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The City College of New York
Pengantar Robotika – Universitas Gunadarma 1
Mohammad IqbalBased-on Slide Jizhong Xiao
Robot Kinematics II
Introduction to ROBOTICS
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The City College of New York
Pengantar Robotika – Universitas Gunadarma 2
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
– Precision
– Repeatability
How accurately a specified point can be reached
How accurately the same position can be reached
if the motion is repeated many times
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Review• Manipulators:
Cartesian: PPP Cylindrical: RPP Spherical: RRP
SCARA: RRP
(Selective Compliance
Assembly Robot Arm)
Articulated: RRR
Hand coordinate:
n: normal vector; s: sliding vector;
a: approach vector, normal to the
tool mounting plate
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Review•
Basic Rotation Matrix
uvw xyz RP P
xyz uvw QP P
T R RQ 1
uvw
w
v
u
z
y
x
xyz RP
p
p
p
p
p
p
P
wzvzuz
wyvyuy
wxvxux
k k jk ik
k j j ji j
k i jiii
x
z
y
v
w
P
u
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Basic Rotation Matrices – Rotation about x-axis with
– Rotation about y-axis with
– Rotation about z-axis with
uvw xyz RP P
C S
S C x Rot
0
0
001
),(
0
010
0
),(
C S
S C
y Rot
100
0
0
),(
C S
S C
z Rot
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Review•
Coordinate transformation from { B
} to { A
}
• Homogeneous transformation matrix
'o A P B
B
A P A r r Rr
1101 31
' P Bo A
B
A P A
r r Rr
1010
1333
31
' P Rr RT
o A
B
A
B
A
Position
vector
Rotation
matrix
Scaling
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Review•
Homogeneous Transformation – Special cases
1. Translation
2. Rotation
10
0
31
13 B
A
B
A RT
10 31
'
33
o A
B A r I T
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Review
• Composite Homogeneous TransformationMatrix
• 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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Review• Homogeneous Representation
– A point in space
– A frame in space
3 R
1000
1000 z z z z
y y y y
x x x x
pa sn
pa sn
pa sn
P a sn F
x
y
z ),,( z y x p p p P
n
sa
1
z
y
x
p
p
p
P Homogeneous coordinate of P w.r.t. OXYZ
3 R
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Review
• Orientation Representation(Euler Angles)
– Description of Yaw, Pitch, Roll
•
A rotation of about the OXaxis ( ) -- yaw
• A rotation of about the OY
axis ( ) -- pitch
•
A rotation of about the OZaxis ( ) -- roll
X
Y
Z
, x R
, y R
, z R
yaw
pitch
roll
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Model Kinematik•
Forward (direct) Kinematics
• Inverse Kinematics
),,( 21 nqqqq
),,,,,( z y xY
x
y
z
Direct Kinematics
Inverse Kinematics
Position and Orientation
of the end-effector
Joint
variables
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Robot Link dan Joint
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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 at the supporting base
with 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 X i axis. Establish or along the
common normal between the Zi-1 & Zi axes when they are parallel.
• Establish Y i axis. Assign to complete the
right-handed coordinate system.
• Find the link and joint parameters
),,( 000 Z Y X
iiiii Z Z Z Z X 11 /)(
iiiii X Z X Z Y /)(
0 Z
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Contoh I
• 3 Revolute Joints
a0 a1
Z0
X0
Y0
Z3
X2
Y1
X1
Y2
d2
Z1
X3 3O
2O1O0O
Joint 1
Joint 2
Joint 3
Link 1 Link 2
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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;
Link 1 Link 2
a0 a1
Z0
X0
Y0
Z3
X2
Y1
X1
Y2
d2
Z1
X3 3O
2O1O0OJoint 1
Joint 2
Joint 3
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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 , (if Zi-1 is
parallel to Zi , then Xi is specified arbitrarily, subject only
to Xi being perpendicular to Zi );
iiiii Z Z Z Z X 11 /)(
a0 a1
Z0
X0
Y0
Z3
X2
Y1
X1
Y2
d2
Z1
X3 3O
2O1O0OJoint 1
Joint 2
Joint 3
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Link Coordinate Frames – Assign to complete the right-
handed coordinate system.
