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