Part 3 section B Kinamatics V20. And rotation matrix pdf

This document discusses kinamatics in robotics, specifically forward kinematic , rotation matrices ,pose
Topics Covered:

Definition of Robotics
What is a robot?
Difference between robots, automation, and AI.
Characteristics of robotic systems.
History and Evolution of Robotics
Early concepts and automata.
The rise of industrial robotics.
Modern advancements, including AI and autonomous systems.
Applications of Robotics
Industrial robotics (manufacturing, assembly lines).
Medical robotics (surgery, prosthetics).
Service and consumer robots (vacuum cleaners, personal assistants).
Military and space robotics.
Autonomous vehicles and drones. Basic Components of a Robot
Sensors (vision, proximity, touch).
Actuators (motors, servos, hydraulic systems).
Controllers and microcontrollers.
Power sources and communication systems.
Types of Robots
Mobile robots (wheeled, legged, drones).
Manipulators (robotic arms).
Humanoid robots.
Swarm robotics and collaborative robots (cobots).
The Role of AI in Robotics
Machine learning and decision-making.
Path planning and autonomous navigation.
Computer vision and object recognition.
Ethical and Societal Impacts of Robotics
Automation and job displacement.
Ethical concerns in AI-driven robots.
Laws and regulations governing robotics.

1. F) CompositionofRotations
•Considerapointin3frames:{A},{B},{C}
•PointPinframe{A}:
A
p
•PointPinframe{B}:
B
p
•Point Pinframe{C}:
C
p
1
•Relations:
•Pointfromframe{B}toframe{A}:
•Pointfromframe{C}toframe{B}:
•Pointfromframe{C}inframe{A}:
•Compositionofrotations:
Interpretationsof
A
R:
C
1)ItrotatespointPfrom {C}to{B}andthenfrom {B}to{A}
2)Itstartswith frame{A},thenmakes itcoincidentwith {B},andthenwith{C}
B
A
p
A
R
C
B
p
B
R
B
p
C
p

B
B C
A
p
A
RR
C
p
C B
C
B
R
A
R
A
R

2. CompositionofRotations
•RotationwithRespecttotheCurrent(mobile)System
•Apost-multiplicationwitharotationmatrixisneeded
•Example1
Initialframe:{0}
1º:Rotateθ
1aboutaxisy
0:R
y(θ
1)→weobtainframe{1}
2º:Rotateθ
2aboutaxisz
1(ofthecurrentsystem{1}):R
z(θ
2)→weobtain
frame {2}
Resultingrotation:R=R
y(θ
1)R
z(θ
2)
x
0
0
z
0
??????
1
y
1
y
z
1
1
x
1
y
z
1
x
1
2
x
2
y
z
2
??????
2
x
0
0
y
z
0
x
2
y
2
2
z
R
y(
1)R
z(
2)
2
1° 2° Final:
R
y(
1) R
z(
2)
Frame {1} was rotated acourdingto frame {0}. The rotation was θ
1
aboutaxisy
0. We rotate θ
2 aboutaxisz
1ofthecurrentsystem. 
R
y(θ
1)R
z(θ
2)
ALL ROTATION ARE DONE
RESPECT TO THE NEW FRAME
“ THEN”
Notes :
•initial, fixed, thenPre
•new , current post

3. CompositionofRotations
•RotationwithRespecttotheFixedFrame
•Apre-multiplicationwitharotationmatrixisneeded
•Example2
Initialframe:{0}
1º:Rotateθ
1aboutframey
0:R
y(θ
1)→weobtainframe{1}
2º:Rotateθ
2aboutframez
0(oftheinitial,fixed{0}):R
z(θ
2)→weobtain
frame {2}
Resultingrotation:R=R
z(θ
2)R
y(θ
1)
z2
R()
x
0
y
0
z
0
??????
1
y
1
1
z
x
1
??????
2
z
2
2
x
1°
3
2°
x
0
y
2
y
0
z
0
y1
R()
Frame {1} was initialycoinsidewith frame {0}. Axisy
0 was rotated aboutangle θ
1. Then we
rotate θ
2 aboutaxisz
1oftheinitialframe. R
z(θ
2)R
y(θ
1)
ALL ROTATION ARE DONE RESPECT TO
THE NEW FRAME “ THEN”
Notes :
•initial, fixed, thenPre
•new , current, mobilepost

