![Rotation Matrix
• Stack three unit vectors to form Rotation Matrix
• describes {B} with respect to {A}
• Each vector in can be written as dot product
of pair of unit vectors: cosine matrixcosine matrix
• Rows of : unit vectors of {A} with respect to
{B}
• What is ? What is det( )?
• Position + orientation = Frame
RA
B
[ ]B
A
B
A
B
AA
B ZYXR ˆˆˆ=
RA
B
RA
B
1−
RA
B RA
B](https://image.slidesharecdn.com/roboticspositionandorientation-171209102541/85/Robotics-position-and-orientation-7-320.jpg)
![Rotated Frames
YA
XA
ZA
XB
ZB
YB
B
P
BzByBx
B
ZpYpXpP ++=
B
A
zB
A
yB
A
x
A
ZpYpXpP ++=
[ ]
=
z
y
x
B
A
B
A
B
AA
p
p
p
ZYXP
PRP BA
B
A
=
Description of Rotation = Rotation Matrix](https://image.slidesharecdn.com/roboticspositionandorientation-171209102541/85/Robotics-position-and-orientation-10-320.jpg)
Robotics position and orientation
- 1. CL-7004 – Robotics and Control - Robotics Position and Orientation Dr.B.Arunadevi Associate Professor /ECE Dr.N.G.P Institute of Technology Coimbatore
- 2. Objectives of the Lecture • Learn to represent position and orientation • Be able to transform between coordinate systems. • Use frames and homogeneous coordinates • Describe Euler angles
- 3. A robot may not injure a human being, or, through inaction, allow a human being to come to harm. A robot must obey the orders given it by human beings except where such orders would conflict with the First Law. A robot must protect its own existence as long as such protection Three Laws of Robotics
- 4. Description of a Position • point = position vector YA XA ZA }A{ A P = z y x A p p p P a c o o r d i na t e s y s t e m
- 5. Description of an Orientation • In the example, a description of {B} with respect to {A} suffices to give orientation • Orientation = System of Coordinates • Directions of {B}: XB, YB and ZB • In {A} coord. system: A XB, A YB and A ZB x0 x1 z0 y0 z1 y1 θ }A{ }B{ Often a point is not enough: need orientation
- 6. From {A} to {B{ }A{ XB aX aY aZ( ) BAX XX ⋅=αcos ( ) BAY XY ⋅=αcos ( ) BAZ XZ ⋅=αcos We conclude: ⋅ ⋅ ⋅ = BA BA BA B A XZ XY XX X
- 7. Rotation Matrix • Stack three unit vectors to form Rotation Matrix • describes {B} with respect to {A} • Each vector in can be written as dot product of pair of unit vectors: cosine matrixcosine matrix • Rows of : unit vectors of {A} with respect to {B} • What is ? What is det( )? • Position + orientation = Frame RA B [ ]B A B A B AA B ZYXR ˆˆˆ= RA B RA B 1− RA B RA B
- 8. Description of a Frame • Frame: set of four vectors giving position + orientation • Description of a frame: position + rotation matrix • Ex.: x0 x1 z0 y0 z1 y1 θ }A{ }B{ { }BORG AA B PRB ,}{ = • position: frame with identity as rotation • orientation: frame with zero position
- 9. Mapping: from frame 2 frame • If {A} has same orientation as {B}, then {B} differs from {A} in a translation: A PBORG A P = B P + A PBORG • Mapping: change of description from one frame to another. The vector A PBORGdefines the mapping. XA ZA YA {A} ZB YB {B} XB A P A PBORG B P Translated Frames
- 10. Rotated Frames YA XA ZA XB ZB YB B P BzByBx B ZpYpXpP ++= B A zB A yB A x A ZpYpXpP ++= [ ] = z y x B A B A B AA p p p ZYXP PRP BA B A = Description of Rotation = Rotation Matrix
- 11. Rotated Frame (cont.) • The previous expression can be written as • The rotation mapping changes the description of a point from one coordinate system to another • The point does not change! only its description PRP BA B A =
- 13. General Frame Mapping }A{ XB YB ZB B P XA YA ZA A P A PBORG BORG ABA B A PPRP += Replace by the more appealing equation: = 110001 PPRP B BORG AA B A A “1” added here A row added here
- 14. Homogeneous Coords • Homogeneous coordinates: embed 3D vectors into 4D by adding a “1” • More generally, the transformation matrix T has the form: = FactorScalingTrans.Perspect. VectorTrans.MatrixRot. T
- 15. Operators: Translation, Rotation and General Transformation • Translation Operator: ( )1 12 |)|,ˆ( PQQTRANS QPP AA AAA = += 1P 2P Q
- 16. Translation Operator • Translation Operator: • Only one coordinate frame, point moves • Equivalent to mapping point to a 2nd frame • Point Forward = Frame Backwards • How does TRANS look in homogeneous coordinates?
