Jane Li. Assistant Professor Mechanical Engineering Department, Robotic Engineering Program Worcester Polytechnic Institute

Transcription

1 Jane Li Assistant Professor Mechanical Engineering Department, Robotic Engineering Program Worcester Polytechnic Institute

2 What are the DH parameters for describing the relative pose of the two frames? The DH parameter table describes frame transforms (a) about a fixed frame or (b) about a moving frame? Why the forward kinematics for a parallel mechanism is harder to solve than its inverse kinematics? Explain via a 3 DOF parallel mechanism example? 2

3 Link twist a Link length a Joint angle q Link offset d 3

4 g b a 4

5 a b g 5

6 Rotate about a fixed frame Rotate about a moving frame 6

7 Determine the end-effector position P Inverse Kinematics Forward Kinematics 7

8 You can work in team, but not more than four Everyone needs to contribute an effort Shared credit 8

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10 Inverse kinematics Exact solutions Approximate solutions IK solvers 10

11 Joint space Task space FK = Joint space Task space IK = Task space Joint space End-Effector pose* (point is selected arbitrarily) Joint Link 11

12 Baxter 7 DOF (arm) A manipulator robot at singularity Puma DOF 12

13 Joint position level Joint angle end-effector pose First order derivative Joint velocity end-effector velocity Second order derivative Joint acceleration end-effector acceleration (inverse dynamics) 13

14 Exact solutions Analytical solution Algebraic solution Inverse and Pseudo-inverse Approximate solution Iterative Jacobian Pseudo-Inverse 14

15 All about geometry Need a cheat sheet? If a non-singular solution exists 15

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17 EE position Joint angle 17

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23 Forget about geometry, look at your homogeneous transform matrix 23

24 I know you love math 12 Equations 24

25 Let s do it then 25

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28 You have the template for each frame transform 28

29 These are useful 29

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33 If we square and add the following equations 33

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37 Resolve all the 6 joint angles Program in Matlab Your hint 37

38 x FK (q) dx d[ FK( q)] dfk( q) dt dt dq dq dt dx dt J( q) dq dt dfk( q) dq J ( q) 38

39 9/13/2017 RBE 595 Synergy of Human and Robotic Systems Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 39 q q J x ) ( q z q z q z q q x q q x q q x q J n n n Revolute Joint k 1 Prismatic Joint k 0 k Revolute joint Prismatic joint

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41 No problem, pseudo-inverse 41

42 ሶ ሶ ሶ ሶ ሶ While true end X current = FK(q current ) x = (x target - x current ) error = x ሶ If error < threshold return Success q = J q + x If( q ሶ > a) q = a( q ሶ / q ሶ ) q current = q current - q Iterative Jacobian Pseudo-Inverse F 0 Configuration q current X current X target 42

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44 Most of the IK solvers are not very much more than using Jacobian pseudo-inverse For kinematically redundant robots, numerical IK solvers will require you to specify constraints or optimization objectives, in order to find a unique solution 44

45 Pseudo-inverse can find a unique solution for a redundant manipulator robot, without asking you how to choose Hidden objective function? Pseudo-inverse = least norm solution Magic? 45

46 Analytical solution? Fast & accurately Need to solve a math puzzle (Pseudo)-Inverse Jacobian Kinetically redundant? Good luck Iterative Inverse Jacobian Approximate solution Not as fast 46

47 Specify your own optimization objectives or constraints? Yes, you can. I don t like pseudoinverse because I want to be more human-like But what are they? 47

48 Don t forget finish resolving all the 6 joint angles for puma Using algebraic method Code in Matlab Implement an iterative Jacobian pseudo-inverse solver 48