Mehdi Ghayoumi MSB rm 160 [email protected] Ofc hr: Thur, 11-12:30a Robotic Concepts.
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Transcript of Mehdi Ghayoumi MSB rm 160 [email protected] Ofc hr: Thur, 11-12:30a Robotic Concepts.
Mehdi Ghayoumi
MSB rm 160
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
Robotic Concepts
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.
Robotic Concepts
Robotic Concepts
Robotic Concepts
Robotic Concepts
Robotic Concepts
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.
Robotic Concepts
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
Robotic Concepts
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]
Robotic Concepts
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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Robotic Concepts
• 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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Robotic Concepts
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
Robotic Concepts
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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Robotic Concepts
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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Robotic Concepts
Robotic Concepts
Seiko RT3300 Robot
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Robotic Concepts
Thank you!