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More details and examples on robot arms and kinematics

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Title: More details and examples on robot arms and kinematics


1
More details and examples on robot arms and
kinematics
  • Denavit-Hartenberg Notation

2
INTRODUCTION
?Forward Kinematics to determine where
the robots hand is?
(If all joint variables are known) ?Inverse
Kinematics to calculate what each joint
variable is?
(If we desire that the hand be
located at a particular
point)
3
Direct Kinematics
4
Direct Kinematics with no matrices
Direct Kinematics HERE!
5
Direct Kinematics
  • Position of tip in (x,y) coordinates

6
Direct Kinematics Algorithm
Often sufficient for 2D
  • 1) Draw sketch
  • 2) Number links. Base0, Last link n
  • 3) Identify and number robot joints
  • 4) Draw axis Zi for joint i
  • 5) Determine joint length ai-1 between Zi-1 and
    Zi
  • 6) Draw axis Xi-1
  • 7) Determine joint twist ?i-1 measured around
    Xi-1
  • 8) Determine the joint offset di
  • 9) Determine joint angle ?i around Zi
  • 1011) Write link transformation and concatenate

7
Kinematic Problems for Manipulation
  • Reliably position the tip - go from one position
    to another position
  • Dont hit anything, avoid obstacles
  • Make smooth motions
  • at reasonable speeds and
  • at reasonable accelerations
  • Adjust to changing conditions -
  • i.e. when something is picked up respond to the
    change in weight

8
ROBOTS AS MECHANISMs
9
Robot Kinematics ROBOTS AS MECHANISM
?Multiple type robot have multiple DOF. (3
Dimensional, open loop, chain mechanisms)
Fig. 2.1 A one-degree-of-freedom closed-loop
four-bar mechanism
Fig. 2.2 (a) Closed-loop versus (b) open-loop
mechanism
10
Chapter 2Robot Kinematics Position Analysis
Representation of a Point in Space
?A point P in space 3 coordinates
relative to a reference frame
Fig. 2.3 Representation of a point in space
11
Chapter 2Robot Kinematics Position Analysis
Representation of a Vector in Space
?A Vector P in space 3
coordinates of its tail and of its head
Fig. 2.4 Representation of a vector in space
12
Chapter 2Robot Kinematics Position Analysis
Representation of a Frame at the Origin of a
Fixed-Reference Frame
?Each Unit Vector is mutually perpendicular.
normal, orientation, approach
vector
Fig. 2.5 Representation of a frame at the origin
of the reference frame
13
Chapter 2Robot Kinematics Position Analysis
Representation of a Frame in a Fixed Reference
Frame
?Each Unit Vector is mutually perpendicular.
normal, orientation, approach
vector
The same as last slide
Fig. 2.6 Representation of a frame in a frame
14
Chapter 2Robot Kinematics Position Analysis
Representation of a Rigid Body
?An object can be represented in space by
attaching a frame to it and representing the
frame in space.
Fig. 2.8 Representation of an object in space
15
Chapter 2Robot Kinematics Position Analysis
HOMOGENEOUS TRANSFORMATION MATRICES
  • ?A transformation matrices must be in square
    form.
  • It is much easier to calculate the inverse of
    square matrices.
  • To multiply two matrices, their dimensions must
    match.

16
Representation of Transformations of rigid
objects in 3D space
17
Chapter 2Robot Kinematics Position Analysis
Representation of a Pure Translation
  • ?A transformation is defined as making a movement
    in space.
  • A pure translation.
  • A pure rotation about an axis.
  • A combination of translation or rotations.

Same value a
identity
Fig. 2.9 Representation of an pure translation in
space
18
Chapter 2Robot Kinematics Position Analysis
Representation of a Pure Rotation about an Axis
x,y,z ? n, o, a
?Assumption The frame is at the origin of the
reference frame and parallel to it.
Projections as seen from x axis
Fig. 2.10 Coordinates of a point in a rotating
frame before and after rotation around axis x.
Fig. 2.11 Coordinates of a point relative to the
reference frame and rotating frame as viewed
from the x-axis.
19
Representation of Combined Transformations
?A number of successive translations and
rotations.
T1
ai

Fig. 2.13 Effects of three successive
transformations
oi
ni
T2
T3
x,y,z ? n, o, a
Order is important
20
x,y,z ? n, o, a
translation
Order of Transformations is important
Fig. 2.14 Changing the order of transformations
will change the final result
21
Chapter 2Robot Kinematics Position Analysis
Transformations Relative to the Rotating Frame
?Example 2.8
translation
rotation
Fig. 2.15 Transformations relative to the current
frames.
22
MATRICES FORFORWARD AND INVERSE KINEMATICS OF
ROBOTS
  • For position
  • For orientation

