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Manipulator Dynamics 2

Manipulator Dynamics 2. Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA. Forward Dynamics. Problem Given: Joint torques and links geometry, mass, inertia, friction

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Manipulator Dynamics 2

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  1. Manipulator Dynamics 2 Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

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

  4. Dynamics - Newton-Euler Equations • To solve the Newton and Euler equations, we’ll need to develop mathematical terms for: • The linear acceleration of the center of mass • The angular acceleration • The Inertia tensor (moment of inertia) - The sum of all the forces applied on the center of mass - The sum of all the moments applied on the center of mass Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  5. Iterative Newton-Euler Equations - Solution Procedure • Step 1 - Calculate the link velocities and accelerations iteratively from the robot’s base to the end effector • Step 2 - Write the Newton and Euler equations for each link. Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  6. Iterative Newton-Euler Equations - Solution Procedure • Step 3 - Use the forces and torques generated by interacting with the environment (that is, tools, work stations, parts etc.) in calculating the joint torques from the end effector to the robot’s base. Instructor: Jacob Rosen Advanced Robotic - MAE 263D - 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 . 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 263D - Department of Mechanical & Aerospace Engineering - UCLA

  8. Moment of Inertia / Inertia Tensor Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  9. Moment of Inertia – Intuitive Understanding Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  10. Moment of Inertia – Intuitive Understanding Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  11. Moment of Inertia – Intuitive Understanding Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  12. Moment of Inertia – Intuitive Understanding Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  13. Moment of Inertia – Particle – WRT Axis Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  14. Moment of Inertia – Solid – WRT Axis Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  15. Moment of Inertia – Solid – WRT Frame Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  16. Moment of Inertia – Solid – WRT an Arbitrary Axis Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  17. Moment of Inertia – Solid – WRT an Arbitrary Axis Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  18. Moment of Inertia – Solid – WRT an Arbitrary Axis Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  19. Moment of Inertia – Solid – WRT an Arbitrary Axis • For a rigid body that is free to move in a 3D space there are infinite possible rotation axes • The intertie tensor characterizes the mass distribution of the rigid body with respect to a specific coordinate system Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  20. Inertia Tensor • For a rigid body that is free to move in a 3D space there are infinite possible rotation axes • The intertie tensor characterizes the mass distribution of the rigid body with respect to a specific coordinate system • The intertie Tensor relative to frame {A} is express as a matrix Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  21. Inertia Tensor Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  22. Tensor of Inertia – Example • This set of six independent quantities for a given body, depend on the position and orientation of the frame in which they are defined • We are free to choose the orientation of the reference frame. It is possible to cause the product of inertia to be zero • The axes of the reference frame when so aligned are called the principle axes and the corresponding mass moments are called the principle moments of intertie Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  23. Tensor of Inertia – Example Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  24. Tensor of Inertia – Example Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  25. Tensor of Inertia – Example Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  26. Tensor of Inertia – Example Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  27. Tensor of Inertia – Example Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  28. Parallel Axis Theorem – 1D • The inertia tensor is a function of the position and orientation of the reference frame • Parallel Axis Theorem – How the inertia tensor changes under translation of the reference coordinate system Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  29. Parallel Axis Theorem – 3D Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  30. Parallel Axis Theorem – 3D Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  31. Inertia Tensor Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  32. Tensor of Inertia – Example Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  33. Rotation of the Inertia Tensor • Given: • The inertia tensor of the a body expressed in frame A • Frame B is rotate with respect to frame A • Find • The inertia tensor of frame B • The angular Momentum of a rigid body rotating about an axis passing through is • Let’s transform the angular momentum vector to frame B • Plugging the expression of in the expression of results in Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  34. Rotation of the Inertia Tensor • Expend the expression • The relationship between the tensor of inertia expressed in frame A to the tensor of inertia expressed in frame B is given by Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  35. Inertia Tensor 2/ • The elements for relatively simple shapes can be solved from the equations describing the shape of the links and their density. However, most robot arms are far from simple shapes and as a result, these terms are simply measured in practice. Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  36. Inertia Tensor 2/ Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  37. Inertia Tensor 2/ Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  38. Inertia Tensor – Robotic Links Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  39. Inertia Tensor – Robotic Links Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  40. Inertia Tensor – Robotic Links Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  41. Inertia Tensor – Robotic Links Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  42. Inertia Tensor – Robotic Links Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  43. Inertia Tensor – Robotic Links Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  44. Inertia Tensor – Robotic Links Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  45. Inertia Tensor – Robotic Links Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  46. Inertia Tensor – Robotic Links Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  47. Inertia Tensor – Robotic Links Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

  48. Inertia Tensor – Robotic Links Instructor: Jacob Rosen Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

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