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Using Mathematica for Matrix Math -- as it Applies in Robotic Kinematics

Using Mathematica for Matrix Math -- as it Applies in Robotic Kinematics. A Lecture Supplement R. Lindeke, Ph. D. UMD - MIE. Starting A worksheet -- . Use Subscript input Icon Matrices are A i. Choose Create Table/Matrix … to add one. We need 4x4 matrices – this can be set!.

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Using Mathematica for Matrix Math -- as it Applies in Robotic Kinematics

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  1. Using Mathematica for Matrix Math -- as it Applies in Robotic Kinematics A Lecture Supplement R. Lindeke, Ph. D. UMD - MIE

  2. Starting A worksheet -- Use Subscript input Icon Matrices are Ai

  3. Choose Create Table/Matrix …to add one We need 4x4 matrices – this can be set!

  4. Let’s enter the 5 Ai Matrices for this device (1 to 4 here)

  5. #5 here: • Then we will Form the FKS: T0n = A1*A2*A3*A4*A5 • I’ll use a software: Mathematica (or DERIVE5/6 on another day!)

  6. After Entering Matrix – Shift+Enter

  7. The Next Matrix Similarly (then all the rest)

  8. And the Rest of Them Yields:

  9. After All the Matrices are entered: • Lets Verify the accuracy of our input – We will set each angle and length (temporally to 0 units – degrees or mm)

  10. Physical Verification (A1) 4th Column states O1 same place as O0 Implies X1 parallel to X0 Implies Y1 ‘anti-parallel’ to Z0 States, Z1 Parallel to Y0

  11. Similarly for A2 to A5

  12. Before Proceeding we must Unset all the variables (angles and length) This Symbol (equal period) is “UnSet” – returns variables to original condition for symbolic math (after Physical Verification is complete)

  13. After Variables are “UnSet” • We must “Pre-process” Parallel Z –Revolute – Consecutive Joint Matrices using FullSimplify • Then we Develop FKS solutions with the proper order of the Ai’s

  14. Simplifying A2A3 Great News: Simplify is NOW SMART Too! Use FullSimplify to capture Proper Solution … And MatrixForm to display! Period to indicate Multiply

  15. Build FKS – correct order – Then Simplify opps – cant read it! Again with the MatrixForm!

  16. Physical Verification of FKS (TOO!) : It Agrees with my Model (at least at the Home Position! (Xn  X0; YnY0; ZnZ0; Origin is at:6,-.5, 4.25)

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