• The hand coordinate frame is specified by the geometry
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.
iiiii X Z X Z Y /)(
nO
a0 a1
Z0
X0
Y0
Z3
X2
Y1
X1
Y2
d2
Z1
X3 3O
2O1O0OJoint 1
Joint 2
Joint 3
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Parameter Link dan Joint• Joint angle : the angle of rotation from the X
i-1
axis to
the Xi axis about the Zi-1 axis. It is the joint variable if joint iis rotary.
• Joint distance : 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 iis prismatic.
• Link length : 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 : the angle of rotation from the Zi-1 axisto the Zi axis about the Xi axis.
i
id
ia
i
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Contoh I
Joint i i a i d i i
1 0 a0 0 0
2 -90 a1 0 1
3 0 0 d2 2
D-H Link Parameter Table
: rotation angle from Xi-1
to Xi about Z
i-1
i
: distance from origin of (i-1) coordinate to intersection of Zi-1
& Xi along Z
i-1
: distance from intersection of Zi-1 & Xi to origin of i coordinate along X
i
id
: rotation angle from Zi-1
to Zi about X
i
ia
i
a0 a1
Z0
X0
Y0
Z3
X2
Y1
X1
Y2
d2
Z1
X3 3O
2O1O0OJoint 1
Joint 2
Joint 3
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Contoh II: PUMA 260
iiiii Z Z Z Z X 11 /)(
iiiii X Z X Z Y /)(
1 2
3
4
5 6
0 Z
1 Z
2 Z
3 Z
4 Z 5 Z
1O
2O
3O
5O4O
6O
1 X 1Y
2 X 2Y
3 X
3Y
4 X
4Y
5 X 5Y
6 X
6Y
6 Z
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 )
5. Establish Xi,Yi
PUMA 260
t
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Link Parameters
1 2
3
4
5 6
0 Z
1 Z
2 Z
3 Z
4 Z 5 Z
1O
2O
3O
5O4O
6O
1 X 1Y
2 X 2Y
3 X
3Y
4 X
4Y
5 X 5Y
6 X
6Y
6 Z
: angle from Zi-1
to Zi
about Xi
: distance from intersectionof Z
i-1& X
ito Oi along X
i
Joint distance : distance from Oi-1 to intersection of Zi-1
& Xi along Z
i-1
: angle from Xi-1
to Xi
about Zi-1
i
i
ia
id
t006
0090580-904
0
0903
802
130-901
J i
1
4
2
3
6
5
i ia id
-
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Transformasi antara i-1 dan i
•
Four successive elementary transformationsare 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 intocoincidence.
– Rotate about the Xi axis an angle of αi ( in the right-
handed sense), to bring the two coordinates into
coincidence.
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Transformasi antara i-1 dan 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:
1000
0
),(),(),(),( 111
iii
iiiiiii
iiiiiii
iiiiiiiiii
d C S
S aC S C C S
C aS S S C C
x Ra xT z Rd z T T
Source coordinate
ReferenceCoordinate
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Persamaan Kinematik• Forward Kinematics
– Given joint variables
– End-effector position & orientation
• Homogeneous matrix
– specifies the location of the ith coordinate frame w.r.t.the base coordinate system
– chain product of successive coordinate transformation
matrices of
100010
000
1
2
1
1
00
nnn
n
n
n
P a sn P R
T T T T
),,(21 n
qqqq
),,,,,( z y xY
iiT 1
nT 0
Orientation
matrix
Position
vector
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Persamaan Kinematik
• Other representations – reference from, tool frame
– Yaw-Pitch-Roll representation for orientation
tool
n
n
ref
tool
ref H T BT 0
0
,,, x y z R R RT
1000
0100
00
00
C S
S C
1000
00
0010
00
C S
S C
1000
00
00
0001
C S
S C
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Representasi forward kinematics
• Forward kinematics
z
y
x
p p
p
6
5
4
3
2
1
1000
z z z z
y y y y
x x x x
pa sn
pa sn pa sn
T
• Transformation Matrix
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Representasi forward kinematics
•
Yaw-Pitch-Roll representation for orientation
1000
0
z
y
x
n
pC C S C S
pS C C S S C C S S S C S
pS S C S C C S S S C C C
T
1000
0
z z z z
y y y y
x x x x
n
pa sn
pa sn
pa sn
T
)(sin 1
z n
)
cos
(cos 1
z a
)cos
(cos 1
xn
Problem? Solution is inconsistent and ill-conditioned!!