4. CompositionofRotations
Example3
Considerthefollowingrigidbody
First,applyarotationof90°aboutz.Then(2cases):
a)Applyarotationof90°aboutthecurrent(“new”)yaxis
b)Applyarotationof90°aboutthefixedyaxisR
y(90)
a)currenty
R
y(90)
b)fixedy
RR
z(90)R
y(90)
RR
y(90)R
z(90)
R
z(90)
5
Frame O` was rotated with respect to
frame O. The rotation was 90about
axisZ
. We rotate 90aboutaxisyof
thecurrentsystem. R
z(90)R
y(90)
Frame O` was rotated with respect to
frame O. The rotation was 90aboutaxis
Z
. Then we rotate 90aboutaxisyofthe
initial/ fixedframe. R
y(90)R
z(90)

5. CompositionofRotations
Example4
Considerthefollowingrigidbody
First,apply arotationof90°abouty.Then(2cases):
a)Applyarotationof90°aboutthecurrent(“new”)z axis
b)Applyarotationof90°aboutthefixed(“old”) zaxis
R
z(90)
b)fixedz
R
y(90)
RR
y(90)R
z(90)
RR
z(90)R
y(90)
6
R
z(90)
a)currentz Frame O` was rotated with respect to
frame O. The rotation was 90aboutaxisY
.
We rotate 90aboutaxisZofthecurrent
system. R
z(90)R
y(90)
Frame O` was rotated with respect to frame
O. The rotation was 90aboutaxisZ
. Then we
rotate 90aboutaxisyoftheinitial/ fixed
frame. R
y(90)R
z(90)

6. The rotation from frame A to C (that is, the composition ofR
1
andR
2
) can be calculated by
multiplying the two matrices.Theorder of multiplication changes depending on howR
2
applies the
rotation on frame B:
•Post-multiply (R
1
R
2
): R
2
is a rotation of frame B with respect to frame A, the intermediate frame.
This is called intrinsic rotation.
•Pre-multiply (R
2
R
1
): R
2
is a rotation of frame B with respect to frame A, the original frame. This
is called extrinsic rotation.
The order of multiplication and its interpretation applies equally with quaternions.
Example
•R
1
R
2
•R
2
R
1

7. CompositionofRotations
Example5
Weapplythefollowing rotationstoareferenceframeinthefollowing
order:
1.Rotationof θaboutthecurrentxaxis
2.Rotationof ϕaboutthecurrentzaxis
3.Rotation ofαaboutthefixedzaxis
4.Rotationof βaboutthecurrentyaxis
5.Rotationof γaboutthefixedxaxis
Writetheexpressionoftheresultingrotationmatrix
R
x()
R
x()R
z()
R
z()R
x()R
z()
R
z()R
x()R
z()R
y()
R R
x()R
z()R
x()R
z()R
y()
8

8. ALL ROTATION ARE DONE RESPECT TO
THE NEW FRAME “ THEN”
Notes :
•initial, fixed, thenPre
•new , current post
-----------------------------------------------
•B initially with A From A to B.
Calculate angle from A to B
•Then B Z 60 post
•Then B Y 45 post
�
??????45�
??????60
•Then B X 30 post
�
??????(30)�
??????45�
??????60
�
??????(60)

9. 1 0 0
0 ൗ
1
2
ൗ
3
2
0−ൗ
3
2
ൗ
1
2
00−1
010
100
=
0 0 −1
ൗ
3
2
ൗ
1
2
0
ൗ
1
2
−ൗ
3
2
0
2
�
3=
2
�
1
1
�
3

11. Example
A frame F has been moved 4 units along the X axis and -2 along the Y axis an d 0 along the Z
axis.
=
1004
010−2
0010
0001A

12. B
A
p
A
t
B
T*
B

A
p
A
I
A
t
B
T

B

0
A
T
1
A
p
1 1
B
p
B
A
p
A
t
B
p
B

A
p
A
I
A
t
B
p

B

0
A
T
1
A
p
1 1
B
p

13. •PointPof frame{B}inframe {A}:
•Inamorecompact way:
x
A
y
A
z
A
B
y
z
B
x
B
P
B
p
A
p
B
A
t
B
A
p
A
t
B
p*

A
p
A
t
B
p

B


:Originofframe B with
respectto frameA
B
A
t
A
T
B
transformationmatrix
1
A
p
1 1
B
p
14
Moved
5 on y
2 on z
-3 on x
And the P (1,0,1)
100−3
0105
0
0
0
0
1
0
2
1
×
1
0
1
1
=
−2
5
3
1

14. InterpretationsoftheRotationMatrix
b)Mappingbetweenframes
Example
a)Determinebyinspectionthecoordinatesof
pointPinframe B
d
2
b)StartingwithpointPinframeB,determine
itscoordinatesinframeA
010
1
d
P
2
15
d

