Links and Joints

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Links and Joints
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Links and Joints
End Effector
Robot Basis
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(No Transcript)
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Denavit Hartenberg details and examples
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Chapter 2Robot Kinematics Position Analysis

• DENAVIT-HARTENBERG REPRESENTATION
• Symbol Terminologies

? ? 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.
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Z-axis aligned with joint
Joints U Links S
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X-axis aligned with outgoing limb
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Y-axis is orthogonal
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Joints are numbered to represent hierarchy Ui-1
is parent of Ui
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Parameter ai-1 is outgoinglimb length of joint
Ui-1
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Joint angle, qi, is rotation of xi-1 about zi-1
relative to xi
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Link twist, ai-1, is the rotation of ith z-axis
about xi-1-axis relative to z-axis of i-1th frame
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Link offset, di-1, specifies the distance along
the zi-1-axis (rotated by ai-1) of the ith frame
from the i-1th x-axis to the ith x-axis
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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.

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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.).

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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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Denavit - Hartenberg Parameters a general
explanation
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Denavit-Hartenberg Notation
Only ? and d are joint variables
Z(i - 1)
Y(i -1)
Y i
Z i
X i
a i
a(i - 1 )
d i
X(i -1)
? i
?( i - 1)

• IDEA Each joint is assigned a coordinate frame.
  
• Using the Denavit-Hartenberg notation, you need 4
  parameters to describe how a frame (i) relates to
  a previous frame ( i -1 ).
• THE PARAMETERS/VARIABLES ?, a , d, ?

? ? 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)
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The a(i-1) Parameter
You can align the two axis just using the 4
parameters
Z(i - 1)
Y(i -1)
Y i
Z i
X i
a i
a(i - 1 )
d i
X(i -1)
? i
?( i - 1)

• 1) a(i-1)
• Technical Definition a(i-1) is the length of
  the perpendicular between the joint axes.
• The joint axes are the axes around which
  revolution takes place which are the Z(i-1) and
  Z(i) axes.
• These two axes can be viewed as lines in space.
• The common perpendicular is the shortest line
  between the two axis-lines and is perpendicular
  to both axis-lines.

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The alpha a(i-1) Parameter
a(i-1) cont... Visual Approach - A way to
visualize the link parameter a(i-1) is to imagine
an expanding cylinder whose axis is the Z(i-1)
axis - when the cylinder just touches the joint
axis i the radius of the cylinder is equal to
a(i-1). (Manipulator Kinematics)
? ? 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)
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• Its Usually on the Diagram Approach -
• If the diagram already specifies the various
  coordinate frames, then the common perpendicular
  is usually the X(i-1) axis.
• So a(i-1) is just the displacement along the
  X(i-1) to move from the (i-1) frame to the i
  frame.
• If the link is prismatic, then a(i-1) is a
  variable, not a parameter.

? ? 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)
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The ?(i-1) Parameter
2)?(i-1) Technical Definition Amount of
rotation around the common perpendicular so that
the joint axes are parallel. i.e. How much you
have to rotate around the X(i-1) axis so that the
Z(i-1) is pointing in the same direction as the
Zi axis. Positive rotation follows the right
hand rule.
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The d(i-1) Parameter
3) d(i-1) Technical Definition The displacement
along the Zi axis needed to align the a(i-1)
common perpendicular to the ai common
perpendicular. In other words, displacement
along the Zi to align the X(i-1) and Xi axes. 4)
? i Amount of rotation around the Zi axis needed
to align the X(i-1) axis with the Xi axis.
The ?i Parameter
The same table as last slide
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The Denavit-Hartenberg Matrix
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
? ? 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)
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ExampleCalculating the final DH matrix with the
DH Parameter Table
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The DH Parameter Table
Example with three Revolute Joints
Denavit-Hartenberg Link Parameter Table
Notice that the table has two uses 1) To
describe the robot with its variables and
parameters. 2) To describe some state of the
robot by having a numerical values for the
variables.
We calculate with respect to previous ?
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Example with three Revolute Joints
Denavit-Hartenberg Link Parameter Table
Notice that the table has two uses 1) To
describe the robot with its variables and
parameters. 2) To describe some state of the
robot by having a numerical values for the
variables.
The same table as last slide
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The same table as last slide
Note T is the D-H matrix with (i-1) 0 and i
1.
World coordinates
tool coordinates
These matrices T are calculated in next slide
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The same table as last slide
This is just a rotation around the Z0 axis
This is a translation by a1 and then d2 followed
by a rotation around the X2 and Z2 axis
This is a translation by a0 followed by a
rotation around the Z1 axis
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Conclusions
World coordinates
tool coordinates
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Forward Kinematics
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Forward Kinematics Problem
The Situation You have a robotic arm that
starts out aligned with the xo-axis. You tell the
first link to move by ?1 and the second link to
move by ?2. The Quest What is the position of
the end of the robotic arm?
Solution 1. Geometric Approach This might be
the easiest solution for the simple situation.
However, notice that the angles are measured
relative to the direction of the previous link.
(The first link is the exception. The angle is
measured relative to its initial position.) For
robots with more links and whose arm extends into
3 dimensions the geometry gets much more tedious.
2. Algebraic Approach Involves coordinate
transformations.
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• Example Problem with H matrices
• You have a three link arm that starts out aligned
  in the x-axis.
• Each link has lengths l1, l2, l3, respectively.
• You tell the first one to move by ?1 , and so on
  as the diagram suggests.
• Find the Homogeneous matrix to get the position
  of the yellow dot in the X0Y0 frame.

