Links and
Joints
Links and Joints
Links
Joints:
End Effector
2 DOF’s
Robot Basis
Denavit –
Hartenberg
details and
examples
DENAVIT-HARTENBERG REPRESENTATION
Chapter 2
Symbol Terminologies :
Robot Kinematics: Position Analysis
⊙
 : 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.
Joints U
Links S
Z-axis aligned with joint
X-axis aligned with outgoing
limb
Y-axis is orthogonal
Joints are numbered to represent
hierarchy
Ui-1 is parent of Ui
Parameter ai-1 is outgoing
limb length of joint Ui-1
Joint angle, i, is rotation of xi-1 about zi1 relative to xi
Link twist, i-1, is the rotation of ith z-axis about
xi-1-axis relative to z-axis of i-1th frame
Link offset, di-1, specifies the distance along the zi-1-axis (rotated
by i-1) of the ith frame from the
i-1th x-axis to the ith x-axis
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.
DENAVIT-HARTENBERG REPRESENTATION
Procedures for assigning a local reference frame to each joint:
1. ‫ ٭‬All joints are represented by a z-axis.
•
(right-hand rule for rotational joint, linear movement for prismatic
joint)
2. The common normal is one line mutually perpendicular to any two
skew lines.
3.
Parallel z-axes joints make a infinite number of common normal.
4. Intersecting z-axes of two successive joints make no common
normal between them(Length is 0.).
DENAVIT-HARTENBERG
ChapterREPRESENTATION
2
Robot Kinematics:
Positionto
Analysis
The necessary
motions
transform from
one reference frame to the next.
(I)
Rotate about the zn-axis an able of n+1. (Coplanar)
(II) Translate along zn-axis a distance of dn+1 to make xn and xn+1
colinear.
(III) Translate along the xn-axis a distance of an+1 to bring the origins
of xn+1 together.
(IV) Rotate zn-axis about xn+1 axis an angle of n+1 to align zn-axis
with zn+1-axis.
Denavit Hartenberg
Parameters –
a general explanation
Denavit-Hartenberg Notation
Only  and d are joint variables
Z(i - 1)
Y(i -1)
Yi
Zi
Xi
a(i - 1 )
X(i -1)
ai
di
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 ).
⊙  : A rotation about the z-axis.
• THE PARAMETERS/VARIABLES:
⊙ 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)
, a , d,
The a(i-1) Parameter
Z(i - 1)
Y(i -1)
X(i -1)
• 1) a(i-1)
( i - 1)
Yi
a(i - 1 )
Zi
Xi
di
ai
You can
align the
two axis
just using
the 4
parameters
i
• 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.
a(i-1) cont...
The alpha a(i-1) Parameter
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)
Z(i - 1)
Y(i -1)
Yi
Zi
Xi
a(i - 1 )
X(i -1)
⊙  : A rotation about the z-axis.
( i - 1)
⊙ 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)
ai
di
i
• It’s 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.
Z(i - 1)
Y(i -1)
X(i -1)
( i - 1)
⊙  : 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)
Yi
a(i - 1 )
Zi
Xi
di
ai
i
2)(i-1)
The (i-1) Parameter
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.
Z(i - 1)
Y(i -1)
X(i -1)
( i - 1)
Yi
a(i - 1 )
Zi
Xi
di
ai
i
3) d(i-1)
The d(i-1) Parameter
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.
The i Parameter
4)  i
Amount of rotation around the Zi axis needed to align the X(i-1) axis with the Xi
axis.
Z(i - 1)
Y(i -1)
X(i -1)
( i - 1)
The same table as last slide
Yi Z
i
a(i - 1 )
Xi
di
ai
i
The Denavit-Hartenberg Matrix
cosθ i
 sinθ i

sinθ cosα
cosθ i cosα (i 1)
i
(i 1)

 sinθ i sinα (i 1) cosθ i sinα (i 1)

0
0

0
 sinα (i 1)
cosα (i 1)
0
a(i 1)

 sinα (i 1)d i 
cosα (i 1)d i 

1

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
Z(i -
Y(i -
1)
1)
X(i -
⊙  : 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)
( i - 1)
1)
Y Z
i
a(i -
d
1)
i
i
X a
i
i
i
Example:
Calculating the
final DH matrix
with the DH
Parameter Table
Example with three
Revolute Joints
The DH
Parameter
Table
Y2
Z1
Z0
X2
X0
d2
X1
Y0
Y1
a0
Denavit-Hartenberg Link
Parameter Table
a1
i
(i-1)
a(i-1)
di
i
1) To describe the robot with its
variables and parameters.
0
0
0
0
0
2) To describe some state of the
robot by having a numerical values
for the variables.
1
0
a0
0
1
2
-90
a1
d2
2
Notice that the table has two uses:
We calculate with
respect to previous 
Example with three
Revolute Joints
Y2
Z1
Z0
X2
X0
d2
X1
Y0
Y1
a0
Denavit-Hartenberg Link
Parameter Table
a1
i
(i-1)
a(i-1)
di
i
1) To describe the robot with its
variables and parameters.
0
0
0
0
0
2) To describe some state of the
robot by having a numerical values
for the variables.
1
0
a0
0
1
2
-90
a1
d2
2
Notice that the table has two uses:
The same table as last slide
Y2
Z1
Z0
X2
X0
X1
Y0
Y1
a0
i
(i-1)
a(i-1)
di
i
0
0
0
0
0
1
0
a0
0
1
2
-90
a1
d2
2
d2
a1
The same table as last slide
V
X 0 Y0 Z 0
World coordinates
V X 2 
 Y2 
V 

