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

2. INTRODUCTION Forward Kinematics:
to determine where the robot’s 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
Where is my hand?
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. Base=0, 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
10+11) Write link transformation and concatenate

7. Kinematic Problems for Manipulation
Reliably position the tip - go from one position to another position
Don’t 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 2 Robot Kinematics: Position Analysis
Representation of a Point in Space
Chapter 2 Robot Kinematics: Position Analysis
A point P in space :
3 coordinates relative to a reference frame
Fig. 2.3 Representation of a point in space

11. Chapter 2 Robot Kinematics: Position Analysis
Representation of a Vector in Space
Chapter 2 Robot Kinematics: Position Analysis
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 2 Robot Kinematics: Position Analysis
Representation of a Frame at the Origin of a Fixed-Reference Frame
Chapter 2 Robot Kinematics: Position Analysis
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 2 Robot Kinematics: Position Analysis
Representation of a Frame in a Fixed Reference Frame
Chapter 2 Robot Kinematics: Position Analysis
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 2 Robot Kinematics: Position Analysis
Representation of a Rigid Body
Chapter 2 Robot Kinematics: Position Analysis
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 2 Robot Kinematics: Position Analysis
HOMOGENEOUS TRANSFORMATION MATRICES
Chapter 2 Robot Kinematics: Position Analysis
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 2 Robot Kinematics: Position Analysis
Representation of a Pure Translation
Chapter 2 Robot Kinematics: Position Analysis
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 2 Robot Kinematics: Position Analysis
Representation of a Pure Rotation about an Axis
Chapter 2 Robot Kinematics: Position Analysis
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 Coordinates of a point in a rotating frame before and after rotation around axis x.
Fig 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…. ai T1
Fig Effects of three successive transformations oi ni T2 T3
x,y,z  n, o, a
Order is important

20. Order of Transformations is important
x,y,z  n, o, a
translation
Order of Transformations is important
Fig Changing the order of transformations will change the final result

21. Chapter 2 Robot Kinematics: Position Analysis
Transformations Relative to the Rotating Frame
Chapter 2 Robot Kinematics: Position Analysis
Example 2.8
translation
rotation
Fig Transformations relative to the current frames.

22. MATRICES FOR FORWARD AND INVERSE KINEMATICS OF ROBOTS
For position
For orientation

23. Chapter 2 Robot Kinematics: Position Analysis
FORWARD AND INVERSE KINEMATICS OF ROBOTS
Chapter 2 Robot Kinematics: Position Analysis
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 The hand frame of the robot relative to the reference frame.

24. Chapter 2 Robot Kinematics: Position Analysis
Forward and Inverse Kinematics Equations for Position
Chapter 2 Robot Kinematics: Position Analysis
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 2 Robot Kinematics: Position Analysis
Forward and Inverse Kinematics Equations for Position
(a) Cartesian (Gantry, Rectangular) Coordinates
Chapter 2 Robot Kinematics: Position Analysis
IBM 7565 robot
• All actuator is linear.
• A gantry robot is a Cartesian robot.
Fig Cartesian Coordinates.

26. Chapter 2 Robot Kinematics: Position Analysis
Forward and Inverse Kinematics Equations for Position:
Cylindrical Coordinates
Chapter 2 Robot Kinematics: Position Analysis
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 Cylindrical Coordinates.

27. Chapter 2 Robot Kinematics: Position Analysis
Forward and Inverse Kinematics Equations for Position
(c) Spherical Coordinates
Chapter 2 Robot Kinematics: Position Analysis
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 Spherical Coordinates.

28. Chapter 2 Robot Kinematics: Position Analysis
Forward and Inverse Kinematics Equations for Position
(d) Articulated Coordinates
Chapter 2 Robot Kinematics: Position Analysis
3 rotations -> Denavit-Hartenberg representation
Fig Articulated Coordinates.

29. Chapter 2 Robot Kinematics: Position Analysis
Forward and Inverse Kinematics Equations for Orientation
Chapter 2 Robot Kinematics: Position Analysis
 Roll, Pitch, Yaw (RPY) angles
 Euler angles
 Articulated joints

30. Chapter 2 Robot Kinematics: Position Analysis
Forward and Inverse Kinematics Equations for Orientation
(a) Roll, Pitch, Yaw(RPY) Angles
Chapter 2 Robot Kinematics: Position Analysis
Roll: Rotation of about -axis (z-axis of the moving frame)
Pitch: Rotation of about -axis (y-axis of the moving frame)
Yaw: Rotation of about -axis (x-axis of the moving frame)
Fig RPY rotations about the current axes.

31. Chapter 2 Robot Kinematics: Position Analysis
Forward and Inverse Kinematics Equations for Orientation
(b) Euler Angles
Chapter 2 Robot Kinematics: Position Analysis
Rotation of about -axis (z-axis of the moving frame) followed by
Rotation of about -axis (y-axis of the moving frame) followed by
Rotation of about -axis (z-axis of the moving frame).
Fig Euler rotations about the current axes.

32. Chapter 2 Robot Kinematics: Position Analysis
Forward and Inverse Kinematics Equations for Orientation
Chapter 2 Robot Kinematics: Position Analysis
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 The Universe, robot, hand, part, and end effecter frames.

35. Identity Transformations

36. We often need to calculate INVERSE MATRICES
It is good to reduce the number of such operations
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 c 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

46. Denavit – Hartenberg idea

47. DENAVIT-HARTENBERG REPRESENTATION OF
FORWARD KINEMATIC EQUATIONS OF ROBOT
Denavit-Hartenberg Representation :
@ Simple way of modeling robot links and
joints for any robot configuration,
regardless of its sequence or complexity.
Fig A D-H representation of a general-purpose joint-link combination
@ Transformations in any coordinates
is possible.
@ Any possible combinations of joints
and links and all-revolute articulated
robots can be represented.

48. Chapter 2 Robot Kinematics: Position Analysis
DENAVIT-HARTENBERG REPRESENTATION
Symbol Terminologies :
Chapter 2 Robot Kinematics: Position Analysis
⊙  : 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 rotation
Only translation
Only offset
Only offset
Only rotation
Axis alignment

51. DENAVIT-HARTENBERG REPRESENTATION for each link

52. 4 link parameters

53. Chapter 2 Robot Kinematics: Position Analysis
DENAVIT-HARTENBERG REPRESENTATION
Symbol Terminologies :
Chapter 2 Robot Kinematics: Position Analysis
⊙  : 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. 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
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. Put the transformation here for every link
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.
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 for every link

61. In DENAVIT-HARTENBERG REPRESENTATION we must be able to find parameters for each link
So we must know link types

63. Links between revolute joints

65. Type 3 Link
Joint n+1 xn
Link n
ln=0
dn=0
Joint n
xn-1

66. Type 4 Link Origins coincide n-1 Link n n Joint n+1 Joint n xn xn-1
ln=0
dn=0
Part of dn-1
Joint n n xn
Origins coincide
xn-1
yn-1

67. Links between prismatic joints

70. Forward and Inverse Transformations on Matrices

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

75. 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.