Coordinate Frames

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Coordinate Frames Transformations
CSE398/498 04 Feb 05
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Administration

• Todays lab is cancelled
• Lab sessions will resume (most likely) next
  Monday
• We will be migrating (most likely) to Linux
• I will make the lab available for teams to make
  up these hours

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Robot Safety

• Know where your dog is at all times. If you
  tread on its ankle while stepping backwards, it
  will hurt.

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References

• Essential Kinematics for Autonomous Vehicles,
  Alonzo Kelly, TR CMU-RI-TR-94-14,
• http//www.frc.ri.cmu.edu/alonzo/pubs/reports/pdf
  _files/kinematics.pdf

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Background

• Ultimately, each Aibo will need to estimate the
  relative position of items of interest (the ball,
  the goal, teammates, etc.) from a camera image
• Let us assume that using its camera alone, the
  Aibo is able to infer the relative distance to
  the ball
• How could it do this???
• The Aibos head has 2 degrees of freedom
  (ignoring the lower neck joint for now)
• It can pan left and right (yaw)
• It can tilt up and down (pitch)

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Todays Question

• The Aibo sees the soccer ball in the center of
  the camera image
• The ball is estimated to be 1 meter away
• Its head is panned to the left 45 degrees
• Its head is tilted down 20 degrees
• Q Where is the ball?
• A Relative to what?

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Defining a Reference Frame

• Prior to any position estimation, the coordinate
  frame that we are measuring relative to must be
  defined
• Q1 Where is the ball?
• A1 Relative to what?
• Q2 OK Where is the ball relative to the
  reference frame below?

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Defining a Reference Frame (contd)

• A2 1 a nose, 0, 0T
• OK, maybe that wasnt such a good choice
• Lets try a body fixed frame

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Defining a Reference Frame (contd)

• Or better still, why not define several relevant
  coordinate frames

Body Frame
Sensor Head Frame
Sensor (Camera) Frame
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Todays Question

• The Aibo sees the soccer ball in the center of
  the camera image
• The ball is estimated to be 1 meter away from the
  camera frame
• Its head is tilted up 20 degrees
• Its head is panned to the left 45 degrees
• Its neck is tilted down 30 degrees
• Q What is the balls position in the body
  frame?
• A Lets look at coordinate transformations
  first

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Coordinate Transformations

• A Coordinate Transformation relates the position
  vector of any point in coordinate frame 2 to the
  same point in coordinate frame 1
• We are interested in Rigid Transformations which
  reflect the relative position and orientation of
  1 coordinate frame with respect to another
• Here ix denotes the position vector for point x
  as viewed from coordinate frame i, iRj denotes a
  rotation matrix which describes the rotation
  necessary to align the axis of coordinate frame i
  to j, and itj the translation from the origin of
  frame i to j
• The rigid transformation necessary to align
  coordinate frame i with j has the opposite effect
  of translating points from frame j to frame i

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Rotation Matrices

• A rotation matrix R rotates position vectors in
  reference frame 2 (F2) to position vectors in F1
• In two dimensions
• where ? corresponds to the relative difference
  in orientation of F2 with respect to F1
• In this definition, R transforms a position
  vector in F2 to how the corresponding position
  vector would appear in frame F1.

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Coordinate Transformationfrom Pure Rotation
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Coordinate Transformationfrom Pure Rotation
Example
Example Lets say that a point in frame F2 is
1,0T , and ?30o. What are the points
coordinates in frame F1?
p
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Some Properties of Rotation Matrices

• det(R) 1, RTR-1
• RI denotes no rotation
• The product of 2 rotation matrices is a rotation
  matrix
• In three dimensions
• where xyz rotations correspond to roll, pitch,
  and yaw

NOTE Order is important, as ABC ? ACB in
general
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Pure Translations

• A pure translation corresponds to a
  transformation when RI. The equation for a
  rigid transformation then reduces to

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Pure Translations
Example Lets say that a point in frame F2 is
4,3T , t10,5T and ?0o. What are the
points coordinates in frame F1?
2p
1p
1t2
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General Rigid Transformation
1p
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Homogeneous Coordinates

• There is no nice way to represent translations
  using a 2x2 matrix in R2 (or 3x3 matrix in R3)
• Homogeneous Coordinates provide a convenient
  means for representing and composing multiple
  rigid transformations
• The dimension of each coordinate is increased by
  1

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Homogeneous Coordinates

• There is no nice way to represent translations
  using a 2x2 matrix in R2 (or 3x3 matrix in R3)
• Homogeneous Coordinates provide a convenient
  means for representing and composing multiple
  rigid transformations
• The dimension of each coordinate is increased by
  1
• Last coordinate normally normalized to 1

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Homogeneous Coordinates (contd)

• An important property of homogenous coordinates
  is that two are equivalent if they are a scalar
  multiple of one another
• HC also provides a convenient representation for
  points at infinity

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Homogeneous Transformation Matrices

• With this representation, we can compose a rigid
  transformation into a single, linear
  transformation matrix.
• In two dimensions, the representation is as
  follows
• Note that this is a rotation followed by a
  translation, which is NOT the same as the reverse!

1t2
1R2
constant
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Homogeneous Transformation Matrices in Three
Dimensions

• The representation readily extends to three
  dimensions

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Pop Quiz ?

• The distance from the camera in the dogs nose to
  the origin of the head frame is 8 cm, as is the
  body frame from the head frame. If only the neck
  joint is tilted by an angle f, write the
  homogenous transform relating the nose position
  in the head frame with the body frame.

Nose
Neck Joint
Head Frame
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Composing Homogenous Transformations

• Perhaps the strongest point for the homogeneous
  transformation representation is the ability to
  compose multiple transformations across multiple
  frames
• Suppose we would like to estimate the position of
  a point that has seen coordinate transformations
  across 2 frames

3p
1A2
3y
3x
2A3
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Composing Homogenous Transformations
2p
3p
1A2
3y
1p
3x
2A3

• This generalizes for n frames
• NOTE The transformations are done LOCAL to the
  current frame

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Todays Homework

• The Aibo sees the soccer ball in the center of
  the camera image
• The ball is estimated to be 1 meter away from the
  camera frame
• Its head is tilted up 20 degrees
• Its head is panned to the left 45 degrees
• Its neck is tilted down 30 degrees
• Q1 What are the necessary homogeneous
  transformations to calculate this?
• Q2 What is the balls position in the body
  frame?

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Summary

• Points are defined with respect to a specific
  coordinate frame
• Often, it is convenient to measure a point with
  respect to one frame (e.g. an objects position
  in the sensor frame), but it must be transformed
  to another frame for other reasons (e.g.
  navigational convenience)
• Coordinate transformations provide this mechanism
• The transformation necessary to align coordinate
  frame F1 with frame F2 is also the same
  transformation necessary to convert points from
  frame F2 to frame F1
• Homogeneous coordinates provide a convenient
  means for representing and composing rigid
  transformations