MECH572A Introduction To Robotics Lecture 5 Dept. Of Mechanical Engineering

Midterm Exam Date & Time: 19: :00,Oct 25, 2004 Open Book Chapters 2 & 3 of the text book Note: Regular lecture will take place 18:00 –18:45 on Oct 25

Review New concepts Twist of rigid body Wrench (static analysis) Instantaneous Screw of rigid-body motion –Define by direction + one point Similarity between Velocity and Force/Moment Analysis –Screw-like force and moment property: Wrench axis

Review Acceleration Analysis –Fixed reference frame: –Moving Reference frame Corilios term in the expression Basics in Rigid Body Dynamics Mass properties - Mass 1 st & 2 nd Moment; Parallel Axes Theorem; Principle Axes/Moments (Eigenvectors/values) Equation of Motion – Newton-Euler Equations Acceleration tensor

Robotic Kinematics Overview Basic Concepts Robot Kinematics - Study robot motion without resorting to force and mass properties. Dealing with position, velocity and acceleration Kinematic Chain - A set of rigid bodies connected by kinematic pairs Kinematic Pairs Upper Pair - Line/point contact (gear, cam-follower) Lower Pair - Surface contact (revolute, prismatic)

Robotic Kinematics Overview Basic Concepts (cont'd) Typical Lower Kinematic Pairs Revolute (R) - 1 Dof (Rotation) Prismatic (P) - 1 Dof (Translation) Cylindrical (C) - 2 Dof (Rotation + Translation) Helical (H) - 1 Dof (Coupled Rotation/Translation) Planar (E) - 2 Dof (Translation in 2 directions) Spherical (S) - 3 Dof (Rotation in 3 directions)

Robotic Kinematics Overview Basic Concepts (cont'd) Two Basic Types of Kinematic Pairs - R & P All six lower pairs can be produced from Revolute (R) and Prismatic (P) Rotating pair – Revolute (R) Sliding pair – Prismatic (P)

Robot Kinematics Overview Robot Manipulators Kinematic Chains : Link + Joint Rigid bodies Kinematic Pairs Basic Topologies of Kinematic Chain Open ChainTree Necklace

Robot Kinematics Overview Basic Problems in Robotic Kinematics Direct Kinematics Inverse Kinematics... X Y Z O Base End Effector 11 22 ii nn p x,, p y, p z  Joint Variables Cartesian Variables Linear relationship between Cartesian rate of EE and joint rates  x Direct Inverse (Joint) (Cartesian)

Denavit-Hartenberg Notation Purpose –To uniquely define architecture of robot manipulator (Kinematic chains) Assumptions –Links : 0, 1, …, n - n+1 links –Pairs: 1, 2, …, n - n pairs –Frame F i (O i - X i -Y i -Z i ) is attached to (i-1)st frame (NOT ith frame)

Denavite-Hartenberg Notation Definition of Axes –Z i - Axes of the pair (Rotational/translational) ZiZi ZiZi

Denavite-Hartenberg Notation Definition of Axes (cont'd) –X i - Common perpendicular to Z i+1 and Z i directed from Z i+1 to Z i (Follow right hand rule if intersect) –Y i = Z i  X i (d) Z i-1 ZiZi X i undefined

DH Notation Joint Parameters & Joint Variables –a i - Distance between Z i and Z i+1 –b i - Z-coordinate of Z i and X i+1 intersection point (absolute value = distance between X i and X i+1 ) –  i - Angle between Z i and Z i+1 along +X i+1 (R.H.R) –  i - Angle between X i and X i+1 along +Z i (R.H.R) –Joint Variables  i - R joint b i - P joint

DH Notation Summary O i-1 OiOi O i+1 Z i-1 ZiZi Z i+1 i-1 i i+1 X i-1 XiXi X i+1 Revolute joints b i-1 bibi ii a i-1 aiai ii  i-1  i-1

DH Notation Summary – Prismatic joint X i+1 i - 1 i XiXi ZiZi bibi ii b i – Variable  i - Constant

DH Notation Example - PUMA

DH Notation Example - PUMA

DH Notation Example – PUMA DH Parameters of PUMA Robot iaiai bibi ii ii 10b1b1 90° 11 2a2a2 00 22 3a3a3 b3b3 33 40b4b4 44 500 55 60b6b6 66

DH Notation Example - Stanford Arm

DH Notation Example - Stanford Arm X1X1 Y1Y1 Z1Z1 X2X2 Z2Z2 X3X3 Z3Z3 X4X4 X5X5 X6X6 Z4Z4 Z5Z5 Z6Z6 X7X7 Z7Z7

DH Notation Example - Stanford Arm (cont'd) DH Parameters of Stanford Arm iaiai bibi ii ii 10b1b1 90° 11 20b2b2 22 30b 3 ( var )90° 400 44 50b5b5 0°0° 55 60b6b6 0 66

DH Notation Summary ith pairR jointP joint Number of parameters/variable Joint Parameters (Constant) a i, b i,  i a i,  i,  i 3 Joint Variable (Changing) ii bibi 1 If there are n joint, there will be 3n joint parameters and n joint variables

DH Notation Relative Orientation and Position Analysis –Orientation Xi'Xi' Yi'Yi' Zi'Zi' X i+1 Y i+1 Z i+1 XiXi Xi'Xi' Yi'Yi' ZiZi Zi'Zi' YiYi  i about Z i  i about X i ' Rotation Decomposition (a) & (b) (a) (b)

DH Notation Relative Orientation and Position Analysis –Orientation (cont'd) (a)(X i, Y i, Z i ) (X i ', Y i ', Z i ' ) (b) (X i ', Y i ', Z i ' ) (X i+1, Y i+1, Z i+1 )

DH Notation Relative Orientation and Position Analysis –Orientation (cont'd)

DH Notation Relative Orientation and Position Analysis –Position To find the position vector a i in F i frame (position vector connecting O i to O i+1

DH Notation Relative Orientation and Position Analysis –Position –Observation: Changing Constant

DH Notation Relative Orientation and Position Analysis –Summary Orientation Position

Direct Kinematics 6-R Serial Manipulator Problem description: Known  1 …  n, find Q and p in the base frame

Direct Kinematics 6-R Serial Manipulator 1. Orientation With DH Parameter defined, Q 1, … Q 6 can be calculated. Similarity transformation to individual frame Abbreviated notation Q i = [Q i ] i

Direct Kinematics 6-R Serial Manipulator 2. Position 3. Homogeneous form (position + orientation)

Direct Kinematics Some useful properties of Q i