Robotic Manipulator Dynamics Examples

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Explore iterative Newton-Euler equations in robotics with examples on forward dynamics, inverse dynamics, and the algorithm steps for iterative calculations provided by Jacob Rosen, an instructor at UCLA.

  • Robotics
  • Manipulator Dynamics
  • Newton-Euler Equations
  • Advanced Robotics
  • Iterative Algorithm
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  1. Manipulator Dynamics 3 Iterative Newton Euler Equations 2R Example Instructor: Jacob Rosen Advanced Robotic - MAE 263 - Department of Mechanical & Aerospace Engineering - UCLA

  2. Forward Dynamics ?? ?????(?,?,?) ?? ?? ??? ?? ?? Problem Given:Joint torques and links geometry, mass, inertia, friction, joint torques Compute: Angular acceleration of the links (solve differential equations) Solution solve a set of differential equations Dynamic Equations - Newton-Euler method or Lagrangian Dynamics ? = ?( ) + ?( , ) + ?( ) + ?( , ) Instructor: Jacob Rosen Advanced Robotic - MAE 263 - Department of Mechanical & Aerospace Engineering - UCLA

  3. Inverse Dynamics ?????(?,?,?) ?? ?? ??? ?? ?? Problem Given:Angular acceleration, velocity and angels of the linksin addition to the links geometry, mass, inertia, friction Compute: Joint torques ? Solution Solve a set of algebraic equations Dynamic Equations - Newton-Euler method or Lagrangian Dynamics ? = ?( ) + ?( , ) + ?( ) + ?( , ) Instructor: Jacob Rosen Advanced Robotic - MAE 263 - Department of Mechanical & Aerospace Engineering - UCLA

  4. Iterative Newton Euler Equations Steps of the Algorithm (1) Outward Iterations Starting With velocities and accelerations of the base (? = 0 ? 1) (? = 0 ? 1) ? ? ? ??? ? ? 0?0= 0, 0?0= 0, 0?0= 0, 0?0= +? ? 0?0= 0 0?0= 0 0?0= 0 0?0= +? ? Calculate velocities accelerations, along with forces and torques (at the CM) ?, ?, ?, ???,?,? (? = ? 1) (2) Inward Iteration Starting with forces and torques (at the CM) (? = ? 1) ? ? ?,? Calculate forces and torques at the joints ?,? Instructor: Jacob Rosen Advanced Robotic - MAE 263B - Department of Mechanical & Aerospace Engineering - UCLA

  5. Iterative Newton-Euler Equations - Solution Procedure Phase 1: Outward Iteration Outward Iteration: ? : 0 5 Calculate the link velocities and accelerations iteratively from the robot s base to the end effector ?+1? ???+ ??+1 ?+1 ??+1 ?+1??+1= ? ?+1? ??? ??+1 ?+1 ??+1+ ??+1 ?+1? ? ??+ ?+1 ??+1 ?+1 ??+1= ? ? ?+1?( ?+1 ??+1= ???+1+ ??? ( ??? ???+1) + ? ??) ??? ? ?+1 ???+1= ?+1 ??+1 ?+1???+1+ ?+1??+1 ( ?+1??+1 ?+1???+1) + ?+1 ??+1 Calculate the force and torques applied on the CM of each link using the Newton and Euler equations ?+1??+1= ??+1 ?+1 ???+1 C C + + + + + + = + 1 1 1 1 1 1 i i i i i i N I I + + + + + + 1 1 1 1 1 1 i i i i i i Instructor: Jacob Rosen Advanced Robotic - MAE 263B - Department of Mechanical & Aerospace Engineering - UCLA

  6. Iterative Newton-Euler Equations - Solution Procedure Phase 2: Inward Iteration Inward Iteration: ? : 6 1 Use the forces and torques generated at the joints starting with forces and torques generating by interacting with the environment (that is, tools, work stations, parts etc.) at the end effector all the way the robot s base. ?? ???=?+1 ?+1??+1+??? ?? ?? ???= ???+?+1 ?+1??+1+ ???? ???+ ???+1 ?+1 ?+1??+1 ?+1 ?+1?? ??? ??= Instructor: Jacob Rosen Advanced Robotic - MAE 263B - Department of Mechanical & Aerospace Engineering - UCLA

