1. Chapter 3. Kinematic analysis

2. Table of Contents
Kinematics
Velocity Kinematics

3. 3.1 Kinematics Kinematics is describing the geometrical motion of the
robot without considering the force/torque or inertia.
Forward kinematics: Calculating the position/orientation of the
end-effector based upon the joint angles.
Inverse kinematics: Calculating the joint angles for the desired
end-effector position/orientation.

4. Analysis of manipulators
Manipulator: Joint i + Link i + Joint i+1 + Link i+1 Structure.
The links are connected in serial generally, which have linear or angular motions by the actuator.
Analysis of manipulator motion:
The relationship of the moving frame at the end of the link with respect to
the reference frame.

5. General representations
Robot has n+1 links: from 0 to n from the base of the robot to
the end-effector.
Joint number: 1  n , Joint i is located at the beginning of link i .
ith Joint:
Revolute joint:
Prismatic joint:
Frames: at the end of each link.

6. Example 3.1
[Fig. 3-1] 3 DOF Cylindrical manipulator

7. [Table 3-1] Link parameter of 3-DOF manipulator
Link Parameter Table
Following the D-H constraints, the table is obtained as follows:
-90
Link 1 2 3
[Table 3-1] Link parameter of 3-DOF manipulator

8. Homogeneous Transformation Matrix
(1. 1)
(1. 2)
(1. 3)

9. Composition
(1. 4)

10. Example 3.2
[Fig. 3-2] 6 DOF PUMA 560 Manipulator

11. D-H Frames
[Fig. 3-3] Frames for PUMA 560 Manipulator

12. [Table 3-2] Link parameter of 6 DOF PUMA 560 manipulator
Link Parameter Table
Link 1 2
-90 3 4 5 90 6
[Table 3-2] Link parameter of 6 DOF PUMA 560 manipulator

13. Homogeneous Transformation Matrix
(1. 5)
(1. 6)
(1. 7)

14. Homogeneous Transformation Matrix
(1. 8)
(1. 9)
(1.10)

15. Composition
(1. 11)
(1. 12)

16. Composition
(1.13)

17. Composition
(1.14)

18. Composition
(1.15)
(1.16)
(1.17)
(1.18)

19. Composition
Finally for the six links,
(1.19)
(1.20)

20. Composition
(1.21)
(1.22)
(1.23)

21. 3.2 Velocity kinematics
When a matrix, S, satisfies the following conditions:
where the matrix S is skew symmetric.
(1.24)
(1.25)

22. Skew symmetric
(1.26)
(1.27)
(1.28)
(1.29)
where,

23. Properties of skew symmetric matrix
Vectors and are in ; and are scalars,
(1.30)
For a vector,
(1.31)

24. Cross Product When and vectors are in (1.32)
is satisfied and R is Orthogonal.
(1.32)

25. Cross Product
When and vectors are in
(1.33)
where R is Orthogonal.

26. Cross Product
(1.34)

27. Proof
Using the chain rule, derivative of w.r.t
(1.35)
(1.36)
(1.37)

28. Skew symmetric (1.38) Matrix is skew symmetric.
Multiplying at both side of , we have
(1.39)

29. Angular velocity
(1.40)

30. Derivative of Velocity
The vector can be represented w.r.t the frame as,
Then becomes,
(1.41)
(1.42)

31. 3.2.1 Velocity propagation
[Fig. 3-4] Vector representation
In Fig. 3-4 , how the vector will be changed?

32. Derivative of Motion Vector
can be differentiated w.r.t time with fixed
(1.43)
where
is a vector.

33. Proof
Show that
[Fig. 3-5] Rotation about an arbitrary axis k .

34. Composition results (1.44)
can be obtained by differentiating each element as follows:

35. Motion Vector Differentiation
the first element is
(1.45)
where is scalar.
(1.46)

36. Obtaining
Let’s obtain
(1.47)
(1.48)

37. .

38. Derivation
(1.49)
(1.50)

39. Derivation
(1.51)
Therefore .
(1.52)

40. Derivation
Since ,
(1.53)
(1.54)
(1.55)

41. 3.2.2 Propagation of angular velocity
Vector can be represented by as 
(1.56)

42. Angular velocity
(1.57)
(1.58)

43. Angular velocity
(1.59)
(1.60)

44. Addition of Angular Velocity
(1.61)
(1.62)

45. Addition of Angular Velocity
(1.63)
To generalize this equation, is adopted.
(1.64)

46. 3.2.3 Jacobian
Robot generates motion at each joint, which results motion at the end-effector w.r.t frame {0}. The motions are related by J(Jacobian) as
(1.65)

47. Obtaining Jacobian . Using in can be obtained w.r.t frame {0}
That is, .
(1.66)

48. Jacobian 
(1.67)
(1.68)
(1.69)

49. Obtaining Jacobian
where is the rotation axis of th joint. When the th joint is linear,
The rotational motion at the end-effector can be represented as the summation of the rotation motions w.r.t {0} frame.
(1.70)

50. Jacobian The rotation axis is , that is, Therefore (1.71)
gives easily.

51. Jacobian
Finally
(1.72)
Generally , is named as Jacobian.

52. 3.2.4 Acceleration Propagation
[Fig. 3-6] Vector representation

53. Acceleration is time-varying, then (1.74) (1.75)
Taking the derivative at both sides,

54. Acceleration (1.76) First term: Transverse acceleration
Second term: Centripetal acceleration
Third Term: Coriolis acceleration
Fourth Term: Linear acceleration

55. Acceleration Finally, it can be represented as (1.77)
It is very complex.
Generally,
is used to represent the accelerations by taking
derivative at both sides.

56. [Fig. 3-7] 2 Link manipulator
Example 3.3
Jacobian matrix
[Fig. 3-7] 2 Link manipulator

57. [Table 3-3] Link parameter of 2 Link manipulator
Link Parameter Table 2 1
Link
[Table 3-3] Link parameter of 2 Link manipulator

58. Homogeneous matrix
(1.78)
(1.79)

59. Composition
(1.80)
(1.81)

60. Obtaining Jv
(1.82)
(1.83)

61. Obtaining Jω
(1.84)
(1.85)

62. Final Jacobian matrix
(1.86)

63. Example 3.4
[Fig. 3-8] SCARA robot

64. [Table 3-4] Link parameter of SCARA robot
Link Parameter Table
Link 1 2
180 3 4
[Table 3-4] Link parameter of SCARA robot

65. Homogeneous matrix
(1.87)
(1.88)
(1.89)

66. Composition
(1.90)
(1.91)

67. Each element
(1.92)
(1.93)
(1.94)
(1.95)

68. Obtaining Jv
(1.96)
(1.97)
(1.98)
(1.99)
(1.100)
(1.101)

69. Obtaining Jv
(1.102)

70. Obtaing Jω (Method 1)
Method 1.
Using
(1.103)

71. Obtaining Jω (Method 2)
Using
Method 2.
(1.104)

72. Obtaining Jω (Method 2)
(1.105)
(1.106)
(1.107)

73. Cont.
can be obtained
(1.108)

74. Cont.
(1.109)
(1.110)

75. Cont.
(1.111)
Plugging into

76. Cont.
(1.112)
Finally, the Jacobian matrix is
(1.113)

77. Final Jacobian matrix
(1.114)

78. Homework (Homepage)