Transcript Introduction - Electrical Engineering

Introduction to ROBOTICS
Robot Kinematics II
Jizhong Xiao
Department of Electrical Engineering
City College of New York
[email protected]
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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
 pz  k z  i u
Pxyz  RPuvw
i x  jv
jy  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}
r  ARB B r P  Ar o '
A P
 A r P   A RB


 1   013
B P


r
r 


1  1 
A o'
• Homogeneous transformation matrix
 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 R3 space
 px 
p 
P   y   Homogeneous coordinate of P w.r.t. OXYZ
 pz 
 
P
(
p
,
z
x
1
 
a
– A frame in R space
3
 nx

n s a P  n y
F 


0 0 0 1   nz

0
sx
ax
sy
ay
sz
az
0
0
px 
p y 
pz 

1
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p y , pz )
s
n
y
x
10
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 ( R y , ) -- pitch
• A rotation of  about the OZ
axis ( Rz , ) -- roll
Z
 roll
yaw


Y
pitch
X
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Quiz 1
• How to get the resultant rotation matrix for YPR?
T  Rz , R y , Rx ,
C
 S

0

0
 S
C
0
0
0 0   C
0 0  0
1 0    S

0 1  0
Z

X
0
S
1
0
0
0
0

1
0 C
0
0
1 0
 0 C

0 S

0 0
0
 S
C
0
0
0
0

1


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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
1
is equivalent to its transpose
T 
 Position of the origin of OXYZ
1  reference frame w.r.t. OUVW frame
0
T

R
1
T T 
0
 RT P  R P  RT 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 , Z 0 ) 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 righthanded coordinate system.
• Find the link and joint parameters
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Example I
• 3 Revolute Joints
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
• 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 d i : 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 ai : 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
ai : distance from intersection of Zi-1 & Xi
D-H Link Parameter Table
Joint i
i
ai
di
i
1
0
a0
0
0
2
-90
a1
0
1
3
0
0
d2
2
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
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
Z2 Z6
5.
Y1
O
Y3 2
Z5
Z4
Y6
O3

X
6
Y2 2
5
O
6
Z0
Y5
Y
X3 4
t
O5
X5 X6
O4 Z 3
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
i
1
2
3
4
5
6
i
ai d i
-90 0
13
0
8
0
90
0
-l
8
4
-90 0
X1
Z2 Z6
Y1
5
90 0 0
Y3O2
Z5
Z4 6
Y6
0
0 t
O3

X
6
2
Y2
5  : angle from X to X
O
i-1
i
6
Z0
i
Y5
about Zi-1
X 3 Y4
O5
X5 X6
 i : angle from Zi-1 to Zi
O4 Z 3
about X
i
X4
Joint distance
4 ai : 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
C i
0
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,  , , )
n
• Homogeneous matrix T0
– 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 , R y , Rx ,
C
 S

0

0
 S
C
0
0
0 0   C
0 0  0
1 0    S

0 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
1 
 
 2
 3 
 
 4 
 5 
 
 6 
 px 
p 
 y
 pz 
 
 
 
 
 
• Transformation Matrix
 nx
n
T  y
 nz

0
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sx
ax
sy
ay
sz
az
0
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
ax
sy
ay
sz
az
0
0
px 
p y 
pz 

1
CSC  SS
SSC  CS
CC
0
  sin 1 (nz )
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)  


for  x and  y
 180    90
  90    0
for  x and  y
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Yaw-Pitch-Roll Representation
T  Rz , R y , Rx ,
C
 S

0

0
 S
 nx
n
 y
 nz

0
sx
ax
sy
ay
sz
az
0
0
C
0
0
0 0
0 0
1 0

0 1
 C
 0

  S

 0
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
0
0
0

1
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Yaw-Pitch-Roll Representation
1
z ,
R T  Ry , Rx,
 C
 S

 0

 0
 C
 0

 S

 0
S
C
0
0
0 0  n x
0 0 n y

1 0  n z
 
0 1  0
0
S
1
0
0 C
0
0
0
0
0

1
sx
ax
sy
ay
sz
az
0
0
1 0
 0 C

0 S

0 0
0
 S
C
0
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0
0
0

1
(Equation A)
0
0
0

1
33
Yaw-Pitch-Roll Representation
• Compare LHS and RHS of Equation A, we have:
 sin   nx  cos   n y  0
  a tan 2(n y , nx )
cos   nx  sin   n y  cos 

nz   sin 

  a tan 2(nz , cos   nz  sin   n y )
  sin   sx  cos   s y  cos

 sin   ax  cos   a y   sin 
  a tan 2(sin   a x  cos   a y , sin   s x  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
0
2
-90
a1
0
1
3
0
0
d2
2
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Example
Joint i
i
ai
di
i
1
0
a0
0
0
i
i 1
T
2
-90
a1
0
1
3
0
0
d2
2
C i
 S
 i
 0

 0
 C i S i
S i S i
C i C i
 S i C i
S i
C i
0
0
T03  (T 01)(T 21)(T 23)
ai C i 
ai S i 
di 

1 
cosθ 0

1  sinθ 0
T 0
 0

 0
cosθ1

2  sinθ1
T 1
 0

 0
cosθ 2

3  sin  2
T 2
 0

 0
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0 a0 cos  0 
0 a0 sin  0 
1
0 

0
1 
 sinθ 0
cosθ 0
0
0
0
 sin 1
0
cos 1
1
0
0
0
 sinθ 2
cos  2
0
0
a1 cos 1 
a1 sin 1 
0 

1 
0
0 0 
1 d2 

0 1
0
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 endeffector
z
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
ax
sy
ay
sz
az
0
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.
Due: Sept. 23, 2008
Next class: Inverse Kinematics, Jocobian
Matrix, Trajectory planningz
z
z
y
y
x
z
x
y
x
y
x
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