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# Robot Kinematics II

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1. Introduction to ROBOTICS Robot Kinematics II Jizhong Xiao Department of Electrical Engineering City College of New York jxiao@ccny.cuny.edu

2. Outline • Review • Manipulator Specifications • Precision, Repeatability • Homogeneous Matrix • Denavit-Hartenberg (D-H) Representation • Kinematics Equations • Inverse Kinematics

3. Review • Manipulator, Robot arms, Industrial robot • A chain of rigid bodies (links) connected by joints (revolute or prismatic) • Manipulator Specification • DOF, Redundant Robot • Workspace, Payload • Precision • Repeatability How accurately a specified point can be reached How accurately the same position can be reached if the motion is repeated many times

4. Review • Manipulators: Cartesian: PPP Cylindrical: RPP Spherical: RRP Hand coordinate: n: normal vector; s: sliding vector; a: approach vector, normal to the tool mounting plate SCARA: RRP (Selective Compliance Assembly Robot Arm) Articulated: RRR

5. Review • Basic Rotation Matrix

6. Basic Rotation Matrices • Rotation about x-axis with • Rotation about y-axis with • Rotation about z-axis with

7. Review • Coordinate transformation from {B} to {A} • Homogeneous transformation matrix Rotation matrix Position vector Scaling

8. Review • Homogeneous Transformation • Special cases 1. Translation 2. Rotation

9. Review • Composite Homogeneous Transformation Matrix • Rules: • Transformation (rotation/translation) w.r.t. (X,Y,Z) (OLD FRAME), using pre-multiplication • Transformation (rotation/translation) w.r.t. (U,V,W) (NEW FRAME), using post-multiplication

10. Review • Homogeneous Representation • A point in space • A frame in space •  Homogeneous coordinate of P w.r.t. OXYZ

11. Review • Orientation Representation (Euler Angles) • Description of Roll-Pitch-Yaw • A rotation of about the OX axis ( ) -- yaw • A rotation of about the OY axis ( ) -- pitch • A rotation of about the OZ axis ( ) -- roll Z Y X

12. Quiz 1 • How to get the resultant rotation matrix for YPR? Z Y X

13. Quiz 2 • Geometric Interpretation? Orientation of OUVW coordinate frame w.r.t. OXYZ frame Position of the origin of OUVW coordinate frame w.r.t. OXYZ frame Inverse Homogeneous Matrix? Inverse of the rotation submatrix is equivalent to its transpose Position of the origin of OXYZ reference frame w.r.t. OUVW frame

14. Kinematics Model • Forward (direct) Kinematics • Inverse Kinematics

16. Denavit-Hartenberg Convention • Number the joints from 1 to n starting with the base and ending with the end-effector. • Establish the base coordinate system. Establish a right-handed orthonormal coordinate system at the supporting base with axis lying along the axis of motion of joint 1. • Establish joint axis. Align the Zi with the axis of motion (rotary or sliding) of joint i+1. • Establish the origin of the ith coordinate system. Locate the origin of the ith coordinate at the intersection of the Zi & Zi-1 or at the intersection of common normal between the Zi & Zi-1 axes and the Zi axis. • Establish Xi axis. Establish or along the common normal between the Zi-1 & Zi axes when they are parallel. • Establish Yi axis. Assign to complete the right-handed coordinate system. • Find the link and joint parameters

17. Z3 Z1 Z0 Joint 3 X3 Y0 Y1 Z2 d2 Joint 1 X0 X1 X2 Joint 2 Y2 a0 a1 Example • 3 Revolute Joints Link 1 Link 2

18. Z3 Z1 Z0 Joint 3 X3 Y0 Y1 Z2 d2 Joint 1 X0 X1 X2 Joint 2 Y2 a0 a1 Link Coordinate Frames • Assign Link Coordinate Frames: • To describe the geometry of robot motion, we assign a Cartesian coordinate frame (Oi, Xi,Yi,Zi) to each link, as follows: • establish a right-handed orthonormal coordinate frame O0 at the supporting base with Z0 lying along joint 1 motion axis. • the Ziaxis is directed along the axis of motion of joint (i + 1), that is, link (i + 1) rotates about or translates along Zi; Link 1 Link 2

