rotational matrix.pdf

the required presentation of rotate matrix s given under this presentation

1. Transformation Matrices
Dr.-Ing. John Nassour
Artificial Intelligence & Neuro Cognitive Systems
FakultätfürInformatik
�
0
�
0
�
1
�
1
�
1
�
�
2
�
2
�
2
�
??????
�
??????
�
??????

2. Suggested literature
•Robot Modeling andControl
•Robotics: Modelling, Planning and Control
10/17/2017

3. 10/17/2017
Motivation
Alargepartofrobotkinematicsisconcernedwiththe
establishmentofvariouscoordinatesystemstorepresentthe
positionsandorientationsofrigidobjects,andwith
transformationsamongthesecoordinatesystems.
Indeed,thegeometryofthree-dimensionalspaceandofrigid
motionsplaysacentralroleinallaspectsofroboticmanipulation.
Arigidmotionistheactionoftakinganobjectandmovingittoa
differentlocationwithoutalteringitsshapeorsize

4. 10/17/2017
Transformation
TheoperationsofROTATIONandTRANSLATION.
IntroducethenotionofHOMOGENEOUS
TRANSFORMATIONS(combiningtheoperations
ofrotationandtranslationintoasinglematrix
multiplication).

5. 10/17/2017
Representing Positions
Thecoordinatevectorsthat
representthelocationofthe
point�inspacewithrespectto
coordinateframes�
��
��
�and
�
��
��
�,respectivelyare:

6. 10/17/2017
Representing Positions
Letsassigncoordinatesthat
representthepositionoftheorigin
ofonecoordinatesystem(frame)
withrespecttoanother.

7. 10/17/2017
Representing Positions
Whatisthedifference
betweenthegeometric
entitycalled�andany
particularcoordinate
vector??????thatisassigned
torepresent�?
�isindependentofthechoiceofcoordinatesystems.
??????dependsonthechoiceofcoordinateframes.

8. 10/17/2017
Representing Positions
Apointcorrespondstoaspecific
locationinspace.
Avectorspecifiesadirectionand
amagnitude(e.g.displacements
orforces).
Thepoint�isnotequivalentto
thevector??????
�,thedisplacement
fromtheorigin�
�tothepoint�
isgivenbythevector??????
�.
We will use the term vectorto refer to what are sometimes called free vectors, i.e.,
vectors that are not constrained to be located at a particular point in space.

9. 10/17/2017
Representing Positions
Whenassigningcoordinatesto
vectors,weusethesamenotational
conventionthatweusedwhen
assigningcoordinatestopoints.
Thus,??????
�and??????
�aregeometric
entitiesthatareinvariantwith
respecttothechoiceofcoordinate
systems,buttherepresentationby
coordinatesofthesevectors
dependsdirectlyonthechoiceof
referencecoordinateframe.

10. 10/17/2017
Coordinate Convention
Inordertoperformalgebraicmanipulationsusing
coordinates,itisessentialthatallcoordinatevectors
bedefinedwithrespecttothesamecoordinate
frame.
Inthecaseoffreevectors,itisenoughthattheybe
definedwithrespecttoparallelcoordinateframes.

11. 10/17/2017
Coordinate Convention
An expression of the form:
is not defined since the frames
�
��
��
�and�
��
��
�are not
parallel.

12. 10/17/2017
Coordinate Convention
An expression of the form:
is not defined since the frames
�
��
��
�and�
��
��
�are not
parallel.
Thus,weseeaclearneed,notonlyforarepresentationsystemthatallowspointsto
beexpressedwithrespecttovariouscoordinatesystems,butalsoforamechanism
thatallowsustotransformthecoordinatesofpointsthatareexpressedinone
coordinatesystemintotheappropriatecoordinateswithrespecttosomeother
coordinateframe.

13. 10/17/2017
Representing Rotations
Inordertorepresenttherelative
positionandorientationofonerigid
bodywithrespecttoanother,wewill
rigidlyattachcoordinateframesto
eachbody,andthenspecifythe
geometricrelationshipsbetween
thesecoordinateframes.

14. 10/17/2017
Representing Rotations
Inordertorepresenttherelative
positionandorientationofonerigid
bodywithrespecttoanother,wewill
rigidlyattachcoordinateframesto
eachbody,andthenspecifythe
geometricrelationshipsbetween
thesecoordinateframes.

15. 10/17/2017
Representing Rotations
Inordertorepresenttherelative
positionandorientationofonerigid
bodywithrespecttoanother,wewill
rigidlyattachcoordinateframesto
eachbody,andthenspecifythe
geometricrelationshipsbetween
thesecoordinateframes.
How to describe the orientation of one coordinate frame relative to another frame?

16. 10/17/2017
Rotation In The Plane

17. 10/17/2017
Rotation In The Plane
Fram�
��
��
�isobtainedbyrotating
frame�
��
��
�byanangle??????.
Thecoordinatevectorsfortheaxesof
frame�
��
��
�withrespecttocoordinate
frame�
��
��
�aredescribedbyarotation
matrix:
where�
�
�
and �
�
�
are the coordinates in frame �
��
��
�of unit
vectors �
�and �
�, respectively.

18. 10/17/2017
Rotation In The Plane
�
�
�
is a matrix whose column vectors are the coordinates of the (unit vectors along the)
axes of frame �
��
��
�expressed relative to frame �
��
��
�.

19. 10/17/2017
Rotation In The Plane
�
�
�
describes the orientation of frame �
��
��
�with respect to the frame �
��
��
�.
The dot product of two unit vectors gives the
projection of one onto the other
�
�
�
=?

20. 10/17/2017
Rotation In The Plane
The dot product of two unit vectors gives the
projection of one onto the other
The orientation of frame �
��
��
�with
respect to the frame �
��
��
�.
Since the inner product is commutative

21. 10/17/2017
Rotations In Three Dimensions
Each axis of the frame �
��
��
��
�is projected onto coordinate frame �
��
��
��
�.
The resulting rotation matrix is given by:

22. 10/17/2017
Rotation About �
�By An Angle ??????

23. 10/17/2017
Rotation About �
�By An Angle ??????
Called a basic rotation matrix (about the z-axis)�
�,??????

24. 10/17/2017
Basic Rotation Matrix About The Z-axis

25. 10/17/2017
Basic Rotation Matrix About The X-axis
�
�,??????

26. 10/17/2017
Basic Rotation Matrix About The Y-axis
�
�,??????

27. 10/17/2017
Example
Find the description of frame �
��
��
��
�with respect to the frame �
��
��
��
�.

28. 10/17/2017
Example
The coordinates of �
�are
The coordinates of �
�are
The coordinates of �
�are
Find the description of frame�
��
��
��
�with respect to the frame �
��
��
��
�.

29. 10/17/2017
Example
Find the description of frame�
��
��
��
�with respect to the frame �
��
��
��
�.
The coordinates of �
�are
The coordinates of �
�are
The coordinates of �
�are

30. 10/17/2017
Rotational Transformations
�is a rigid object to which a coordinate ������
is attached.
Given�
�
of the point �, determine the coordinates
of �relative to a fixed reference ������.
The projection of the point �onto the
coordinate axes of the ������:

31. 10/17/2017
Rotational Transformations

32. 10/17/2017
Rotational Transformations
Thus,therotationmatrix�
�
�
canbeusednot
onlytorepresenttheorientationofcoordinate
frame�
��
��
��
�withrespecttoframe
�
��
��
��
�,butalsototransformthe
coordinatesofapointfromoneframeto
another.
Ifagivenpointisexpressedrelativeto
�
��
��
��
�bycoordinates�
�
,then�
�
�
�
�
representsthesamepointexpressedrelative
totheframe�
��
��
��
�.

