Introduction to ROBOTICS
Kinematics of Robot Manipulator
Dr. Jizhong Xiao
Department of Electrical Engineering
City College of New York
jxiao@ccny.cuny.edu
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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:
Articulated: RRR
SCARA: RRP
n: normal vector; s: sliding vector;
(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, controlled-path 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
how accurately a specified point
can be reached
how accurately the same position
can be reached if the motion is
Which one is more important?
repeated many times
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What is Kinematics
• Forward kinematics
z
Given joint variables
q  (q1 , q2 , q3 , q4 , q5 , q6 ,qn )
Y  ( x, y, z, O, A, T )
y
x
End-effector position and orientation, -Formula?
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What is Kinematics
• Inverse kinematics
End effector position
and orientation
z
( x, y, z, O, A, T )
y
q  (q1 , q2 , q3 , q4 , q5 , q6 ,qn )
x
Joint variables -Formula?
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Example 1
Forward kinemat ics
x1  l cos
y0
y1
x1
y1  l sin 
l
Inversekinemat ics
  cos ( x1 / l )
1
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
x0
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Preliminary
• Robot Reference Frames
– World frame
– Joint frame
– Tool frame
z
z
y
x
y
W
R
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T
P
x
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Preliminary
• Coordinate Transformation
– Reference coordinate frame
OXYZ
– Body-attached frame O’uvw
z
P
Point represented in OXYZ:
Pxyz  [ px , py , pz ]

Pxyz  px i x  p y jy  pz k z
T
Point represented in O’uvw:

Puvw  pu i u  pv jv  pwk w
Two frames coincide ==>
y
w
v
O, O’
u
x
pu  px pv  py pw  pz
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Preliminary
Properties: Dot Product
3
Let x and y be arbitrary vectors in R and  be
the angle from x to y , then
x  y  x y cos
Properties of orthonormal coordinate frame
• Mutually perpendicular
 
i  j 0
 
i k  0
 
k j 0
• Unit vectors
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
| i | 1

| j | 1

| k | 1
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Preliminary
• Coordinate Transformation
z
– Rotation only

Pxyz  px i x  p y jy  pz k z

Puvw  pu i u  pv jv  pwk w
P
y
w
v
Pxyz  RPuvw
u
x
How to relate the coordinate in these two frames?
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Preliminary
• 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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Preliminary
• 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 
1 0
Rot( x, )  0 C
0 S
0 
 S 
C 
z
w
P
v

u
y
x
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Preliminary
• Is it True?
– Rotation about x axis with
0
 px  1
 p   0 cos
 y 
 pz  0 sin 
  pu 



 sin    pv 
cos   pw 
0
z
w
P
p x  pu

p y  pv cos  pw sin 
p z  pv sin   pw cos
v
u
y
x
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Basic Rotation Matrices
– Rotation about x-axis with 
1 0
Rot( x, )  0 C
0 S
0 
 S 
C 
– Rotation about y-axis with 
 C
Rot( y,  )   0
 S
– Rotation about z-axis with 
Pxyz  RPuvw
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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Preliminary
• Basic Rotation Matrix
 i x  i u i x  jv i x  k w 
Pxyz  RPuvw
R   jy  i u jy  jv jy  k w 
k z  i u k z  jv k z  k w 
– Obtain the coordinate of Puvw from the coordinate
Dot products are commutative!
of Pxyz
 pu   i u  i x
 p    j i
 v  v x
 pw  k w  i x
i u  jy
jv  j y
k w  jy
i u  k z   px 

jv  k z   p y 
k w  k z   p z 
QR  RT R  R1R  I3
Puvw  QPxyz
Q  R1  RT
<== 3X3 identity matrix
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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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Example 3
• A point axyz  (4,3,2) is the coordinate w.r.t. the
reference coordinate system, find the
corresponding point auvw w.r.t. the rotated
OU-V-W coordinate system if it has been
rotated 60 degree about OZ axis.
auvw  Rot( z ,60)T a xyz
0.866 0 4  4.598 
 0.5
  0.866 0.5 0 3   1.964
 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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Example 4
• 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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Coordinate Transformations
• position vector of P
in {B} is transformed
to position vector of P
in {A}
• description of {B} as
seen from an observer
in {A}
Rotation of {B} with respect to {A}
Translation of the origin of {B} with respect to origin of {A}
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Coordinate Transformations
• Two Special Cases
A P A
B P A o'
r  RB r  r
1. Translation only
– Axes of {B} and {A} are
parallel
A
RB  1
2. Rotation only
– Origins of {B} and {A}
are coincident
r 0
A o'
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Homogeneous Representation
• Coordinate transformation from {B} to {A}
A P A
r  RB B r P  Ar o'
A
A o'

 r 
RB
r  B r P 




1  1 
 1   013
• Homogeneous transformation matrix
A P
 RB
TB  
 013
A
A
r   R33

1   0
A o'
P31 

1 
Scaling
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Rotation
matrix
Position
vector
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Homogeneous Transformation
• Special cases
1. Translation
 I 33
A
TB  
013
r 

1 
A o'
2. Rotation
A

RB
A
TB  
 013
031 

1 
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Example 5
• Translation along Z-axis with h:
1
0
Trans( z , h)  
0

0
0
1
0
0
0
0
1
0
 x  1
 y  0
 
 z  0
  
 1  0
0
0
h

1
z
0
1
0
0
0
0
1
0
0  pu   pu 
0  pv   pv 

h   pw   pw  h 
  

1  1   1 
z
P
P
y
w
y
w
v
v
O, O’
u
x
h
O, O’
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u
x
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Example 6
• Rotation about the X-axis by
1 0
0 C
Rot( x, )  
0 S