23
Chapter 2Robot Kinematics Position Analysis
FORWARD AND INVERSE KINEMATICS OF ROBOTS
?Forward Kinematics Analysis Calculating
the position and orientation of the hand of the
robot. If all robot joint variables are
known, one can calculate where the robot is
at any instant. .
Fig. 2.17 The hand frame of the robot relative to
the reference frame.
24
Chapter 2Robot Kinematics Position Analysis
Forward and Inverse Kinematics Equations for
Position
?Forward Kinematics and Inverse Kinematics
equation for position analysis (a)
Cartesian (gantry, rectangular) coordinates.
(b) Cylindrical coordinates. (c) Spherical
coordinates. (d) Articulated
(anthropomorphic, or all-revolute) coordinates.
25
Chapter 2Robot Kinematics Position Analysis
Forward and Inverse Kinematics Equations for
Position (a) Cartesian (Gantry, Rectangular)
Coordinates
?IBM 7565 robot All actuator is linear.
A gantry robot is a Cartesian robot.
Fig. 2.18 Cartesian Coordinates.
26
Chapter 2Robot Kinematics Position Analysis
Forward and Inverse Kinematics Equations for
Position Cylindrical Coordinates
?2 Linear translations and 1 rotation
translation of r along the x-axis rotation
of ? about the z-axis translation of l
along the z-axis
cosine
sine
Fig. 2.19 Cylindrical Coordinates.
27
Chapter 2Robot Kinematics Position Analysis
Forward and Inverse Kinematics Equations for
Position (c) Spherical Coordinates
?2 Linear translations and 1 rotation
translation of r along the z-axis rotation
of ? about the y-axis rotation of ? along
the z-axis
Fig. 2.20 Spherical Coordinates.
28
Chapter 2Robot Kinematics Position Analysis
Forward and Inverse Kinematics Equations for
Position (d) Articulated Coordinates
?3 rotations -gt Denavit-Hartenberg representation

Fig. 2.21 Articulated Coordinates.
29
Chapter 2Robot Kinematics Position Analysis
Forward and Inverse Kinematics Equations for
Orientation
? Roll, Pitch, Yaw (RPY) angles ? Euler angles ?
Articulated joints
30
Chapter 2Robot Kinematics Position Analysis
Forward and Inverse Kinematics Equations for
Orientation (a) Roll, Pitch, Yaw(RPY) Angles
Fig. 2.22 RPY rotations about the current axes.
31
Chapter 2Robot Kinematics Position Analysis
Forward and Inverse Kinematics Equations for
Orientation (b) Euler Angles
Fig. 2.24 Euler rotations about the current axes.
32
Chapter 2Robot Kinematics Position Analysis
Forward and Inverse Kinematics Equations for
Orientation
Roll, Pitch, Yaw(RPY) Angles
  • Assumption Robot is made of a Cartesian and an
    RPY set of joints.
  • Assumption Robot is made of a Spherical
    Coordinate and an Euler angle.

Another Combination can be possible
Denavit-Hartenberg Representation
33
Forward and Inverse Transformations for robot arms
34
INVERSE OF TRANSFORMATION MATRICES
  • ?Steps of calculation of an Inverse matrix
  • Calculate the determinant of the matrix.
  • Transpose the matrix.
  • Replace each element of the transposed
    matrix by its own minor (adjoint matrix).
  • Divide the converted matrix by the
    determinant.

Fig. 2.16 The Universe, robot, hand, part, and
end effecter frames.
35
Identity Transformations
36
  1. We often need to calculate INVERSE MATRICES
  2. It is good to reduce the number of such
    operations
  3. We need to do these calculations fast

37
How to find an Inverse Matrix B of matrix A?
38
Inverse Homogeneous Transformation
39
Homogeneous Coordinates
  • Homogeneous coordinates embed 3D vectors into 4D
    by adding a 1
  • More generally, the transformation matrix T has
    the form

a11 a12 a13 b1 a21 a22 a23 b2 a31 a32 a33
b3 c1 c2 c3 sf
It is presented in more detail on the WWW!
40
For various types of robots we have different
maneuvering spaces
41
For various types of robots we calculate
different forward and inverse transformations
42
For various types of robots we solve different
forward and inverse kinematic problems
43
Forward and Inverse Kinematics Single Link
Example
44
Forward and Inverse Kinematics Single Link
Example
easy
45
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46
Denavit Hartenberg idea
47
DENAVIT-HARTENBERG REPRESENTATION OF
FORWARD KINEMATIC EQUATIONS OF ROBOT
  • Denavit-Hartenberg Representation