33. 10/17/2017
Rotational Transformations
It is possible to derive the coordinates for �
�given only the coordinates for
�
�and the rotation matrix that corresponds to the rotation about �
�.
Suppose that a coordinate frame �
��
��
��
�is rigidly attached to the block.
After the rotation by ??????, the block’s coordinate frame, which is rigidly attached
to the block, is also rotated by ??????.
Rotation matrices to represent rigid
motions

34. 10/17/2017
Rotational Transformations
The coordinates of �
�with respect to the reference frame �
��
��
��
�:
Rotation matrices to represent rigid
motions

35. 10/17/2017
Rotational Transformations
Rotation matrices to represent vector rotation with
respect to a coordinate frame.
Reminder:

36. 10/17/2017
Summary: Rotation Matrix
1.It represents a coordinate transformation
relating the coordinates of a point p in two
different frames.
2. It gives the orientation of a transformed
coordinate frame with respect to a fixed
coordinate frame.
3. It is an operatortaking a vector and rotating it
to a new vector in the same coordinate system.

37. 10/17/2017
Similarity Transformations
The matrix representation of a general linear transformation is transformed from one
frame to another using a so-called similarity transformation.
For example, if �is the matrix representation of a given linear transformation in
�
��
��
��
�and �is the representation of the same linear transformation in
�
��
��
��
�then �and �are related as:
where �
�
�
is the coordinate transformation between frames �
��
��
��
�and
�
��
��
��
�. In particular, if �itself is a rotation, then so is �, and thus the use of
similarity transformations allows us to express the same rotation easily with respect to
different frames.

38. 10/17/2017
Example
Suppose frames �
��
��
��
�and �
��
��
��
�are related
by the rotation
If �=�
�relative to the frame �
��
��
��
�, then, relative to frame �
��
��
��
�we have
�is a rotation about the ��−����but expressed relative to the frame �
��
��
��
�.

39. 10/17/2017
Rotation With Respect To The Current
Frame
Thematrix�
�
�
representsarotationaltransformationbetweentheframes�
��
��
��
�
and�
��
��
��
�.
Supposewenowaddathirdcoordinateframe�
��
��
��
�relatedtotheframes
�
��
��
��
�and�
��
��
��
�byrotationaltransformations.
Agivenpoint�canthenberepresentedbycoordinatesspecifiedwithrespecttoanyof
thesethreeframes:�
�
,�
�
and�
�
.
The relationship among these representations of �is:
where each�
�
�
is a rotation matrix

40. 10/17/2017
Inordertotransformthecoordinatesofapoint�fromits
representation�
�
intheframe�
��
��
��
�toitsrepresentation
�
�
intheframe�
��
��
��
�,wemayfirsttransformtoits
coordinates�
�
intheframe�
��
��
��
�using�
�
�
andthen
transform�
�
to�
�
using�
�
�
.
Composition Law for Rotational
Transformations

41. 10/17/2017
Composition Law for Rotational
Transformations
Coincident: lie exactly on top of each other
Suppose initially that all three of the coordinate frames are coincide.
We first rotate the frame �
��
��
��
�relative to �
��
��
��
�according to the
transformation �
�
�
.
Then, with the frames �
��
��
��
�and �
��
��
��
�coincident, we rotate�
��
��
��
�
relative to �
��
��
��
�according to the transformation �
�
�
.
In each case we call the frame relative to which the rotation occurs the current frame.

42. 10/17/2017
Example
Suppose a rotation matrix R represents
•a rotation of angle ??????about the current �−����followed by
•a rotation of angle ??????about the current �−����.

43. 10/17/2017
Example
Suppose a rotation matrix R represents
•a rotation of angle ??????about the current �−����followed by
•a rotation of angle ??????about the current �−����.

44. 10/17/2017
Example
Suppose a rotation matrix R represents
•a rotation of angle ??????about the current �−����followed by
•a rotation of angle ??????about the current �−����.

45. 10/17/2017
Example
Suppose a rotation matrix R represents
•a rotation of angle ??????about the current �−����followed by
•a rotation of angle ??????about the current �−����.

46. 10/17/2017
Example
Suppose a rotation matrix R represents
•a rotation of angle ??????about the current �−����followedby
•a rotation of angle ??????about the current �−����

47. 10/17/2017
Example
Suppose a rotation matrix R represents
•a rotation of angle ??????about the current �−����followedby
•a rotation of angle ??????about the current �−����

48. 10/17/2017
Example
Suppose a rotation matrix R represents
•a rotation of angle ??????about the current �−����followedby
•a rotation of angle ??????about the current �−����
Rotational transformations do not commute

49. 10/17/2017
Rotation With Respect To The Fixed
Frame
Performing a sequence of rotations, each about a given fixed coordinate frame, rather
than about successive current frames.
For example we may wish to perform a rotation about �
�followed by a rotation about
�
�(and not �
�!). We will refer to �
��
��
��
�as the fixed frame. In this case the
composition law given before is not valid.
The composition law that was obtained by multiplying the successive rotation matrices
in the reverse order from that given by is not valid.

50. 10/17/2017
Rotation with Respect to the Fixed Frame
Suppose we have two frames �
��
��
��
�and
�
��
��
��
�related by the rotational transformation �
�
�
.
If �represents a rotation relative to �
��
��
��
�, the
representation for �in the current frame �
��
��
��
�is given
by:
Similarity Transformations
With applying the composition law for rotations about the
current axis:
composition law for
rotations about the
current axis
Reminder:

51. 10/17/2017
Example
Suppose a rotation matrix R represents
•a rotation of angle ??????about �
�−����followedby
•a rotation of angle??????about the fixed �
�−����
Similarity Transformations
Reminder:
composition law for
rotations about the
current axis
composition law for
rotations about the fixed
axis
The secondrotation about the fixed axis is given by
which is the basic rotation about the z-axis expressed relative
to the frame �
��
��
��
�using a similarity transformation.

52. 10/17/2017
Example
Suppose a rotation matrix R represents
•a rotation of angle ??????about �
�−����followedby
•a rotation of angle??????about the fixed �
�−����
Similarity Transformations
Reminder:
composition law for
rotations about the
current axis
composition law for
rotations about the fixed
axis
Therefore, the composition rule for rotational transformations

53. 10/17/2017
Example
Suppose a rotation matrix R represents
•a rotation of angle ??????about �
�−����followedby
•a rotation of angle??????about the fixed �
�−����

54. 10/17/2017
Example
Suppose a rotation matrix R represents
•a rotation of angle ??????about �
�−����followedby
•a rotation of angle??????about the fixed�
�−����
Suppose a rotation matrix R represents
•a rotation of angle ??????about the current�−����followed by
•a rotation of angle ??????about the current�−����.

55. 10/17/2017
Summary
To note that we obtain the same basic rotation matrices, but in the reverse order.
Rotation with Respect to
the Fixed Frame
Rotation with Respect to
the Current Frame

56. 10/17/2017
Rules for Composition of Rotational
Transformations
Wecansummarizetheruleofcompositionofrotationaltransformationsby:
Givenafixedframe�
��
��
��
�acurrentframe�
��
��
��
�,togetherwithrotation
matrix�
�
�
relatingthem,ifathirdframe�
��
��
��
�isobtainedbyarotation�
performedrelativetothecurrentframethenpost-multiply�
�
�
by�=�
�
�
toobtain
In each case �
�
�
represents the transformation between the frames �
��
��
��
�and
�
��
��
��
�.
If the second rotation is to be performed relative to the fixed frame then it is both
confusing and inappropriate to use the notation �
�
�
to represent this rotation. Therefore,
if we represent the rotation by �, we pre-multiply �
�
�
by �to obtain

57. 10/17/2017
Example
Find Rfor the following sequence of basic rotations:
1.A rotation of ϴabout the current x-axis
2.A rotation of φabout the current z-axis
3.A rotation of αabout the fixed z-axis
4.A rotation of βabout the current y-axis
5.A rotation of δabout the fixed x-axis
Rotation with Respect to
the Fixed Frame
Rotation with Respect to
the Current Frame
Reminder:

