A1.
1a [code]
1b [code]
1c [code]
1d [code]
A2.
2a [code]
2b [code]
2c [code]
B1.
The arm of the robot is seen to move with 4 degrees of freedom. Two about the
shoulder, two about the ankle ; rotations in the vertical and horizontal planes each.
However, the most likely dimension observed from the residual
error curve [code] (using the eucleidian distance) is three.
B2.
As given, the number of degrees of freedom of the given system is four.
Using the eucledian distance to plot the isomap results in a
residual error curve [code]
that gives an uncertainity in the dimensionality as it is continuously decreasing.
However, by plotting the isomap using geodesic distances
[code] gives a close approximation of three.
B3.
diagram
theta1 = 144.1 deg
theta2 = 48.2 deg
theta3 = 81.46 deg
theta4 = 109.48 deg
B4.
Applying isomap on the set of images, the residual error
curve [code] gives a dimensionality of one.
Comparing to the case in B2, there is a decrease in dimensionality.
This can be accounted to the constrained motion described by the system
in the given series of images. The box moves along a horizontal line,
restricting the vector (theta1, theta2, theta3, theta4) to a single possibility.
So, the entire system is dependent only upon the position of the box in the
horizontal path.
