Yasser F. O. Mohammad REMINDER 1: The whole Jacobian (METHOD 1) - - PowerPoint PPT Presentation

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Oct 19, 2023 765 likes •1.02k views

Yasser F. O. Mohammad REMINDER 1: The whole Jacobian (METHOD 1) Revolute Joint Prismatic Joint REMINDER 2: Wrist Singularities Whenever z 3 and z 5 are aligned Prove it REMINDER 3: Elbow Singularity REMINDER 4: Resolved Motion

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Yasser F. O. Mohammad

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REMINDER 1: The whole Jacobian (METHOD 1)

 Revolute Joint  Prismatic Joint

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REMINDER 2: Wrist Singularities

 Whenever z3 and z5 are aligned  Prove it

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REMINDER 3: Elbow Singularity

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REMINDER 4: Resolved Motion Rate Control (Whitney 1972)

From Osama Khatib

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REMINDER 5: How to calculate J+

 Most difficult method (from definition):  Simplest Method (SVD):

 

1 T T

J J J J

  

T T

J U V J U V

 

   

1/ , when

ij ij ij

  

 



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REMINDER 6: Manipulability

1 m ii i

 





If the robot is not redundant (n<=6)

det J  

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REMINDER 7: Velocity Force Duality

T

J J F f F n            

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REMINDER 8: How to do the elimination

From Osama Khatib

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What is dynamics

 Relationship between forces and motion  Two approaches:

 Traditional Newton laws approach  Eular-Lagrange formulation

 In

robotics we use Eular-Lagrange formulation because it is MUCH easier for complex objects.

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Motivation (1D system)

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Eular- Lagrange Equation

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Kinetic Energy

Translation Rotation Inertia Tensor

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Calculating Inertia Tensor in Fixed Frame

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Example (Uniform Solid)

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Moving Inertia Tensor Between Frames

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Kinetic Energy for N Links Robot

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Potential Energy for N Link Robot

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Eular-Lagrange Equation of a Robot

Christoffel Symbols of the first order

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Example 2D Cartesian (PP)

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Planar Elbow

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Planar Elbow

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Planar Elbow

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