Vectors, Matrices, Rotations Robert Platt Northeastern University - PowerPoint PPT Presentation

Vectors, Matrices, Rotations Robert Platt Northeastern University

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? Visual perception tells Joint encoders tell us head angle us object position and orientation (pose) Need to know where hand is... Need to tell the hand where to move! KIT Humanoid

Why are we studying this?

Representing Position: Vectors

Representing Position: vectors y (“column” vector) p 2 (“row” vector) x 5 y p x z

Representing Position: vectors ˆ y b • Vectors are a way to transform between ˆ y two different reference frames w/ the p same orientation p • The prefix superscript denotes the ˆ x 2 p reference frame in which the vector should be understood b ˆ x b 5 Same point, two different reference frames

Representing Position: vectors ˆ y b ˆ y p • Note that I am denoting the axes as orthogonal unit basis vectors p ˆ x 2 p This means “perpendicular” b ˆ x b 5 ˆ x A vector of length one pointing in b the direction of the base frame x axis ˆ y y axis b ˆ y p frame y axis p

What is a unit vector? These are the elements of a : b ˆ y Vector length/magnitude: 2 b ^ x 5 Definition of unit vector: How convert a non-zero vector a into a unit vector pointing in the same direction?

Orthogonal vectors    a b a b a b First, define the dot product: x x y y Under what conditions is the dot product zero? b  ˆ a

A couple of other random things b ˆ y ˆ ˆ   p 5 x 2 y b b b 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 b b b   x x x b x object eff err eff b x err b x object The eff needs to make a Cartesian displacement of this much to reach the object

The importance of differencing two vectors ˆ y object object ˆ x ˆ y object b ˆ y eff err b b b   x x x object eff err eff ˆ x eff b ˆ x b The eff 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 Question:

and matrix multiplication… Can represent dot product as a matrix multiply:

Same point - different reference frames a ˆ y b ˆ y • for the moment, assume that there is no difference in 3 . 8 position… p 2 a ˆ x 5 3 . 8 b ˆ x

Another important use of the dot product: projection    a b a b a b x x y y b  cos(  a b )  ˆ a l ˆ ˆ       l a b a b cos( ) b cos( )

Think-pair-share 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 Calculate given

Same point - different reference frames Rotation matrix Where:

Rotation matrices A rotation matrix is a 2x2 or 3x3 matrix R such that: 1. Rows and columns are unit length and 2. orthogonal Right handed coordinate frame Rotation matrix inverse equals transpose:

Rotation matrices A rotation matrix is a 2x2 or 3x3 matrix R such that: 1. Rows and columns are unit length and 2. orthogonal Right handed coordinate frame Unit vectors and orthogonal to each other

Rotation matrices A rotation matrix is a 2x2 or 3x3 matrix R such that: 1. Rows and columns are unit length and 2. orthogonal Right handed coordinate frame Unit vectors and orthogonal to each other

Rotation matrices By convention: where Similarly: Notice: Because of properties of rotation matrix

Think-pair-share Given: Calculate:

Example 1: rotation matrix a ˆ y b ˆ y b ˆ x  a ˆ  x x b = ( sin ( θ ) ) cos ( θ ) a ^ y b ) = ( cos ( θ ) ) cos ( θ ) − sin ( θ ) a ^ a ^ a R b = ( x b sin ( θ ) b R a = ( cos ( θ ) ) y b = ( cos ( θ ) ) cos ( θ ) sin ( θ ) − sin ( θ ) a ^ − sin ( θ )

Example 2: rotation matrix A y B z A x  45 A z B x B y 1. Calculate: 2. What’s the magnitude of this rotation?

Example 3: rotation matrix ˆ z ˆ x a b ˆ z b ˆ y ˆ y b a   ˆ x a a R c = ( 2 ) − s θ c θ c φ c θ c φ + π = ( c φ ) − s θ − c θ s φ c θ c φ 2 − s θ s φ s θ c φ c θ s θ c φ + π s θ c φ c θ 2 s φ 0 s φ 0 s φ + π

Rotations about x, y, z R z ( α ) = ( 1 ) cos ( α ) − sin ( α ) 0 sin ( α ) cos ( α ) 0 0 0 R y ( β ) = ( cos ( β ) ) cos ( β ) sin ( β ) 0 0 1 0 − sin ( β ) 0 R x ( γ ) = ( cos ( γ ) ) 1 0 0 cos ( γ ) − sin ( γ ) 0 sin ( γ ) 0 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…           sin     sin  cos   cos  sin                     cos cos cos sin sin