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Introduction to ROBOTICS Kinematics of Robot Manipulator Jizhong Xiao Department of Electrical Engineering City College of New York jxiao_at_ccny.cuny.edu – PowerPoint PPT presentation

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Title: Jizhong Xiao


1
Kinematics of Robot Manipulator
Introduction to ROBOTICS
  • Jizhong Xiao
  • Department of Electrical Engineering
  • City College of New York
  • jxiao_at_ccny.cuny.edu

2
Outline
  • Review
  • Robot Manipulators
  • Robot Configuration
  • Robot Specification
  • Number of Axes, DOF
  • Precision, Repeatability
  • Kinematics
  • Preliminary
  • World frame, joint frame, end-effector frame
  • Rotation Matrix, composite rotation matrix
  • Homogeneous Matrix
  • Direct kinematics
  • Denavit-Hartenberg Representation
  • Examples
  • Inverse kinematics

3
Review
  • What is a robot?
  • By general agreement a robot is
  • A programmable machine that imitates the actions
    or appearance of an intelligent creatureusually
    a human.
  • To qualify as a robot, a machine must be able to
  • 1) Sensing and perception get information from
    its surroundings
  • 2) Carry out different tasks Locomotion or
    manipulation, do something physicalsuch as move
    or manipulate objects
  • 3) Re-programmable can do different things
  • 4) Function autonomously and/or interact with
    human beings
  • Why use robots?
  • Perform 4A tasks in 4D environments

4A Automation, Augmentation, Assistance,
Autonomous
4D Dangerous, Dirty, Dull, Difficult
4
Manipulators
  • Robot arms, industrial robot
  • Rigid bodies (links) connected by joints
  • Joints revolute or prismatic
  • Drive electric or hydraulic
  • End-effector (tool) mounted on a flange or plate
    secured to the wrist joint of robot

5
Manipulators
  • Robot Configuration

Cartesian PPP
Cylindrical RPP
Spherical RRP
Hand coordinate n normal vector s sliding
vector a approach vector, normal to the tool
mounting plate
SCARA RRP (Selective Compliance Assembly Robot
Arm)
Articulated RRR
6
Manipulators
  • Motion Control Methods
  • Point to point control
  • a sequence of discrete points
  • spot welding, pick-and-place, loading unloading
  • Continuous path control
  • follow a prescribed path, controlled-path motion
  • Spray painting, Arc welding, Gluing

7
Manipulators
  • Robot Specifications
  • Number of Axes
  • Major axes, (1-3) gt Position the wrist
  • Minor axes, (4-6) gt Orient the tool
  • Redundant, (7-n) gt reaching around obstacles,
    avoiding undesirable configuration
  • Degree of Freedom (DOF)
  • Workspace
  • Payload (load capacity)
  • Precision v.s. Repeatability

Which one is more important?
8
What is Kinematics
  • Forward kinematics
  • Given joint variables
  • End-effector position and orientation, -Formula?

9
What is Kinematics
  • Inverse kinematics
  • End effector position
  • and orientation
  • Joint variables -Formula?

10
Example 1
11
Preliminary
  • Robot Reference Frames
  • World frame
  • Joint frame
  • Tool frame

T
P
W
R
12
Preliminary
  • Coordinate Transformation
  • Reference coordinate frame OXYZ
  • Body-attached frame Ouvw

Point represented in OXYZ
Point represented in Ouvw
Two frames coincide gt
13
Preliminary
Properties Dot Product Let and be
arbitrary vectors in and be the angle
from to , then
Properties of orthonormal coordinate frame
  • Mutually perpendicular
  • Unit vectors

14
Preliminary
  • Coordinate Transformation
  • Rotation only

How to relate the coordinate in these two frames?
15
Preliminary
  • Basic Rotation
  • , , and represent the projections
    of onto OX, OY, OZ axes, respectively
  • Since

16
Preliminary
  • Basic Rotation Matrix
  • Rotation about x-axis with

17
Preliminary
  • Is it True?
  • Rotation about x axis with

18
Basic Rotation Matrices
  • Rotation about x-axis with
  • Rotation about y-axis with
  • Rotation about z-axis with

19
Preliminary
  • Basic Rotation Matrix
  • Obtain the coordinate of from the
    coordinate of

Dot products are commutative!
lt 3X3 identity matrix
20
Example 2
  • A point is attached to a
    rotating frame, the frame rotates 60 degree about
    the OZ axis of the reference frame. Find the
    coordinates of the point relative to the
    reference frame after the rotation.

