Assignment 3

Part A

• The dimentionality can be justified at 2 as the elbow in residual variance vs dimentionality plot shown above appears at 2.

Part B

• The 2-dimensional manifold of the given robot system is shown in the first plot
• Some images are too superimposed in the second plot

Part C

• From the 2-D isomap below, we can clearly see that :-

  theta1 changes as we rotate around this torus figure in Isomap and theta2 varies in radial direction. It can be thought of as a projection of 3D torus on a plane where each point corresponds to unique (theta1,theta2).

  Matlab Figure file for Part C

  Part D

  • 3-D embedding of points is below at different view angles
  

  

  

  Part E

  • using Neural Network of 8 Neurons: Converges in 170 iterations (Map from angles to SinesCosines)
  
  • The path from start ie 1.png to 161.png as embedded on 2-D isomap gives
    

    Part F

    • If we consider obstacle 1, then even red rod would not rotate fully
      • the torus should have a cut in itself
      • path should not pass through this cut and should come from other way round

    

    Part G

    • If we consider obstacle 2, then red rod can rotate fully, but the blue one may collide with obstacle
      • the torus should have a slice cut off somewhere
      • the path may pass through this narrow gap left after slice