1

0

B
p

d

B
A
R 

100

 
001
B
01
A
p
A
R
B
p

10

00
0d
2d
1
0

d



d

 
1
2 
100
-Rotationmatrix:
-Transformed point:
z
cossin0
cos
0
0

R()

sin

0

1
??????
??????
??????
Rotationaboutthezaxis
z
cos90sin90
Cos900

R(9)

sin90

0

10
Cos90 =0
Sin90 = 1
Rotation
from A to B

15. HomogeneousTransformations
•PointPof frame{B}inframe {A}:
x
A
y
A
z
A
B
y
z
B
x
B
P
B
p
A
p
B
A
t
16
Moved
5 on y
2 on z
3 on x
On the P (1,0,1)
With rotation 1011
A
B
AB
B
A
pRtp 
 
  A
B
T
Homogeneous
transformation matrixA
p B
p B
A A A B
B
ptRp A
p B
p
Homogeneous
coordinates
with a tilde:
A
B
t
Origin of frame B
with respect to
frame A

16. Homogeneous Transformations
•They represent the position and orientation(pose) of a frame with respect
to another frame
•Parts
•Mathematically, they belong to SE(3): ℝ
3
�??????3
•They can also be viewed from the perspective of projective geometry
•Pure transformations
(SE= Special Euclidean)11 12 13
21 22 23
31 32 33
1
0001
x
y
z
rrrt
rrrtR
T
rrrt









t
0
Rotation
(orientation)
translation
(position)1
R
T




0
0 1
I
T




t
0
Pure rotation Pure translation

17. PureTransformations
x
18
0

Rot()
0 0 0
cossin

0sincos0

00 0 1

 
Purerotations:
1

0
y
cos

0

Rot()
0sin0
0 10

0cos0
0 00
sin

1

 
0 0 0
z
Rot()

1

 
cossin00 1000

sincos00
 
0
Trans (d)100


0 0 10
z
001d
x
0

Trans(d) 
0
1

 
Puretranslations:
100d

010
001

000
y
d

Trans(d)
1000

010

0010

0001

 

0001

 

18. Composition
Decomposition in pure transformations
Any homogeneous transformation can be decomposed in 2
components:
Interpretations of the (de)composition of T
Interpretation1:
1.First, it applies a translation of tunits
2.Then, it applies a rotation Rwith respect to the new frame
Interpretation2:
1.First, it applies a rotation R
2.Then, it applies a translation of tunits with respect to the fixed (initial)
frame1
R
T




t
0 11
IR



t0
00
translationtrotationR

19. Composition
1.Whenatransformationisappliedwithrespecttothefixedframe:
-Apre-multiplicationisused
2.When atransformationisappliedwithrespecttothemobile (current,new)
frame
-Apost-multiplicationisused
•Example
Aframe{A}isrotated90°aboutx, andthenitistranslatedavector(6,
-2,10)withrespecttothefixed(initial)frame.Findthehomogeneous transformationthat
describes{B}withrespectto{A}
201 0 0 6 1 0 0 0
0 1 0 2 0 cos90 sin90 0
0 0 1 10 0 sin90 cos90 0
0 0 0 1 0 0 0 1
   
   
   
   
    
   
    (6, 2,10)(90 )
A
Bx
T Trans Rot  1 0 0 6
0 0 1 2
0 1 0 10
0 0 0 1







ALL ROTATION ARE DONE RESPECT TO
THE NEW FRAME “ THEN”
•initial, fixed, thenPre
•new , current, mobilepost

20. Uses
21
1.They represent the pose(position+orientation) of a frame (rigid body)with
respect to another frame
2.They change the reference frame in which a point is represented (using a linear
relation):
Note: the point must be represented using homogeneous coordinates (its
notation uses ~)
3.They apply a transformation (rotation + translation)to a point in the same
reference frame
B
A A B
pTp

21. Uses
Example 1
Consider frame {A} and {B}. Point Pin frame {B}is given by (2, 0, 1), find its
coordinates with respect to frame {A}using a homogeneous transformation matrix
x
A
z
A
P
y
B
z
B
x
B
6
y
A
4
221 0 1 1
A
B
AB
B
A
p R t p     
     
      010424
100604
0010111
000111
A
p
 
 

 
 

 
  4
4
1
A
p






Solutioncossin00
sincos00
0010
0001
()
z
Rot









Rotz(-90)

22. Uses
•Example2
Aframe{A}isrotated90°aboutx, andthenitistranslatedavector(6,
-2,10)withrespecttothefixed(initial)frame. Considerapointp=(-
5,2,-12)withrespecttothenewframe{B}.Determinethe
coordinatesofthatpointwithrespecttotheinitialframe
Solution1000
0cossin0
0sincos0
0001
()
x
Rot









 (6,2,10)(90)
A
Bx
TTrans Rot  10061000
01020cos90sin900
001100sin90cos900
00010001
  