21
Example 3
  • A point is the coordinate w.r.t.
    the reference coordinate system, find the
    corresponding point w.r.t. the rotated
    OU-V-W coordinate system if it has been rotated
    60 degree about OZ axis.

22
Composite Rotation Matrix
  • A sequence of finite rotations
  • matrix multiplications do not commute
  • rules
  • if rotating coordinate O-U-V-W is rotating about
    principal axis of OXYZ frame, then Pre-multiply
    the previous (resultant) rotation matrix with an
    appropriate basic rotation matrix
  • if rotating coordinate OUVW is rotating about its
    own principal axes, then post-multiply the
    previous (resultant) rotation matrix with an
    appropriate basic rotation matrix

23
Example 4
  • Find the rotation matrix for the following
    operations

Pre-multiply if rotate about the OXYZ axes
Post-multiply if rotate about the OUVW axes
24
Coordinate Transformations
  • position vector of P in B is transformed to
    position vector of P in A
  • description of B as seen from an observer in
    A

Rotation of B with respect to A
Translation of the origin of B with respect to
origin of A
25
Coordinate Transformations
  • Two Special Cases
  • 1. Translation only
  • Axes of B and A are parallel
  • 2. Rotation only
  • Origins of B and A are coincident

26
Homogeneous Representation
  • Coordinate transformation from B to A
  • Homogeneous transformation matrix

Rotation matrix
Position vector
Scaling
27
Homogeneous Transformation
  • Special cases
  • 1. Translation
  • 2. Rotation

28
Example 5
  • Translation along Z-axis with h

29
Example 6
  • Rotation about the X-axis by

30
Homogeneous Transformation
  • Composite Homogeneous Transformation Matrix
  • Rules
  • Transformation (rotation/translation) w.r.t
    (X,Y,Z) (OLD FRAME), using pre-multiplication
  • Transformation (rotation/translation) w.r.t
    (U,V,W) (NEW FRAME), using post-multiplication

31
Example 7
  • Find the homogeneous transformation matrix (T)
    for the following operations

32
Homogeneous Representation
  • A frame in space (Geometric Interpretation)

(z)
(y)
(X)
Principal axis n w.r.t. the reference coordinate
system
33
Homogeneous Transformation
  • Translation

34
Homogeneous Transformation
Composite Homogeneous Transformation Matrix
Transformation matrix for adjacent coordinate
frames
Chain product of successive coordinate
transformation matrices
35
Example 8
  • For the figure shown below, find the 4x4
    homogeneous transformation matrices and
    for i1, 2, 3, 4, 5

Can you find the answer by observation based on
the geometric interpretation of homogeneous
transformation matrix?
36
Orientation Representation
  • Rotation matrix representation needs 9 elements
    to completely describe the orientation of a
    rotating rigid body.
  • Any easy way?

Euler Angles Representation
37
Orientation Representation
  • Euler Angles Representation ( , , )
  • Many different types
  • Description of Euler angle representations

38
Euler Angle I, Animated
z
w'
w"
w'"
f
v'"
v "
?
v'
y
u'"
?
u'
u"
x
39
Orientation Representation
  • Euler Angle I

40
Euler Angle I
Resultant eulerian rotation matrix
41
Euler Angle II, Animated
z
w'
w"
w"'
?
v"'
?
v'
v"
?
y
u"'
u"
Note the opposite (clockwise) sense of the third
rotation, f.
u'
x
42
Orientation Representation
  • Matrix with Euler Angle II

Quiz How to get this matrix ?
43
Orientation Representation
  • Description of Roll Pitch Yaw

Z
Y
X
Quiz How to get rotation matrix ?
44
Thank you!
Homework 1 is posted on the web. Due Sept. 16,
2008, before class
Next class kinematics II
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