Mathematical tools for referencing a frame and transforming

Robotics presentation number 3

1. Robotics
MATHEMATICAL TOOLS AND REFERENCE FRAME
TRANSFORMATIONS

2. So far…
Descriptionofa position: Vector
Descriptionofanorientation: Attached
coordinateframeand a rotationmatrix
DescriptionofPose: HomogeneousTransformation
Matrix (MTH)
schedule

3. Forward kinematics
•
How does the tool point
movegiven the dimensions
of the robotand the joint
angles?
https://saurabhpalan.files.wordpress.com/2010/02/i
arm.jpg
•
Howdoesthemobilerobot
movegivenitsgeometryand
thespeedofitswheels?
https://www.robotsinsearch.com/products/tetra-ds-iv-mobile-robot-
platform

4. Description of position
A descriptionisusedtospecifyattributesofvarious
objectswithwhicha manipulationsystemdeals.
●
Theseobjectsare parts, tools, and themanipulatoritself.
Descriptionofa position: Once a coordinatesystemis
established, wecan locate anypointin theuniverse
witha 3 x 1 position vector.

5. Descriptionoforientation
Wewillfinditnecessarynotonlytorepresenta point
in spacebutalsotodescribe theorientationofa body
in space
In ordertodescribe theorientationofa body, wewill
attacha coordinatesystemtothebodyand thengive
a descriptionofthiscoordinatesystemrelative tothe
referencesystem

6. Descriptionofposition and orientation
Firststep: Attacha frame(coordinatesystem) totheobject

7. Add coordinate systems
Adda coordinatesystemtoeachimportant
physicalpartofyourrobot

9. A memory pill: Vector projection
https://en.wikipedia.org/wiki/Vector_projectio
n

10. Description of orientation
One way to describe the body attached coordinate system,
{B}, is to write the unit vectors of its three principal axes in
terms of the coordinate system {A}: A rotation matrix!
Further inspection shows that the rows of the matrix are the
unit vectors of {A} expressed in {B}. Hence, the description
of frame {A} relative to {B}, is given by
The inverse of a rotation matrix is equal to its transpose!

11. Rotation matrix
Used to represent the orientation of a body
Properties:

12. So far…
Descriptionofa position: Vector
Descriptionofanorientation: Attached
coordinateframeand a rotationmatrix
Whereastheposition ofa pointisrepresented
witha vector, theorientationofa bodyis
representedwitha matrix.

13. Description of a frame
A frame:
●
A coordinatesystem
describedbya
position vectorand a
rotationmatrix
Arrow between frames: {C} is
known relative {A} and not
viceversa

14. Mappings: Changing descriptions from
frame to frame
We will be concerned with expressing the same quantity
in terms of various reference coordinate systems.
Given P
B
, compute P
A
: Rotation and translation

15. Transformation matrices
•
Thechangein position and orientationbetween:
•
Joints,
•
mobilerobot locations,
•
Sensors,
•
robot and otherobjects,
•
isdescribedusingtransformationmatrices
•
Anyframecan serve as a referencesystem: transforming
fromoneframetoanother

16. Mappings: Changing descriptions from
frame to frame
Mappingas anoperatorin matrixform:
HomogeneousTransformationMatrix (HTM):●
A "1" isaddedas thelastelementofthe4 x 1 vectors
●
row"[0 0 0 1]" isaddedas thelastrowofthe4 x 4
matrix.

17. Homogeneous transform -
Example

18. Operators: translations, rotationsand
transformations
Translations
Rotations
What about rotations about other axes?

19. Transformationoperators
ThetransformthatrotatesbyR and translatesbyQ
isthesameas thetransformthatdescribesa frame
rotatedbyR and translatedbyQ relative tothe
referenceframe.
Example2.4 –Page 33 (Craig)

20. Interpretations of a homogeneous
transform (T)

21. Transformationarithmetic
Compoundtransformations
C
P
A
P ?

22. Transformation arithmetic
Invertinga transfrom: Calculatetheinverse?
A computationallysimplermethod(OWb):
Example2.5, page 36

23. Transformequations
?

24. Transform equations
?

25. Example2.6 (Craig)

26. Homework
Read
and solveexercis
esin
Chapter2 of
Craig’sbook

27. Themanipulationofthepiececarriedoutbytherobot impliesthespatialmovementofitsend
In orderfortherobot tobe abletomanipulatea piece, itisnecessarytoknowitsLOCATION,
thatis, itspositionand orientationwithrespecttotherobot base.
Introduction
Needingofa matematicalstooltospecifythepositionand orientationofendeffectorwith
respecttotherobot base

28. Thereferencecoordinatesystemlocatedfromrobot base named{S} and itsassociatedaxesXYZ
configuringOXYZ system
Thecoordinatesystemlocatedontherobot'swristisnamed{S'} and itsassociatedUVW axesby
configuringtheO'UVW system
Thematrix??????
??
?
mathematicallyrelates thesystemS totheS'

29. SPACIAL LOCALIZATION
Rpresentationof:
•Cartesiancoordinates
•Cylindricalcoordinate
•Sphericalcoordinates
Position
•Rotationmatrices
•Euler angles
•Rotationtorque
Orientation
•Homogeneouscoordinates
•Homogeneoustransformationmatrices (HMT)
Localization(position + orientation)