  7. Iterative Newton-Euler Equations - Solution Procedure Error Checking - Check the units of each term in the resulting equations Gravity Effect - The effect of gravity can be included by setting 0?0= ? . This is the equivalent to saying that the base of the robot is accelerating upward at 1 g. The result of this accelerating is the same as accelerating all the links individually as gravity does. Instructor: Jacob Rosen Advanced Robotic - MAE 263 - Department of Mechanical & Aerospace Engineering - UCLA

  8. Iterative Newton-Euler Equations - 2R Robot Example Instructor: Jacob Rosen Advanced Robotic - MAE 263 - Department of Mechanical & Aerospace Engineering - UCLA

  9. Iterative Newton-Euler Equations - 2R Robot Example ?1 ?1 0 0 ?2 ?2 0 0 ?1 ?1 0 0 ?2 ?2 0 0 0 0 1 0 0 0 1 0 0 0 0 1 ?1 0 0 1 ?2 0? = ?2 1 ?2 ?2 ?2 ?0 1? = ?1 2 ?1 ?1 ?1 ?1 ?0 Vectors locates the center of mass for each link ?1 0 0 ?2 0 0 ?1?1= 0 ?2?2= 0 1??1= ?1 ?1= 2??2= ?2 ?2= Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  10. Iterative Newton-Euler Equations - 2R Robot Example No External force/torque on the end effector ?3= 0 ?3= 0 The base of the robot is not rotating ?0= 0 ?0= 0 To include gravity 0 ? 0 0 ?0= ? ?0= Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  11. Iterative Newton-Euler Equations - Solution Procedure Phase 1: Outward Iteration Outward Iteration: ? = 0 Calculate the link velocities and accelerations iteratively from the robot s base to the end effector ?+1? ???+ ??+1 ?+1 ??+1 ?+1??+1= ? ?+1? ??? ??+1 ?+1 ??+1+ ??+1 ?+1? ? ??+ ?+1 ??+1 ?+1 ??+1= ? ? ?+1?( ?+1 ??+1= ???+1+ ??? ( ??? ???+1) + ? ??) ??? ? ?+1 ???+1= ?+1 ??+1 ?+1???+1+ ?+1??+1 ( ?+1??+1 ?+1???+1) + ?+1 ??+1 Calculate the force and torques applied on the CM of each link using the Newton and Euler equations ?+1??+1= ??+1 ?+1 ???+1 C C + + + + + + = + 1 1 1 1 1 1 i i i i i i N I I + + + + + + 1 1 1 1 1 1 i i i i i i Instructor: Jacob Rosen Advanced Robotic - MAE 263B - Department of Mechanical & Aerospace Engineering - UCLA

  12. Iterative Newton-Euler Equations - 2R Robot Example Outward Iteration ? = 0 ?+1? ???+ ??+1 ?+1 ??+1 ?+1??+1= ? 0 0 0 ?1 0?0+ ?1 ?1= 1? 1?1= 0 Instructor: Jacob Rosen Advanced Robotic - MAE 263 - Department of Mechanical & Aerospace Engineering - UCLA

  13. Iterative Newton-Euler Equations - 2R Robot Example Outward Iteration ? = 0 ?+1? ??? ??+1 ?+1 ??+1+ ??+1 ?+1? ? ??+ ?+1 ??+1 ?+1 ??+1= ? ? 0 0 0 0 ?1 0 0 1 0?0 ?1 1 ?1+ ?1 1 ?1= ?1 1? 1? 1 ?1=0 0 ?0+0 = Instructor: Jacob Rosen Advanced Robotic - MAE 263 - Department of Mechanical & Aerospace Engineering - UCLA

