Robotics kinematics: D-H Approach

1. Robotics kinematics: D-H Approach Handout 4

2. General idea for robot kinematics: revisit Handout 4

3. End-effector World frame Handout 4

4. Robotics kinematics: Definition, Motor and End-effector • Each component has a coordinate system or frame. • Kinematics reduces to the relationship between the frames. • Further, if one frame is set up on the ground called world frame, the “absolute” position and orientation of the end-effector is known. The relationship between different frames = kinematics Handout 4

5. How to set up or assign a local frame to each component of the robot? What is called a component? What is called a joint? Handout 4

6. Robot Kinematics: Logics of presentation: Kinematics: what Coordinate system: way to describe motion Relation between two coordinate systems Definition of component and joint: robot structure Handout 4

7. World frame Handout 4

8. Link: Component with only considering its joint line but neglecting its detailed shape. Next slide (Fig. 2-21) shows various types of joints Handout 4

9. Fig. 2-21 Joint types Kinematic pair types Neglecting the details of the joint but relative motions or relative constraints between two connected links Degrees of freedom of joint: the number of relative motions between two links that are in connection Handout 4

10. General configuration of link Fig.2-22 Handout 4

11. From axis i-1 to axis i The geometrical parameters of the general link are: - The mutual perpendicular distance: a i-1 - The link twist: i-1 From axis i-1 to axis i Fig. 2-23 shows two links that are connected, which leads to the following geometrical parameters: - d i link offset - joint angle From axis a(i-1) to axis a(i) along axis i From axis a(i-1) to axis a(i) Handout 4

12. Fig. 2-23 Handout 4

13. Denavit-Hartenberg (D-H) notation for describing robot kinematic geometry. • It has the benefit that only four parameters describe completely robot kinematic geometry. The above four parameters define the geometry of Link (i-1). • The shortcoming is that the four parameters defined across two links, e.g., for Link (i-1) in the above, the four parameters are defined based on Link (1-1) and Link (i). Handout 4

14. Alternative way to define D-H parameters Definition of DH parameters for Link (i) will cross two links as well, that is, Link (i) and Link (i-1). In this class, we take the previous one. Handout 4

15. Labeling of links: towards a unified representation • The base link or ground 0. • The last link n. • For other links, 1, 2, …., n-1. Handout 4

16. Robot Kinematics: Logics of presentation: Kinematics: what Coordinate system: way to describe motion Relation between two coordinate systems Definition of component and joint: robot structure Assign a local frame to each link (D-H notation) Handout 4

17. Rule to assign a frame to each link (intermediate links) • The Z-axis of frame (i), Zi is coincident with the joint axis i. The origin of frame (i) is located at the intersection point on axis i of the common perpendicular line between axis i and axis i+1. • Xi points along the common perpendicular line between axis i and axis I+1, particularly directed from axis i to axis i+1. In the case that the common perpendicular distance is zero, Xi is normal to the plane which is spanned by axis Zi and Zi+1. • Yi is formed by the right-hand rule based on Xi and Zi. Handout 4

18. Fig. 2-24 An example for link i-1 and link i Handout 4

19. The rule for the D-H coordinates of frame (0) and frame (n): • For frame (0), the rule is as follows: • Define Z0 coincident with Z1 such that ao= 0.0. • Define X0 such that αo = 0.0. • Additionally, define the origin of frame (0) such that d1= 0.0 if joint 1 is revolute, or θ1 = 0.0 if joint 1 is prismatic. • For frame (n), the rule is as follows: • Define Xn such that αn = 0.0. • Z axes are all normal to the paper plane, including Z0. • Z0 is coincident with Z0 and Z1, so a0=0.0, α0 = 0.0. • X0 is set such that d0=0.0. • X3 is set such that d3=0. Handout 4

20. Summaryofthe D-H parameters • If the link frames have been attached to the links following the foregoing convention, the definitions of the link parameters are (for link i): • ai : the distance from Zi to Zi+1, measured along Xi. • di : the distance from Xi to Xi-1 measured along Zi. • αi : the angle between Zi, and Z i+1measured about Xi. • Θi : the angle between X i-1 and Xi, measured about Zi. • Remark: Choose ai> 0 since it corresponds to the distance; however, other three parameters could be a number with signs (plus, minus). Handout 4

21. A note about non-uniqueness in assignment of D-H frames: The convention outlined above may not result in a unique assignment of the frame to the link. There are two choices of the direction of Zi when defining Zi axis with joint axis i. When axes i and i+1 are parallel, there are multiple choices of the location of the origin for frame (i). When axes i and i+1 are in intersection, there are two choices of the direction of Xi. When Zi and Zi+1 are coincident, there are multiple choices for Xi as well as for the location of the origin of frame (0). Handout 4

22. Example 1 Fig.2-25 Handout 4

23. Fig.2-26 Link 0 Link 1 Link 2 Handout 4

24. Example 2 Handout 4

25. Handout 4

26. Parameter table to be given in the classroom Handout 4

27. Summary Link and joint concept. D-H notation for link. Assign frames to links based on D-H. Benefit of D-H: a minimum number of parameters to describe links and joints. Shortcoming of D-H: parameters must cross two consecutively connected links. Handout 4

28. Robot Kinematics: Logics of presentation: Kinematics: what Coordinate system: way to describe motion Relation between two coordinate systems Definition of component and joint: robot structure Assign a local frame to each link (D-H notation) Kinematic equation Handout 4

29. Robot kinematics: The relationship among the D-H frames In the previous discussion, D-H frames are established, that is, we have 0, 1, 2, …, n frames established based on the D-H notation and rule. In this slide, we discuss the mathematical representation for this relationship. Handout 4

30. The goal is to find the relationship matrix for frame i-1 and frame i Handout 4

31. The idea is to put a series of frames between them, denoting them as FR, FQ, FP. As such, frame i-1  FR  FQ  FP  frame i. Handout 4

32. The idea is to put a series of frames between them, denoting them as FR, FQ, FP. As such, frame i-1  FR  FQ  FP  frame i. T from i-1 to FR T from FQ to FP T from FR to FQ T from FP to i Handout 4

33. Transformation matrix between two DH frames Handout 4

34. Forward kinematics • General idea: suppose that we have n moving likes. • Solution: forward kinematics, given the motor’s motion, to find the position and orientation of the end-effector. • The position of the end-effector and the orientation of the end-effector completely describe the end-effector. • The position of the end-effector can be the position of the origin of the frame (on the end-effector). • The orientation of the end-effector is represented in the R matrix between the frame (on the end-effector) and the world frame or frame to the ground, i.e., {0}. Handout 4

35. The problem is: known the right side variable to find the left side variable. Handout 4

36. Inverse kinematics Handout 4

37. The problem is: known the left side variable to find the right side variable. Handout 4

38. Example 1 Handout 4

39. Handout 4

40. T matrix here Handout 4

41. Forward kinematics Handout 4

42. Inverse kinematics Handout 4

43. Summary Transformation matrix between two DH frames. General equations for forward kinematics. General equations for inverse kinematics. Handout 4