Kinematics: Pose position and orientation of a Rigid Body

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Kinematics Pose (position and orientation) of a
Rigid Body
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Notes

• The syllabus is on the course website at
  owlnet.rice.edu/mech498
• If you have questions, the best way to reach me
  is by email at robert (dot) platt-1 (at) nasa.gov
• Office hours are by appointment Ill probably
  want to schedule a meeting time before class.

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Why are we studying pose?

• You want to put your hand on the cup
• Suppose your eyes tell you where the mug is and
  its orientation in the robot base frame (big
  assumption)
• In order to put your hand on the object, you want
  to align the coordinate frame of your hand w/
  that of the object
• This kind of problem makes representation of pose
  important...

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Representing Position Vectors
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Representing Position vectors
(column vector)
(row vector)
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Representing Position vectors

• Vectors are a way to transform between two
  different reference frames w/ the same
  orientation
• The prefix superscript denotes the reference
  frame in which the vector should be understood

Same point, two different reference frames
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Representing Position vectors

• Note that I am denoting the axes as orthogonal
  unit basis vectors

This means perpendicular
A vector of length one pointing in the direction
of the base frame x axis
y axis
p frame y axis
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What is this unit vector you speak of?
These are the elements of a
Vector length/magnitude
Definition of unit vector
You can turn a into a unit vector of the same
direction this way
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And what does orthogonal mean?
First, define the dot product
when
or,
or,
Unit vectors are orthogonal iff (if and only if)
the dot product is zero
is orthogonal to
iff
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A couple of other random things
Vectors are elements of
right-handed coordinate frame
left-handed coordinate frame
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The importance of differencing two vectors
The eff needs to make a Cartesian displacement of
this much to reach the object
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The importance of differencing two vectors
The eff needs to make a Cartesian displacement of
this much to reach the object
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Representing Orientation Rotation Matrices

• The reference frame of the hand and the object
  have different orientations
• We want to represent and difference orientations
  just like we did for positions

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Before we go there review of matrix transpose
Important property
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and matrix multiplication
Can represent dot product as a matrix multiply
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Same point - different reference frames

• for the moment, assume that there is no
  difference in position

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Another important use of the dot product
projection
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Same point - different reference frames
B-frames y axis written in A frame
B-frames x axis written in A frame
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Same point - different reference frames
B-frames y axis written in A frame
B-frames x axis written in A frame
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Same point - different reference frames
B-frames y axis written in A frame
B-frames x axis written in A frame
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Same point - different reference frames
Rotation matrix
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The rotation matrix
From last page
By the same reasoning
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The rotation matrix
and

• The rotation matrix can be understood as
• Columns of vectors of B in A reference frame, OR
• Rows of column vectors A in B reference frame

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The rotation matrix
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Example 1 rotation matrix
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Example 2 rotation matrix
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Example 3 rotation matrix
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Rotations about x, y, z
These rotation matrices encode the basis vectors
of the after-rotation reference frame in terms of
the before-rotation reference frame
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Remember those double-angle formulas
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Example 1 composition of rotation matrices
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Example 2 composition of rotation matrices
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Example 2 composition of rotation matrices
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Recap of rotation matrices