58. 10/17/2017
Example
Find Rfor the following sequence of basic rotations:
1.A rotation of ϴabout the current x-axis
2.A rotation of φabout the current z-axis
3.A rotation of αabout the fixed z-axis
4.A rotation of βabout the current y-axis
5.A rotation of δabout the fixed x-axis
Rotation with Respect to
the Fixed Frame
Rotation with Respect to
the Current Frame
Reminder:

59. 10/17/2017
Example
Find Rfor the following sequence of basic rotations:
1.A rotation of ϴabout the current x-axis
2.A rotation of φabout the current z-axis
3.A rotation of αabout the fixed z-axis
4.A rotation of βabout the current y-axis
5.A rotation of δabout the fixed x-axis
Rotation with Respect to
the Fixed Frame
Rotation with Respect to
the Current Frame
Reminder:

60. 10/17/2017
Example
Find Rfor the following sequence of basic rotations:
1.A rotation of ϴabout the current x-axis
2.A rotation of φabout the current z-axis
3.A rotation of αabout the fixed z-axis
4.A rotation of βabout the current y-axis
5.A rotation of δabout the fixed x-axis
Rotation with Respect to
the Fixed Frame
Rotation with Respect to
the Current Frame
Reminder:

61. 10/17/2017
Example
Find Rfor the following sequence of basic rotations:
1.A rotation of ϴabout the current x-axis
2.A rotation of φabout the current z-axis
3.A rotation of αabout the fixed z-axis
4.A rotation of βabout the current y-axis
5.A rotation of δabout the fixed x-axis
Rotation with Respect to
the Fixed Frame
Rotation with Respect to
the Current Frame
Reminder:

62. 10/17/2017
Example
Find Rfor the following sequence of basic rotations:
1.A rotation of ϴabout the current x-axis
2.A rotation of φabout the current z-axis
3.A rotation of αabout the fixed z-axis
4.A rotation of βabout the current y-axis
5.A rotation of δabout the fixed x-axis
Rotation with Respect to
the Fixed Frame
Rotation with Respect to
the Current Frame
Reminder:

63. 10/17/2017
Example
Find Rfor the following sequence of basic rotations:
1.A rotation of δabout the fixed x-axis
2.A rotation of βabout the current y-axis
3.A rotation of αabout the fixed z-axis
4.A rotation of φabout the current z-axis
5.A rotation of ϴabout the current x-axis
Rotation with Respect to
the Fixed Frame
Rotation with Respect to
the Current Frame
Reminder:

64. 10/17/2017
Rotations in Three Dimensions
Each axis of the frame �
��
��
��
�is projected onto the coordinate frame
�
��
��
��
�.
The resulting rotation matrix is given by
Reminder:
The nine elements �
��in a general rotational transformation Rare not independent
quantities.
�??????���
Where ��(n)denotes the Special Orthogonal group of order n.

65. 10/17/2017
Rotations In Three Dimensions
For any �??????�??????�The following properties hold
•�
??????
=�
−1
??????�??????�
•The columns and the rows of �are mutually orthogonal
•Each column and each row of �is a unit vector
•det�=1(the determinant)
Where �??????(n) denotes the Special Orthogonal group of order n.
Example for �
�
=�
−�
??????���:

66. 10/17/2017
Parameterizations Of Rotations
As each column of �is a unit vector, then we can
write:
As the columns of �are mutually orthogonal, then
we can write:
Together, these constraints define six independent equations with nine unknowns, which
implies that there are three free variables.
The nine elements �
��in a general rotational transformation R are not independent
quantities.
�??????�??????3
Where �??????(n) denotes the Special Orthogonal group of order n.

67. 10/17/2017
We present three ways in which an arbitrary rotation can be represented using only three
independent quantities:
•Euler Angles representation
•Roll-Pitch-Yawrepresentation
•Axis/Anglerepresentation
Parameterizations of Rotations

68. 10/17/2017
Euler Angles Representation
We can specify the orientation of the frame �
��
��
��
�relative to the frame�
��
��
��
�by
three angles (??????,??????,??????), known as Euler Angles, and obtained by three successive rotations as
follows:
1.rotation about thez-axisby the angle ??????
2.rotationaboutthecurrenty−axisbytheangle??????
3.rotationaboutthecurrentz−axisbytheangle??????

69. 10/17/2017
Euler Angles Representation

70. 10/17/2017

71. 10/17/2017
Trigonometry (Atanvs. Atan2)
Reminder:

72. Trigonometry (Atanvs. Atan2)
Reminder:
10/17/2017

73. 10/17/2017
Trigonometry (Atanvs. Atan2)
Reminder:
tan(angle) = opposite/adjacent
atan(opposite/adjacent) = angle

74. 10/17/2017
Euler Angles Representation
Given a matrix �??????�??????3
Determinea set of Euler angles??????,??????,and??????so that�=�
���
If �
��≠�and�
��≠�,
itfollowthat:??????=Atan2(�
��,�
��)
where the function ??????�??????�2(�,�)computes the arctangent of the ration �/�.
Then squaring the summing of the elements (1,3) and (2,3) and using the element (3,3)
yields:
??????=Atan2(+�
��
�
+�
��
�
,�
��) or??????=Atan2(−�
��
�
+�
��
�
,�
��)
If we consider the first choice then ���(??????)>�then: ??????=Atan2(�
��,−�
��)
If we consider the second choice then &#3627408532;&#3627408522;&#3627408527;??????<&#3627409358;then: ??????=Atan2(−&#3627408531;
&#3627409361;&#3627409360;,&#3627408531;
&#3627409361;&#3627409359;)
and ??????=Atan2(−&#3627408531;
&#3627409360;&#3627409361;,−&#3627408531;
&#3627409359;&#3627409361;)

75. 10/17/2017
Euler Angles Representation
Given a matrix &#3627408453;??????&#3627408454;??????3
Determinea set of Euler angles??????,??????,and??????so that&#3627408505;=&#3627408505;
&#3627408513;&#3627408512;&#3627408513;
If &#3627408531;
&#3627409359;&#3627409361;=&#3627408531;
&#3627409360;&#3627409361;=&#3627409358;,thenthe fact that R is orthogonal implies that &#3627408531;
&#3627409361;&#3627409361;=±&#3627409359;and that &#3627408531;
&#3627409361;&#3627409359;
=&#3627408531;
&#3627409361;&#3627409360;=&#3627409358;thus R has the form:
If &#3627408531;
&#3627409361;&#3627409361;=+&#3627409359;then &#3627408464;
??????=1and &#3627408480;
??????=0, so that??????=0.
Thus, the sum ??????+??????can be determined as ??????+??????= Atan2(&#3627408531;
&#3627409360;&#3627409359;,&#3627408531;
&#3627409359;&#3627409359;) = Atan2(−&#3627408531;
&#3627409359;&#3627409360;,&#3627408531;
&#3627409360;&#3627409360;)
There is infinity of solutions.

76. 10/17/2017
Euler Angles Representation
Given a matrix &#3627408453;??????&#3627408454;??????3
Determinea set of Euler angles??????,??????,and??????so that&#3627408505;=&#3627408505;
&#3627408513;&#3627408512;&#3627408513;
If &#3627408531;
&#3627409359;&#3627409361;=&#3627408531;
&#3627409360;&#3627409361;=&#3627409358;,thenthe fact that R is orthogonal implies that &#3627408531;
&#3627409361;&#3627409361;=±&#3627409359;and that &#3627408531;
&#3627409361;&#3627409359;
=&#3627408531;
&#3627409361;&#3627409360;=&#3627409358;thus R has the form:
If &#3627408531;
&#3627409361;&#3627409361;=−&#3627409359;then &#3627408464;
??????=−1and &#3627408480;
??????=0, so that??????=π.
Thus, the ??????−??????can be determined as ??????−??????= Atan2(−&#3627408531;
&#3627409359;&#3627409360;,−&#3627408531;
&#3627409359;&#3627409359;) = Atan2(&#3627408531;
&#3627409360;&#3627409359;,&#3627408531;
&#3627409360;&#3627409360;)
As before there is infinity of solutions.