0 0
0
0
0

1
0
 S
C
0
z
w
 x  1 0
 y  0 C
 
 z  0 S
  
 1  0 0
P
u
x
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0
 S
C
0
0  pu 
0  pv 
0  p w 
 
1  1 
v
y
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Homogeneous Transformation
• Composite Homogeneous Transformation
Matrix
• Rules:
– Transformation (rotation/translation) w.r.t
(X,Y,Z) (OLD FRAME), using premultiplication
– Transformation (rotation/translation) w.r.t
(U,V,W) (NEW FRAME), using postmultiplication
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Example 7
• Find the homogeneous transformation matrix
(T) for the following operations:
Rotation about OX axis
T ranslation of a alongOX axis
T ranslation of d alongOZ axis
Rotationof  about OZ axis
T  Tz , Tz ,dTx,aTx, I 44
Answer :
C
 S

 0

 0
 S
C
0
0
0
0
1
0
0 1
0 0
0  0

1  0
0
1
0
0
0
0
1
0
0  1
0  0
d  0

1  0
0
1
0
0
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0
0
1
0
a  1 0
0  0 C
0  0 S

1  0 0
0
 S
C
0
0
0
0

1
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Homogeneous Representation
• A frame in space (Geometric
Interpretation)
z
 R33 P31 
F 

1 
 0
 nx
n
F  y
 nz

0
sx
sy
sz
0
ax
ay
az
0
P( px , py , pz )
a (z’)
s(y’)
n (X’)
px 
p y 
pz 

1
y
x
Principal axis n w.r.t. the reference coordinate system
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Homogeneous Transformation
• Translation
1
0
Fnew  
0

0
 nx
n
 y
 nz

0
a
0 0 d x   nx s x
1 0 d y  n y s y


0 1 d z  nz s z
 
0 0 1 0 0
sx ax px  d x 
s y a y p y  d y 
sz az pz  d z 

0 0
1 
ax
ay
az
0
px 
p y 
pz 

1
z
s
n
a
s
n
y
Fnew  Trans(d x , d y , d z )  Fold
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Homogeneous Transformation
Composite Homogeneous Transformation Matrix
z1
z2
y2
z0
0
y1
A1
y0
1
A2
x1
x0
?
i 1
0
x2
A2 0A11 A2
Ai
Transformation matrix for
adjacent coordinate frames
Chain product of successive
coordinate transformation matrices
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Example 8
a
• For the figure shown below, find the 4x4 homogeneous transformation
matrices i 1 A and 0 Ai for i=1, 2, 3, 4, 5
i
0 
 1 0 0
n
s
a
p


x
x
x
x
c
 0 0 1 e  c 
n s a p  0

y
y
y
A1  
z3
F  y
 0 1 0 a  d 
b
 nz s z a z p z 
y3 x


3


0
0
0
1
d
z5


0 0 0 1 
x5
b 
0  1 0
y5
0 0  1 a  d 
z4
e
1

A2  
1 0 0
y4
0 
z2
x4


x2
0
0
0
1


y
z1
z0
x0
x1
2
y1
y0
Can you find the answer by observation
based on the geometric interpretation of
homogeneous transformation matrix?
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0
 1
0
A2  
0

0
1
0
0
0
0 b 
0 e  c 
1
0 

0
1 
35
Orientation Representation
 R33
F 
 0
P31 

1 
• Rotation matrix representation needs 9
elements to completely describe the
orientation of a rotating rigid body.
• Any easy way?
Euler Angles Representation
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Orientation Representation
• Euler Angles Representation (  , ,  )
– Many different types
– Description of Euler angle representations
Euler Angle I
Sequence

of
 about OU axis
about OZ axis
Rotations  about OW axis
Euler Angle II

Roll-Pitch-Yaw
about OZ axis
 about OX axis
 about OV axis
 about OY axis
 about OW axis

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about OZ axis
37
Euler Angle I, Animated
w'= z
w'"= w"


v'"
v"
v'

y
u'"
u' =u"
x
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Orientation Representation
• Euler Angle I
 cos  sin 

Rz   sin 
cos
Rw''
 0

 cos

  sin 
 0

0
 sin 
cos
0
0
0
1


0 , Ru '   0 cos
 0 sin 
1 

0

0
1 
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

 sin  ,
cos 
0
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Euler Angle I
Resultant eulerian rotation matrix:
R  Rz Ru ' Rw''
 cos cos

  sin  sin  cos


 sin  cos
  cos sin  cos



sin  sin 

 cos sin 
 sin  cos cos
 sin  sin 
 cos cos cos
cos sin 
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
sin  sin  



 cos sin  




cos

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Euler Angle II, Animated
w'= z
w"'= w"

v"'

 v' =v"
y
u"'
Note the opposite
(clockwise) sense of the
third rotation, .
u'
u"
x
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Orientation Representation
• Matrix with Euler Angle II
  sin  sin 

  cos cos cos


 cos sin 
  sin  cos cos


  cos sin 

 sin  cos
 sin  cos cos
cos cos
 sin  cos cos
sin  sin 

cos sin  



sin  sin  


cos 

Quiz: How to get this matrix ?
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Orientation Representation
• Description of Roll Pitch Yaw
Z



Y
X
Quiz: How to get rotation matrix ?
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43
Thank you!
Homework 1 is posted on the web.
Next class: kinematics II
z
z
z
y
y
x
z
x
y
x
y
x
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