  14. Iterative Newton-Euler Equations - 2R Robot Example Outward Iteration ? = 0 ?+1?( ?+1 ??+1= ???+1+ ??? ( ??? ???+1) + ? ??) ??? ? 0 0 0 ??1 ??1 0 ?1 ?1 0 ?1 ?1 0 0 0 1 0 ? 0 1? 1 ?1=0 0?1+ 0?0 0?0 0?1 + 0 ?0 = = 0?0 Instructor: Jacob Rosen Advanced Robotic - MAE 263 - Department of Mechanical & Aerospace Engineering - UCLA

  15. Iterative Newton-Euler Equations - 2R Robot Example Outward Iteration ? = 0 ?+1 ???+1= ?+1 ??+1 ?+1???+1+ ?+1??+1 ( ?+1??+1 ?+1???+1) + ?+1 ??+1 1 ??1= 1 ?1 ? 0 0 ??1 ??1 0 ??1 ??1 0 1??1+ ? ?1 0 1?1 ??1 ??1 0 1?1 1??1 1 ?1= + ? 0 ?1 ? 0 0 ? ?1 0 ? 0 ?1 1?1 + = + 0 0 + 1?1 + = ?1 ?1 0 0 ?1 ?1 0 ?1 ?1 0 ? 0 0 ? 0 ? ?1 0 2+ ??1 2+ ??1 0 ??1 ??1 0 2 ?1 ?1 ?1 ?1 0 ?1 ?1 0 0 + + = + + = ?1 ?1 0 ?1 ?1 Instructor: Jacob Rosen Advanced Robotic - MAE 263 - Department of Mechanical & Aerospace Engineering - UCLA

  16. Iterative Newton-Euler Equations - Solution Procedure Phase 1: Outward Iteration Outward Iteration: ? : 0 Calculate the link velocities and accelerations iteratively from the robot s base to the end effector ?+1? ???+ ??+1 ?+1 ??+1 ?+1??+1= ? ?+1? ??? ??+1 ?+1 ??+1+ ??+1 ?+1? ? ??+ ?+1 ??+1 ?+1 ??+1= ? ? ?+1?( ?+1 ??+1= ???+1+ ??? ( ??? ???+1) + ? ??) ??? ? ?+1 ???+1= ?+1 ??+1 ?+1???+1+ ?+1??+1 ( ?+1??+1 ?+1???+1) + ?+1 ??+1 Calculate the force and torques applied on the CM of each link using the Newton and Euler equations ?+1??+1= ??+1 ?+1 ???+1 C C + + + + + + = + 1 1 1 1 1 1 i i i i i i N I I + + + + + + 1 1 1 1 1 1 i i i i i i Instructor: Jacob Rosen Advanced Robotic - MAE 263B - Department of Mechanical & Aerospace Engineering - UCLA

  17. Iterative Newton-Euler Equations - 2R Robot Example Outward Iteration ? = 0 ?+1??+1= ??+1 ?+1 ???+1 1?1? 2+ ??1 2+ ?1??1 ?1 ?1 ?1 ?1+ ??1 0 ?1?1 ?1 ?1?1 ?1+ ?1??1 0 1?1= ?1 1 ??1= = 1?1? Instructor: Jacob Rosen Advanced Robotic - MAE 263 - Department of Mechanical & Aerospace Engineering - UCLA

  18. Iterative Newton-Euler Equations - 2R Robot Example Outward Iteration ? = 0 ?+1??+1= ??+1??+1 ?+1??+1 ??+1??+1 ?+1??+1 ?+1??+1+ 0 0 1?1= ?1?1 1?1 ?1?1 1?1 1?1+ Instructor: Jacob Rosen Advanced Robotic - MAE 263 - Department of Mechanical & Aerospace Engineering - UCLA

