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

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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

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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.

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Problem definition

In our Project we deal with the Kinematics , Dynamics and Control of

an industrial serial robot.

We select the Puma 762

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Puma Robot Presentation

High-precision material handling

Machine loading Inspection

Testing Joining Assembly in medium and heavier weight applications

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Puma Robot Presentation

Robot Arm Joint Identification and Axes of Rotation

Robot Arm Joint Axes and Ranges of Rotation

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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

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Puma Robot Presentation

Dimensions of Robot Arm

Weight 590 kg (1298 lb)

Static Load 20 kg (44. 1 Ib)

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Puma Robot Presentation

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The forward kinematics

D-H Convention

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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

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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

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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

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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

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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

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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

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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

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The Jacobian of the manipulator and its singularities

SINGULARITIES

Forearm boundary singularity of Puma 762 robot

3cos( ) 0

The elbow is fully extended.

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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

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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.

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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

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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

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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

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. .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

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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

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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)

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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)

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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)

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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)

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PD control with simulations and animation

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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

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D-2C Skew Symmetric??

Symmetric??D