Y3
? 3
l2
l3
X3
Y2
? 2

• H Rz(? 1 ) Tx1(l1) Rz(? 2 ) Tx2(l2)
  Rz(? 3 )
• Rotating by ?1 will put you in the X1Y1 frame.
• Translate in the along the X1 axis by l1.
• Rotating by ? 2 will put you in the X2Y2 frame.
• and so on until you are in the X3Y3 frame.
• The position of the yellow dot relative to the
  X3Y3 frame is
• (l3, 0).
• Multiplying H by that position vector will give
  you the
• coordinates of the yellow point relative the X0Y0
  frame.

X2
Y0
l1
X1
? 1
Y1
X0
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Slight variation on the last solution Make the
yellow dot the origin of a new coordinate X4Y4
frame
Y3
Y4
?3
2
3
X3
Y2
?2
added
X2
X4
H Rz(?1 ) Tx1(l1) Rz(?2 ) Tx2(l2) Rz(?3
) Tx3(l3) This takes you from the X0Y0 frame
to the X4Y4 frame. The position of the yellow
dot relative to the X4Y4 frame is (0,0).
Y0
1
X1
?1
Y1
X0
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THE INVERSE KINEMATIC SOLUTION OF A ROBOT
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THE INVERSE KINEMATIC SOLUTION OF ROBOT

• Determine the value of each joint to place the
  arm at a
• desired position and orientation.

RHS
Multiply both sides by A1 -1
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THE INVERSE KINEMATIC SOLUTION OF ROBOT
A1 -1
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THE INVERSE KINEMATIC SOLUTION OF ROBOT
We calculate all angles from px, py, a1, a2, ni,
oi, etc
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INVERSE KINEMATIC PROGRAM a predictable path on
a straight line

• A robot has a predictable path on a straight
  line,
• Or an unpredictable path on a straight line.

? A predictable path is necessary to recalculate
joint variables. (Between 50 to 200 times a
second) ? To make the robot follow a straight
line, it is necessary to break the line into
many small sections. ? All unnecessary
computations should be eliminated.
Fig. 2.30 Small sections of movement for
straight-line motions
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PROBLEMS with DH
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DEGENERACY AND DEXTERITY

• Degeneracy The robot looses a degree of freedom
• and thus cannot perform
  as desired.

? When the robots joints reach their physical
limits, and as a result, cannot move any
further. ? In the middle point of its workspace
if the z-axes of two similar joints becomes
collinear.

• Dexterity The volume of points where one can
  
• position the robot as desired,
  but not
• orientate it.

Fig. 2.31 An example of a robot in a degenerate
position.
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THE FUNDAMENTAL PROBLEM WITH D-H REPRESENTATION

• Defect of D-H presentation D-H cannot represent
  any motion about
• the y-axis, because all motions are about the
  x- and z-axis.

TABLE 2.3 THE PARAMETERS TABLE FOR THE
STANFORD ARM
? d a ?
1 ?1 0 0 -90
2 ?2 d1 0 90
3 0 d1 0 0
4 ?4 0 0 -90
5 ?5 0 0 90
6 ?6 0 0 0
Fig. 2.31 The frames of the Stanford Arm.