77. 10/17/2017
Yaw-Pitch-RollRepresentation
A rotation matrix &#3627408505;can also be described as a product of successive rotations about the
principal coordinate axes&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;taken in a specific order. These rotations define
the roll, pitch, and yawangles, which we shall also denote (??????,??????,??????)
We specify the order in three successive rotations as follows:
1.Yaw rotation about &#3627408537;
&#3627409358;−axisby the angle??????
2.Pitchrotationabout&#3627408538;
&#3627409358;−axisbytheangle??????
3.Roll rotationabout&#3627408539;
&#3627409358;−axisbytheangle??????
Since the successive rotations are relative to the fixed
frame, the resulting transformation matrix is given by:

78. 10/17/2017
Yaw-Pitch-Roll Representation
A rotation matrix &#3627408505;can also be described as a product of successive rotations about the
principal coordinate axes&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;taken in a specific order. These rotations define
the roll, pitch, and yawangles, which we shall also denote (??????,??????,??????)
We specify the order in three successive rotations as follows:
1.Yaw rotation about &#3627408537;
&#3627409358;−axisby the angle??????
2.Pitchrotationabout&#3627408538;
&#3627409358;−axisbytheangle??????
3.Roll rotationabout&#3627408539;
&#3627409358;−axisbytheangle??????
Since the successive rotations are relative to the fixed
frame, the resulting transformation matrix is given by:

79. 10/17/2017
Yaw-Pitch-Roll Representation
A rotation matrix &#3627408505;can also be described as a product of successive rotations about the
principal coordinate axes&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;taken in a specific order. These rotations define
the roll, pitch, and yawangles, which we shall also denote (??????,??????,??????)
We specify the order in three successive rotations as follows:
1.Yaw rotation about &#3627408537;
&#3627409358;−axisby the angle??????
2.Pitchrotationabout&#3627408538;
&#3627409358;−axisbytheangle??????
3.Roll rotationabout&#3627408539;
&#3627409358;−axisbytheangle??????
Since the successive rotations are relative to the fixed
frame, the resulting transformation matrix is given by:

80. 10/17/2017
Yaw-Pitch-Roll Representation
A rotation matrix &#3627408505;can also be described as a product of successive rotations about the
principal coordinate axes&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;taken in a specific order. These rotations define
the roll, pitch, and yawangles, which we shall also denote (??????,??????,??????)
We specify the order in three successive rotations as follows:
1.Yaw rotation about &#3627408537;
&#3627409358;−axisby the angle??????
2.Pitchrotationabout&#3627408538;
&#3627409358;−axisbytheangle??????
3.Roll rotationabout&#3627408539;
&#3627409358;−axisbytheangle??????
Since the successive rotations are relative to the fixed
frame, the resulting transformation matrix is given by:

81. 10/17/2017
Yaw-Pitch-Roll Representation
A rotation matrix &#3627408505;can also be described as a product of successive rotations about the
principal coordinate axes&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;taken in a specific order. These rotations define
the roll, pitch, and yawangles, which we shall also denote (??????,??????,??????)
We specify the order in three successive rotations as follows:
1.Yaw rotation about &#3627408537;
&#3627409358;−axisby the angle??????
2.Pitchrotationabout&#3627408538;
&#3627409358;−axisbytheangle??????
3.Roll rotationabout&#3627408539;
&#3627409358;−axisbytheangle??????
Since the successive rotations are relative to the fixed
frame, the resulting transformation matrix is given by:

82. 10/17/2017
Yaw-Pitch-Roll Representation
A rotation matrix &#3627408505;can also be described as a product of successive rotations about the
principal coordinate axes&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;taken in a specific order. These rotations define
the roll, pitch, and yawangles, which we shall also denote (??????,??????,??????)
We specify the order in three successive rotations as follows:
1.Yaw rotation about &#3627408537;
&#3627409358;−axisby the angle??????
2.Pitchrotationabout&#3627408538;
&#3627409358;−axisbytheangle??????
3.Roll rotationabout&#3627408539;
&#3627409358;−axisbytheangle??????
Since the successive rotations are relative to the fixed
frame, the resulting transformation matrix is given by:
Instead of yaw-pitch-roll relative to the fixedframes we could
also interpret the above transformation as roll-pitch-yaw, in
that order, each taken with respect to the currentframe.
The end result is the same matrix.
??????
??????
??????

83. 10/17/2017
Yaw-Pitch-Roll Representation
Find the inverse solution to a given rotation matrix R.
Determinea set of Roll-Pitch-Yaw angles??????,??????,and??????so that&#3627408505;
=&#3627408505;
&#3627408511;&#3627408512;&#3627408513;
??????
??????
??????

84. 10/17/2017
Axis/Angle Representation
Rotations are not always performed about the principal coordinate axes. We are often
interested in a rotation about an arbitrary axis in space. This provides both a convenient
way to describe rotations, and an alternative parameterization for rotation matrices.
Let &#3627408524;=[&#3627408524;
&#3627408537;,&#3627408524;
&#3627408538;,&#3627408524;
&#3627408539;]
&#3627408507;
, expressed in the frame &#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;, be
a unit vector defining an axis. We wish to derive the rotation
matrix &#3627408505;
&#3627408524;,??????representing a rotation of ??????about this axis.
A possible solution is to rotate first &#3627408524;by the angles necessary
to align it with &#3627408539;, then to rotate by ??????about &#3627408539;, and finally to
rotate by the angels necessary to align the unit vector with
the initial direction.

85. 10/17/2017
Axis/Angle Representation
Rotations are not always performed about the principal coordinate axes. We are often
interested in a rotation about an arbitrary axis in space. This provides both a convenient
way to describe rotations, and an alternative parameterization for rotation matrices.
Let &#3627408524;=[&#3627408524;
&#3627408537;,&#3627408524;
&#3627408538;,&#3627408524;
&#3627408539;]
&#3627408507;
, expressed in the frame &#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;, be
a unit vector defining an axis. We wish to derive the rotation
matrix &#3627408505;
&#3627408524;,??????representing a rotation of ??????about this axis.
The sequence of rotations to be made with respect to axes of
fixed frame is the following:
•Align &#3627408524;with &#3627408539;(which is obtained as the sequence of a
rotation by −&#3627409206;about &#3627408539;and a rotation of −&#3627409207;about &#3627408538;).
•Rotate by ??????about &#3627408539;.
•Realignwith the initial direction of &#3627408524;, which is obtained as
the sequence of a rotation by &#3627409207;about &#3627408538;and a rotation by
&#3627409206;about &#3627408539;.
&#3627408505;
&#3627408524;,??????=&#3627408505;
&#3627408539;,&#3627409206;&#3627408505;
&#3627408538;,&#3627409207;&#3627408505;
&#3627408539;,??????&#3627408505;
&#3627408538;,−&#3627409207;&#3627408505;
&#3627408539;,−&#3627409206;

86. 10/17/2017
Axis/Angle Representation
&#3627408505;
&#3627408524;,??????=&#3627408505;
&#3627408539;,&#3627409206;&#3627408505;
&#3627408538;,&#3627409207;&#3627408505;
&#3627408539;,??????&#3627408505;
&#3627408538;,−&#3627409207;&#3627408505;
&#3627408539;,−&#3627409206;

87. 10/17/2017
Axis/Angle Representation
Rotations are not always performed about the principal coordinate axes. We are often
interested in a rotation about an arbitrary axis in space. This provides both a convenient
way to describe rotations, and an alternative parameterization for rotation matrices.
&#3627408505;
&#3627408524;,??????=&#3627408505;
&#3627408539;,&#3627409206;&#3627408505;
&#3627408538;,&#3627409207;&#3627408505;
&#3627408539;,??????&#3627408505;
&#3627408538;,−&#3627409207;&#3627408505;
&#3627408539;,−&#3627409206;

88. 10/17/2017
Axis/Angle Representation
Any rotation matrix &#3627408453;??????&#3627408454;??????3can be represented by a single
rotation about a suitable axis in space by a suitable angle.
&#3627408505;=&#3627408505;
&#3627408524;,??????
where &#3627408524;is a unit vector defining the axis of rotation, and ??????is
the angle of rotation about &#3627408524;.
The matrix &#3627408505;
&#3627408524;,??????is called the axis/angle representation of &#3627408505;.
Given &#3627408505;find ??????and &#3627408524;:
Reminder:

89. 10/17/2017
Axis/Angle Representation
The axis/angle representation is not unique since
a rotation of −??????about −&#3627408524;is the same as a
rotation of ??????about &#3627408524;.
&#3627408505;
&#3627408524;,??????=&#3627408505;
−&#3627408524;,−??????
If??????=&#3627409358;then &#3627408505;istheidentitymatrixandthe
axisofrotationisundefined.

