stator is same as induction motor
• Rotor is permanent magnet
• Since stator magnetic field rotates 
Rotor rotates
• Speed is same as supply frequency
• Used for precise speed requirement
• Not self-starting

53
AC vs. DC Motors
• Cheaper, rugged, reliable,
maintenance free
• Speed control is more complex
• Speed-controlled DC drive (stator
voltage) is cheaper than speed-
controlled AC drive (Variable
Frequency Drive)
• Price of VFD is steadily reducing

54
Motor Selection
• For robot applications
– Positioning accuracy, reliability, speed of
operation, cost, etc.
• Electric is clean + Capable of high
precision
• Electronics is cheap but more heat
• Pneumatics are not for high precision
for continuous path

55
Motor Selection (contd.)
• Hydraulics can generate more power
in compact volume
• Capable of high torque + Rapid
operations
• Power for electro-hydraulic valve is
small but expensive
• All power can be from one powerful
hydraulic pump located at distance

56
Thumb Rule for Motor Selection
• Rapid movement with high torques (>
3.5 kW): Hydraulic actuator
• < 1.5 kW (no fire hazard): Electric
motors
• 1-5 kW: Availability or cost will
determine the choice

57
Sample Calculations
Two meter robot arm to lift 25 kg mass
at 10 rpm
• Force = 25 x 9.81 = 245.25 N
• Torque = 245.25 x 2 = 490.5 Nm
• Speed = 2 x 10/60 = 1.047 rad/sec
• Power = Torque x Speed = 0.513 kW
• Simple but sufficient for approximation

58
Summary
• DC motors
– Permanent Magnet (PM)
– Brushless PM
– Their construction + advantages, etc.
• AC motors
– Single-phase: Induction vs. Synchronous
– Three-phase
• Selection of motors in practical
applications

1
Denavit-Hartenberg (DH)
Parameters
• Four parameters
– Joint offset (b)
– Joint angle ()
– Link length (a)
– Twist angle ()

2
Three-link Planar Arm
Ti =











 
1000
0100
0
0
iiii
iiii
SθaC θSθ
CaSθC θ
• DH-parameters
, for i=1,2,3
Link bi i ai i
1 0 1 (JV) a1 0
2 0 2 (JV) a2 0
3 0 3 (JV) a3 0
• Frame transformations
(Homogeneous)

3
Revolute-Prismatic Planar Arm
• DH-parameters
• Frame transformations (Homogeneous)
1 0 1 (JV) 0 /2
2 b2 (JV) 0 0 0o
T1 =














1000
0010
00
00
11
11
CSθ
SC θ
T2 =












1000
100
0010
0001
2b

4
• DH-parameters
T1 = T2 =
1 b1 (JV) -/2 0 /2
2 0 2 (JV) a2 0













1000
010
0100
0001
1
b











 
1000
0100
0
0
2222
2222
SθaCθSθ
CaSθCθ
Prismatic-Revolute Planar Arm

5
Spherical-type Arm
• DH-parameters
1 0 1 (JV) 0 /2
2 b2 2 (JV) 0 /2
3 b3
(JV)
0 0 0

6
T1 =













1000
0010
00
00
11
11
C θSθ
SθC θ
T2 =














1000
010
00
00
2
22
22
b
CθSθ
SθCθ
T3 =












1000
100
0010
0001
3b
• Frame transformations for Spherical Arm

7
In Summary
• Denavit-Hartenberg (DH) parameters
– DH frames
– Definitions
• DH frame transformations
• Examples
– Three-link planar arm
– RP and PR arms
– Spatial arm

1
Transformations

2
Outline
• Links and Joints
• Kinematic chain
• Degrees-of-freedom (DOF)
• Pose ( Configuration)
• Denavit-Hartenberg (DH) Parameters
• Homogeneous transformation
• Examples

3
Transformations
• To control robot
– Relationship between joint motion (input)
and end-effector motion (output) is required
– Transformations between different
coordinate frames are required
• Robot Architecture
– Links: A rigid body with 6-DOF
– Joints: Couples 2 bodies. Provide
restrictions

4
Joints or Kinematic Pairs
• Lower Pair
– Surface contact: Hinge joint of a door
• Higher pair
– Line or point contact: Roller or ball rolling

