Robotics Project 000
-
Upload
imen-mnejja-hbiri -
Category
Documents
-
view
94 -
download
5
Transcript of Robotics Project 000
1
Robotics Project
Prepared by :
Naifar Slim
Sonya Bradai
Hbiri Imen
MSPR3: Introduction to Robotics:Kinematics, Dynamics, Control and Vision (Part 1)
PUMA 762
ACADEMIC YEAR 2011/2012
2
Outline
1. Introduction
2. Problem definition
3. Puma Robot Presentation
4. The forward kinematics
5. The inverse kinematics
6. The Jacobian of the manipulator and its singularities
7. Dynamical equations
8. PD control with simulations and animation
9. Conclusion
3
Introduction
A robot is a manipulator which can modify its environment.
This Figure shows a serial link manipulator (left) and a parallel link manipulator (right). A parallel robot, by definition, contains two or more independent serial link chains.
We will focus our discussion on serial link manipulators with revolute or prismatic joints.
4
Problem definition
In our Project we deal with the Kinematics , Dynamics and Control of
an industrial serial robot.
We select the Puma 762
5
Puma Robot Presentation
High-precision material handling
Machine loading Inspection
Testing Joining Assembly in medium and heavier weight applications
6
Puma Robot Presentation
Robot Arm Joint Identification and Axes of Rotation
Robot Arm Joint Axes and Ranges of Rotation
7
Puma Robot Presentation
Joint 1 2 3 4 5 6
Software Movement Limits (deg) 320 220 270 532 200 600
Robot Arm Joint Axes and Ranges of Rotation
8
Puma Robot Presentation
Dimensions of Robot Arm
Weight 590 kg (1298 lb)
Static Load 20 kg (44. 1 Ib)
9
Puma Robot Presentation
10
The forward kinematics
D-H Convention
11
Z0
Z2
Z3
Z4
Z5, Z6
X5
X4
X3X2
X1
X0
O0
O1
O2
O3
O4=O5 O6
Z1
The forward kinematics
D-H Convention
12
The forward kinematics
Link ai (mm) di (mm) αi θi
1 0 1120 -90 θ1
2 650 263 0 θ2
3 0 73 90 θ3
4 0 600 -90 θ4
5 0 0 90 θ5
6 0 125 0 θ6
Mechanism parameters of the Puma 762 robot
D-H Convention
13
The forward kinematics
1 1
1 11
cos( ) 0 sin( ) 0
sin( ) 0 cos( ) 0A
0 1 0 1120
0 0 0 1
2 2 2
2 2 22
cos( ) sin( ) 0 650cos( )
sin( ) cos( ) 0 650sin( )A
0 0 1 0
0 0 0 1
3 3
3 33
cos( ) 0 sin( ) 0
sin( ) 0 cos( ) 0A
0 1 0 165
0 0 0 1
4 4
4 44
cos( ) 0 sin( ) 0
sin( ) 0 cos( ) 0A
0 1 0 500
0 0 0 1
5 5
5 55
cos( ) 0 sin( ) 0
sin( ) 0 cos( ) 0A
0 1 0 0
0 0 0 1
6 6
6 66
cos( ) sin( ) 0 0
sin( ) cos( ) 0 0A
0 0 1 125
0 0 0 1
6 1 2 3 4 5 6T = A .A .A .A .A .A
The forward kinematic equations are therefore given by
D-H Convention
14
The inverse kinematics
Objective: Find the joint variables in terms of the end-effector position
and orientation.
X = 837.66
Y = 996.60
Z = 1451.25
15
The Jacobian of the manipulator and its singularities
Objective: Relate the linear and angular velocity of the end-effector to
the vector of joint velocities .
0 6 0 1 6 1 2 6 2 3 6 3 4 6 4 5 6 5
0 1 2 3 4 5
( ) ( ) ( ) ( ) ( ) ( )Z O O Z O O Z O O Z O O Z O O Z O OJ
Z Z Z Z Z Z
The Manipulator Jacobian or Jacobian J for short is given by
DERIVATION OF THE JACOBIAN
16
The Jacobian of the manipulator and its singularities
DERIVATION OF THE JACOBIAN
1 0 6 0( )
25. 1(26. 2 (24 5. 5). 23 5. 23. 4. 5) 1(336 125. 4. 5)
25. 1(26. 2 (24 5. 5). 23 5. 23. 4. 5) 1(336 125. 4. 5)
0
Jv Z O O
s c c s c c s c s s
c c c s c c s s s s
2 1 6 1( )
25. 1( 23.(24 5. 5) 26. 2 5. 4. 23. 5)
25. 1( 23.(24 5. 5) 26. 2 5. 4. 23. 5)
25( 2(26 (24 5. 5). 3 5. 3. 4. 5) 2( 3(24 5. 5) 5. 4. 2. 5))
Jv Z O O
c c c s c s s
s c c s c s s
c c s c c s s c c c s s
3 2 6 2( )
25. 1( 23.(24 5. 5) 5. 4. 23. 5)
25. 1( 23.(24 5. 5) 5. 4. 23. 5)
25( 3((24 5. 5). 2 5. 2. 4. 5) 3( 2(24 5. 5) 5. 4. 2. 5))
Jv Z O O
c c c c s s
s c c c s s
c c s c c s s c c c s s
4 3 6 3( )
125( 4. 1 1. 23. 4). 5
125( 1. 4 23. 1. 4). 5
125. 23. 4. 5
Jv Z O O
c s c s s s
c c c s s s
s s s
5 4 6 4( )
125( 5. 1. 4 1( 2( 4. 5. 3 3. 5) 2( 3. 4. 5 3. 5)))
125( 5. 1. 4. 2. 3 1. 5. 4 3. 1. 2. 5 2. 1( 3. 4. 5 3 5))
125( 3( 4. 5. 2 2. 5) 3.( 2. 4. 5 2. 5))
Jv Z O O
c s s c s c c s c s s c c c s s
c s c s s c c s c s s s c s c c c s s
c c c s c s s c c c s s
6 5 6 5
0
( ) 0
0
Jv Z O O
17
A robot is defined to be in a singular configuration when the determinant of the Jacobian is equal to 0.