90. 10/17/2017
Example
Suppose &#3627408505;is generated by a rotation of 90
°
about &#3627408487;
0followed by a rotation of 30
°
about
&#3627408486;
0followed by a rotation of 60
°
about &#3627408485;
0. Find the axis/angle representation of &#3627408505;
Reminder:
The axis/angle representation of &#3627408505;

91. 10/17/2017
Example
Suppose &#3627408505;is generated by a rotation of 90
°
about &#3627408487;
0followed by a rotation of 30
°
about
&#3627408486;
0followed by a rotation of 60
°
about &#3627408486;
0. Find the axis/angle representation of &#3627408505;
Reminder:
The axis/angle representation of &#3627408505;

92. 10/17/2017
Example
Suppose &#3627408505;is generated by a rotation of 90
°
about &#3627408487;
0followed by a rotation of 30
°
about
&#3627408486;
0followed by a rotation of 60
°
about &#3627408486;
0. Find the axis/angle representation of &#3627408505;
Reminder:
The axis/angle representation of &#3627408505;

93. 10/17/2017
Example
Suppose &#3627408505;is generated by a rotation of 90
°
about &#3627408487;
0followed by a rotation of 30
°
about
&#3627408486;
0followed by a rotation of 60
°
about &#3627408486;
0. Find the axis/angle representation of &#3627408505;
Reminder:
The axis/angle representation of &#3627408505;

94. 10/17/2017
Example
Suppose &#3627408505;is generated by a rotation of 90
°
about &#3627408487;
0followed by a rotation of 30
°
about
&#3627408486;
0followed by a rotation of 60
°
about &#3627408485;
0. Find the axis/angle representation of &#3627408505;
Reminder:
The axis/angle representation of &#3627408505;

95. 10/17/2017
Axis/Angle Representation
Theaboveaxis/anglerepresentationcharacterizesagiven
rotationbyfourquantities,namelythethreecomponents
oftheequivalentaxis&#3627408524;andtheequivalentangle??????.
However,sincetheequivalentaxis&#3627408524;isgivenasaunit
vectoronlytwoofitscomponentsareindependent.
Thethirdisconstrainedbytheconditionthat&#3627408524;isofunit
length.
Therefore,onlythreeindependentquantitiesare
requiredinthisrepresentationofarotation&#3627408505;.Wecan
representtheequivalentaxis/anglebyasinglevector&#3627408531;
as:
since&#3627408524;isaunitvector,thelengthofthevector&#3627408531;isthe
equivalentangle??????andthedirectionof&#3627408531;istheequivalent
axis&#3627408524;.

96. 10/17/2017
Rigid Motions
Arigidmotionisapuretranslationtogetherwithapurerotation.
Arigidmotionisanorderedpair(&#3627408517;,&#3627408505;)where&#3627408517;∈ℝ
&#3627409361;
and&#3627408505;
∈&#3627408506;&#3627408502;&#3627409361;.ThegroupofallrigidmotionsisknownastheSpecial
EuclideanGroupandisdenotedby&#3627408506;??????&#3627409361;.Weseethenthat
&#3627408506;??????&#3627409361;=ℝ
&#3627409361;
×&#3627408506;&#3627408502;&#3627409361;.

97. 10/17/2017
One Rigid Motion
Ifframe&#3627408528;
&#3627409359;&#3627408537;
&#3627409359;&#3627408538;
&#3627409359;&#3627408539;
&#3627409359;isobtainedfromframe&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;byfirstapplyingarotation
specifiedby&#3627408505;
&#3627409359;
&#3627409358;
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408486;
1
&#3627408485;
1
&#3627408487;
1

98. 10/17/2017
One Rigid Motion
Ifframe&#3627408528;
&#3627409359;&#3627408537;
&#3627409359;&#3627408538;
&#3627409359;&#3627408539;
&#3627409359;isobtainedfromframe&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;byfirstapplyingarotation
specifiedby&#3627408505;
&#3627409359;
&#3627409358;
followedbyatranslationgiven(withrespectto&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;)by&#3627408517;
&#3627409359;
&#3627409358;
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408505;
&#3627409359;
&#3627409358;
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408486;
1
&#3627408485;
1
&#3627408487;
1

99. 10/17/2017
One Rigid Motion
Ifframe&#3627408528;
&#3627409359;&#3627408537;
&#3627409359;&#3627408538;
&#3627409359;&#3627408539;
&#3627409359;isobtainedfromframe&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;byfirstapplyingarotation
specifiedby&#3627408505;
&#3627409359;
&#3627409358;
followedbyatranslationgiven(withrespectto&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;)by&#3627408517;
&#3627409359;
&#3627409358;
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408486;
1
&#3627408485;
1
&#3627408487;
1
&#3627408517;
&#3627409359;
&#3627409358;
&#3627408505;
&#3627409359;
&#3627409358;
&#3627408529;
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408486;
1
&#3627408485;
1
&#3627408487;
1

100. 10/17/2017
One Rigid Motion
Ifframe&#3627408528;
&#3627409359;&#3627408537;
&#3627409359;&#3627408538;
&#3627409359;&#3627408539;
&#3627409359;isobtainedfromframe&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;byfirstapplyingarotation
specifiedby&#3627408505;
&#3627409359;
&#3627409358;
followedbyatranslationgiven(withrespectto&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;)by&#3627408517;
&#3627409359;
&#3627409358;
,
thenthecoordinates&#3627408529;
&#3627409358;
aregivenby:
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408486;
1
&#3627408485;
1
&#3627408487;
1
&#3627408517;
&#3627409359;
&#3627409358;
&#3627408505;
&#3627409359;
&#3627409358;
&#3627408529;
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408486;
1
&#3627408485;
1
&#3627408487;
1
&#3627408529;
&#3627409358;
&#3627408529;
&#3627409359;

101. 10/17/2017
One Rigid Motion
Ifframe&#3627408528;
&#3627409359;&#3627408537;
&#3627409359;&#3627408538;
&#3627409359;&#3627408539;
&#3627409359;isobtainedfromframe&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;byfirstapplyingarotation
specifiedby&#3627408505;
&#3627409359;
&#3627409358;
followedbyatranslationgiven(withrespectto&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;)by&#3627408517;
&#3627409359;
&#3627409358;
,
thenthecoordinates&#3627408529;
&#3627409358;
aregivenby:
&#3627408529;
&#3627409358;
=&#3627408505;
&#3627409359;
&#3627409358;
&#3627408529;
&#3627409359;
+&#3627408517;
&#3627409359;
&#3627409358;
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408486;
1
&#3627408485;
1
&#3627408487;
1
&#3627408517;
&#3627409359;
&#3627409358;
&#3627408505;
&#3627409359;
&#3627409358;
&#3627408529;
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408486;
1
&#3627408485;
1
&#3627408487;
1
&#3627408529;
&#3627409358;
&#3627408529;
&#3627409359;

