; Copyright © 1999, 2000, 2001 rht+sg Introduction to Vectors and Frames CIS Russell Taylor Sarah Graham

; Copyright © 1999, 2000, 2001 rht+sg x x x CT image Planned hole Pins Femur Tool path COMMON NOTATION: Use the notation F obj to represent a coordinate system or the position and orientation of an object (relative to some unspecified coordinate system). Use F x,y to mean position and orientation of y relative to x.

; Copyright © 1999, 2000, 2001 rht+sg x x x CT image Planned hole Pins Femur Tool path

; Copyright © 1999, 2000, 2001 rht+sg CT image Pin 1Pin 2Pin 3 Planned hole Tool path Femur Assume equal x x x

; Copyright © 1999, 2000, 2001 rht+sg

x x x x x x

Base of robot CT image Pin 1Pin 2Pin 3 Planned hole Tool holder Tool tipTool path Femur Assume equal Can calibrate (assume known for now) Can control Want these to be equal

; Copyright © 1999, 2000, 2001 rht+sg Base of robot Tool holder Tool tipTarget

; Copyright © 1999, 2000, 2001 rht+sg CT image Pin 1Pin 2Pin 3 Planned hole Tool path Femur Assume equal

; Copyright © 1999, 2000, 2001 rht+sg CT image Pin 1Pin 2Pin 3 Tool path Base of robot Tool holder Tool tip

; Copyright © 1999, 2000, 2001 rht+sg CT image Pin 1Pin 2Pin 3 Tool path Base of robot Tool holder Tool tip But: We must find F CT … Let’s review some math

; Copyright © 1999, 2000, 2001 rht+sg x0x0 y0y0 z0z0 x1x1 y1y1 z1z1 F Coordinate Frame Transformation

; Copyright © 1999, 2000, 2001 rht+sg

b F = [R,p]

; Copyright © 1999, 2000, 2001 rht+sg b F = [R,p]

; Copyright © 1999, 2000, 2001 rht+sg b F = [ I,0]

; Copyright © 1999, 2000, 2001 rht+sg b F = [R,0]

; Copyright © 1999, 2000, 2001 rht+sg b F = [R,p]

; Copyright © 1999, 2000, 2001 rht+sg Coordinate Frames b F = [R,p]

; Copyright © 1999, 2000, 2001 rht+sg

Forward and Inverse Frame Transformations Forward Inverse

; Copyright © 1999, 2000, 2001 rht+sg Composition

; Copyright © 1999, 2000, 2001 rht+sg Vectors v x y z w vw u = v x w

; Copyright © 1999, 2000, 2001 rht+sg Vectors as Displacements v z w v+w x y v w v-w x y w

; Copyright © 1999, 2000, 2001 rht+sg Vectors as Displacements Between Parallel Frames v0v0 x0x0 y0y0 z0z0 x1x1 y1y1 z1z1 v1v1 w

; Copyright © 1999, 2000, 2001 rht+sg Rotations: Some Notation

; Copyright © 1999, 2000, 2001 rht+sg Rotations: A few useful facts

; Copyright © 1999, 2000, 2001 rht+sg Rotations: more facts

; Copyright © 1999, 2000, 2001 rht+sg Rotations in the plane

; Copyright © 1999, 2000, 2001 rht+sg Rotations in the plane

; Copyright © 1999, 2000, 2001 rht+sg 3D Rotation Matrices

; Copyright © 1999, 2000, 2001 rht+sg Inverse of a Rotation Matrix equals its transpose: R -1 = R T R T R=R R T = I The Determinant of a Rotation matrix is equal to +1: det(R)= +1 Any Rotation can be described by consecutive rotations about the three primary axes, x, y, and z: R = R z,  R y,  R x,  Properties of Rotation Matrices

; Copyright © 1999, 2000, 2001 rht+sg Canonical 3D Rotation Matrices Note: Right-Handed Coordinate System