30. Position representation
Somerobots onlyneedtoposition theirendeffector

31. Position representation
a point can be positioned in the plane or in space
Position in a PLANE
Positioning by 2 DoFthrough 2 independent
components.
Coordinate vectors: OX and OY in the OXY
coordinate reference system.
Position in theSPACE
Positioning by 3 DoFthrough 3 independent
components.
Coordinate vectors: OX , OY and OZ in the
OXYZ coordinate reference system.
There are several coordinates system to position a point,
they are:Cartesian, cylindrical, polar and spherical

32. Position representation
Positioning by Cartesian coordinates
•Position vector p(x,y)
•(x,y):Cartesiancoordinates,where
(x,y)arethep vector projection in the
axes OX and OY respectively
•Position vector p(x,y,z)
•(x,y,z): Cartesiancoordinates,where
(x,y,z) arethep vector projection in the
axes OX, OY and OZ respectively

33. Position representation
Positioning by Polarand Cylindrical coordinates
•Position vector p(r,Ɵ)
•(r,Ɵ): Polar coordinates, where r is
the distance from origin to “a” point and
Ɵ is the p angle from OX axe.
•Position vector p(r,Ɵ,z)
•(r,Ɵ,z): Cylindrical coordinates,
where r is the distance from origin to
“a” point,Ɵis the p angle from OXY
axe and z is the p projection from OY
axe.
Polar coordinates
Cylindrical coordinates

34. Position representation
Positioning by SPHERICAL coordinates
•Position vector p(r,Ɵ,ϕ)
•(r,Ɵ,ϕ ): Spherical coordinates,
where r is the distance from origin
to “a” point or extremof p vector, Ɵ
is the p angle from OXY axe and ϕ
is the p angle from OZ axe.

35. Orientationrepresentation

36. Orientationrepresentation
2D Rotationmatrices
??????=??????cos??????
??????=??????sin??????
??????
?
=??????cos??????+??????=??????cos??????cos??????−sin??????sin??????
=??????cos??????−??????sin??????
??????
?
=??????sin??????+??????= ??????sin??????cos??????+cos??????sin??????
= ??????sin??????+??????cos??????
Thematrixrepresentationis:
??????
?
??????
?
=
cos??????−sin??????
sin??????cos??????
??????
??????
??????
?
?
?
?
=????????????
??
??????isrotationmatrix

37. Orientationrepresentation
3D Rotationmatrices (OX)

38. Orientationrepresentation
3D Rotationmatrices (OY)

39. Orientationrepresentation
3D Rotationmatrices (OZ)

40. Orientationrepresentation
Signsoftwists

41. Orientationrepresentation
Signsoftwists

42. Orientationrepresentation
Compositionofrotationmatrices
Torepresenta finite sequenceofrotationswithrespecttotheprincipal
axis oftheOXYZ coordinatesystem, thebasicrotationmatrices are
multiplied.
but,it must be taken into accountthat:
•Matrix multiplication is not commutative.
•And therefore, the order in which the rotations are carried out is
important.
Basic rotations with respect to the principal axes can also be chained
together of the coordinate systems obtained after a rotation

43. Orientationrepresentation
Compositionofrotationmatrices
Rule for orderly composition of rotations:
Initially both coordinate systems are assumed to coincide. initial rotation matrix
=I initial rotation matrix =I.
If the moving system rotates with respect to one of the principal axes of the
fixed system fixed system, premultiplythe previous rotation matrix by the
corresponding elementary rotation matrix.
If the moving system rotates about one of its own principal axes, postmultiply
the previous rotation matrix by the corresponding elementary rotation matrix.

44. Orientationrepresentation
Compositionofrotationmatrices
EXERCISE 1
Obtaintherotationmatrixofa stationarysystem{A} thatrotates3 times untiltobecomethesystem{B}
RotationofanƟ angle abouttheOZ axis red system1
Rotationofanϕangle abouttheOY axis greensystem2
Rotationofanαangle abouttheOX axis blue system3

45. Orientationrepresentation
Compositionofrotationmatrices
EXERCISE 1
Obtaintherotationmatrixofa stationarysystem{A} thatrotates3 times untiltobecomethesystem{B}
RotationofanƟ angle abouttheOZ axis red system1
Rotationofanϕangle abouttheOY axis greensystem2
Rotationofanαangle abouttheOX axis blue system3

46. Orientationrepresentation
Compositionofrotationmatrices
EXERCISE 1
Obtaintherotationmatrixofa stationarysystem{A} thatrotates3 times untiltobecomethesystem{B}
RotationofanƟ angle abouttheOZ axis red system1
Rotationofanϕangle abouttheOY axis greensystem2
Rotationofanαangle abouttheOX axis blue system3

47. Orientationand position representation
Homogeneouscoordinates

60. References
1.https://awseducate.instructure.com/courses/737/pages
/motivation?module_item_id=12841
2.Introduction to Autonomous Mobile Robots (Intelligent
Robotics and Autonomous Agents series). Roland
Siegwart, Illah Reza Nourbakhsh, Davide Scaramuzza.