19. Z3 Z1 Z0 Joint 3 X3 Y0 Y1 Z2 d2 Joint 1 X0 X1 X2 Joint 2 Y2 a0 a1 Link Coordinate Frames • Locate the origin of the ith coordinate at the intersection of the Zi & Zi-1 or at the intersection of common normal between the Zi & Zi-1 axes and the Zi axis. • the Xiaxis lies along the common normal from the Zi-1 axis to the Ziaxis , (if Zi-1 is parallel to Zi, then Xiis specified arbitrarily, subject only to Xibeing perpendicular to Zi);

20. Z3 Z1 Z0 Joint 3 X3 Y0 Y1 Z2 d2 Joint 1 X0 X1 X2 Joint 2 Y2 a0 a1 Link Coordinate Frames • Assign to complete the right-handed coordinate system. • The hand coordinate frame is specified by the geometry of the end-effector. Normally, establish Zn along the direction of Zn-1 axis and pointing away from the robot; establish Xn such that it is normal to both Zn-1 and Zn axes. Assign Yn to complete the right-handed coordinate system.

21. Link and Joint Parameters • Joint angle : the angle of rotation from the Xi-1 axis to the Xi axis about the Zi-1 axis. It is the joint variable if joint i is rotary. • Joint distance : the distance from the origin of the (i-1) coordinate system to the intersection of the Zi-1 axis and the Xi axis along the Zi-1 axis. It is the joint variable if joint i is prismatic. • Link length : the distance from the intersection of the Zi-1 axis and the Xi axis to the origin of the ith coordinate system along the Xi axis. • Link twist angle : the angle of rotation from the Zi-1 axis to the Zi axis about the Xi axis.

22. Z3 Z1 Z0 Joint 3 X3 Y0 Y1 Z2 d2 Joint 1 X0 X1 X2 Joint 2 Y2 a0 a1 Example D-H Link Parameter Table : rotation angle from Zi-1 to Zi about Xi : distance from intersection of Zi-1 & Xi to origin of i coordinate along Xi : distance from origin of (i-1) coordinate to intersection of Zi-1 & Xi along Zi-1 : rotation angle from Xi-1to Xi about Zi-1

23. Transformation between i-1 and i • Four successive elementary transformations are required to relate the i-th coordinate frame to the (i-1)-th coordinate frame: • Rotate about the Z i-1 axis an angle of i to align the X i-1 axis with the X iaxis. • Translate along the Zi-1 axis a distance of di, to bring Xi-1 and Xi axes into coincidence. • Translate along the Xiaxis a distance of aito bring the two origins Oi-1 and Oi as well as the X axis into coincidence. • Rotate about the Xiaxis an angle of αi( in the right-handed sense), to bring the two coordinates into coincidence.

24. Transformation between i-1 and i • D-H transformation matrix for adjacent coordinate frames, i and i-1. • The position and orientation of the i-th frame coordinate can be expressed in the (i-1)th frame by the following homogeneous transformation matrix: Source coordinate Reference Coordinate

25. Kinematic Equations • Forward Kinematics • Given joint variables • End-effector position & orientation • Homogeneous matrix • specifies the location of the ith coordinate frame w.r.t. the base coordinate system • chain product of successive coordinate transformation matrices of Position vector Orientation matrix

26. Kinematics Equations • Other representations • reference from, tool frame • Roll-Pitch-Yaw representation for orientation

27. Solving forward kinematics • Forward kinematics • Transformation Matrix

28. Solving forward kinematics • Roll-Pitch-Yaw representation for orientation Problem? Solution is inconsistent and ill-conditioned!!

29. y x atan2(y,x)

30. Roll-Pitch-Yaw Representation

31. Roll-Pitch-Yaw Representation (Equation A)

32. Roll-Pitch-Yaw Representation • Compare LHS and RHS of Equation A, we have:

33. Kinematic Model • Steps to derive kinematics model: • Assign D-H coordinates frames • Find link parameters • Transformation matrices of adjacent joints • Calculate Kinematics Matrix • When necessary, Euler angle representation

34. Z3 Z1 Z0 Joint 3 X3 Y0 Y1 Z2 d2 Joint 1 X0 X1 X2 Joint 2 Y2 a0 a1 Example

35. Example

36. Example: Puma 560

37. Example: Puma 560