Transcript Lecture 2 , Sep

Robotic Concepts
Mehdi Ghayoumi
MSB rm 160
[email protected]
Ofc hr: Thur, 11-12:30a
Robotic Concepts
Announcements:
•
•
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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
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Robot 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
A matrix is any doubly subscripted array of elements arranged
in rows and columns.
a11 ,, a1n 
a 21 ,, a 2 n 
  Aij
A
 


am1 ,, amn
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Row Vector
[1 x n] matrix
A a1 a2 ,, an  aj
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Column Vector
[m x 1] matrix
a1 
a 2 
A     ai
 
 
am 
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Square Matrix
Same number of rows and columns
5 4 7
B  3 6 1 


2 1 3 
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Identity Matrix
Square matrix with ones on the diagonal and zeros elsewhere.
1
0
I  
0
0

0 0 0 
1 0 0 
0 1 0 

0 0 1
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Transpose Matrix
Rows become columns and columns become rows
a11 a 21 ,, am1 
a12 a 22 ,, am 2 
A'  
     
a1n a 2n ,, amn 


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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
If
a11 a12 
A 

a 21 a 22 

and
b11 b12 
B 

b 21 b 22

then
a11  b11 a12  b12 
C 

a 21  b 21 a 22  b22 

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Matrix Addition Example
3
A B 
5
4 

6 
1 2 

3 4 
4 6 
 C
8 10 
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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
a11 a12 
[2 x 2]
A 

a 21 a 22 

b11 b12 b13 
[2 x 3]
B 

b 21 b 22 b 23

a11b11  a12b21 a11b12  a12b22 a11b13  a12b23 
C

a 21b11  a 22b21 a 21b12  a 22b22 a 21b13  a 22b23
[2 x 3]
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2 3
1 1 1 
A  1 1  and B 


1
0
2




1 0 
[3 x 2]
[2 x 3]
A and B can be multiplied
2 *1  3 *1  5 2 *1  3 * 0  2 2 *1  3 * 2  8 5 2 8
C  1*1  1*1  2 1*1  1* 0  1 1*1  1* 2  3   2 1 3 
1*1  0 *1  1 1*1  0 * 0  1 1*1  0 * 2  1  111 
[3 x 3]
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Matrix Inversion
1
1
B B  BB
Like a reciprocal
in scalar math
 I
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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det(A) =
[ ]
a
c
b = ad - bc
d
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• X =A-1B
• To find A-1


d
b
1
A 1 


det( A) c a 
• Need to find determinant of matrix A
• From earlier

a b
det( A) 
 ad  bc
c d
(2 -2) – (3 1) = -4 – 3 = -7
• So determinant is -7
2 3 
1  2 


Linear Algebra & Matrices, MfD 2009
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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,
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A 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
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Robot arms
On the market we find 5 commercially available structures
of robot arms:
• Anthropomorphic,
• Spherical,
• SCARA,
• Cylindrical,
• Cartesian.
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Robot 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.
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Robot 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.
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Robot 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.
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Robot 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.
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Robot 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!