; Copyright © 1999, 2000, 2001 rht+sg Homogeneous Coordinates Widely used in graphics, geometric calculations Represent 3D vector as 4D quantity For our purposes, we will keep the “scale” s = 1

; Copyright © 1999, 2000, 2001 rht+sg Representing Frame Transformations as Matrices

; Copyright © 1999, 2000, 2001 rht+sg x x x x x x

CT image Pin 1Pin 2Pin 3 Base of robot Tool holder Tool tip

; Copyright © 1999, 2000, 2001 rht+sg CT image Pin 1Pin 2Pin 3 Base of robot Tool holder Tool tip

; Copyright © 1999, 2000, 2001 rht+sg CT image Pin 1Pin 2Pin 3 Base of robot Tool holder Tool tip

; Copyright © 1999, 2000, 2001 rht+sg Frame transformation from 3 point pairs x x x x x x

; Copyright © 1999, 2000, 2001 rht+sg Frame transformation from 3 point pairs x x x

; Copyright © 1999, 2000, 2001 rht+sg Frame transformation from 3 point pairs x x x x x x x x Solve These!!

; Copyright © 1999, 2000, 2001 rht+sg Rotation from multiple vector pairs

; Copyright © 1999, 2000, 2001 rht+sg Renormalizing Rotation Matrix

; Copyright © 1999, 2000, 2001 rht+sg Calibrating a pointer b tip F ptr But what is b tip ??

; Copyright © 1999, 2000, 2001 rht+sg Calibrating a pointer b tip F ptr

; Copyright © 1999, 2000, 2001 rht+sg Calibrating a pointer b tip F ptr b tip F ptr b tip F ptr b tip F ptr

; Copyright © 1999, 2000, 2001 rht+sg Kinematic Links FkFk LkLk F k-1 kk

; Copyright © 1999, 2000, 2001 rht+sg Kinematic Links Base of robot End of link k-1 End of link k

; Copyright © 1999, 2000, 2001 rht+sg Kinematic Chains L3L3 L2L2 22 F0F0 L1L1 11 33 F1F1 F2F2 F3F3

; Copyright © 1999, 2000, 2001 rht+sg Kinematic Chains L3L3 L2L2 22 F0F0 L1L1 11 33 F3F3

; Copyright © 1999, 2000, 2001 rht+sg Kinematic Chains

; Copyright © 1999, 2000, 2001 rht+sg “Small” Frame Transformations

; Copyright © 1999, 2000, 2001 rht+sg Small Rotations

; Copyright © 1999, 2000, 2001 rht+sg Approximations to “Small” Frames

; Copyright © 1999, 2000, 2001 rht+sg Errors & sensitivity

; Copyright © 1999, 2000, 2001 rht+sg x x x F = [R,p]

; Copyright © 1999, 2000, 2001 rht+sg x x x

Errors & Sensitivity

; Copyright © 1999, 2000, 2001 rht+sg Errors & Sensitivity

; Copyright © 1999, 2000, 2001 rht+sg Digression: “rotation triple product”

; Copyright © 1999, 2000, 2001 rht+sg Errors & Sensitivity

; Copyright © 1999, 2000, 2001 rht+sg Errors & Sensitivity

; Copyright © 1999, 2000, 2001 rht+sg Error Propagation in Chains FkFk LkLk F k-1 kk

; Copyright © 1999, 2000, 2001 rht+sg Exercise L3L3 L2L2 22 F0F0 L1L1 11 33 F1F1 F2F2 F3F3

; Copyright © 1999, 2000, 2001 rht+sg Exercise L3L3 L2L2 22 F0F0 L1L1 11 33 F1F1 F2F2 F3F3

; Copyright © 1999, 2000, 2001 rht+sg Parametric Sensitivity

; Copyright © 1999, 2000, 2001 rht+sg Parametric Sensitivity Grinding this out gives: