Introduction to ROBOTICS
Robot Kinematics II
Dr. Jizhong Xiao
Department of Electrical Engineering
City College of New York
jxiao@ccny.cuny.edu
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Outline
• Review
– Manipulator Specifications
• Precision, Repeatability
– Homogeneous Matrix
• Denavit-Hartenberg (D-H)
Representation
• Kinematics Equations
• Inverse Kinematics
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Review
• Manipulator, Robot arms, Industrial robot
– A chain of rigid bodies (links) connected by
joints (revolute or prismatic)
• Manipulator Specification
– DOF, Redundant Robot
– Workspace, Payload
How accurately a specified point can be reached
– Precision
accurately the same position can be reached
– Repeatability How
if the motion is repeated many times
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Review
• Manipulators:
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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Review
• Basic Rotation Matrix
 px   i x  i u
Pxyz  p y    jy  i u
 p z  k z  i u
Pxyz  RPuvw
i x  jv
j y  jv
k z  jv
i x  k w   pu 
jy  k w   pv   RPuvw
k z  k w   pw 
z
w
P
Puvw  QPxyz
Q  R1  RT
u
v
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
6
Review
• 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
7
Review
• Homogeneous Transformation
– Special cases
1. Translation
 I 33
A
TB  
013
A o'
r 

1 
A

RB
A
TB  
 013
031 

1 
2. Rotation
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Review
• 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 postmultiplication
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Review
• Homogeneous Representation
– A point in R 3 space
 px 
p 
P   y   Homogeneous coordinate of P w.r.t. OXYZ
 pz 
 
P( px ,
z
1
 
a
– A frame in R space
3
 nx

n s a P  n y
F 


0 0 0 1   nz

0
sx
sy
sz
0
ax
ay
az
0
px 
p y 
pz 

1
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py , pz )
s
n
y
x
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Review
• Orientation Representation
(Euler Angles)
– Description of Yaw, Pitch, Roll
• A rotation of  about the OX
axis ( Rx, ) -- yaw
• A rotation of  about the OY
axis ( Ry , ) -- pitch
yaw
• A rotation of  about the OZ 
axis ( Rz , ) -- roll
Z
 roll

Y
pitch
X
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Quiz 1
• How to get the resultant rotation matrix for YPR?
T  Rz , Ry , Rx,
C
 S

0

0
 S
C
0
0
0
0
1
0
0  C
0  0
0  S

1  0
Z

X
0 S
1 0
0 C
0 0
0
0
0

1
1 0
 0 C

0 S

0 0
0
 S
C
0
0
0
0

1


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Y
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Quiz 2
• Geometric Interpretation?
 R33
T 
 0
P31 

1 
Orientation of OUVW coordinate
frame w.r.t. OXYZ frame
Position of the origin of OUVW
coordinate frame w.r.t. OXYZ frame
• Inverse Homogeneous Matrix?
T
T
Inverse of the rotation submatrix


R

R
P
is equivalent to its transpose
T 1  
 Position of the origin of OXYZ
1  reference frame w.r.t. OUVW frame
0
T