  19. Iterative Newton-Euler Equations - Solution Procedure Phase 1: Outward Iteration Outward Iteration: ? = 1 Calculate the link velocities and accelerations iteratively from the robot s base to the end effector ?+1? ???+ ??+1 ?+1 ??+1 ?+1??+1= ? ?+1? ??? ??+1 ?+1 ??+1+ ??+1 ?+1? ? ??+ ?+1 ??+1 ?+1 ??+1= ? ? ?+1?( ?+1 ??+1= ???+1+ ??? ( ??? ???+1) + ? ??) ??? ? ?+1 ???+1= ?+1 ??+1 ?+1???+1+ ?+1??+1 ( ?+1??+1 ?+1???+1) + ?+1 ??+1 Calculate the force and torques applied on the CM of each link using the Newton and Euler equations ?+1??+1= ??+1 ?+1 ???+1 C C + + + + + + = + 1 1 1 1 1 1 i i i i i i N I I + + + + + + 1 1 1 1 1 1 i i i i i i Instructor: Jacob Rosen Advanced Robotic - MAE 263B - Department of Mechanical & Aerospace Engineering - UCLA

  20. Iterative Newton-Euler Equations - 2R Robot Example Outward Iteration ? = 1 ?+1? ???+ ??+1 ?+1 ??+1 ?+1??+1= ? 0 0 ?1 0 0 ?2 0 0 ?2 ?2 0 ?2 ?2 0 0 0 1 1?1+ ?2 2 ?2= 2? 2?2=2 + = ?1+ ?2 Instructor: Jacob Rosen Advanced Robotic - MAE 263 - Department of Mechanical & Aerospace Engineering - UCLA

  21. Iterative Newton-Euler Equations - 2R Robot Example Outward Iteration ? = 1 ?+1? ??? ??+1 ?+1 ??+1+ ??+1 ?+1? ? ??+ ?+1 ??+1 ?+1 ??+1= ? ? 2? ??? ?2 0 0 1 ?1 2 ?2+ ?2 0 0 ?2 2 ?2 0 0 ?2 2? ?2 ?2 0 2 ?2=1 0 0 ?1 1 ?1+1 ?2 ?2 0 0 0 ?2 ?2 0 0 0 ?1 ?2 ?2 0 0 0 1 + + = ? 0 0 ? 0 0 ? ?1 ?2 0 0 ?2 0 0 + = ?1+ ?2 Instructor: Jacob Rosen Advanced Robotic - MAE 263 - Department of Mechanical & Aerospace Engineering - UCLA

  22. Iterative Newton-Euler Equations - 2R Robot Example Outward Iteration ? = 1 ?+1?( ?+1 ??+1= ???+1+ ??? ( ??? ???+1) + ? ??) ??? ? 2? 2 ?2=1 1?2+ 1?1 1?1 1?2 + 1 ?1= 1?1 ? 0 ?2 ? 0 0 ? ?1 0 ? 0 0 ?1 ?1 ? 0 ?1 ? 0 0 ??1 ??1 0 ? ?1 0 0 0 ?1 ? 0 ??1 ??1 0 ?2 ?2 0 ?2 ?2 0 ?2 ?2 0 ?2 ?2 0 ?2 ?2 0 ?2 ?2 0 0 0 1 0 0 1 0 0 1 = + + ? ?1 0 ??1 ??1 0 0 = + + ?1 ?1 0 0 ?1 ?1 0 ?1 ?2 2 2+ ??1 2?2+ ??12 2?2+ ??12 ?1 ?1 ?1 ?1+ ??1 0 ?1 ?1?2 ?1 ?1 ?1 ?1?2+ ?1 ?1 ?2 ?2 0 ?2 ?2 0 0 0 1 = = + + 0 0 0 Instructor: Jacob Rosen Advanced Robotic - MAE 263 - Department of Mechanical & Aerospace Engineering - UCLA