The Jacobian of the manipulator and its singularities
SINGULARITIES
3 2 3 1 3 1 2 3 3
1 3 2 3 2 3 1 2 3 1 2 3
1 2 3 1 2 3
Det Jt 2437500cos( )cos( ) 10 26cos ( ) 26cos( 2 ) 52cos( )
26cos( ) 10cos(2( )) 52cos(2 ) 26cos( 2 ) 5cos( 2( ))
5cos( 2( )) 48sin( ) 48sin(2( )) 24s
1 2 3 1 2 3in( 2( )) 24sin( 2( ) )
Det Jt 0
18
The Jacobian of the manipulator and its singularities
SINGULARITIES
Forearm boundary singularity of Puma 762 robot
3cos( ) 0
The elbow is fully extended.
19
The Jacobian of the manipulator and its singularities
SINGULARITIES
Forearm interior singularity of Puma 762 robot
2 3cos( ) 0
For this second singularity , the wrist center intersects the axis of the base rotation
20
1 3 1 2 3 3 1 3 2 3
2 3 1 2 3 1 2 3 1 2 3
1 2 3 1 2 3 1 2
10 26cos ( ) 26cos( 2 ) 52cos( ) 26cos( ) 10cos(2( ))
52cos(2 ) 26cos( 2 ) 5cos( 2( )) 5cos( 2( ))
48sin( ) 48sin(2( )) 24sin( 2( )) 24sin( 2(
3)) 0
The Jacobian of the manipulator and its singularities
SINGULARITIES
The interpretation of this term is complicated.
21
Dynamical equations
The equation of motion of the robot is given by
.. . .
( ) ( , ) ( ) D q q C q q q g q
Where:D(q) : Inertia matrixC: Centrifugal and Coriolis matrixg(q): Gravity vector
22
DETERMINATION OF THE INERTIA MATRIX
Dynamical equations
( . . . . . . )1
vci vci wi wi
n t t tD m J J J JR I Ri i iii
1[ ( - ) ]0 jJ z O Ovcij ci
Where [ ]1 2 3[ ( - ) 0 0]1 0 1 0[ ( - ) ( - ) 0]2 0 2 0 0 2 1[ ( - ) ( - ) ( - )]3 0 3 0 0 3 1 0 3 2[ ( - ) ( - ) ( - )]4 0 4 0 0 4 1 0 4 2
56 4
J J J Jvc vc vc vcJ z O Ovc cJ z O O z O Ovc c cJ z O O z O O z O Ovc c c cJ z O O z O O z O Ovc c c cJ J Jvcvc vc
23
Dynamical equations
[ 0 0]1 0[ 0]2 0 1[ ]3 0 1 2
56 4 3
J zwJ z zwJ z z zwJ J J Jww w w
( . . . . . . )1
vci vci wi wi
n t t tD m J J J JR I Ri i iii
1 2 3 2 3 3
2 3 2 3 3
3 3 3
I I I I I I
I I I I I I
I I I
24
. .1( )
21 1
d ddn n kj ijkic c q q qkj ijk q q qi i i j k
( )
k
Vg q
q
322 2 3 2 2 3 2 3. . .sin( ) . . .sin( ) . . .sin( )
2 2
ddV m g m d g m g
Dynamical equations
DETERMINATION OF CORIOLIS AND CENTRIFUGAL TERM AND THE GRAVITY TERM
25
PD control with simulations and animation
~ .
( )P DK q K q q
.. . .
( ) ( , ) ( ) D q q C q q q g q.. . . ~ .
( ) ( , ) P DD q q C q q q K q K q
26
PD control with simulations and animation
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
Time
posi
tion
q1(t), q2(t) q3(t)
q1(t)
27
PD control with simulations and animation
0 5 10 15 20 25 30 35 40 45 50-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
Time
posi
tion
q1(t), q2(t) q3(t)
q2(t)
28
PD control with simulations and animation
0 5 10 15 20 25 30 35 40 45 50-1.5
-1
-0.5
0
0.5
1
1.5
2
Time
posi
tion
q1(t), q2(t) q3(t)
q3(t)
29
PD control with simulations and animation
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
Time
position
q1(t), q2(t) q3(t)
q1(t)
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
Time
position
q1(t), q2(t) q3(t)
q1(t)
0 5 10 15 20 25 30 35 40 45 50-1.5
-1
-0.5
0
0.5
1
1.5
2
Time
position
q1(t), q2(t) q3(t)
q3(t)
30
PD control with simulations and animation
31
Conclusion
We have study Puma Robot six links.
We have deriving the forward kinematics’ equation and
studying the problem of inverse kinematics.
Then, we have determinated the Jacobian matrix and discuss the
singularities.
After that, we have computed the dynamical equation of motion
of the manipulator.
Finally, we have studied the control of the robot.
Robotics Project PUMA 762
Slim Naifar & Sonya Bradai & Imen Hbiri
Thank You For Your Attention
33
D-2C Skew Symmetric??
Symmetric??D