- 17. • Rotational Operator Operators (cont.) A P1 A P2 K θ Rotation around axis:
- 18. Rotation Operator • Rotational Operator The rotation matrix can be seen as rotational operator • Takes A P1 and rotates it to A P2=R A P1 • A P2=ROT(K, q)(A P2) • Write ROT for a rotation around K
- 19. Operators (Cont.) • Transformation Operators * A transformation mapping can be viewed as a transformation operator: map a point to any other in the same frame * Transform that rotates by R and translates by Q is the same a transforming the frame by R & Q
- 20. Compound Transformation If {C} is known relative to {B}, and {B} is known relative to {A}. We want to transform P from {C} to {A}: Write down the compound in homog. coords TTT PTP PTP PTP B C A B A C BA B A CA C A CB C B = = =⇒ = definehereFrom
- 21. More on Rotations • We saw that a rotation can be represented by a rotation matrix • Matrix has 9 variables and 6+ constraints (which?) • Rotations are far from intuitive: they do not commute! • Rotation matrix can be parameterized in different manners: —Roll, pitch and yaw angles —Euler Angles —Others
- 22. Euler’s Theorem • Euler’s Theorem: Any two independent orthonormal coordinate frames can be related by a sequence of rotations (not more than three) about coordinate axes, where no two successive rotations may be about the same axis. • Not to be confused with Euler angles, Euler integration, Newton-Euler dynamics, inviscid Euler equations, Euler characteristic… • Leonard Euler (1707-1783)
- 23. XYZ Fixed
- 24. Euler Angles • This means that we can represent an orientation with 3 numbers • A sequence of rotations around principle axes is called an Euler Angle Sequence • Assuming we limit ourselves to 3 rotations without successive rotations about the same axis, we could use any of the following 12 sequences: XYZ XZY XYX XZX YXZ YZX YXY YZY ZXY ZYX ZXZ ZYZ
- 25. ZYX Euler
- 26. Euler Angles to Matrix Conversion • To build a matrix from a set of Euler angles, we just multiply a sequence of rotation matrices together: −+ +− − = yxzxzyxzxzyx yxzxzyxzxzyx yzyzy cccssscsscsc csccssssccss ssccc −⋅ − ⋅ − =⋅⋅ 100 0 0 0 010 0 0 0 001 zz zz yy yy xx xxzyx cs sc cs sc cs scRRR
- 27. Euler Angle Order • As matrix multiplication is not commutative, the order of operations is important • Rotations are assumed to be relative to fixed world axes, rather than local to the object • One can think of them as being local to the object if the sequence order is reversed • This gives us 12 redundant ways to store an orientation using Euler angles
- 28. x y z u' v' φ θ v " w" w'= =u" v'" ϕ u'" w'"= Euler Angle I, Animated
- 29. Orientation Representation • Euler Angle I − = −= − = 100 0cossin 0sincos , cossin0 sincos0 001 , 100 0cossin 0sincos '' ' ϕϕ ϕϕ θθ θθφφ φφ ϕ θφ w uz R RR
- 30. Euler Angle I − + − + − − − = θθϕθϕ θφ θϕφ ϕφ θϕφ ϕφ θϕ θϕφ ϕφ θϕφ ϕφ ϕθφ cossincossinsin sincos coscoscos sinsin cossincos cossin sinsin coscossin sincos cossinsin coscos ''' wuz RRRR Resultant eulerian rotation matrix:
- 31. Euler Angle II, Animated x y z u' v' φ θ =v" w" w'= u" v"'ϕ u"' w"'= Note the opposite (clockwise) sense of the third rotation, φ.
- 32. Orientation Representation • Matrix with Euler Angle II − −+ − − + − θ θϕθϕ θϕ θϕφ ϕφ θϕφ ϕφ θφ θϕφ ϕφ θϕφ ϕφ cos sinsinsincos sinsin coscossin coscos coscossin sincos sincos coscossin cossin coscoscos sinsin Quiz: How to get this matrix ?
- 33. Orientation Representation • Description of Roll Pitch Yaw X Y Z ϕ θ φ Quiz: How to get rotation matrix ?
- 34. Vehicle Orientation • Generally, for vehicles, it is most convenient to rotate in roll (z), pitch (x), and then yaw (y) • In situations where there is a definite ground plane, Euler angles can actually be an intuitive representation x y z front of vehicle
- 35. Euler Angles - Summary • Euler angles are used in a lot of applications, but they tend to require some rather arbitrary decisions • They also do not interpolate in a consistent way (but this isn’t always bad) • There is no simple way to concatenate rotations • Conversion to/from a matrix requires several trigonometry operations • They are compact (requiring only 3 numbers)