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Representasi Yaw-Pitch-Roll
,,, x y z
R R RT
1000
0100
00
00
C S
S C
1000
00
0010
00
C S
S C
1000
00
00
0001
C S
S C
1000
0
0
0
z z z
y y y
x x x
a sn
a sn
a sn
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Representasi Yaw-Pitch-Roll
,,
1
, x y z R RT R
1000
0100
00
00
C S
S C
1000
00
0010
00
C S
S C
1000
00
00
0001
C S
S C
1000
0
0
0
z z z
y y y
x x x
a sn
a sn
a sn
(Equation A)
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Representasi Yaw-Pitch-Roll
0cossin y x nn
sincossincos
z
y x
nnn
sincossin
coscossin
y x
y x
aa
s s
• Compare LHS and RHS of Equation A, we have:
),(2tan x y nna
)sincos,(2tan y z z nnna
)cossin,cos(sin2tan y x y x s saaa
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Model Kinematic
• 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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Contoh
Joint i i a i d i i
1 0 a0 0 0
2 -90 a1 0 1
3 0 0 d2 2
a0 a1
Z0
X0
Y0
Z3
X2
Y1
X1
Y2
d2
Z1
X3 3O
2O1O0OJoint 1
Joint 2
Joint 3
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Contoh Joint i i a i d i i
1 0 a0 0 0
2 -90 a1 0 1
3 0 0 d2 2
1000
0100
0cosθsinθ
0sinθcosθ
00
00
0
00
00
1sin
cos
a
a
T
1000
000
sinθcosθ
1
1
11
sincos0cossin0
111
111
2
aa
T
))()(( 210
3213
0 T T T T
1000
0sinθcosθ 22
2
2
223
100
00cossin
0
d T
1000
01
iii
iiiiiii
iiiiiii
ii
d C S
S aC S C C S
C aS S S C C
T
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Contoh: Puma 560
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Contoh : Puma 560
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Parameter Koordinat Link
Joint i i i a i (mm) d i (mm
1 1 -90 0 0
2 2 0 431.8 149.09
3 3 90 -20.32 0
4 4 -90 0 433.07
5 5 90 0 0
6 6 0 0 56.25
PUMA 560 robot arm link coordinate parameters
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Contoh : Puma 560
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Contoh : Puma 560
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Inverse Kinematics• Given a desired position (P)
& orientation (R) of the end-
effector
• Find the joint variables
which can bring the robot
the desired configuration
),,( 21 nqqqq
x
y
z
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Inverse Kinematics
• More difficult – Systematic closed-form
solution in general is not
available
– Solution not unique
• Redundant robot
• Elbow-up/elbow-down
configuration – Robot dependent
(x , y)
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Inverse Kinematics
6
5
4
3
2
1
6
5
5
4
4
3
3
2
2
1
1
0
1000
T T T T T T pa sn
pa sn
pa sn
T z z z z
y y y y
x x x x
• Transformation Matrix
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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Terima Kasih
x
y z
x
y z
x
y z
x
z
y
Materi Selanjutnya : Mobot