  23. Iterative Newton-Euler Equations - 2R Robot Example Outward Iteration ? = 1 ?+1 ???+1= ?+1 ??+1 ?+1???+1+ ?+1??+1 ( ?+1??+1 ?+1???+1) + ?+1 ??+1 2 ??2= ? 0 ?2 2 ?2 2??2+ 2?2 0 0 ?1+ ?2 2?2 ? 0 ?2 2??2 ? 0 0 ? 2 ?2= + ? ? 0 0 0 ? 2?2+ ??12 2?2+ ??12 ?1 ?1?2 ?1 ?1 ?1 ?1?2+ ?1 ?1 ?1+ ?2 0 ?1+ ?2 0 + + 0 ? 0 0 ? 0 2?2+ ??12 2?2+ ??12 ?1 ?1?2 ?1 ?1 ?1 ?1?2+ ?1 ?1 ?1+ ?2 0 ?2( ?1+ ?2) 0 + = + ?2( ?1+ ?2) ?2( ?1+ ?2) 2 0 0 0 2+ ??12 ?1+ ?2 2?2 ?2 2?2+ ?2( ?1+ ?2) + ??12 0 2?2+ ??12 2?2+ ??12 ?1 ?1?2 ?1 ?1 0 ?1 ?1?2 ?1 ?1 ?1 ?1?2+ ?1 ?1 ?2( ?1+ ?2) 0 + = + ?1 ?1?2+ ?1 ?1 0 Instructor: Jacob Rosen Advanced Robotic - MAE 263 - Department of Mechanical & Aerospace Engineering - UCLA

  24. Iterative Newton-Euler Equations - Solution Procedure Phase 1: Outward Iteration Outward Iteration: ? : 1 Calculate the link velocities and accelerations iteratively from the robot s base to the end effector ?+1? ???+ ??+1 ?+1 ??+1 ?+1??+1= ? ?+1? ??? ??+1 ?+1 ??+1+ ??+1 ?+1? ? ??+ ?+1 ??+1 ?+1 ??+1= ? ? ?+1?( ?+1 ??+1= ???+1+ ??? ( ??? ???+1) + ? ??) ??? ? ?+1 ???+1= ?+1 ??+1 ?+1???+1+ ?+1??+1 ( ?+1??+1 ?+1???+1) + ?+1 ??+1 Calculate the force and torques applied on the CM of each link using the Newton and Euler equations ?+1??+1= ??+1 ?+1 ???+1 C C + + + + + + = + 1 1 1 1 1 1 i i i i i i N I I + + + + + + 1 1 1 1 1 1 i i i i i i Instructor: Jacob Rosen Advanced Robotic - MAE 263B - Department of Mechanical & Aerospace Engineering - UCLA

  25. Iterative Newton-Euler Equations - 2R Robot Example Outward Iteration ? = 1 ?+1??+1= ??+1 ?+1 ???+1 2+ ??12) ?1+ ?2 2?2 ?1 2?2+ ?2( ?1+ ?2) + ??12) 0 ?2(?1 ?1?2 ?1 ?1 ?2(?1 ?1?2+ ?1 ?1 2?2= ?22 ??2= Instructor: Jacob Rosen Advanced Robotic - MAE 263 - Department of Mechanical & Aerospace Engineering - UCLA

  26. Iterative Newton-Euler Equations - 2R Robot Example Outward Iteration ? = 1 ?+1??+1= ??+1??+1 ?+1??+1 ??+1??+1 ?+1??+1 ?+1??+1+ 0 0 0 0 0 2?2= ?2?2 2?2 ?2?2 2?2= 2?2+ Instructor: Jacob Rosen Advanced Robotic - MAE 263 - Department of Mechanical & Aerospace Engineering - UCLA

  27. Iterative Newton-Euler Equations - Solution Procedure Phase 2: Inward Iteration Inward Iteration: ? = 2 Use the forces and torques generated at the joints starting with forces and torques generating by interacting with the environment (that is, tools, work stations, parts etc.) at the end effector all the way the robot s base. ?? ???=?+1 ?+1??+1+??? ?? ?? ???= ???+?+1 ?+1??+1+ ???? ???+ ???+1 ?+1 ?+1??+1 ?+1 ?+1?? ??? ??= Instructor: Jacob Rosen Advanced Robotic - MAE 263B - Department of Mechanical & Aerospace Engineering - UCLA

