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# CS I400/B659: Intelligent Robotics - PowerPoint PPT Presentation

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

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### CS I400/B659: Intelligent Robotics

Transformations and Matrix Algebra

• Principles, Ch. 3.5-8

q2

q1

Articulated Robot

• Robot: usually a rigid articulated structure

• Geometric CAD models, relative to reference frames

• A configuration specifies the placement of those frames

workspace

Frame T0

robot

reference direction

q

ty

reference point

tx

Robot R0R2 given in reference frame T0

Located at configuration q = (tx,ty,q) with q [0,2p)

workspace

Frame T0

robot

reference direction

P

P

q

ty

reference point

tx

Robot R0R2 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?

• 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)

-sin q

• X axis of T0 gets coords, Y axis gets

cos q

cos q

q

sin q

-sin q

• X axis of T0 gets coords, Y axis gets

• gets rotated to coords

cos q

cos q

q

sin q

px

py

-sin q

• X axis of T0 gets coords, Y axis gets

• gets rotated to coords

cos q

cos q

q

sin q

pxcosq -pysinq

px

pxsinq +pycosq

py

• 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)

• In 2D:

• In 3D:

• In n-D:

• Key properties:

• Symmetric

• 0 only if and are perpendicular (orthogonal)

• In 2D:

• In 3D:

• In n-D:

• Key properties:

• Symmetric

• 0 only if and are perpendicular (orthogonal)

If is a unit vector () then is the length of the projection of onto .

• In 2D:

• In 3D:

• In n-D:

• Key properties:

• Symmetric

• 0 only if and are perpendicular (orthogonal)

If and are unit vectors with inner angle then =

• For any P=(px,py) rotated by any q, we have the new coordinates

• We can express this as a matrix-vector product:

• 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

• 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

• 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

• A has dimensions m x n, has m entries, has n entries

• for each i=1,…,m

• 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 ?

• 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 ?

• 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 ?

• 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 ?

• somust be 2x2

• somust be 2x2

Row 1

Column 1

Entry (1,1)

• somust be 2x2

Row 1

Column 2

Entry (1,2)

• somust be 2x2

Column 1

Entry (2,1)

Row 2

• somust be 2x2

Column 2

Entry (2,2)

Row 2

• somust be 2x2

• somust be 2x2

• Verify that

• somust be 2x2

• somust be 2x2

• somust be 2x2

• So,

• 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

• An nxnidentity matrix has 1’s on its diagonals and 0s everywhere else

• for all vectors

• for all nxm matrices

• for all mxnmatrices

• 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 mxnmatrix is the nxm matrix formed swapping its rows and columns. It is denoted .

• i.e.,

• Since…

• (the identity matrix)

• But

• …so a rotation matrix’s inverse is its transpose.

q = (tx,ty,q) with q [0,2p)

Robot R0R2 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

Rigid Transformation in 2D

cos θ -sin θ

sin θ cos θ

x

y

tx

ty

Tq(x,y) =

+

2D rotation matrix

Affine translation

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!

Note: transforming points vs directional quantities

cos θ -sin θ

sin θ cos θ

x

y

tx

ty

Tq(x,y) =

+

cos θ -sin θ

sin θ cos θ

vx

vy

Rq(vx,vy) =

• Optional: A Mathematical Introduction to Robotic Manipulation, Ch. 2.1-3

• http://www.cds.caltech.edu/~murray/mlswiki/?title=First_edition