The document outlines key concepts in robotics kinematics, including rigid body poses, rotation matrices, and direct and inverse kinematics. It details manipulator structures and the Denavit-Hartenberg convention for describing robot movements, with examples including three-link planar and spherical arms. Additionally, the document addresses kinematic calibration techniques to estimate parameters and discusses the complexity of inverse kinematics compared to direct kinematics.

1. D R . T A R E K A . T U T U N J I
M E C H A T R O N I C S E N G I N E E R I N G D E P A R T M E N T
P H I L A D E L P H I A U N I V E R S I T Y , J O R D A N
2018
Robotics: Kinematics

2. Outline
 Description and Transformation
 Rigid Body Pose
 Rotation Matrix
 Euler Angles
 Homogeneous Transformation
 Direct Kinematics
 DH Convention
 Manipulator Kinematics
 Work Space
 Inverse Kinematics

3. Rigid Body Pose

4. Rotation Matrix

5. Elementary Rotations
Rotated by angle a about the z-axis

6. Rotation Matrices

7. Vector Representation

8. Example

9. Rotation Matrices Composition

10. Successive Rotations

11. Fixed Frame (ZYX)

12. Inverse Problem

13. Rotating Frame (Euler Angles): ZYZ
https://www.youtube.com/watch?v=q0jgqeS_ACM

14. Homogeneous Transformations

15. Kinematics
 Manipulator consists of a
series of links connected
by means of kinematic
pairs of joints
 Base
 End-Effector
 Degrees of Freedom
(DoF)

16. Revolt vs. Prismatic Joints

17. Direct Kinematics
The objective of direct kinematics is to
compute the position and orientation of the
end-effector as a function of joint variables.
https://www.youtube.com/watch?v=jTAtaPrN2CI

18. Direct Kinematics

19. Example

20. Open vs. Closed Chain
 Open-chain
manipulators have n+1
links connected by n
joints. Each joint
provides a single DOF
 Closed-chain
manipulators contain
at least one loop and
therefore the number of
joints l is equal or greater
than number of links n.

21. KUKA KR-270

22. Denavit-Hartenberg (DH) Convention

23. Denavit-Hartenberg Convention
https://www.youtube.com/watch?v=rA9tm0gTln8

24. Denavit-Hartenberg Convention

25. DH Classical vs. Modified

26. Kinematics of Typical Manipulator Structures
1. Three-Link Planar Arm
2. Parallelogram Arm
3. Spherical Arm
4. Anthropomorphic Arm
5. Spherical Wrist
6. Spherical Arm + Spherical wrist (Stanford Manipulator)
7. Anthropomorphic arm + Spherical Wrist
8. DLR Manipulator
9. Humanoid Manipulator

27. Example: Three-Link Planar Arm

28. Example: Three-Link Planar Arm

29. Example: Three-Link Planar Arm

30. Example: Spherical Arm

31. Example: Spherical Arm

32. Example: Anthropomorphic Arm

33. Example: Spherical Wrist

34. Example:
Anthropomorphic Arm with Spherical Wrist

35. Joint Space and Operational Space
 The direct kinematics equation of a manipulator allows the
position and orientation of the end-effector frame to be
expressed as a function of the joint variables w.r.t. the base
frame.
 The position can be given by the coordinates with regard to the
geometry structure while the orientation can be specified in
terms of the Euler angles
Position
Orientation
• It can also be written in a form of the joint space
where
Is the angle for revolt joint and distance for prismatic joint

36. Workspace
Workspace is the reachable configurations of the end effector

37. Kinematic Calibration
 Kinematic calibration techniques are devoted to find
accurate estimates of DH parameters from a series of
measurements on the manipulator’s end effector
pose.

38. Inverse Kinematics
Objective is the determination of the joint
variables corresponding to a given end-
effector position and orientation.
 This is a much more complex problem when
compared to direct kinematics

39. Inverse Kinematic Examples
1. Three-link Planar Arm
2. Manipulators with Spherical Wrists
3. Spherical Arm
4. Anthropomorphic Arm
5. Spherical Wrist

40. Example: Three-link Planer Arm

41. Example: Three-link Planer Arm

42. Example: Three-link Planer Arm
And finally
Gives

43. Example: Three-link Planer Arm

44. Reference
 Siciliano, Sciavicco, Villani, and Oriolo. Robotics:
Modeling, Planning, and Control. Advanced
Textbooks in Control and Signal Processing.
Springer 2009