CS365: Assignment-3

submitted by

Shubham Gupta (10699) and Vedant Mishra (10792)

PART A

1.Graph of 'Residual Variance' vs 'Isomap Dimensionality' value (image)

It can be seen from the residual variance plot that there is a sudden decrease in the residual variance as we move from dimensionality 1 to 2 and then on further increase it is fairly constant.
Hence it can be said that the dimensionality of manifold is best explained at 2.

PART B

Two dimensional embeddings(with images superimposed)(image)

PART C

1.Variation of Theta1(image)
At any point,Upper value denotes theta1
Lower value denotes theta 2

It is seen from the plot that on moving round the manifold(ring) at a given distance from centre, Theta 1 varies from 0 to 360 in one complete rotation. Theta 2 is fairly constant given the distance from centre is same.

2.Variation of Theta2(Zoomed in image)(image)

For a given theta1, it can be interpreted theta2 varies from 0 to 360.
This topology shows the variation of theta1 quite nicely.But variation of theta2 is not so easily understood.

PART D

1.Embedding in 3D space(image)

Topology is a TORUS.

PART E

1.Results for learning a map from (v1,v2) to (q1,q2,q3,q4) (image)


2.Results for learning a map from images to (q1,q2,q3,q4) (image)

When (v1,v2) are used as the input, the algorithm converges faster as compared to when images are taken as the input.This is because the dimensionality of images is way higher than that of (v1,v2).
Also there is no loss in the information when the dimensionality is reduced to 2.So for a very very feww input set we can get good results in lesser time.

3.Path planning from 00001.png to 00161.png (image)
Blue line shows the path

PART F

1.Two dimensional embedding with obstacle 1 (image)

2.Path planning with obstacle1 from 00001.png to 00161.png (image)
Blue line shows the path

PART G

Path planning with obstacle2 from 00001.png to 00161.png (image)
Blue line shows the path

Difference between PART F and PART G For obstacle 1, most of the confgurations which are a part of path planning with no obstacle are restricted due to collision.So all those configuratons had to be removed.
The arm has to thus follow a longer path to change its configuration according to the given specifications. But for obstacle 2,most of the configurations are more or less the same and hence the path is nearly the same as the path with no obstacles.
This difference is due to the position of obstacle w.r.t the arm.

Image for obstacle 1
Image for obstacle 2
source codes