  
 
  
   
  
   1006
0012
01010
0001






 A AB
B
pTp 100651
0012210
010101212
000111
 
 

 
  
 
 
-Homogeneous transformation
-Point after transformation:

23. Uses
24
•Example3
Aframe{A}istranslatedavector(6,-2,10)andthenitisrotated90°aboutaxisxof
thefixed(initial)frame.Considerapointwith coordinates(-5,2,-12)withrespect
tothenewframe({B}).Findthe coordinatesofthatpointwithrespecttotheinitial
frame1000
0cossin0
0sincos0
0001
()
x
Rot









 (90 (6,2,10) )
x
A
B
TRotTrans  10001006
0cos90sin9000102
0sin90cos90000110
00010001
  
  
 
  
  
  
   1006
00110
0102
0001




 

 A AB
B
pTp 100651
0011022
0102120
000111
 
 

 
 
 
 
-Homogeneous transformation
-Transformed point:
Solution

24. Show the steps of this transformations?

25. Show the steps of this transformations?

26. 27
Properties
Inverseof a homogeneous transformation:
Why?
Productof homogeneous transformations:1
R
T




t
0 1
1
TT
RR
T




t
0 B
A A A B
B
p t R p TT
B BB
B A A A A
pRpRt
B
A A B
p T p 1

B
B A A
pTp


BA
A
Tp 11
1
1
R
T




t
0 22
2
1
R
T




t
0 12 12 1
12
1
RR
TT
R



tt
0
It is not commutative
Solvingfor
B
P

27. •Given a transformation matrix, Finditsinverse.
T-11
R
T




t
0 1
1
TT
RR
T




t
0

28. Review Links
https://www.youtube.com/watch?v=vlb3P7arbkU
https://www.youtube.com/watch?v=cEE0oMae89M
0
3
1
=
0
1
3
10 0
0 ൗ
3
2
−ൗ
1
2
0ൗ
1
2
ൗ
3
2

29. Homework
Rot
x()Trans
x(b)Trans
z(d)Rot
z()
30
B
A
T
•Example
Findthehomogeneous transformationmatrixthatrepresentsa rotationofanangle
αaboutthexaxis,followedbyatranslationofb unitsalongthenewxaxis,
followedbyatranslationofdunitsalong thenewz axis,followedbyarotationof
anangleθaboutthenewz axis
Solution
cos

cossin
0 1
b 
dsin

sin 0
coscossin

sincoscosdcos
0 0

sinsin
 
 1000
0cossin0
0sincos0
0001
()
x
Rot










T=
10 0 0
0cos�−sin�0
0sin�cos�0
00 0 1
* Trans
x(b) Trans
z(d)
10 0 0
0cos??????−sin??????0
0sin??????cos??????0
00 0 1
T=
10 0 0
0cos�−sin�0
0sin�cos�0
00 0 1
100�
0100
0010
0001
1000
0100
001??????
0001
10 0 0
0cos??????−sin??????0
0sin??????cos??????0
00 0 1

30. Extra class
1 0 0
0 ൗ
1
2
ൗ
3
2
0−ൗ
3
2
ൗ
1
2
00−1
010
100
=
0 0 −1
ൗ
3
2
ൗ
1
2
ൗ
3
2
ൗ
1
2
−ൗ
3
2
0
2
�
3=
2
�
1
1
�
3

32. A camera is attached to the robot end-
effector to observe the object and position
the end-effector in the right position. Four
reference frames are attached to different
elements in the robot’s workspace, as
shown in the figure. The configuration of
one frame relative to the other can be
represented by a homogenous
transformation matrix.
Four frames are attached to different
elements in the robot’s workspace, as
shown above. {a} is the frame coincident
with the space frame {s}, {b} is the gripper
frame, {c} is the camera frame, and {d} is
the workpiece frame

33. First, we want to calculate
the configuration of the
workpiece frame {d} relative
to the frame {a} and the
camera frame {c}. To this
end, we just need to
calculate the homogenous
transformation matrices,
T
adand T
cd.

37. Consider the scene in Figure 3.5 of a once peaceful park overrun by robots. Frames
are shown attached to the tree {t}, robot chassis {c}, manipulator {m}, and
quadcopter {q}. The distances shown in the figure ared1 =4m,d2 =3m,d3 =6m,d4
=5m,d5 =3m. Themanipulatorisataposition pcm= (0, 2, 1) m relative to the chassis
frame {c}, and {m} is rotated from {c} by 45 degrees about the xˆc-axis.
Give the transformation matrices representing the quadcopter frame {q}, chassis
frame {c}, and manipulator frame {m} in the tree frame {t}.