5
Lower Pair: Revolute Joint
Turning pair or a
hinge or a pin joint

6
Lower Pair: Prismatic Joint
Sliding pair

7
Lower Pair: Helical Joint

8
Lower Pair: Cylindrical Joint

9
Lower Pair: Spherical Joint

10
Lower Pair: Planar Joint

11
Lower Pair: Universal Joint

12
Kinematic Chain
• Series of links connected by joints
• Simple Kinematic Chain: When each
and every link is coupled to at most
two other links
– Closed: If each and every link coupled to
two other links  Mechanism
– Open: If it contains only two links (end
ones) that are connected to only one link
 Manipulator

13
Closed-chain

14
Open-chain

15
Degrees of Freedom (DOF)
• Number of independent (or minimum)
coordinates required to fully describe
pose or configuration (position + rotation)
– A rigid body in 3D space has 6-DOF
• Use Grubler formula (1917) for planar
mechanisms
• Use Kutzbach formula (1929) for spatial
mechanisms

16
n = s (r  1)  c, c  . . . (5.1)
i
1
c
p
i

Grubler-Kutzbach Criterion
s : dimension of working space
(Planar, s = 3; Spatial, s = 6);
r : no. of rigid bodies or links in the system;
p : no. of kinematic pairs or joints in the system;
ci : no. of constraints imposed by each joint;
c : total no. of constraints imposed by p joints;
ni : relative DOF of each joint;
n : DOF of the whole system.

17
Note that,
i
p
1i
i
p
1i
i
p
1i
nps)n(scc


i
p
i
n1)ps(rn 
. . . (5.2)
ii ncs 
. . . (5.3)
Substituting eq. (5.2) into eq. (5.1) 

18
DOF of a Four-bar Mechanism
Four-bar Mechanism,
n = 3 (4  4  1) + (1 + 1 + 1 + 1)
= 1 . . . (5.4)
i
p
i
n1)ps(rn 

19
Six-DOF Manipulator
n = 6 (7  6  1) + 6  1 = 6 . . . (5.5)
DOF of a Robot Manipulator
i
p
i
n1)ps(rn 

20
Five-bar mechanism
n = 3 (5  5  1) + 5  1
= 2 . . . (5.6)
Double parallelogram
n = 3 (5  6  1) + 6  1
= 0 . . . (5.7)
i
p
i
n1)ps(rn 

21
In Summary
• Links and joint were introduced
• Kinematic chain and DOF were defined
• Formulae for finding DOF
• Examples

22
Pose  Configuration
• Rigid-body motion
– Translation
– Rotation
• Translation: Three position coordinates
• Rotation: Three angular coordinates
• Total: Six coordinates
• A fixed-coordinate. A coordinate frame on
moving body  ‘Pose’ or ‘Configuration’

23
F
p΄
o
p
U
MOM
V
P
W
O
X
Z
Y
Moving Frame M with respect to Fixed frame F
Pose  Position + Rotation

24











xp
zp
ypF][p . . . (5.8)
[ ] , [ ] , and [ ]
1 0 0
0 1 0
00 1
F F Fx y z
    
    
      
    
         
. . . (5.10)
Position Description
p = px x + py y + pz z . . . (5.9)

25
Orientation Description
• Direction cosine representation
• Euler angles representation
• Euler parameters representation, etc.
We will study first two only

26
u = ux x + uy y + uz z
. . . (5.11a)
v = vx x + vy y + vz z
. . . (5.11b)
w = wx x + wy y + wz z
. . . (5.11c)
Direction Cosine Representation
Refer to Fig. 5.12
p = puu + pvv + pww . . . (5.12)

27
p = (puux + pvvx + pwwx)x + (puuy + pvvy + pwwy)y
+ (puuz + pvvz + pwwz)z . . . (5.13)
px = uxpu + vxpv + wxpw . . . (5.14a)
py = uypu + vypv + wypw . . . (5.14b)
pz = uzpu + vzpv + wzpw . . . (5.14c)
Substitute eqs. (5.11a-c) into eq. (5.12)
[p]F = Q [p]M . . . (5.15)

28
[p]F = Q [p]M . . . (5.15)




























































xwxvxu
zwzvzu
ywyvyuQpp
TTT
TTT
TTT,][,][
x
w
x
v
x
u
z
w
z
v
z
u
y
w
y
v
y
u
F
u
p
w
p
v
p
x
p
z
p
y
p M
.. . (5.16)
uTu = vTv = wTw = 1, and
uTv(vTu) = uTw(wTu) = vTw(wTv) = 0 … (5.17)
Q is called Orthogonal

