Mechatronics 1
Week 3 & 4
Learning Outcomes
• By the end of week 3-4 session, students
will understand kinematics of industrial
robots.
Course Outline
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Forward Kinematics
Homogeneous Transformation Matrix.
Denavit Hartenberg (D-H) parameters.
D-H Transformation Matrix
Interpretation and application of
homogeneous tranformation matrix
• Inverse kinematics.
Homogenous Transformation
Matrix
• It provides information on position and
orientation in a 4x4 matrix.
 px
p
T  y
 pz

0
R11
R12
R21
R22
R31
R32
0
0
R13 

R23 
R33 

1 
Coordinate Frame Placement
(Denavit Hartenberg Concept)
Joint i axis
Joint i-1 axis
Joint i+1 axis
Link i-1
Link i
i
Yi
Zi'
Yi'
ai
Xi'
Zi
Xi
i
Zi-1
di
Yi-1
a i-1
Xi-1
ai : length of link i
di : offset distance at joint i
 i : joint angle of link i
Ref. Lee, Fu, Gonzalez, 1987
 i : twist angle of link i
Rotation Matrix with D-H
Parameters
cos  i
i 1
R i   sin  i
 0
cos  i
i
R i 1   sin  i
 0
 cos  i sin  i
cos  i cos  i
sin  i
 cos  i sin  i
cos  i cos  i
sin  i
sin  i sin  i 
 sin  i cos  i 

cos  i
sin  i sin  i 
 sin  i cos  i 

cos  i
Homogenous Transformation
Matrix with D-H Parameters
cos  i
 sin 
i
i 1
Ai  
 0

 0
cos  i
 sin 
i
i
A i 1  
 0

 0
 cos  i sin  i
sin  i sin  i
cos  i cos  i
 sin  i cos  i
sin  i
cos  i
0
0
 cos  i sin  i
sin  i sin  i
cos  i cos  i
 sin  i cos  i
sin  i
cos  i
0
0
ai cos  i 
ai sin  i 
di 

1 
ai cos  i 
ai sin  i 
di 

1 
Coordinate Frame Placement (1)
d2
Z1
Z2
X1
Y1
a2
Z0
Y2
Z3
Y3
Y0
X0
Gravity
X2
X3
Coordinate Frame Placement (2)
a2
Z1
2
Z2
X2
X1
Y1
Y2
Z0
1
Y0
Z3
Y3
X0
Gravity
X3
3
Homogenous Transf Matrix
{B}
{W}
{T}
{G}
{S}
Ref. Craig, 1987
Kinematics & Inverse Kinematics
Link Parameters
Joint Coordinates
Direct Kinematics
Link Parameters
Joint Coordinates
Inverse Kinematics
Position & Orientation of
End Effector
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 to
the desired posture.
y
x
Inverse Kinematics
• More difficult
• Solution not unique
• Redundant robot
• Elbow-up/elbow-down
configuration
A shorter path is generally
desirable
A
B
1
2
Solving The Problems
• Geometric approach
• Algebraic approach
• Numeric approach
Upper & Lower Configuration
(Condition arises in Inverse Kinematics
problems)
A
B
1
2
L2
Kinematics
L3
J3
J2
(Mitsubishi RV-M1)
Assumed position of end
effector
J1
z0
Base
y0 z1
z2
z3
x0 x1
x2
x3
y1
y2
y3
Kinematics (RV-M1)
Limitation
on
Movement
of Robot
RV-M1
Total Span
Min
Max
Waist
300o
-150o
150o
Shoulder
130o
-100o
30o
Elbow
110o
0o
110o
i
ai
αi
di
θi
1
0
-90º
0
θ1
2
a2 = L2
0
0
θ2
3
a3 = L3
0
0
θ3
Work Space
z0
Direction of Joint
Movements
Arah gerak
positif joint 1
y0 z1
Arah gerak
positif joint 2
z2
Arah gerak positif
joint 3
z3
x0 x1
x2
x3
y1
y2
y3
Joint
Direction (+)
Direction (–)
1
150o
150o
2
30o
100o
3
110o
-