T Z
V 2 


 1 
tool coordinates
T ( 0T)( 01T)( 12T)
Note: T is the D-H matrix with (i-1) = 0 and i = 1.
These matrices T are
calculated in next slide
The same table as last slide
i
(i-1)
a(i-1)
di
i
0
0
0
0
0
1
0
a0
0
1
2
cosθ1
 sinθ
1
0

T

1
 0

 0
-90
a1
 sinθ1
cosθ1
0
0
d2
2
0 a0 
0 0 
0 0

0 1
This is a translation by a0 followed by a
rotation around the Z1 axis
cosθ 0  sinθ 0
 sinθ
cosθ 0
0

0T 
 0
0

0
 0
0 0
0 0
1 0

0 1
This is just a rotation around the Z0 axis
 cosθ 2
 0
1

2T 
 sinθ 2

 0
 sinθ 2
0
 cosθ 2
0
0 a1 
1 d 2 
0 0

0 1
This is a translation by a1 and then d2
followed by a rotation around the X2 and
Z2 axis
T ( 0T)( 01T)( 12T)
Conclusions
V X 0 Y0 Z 0
World coordinates
V X 2 
 Y2 
V 

T Z
V 2 


 1 
tool coordinates
T ( 0T)( 01T)( 12T)
Forward
Kinematics
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 it’s
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.
Example Problem with H matrices:
2.
3.
4.
1. 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
Y2
2
l2
X3
l3
X2
H = Rz( 1 ) * Tx1(l1) * Rz( 2 ) * Tx2(l2) * Rz( 3 )
Y0
1.
2.
3.
4.
l1
1
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.
X0
• 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.
Slight variation on the last solution:
Make the yellow dot the origin of a new coordinate X4Y4 frame
Y3
Y4
3
2
Y2
2
3
X3
added
X2
X4
H = Rz(
Y0
1
1
) * Tx1(l1) * Rz(
2
) * Tx2(l2) * Rz(
3
) * Tx3(l3)
This takes you from the X0Y0 frame to the X4Y4 frame.
1
X0
 X
0 
Y 
0 
   H 
Z
0 
 
 
1
 
1 
The position of the yellow dot relative to the X4Y4 frame
is (0,0).
THE INVERSE
KINEMATIC
SOLUTION OF A
ROBOT
THE INVERSE KINEMATIC SOLUTION
OF ROBOT
 Determine the value of each joint to place the arm at a
desired position and orientation.
TH  A1 A2 A3 A4 A5 A6
R
C1 (C234C5C6  S 234S 6 )
 S S C
 1 5 6
S1 (C234C5C6  S 234S 6 )

  C1S5C6
 S 234C5C6  C234S 6


0
 nx
n
y

 nz

0
ox a x p x 
o y a y p y 
oz a z p z 

0 0 1
RHS
C1 (C234C5C6  S 234C6 )

C1 (C234S5 )  S1C5 C1 (C234a4  C23a3  C2 a2 )
 S1S5C6

S1 (C234C5C6  S 234C6 )
S1 (C234S5 )  C1C5 S1 (C234a4  C23a3  C2 a2 ) 

 C1S5C6
 S 234C5C6  C234C6
S 234S5
S 234a4  S 23a3  S 2 a2 


0
0
1
Multiply both sides by A1
-1
THE INVERSE KINEMATIC SOLUTION OF ROBOT
 nx
n
1  y
A1 
 nz

0
ox a x p x 

oy a y p y 
 A11[ RHS ]  A2 A3 A4 A5 A6
oz a z p z 

0 0 1
C1 S1
0 0

 S1  C1

0 0
A1
-1
0  n x


0  n y

0  n z
 
0 1  0
0
1
0
ox a x p x 

oy a y p y 
 A2 A3 A4 A5 A6
oz a z p z 

0 0 1
THE INVERSE KINEMATIC SOLUTION OF ROBOT
We calculate all angles from
px, py, a1, a2, ni, oi, etc
 py 

 px 
1  tan 1 
 2  tan 1
(C3a3  a2 )( pz  S 234a4 )  S3a3 ( pxC1  p y S1  C234a4 )
(C3a3  a2 )( pxC1  p y S1  C234a4 )  S3a3 ( Pz  S 234a4 )
 S3 

C
3
 
 3  tan 1 
 4   234   2  3
 5  tan 1
C234 (C1ax  S1a y )  S 234az
S1ax  C1a y
 6  tan 1
 S 234 (C1nx  S1n y )  S 234nz
 S 234 (C1ox  S1o y )  C234oz
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 straightline motions
PROBLEMS
with DH
DEGENERACY AND DEXTERITY
Degeneracy : The robot looses a degree of freedom
and thus cannot perform as desired.
‫ ٭‬When the robot’s 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.
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
Fig. 2.31 The frames of 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