29
u  v = w, v  w = u, and w  u = v . . . (5.18)
QTQ = QQT = 1 ; det (Q) = 1; Q1 = QT . . . (5.19)
Due to orthogonality

30
[ ,
[ ] ,
[ ]
0
0
0
0
1
u]
v
w
F
F
F
Cα
Sα
Sα
Cα
 
 
  
 
  
 
 
  
 
  
 
 
  
 
  

. . . (5.20)
Example 5.6 Elementary Rotations (Fig. 5.13a)

31













100
0
0
CS
SC
ZQ . . . (5.21)



























CS
SC
CS
SC
XY
0
0
001
;
0
010
0
QQ
. . . (5.22)

32
pz = pw . . . (5.25)
[p]F = QZ [p]M . . . (5.26)
py = pu S + pv C . . . (5.24)
px = pu C  pv S . . . (5.23)
Example 5.8 Coordinate Transformation (Fig. 5.13b)

33
px = px C  py S . . . (5.27)
py = px S + py C . . . (5.28)
pz = pz . . . (5.29)
Example 5.9 Vector Rotation (Fig. 5.13c)
[p]F = QZ [p]F … (5.30)

34
…(5.31a)
Euler Angle Representation (ZYZ)









 

100
0
0
Z 

CS
SC
Q

35
…(5.31b)
Euler Angle Representation (contd.)














CS
SC
0
010
0
Y'Q

36









 

100
0
0
'Z' 

CS
SC
Q
…(5.31c)
Euler Angle Representation (contd.)

37
Q = QZQY’QZ’’ . . . (5.31d)







 





SCCSSCCSSCCC
CSSCS
SSCCSCSSCCCSQ
. . . (5.31e)











333231
232221
131211
qqq
qqq
qqq
Q …(5.32a)
For extraction purpose, say, input is given by
),(2tan 1323


S
q
S
q
a …(5.32b)
Cannot find  when S = 0 or 

38
W.R.T. fixed frame: QZY = QYQZ =










010
001
100
























001
010
100
90090
010
90090
Y
oo
oo
CS
SC
Q
But, QYZ = QZQY =












001
100
010
Non-commutative Property









 










 

100
001
010
100
09090
09090
Z
oo
oo
CS
SC
Q
Hence, QZY  QYZ

39
Non-commutative Property: An Illustration

40
Non-commutative Property (contd.)

41
In Summary
• Pose or configuration was defined
• Position description was given
• Orientation description was explained
– Direction cosine
– Euler angles
• Examples were shown
• Euler angle representation
– 12 combinations, ZYZ shown
• Non-commutative property of rotation

42
Coordinate Transformation
F
p΄
o
p
U
MOM
V
P
W
O
X
Z
Y
Task: Point P is known in moving frame M. Find P in fixed frame F.

43
p = o + p . . . (5.34)
[p]F = [o]F + Q[p’]M . . . (5.35)





 












1
][
1
][
1
][
T
F MF poQp
0
. . . (5.36)
MF ][][ pTp  . . . (5.37)
Homogenous Transformation

44
TTT  1 or T1  TT . . . (5.38)







 

1
][
T
TT
1
0
oQQ
T F
. . . (5.39)













1000
1100
2010
0001
T
. . . (5.40)
Example 5.10 Pure Translation
T: Homogenous transformation matrix (4  4)
Fig. 5.19 (a)

45
. . . (5.41)
Example 5.11 Pure Rotation
30 30 0 0
30 30 0 0
0 0 1 0
0 0 0 1
3 1
0 0
2 2
1 3
0 0
2 2
0 0 1 0
0 0 0 1
T
o o
o o
C S
S C
 
 
 
 
 
  
 
 
 
 
  
 
 
 
  
Fig. 5.19 (b)

46
rt TTT  . . . (5.42)
30 30 0 2
30 30 0 1
0 0 1 0
0 0 0 1
3 1
0 2
2 2
1 3
0 1
2 2
0 0 1 0
0 0 0 1
T
o o
o o
C S
S C
 
 
 
 
 
  
 
 
 
 
  
 
 
 
 
. . . (5.43)
Example 5.12 General Motion
Fig. 5.19 (c)

47
Like rotation matrices homogeneous transformation
matrices are non-commutative, i. e.,
Non-commutative Property
TATB  TBTA

48
Denavit and Hartenberg (DH)
Parameters—Frame Allotment
• Serial chain
- Two links connected
by revolute joint, or
- Two links connected
by prismatic joint

49
Connection with a Revolute Joint
Fig. 5.23

50
Connection with a Prismatic Joint

51
• Let axis i denotes the axis of the joint connecting
link (i 1) to link i.
• A coordinate system Xi, Yi, Zi is attached to the
end of the link (i 1)  not to the link i  for i =
1, . . . n+1.
• Choose axis Zi along the axis of joint i, whose
positive direction can be taken towards either
direction of the axis.

