1. Chapter 2 Mathematical Analysis for Kinematics

2. Table of Contents Mathematical background
Representation of coordinates
Rotation matrix
Multiple rotations
Homogeneous Transformation
Denavit-Hartenberg Coordinates
Forward kinematics with HT

3. 2.1 Mathematical Background
To bring the end-effector to a desired posture, the end-effector coordinates need to be represented with respect to the reference frame.
Cartesian coordinates, cylindrical coordinates, and spherical coordinates are generally used. In this chapter, only Cartesian coordinates are handled.

4. Representations and Inner Product
Types
Representations
Examples
Scalar
Small Italic
x, y
Vector
Small Gothic
Matrix, Set
Capital
X, Y
대문자
[Table 2-1] Representations

5. Properties of Inner Product
[Definition 1] Property of inner product:
(2.1)
Prove this property using x =[2 3 4]T and y = [3 1 2]T

6. 2.2 Representation of Coordinates
Position: With respect to the reference coordinates {A}, a given position a can be represented as
Orientation: {B} w.r.t. {A}
(2.2)
(2.3)

7. 2.3 Rotation Matrix
[Fig. 2-1] Two coordinates for the same vector

8. Vector Representation
The vector can be represented in {0} frame and {1} frame as follows:
are unit vectors along axes, respectively.
(2.4)

9. Transformation
(2.5)

10. Transformation Matrix
This can be represented in a matrix equation as
where
This 3 X 3 matrix transforms from coordinates to
(2.6)

11. Rotation Matrix As the same way, (2.7)
This can be represented in a matrix equation as
where
(2.8)

12. Orthogonal Matrix Matrix is the inverse of .
Since , it is easy to show that
This matrix is named as orthogonal since the inverse is the same as
the transpose. Also the columns are unit and orthogonal to each other.
Therefore, is called as orthonormal matrix .
(2.9)

13. Properties of Rotation Matrix
When the frames are determined by the right hand rule,
Simply, rotation matrices have unit determinant and they are orthogonal.
(2.10)

14. Basic Rotation Matrix
[Fig. 2-2] rotation about X axis.

15. Basic Rotation Matrix From [Fig. 2-2], we know that Therefore, (2.11)
(2.12)

16. Basic Rotation Matrix
For y and z axes, the rotation matrices are determined as
(2.13)

17. Composition of Rotation Matrices
The same vector is represented w.r.t. the different frames as,
(2.14)
Be careful not to change the multiplication order.

18. Composition of Rotation Matrices
Instead of , when we rotate about the fixed axis, 
(2.15)

19. Composition of Rotation Matrices
Rotate about y axis, and rotate about z axis,
the rotation matrix R becomes
(2.16)

20. Properties of Rotation Matrix
(2.17)

21. Fixed Axis Rotation Rotate about the fixed axis, and rotate
about fixed axis, the resulting rotation matrix is different.

22. Rotation about Fixed AxisP
[Fig. 2-3] rotation about y and z axis.

23. Rotation around a Fixed Axis
About axis, rotate, 
about axis, rotate 
In this case, is a fixed frame.
The multiplication sequence will be reversed in this case.
When , , and are the same vector ,
there is the relations:

24. Second Rotation
The second rotation along z axis is done w.r.t. the fixed frame,
not w.r.t. the current frame,
Therefore,
implies the rotation about axis.

25. Undo the Previous Rotation
To solve this problem, can be matched to .
By rotating about y axis through - ,
Now the rotation about z axis will be done through .
The inverse operation will be followed by rotating about y axis.
That is,
(2.18)

26. Reverse Order
Finally the rotation w.r.t. the fixed frame results the reverse order
multiplication of the rotation matrices.
(2.19)

27. Example 2.2 1) can be represented in {A} as using
For fixed frame, {A} and rotated frame {B} which is rotated about
Through ,
1) can be represented in {A} as using
(2.20)

28. Sol)
As follows:
(2.21)

29. Sol)
2) If , represent w.r.t {B}?
sol) Since ,
(2.22)

30. 2.4 Composition of Rotation
Any frames can be represented w.r.t. a reference frame by consecutive
rotations.
The total rotation matrix is composed of basic rotation matrices.
Note that AB ≠ BA.

31. Euler Angles
Rotation matrix for a wrist manipulator which has Roll –Pitch – Roll type
can be represented by Euler Angles.
PUMA-560 is a typical example of wrist manipulator with Euler angles. The
relation between two frames can be represented by three angles about three
axes, respectively.

