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

Vectors, Matrices, Rotations

Why are we studying this? 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...

Why are we studying this? Puma 500/560

Why are we studying this?

Why are we studying this?

Representing Position: Vectors

Representing Position: vectors p = [ 2 ] y 5 (“column” vector) p 2 [ ] = p 2 5 (“row” vector) x 5 y 2 p     = p 5   x   2   z

Representing Position: vectors The “a” reference frame p 2 5 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? These are the elements of p : b ˆ y Vector length/magnitude: 2 b ˆ x 5 Definition of unit vector: You can turn an arbitrary vector p into a unit vector of the same direction this way:

And what does orthogonal mean? ⋅ = + a b a b a b First, define the dot product: x x y y = cos( θ a b ) b ⋅ b = = a 0 a 0 when: = or, b 0 θ ( ) ˆ a θ = cos 0 or, Unit vectors are orthogonal iff (if and only if) the dot product is zero: is orthogonal to iff

A couple of other random things b ˆ y n R Vectors are elements of 2 b ˆ b x 5 y y z x x z right-handed left-handed coordinate frame coordinate frame

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 object have different orientations • We want to represent and difference orientations just like we did for positions…

Before we go there – review of matrix transpose a a a a a a     11 12 13 11 21 31     = A a a a T = A a a a   21 22 23   12 22 32   a a a     a a a   31 32 33 13 23 33 a a a   11 12 13   a a a   21 22 23   a a a   31 32 33 5   ( ) T T T [ ] = A B BA Important property: = 2 T p = p 5 2    

and matrix multiplication… a a   b b   11 12 = A 11 12 = B     a a b b     21 22 21 22 + + a a b b a b a b a b a b       11 12 11 12 11 11 12 21 11 12 12 22 = = AB       + + a a b b a b a b a b a b       21 22 21 22 21 11 22 21 21 12 22 22 Can represent dot product as a matrix multiply: b   [ ] x T ⋅ = + = = a b a b a b a a a b   x x y y x y b   y

Same point - different reference frames a ˆ y b ˆ y 3 . 8 p 2 a ˆ x 5 3 . 8 b ˆ x 5   3 . 8   a = 2 p b = p     3 . 8    

Another important use of the dot product: projection b θ ˆ a l = ˆ ⋅ = ˆ θ = θ l a b a b cos( ) b cos( )

Another important use of the dot product: projection b Another way of writing the dot product θ ˆ a l = ˆ ⋅ = ˆ θ = θ l a b a b cos( ) b cos( )

Same point - different reference frames a y a y ˆ B-frame’s y axis written b in A frame 3 . 8 p 2 a x 5 3 . 8 a x ˆ b B-frame’s x axis written in A frame

Same point - different reference frames a y a y ˆ B-frame’s y axis written b in A frame 3 . 8 a p 2 a x θ 5 3 . 8 a x ˆ b B-frame’s x axis written in A frame

Same point - different reference frames a y a y ˆ B-frame’s y axis written b in A frame 3 . 8 a p 2 a x 5 3 . 8 a x ˆ b B-frame’s x axis written in A frame

Same point - different reference frames A y A ˆ where: y B 3 . 8 or p 2 A x 5 3 . 8 A x ˆ B

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

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

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 The same matrix can be understood both ways! But: So, the rows are orthogonal too!

Example 1: rotation matrix a ˆ y b ˆ y b ˆ x θ a ˆ θ x ( ) θ cos   ( ) ( ) θ − θ  cos sin  ( ) a x ˆ b =    )  ( a a a   = ˆ ˆ = R x y θ sin  )  ( ) (   b b b θ θ sin cos   ( ) ( ) ( ) θ θ cos sin − θ   sin  a y   b R ˆ b = =    )   )  ( ( ) ( θ a cos − θ θ sin cos    

Example 2: rotation matrix ( ) A A A A = ˆ ˆ ˆ R x y z A y B B B B       0   1 1       2   2 A R =  −  0 1 0       B z B           − 0 1 1   A     x   2 2  45   0 1 1   A z 2 2 B x A R = − 0 1 0   B y B   − 0 1 1   2 2

Example 3: rotation matrix ˆ z ˆ x a b ˆ z b ˆ y ˆ y b a φ θ ˆ x a     − − − c c s c c c c s c s     θ φ θ θ θ φ θ θ φ π φ + 2   a = = − R c s c c s c  s c c s s  θ φ θ θ θ φ θ θ φ π φ +     2   s 0 s s 0 c     φ φ φ π φ + 2

Rotations about x, y, z ( ) ( ) α − α cos sin 0     ( ) ( ) ( ) α = α α R sin cos 0   z   0 0 1   ( ) ( ) β β cos 0 sin     ( ) β = R 0 1 0   y  )  ( ) ( − β β sin 0 cos   1 0 0     ( ) ( ) ( ) γ = γ − γ R 0 cos sin   x  ( )  ( ) γ γ 0 sin cos   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… ( ) ( ) ( ) ( ) ( ) θ ± φ = θ φ ± θ φ sin sin cos cos sin ( ) ( ) ( ) ( ) ( ) θ ± φ = θ φ  θ φ cos cos cos sin sin

Example 1: composition of rotation matrices a ˆ y b ˆ y c ˆ y θ θ 1 2 p a ˆ x A A B R = R R C B C b ˆ x c ˆ x ( ) ( ) ( ) ( ) θ − θ θ − θ − − −  cos sin   cos sin   c c s s c s s c  1 1 2 2 1 2 1 2 1 2 1 2 a       = = R c       ( ) ( ) ( ) ( ) θ θ θ θ + − sin cos sin cos s c c s c c s s       1 1 2 2 1 2 1 2 1 2 1 2 −  c s  12 12 =     s c   12 12

Example 2: composition of rotation matrices ˆ z ˆ x a c ˆ y a φ ˆ x b θ ˆ x a − c s 0      −  c 0 s c 0 s θ θ       − φ − φ φ φ a = R b s c 0 b   = = R c 0 1 0 0 1 0     θ θ       0 0 1   − s 0 c s 0 c     − φ − φ φ φ

Example 2: composition of rotation matrices ˆ z ˆ x a c ˆ y a φ ˆ x b θ ˆ x a     − − − − c s 0 c 0 s c c s c s       θ θ φ φ θ φ θ θ φ   a a b = = = − R R R s c 0  0 1 0   s c c s s    c b c θ θ θ φ θ θ φ       0 0 1 s 0 c s 0 c       φ φ φ φ