CPSC 452: Lecture 1 Introduction, Homogeneous transformations and Coordinate frames Introduction Robots in movie 2 Modern Robots Robot in life Industry Medicare 3 Modern Robots Robot in life Home/Entertainment 4 Modern Robots Robots in life Military/Unmanned Vehicle 5 What is a robot A robot is a reprogrammable multifunctional manipulator designed to
move material, parts, tools, or specialized devices through variable programmed motions for the performance of a variety of tasks by Robot Institute of America 6 Scope of CPSC 452 Planning Sensing Control Dynamics Kinematics Rigid body mechanics 7 Scope of CPSC 452 Planning Sensing Control Dynamics Kinematics Rigid body mechanics
8 Intro Space Type Physical, Geometry, Functional Dimension Direction Basis vectors Distance Norm Description Coordinate System Matrix 9 A review of vectors and matrix Vectors Column vector and row vector v1 v v 2 vn v v1 v2 vn Norm of a vector v v12 v22 ... vn2 10
Dot product of two vectors Vector v and w v w | v || w | cos If |v|=|w|=1, v w cos v w 11 Position Description Coordinate System A Z A x A A P Py z YA X A 12
Orientation Description Coordinate System A Z A A P x A Py z YA X A 13 Orientation Description Coordinate System A Attach Frame Coordinate System B x A Py Z A X B A
P Z B YB z YA X A 14 Orientation Description Coordinate System A Attach Frame Coordinate System B Rotation matrix A B R X B A r11 r12 r21 r22 r31 r32 YB
A r13 r23 r33 A Z B r31 X B Z A X A Z A A P X B r21 X B YA Z B YB YA r11 X B X A
15 Rotation matrix A B R A X B YB A A Z B X B X A YB X A X B YA YB YA YB Z A Directional X B Z A Cosines Directional Cosines
B Z B X A Z B YA Z B Z A T XA BT B Y A X A B Z AT X T A B A X B YA
B B T B T Z A A R 16 Rotation matrix A B RABR T A B R ABR I BAR ABR 1 A B T B
A R R 1 For matrix M, If M-1 = MT , M is orthogonal matrix BA R is orthogonal!! 17 Orthogonal Matrix A B R X B A YB A A Z B X B X A YB X A X B YA YB YA
X B Z A YB Z A Z B X A Z B YA Z B Z A 9 Parameters to describe orientation! 18 Description of a frame Position + orientation Z A {B} {BA R, APBORG } Z B X B A PBORG YB YA X A 19
Graphical representation { A} {Au R,u PAORG } {B} {Bu R,u PBORG } {BA R, APBORG } Z A u Z u X B {U} Yu X u {A} X A PBORG A Z B {B} PBORG YB YA 20 Mapping Translation Difference Z A A
PBORG Z B B X B A X A A P YB P APBORG BP AP A B P BT 1 1 P YA I A
BT 0 0 0 A PBORG 1 21 Mapping rotation difference A Z A Z B P Py Px X B PyYB Pz Z B A B P P Az P B B P BAPRx BP
B P YB BT A P1 P A X 1P AY x B y B Pz A Z B B YA X A X B [ A X B A BT
BAR BP 0 YB A A B R 0 A Z B ] 0 0 1 Px P y Pz
22 Example 1 B P 2 AP ? 1 YA YB B P X A 30 30 Z B Z A A B R A X B YB
A A Z B X B X A YB X A X B YA YB YA X B Z A YB Z A X B Z B X A cos 30 cos 60 cos 90 Z B YA cos120 cos 30 cos 90 Z B Z A cos 90 cos 90 cos 0 23 Rotation + Translation Difference Z A Z B A A B Y
RBBP B PBORG A P X B YA A X A P 24 Homogeneous Transformation A P APBORG BAR BP A Px A Py A Pz
1 0 A A B PBORG _ x B Px A PBORG _ y B Py A PBORG _ z B Pz 1 1 R 0 0 A B
T 25 Mapping Translation Difference Z A Z B A PBORG X B A X A P APBORG BP AP A B P BT 1 1 B P A YB
P YA I A BT 0 0 0 A PBORG 1 26 Translation Operator Translation operator A Z A A P1 A
Q X A P2 1 0 DQ ( q) 0 0 0 1 0 0 0 qx 0 qy 1 qz 0 1 YA A A P2 AP1 AQ
P2 DQ ( q) AP1 2 2 q q x q y qz 27 2 Mapping rotation difference A Z A A Z B B P YB YA X A P BAR AP X B
B P A P BT 1 1 A BT 0 A R 0 B 0 0 1 28 Rotation Operator YA YB
B P1 AP2 A P1 A X B Z A P2 X A A B R A X B YB A A
Z B X B X A YB X A X B YA YB YA X B Z A YB Z A Z B X A Z B YA Z B Z A cos cos(90 ) cos 90 cos(90 ) cos cos 90 cos 90 cos 90 cos 0 cos sin 0 sin cos 0 0
1 0 29