R
1
T T 
0
 R T P   R P   R T R 0


  I 44

1   0 1   0 1
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Kinematics Model
• Forward (direct) Kinematics
q  (q1 , q2 ,qn )
Joint
variables
Direct Kinematics
z
Position and Orientation
of the end-effector
y
x
Inverse Kinematics
Y  ( x, y, z,  , , )
• Inverse Kinematics
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Robot Links and Joints
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Denavit-Hartenberg Convention
• Number the joints from 1 to n starting with the base and ending with
the end-effector.
• Establish the base coordinate system. Establish a right-handed
orthonormal coordinate system ( X 0 , Y0 , Z0 ) at the supporting base
with Z 0 axis lying along the axis of motion of joint 1.
• Establish joint axis. Align the Zi with the axis of motion (rotary or
sliding) of joint i+1.
• Establish the origin of the ith coordinate system. Locate the origin of
the ith coordinate at the intersection of the Zi & Zi-1 or at the
intersection of common normal between the Zi & Zi-1 axes and the Zi
axis.
• Establish Xi axis. Establish X i  (Zi1  Zi ) / Zi1  Zi or along the
common normal between the Zi-1 & Zi axes when they are parallel.
• Establish Yi axis. Assign Yi  (Zi  X i ) / Zi  X i to complete the
right-handed coordinate system.
• Find the link and joint parameters
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Example I
• 3 Revolute Joints
Z3 end-effector frame
Z1
Z0
Y0
O3
Y1
Link 1
Joint 1
Joint 3
O0 X0
Joint 2
Link 2
X3
d2
O1 X1 O2 X2
Y2
a0
a1
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Link Coordinate Frames
• Assign Link Coordinate Frames:
– To describe the geometry of robot motion, we assign a Cartesian
coordinate frame (Oi, Xi,Yi,Zi) to each link, as follows:
• establish a right-handed orthonormal coordinate frame O0 at
the supporting base with Z0 lying along joint 1 motion axis.
• the Zi axis is directed along the axis of motion of joint (i + 1),
that is, link (i + 1) rotates about or translates along Zi;
Z3
Z1
Z0
Y0
O3
Y1
Link 1
Joint 1
Joint 3
O0 X0
Joint 2
Link 2
X3
d2
O1 X1 O2 X2
Y2
a0
a1
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Link Coordinate Frames
– Locate the origin of the ith coordinate at the intersection
of the Zi & Zi-1 or at the intersection of common normal
between the Zi & Zi-1 axes and the Zi axis.
– the Xi axis lies along the common normal from the Zi-1
axis to the Zi axis X i  (Zi1  Zi ) / Zi1  Zi , (if Zi-1 is
parallel to Zi, then Xi is specified arbitrarily, subject only
Z3
to Xi being perpendicular to Zi);
Z1
Z0
Y0
Joint 3
O3
Y1
X3
d2
Joint 1
O0 X0
Joint 2
O1 X1 O2 X2
Y2
a0
a1
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Link Coordinate Frames
– Assign Yi  (Zi  X i ) / Zi  X i to complete the righthanded coordinate system.
• The hand coordinate frame is specified by the geometry
On
of the end-effector. Normally, establish Zn along the
direction of Zn-1 axis and pointing away from the robot;
establish Xn such that it is normal to both Zn-1 and Zn
axes. Assign Yn to complete the right-handed coordinate
system.
Z3
Z1
Z0
Y0
Joint 3
O3
Y1
X3
d2
Joint 1
O0 X0
Joint 2
O1 X1 O2 X2
Y2
a0
a1
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Link and Joint Parameters
• Joint angle  i : the angle of rotation from the Xi-1 axis to
the Xi axis about the Zi-1 axis. It is the joint variable if joint i
is rotary.
• Joint distance di : the distance from the origin of the (i-1)
coordinate system to the intersection of the Zi-1 axis and
the Xi axis along the Zi-1 axis. It is the joint variable if joint i
is prismatic.
• Link length a i : the distance from the intersection of the Zi-1
axis and the Xi axis to the origin of the ith coordinate
system along the Xi axis.
• Link twist angle  i : the angle of rotation from the Zi-1 axis
to the Zi axis about the Xi axis.
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Example I
Z3
Z1
Z0
Y0
Joint 3
O3
Y1
X3
d2
Joint 1
O0 X0
Joint 2
O1 X1 O2 X2
Y2
a0
a1
i : rotation angle from Zi-1 to Zi about Xi
a i : distance from intersection of Zi-1 & Xi
D-H Link Parameter Table
Joint i
i
ai
di
i
1
0
a0
0
1
2
-90
a1
0
2
3
0
0
d2
3
to origin of i coordinate along Xi
di
: distance from origin of (i-1) coordinate to intersection of Zi-1 & Xi along Zi-1
i
: rotation angle from Xi-1 to Xi about Zi-1
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Example II: PUMA 260
1
2
Z1
3
O1
Y1
Z0
1.
Number the joints
2.
Establish base frame
3.
Establish joint axis Zi
4.
Locate origin, (intersect.
of Zi & Zi-1) OR (intersect
of common normal & Zi )
X1
Z 2 Z6
5.
O2
Y3
Z4
Z
X 2 5 6 Y6
O3
Y2

