ECE 450 Introduction to Robotics

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ECE 450 Introduction to Robotics. Section: 50883 Instructor: Linda A. Gee 9/07/99 Lecture 03. Vectors. 2-D. Y. F X = F cos . F Y = F sin . F. F = F X 2 +F Y 2. tan  = F Y /F X. X. Vectors cont’d. 3-D. Y. F Y = F cos . F h = F sin . F.  Y.

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### ECE 450 Introduction to Robotics

Section: 50883

Instructor: Linda A. Gee

9/07/99

Lecture 03

Vectors
• 2-D

Y

FX = F cos

FY = F sin

F

F = FX2 +FY2

tan = FY/FX

X

Vectors cont’d
• 3-D

Y

FY = F cos

Fh = F sin

F

Y

FX = Fh cos = F siny cos

FY = Fh sin = F siny sin

X

Fh

F = FXi + FYj + FZk

Z

Vectors cont’d
• Cross Product

A x B = AB sin 

Let V = A x B

V =

i j k

AX AY AZ

BX BY BZ

V = (AY BZ - AZ BY)i

- j(AX BZ - AZ BX)

+ k(AX BY - AY BX)

Vectors cont’d
• Dot Product

AB = AB cos

AB = AX BX + AY BY + AZ BZ

Robot Arm Kinematics
• Direct/Forward Kinematics

Use this approach to find position and orientation of the end-effector of the manipulator with respect to the reference coordinate system

Robot Arm Kinematics cont’d
• Inverse Kinematics

Use this approach to determine whether it is possible for the manipulator and end effector to reach a desired position and orientation

Solving the Direct Kinematics Problem
• Reduce the problem to finding the transformation matrix that relates the body-attached coordinate frame to the reference frame
• Rotation matrix is a 3x3 matrix that relates rotational information; can be extended to include translation as a 4x4 matrix
History of Matrix Representation
• Method introduced in 1955 by Denavit and Hartenburg
• Matrix representation is known as the Denavit-Hartenburg (D-H) representation of linkages
Rotation Matrix
• Rotation matrix is a transformation matrix that operates on a position vector and maps coordinates into a rotated coordinate system OUVW to OXYZ
• Origins O are coincident
• R represents the rotation matrix
• PXYZ = R PUVW
Point Representation
• A point, P, can be represented in both coordinate systems OUVW and OXYZ
• PUVW = (PU PV PW)T
• PXYZ = (PX PY PZ)T
Solving for a Transformation Matrix
• Find a transformation matrix (3x3) to transform the coordinates of PUVW to the OXYZ coordinates
• Rewriting, using (i,j,k) unit vectors

PUVW = PUiU + PVjV + PWkW

PXYZ = PXiX + PYjY + PZkZ

Transformation Matrix cont’d
• PX = iX P
• PY = jY P
• PZ = kZ P

PU

PX

iX

iX

jV

iX

iU

kW

PV

=

jY

PY

iU

jY

jY

jV

kW

PW

iU

kZ

jV

kZ

PZ

kW

kZ

Basic Rotation Matrices

=

R X,

1 0 0

angle 

0 cos -sin

0 sin cos

cos 0 sin

=

R Y,

angle 

0 1 0

-sin 0 cos

Basic Rotation Matrices cont’d

cos -sin 0

angle 

R Z,

=

sin cos 0

0 0 1

Coordinate Transformation
• PUVW = Q PXYZ
• Q = R-1 = RT
• QR = RTR = R-1R = I3
Examples
• Coordinate transformation to the reference frame
• Coordinate transformation from the reference frame