mehdi ghayoumi msb rm 160 mghayoum@kent.edu ofc hr: thur, 11-12:30a robotic concepts

Mehdi Ghayoumi

MSB rm 160

mghayoum@kent.edu

Ofc hr: Thur, 11-12:30a

Robotic Concepts

Robotic ConceptsAnnouncements:

• Today we talk about introduction in robotic

• HW #2 is available now due to Monday Sep-07• Office Hours: Tur: 11-12:30• Room 160 MSB

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Robotic ConceptsRobot kinematics 

Robot kinematics studies the relationship between the

dimensions and connectivity of kinematic chains and the

position, velocity and acceleration of each of the links in the

robotic system, in order to plan and control movement and to

compute actuator forces and torques.

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Matrix

ij

mnm

n

n

A

aa

aa

aa

,,

,,

,,

1

221

111

A

A matrix is any doubly subscripted array of elements arranged

in rows and columns.

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Row Vector

[1 x n] matrix 

jn aaaaA ,, 2 1

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Column Vector

i

m

a

a

a

a

A 2

1

[m x 1] matrix

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Square Matrix

B

5 4 7

3 6 1

2 1 3

Same number of rows and columns

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Identity Matrix

I

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

Square matrix with ones on the diagonal and zeros elsewhere.

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Transpose Matrix

A'

a11 a21 ,, am1

a12 a22 ,, am 2

a1n a2n ,, amn

Rows become columns and columns become rows

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Matrix Addition and Subtraction

A new matrix C may be defined as the additive combination of

matrices A and B where: C = A + B is defined by: 

Cij Aij Bij Note: all three matrices are of the same dimension

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Addition

A a11 a12

a21 a22

B b11 b12

b21 b22

C a11 b11 a12 b12

a21 b21 a 22 b22

If

and

then

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Matrix Addition Example

A B 3 4

5 6

1 2

3 4

4 6

8 10

C

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Matrix Subtraction

C = A - B Is defined by

Cij Aij Bij

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Matrix Multiplication

[r x c] and [s x d]

c = s

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Computation: A x B = C

A a11 a12

a21 a22

B b11 b12 b13

b21 b22 b23

232213212222122121221121

2312131122121211 21121111

babababababa

babababababaC

[2 x 2]

[2 x 3]

[2 x 3]

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A

2 3

1 1

1 0

and B

1 1 1

1 0 2

[3 x 2] [2 x 3]A and B can be multiplied

1 1 1

3 1 2

8 2 5

12*01*1 10*01*1 11*01*1

32*11*1 10*11*1 21*11*1

82*31*2 20*31*2 51*31*2

C [3 x 3]

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Matrix Inversion

B 1B BB 1 I

Like a reciprocal in scalar math

Like the number one in scalar math

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• For a XxX square matrix:

• The inverse matrix is:

• E.g.: 2x2 matrix:

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a bc d

det(A) = = ad - bc [ ]

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• X =A-1B• To find A-1

• Need to find determinant of matrix A

• From earlier

(2 -2) – (3 1) = -4 – 3 = -7• So determinant is -7

bcaddc

baA )det(

21

32

Linear Algebra & Matrices, MfD 2009

A 1 1

det(A)

d b c a

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Degree of freedom

The number of degrees of freedom is defined as the

number of independent coordinates which are

necessary for the complete description of the

position of a mass particle. 1. Mass particles

2.Rigid bodies

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Degree of freedom

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Degree of freedom

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Degree of freedom

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Degree of freedom

A rigid body, has six degrees of freedom:

1. Three translations (the position of the body),

2. Three rotations(the orientation of the body).

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Translational transformation

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Translational transformation

d = ai+bj+ck,

Robotic ConceptsA translational displacement of vector q for a distance d is obtained by multiplying the vector q with the matrix H

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Rotational transformation

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Rotational transformation

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Rotational transformation

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Rotational transformation

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Rotational transformation

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we wish to determine the vector w which is obtained

by rotating the vector u = 7i+3j+0k for 90◦ in the

counter clockwise i.e. positive direction around the z

axis.

As cos90◦ = 0 and sin90◦ = 1, it is not difficult to

determine the matrix describing Rot(z,90◦) and

multiplying it by the vector u.

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Pose and displacement

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Robot manipulator

The robot manipulator consists

of :

1.A robot arm,

2.A robot wrist,

3.A robot gripper.

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Robot manipulator

• The task of the robot manipulator is to place an

object grasped by the gripper into an arbitrary

pose.

• The task of the robot arm is to provide the

desired position of the robot end point.

• The task of the robot wrist is to enable the

required orientation of the object grasped by the

robot gripper.

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Robot manipulator

• In robotics the joint angles are denoted by the Greek

letter ϑ.

• The relative position between the two segments is

measured as a distance.

• The distance is denoted by the letter d.

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Robot manipulator

Robotic ConceptsRobot arms

On the market we find 5 commercially available

structures of robot arms:

•Anthropomorphic,

•Spherical,

•SCARA,

•Cylindrical,

•Cartesian.

Robotic ConceptsRobot arms

• Anthropomorphic, The anthropomorphic robot arm

has all three joints of the

rotational type (RRR). Among the

robot arms it resembles the

human arm to the largest extent.

The second joint axis is

perpendicular to the first one,

while the third joint axis is parallel

to the second one.

Robotic ConceptsRobot arms

• Spherical,

The spherical robot arm has two

rotational and one translational

degree of freedom (RRT). The

second joint axis is perpendicular

to the first one and the third axis

is perpendicular to the second

one.

Robotic ConceptsRobot arms

• SCARA,

The SCARA (Selective Compliant

Articulated Robot for Assembly)

robot arm appeared relatively late

in the development of industrial

robotics. It is predominantly aimed

for industrial processes of

assembly. Two joints are rotational

and one is translational (RRT). The

axes of all three joints are parallel.

Robotic ConceptsRobot arms

• Cylindrical,

The cylindrical shape of the

workspace is even more

evident with the cylindrical

robot arm. This robot has one

rotational and two

translational degrees of

freedom (RTT). The axis of the

second joint is parallel to the

first axis, while the third joint

axis is perpendicular to the

second one.

Robotic ConceptsRobot arms

• Cartesian. The cartesian robot arm has all

three joints of the translational

type (TTT). The joint axes are

perpendicular one to another.

Cartesian robot arms are known

for high accuracy, while the

special structure of gantry robots

is suitable for manipulation of

heavy objects.

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Seiko RT3300 Robot

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Thank you!