52
• Locate the origin, Oi, at the intersection of axis Zi
with the common normal to Zi  1 and Zi. Also,
locate Oi on Zi at the intersection of the common
normal to Zi and Zi + 1.
• Choose axis Xi along the common normal to axes
Zi1 and Zi with the direction from former to the
later.
• Choose axis Yi so as to complete a right handed
frame.

53
For Non-unique Cases
• For Frame 1 that is attached to the fixed
base, i.e., link 0, only the direction of axes
Z1 is specified. Then O1 and X1 can be
chosen arbitrarily.
• For the last frame n + 1 the foregoing
convention do not apply since there is no
link n + 1. Thus, frame n + 1 can be
arbitrarily chosen.

54
• When two consecutive axes are parallel,
the common normal between them is not
uniquely defined.
• When two consecutive axes intersect, the
direction of Xi is arbitrary.
• When joint i is prismatic, only the direction
of axis Zi is determined, whereas the
location of Oi is arbitrary.

55
Denavit-Hartenberg (DH)
Parameters
• Four parameters
– Joint offset (b)
– Joint angle ()
– Link length (a)
– Twist angle ()

56
• bi (Joint offset): Length of the intersections of the
common normals on the joint axis Zi, i.e., Oi and Oi. It is
the relative position of links i  1 and i. This is measured
as the distance between Xi and Xi + 1 along Zi.

57
• i (Joint angle): Angle between the orthogonal projections of
the common normals, Xi and Xi + 1, to a plane normal to the
joint axes Zi. Rotation is positive when it is made counter
clockwise. It is the relative angle between links i  1 and i.
This is measured as the angle between Xi and Xi + 1 about Zi.

58
• ai (Link length): Length between the O’i and Oi
+1. This is measured as the distance along the
common normal Xi + 1 between axes Zi and Zi + 1.

59
• i (Twist angle): Angle between the orthogonal
projections of joint axes, Zi and Zi+1 onto a plane
normal to the common normal. This is measured as
the angle between the axes, Zi and Zi + 1, about axis Xi
+ 1 to be taken positive when rotation is made counter
clockwise.

60
Variable DH Parameters
• First two parameters, bi and i, define the
relative position of links i  1 and i
• Last two parameters, ai and i, describe
the size and shape of link i that are always
constant.
• Parameters, bi and i, are variable
– i is variable if joint i is revolute
– bi is variable if joint i is prismatic.

61
Tb =












1000
100
0010
0001
ib
T =











 
1000
0100
00
00
ii
ii
CθSθ
θSCθ
DH Frame Transformations
• Translation along Zi
• Rotation about Zi

62













1000
00
00
0001
ii
ii
CαSα
αSCα
T = . . . (5.49d)
Ta =












1000
0100
0010
001 ia
. . . (5.49c)
• Translation along Xi+1
• Rotation about Xi+1

63
Ti = TbTTaT
Ti =














1000
0 iii
iiiiiii
iiiiiii
bCαSα
SθaSαCθCαCθSθ
CaSαSθCαSθCθ 
• Total transformation from Frame i to Frame i+1

64
Three-link Planar Arm
Ti =











 
1000
0100
0
0
iiii
iiii
SθaCθSθ
CaSθCθ
• DH-parameters
, for i=1,2,3
1 0 1 (JV) a1 0
2 0 2 (JV) a2 0
3 0 3 (JV) a3 0
• Frame transformations
(Homogeneous)

