1. CHAPTER 2
FORWARD KINEMATIC 1

2. 3.1 Denavit-Hartenberg Algorithm
In 1955, Denavit and Hartenberg, pointed out that any neighboring frames can be brought into coincidence by a prescribed sequence of at most two rotations and two translations.
This is the key point of the D-H algorithm, which widely used in robotics for obtaining the kinematic solutions of robot manipulators.
The D-H model of representation is a very simple way of modeling robot links and joints that can be used for any robot configuration, regardless of its sequence or complexity

3. 3.1 Denavit-Hartenberg Algorithm
4 points should be noted:
Assign coordinate frames to all links and the
tool tip of a robot manipulator
Derive a 4x4 HTM to describe the position and the orientation of each link or the tool tip relative to its neighboring link.
Compute the forward kinematics of the robot manipulator using the post-multiplication rule.
Determine the position and orientation of the robot hand with respect to the base frame using forward kinematics.

4. 3.2 Arm Parameters The robot base is defined as link 0
• The joint 1 connects the base (link 0) to link 1.
• We number links outward from link 0 (the base) to link n (the tool).
• The joint i (1 ≤ i ≤ n) connects the link i-1 to the link i, but there is no joint at the end of the last link.

5. 3.2 Arm Parameters
Figure shows a spatial linkage, where the joint i connects the link i to the link i-1.

6. 3.2 Arm Parameters
The basic rules for the assignment of the frame xiyizi to the link i are as follow:
The zi axis lies along the motion axis of the rotary joint i+ 1.
The Xi axis is normal to both zi-1 and zi axes, and points away from the zi-1 axis.
The yi axis is set to form a right-handed frame xiyizi.
Two parameters of the link i are the link length ai and the twist angle.
Two joint parameters are the joint angle and the joint distance d.

7. 3.3 D-H Parameters Link length ai
The link length ai is defined as the common normal between the zi-1 and zi axes.

8. 3.3 D-H Parameters Twist angle
The twist angle αi is defined as the rotational angle of the zi-1 axis about the xi axis, such that the zi-1 axis will parallel to the zi axis after the rotation.

9. 3.3 D-H Parameters Joint angle
The joint angle θi is the rotational angle of the xi-1 about the zi-1 axis, such that the xi-1 axis will parallel with the xi axis after the rotation.

10. 3.3 D-H Parameters Joint distant
The joint distance di is defined as the translation distance of the frame i-1 along the zi-1 axis, from 0i-1, the origin of the frame i-1 to bi (intersection of xi axis and the zi-1.

11. 3.3 D-H Parameters

12. 3.3 D-H Parameters

13. 3.3 D-H Parameters

14. 3.3 D-H Parameters

15. Table 3.1: The four arm parameters
3.3 D-H Parameters
Table 3.1: The four arm parameters

16. 3.4 Procedures of D-H Algorithm
The following shows the detailed procedure to assign a frame to each link or the tool tip of an n-degrees-of-freedom robot manipulator using the D-H algorithm

17. 3.4 Procedures of D-H Algorithm

18. 3.4 Procedures of D-H Algorithm

19. 3.4 Procedures of D-H Algorithm

20. 3.4 Procedures of D-H Algorithm

21. 3.4 D-H Transformation Matrix
Rotate the frame xi-1yi-1zi-1 about the zi axis by an angle θi.
Translate the frame xi-1yi-1zi-1 along the zi axis by di units.
Translate the frame xi-1yi-1zi-1 along the xi axis by ai units.
Rotate the frame xi-1yi-1zi-1 about the xi axis by an angle αi

22. 3.4 D-H Transformation Matrix

23. 3.4 Forward Kinematic Solution
The transformation matrix describes the position and orientation of the ith link with respect to the i-1th link.
To describe the position and orientation of the robot tool frame with respect to the robot base frame, the following is used:
As a conclusion, the D-H algorithm is used to fulfill the following tasks:
Assign coordinate frames to all links and the tool tip of a robot.
Determine the arm parameters for each link.
Derive a transformation matrix for each link and the tool tip.
Compute the solution of the forward kinematics.

24. 3.4 Examples of D-H Algorithm

25. 3.4 Examples of D-H Algorithm# θ d a α 90 r

26. 3.4 Examples of D-H Algorithm

27. 3.4 Examples of D-H Algorithm# θ d a α θ1 90 θ2 e θ3 f

28. 3.4 Examples of D-H Algorithm
A simple 6-DOF articulate robot

29. 3.4 Examples of D-H Algorithm# θ d a α 1 θ1 90 2 θ2 a2 3 θ3 a3 4 θ4 a4
-90 5 θ5 6 θ6

30. 3.4 Examples of D-H Algorithm
A simple 6-DOF articulate robot

31. 3.4 Examples of D-H Algorithm

32. 3.4 Examples of D-H Algorithm# θ d a α θ1 a1
90+θ2 90 θ3 d3

33. 3.4 Examples of D-H Algorithm