  28. Iterative Newton-Euler Equations - 2R Robot Example Inward iteration ? = 2 ?? ???=?+1 ?+1??+1+??? 0 2? 2?2= 3 3?3+ 2?2= 2?2 Instructor: Jacob Rosen Advanced Robotic - MAE 263 - Department of Mechanical & Aerospace Engineering - UCLA

  29. Iterative Newton-Euler Equations - 2R Robot Example Inward iteration ? = 2 ?? ?? ???=???+?+1 ?+1??+1+???? ???+???+1 ?+1 ?+1??+1 0 0 0 2? 2? 2?2= 2?2+3 3?3+ 2??2 2?2+ 2?3 3 3?3 ? ? 0 ? 0 0 0 0 2?2? 0 0 ?2 2?2? = = = 2?2+ ?2( ?1+ ?2) + ??12) 2?2? ?2 ?2?2(?1 ?1?2+ ?1 ?1 2?2 ? Instructor: Jacob Rosen Advanced Robotic - MAE 263 - Department of Mechanical & Aerospace Engineering - UCLA

  30. Iterative Newton-Euler Equations - Solution Procedure Phase 2: Inward Iteration Inward Iteration: ? = 1 Use the forces and torques generated at the joints starting with forces and torques generating by interacting with the environment (that is, tools, work stations, parts etc.) at the end effector all the way the robot s base. ?? ???=?+1 ?+1??+1+??? ?? ?? ???= ???+?+1 ?+1??+1+ ???? ???+ ???+1 ?+1 ?+1??+1 ?+1 ?+1?? ??? ??= Instructor: Jacob Rosen Advanced Robotic - MAE 263B - Department of Mechanical & Aerospace Engineering - UCLA

  31. Iterative Newton-Euler Equations - 2R Robot Example Inward iteration ? = 1 ?? ???=?+1 ?+1??+1+??? 1? ?2 ?2 0 1?1= 2 ?2 ?2 0 2?2+ 0 0 1 1?1= ?2(?1 ?1?2 ?1 ?1 ?2(?1 ?1?2+ ?1 ?1 2+ ??12) 2+ ?1??1 ?1+ ?2 2?2 ?2 2?2+ ?2( ?1+ ?2) + ??12) 0 ?1?1 ?1 ?1?1 ?1+ ?1??1 0 + Instructor: Jacob Rosen Advanced Robotic - MAE 263B - Department of Mechanical & Aerospace Engineering - UCLA

  32. Iterative Newton-Euler Equations - 2R Robot Example Inward iteration ? = 1 ?? ?? ???= ???+?+1 ?+1??+1+ ???? ???+ ???+1 ?+1 ?+1??+1 0 1? 1? ?1 0 0 1?1= ?2 ?2 0 0 0 2?2 ? 1?1+2 ?2 ?2 0 2?2 + 1??1 ? ?1 1?1? 1?1+ ? 0 1?1? 1?2 2 ? 0 0 2?2= 0 0 2?2? 2?2? 0 0 0 1 ?2 ?2 0 ?2 ?2 0 0 0 1 + + = 2?2 ? ? ? 0 ? 0 0 0 0 1?1? + + ?1 ?1 2?2? ?2 2?2? 2?2? + ?2 2?2? ?2 ?2 Instructor: Jacob Rosen Advanced Robotic - MAE 263B - Department of Mechanical & Aerospace Engineering - UCLA

  33. Iterative Newton-Euler Equations - 2R Robot Example Inward iteration ? ? 0 ? 0 0 0 0 1?1? 0 0 1?1 = = ?1 + + 2?2 ? ?1 2?2? ?2 2?2? 2?2? + ?2 2?2? ?2 ?2 = 0 0 0 0 + + 2?2+ ?2( ?1+ ?2) + ??12) ?2?2(?1 ?1?2+ ?1 ?1 ?1(?1?1 ?1+ ?1??1) 0 0 2+ ??12)) + ?2(?2(?1 ?1?2+ ?1 ?1 ?1+ ?2 2?2 ?1 ( ?1+ ?2) + ??12)) 2?2 ?2 ?1(?2(?2(?1 ?1?2 ?1 ?1 Instructor: Jacob Rosen Advanced Robotic - MAE 263B - Department of Mechanical & Aerospace Engineering - UCLA

