Vectors, Matrices, Rotations Why are we studying this? You want to - - PowerPoint PPT Presentation

Vectors, Matrices, Rotations

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
• f pose important...

Why are we studying this?

Why are we studying this?

Puma 500/560

Why are we studying this?

Why are we studying this?

Representing Position: Vectors

Representing Position: vectors

x y p 5 2

p=[ 5 2]

(“column” vector)

[ ]

5 2 = p

(“row” vector)

x y z

          = 2 5 2 p

p

Representing Position: vectors

p 5 2

The “a” reference frame Basis vectors – unit vectors (length of magnitude 1) – orthogonal (perpendicular to each other) Vector p in written in a reference frame

What is this unit vector you speak of?

Vector length/magnitude: Definition of unit vector: These are the elements of p:

5 2

x

b ˆ

y

b ˆ

You can turn an arbitrary vector p into a unit vector of the same direction this way:

y y x x

b a b a b a + = ⋅ ) cos(θ b a =

And what does orthogonal mean?

Unit vectors are orthogonal iff (if and

• nly if) the dot product is zero:

is orthogonal to iff First, define the dot product:

= ⋅b a

when:

= a = b

( )

cos = θ

• r,
• r,

a ˆ b

θ

A couple of other random things

5 2 b

x

b ˆ

y

b ˆ

x y z x y z

right-handed coordinate frame left-handed coordinate frame Vectors are elements of

n

R

The importance of differencing two vectors

The hand needs to make a Cartesian displacement of this much to reach the object

The importance of differencing two vectors

b

The hand needs to make a Cartesian displacement of this much to reach the object

Representing Orientation: Rotation Matrices

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

positions…

          =

33 32 31 23 22 21 13 12 11

a a a a a a a a a A

          =

33 23 13 32 22 12 31 21 11

a a a a a a a a a

T

A          

33 32 31 23 22 21 13 12 11

a a a a a a a a a

      = 2 5 p

[ ]

2 5 =

T

p

Before we go there – review of matrix transpose

( )

T T T

BA B A =

Important property:

      =

22 21 12 11

a a a a A       =

22 21 12 11

b b b b B

and matrix multiplication…

      + + + + =             =

22 22 12 21 21 22 11 21 22 12 12 11 21 12 11 11 22 21 12 11 22 21 12 11

b a b a b a b a b a b a b a b a b b b b a a a a AB

[ ]

b a b b a a b a b a b a

T y x y x y y x x

=       = + = ⋅

Can represent dot product as a matrix multiply:

Same point - different reference frames

x

a ˆ

p 5 2

y

a ˆ

y

b ˆ

x

b ˆ

8 . 3 8 . 3

      = 2 5 p

a

      = 8 . 3 8 . 3 p

b

a ˆ b ) cos( ) cos( ˆ ˆ θ θ b b a b a l = = ⋅ =

θ

Another important use of the dot product: projection

l

a ˆ b ) cos( ) cos( ˆ ˆ θ θ b b a b a l = = ⋅ =

θ

Another important use of the dot product: projection

l

Another way of writing the dot product

p 5 2 8 . 3 8 . 3

B-frame’s x axis written in A frame B-frame’s y axis written in A frame

Same point - different reference frames

x

a

y

a b ax

ˆ

b a y

ˆ

x

a

p

a

5 2

y

a

8 . 3 8 . 3

b ax

ˆ

b a y

ˆ

B-frame’s y axis written in A frame

Same point - different reference frames

θ

B-frame’s x axis written in A frame

5 2 8 . 3 8 . 3

B-frame’s y axis written in A frame

Same point - different reference frames

x

a

p

a

y

a b ax

ˆ

b a y

ˆ

B-frame’s x axis written in A frame

x

A

p 5 2

y

A

8 . 3 8 . 3

B Ax

ˆ

B A y

ˆ

Same point - different reference frames

where:

• r

The rotation matrix

To recap: where:

The rotation matrix

To recap: where: We will write: so: Notice the way the notation “cancels out” But, can we do this: ???

The rotation matrix

Multiply both sides by inverse: But, can we do this: ??? It turns out that: because the columns of are unit, orthogonal

The rotation matrix

Multiply both sides by inverse: But, can we do this: ??? It turns out that: because the columns of are orthogonal This is important!

The rotation matrix

So, if: Then:

The rotation matrix

Both columns are orthogonal But: So, the rows are orthogonal too!

The rotation matrix

Both columns are orthogonal But: So, the rows are orthogonal too!

The same matrix can be understood both ways!

Example 1: rotation matrix

x

a ˆ

y

a ˆ

x

b ˆ

y

b ˆ

θ θ

( )

( ) ( ) ( ) ( ) 

       − = = θ θ θ θ cos sin sin cos ˆ ˆ

b a b a b a

y x R

( ) ( ) 

       = θ θ sin cos ˆb

ax

( ) ( ) 

      − = θ θ cos sin ˆb

a y

( ) ( ) ( ) ( )

       − = θ θ θ θ cos sin sin cos

a bR

Example 2: rotation matrix

y

A

z

A

x

A



45

z

B

x

B

y

B

( )

B A B A B A B A

z y x R ˆ ˆ ˆ =                       −           −           =

2 1 2 1 2 1 2 1

1

B AR

          − − =

2 1 2 1 2 1 2 1

1

B AR

Example 3: rotation matrix

a

x ˆ

a

y ˆ

θ φ

a

z ˆ

b

z ˆ

b

x ˆ

b

y ˆ

          − − − =             − =

+ + + φ φ φ θ θ φ θ φ θ θ φ θ φ φ φ θ θ φ θ φ θ θ φ θ

π π π

c s s s c c s s c s c c s s c s c c s c c s c c Rc

a

2 2 2

Rotations about x, y, z

( ) ( ) ( ) ( ) ( )

          − = 1 cos sin sin cos α α α α α

z

R

( ) ( ) ( ) ( ) ( )

         − = β β β β β cos sin 1 sin cos

y

R

( ) ( ) ( ) ( ) ( ) 

         − = γ γ γ γ γ cos sin sin cos 1

x

R

These rotation matrices encode the basis vectors of the after- rotation reference frame in terms of the before-rotation reference frame

Remember those double-angle formulas…

( ) ( ) ( ) ( ) ( )

φ θ φ θ φ θ sin cos cos sin sin ± = ±

( ) ( ) ( ) ( ) ( )

φ θ φ θ φ θ sin sin cos cos cos  = ±

Example 1: composition of rotation matrices

C B B A C A

R R R =

p

y

a ˆ

x

b ˆ

y

b ˆ

x

a ˆ

y

c ˆ

x

c ˆ 1

θ

2

θ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

        − + − − − =         −         − =

2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2 2 2 1 1 1 1

cos sin sin cos cos sin sin cos s s c c s c c s c s s c s s c c Rc

a

θ θ θ θ θ θ θ θ         − =

12 12 12 12

c s s c

Example 2: composition of rotation matrices

a

x ˆ

a

y ˆ

θ φ

a

z ˆ

b

x ˆ

c

x ˆ

          − = 1

θ θ θ θ

c s s c Rb

a

          − =           − =

− − − − φ φ φ φ φ φ φ φ

c s s c c s s c Rc

b

1 1

Example 2: composition of rotation matrices

a

x ˆ

a

y ˆ

θ φ

a

z ˆ

b

x ˆ

c

x ˆ

          − − − =           −           − = =

φ φ φ θ θ φ θ φ θ θ φ θ φ φ φ φ θ θ θ θ

c s s s c c s s c s c c c s s c c s s c R R R

c b b a c a

1 1