ECE 450 Introduction to Robotics

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

Reference and Coordinate Frame Z P W O V U Y X

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  rotated about x-axis 0 cos -sin 0 sin cos cos 0 sin = R Y, angle  rotated about y-axis 0 1 0 -sin 0 cos

Basic Rotation Matrices cont’d cos -sin 0 angle  rotated about z-axis 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

ECE 450 Introduction to Robotics