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CS I400/B659: Intelligent Robotics. Transformations and Matrix Algebra. Agenda. Principles, Ch. 3.5-8. Rigid Objects. Biological systems, virtual characters. q 2. q 1. Articulated Robot. Robot: usually a rigid articulated structure Geometric CAD models, relative to reference frames - PowerPoint PPT Presentation
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CS I400/B659: Intelligent RoboticsTransformations and Matrix Algebra
Agenda• Principles, Ch. 3.5-8
Rigid Objects
Biological systems, virtual characters
Articulated Robot
• Robot: usually a rigid articulated structure
• Geometric CAD models, relative to reference frames
• A configuration specifies the placement of those frames
q1
q2
reference point
Rigid Transformation in 2D
• Robot R0R2 given in reference frame T0
• Located at configuration q = (tx,ty,q) with q [0,2p)
tx
tyq
robot reference direction
workspaceFrame T0
reference point
Rigid Transformation in 2D
• Robot R0R2 given in reference frame T0
• Located at configuration q = (tx,ty,q) with q [0,2p)
• Point P on the robot (e.g., a camera) has coordinates in frame T0.• What are the coordinates of P in the workspace?
tx
tyq
robot reference direction
workspaceFrame T0
PP
Rigid Transformation in 2D
• Robot at configuration q = (tx,ty,q) with q [0,2p)• Point P on the robot (e.g., a camera) has coordinates in
frame T0.• What are the coordinates of P in the workspace?
• Think of 2 steps: 1) rotating about the origin point by angle q, then 2) translating the reference point to (tx,ty)
• X axis of T0 gets coords , Y axis gets
Rotations in 2D
q
cos q
sin q
-sin q
cos q
• X axis of T0 gets coords , Y axis gets
• gets rotated to coords
Rotations in 2D
q
cos q
sin q
-sin q
cos q
px
py 𝑝 𝑦 ⋅ �⃗� 𝑝𝑥 ⋅ �⃗�
• X axis of T0 gets coords , Y axis gets
• gets rotated to coords
Rotations in 2D
q
cos q
sin q
pxcos q -pysin q
cos q
px
py 𝑝 𝑦 ⋅ �⃗� 𝑝𝑥 ⋅ �⃗�pxsin q +pycos q
-sin q
Dot product
• For any P=(px,py) rotated by any q, we have the new coordinates
• We can express each element as a dot product:
• Definition: • In 3D, • Key properties:
• Symmetric• 0 only if and are perpendicular (orthogonal)
Properties of the dot product• In 2D: • In 3D: • In n-D:• Key properties:
• Symmetric• 0 only if and are perpendicular (orthogonal)
Properties of the dot product• In 2D: • In 3D: • In n-D:• Key properties:
• Symmetric• 0 only if and are perpendicular (orthogonal)
𝑣
�⃗� If is a unit vector () then is the length of the projection of onto .
Properties of the dot product• In 2D: • In 3D: • In n-D:• Key properties:
• Symmetric• 0 only if and are perpendicular (orthogonal)
𝑦
�⃗� If and are unit vectors with inner angle then =
𝜃
cos𝜃
Matrix-vector multiplication
• For any P=(px,py) rotated by any q, we have the new coordinates
• We can express this as a matrix-vector product:
Matrix-vector multiplication
• For any P=(px,py) rotated by any q, we have the new coordinates
• We can express this as a matrix-vector product:
• Or, for A a 2x2 table of numbers• Each entry of is the dot product between the corresponding
row of A and
Matrix-vector multiplication
• For any P=(px,py) rotated by any q, we have the new coordinates
• We can express this as a matrix-vector product:
• Or, for A a 2x2 table of numbers• Each entry of is the dot product between the corresponding
row of A and
Matrix-vector multiplication
• For any P=(px,py) rotated by any q, we have the new coordinates
• We can express this as a matrix-vector product:
• Or, for A a 2x2 table of numbers• Each entry of is the dot product between the corresponding
row of A and
Matrix-vector product examples
General equations
• A has dimensions m x n, has m entries, has n entries• for each i=1,…,m
Matrix-vector product examples
Multiple rotations• Define the 2D rotation matrix • We know that the new coordinates of a point rotated by is
given by • What if we rotate again by ? What are the new coordinates ?
Multiple rotations• Define the 2D rotation matrix • We know that the new coordinates of a point rotated by is
given by • What if we rotate again by ? What are the new coordinates ?
Multiple rotations• Define the 2D rotation matrix • We know that the new coordinates of a point rotated by is
given by • What if we rotate again by ? What are the new coordinates ?
Multiple rotations• Define the 2D rotation matrix • We know that the new coordinates of a point rotated by is
given by • What if we rotate again by ? What are the new coordinates ?
Is it possible to define matrix-matrix multiplication so that ?
Matrix-matrix multiplication• so must be 2x2
Matrix-matrix multiplication• so must be 2x2
Entry (1,1) Row 1 Column 1
Matrix-matrix multiplication• so must be 2x2
Entry (1,2) Row 1 Column 2
Matrix-matrix multiplication• so must be 2x2
Entry (2,1) Row 2
Column 1
Matrix-matrix multiplication• so must be 2x2
Entry (2,2) Row 2
Column 2
Matrix-matrix multiplication• so must be 2x2
Matrix-matrix multiplication• so must be 2x2
• Verify that
Rotation matrix-matrix multiplication• so must be 2x2
Rotation matrix-matrix multiplication• so must be 2x2
Rotation matrix-matrix multiplication• so must be 2x2
• So,
General definition• If A and B are m x p and p x n matrices, respectively, then the
matrix-matrix product is given by the m x n matrix C with entries
Other Fun Facts• An nxn identity matrix has 1’s on its diagonals and 0s
everywhere else• for all vectors • for all nxm matrices • for all mxn matrices
• If A and B are square matrices such that , then B is called the inverse of A (and A is the inverse of B)• Not all matrices are invertible
• The transpose of a matrix mxn matrix is the nxm matrix formed swapping its rows and columns. It is denoted .• i.e.,
Consequence: rotation inverse• Since …• (the identity matrix)
• But • …so a rotation matrix’s inverse is its transpose.
Rigid Transformation in 2D• q = (tx,ty,q) with q [0,2p)
• Robot R0R2 given in reference frame T0
• What’s the new robot Rq? {Tq(x,y) | (x,y) R0}
• Define rigid transformation Tq(x,y) : R2 R2
Tq(x,y) = cos θ -sin θsin θ cos θ
xy
tx
ty+
2D rotation matrix Affine translation
Note: transforming points vs directional quantities• Rigid transform q = (tx,ty,q)
• A point with coordinates (x,y) in T0 undergoes rotation and affine translation
• Directional quantities (e.g., velocity, force) are not affected by the affine translation!
Tq(x,y) = cos θ -sin θsin θ cos θ
xy
tx
ty+
Rq(vx,vy) = cos θ -sin θsin θ cos θ
vx
vy
Next Lecture• Optional: A Mathematical Introduction to Robotic
Manipulation, Ch. 2.1-3• http://www.cds.caltech.edu/~murray/mlswiki/?title=First_edition