102. 10/17/2017
One Rigid Motion
Ifframe&#3627408528;
&#3627409359;&#3627408537;
&#3627409359;&#3627408538;
&#3627409359;&#3627408539;
&#3627409359;isobtainedfromframe&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;byfirstapplyingarotation
specifiedby&#3627408505;
&#3627409359;
&#3627409358;
followedbyatranslationgiven(withrespectto&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;)by&#3627408517;
&#3627409359;
&#3627409358;
,
thenthecoordinates&#3627408529;
&#3627409358;
aregivenby:
&#3627408529;
&#3627409358;
=&#3627408505;
&#3627409359;
&#3627409358;
&#3627408529;
&#3627409359;
+&#3627408517;
&#3627409359;
&#3627409358;
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408486;
1
&#3627408485;
1
&#3627408487;
1
&#3627408517;
&#3627409359;
&#3627409358;
&#3627408505;
&#3627409359;
&#3627409358;
&#3627408529;
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408486;
1
&#3627408485;
1
&#3627408487;
1
&#3627408529;
&#3627409358;
&#3627408529;
&#3627409359;

103. 10/17/2017
Two Rigid Motions
Ifframe&#3627408528;
&#3627409360;&#3627408537;
&#3627409360;&#3627408538;
&#3627409360;&#3627408539;
&#3627409360;isobtainedfromframe&#3627408528;
&#3627409359;&#3627408537;
&#3627409359;&#3627408538;
&#3627409359;&#3627408539;
&#3627409359;byfirstapplyingarotation
specifiedby&#3627408505;
&#3627409360;
&#3627409359;
followedbyatranslationgiven(withrespectto&#3627408528;
&#3627409359;&#3627408537;
&#3627409359;&#3627408538;
&#3627409359;&#3627408539;
&#3627409359;)by&#3627408517;
&#3627409360;
&#3627409359;
.
Ifframe&#3627408528;
&#3627409359;&#3627408537;
&#3627409359;&#3627408538;
&#3627409359;&#3627408539;
&#3627409359;isobtainedfromframe&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;byfirstapplyingarotation
specifiedby&#3627408505;
&#3627409359;
&#3627409358;
followedbyatranslationgiven(withrespectto&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;)by&#3627408517;
&#3627409359;
&#3627409358;
,
findthecoordinates&#3627408529;
&#3627409358;
.
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408486;
1
&#3627408485;
1
&#3627408487;
1
&#3627408529;
&#3627408487;
2
&#3627408486;
2
&#3627408485;
2
For the first rigid motion:
&#3627408529;
&#3627409358;
=&#3627408505;
&#3627409359;
&#3627409358;
&#3627408529;
&#3627409359;
+&#3627408517;
&#3627409359;
&#3627409358;
For the second rigid motion:
&#3627408529;
&#3627409359;
=&#3627408505;
&#3627409360;
&#3627409359;
&#3627408529;
&#3627409360;
+&#3627408517;
&#3627409360;
&#3627409359;
Both rigid motions can be described as one rigid
motion:
&#3627408529;
&#3627409358;
=&#3627408505;
&#3627409360;
&#3627409358;
&#3627408529;
&#3627409360;
+&#3627408517;
&#3627409360;
&#3627409358;
&#3627408529;
&#3627409358;
=&#3627408505;
&#3627409359;
&#3627409358;
&#3627408505;
&#3627409360;
&#3627409359;
&#3627408529;
&#3627409360;
+&#3627408505;
&#3627409359;
&#3627409358;
&#3627408517;
&#3627409360;
&#3627409359;
+&#3627408517;
&#3627409359;
&#3627409358;

104. 10/17/2017
Two Rigid Motions
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408486;
1
&#3627408485;
1
&#3627408487;
1
&#3627408529;
&#3627408487;
2
&#3627408486;
2
&#3627408485;
2
&#3627408505;
&#3627409360;
&#3627409358;
The orientation transformations can simply be multiplied together.
&#3627408517;
&#3627409360;
&#3627409358;
The translation transformation is the sum of:
•&#3627408517;
&#3627409359;
&#3627409358;
the vector from the origin &#3627408476;
0to the origin &#3627408476;
1expressed with respect to &#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;.
•&#3627408505;
&#3627409359;
&#3627409358;
&#3627408517;
&#3627409360;
&#3627409359;
the vector from &#3627408476;
1to &#3627408476;
2expressed in the orientation of the coordinate system
&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;.
&#3627408529;
&#3627409358;
=&#3627408505;
&#3627409359;
&#3627409358;
&#3627408505;
&#3627409360;
&#3627409359;
&#3627408529;
&#3627409360;
+&#3627408505;
&#3627409359;
&#3627409358;
&#3627408517;
&#3627409360;
&#3627409359;
+&#3627408517;
&#3627409359;
&#3627409358;

105. 10/17/2017
Three Rigid Motions
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408486;
1
&#3627408485;
1
&#3627408487;
1
&#3627408529;
&#3627408487;
2
&#3627408486;
2
&#3627408485;
2
&#3627408529;
&#3627409358;
=?
&#3627408487;
3
&#3627408486;
3
&#3627408485;
3

106. 10/17/2017
Homogeneous Transformations
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408486;
1
&#3627408485;
1
&#3627408487;
1
&#3627408529;
&#3627408487;
2
&#3627408486;
2
&#3627408485;
2
&#3627408529;
&#3627409358;
=?
&#3627408487;
??????
&#3627408486;
??????
&#3627408485;
??????
A long sequence of rigid motions, find &#3627408529;
&#3627409358;
.
&#3627408529;
&#3627409358;
=&#3627408505;
&#3627408527;
&#3627409358;
&#3627408529;
&#3627408527;
+&#3627408517;
&#3627408527;
&#3627409358;

107. 10/17/2017
Homogeneous Transformations
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408486;
1
&#3627408485;
1
&#3627408487;
1
&#3627408529;
&#3627408487;
2
&#3627408486;
2
&#3627408485;
2
&#3627408529;
&#3627409358;
=?
&#3627408487;
??????
&#3627408486;
??????
&#3627408485;
??????
A long sequence of rigid motions, find &#3627408529;
&#3627409358;
.
Represent rigid motions in matrix so that
composition of rigid motions can be reduced to
matrix multiplication as was the case for
composition of rotations
&#3627408529;
&#3627409358;
=&#3627408505;
&#3627408527;
&#3627409358;
&#3627408529;
&#3627408527;
+&#3627408517;
&#3627408527;
&#3627409358;

108. 10/17/2017
Homogeneous Transformations
A long sequence of rigid motions, find &#3627408529;
&#3627409358;
.
Represent rigid motions in matrix so that
composition of rigid motions can be reduced to
matrix multiplication as was the case for
composition of rotations
&#3627408529;
&#3627409358;
=&#3627408505;
&#3627408527;
&#3627409358;
&#3627408529;
&#3627408527;
+&#3627408517;
&#3627408527;
&#3627409358;
??????=
&#3627408505;&#3627408517;
;&#3627408517;∈ℝ
&#3627409361;
,&#3627408505;∈&#3627408506;&#3627408502;&#3627409361;

109. 10/17/2017
Homogeneous Transformations
A long sequence of rigid motions, find &#3627408529;
&#3627409358;
.
Represent rigid motions in matrix so that
composition of rigid motions can be reduced to
matrix multiplication as was the case for
composition of rotations
&#3627408529;
&#3627409358;
=&#3627408505;
&#3627408527;
&#3627409358;
&#3627408529;
&#3627408527;
+&#3627408517;
&#3627408527;
&#3627409358;
??????=
&#3627408505;&#3627408517;
&#3627409358;&#3627409359;
;&#3627408517;∈ℝ
&#3627409361;
,&#3627408505;∈&#3627408506;&#3627408502;&#3627409361;
Transformation matrices of the form Hare called homogeneous transformations.
A homogeneous transformation is therefore a matrix representation of a rigid motion.

