Introduction to ROBOTICS
Kinematics of Robot Manipulator
PROF.UJWAL HARODE
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Outline
Review
Robot Manipulators
-Robot Configuration
-Robot Specification
• Number of Axes, DOF
• Precision, Repeatability
Kinematics
-Preliminary
• World frame, joint frame, end-effector frame
• Rotation Matrix, composite rotation matrix
• Homogeneous Matrix
-Direct kinematics
• Denavit-Hartenberg Representation
• Examples
-Inverse kinematics
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Review
What is a robot?
By general agreement a robot is:
A programmable machine that imitates the actions or appearance of an
intelligent creature–usually a human.
To qualify as a robot, a machine must be able to:
1) Sensing and perception: get information from its surroundings
2) Carry out different tasks: Locomotion or manipulation, do something
physical–such as move or manipulate objects
3) Re-programmable: can do different things
4) Function autonomously and/or interact with human beings
Why use robots?
–Perform 4A tasks in 4D
environments
4A: Automation, Augmentation,
Assistance, Autonomous
4D: Dangerous, Dirty, Dull, Difficult
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Manipulators
• Robot arms, industrial
robot
• Rigid bodies (links)
connected by joints
• Joints: revolute or
prismatic
• Drive: electric or
hydraulic
• End-effector (tool)
mounted on a flange or
plate secured to the
wrist joint of robot
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Manipulators
Robot Configuration
Cartesian: PPP
Cylindrical: RPP
Spherical: RRP
Hand coordinate:
n: normal vector; s: sliding vector;
SCARA: RRP
Articulated: RRR
(Selective Compliance Assembly Robot
Arm)
a: approach vector, normal to the
tool mounting plate
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Manipulators
• Motion Control Methods
• Point to point control
• a sequence of discrete points
• spot welding, pick-and-place, loading
& unloading
• Continuous path control
• follow a prescribed path, controlledpath motion
• Spray painting, Arc welding, Gluing
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Manipulators
• Robot Specifications
• Number of Axes
• Major axes, (1-3) => Position the wrist
• Minor axes, (4-6) => Orient the tool
• Redundant, (7-n) => reaching around
obstacles, avoiding undesirable configuration
•
•
•
•
Degree of Freedom (DOF)
Workspace
Payload (load capacity)
Precision v.s. Repeatability
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What is Kinematics
Forward kinematics
z
Given joint variables
q  (q1 , q2 , q3 , q4 , q5 , q6 ,qn )
y
Y  ( x, y, z, O, A, T )
End-effector position and orientation, Formula?
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x
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What is Kinematics
Inverse kinematics
End effector position
and orientation
( x, y, z, O, A, T )
q  (q1 , q2 , q3 , q4 , q5 , q6 ,qn )
Joint variables -Formula?
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Example 1
Forward kinemat ics
x0  l cos
y0
y1
x1
y0  l sin 
l

Inversekinemat ics
x0
  cos ( x0 / l )
1
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Preliminary
• Robot Reference Frames
1. World frame
2. Joint frame
3. Tool frame
z
T
z
y
x
y
W
T
x
R
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Coordinate Transformation
z
Reference coordinate frame –XYZ
Body-attached frame-ijk
Point represented in XYZ
T
P

[
p
,
p
,
p
]
xyz
x
y
z

Pxyz  p x i x  p y jy  p z k z
P
y
w
v
Point represented in uvw:

Puvw  pu i u  pv jv  pw k w
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O, O’
Two frames coincide
x
u
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Properties: Dot Product
Let and be arbitrary vectors in
be the angle from to , then
and
x  y  x y cos
Properties of orthonormal co-ordinate frame
Unit vectors are mutually perpendicular
 
i  j 0
 
i k  0
 
k j 0

| i | 1

| j | 1

| k | 1
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Coordinate Transformation
Reference coordinate frame
OXYZ
Body-attached frame O’uvw
z
P
Pxyz  RPuvw
w
y
v
u
x
O, O’
How to relate the coordinate in these two frames?
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• Basic Rotation
– px , p y, and p z represent the projections of P onto
OX, OY, OZ axes, respectively
– Since
P  pu i u  pv jv  pwk w
px  i x  P  i x  i u pu  i x  jv pv  i x  k w pw
py  jy  P  jy  i u pu  jy  jv pv  jy  k w pw
pz  k z  P  k z  i u pu  k z  jv pv  k z  k w pw
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Basic Rotation Matrix
 px   i x  i u
 p    j i
 y  y u
 p z  k z  i u
i x  jv
j y  jv
k z  jv
i x  k w   pu 



j y  k w   pv 
k z  k w   pw 
Rotation about x-axis with 
0 
1 0
Rot( x, )  0 C  S 
0 S C 
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z
w
P
v

u
x
y
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Rotation about y-axis with  :
 C
Rot( y,  )   0
 S
Rotation about z-axis with  :
C
Rot( z ,  )   S
 0
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S 
1 0 
0 C 
0
 S
C
0
0
0
1
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Example 2
• A point auvw  (4,3,2) is attached to a rotating frame, the
frame rotates 60 degree about the OZ axis of the
reference frame. Find the coordinates of the point
relative to the reference frame after the rotation.
a xyz  Rot( z ,60)auvw
 0.5  0.866 0 4  0.598
 0.866
0.5
0 3   4.964 
 0
0
1 2  2 
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Composite Rotation Matrix
• A sequence of finite rotations
– matrix multiplications do not commute
– rules:
• if rotating coordinate O-U-V-W is rotating about
principal axis of OXYZ frame, then Pre-multiply the
previous (resultant) rotation matrix with an
appropriate basic rotation matrix
• if rotating coordinate OUVW is rotating about its own
principal axes, then post-multiply the previous
(resultant) rotation matrix with an appropriate basic
rotation matrix
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• Find the rotation matrix for the following
operations:
Rotation about OY axis
Rotation about OW axis
Rotation about OU axis
Answer...
R  Rot( y,  ) I 3 Rot( w, ) Rot(u,  )
 C 0
  0 1
- S 0
 CC
  S
 SC
S  C  S 0 1 0
0 
0   S C 0 0 C  S 
C   0
0
1 0 S C 
SS  CSC CSS  SC 

CC
 CS

SSC  CS CC  SSS 
Pre-multiply if rotate about the OXYZ axes
Post-multiply if rotate about the OUVW axes
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Homogeneous Transformation
•
Special cases
1. Translation
 I 33
A
TB  
013
2. Rotation
r 

1 
A o'
A

RB
A
TB  
 013
031 

1 
Homogeneous transformation matrix
 R33 P31 
F 

1 
 0
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Homogeneous Transformation
• Composite Homogeneous Transformation
Matrix
• Rules:
– Transformation (rotation/translation) w.r.t (X,Y,Z)
(OLD FRAME), using pre-multiplication
– Transformation (rotation/translation) w.r.t (U,V,W)
(NEW FRAME), using post-multiplication
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Orientation Representation
• Description of Roll Pitch Yaw
Z

ROLL

PITCH
X

Y
YAW
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