_at_ Simple way of modeling robot links and
joints for any robot configuration,
regardless of its sequence or complexity.
_at_ Transformations in any coordinates is
possible.
_at_ Any possible combinations of joints and
links and all-revolute articulated robots
can be represented.
48
Chapter 2Robot Kinematics Position Analysis
  • DENAVIT-HARTENBERG REPRESENTATION
  • Symbol Terminologies

? ? A rotation angle between two links, about
the z-axis (revolute). ? d The distance
(offset) on the z-axis, between links
(prismatic). ? a The length of each common
normal (Joint offset). ? ? The twist angle
between two successive z-axes (Joint twist)
(revolute) ? Only ? and d are joint variables.
49
? associated with Zi always
Links are in 3D, any shape
50
Only translation
Only offset
Only rotation
Only rotation
Axis alignment
Only offset
51
DENAVIT-HARTENBERG REPRESENTATION for each link
52
4 link parameters
53
Chapter 2Robot Kinematics Position Analysis
  • DENAVIT-HARTENBERG REPRESENTATION
  • Symbol Terminologies

? ? A rotation angle between two links, about
the z-axis (revolute). ? d The distance
(offset) on the z-axis, between links
(prismatic). ? a The length of each common
normal (Joint offset). ? ? The twist angle
between two successive z-axes (Joint twist)
(revolute) ? Only ? and d are joint variables.
54
The DH Parameter Table
Example with three Revolute Joints
Denavit-Hartenberg Link Parameter Table
Apply first
Apply last
55
Denavit-Hartenberg Representation of
Joint-Link-Joint Transformation
56
Notation for Denavit-Hartenberg Representation of
Joint-Link-Joint Transformation
Alpha applied first
57
Four Transformations from one Joint to the Next
  • Order of multiplication of matrices is inverse of
    order of applying them
  • Here we show order of matrices

Joint-Link-Joint
58
Denavit-Hartenberg Representation of
Joint-Link-Joint Transformation
  • Alpha is applied first

How to create a single matrix A n
59
EXAMPLE Denavit-Hartenberg Representation of
Joint-Link-Joint Transformation for Type 1 Link
Final matrix from previous slide
substitute
substitute
Numeric or symbolic matrices
60
The Denavit-Hartenberg Matrix for another link
type
  • Similarity to Homegeneous Just like the
    Homogeneous Matrix, the Denavit-Hartenberg Matrix
    is a transformation matrix from one coordinate
    frame to the next.
  • Using a series of D-H Matrix multiplications and
    the D-H Parameter table, the final result is a
    transformation matrix from some frame to your
    initial frame.

Put the transformation here for every link
61
  1. In DENAVIT-HARTENBERG REPRESENTATION we must be
    able to find parameters for each link
  2. So we must know link types

62
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63
Links between revolute joints
64
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65
Type 3 Link
Joint n1
xn
Link n
ln0
dn0
Joint n
xn-1
66
Type 4 Link
?n-1
Joint n1
Link n
ln0 dn0
Part of dn-1
Joint n
?n
Origins coincide
xn
yn-1
xn-1
67
Links between prismatic joints
68
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69
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70
Forward and Inverse Transformations on Matrices
71
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72
DENAVIT-HARTENBERG REPRESENTATION PROCEDURES
  • Start point
  • Assign joint number n to the first shown
    joint.
  • Assign a local reference frame for each and
    every joint before or
  • after these joints.
  • Y-axis is not used in D-H representation.

73
DENAVIT-HARTENBERG REPRESENTATION Procedures for
assigning a local reference frame to each joint
  • ? All joints are represented by a z-axis.
  • (right-hand rule for rotational joint, linear
    movement for prismatic joint)
  • The common normal is one line mutually
    perpendicular to any two skew lines.
  • Parallel z-axes joints make a infinite number of
    common normal.
  • Intersecting z-axes of two successive joints make
    no common normal between them(Length is 0.).

74
Chapter 2Robot Kinematics Position Analysis
  • DENAVIT-HARTENBERG REPRESENTATION
  • Symbol Terminologies Reminder

? ? A rotation about the z-axis. ? d The
distance on the z-axis. ? a The length of
each common normal (Joint offset). ? ? The
angle between two successive z-axes (Joint
twist) ? Only ? and d are joint variables.
75
Chapter 2Robot Kinematics Position Analysis
  • DENAVIT-HARTENBERG REPRESENTATION
  • The necessary motions to transform from one
    reference frame to the next.

(I) Rotate about the zn-axis an able of ?n1.
(Coplanar) (II) Translate along zn-axis a
distance of dn1 to make xn and xn1
colinear. (III) Translate along the xn-axis a
distance of an1 to bring the origins of
xn1 together. (IV) Rotate zn-axis about xn1
axis an angle of ?n1 to align zn-axis
with zn1-axis.
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