32. Euler Angles
[Fig 2-5] Z-Y-Z Euler Angle

33. Euler Angles

34. Z-Y-Z Euler Angles
(2.23)

35. Roll, Pitch, Yaw Angles Roll-Pitch-Yaw rotation:
Rotation about x axis, about y axis, and about z axis, will complete and
rotational transformation.
[Fig. 2-6] Wrist of Cinccinati Milacron

36. Roll, Pitch, Yaw Angles
Aviation system needs orientation w.r.t., the absolute coordinates. R-P-Y
angles are suitable for the system.
[Fig. 2-7] Roll-Pitch-Yaw Angles

37. Roll, Pitch, and Yaw

38. Roll, Pitch, and Yaw
(2.24)

39. Example 2.5
[Fig. 2-8] Yaw-Pitch-Roll Rotation

40. Sol ) 1) 2) 3)
sol) Yaw, Pitch, and Roll from the right.
(2.25)

41. Rotation about an Arbitrary Axis
[Fig 2-9] Rotation about arbitrary axis k.

42. Rotation about an Arbitrary Axis
is a directional unit vector in frame.
Let’s obtain the rotation matrix, , which is rotated through about
k axis.

43. General Rotation
(2.26)

44. General Rotation
(2.27)
(2.28)

45. 2.5 Homogeneous Transformation
Homogeneous matrix is defined as
A combination of a position vector ,
and orientation matrix,
as,
(2.29)

46. Homogeneous Transformations
[Fig. 2-10] Coordinates transformation

47. Homogeneous Transformations
vector can be represented in frame {0}, as
This can be represented as a homogeneous matrix as,
(2.30)
(2.31a)

48. Homogeneous Transformations
Transforming in frame {2} into frame {1}, we have
If we transform, in frame {2} into frame {0},
(2.32b)

49. Homogeneous Transformations
(2.33)
(2.34)

50. Homogeneous Transformations
Since R is orthogonal,
(2.35)

51. 2.6 Denavit-Hartenberg Coordinates
There are four parameters in D-H representation.
These constraints need to be satisfied:
(DH1): Axis x1 is orthogonal to axis z0.
(DH2) : Axis x1 intersects axis z0 .

52. Denavit-Hartenberg Frame
[Fig. 2-11] Denavit-Hartenberg frame

53. Denavit-Hartenberg Frame
How to obtain ?
Frame {i-1} match  to frame{i}.
: Rotate about , to make parallel to
: Move along to make is on the line of
: Move along to make same as
: Rotate about , to make same as

54. Denavit-Hartenberg Frame
Homogeneous matrix, , is represented as
(2.36)

55. Assignments of Frames
It starts from 0,…,n satisfying DH1 & DH2 conditions
[Fig. 2-12] Assignments of frame

56. Alpha Ⅱ Robotic Arm
[Fig. 2-13] Alpha ⅡRobotic arm & Link coordinates

57. SCARA Robot (Adept One)
[Fig. 2-14] SCARA Robot (Adept One) & Link coordinates

58. 2.7 Forward Kinematics Forward kinemaitcs with D-H representation
Step 1: Locations of joint axes are determined with suitable labels.
Step 2: Assign the reference frame.
The origin on the axis...

59. Repeat step 3: Assign oi at the intersection of zi and zi-1.
Steps 3 – 5 are repeated.
step 3: Assign oi at the intersection of zi and zi-1.
If two are parallel, assign on zi.
step 4: Assign xi at oi vertical to both of zi and zi-1.
step 5: Assign yi following right screw rule.
step 6: Assign frames of the end-effector.

60. Repeat rotation about translation along
Step 7 : Draw the link parameter table
rotation about
translation along
Step 8 : Obtain the homogeneous transformation matrix

61. <Example1> Planar Robot
link 1 a1 2 a2
[Table 2-2] Link parameter of Planar Robot
variable
[Fig. 2-15] Frames of Planar Robot

62. Homogeneous Matrix.
(2.37)
(2.38)

63. Multiplication
(2.39)

64. <Example 2> SCARA Robot
link 1 a1 2 a2
180° 3 4
[Table 2-3] Link parameter of SCARA Robot
[Fig. 2-16] Frames of SCARA Robot
variable

65. Homogeneous Matrix
(2.40)
(2.41)

66. Homogeneous Matrix
(2.42)
(2.43)

67. Resulting Matrix
(2.44)

68. <Example 3> 2 links Cartesian Manipulator1
-90° 2
[Table 2-4] Link parameter of Cartesian manipulator
variable
[Fig. 2-17] Two link Cartesian manipulator.

69. Homogeneous Matrix
(2.45)
(2.46)

70. Multiplication
(2.47)

71. Homework
<Chap 2 >Problems -homepage