110. 10/17/2017
Homogeneous Transformations
A long sequence of rigid motions, find &#3627408529;
&#3627409358;
.
Represent rigid motions in matrix so that
composition of rigid motions can be reduced to
matrix multiplication as was the case for
composition of rotations
&#3627408529;
&#3627409358;
=&#3627408505;
&#3627408527;
&#3627409358;
&#3627408529;
&#3627408527;
+&#3627408517;
&#3627408527;
&#3627409358;
??????=
&#3627408505;&#3627408517;
&#3627409358;&#3627409359;
;&#3627408517;∈ℝ
&#3627409361;
,&#3627408505;∈&#3627408506;&#3627408502;&#3627409361;
??????
−&#3627409359;
=
&#3627408505;
&#3627408507;
−&#3627408505;
&#3627408507;
&#3627408517;
&#3627409358;&#3627409359;
The inverse transformation ??????
−&#3627409359;
is given by

111. 10/17/2017
Ex. :Two Rigid Motions
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408486;
1
&#3627408485;
1
&#3627408487;
1
&#3627408529;
&#3627408487;
2
&#3627408486;
2
&#3627408485;
2
&#3627408505;
&#3627409360;
&#3627409358;
The orientation transformations can simply be multiplied together.
&#3627408517;
&#3627409360;
&#3627409358;
The translation transformation is the sum of:
•&#3627408517;
&#3627409359;
&#3627409358;
the vector from the origin &#3627408476;
0to the origin &#3627408476;
1expressed with respect to &#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;.
•&#3627408505;
&#3627409359;
&#3627409358;
&#3627408517;
&#3627409360;
&#3627409359;
the vector from &#3627408476;
1to &#3627408476;
2expressed in the orientation of the coordinate system
&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;.
&#3627408529;
&#3627409358;
=&#3627408505;
&#3627409359;
&#3627409358;
&#3627408505;
&#3627409360;
&#3627409359;
&#3627408529;
&#3627409360;
+&#3627408505;
&#3627409359;
&#3627409358;
&#3627408517;
&#3627409360;
&#3627409359;
+&#3627408517;
&#3627409359;
&#3627409358;
??????=
&#3627408505;&#3627408517;
&#3627409358;&#3627409359;
;&#3627408517;∈ℝ
&#3627409361;
,&#3627408505;∈&#3627408506;&#3627408502;&#3627409361;
&#3627408505;
&#3627409359;
&#3627409358;
&#3627408517;
&#3627409359;
&#3627409358;
&#3627409358;&#3627409359;
&#3627408505;
&#3627409360;
&#3627409359;
&#3627408517;
&#3627409360;
&#3627409359;
&#3627409358;&#3627409359;
=
&#3627408505;
&#3627409359;
&#3627409358;
&#3627408505;
&#3627409360;
&#3627409359;
&#3627408505;
&#3627409359;
&#3627409358;
&#3627408517;
&#3627409360;
&#3627409359;
+&#3627408517;
&#3627409359;
&#3627409358;
0 1

112. 10/17/2017
Ex. :Two Rigid Motions
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408486;
1
&#3627408485;
1
&#3627408487;
1
&#3627408529;
&#3627408487;
2
&#3627408486;
2
&#3627408485;
2
&#3627408505;
&#3627409360;
&#3627409358;
The orientation transformations can simply be multiplied together.
&#3627408517;
&#3627409360;
&#3627409358;
The translation transformation is the sum of:
•&#3627408517;
&#3627409359;
&#3627409358;
the vector from the origin &#3627408476;
0to the origin &#3627408476;
1expressed with respect to &#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;.
•&#3627408505;
&#3627409359;
&#3627409358;
&#3627408517;
&#3627409360;
&#3627409359;
the vector from &#3627408476;
1to &#3627408476;
2expressed in the orientation of the coordinate system
&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;.
&#3627408529;
&#3627409358;
=&#3627408505;
&#3627409359;
&#3627409358;
&#3627408505;
&#3627409360;
&#3627409359;
&#3627408529;
&#3627409360;
+&#3627408505;
&#3627409359;
&#3627409358;
&#3627408517;
&#3627409360;
&#3627409359;
+&#3627408517;
&#3627409359;
&#3627409358;
??????=
&#3627408505;&#3627408517;
&#3627409358;&#3627409359;
;&#3627408517;∈ℝ
&#3627409361;
,&#3627408505;∈&#3627408506;&#3627408502;&#3627409361;
&#3627408505;
&#3627409359;
&#3627409358;
&#3627408517;
&#3627409359;
&#3627409358;
&#3627409358;&#3627409359;
&#3627408505;
&#3627409360;
&#3627409359;
&#3627408517;
&#3627409360;
&#3627409359;
&#3627409358;&#3627409359;
=
&#3627408505;
&#3627409359;
&#3627409358;
&#3627408505;
&#3627409360;
&#3627409359;
&#3627408505;
&#3627409359;
&#3627409358;
&#3627408517;
&#3627409360;
&#3627409359;
+&#3627408517;
&#3627409359;
&#3627409358;
0 1

113. 10/17/2017
Ex. :Two Rigid Motions
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408486;
1
&#3627408485;
1
&#3627408487;
1
&#3627408529;
&#3627408487;
2
&#3627408486;
2
&#3627408485;
2
&#3627408505;
&#3627409360;
&#3627409358;
The orientation transformations can simply be multiplied together.
&#3627408517;
&#3627409360;
&#3627409358;
The translation transformation is the sum of:
•&#3627408517;
&#3627409359;
&#3627409358;
the vector from the origin &#3627408476;
0to the origin &#3627408476;
1expressed with respect to &#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;.
•&#3627408505;
&#3627409359;
&#3627409358;
&#3627408517;
&#3627409360;
&#3627409359;
the vector from &#3627408476;
1to &#3627408476;
2expressed in the orientation of the coordinate system
&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;.
&#3627408529;
&#3627409358;
=&#3627408505;
&#3627409359;
&#3627409358;
&#3627408505;
&#3627409360;
&#3627409359;
&#3627408529;
&#3627409360;
+&#3627408505;
&#3627409359;
&#3627409358;
&#3627408517;
&#3627409360;
&#3627409359;
+&#3627408517;
&#3627409359;
&#3627409358;
??????=
&#3627408505;&#3627408517;
&#3627409358;&#3627409359;
;&#3627408517;∈ℝ
&#3627409361;
,&#3627408505;∈&#3627408506;&#3627408502;&#3627409361;
&#3627408505;
&#3627409359;
&#3627409358;
&#3627408517;
&#3627409359;
&#3627409358;
&#3627409358;&#3627409359;
&#3627408505;
&#3627409360;
&#3627409359;
&#3627408517;
&#3627409360;
&#3627409359;
&#3627409358;&#3627409359;
=
&#3627408505;
&#3627409359;
&#3627409358;
&#3627408505;
&#3627409360;
&#3627409359;
&#3627408505;
&#3627409359;
&#3627409358;
&#3627408517;
&#3627409360;
&#3627409359;
+&#3627408517;
&#3627409359;
&#3627409358;
0 1

114. 10/17/2017
Ex. :Two Rigid Motions
&#3627408485;
0
&#3627408486;
0
&#3627408487;
0
&#3627408486;
1
&#3627408485;
1
&#3627408487;
1
&#3627408529;
&#3627408487;
2
&#3627408486;
2
&#3627408485;
2
&#3627408505;
&#3627409360;
&#3627409358;
The orientation transformations can simply be multiplied together.
&#3627408517;
&#3627409360;
&#3627409358;
The translation transformation is the sum of:
•&#3627408517;
&#3627409359;
&#3627409358;
the vector from the origin &#3627408476;
0to the origin &#3627408476;
1expressed with respect to &#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;.
•&#3627408505;
&#3627409359;
&#3627409358;
&#3627408517;
&#3627409360;
&#3627409359;
the vector from &#3627408476;
1to &#3627408476;
2expressed in the orientation of the coordinate system
&#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;.
&#3627408529;
&#3627409358;
=&#3627408505;
&#3627409359;
&#3627409358;
&#3627408505;
&#3627409360;
&#3627409359;
&#3627408529;
&#3627409360;
+&#3627408505;
&#3627409359;
&#3627409358;
&#3627408517;
&#3627409360;
&#3627409359;
+&#3627408517;
&#3627409359;
&#3627409358;
??????=
&#3627408505;&#3627408517;
&#3627409358;&#3627409359;
;&#3627408517;∈ℝ
&#3627409361;
,&#3627408505;∈&#3627408506;&#3627408502;&#3627409361;
&#3627408505;
&#3627409359;
&#3627409358;
&#3627408517;
&#3627409359;
&#3627409358;
&#3627409358;&#3627409359;
&#3627408505;
&#3627409360;
&#3627409359;
&#3627408517;
&#3627409360;
&#3627409359;
&#3627409358;&#3627409359;
=
&#3627408505;
&#3627409359;
&#3627409358;
&#3627408505;
&#3627409360;
&#3627409359;
&#3627408505;
&#3627409359;
&#3627409358;
&#3627408517;
&#3627409360;
&#3627409359;
+&#3627408517;
&#3627409359;
&#3627409358;
0 1
We must augment the vectors &#3627408529;
&#3627409358;
,&#3627408529;
&#3627409359;
and &#3627408529;
&#3627409360;
by the addition
of a fourth component of 1:
&#3627408503;
&#3627409358;
=
&#3627408529;
&#3627409358;
1
, &#3627408503;
&#3627409359;
=
&#3627408529;
&#3627409359;
1
, &#3627408503;
&#3627409360;
=
&#3627408529;
&#3627409360;
1