65
Revolute-Prismatic Planar Arm
• DH-parameters
1 0 1 (JV) 0 /2
2 b2 (JV) 0 0 0o
T1 =














1000
0010
00
00
11
11
CSθ
SCθ
T2 =












1000
100
0010
0001
2b

66
• DH-parameters
T1 = T2 =
1 b1 (JV) -/2 0 /2
2 0 2 (JV) a2 0













1000
010
0100
0001
1b











 
1000
0100
0
0
2222
2222
SθaCθSθ
CaSθCθ
Prismatic-Revolute Planar Arm

67
Spherical-type Arm
• DH-parameters
1 0 1 (JV) 0 /2
2 b2 2 (JV) 0 /2
3 b3
(JV)
0 0 0

68
T1 =













1000
0010
00
00
11
11
CθSθ
SθCθ
T2 =














1000
010
00
00
2
22
22
b
CθSθ
SθCθ
T3 =












1000
100
0010
0001
3b
• Frame transformations for Spherical Arm

69
In Summary
– DH frames
– Definitions
• DH frame transformations
• Examples
– Three-link planar arm
– RP and PR arms
– Spatial arm

70
Summary of the Chapter
• Links, joints, kinematic chains, and DOF
were defined
• Pose or configuration was explained
were introduced
• Homegenous transformation matrix was
derived
• Several examples were solved

1
Kinematics
– Forward kinematics
– Inverse Kinematic

2
Kinematics

3
Kinematics
• Forward kinematics
– Admits unique solution
– Requires simple multiplications and
additions
• Inverse kinematics
– Admits many solutions
– Requires solutions of non-linear algebraic
equations

4
Forward Kinematics
• Homogeneous transformation
– Using DH Parameters
• Forward kinematics relation
T = T1 T2 …Tn … (6.1)
• Alternate to 4 x 4 relation
Q = Q1 Q2 …Qn … (6.2)
p = a1 + Q1 a2 + … + Q1 … Qn-1 an … (6.3)
0 1
0 0 0 1
Q a
T
0
i i i i i i i
i i i i i i i i i
i T
i i i
Cθ Sθ Cα Sθ Sα a C
Sθ Cθ Cα Cθ Sα a Sθ
Sα Cα b
 
          
 
 

5
Forward Kinematics (contd.)
• Using 4  4 homogeneous transformations
T = T1 T2 …Tn
• Three-DOF Articulated arm
• Three-DOF Spherical wrist
• PUMA Robot (architecture)
• Stanford arm

6
DH Parameters of Articulated Arm
1 0 1 (JV) 0  π/2
2 0 2 (JV) a2 0
3 0 3 (JV) a3 0

7
Matrices for Articulated Arm
1 1
1 1
1
0 1 0 0
0 0 0 1
c 0 s 0
s 0 c 0
 
 
 
 
 
 
T
2 2 2 2
2 2 2 2
2
c s 0 a c
s c 0 a s
0 0 1 0
0 0 0 1
 
 
 
 
 
 
T
3 3 3 3
3 3 3 3
3
c s 0 a c
s c 0 a s
0 0 1 0
0 0 0 1
 
 
 
 
 
 
T
















1000
sasa0cs
)cac(ascsscs
)cac(acssc-cc
233222323
2332211231231
2332211231231
)(
T … (6.11)

8
DH Parameters of Spherical Wrist
1 0 1(JV) 0 π/2
2 0 2(JV) 0  π/2
3 0 3(JV) 0 0

9
Matrices for Spherical Wrist














1000
0010
0c0s
0s0c
11
11
1T















1000
0010
00
00
22
22
2
cs
sc
T











 

1000
0100
00
00
33
33
3
cs
sc
T
;
;
















1000
0
0
0
23232
213132131321
213132131321
csscs
ssccscsscccs
sccssscssccc
T
… (6.12)

10
DH Parameters of PUMA Robot
i bi i ai i
1 0 1 (JV) [0] 0 -/2
2 b2 2 (JV) [-/2] a2 0
3 0 3 (JV) [/2] a3 /2
4 b4 4 (JV) [0] 0 -/2
5 0 5 (JV) [0] 0 /2
6 b6 6 (JV) [0] 0 0

11
Forward Kinematics Results for
PUMA Robot
;
;























64
and,
bba
b
a
100
010
001
2
2
3
pQ
… (6.14)

12
DH Parameters of Stanford Arm
i bi i ai i
1 b1 1 (JV) [0] 0 -/2
2 b2 2 (JV) [] 0 -/2
3 b3
(JV)
0 0 0
4 b4 4 (JV) [0] 0 /2
5 0 5 (JV) [0] 0 -/2
6 0 6 (JV) [0] 0 0

