1. Links and Joints

2. Links and Joints
Joints:
Links
2 DOF’s
End Effector
Robot Basis

4. Denavit – Hartenberg details and examples

5. Chapter 2 Robot Kinematics: Position Analysis
DENAVIT-HARTENBERG REPRESENTATION
Symbol Terminologies :
Chapter 2 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.

6. Z-axis aligned with joint
Joints U
Links S

7. X-axis aligned with outgoing limb

8. Y-axis is orthogonal

9. Joints are numbered to represent hierarchy Ui-1 is parent of Ui

10. Parameter ai-1 is outgoing limb length of joint Ui-1

11. Joint angle, qi, is rotation of xi-1 about zi-1 relative to xi

12. Link twist, ai-1, is the rotation of ith z-axis about xi-1-axis relative to z-axis of i-1th frame

13. 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

14. 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.

15. 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.).

16. Chapter 2 Robot Kinematics: Position Analysis
DENAVIT-HARTENBERG REPRESENTATION
The necessary motions to transform from one reference frame to the next.
Chapter 2 Robot Kinematics: Position Analysis
(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.

17. Denavit - Hartenberg Parameters –
a general explanation

18. 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)

19. The a(i-1) Parameter 1) a(i-1)
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.

20. 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)
Z(i - 1)
X(i -1)
Y(i -1)
( i - 1)
a(i - 1 )
Z i
Y i
X i
a i
d i
 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)

21. 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)
X(i -1)
Y(i -1)
( i - 1)
a(i - 1 )
Z i
Y i
X i
a i
d i
 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)

22. 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.
The (i-1) Parameter
Z(i - 1)
X(i -1)
Y(i -1)
( i - 1)
a(i - 1 )
Z i
Y i
X i
a i
d i
 i

23. 3) d(i-1) The d(i-1) Parameter The i 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.
4)  i
Amount of rotation around the Zi axis needed to align the X(i-1) axis with the Xi axis.
The d(i-1) Parameter
The i Parameter
Z(i - 1)
X(i -1)
Y(i -1)
( i - 1)
a(i - 1 )
Z i
Y i
X i
a i
d i
 i
The same table as last slide

24. 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.
Z(i - 1)
X(i -1)
Y(i -1)
( i - 1)
a(i - 1 )
Z i
Y i
X i
a i
d i
 i
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)

25. Example: Calculating the final DH matrix with the DH Parameter Table

26. Denavit-Hartenberg Link Parameter Table
The DH Parameter Table
Example with three Revolute Joints Z0 X0 Y0 Z1 X2 Y1 Z2 X1 Y2 d2 a0 a1
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 

27. Denavit-Hartenberg Link Parameter Table
Example with three Revolute Joints Z0 X0 Y0 Z1 X2 Y1 Z2 X1 Y2 d2 a0 a1
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

28. d2 a0 a1 Note: T is the D-H matrix with (i-1) = 0 and i = 1.Z0 X0 Y0 Z1 X2 Y1 Z2 X1 Y2 d2 a0 a1
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

29. This is just a rotation around the Z0 axis
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

30. Conclusions
World coordinates
tool coordinates

31. Forward
Kinematics

32. 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.

33. 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 Y2 X3
 2 X2
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. Y0 l1 X1
 1 Y1 X0

34. Slight variation on the last solution:
Make the yellow dot the origin of a new coordinate X4Y4 frame Y3 Y4 3 2 3 Y2 X3 2 X2
added 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

35. THE INVERSE KINEMATIC SOLUTION OF A ROBOT

36. 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

37. THE INVERSE KINEMATIC SOLUTION OF ROBOT

38. THE INVERSE KINEMATIC SOLUTION OF ROBOT
We calculate all angles from px, py, a1, a2, ni, oi, etc

39. 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 Small sections of movement for straight-line motions

40. PROBLEMS with DH

41. 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 An example of a robot in a degenerate position.

42. 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
-90 2 2 d1 90 3 4 4 5 5 6 6
Fig The frames of the Stanford Arm.