115. 10/17/2017
Homogeneous Transformations
&#3627408503;
&#3627409358;
=??????
&#3627409359;
&#3627409358;
&#3627408503;
&#3627409359;
&#3627408503;
&#3627409358;
=??????
&#3627409360;
&#3627409358;
&#3627408503;
&#3627409360;
…..
&#3627408503;
&#3627409358;
=??????
&#3627408527;
&#3627409358;
&#3627408503;
&#3627408527;
&#3627408486;
0
&#3627408487;
0
&#3627408486;
1
&#3627408485;
1
&#3627408487;
1
&#3627408529;
&#3627408487;
2
&#3627408486;
2
&#3627408485;
2
&#3627408487;
??????
&#3627408486;
??????
&#3627408485;
??????
&#3627408503;
&#3627409358;
=
&#3627408529;
&#3627409358;
1

116. 10/17/2017
Basic Homogeneous Transformations
z

117. 10/17/2017
Homogeneous Transformations
??????
1
0
=
&#3627408475;&#3627408485;&#3627408480;&#3627408485;
&#3627408475;&#3627408486;&#3627408480;&#3627408486;
??????&#3627408485;&#3627408465;&#3627408485;
??????&#3627408486;&#3627408465;&#3627408486;
&#3627408475;&#3627408487;&#3627408480;&#3627408487;
00
??????&#3627408487;&#3627408465;&#3627408487;
01
=
&#3627408475;&#3627408480;
00
??????&#3627408465;
01
&#3627408527;is a vector representing the direction of &#3627408537;
&#3627409359;in the &#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;system
&#3627408532;is a vector representing the direction of &#3627408538;
&#3627409359;in the &#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;system
&#3627408514;is a vector representing the direction of &#3627408539;
&#3627409359;in the &#3627408528;
&#3627409358;&#3627408537;
&#3627409358;&#3627408538;
&#3627409358;&#3627408539;
&#3627409358;system

118. 10/17/2017
Composition Rule For Homogeneous
Transformations
Given a homogeneous transformation ??????
&#3627409359;
&#3627409358;
relating two frames, if a second rigid
motion, represented by ??????is performed relative to the current frame, then:
??????
2
0
=??????
1
0
??????
whereas if the second rigid motion is performed relative to the fixed frame, then:
??????
2
0
=????????????
1
0

119. 10/17/2017
Example
Find ??????for the following sequence of
1.a rotation by&#3627409206;about the current &#3627408537;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;, followed by
2.a translation of &#3627408515;units along the current &#3627408537;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;, followed by
3.a translation of &#3627408517;units along the current &#3627408539;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;,followed by
4.a rotation by angle ??????about the current &#3627408539;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;
Transformationwith
respect to the current
frame
??????
&#3627409360;
&#3627409358;
=??????
&#3627409359;
&#3627409358;
??????
Transformationwith
respect to the fixed
frame
??????
&#3627409360;
&#3627409358;
=????????????
&#3627409359;
&#3627409358;
Reminder:

120. 10/17/2017
Example
Find ??????for the following sequence of
1.a rotation by&#3627409206;about the current &#3627408537;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;, followed by
2.a translation of &#3627408515;units along the current &#3627408537;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;, followed by
3.a translation of &#3627408517;units along the current &#3627408539;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;,followed by
4.a rotation by angle ??????about the current &#3627408539;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;
Transformationwith
respect to the current
frame
??????
&#3627409360;
&#3627409358;
=??????
&#3627409359;
&#3627409358;
??????
Transformationwith
respect to the fixed
frame
??????
&#3627409360;
&#3627409358;
=????????????
&#3627409359;
&#3627409358;
Reminder:

121. 10/17/2017
Example
Find ??????for the following sequence of
1.a rotation by&#3627409206;about the current &#3627408537;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;, followed by
2.a translation of &#3627408515;units along the current &#3627408537;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;, followed by
3.a translation of &#3627408517;units along the current &#3627408539;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;,followed by
4.a rotation by angle ??????about the current &#3627408539;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;
Transformationwith
respect to the current
frame
??????
&#3627409360;
&#3627409358;
=??????
&#3627409359;
&#3627409358;
??????
Transformationwith
respect to the fixed
frame
??????
&#3627409360;
&#3627409358;
=????????????
&#3627409359;
&#3627409358;
Reminder:

122. 10/17/2017
Example
Find ??????for the following sequence of
1.a rotation by&#3627409206;about the current &#3627408537;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;, followed by
2.a translation of &#3627408515;units along the current &#3627408537;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;, followed by
3.a translation of &#3627408517;units along the current &#3627408539;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;,followed by
4.a rotation by angle ??????about the current &#3627408539;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;
Transformationwith
respect to the current
frame
??????
&#3627409360;
&#3627409358;
=??????
&#3627409359;
&#3627409358;
??????
Transformationwith
respect to the fixed
frame
??????
&#3627409360;
&#3627409358;
=????????????
&#3627409359;
&#3627409358;
Reminder:

123. 10/17/2017
Example
Find ??????for the following sequence of
1.a rotation by&#3627409206;about the current &#3627408537;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;, followed by
2.a translation of &#3627408515;units along the current &#3627408537;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;, followed by
3.a translation of &#3627408517;units along the current &#3627408539;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;,followed by
4.a rotation by angle ??????about the current &#3627408539;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;
Transformationwith
respect to the current
frame
??????
&#3627409360;
&#3627409358;
=??????
&#3627409359;
&#3627409358;
??????
Transformationwith
respect to the fixed
frame
??????
&#3627409360;
&#3627409358;
=????????????
&#3627409359;
&#3627409358;
Reminder:

124. 10/17/2017
Example
Find ??????for the following sequence of
1.a rotation by&#3627409206;about the current &#3627408537;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;, followed by
2.a translation of &#3627408515;units along the current &#3627408537;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;, followed by
3.a translation of &#3627408517;units along the current &#3627408539;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;,followed by
4.a rotation by angle ??????about the current &#3627408539;−&#3627408514;&#3627408537;&#3627408522;&#3627408532;
Transformationwith
respect to the current
frame
??????
&#3627409360;
&#3627409358;
=??????
&#3627409359;
&#3627409358;
??????
Transformationwith
respect to the fixed
frame
??????
&#3627409360;
&#3627409358;
=????????????
&#3627409359;
&#3627409358;
Reminder:

125. 10/17/2017
Example
Find the homogeneous transformations ??????
&#3627409359;
&#3627409358;
, ??????
&#3627409360;
&#3627409358;
, ??????
&#3627409360;
&#3627409359;
representing the transformations among the three frames
Shown. Show that ??????
&#3627409360;
&#3627409358;
=??????
&#3627409359;
&#3627409358;
??????
&#3627409360;
&#3627409359;
.