13
Forward Kinematics Results for
Stanford Arm
;
;























43
and,
bbb
b
0
100
01-0
001-
1
2pQ
… (6.15)

14
Inverse Kinematics
• Inverse kinematics of 3-DOF RRR planar arm
• Geometric solution of 3-DOF RRR arm
• Inverse kinematics of 3-DOF articulated arm
• Inverse kinematics of 3-DOF spherical wrist

15
Inverse Kinematics of 3-DOF RRR Arm
321 θθθφ 
123312211 cacacapx 
123312211 sasasapy 
122113 cacac φapw xx 
122113 sasas φapw yy 
… (6.15a)
… (6.15b)
… (6.15c)
… (6.16a)
… (6.16b)

16
w2
x + w2
y = a1
2+ a2
2 + 2 a1a2c2
21
2
2
2
1
22
2
2 aa
aaww
c 21 

2
22 1 cs 
2 = atan2 (s2, c2) . . . (6.18)
2121221 ssa)ccaa(wx 
2121221y sca)sca(aw 
Δ
wsaw)ca(a
s
xy 22221
1


Δ
wsaw)ca(a
c
yx 22221
1


22
221
2
2
2
1 2 yx wwcaaaaΔ 
1 = atan2 (s1, c1) . . . (6.20c)
3 =  - 1  2 . . . (6.21)
… (6.19a)
… (6.19b)
… (6.17a)
… (6.17b,c)
… (6.20a,b)

17
Geometrical Solution
of RRR Arm
Apply cosine theorem
w2
x + w2
y = a1
2+ a2
2 + 2 a1a2c2
Same as obtained algebraically, Hence
… (6.22)
w2
x+w2
y = a2
1+a2
22 a1 a2 cos (2)
Since, cos (  2) =  cos 2 -c2

18
Joint Angles
 = atan2 (wy, wx)
221
22
caacosww yx  
22
12
2
2
2
1
22
1
ywxwa
aaywxw
cos


1 =    . . . (6.25)
2 = cos1 (c2) . . . (6.23)
21
2
2
2
1
22
2
2 aa
aaww
c 21 

… (6.24b)
… (6.24a)

19
Numerical Example





















1000
0100
0
2
1
30
2
1
1
2
3
2
3
2
5
2
3
T
• An RRR planar arm (Example 6.11). Input
where  = 60o, and a1 = a2 = 2 units, and a3 = 1 unit.

20
Using eqs. (6.13b-c), c2 = 0.866, and s2 = 0.5,
Next, from eqs. (6.16a-b), s1 = 0, and c1= 0.866.
Finally, from eq. (6.17) ,
Therefore …(6.22b)
The positive values of s2 was used in evaluating 2 = 30o.
The use of negative value would result in :
…(6.22c)
2 = 30o
1 = 0o.
3 = 30o.
1 = 0o 2 = 30o, and 3 = 30
1 = 30o 2 = -30o, and 3 = 60o

21
Three-DOF Articulated Arm
• Forward kinematics relation
)p,(paθ xy2tan1 
)p,(paπθ xy2tan1 
when 2 is equal to  2
(1), where 2
(1) is one
of the solutions

22
Other Two Joint Solutions
• With 1st joint known, other two joints are
like planar RR arm
)c,(satan2θ 333 
2
33
32
2
3
2
2
2
z
2
y
2
x
3 c1s;
aa2
aappp
c 


)c,(satan2θ 222 
Δ
ppsa)pca(a-
s
2
y
2
x33z332
2


Δ
psapp)ca(a
c
z33
2
y
2
x332
2


;
2 2 2
x y zwhere p p p   

23
For b20, there also exists four solutions
;

24
Three-DOF Spherical Wrist
• For orientation only.
Input
;
11 12 13
21 22 23
31 32 33
1 2 1 1 2
1 2 1 1 2
2 20
Q
q q q
q q q
q q q
c c s c s
s c c s s
s c
 
   
  
  
   
  

25
)q,(qaθ 13231 2tan
 33
2
23
2
132 2tan q,qqaθ 
• Solutions are:
),( 333 12 qqatan2
• Another set of solutions exist for spherical wrist
)q,q(2atan 1323 1
 33
2
23
2
13 q,qq2atan 2
)q,q(tan2a 31323

26
;
• A total of two solutions for spherical wrist

Robotics done

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Robotics done