5
O6
Y5
X 3 Y4
t
O5 O X 5 X 6
4
Z3
X4
Establish Xi,Yi
X i  (Zi 1  Zi ) / Zi 1  Zi
Yi  (Zi  X i ) / Zi  X i
4
PUMA 260
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Link Parameters
1
J
2
1
Z1
2
3
O1
3
X1
Z 2 Z6
Y1
O2
Y3
Z4
Z
X 2 5 6 Y6
O3
Y2

5
O6
Z0
Y5
X 3 Y4
O5 O X 5 X 6
4
Z3
X4
Joint distance
4
4
5
6
i
1
2
3
4
5
6
i
a i di
-90 0
13
0
8
0
90
0
-90 0
-l
8
90
0
0
0
0
t
 i : angle from Xi-1 to Xi
about Zi-1
i : angle from Zi-1 to Zi
about Xi
a i : distance from intersection
of Zi-1 & Xi to Oi along Xi
di : distance from Oi-1 to intersection of Zi-1 & Xi along Zi-1
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Transformation between i-1 and i
• Four successive elementary transformations
are required to relate the i-th coordinate frame
to the (i-1)-th coordinate frame:
– Rotate about the Z i-1 axis an angle of i to align the
X i-1 axis with the X i axis.
– Translate along the Z i-1 axis a distance of di, to bring
Xi-1 and Xi axes into coincidence.
– Translate along the Xi axis a distance of ai to bring
the two origins Oi-1 and Oi as well as the X axis into
coincidence.
– Rotate about the Xi axis an angle of αi ( in the righthanded sense), to bring the two coordinates into
coincidence.
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Transformation between i-1 and i
• D-H transformation matrix for adjacent coordinate
frames, i and i-1.
– The position and orientation of the i-th frame coordinate
can be expressed in the (i-1)th frame by the following
homogeneous transformation matrix:
Source coordinate
Ti i 1  T ( zi 1 , d i ) R ( zi 1 , i )T ( xi , ai ) R ( xi ,  i )
Reference
Coordinate
C i
 S
 i
 0

 0
 C i S i
S i S i
C i C i
 S i C i
S i
0
C i
0
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ai C i 
ai S i 
di 

1 
26
Kinematic Equations
q  (q1 , q2 ,qn )
• Forward Kinematics
– Given joint variables
– End-effector position & orientation Y  ( x, y, z,  , , )
• Homogeneous matrix T0n
– specifies the location of the ith coordinate frame w.r.t.
the base coordinate system
– chain product of successive coordinate transformation
matrices of Ti i 1
T  T T T
n
0
Orientation
matrix
 R0n

0
1 2
0 1
n
n 1
Position
vector
P0n  n s a P0n 


1  0 0 0 1 
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Kinematics Equations
• Other representations
– reference from, tool frame
0
Treftool  Bref
T0n H ntool
– Yaw-Pitch-Roll representation for orientation
T  Rz , Ry , Rx,
C
 S

0

0
 S
C
0
0
0
0
1
0
0  C
0  0
0  S

1  0
0 S
1 0
0 C
0 0
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0
0
0

1
1 0
 0 C

0 S

0 0
0
 S
C
0
0
0
0

1
28
Representing forward kinematics
• Forward kinematics
 px 
1 
p 
 
 y
 2
 pz 
 3 
  
 
 4 
 
 5 
 
 
 6 
 
• Transformation Matrix
 nx
n
y

T 
 nz

0
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sx
sy
sz
0
ax
ay
az
0
px 
p y 
pz 

1
29
Representing forward kinematics
• Yaw-Pitch-Roll representation for orientation
CC CSS  SC
 SC SSS  CC
T0n  
  S
CS

0
 0
 nx
n
T0n   y
 nz

0
Problem?
sx
sy
sz
0
ax
ay
az
0
px 
p y 
pz 

1
CSC  SS
SSC  CS
CC
0
  sin (nz )
1
px 
p y 
pz 

1
az
  cos (
)
cos 
nx
1
  cos (
)
cos 
1
Solution is inconsistent and ill-conditioned!!
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atan2(y,x)
y
x
 0    90
for  x and  y



for  x and  y
 90    180
  a tan 2( y, x)  



180




90
for  x and  y

  90    0
for  x and  y
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Yaw-Pitch-Roll Representation
T  Rz , Ry , Rx,
C
 S