  34. Iterative Newton-Euler Equations - 2R Robot Example Inward iteration 0 0 + ?1?2?2?1 ?2+ ?2?1 ?1 1?1= ?1+ ?2 ?1+ ?2 2?2+ ?2??12 ?2?2 2?2+ ?2??12 ?2?1 ?1?2?2?1 ?2 ?2?1 ?1 Instructor: Jacob Rosen Advanced Robotic - MAE 263B - Department of Mechanical & Aerospace Engineering - UCLA

  35. Iterative Newton-Euler Equations - Solution Procedure Phase 2: Inward Iteration Inward Iteration: ? = 1,2 Use the forces and torques generated at the joints starting with forces and torques generating by interacting with the environment (that is, tools, work stations, parts etc.) at the end effector all the way the robot s base. ?? ???=?+1 ?+1??+1+??? ?? ?? ???= ???+?+1 ?+1??+1+ ???? ???+ ???+1 ?+1 ?+1??+1 ??? ??= ??? ? Instructor: Jacob Rosen Advanced Robotic - MAE 263B - Department of Mechanical & Aerospace Engineering - UCLA

  36. Iterative Newton-Euler Equations - 2R Robot Example Inward iteration ? = 1,2 0 0 + ?1?2?2?1 ?2+ ?2?1 ?1 1?1= ?1+ ?2 ?1+ ?2 2?2+ ?2??12 ?2?2 2?2+ ?2??12 ?2?1 ?1?2?2?1 ?2 ?2?1 ?1 0 0 2?2 = 2?2+ ?2( ?1+ ?2) + ??12) ?2?2(?1 ?1?2+ ?1 ?1 ??? ??=??? ? 2+ ??12)) + ?2(?2(?1 ?1?2+ ?1 ?1 2?2+ ?2( ?1+ ?2) + ??12) ?1+ ?2 2?2+ ?2( ?1+ ?2) + ??12)) 2?2 ?2 ?2= ?2?2(?1 ?1?2+ ?1 ?1 ?1= ?1(?2(?2(?1 ?1?2 ?1 ?1 Instructor: Jacob Rosen Advanced Robotic - MAE 263B - Department of Mechanical & Aerospace Engineering - UCLA

  37. Iterative Newton-Euler Equations - 2R Robot Example ? + ? ?, ? + ? ? ? = ? ? Mass/Inertia Matrix Gravity C ? [ ?2] + ? ? [ ?] Coriolis Centrifugal Instructor: Jacob Rosen Advanced Robotic - MAE 263B - Department of Mechanical & Aerospace Engineering - UCLA

  38. Iterative Newton-Euler Equations - 2R Robot Example 2+ 2?2?1?2?2+ ?1 ?2?2 2?1+ ?2 2+ ?2?1?2?2 2?2?2 ?1 ?2 ?2?2 ?2?2 ?1 ?2 = + 2+ ?2?1?2?2+ ?2?2 2 2 2 2 + 2?2?1?2?2 ?1 ?2 0 ?2?1?2?2 0 [ ?1 ?2] + ?2?1?2?2 0 ?2?2??12+ ?1+ ?2?1??1 ?2?2??12 Instructor: Jacob Rosen Advanced Robotic - MAE 263B - Department of Mechanical & Aerospace Engineering - UCLA

  39. Equation of Motion Non Rigid Body Effects ? + ? ?, ? + ? ? + ? ?, ? ? = ? ? Viscous Friction ?????????= ? ? ?????????= ? sgn( ?) Coulomb Friction ?????????= ? ? + ? sgn( ?) = ?(?, ?) Model of Friction Instructor: Jacob Rosen Advanced Robotic - MAE 263B - Department of Mechanical & Aerospace Engineering - UCLA

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