0

0
 S
C
0
0
 nx
n
 y
 nz

0
sx
sy
sz
0
0
0
1
0
0
0
0

1
ax
ay
az
0
0
0
0

1
 C
 0

 S

 0
0 S
1 0
0 C
0 0
0
0
0

1
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1 0
 0 C

0 S

0 0
0
 S
C
0
0
0
0

1
32
Yaw-Pitch-Roll Representation
1
z ,
R T  Ry, Rx,
 C
  S

 0

 0
 C
 0

 S

 0
S
C
0
0
0
0
1
0
0
0
0

1
 nx
n
 y
 nz

0
0 S
1 0
0 C
0 0
0
0
0

1
1 0
 0 C

0 S

0 0
sx
sy
sz
0
ax
ay
az
0
0
0
0

1
0
 S
C
0
0
0
0

1
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(Equation A)
33
Yaw-Pitch-Roll Representation
• Compare LHS and RHS of Equation A, we have:
 sin   nx  cos  ny  0
  a tan2(ny , nx )
cos   nx  sin   n y  cos 

nz   sin 

  a tan2(nz , cos  nz  sin   ny )
  sin   s x  cos  s y  cos

 sin   ax  cos  a y   sin
  a tan2(sin  ax  cos  ay , sin   sx  cos  s y )
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Kinematic Model
• Steps to derive kinematics model:
– Assign D-H coordinates frames
– Find link parameters
– Transformation matrices of adjacent joints
– Calculate Kinematics Matrix
– When necessary, Euler angle representation
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Example
Z3
Z1
Z0
Y0
Joint 3
O3
Y1
X3
d2
Joint 1
O0 X0
Joint 2
O1 X1 O2 X2
Y2
a0
a1
Joint i
i
ai
di
i
1
0
a0
0
1
2
-90
a1
0
2
3
0
0
d2
3
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Example
Joint i
i
ai
di
i
1
0
a0
0
1
2
-90
a1
0
2
3
0
0
d2
3
Ti i 1
C i
 S
 i
 0

 0
 C i S i
C i C i
S i
0
S i S i
 S i C i
C i
0
T03  (T 01)(T 21)(T 23)
ai C i 
ai S i 
di 

1 
cosθ1
 sinθ
1
T 01 
 0

 0
cosθ 2

2  sinθ 2
T 1
 0

 0
cosθ 3
sin 
3
T 23 
 0

 0
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 sinθ1
cosθ1
0
0
0 a0 cos1 
0 a0 sin 1 
1
0 

0
1 
0  sin  2
0 cos 2
1
0
0
0
 sinθ 3
cos 3
0
0
a1 cos 2 
a1 sin  2 
0 

1 
0 0
0 0 
1 d2 

0 1
37
Example: Puma 560
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Example: Puma 560
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Link Coordinate Parameters
PUMA 560 robot arm link coordinate parameters
Joint i
i
i
1
1
-90
2
2
0
431.8 149.09
3
3
90
-20.32
0
4
5
4
5
-90
90
0
0
433.07
0
6
6
0
0
56.25
ai(mm) di(mm)
0
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0
40
Example: Puma 560
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Example: Puma 560
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Inverse Kinematics
• Given a desired position (P)
& orientation (R) of the endz
effector
q  (q1 , q2 ,qn )
• Find the joint variables
which can bring the robot
the desired configuration
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x
43
Inverse Kinematics
• More difficult
– Systematic closed-form
solution in general is not
available
– Solution not unique
(x , y)
• Redundant robot
• Elbow-up/elbow-down
configuration
– Robot dependent
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Inverse Kinematics
• Transformation Matrix
 nx
n
T  y
 nz

0
sx
sy
sz
0
ax
ay
az
0
px 
p y 
 T01T12T23T34T45T56
pz 

1
1 
 
 2
 3 
 
 4 
 5 
 
 6 
Special cases make the closed-form arm solution possible:
1.
Three adjacent joint axes intersecting (PUMA, Stanford)
2.
Three adjacent joint axes parallel to one another (MINIMOVER)
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Thank you!
Homework 2 posted on the web.
Next class: Inverse Kinematics, Jocobian
Matrix, Trajectory planningz
y
z
z
y
x
z
x
y
x
y
x
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