Scripts Used

Question A

Question B

Part 1: Images of Robot 'nao'

• Isomap
  
• Residual Variance
  
• Explanation:

  The dimensionality reported by the graphs is 2.
  
  This can be well-explained from the fact that the only part changing in the images is the robot's hand, which involves  motion about two axis (the arm and the elbow).
  This motion can be described completely by two angles, thus giving the robot two degrees of freedom. Thus, the expected dimensionality (from the number of degrees of
  freedom) is also 2, in accordance to the image data.

  
  

Part 2: 2arms-random

• Isomap
  
• Residual Variance
  
• Explanation:

  The dimensionality as observed from the graph is 4.
  
  In random motion of the arms of the robot having fixed length of each arm and a fixed distance between their centers, a four dimension vector of angles is required to
  completely specify the motion of the arms. This means that the robot arm motion has four degrees of motion, and thus the dimensionality observed must be four.

  
  

Part 3: Finding Theta

• Solution
• theta1 = 144.116
• theta2 = 48.232
• theta3 = 73.841
• theta4 = 240.02

Part 4: 2arms-box-horiz

• Isomap
  
• Residual Variance
  
• Explanation:

  The dimensionality as observed from the graph is 1.
  
  To describe the horizontal motion of the arms, four angles are required. 
  We get three equations describing the motion of the arms:
  	L*sin(theta1) + L* sin(theta2) = D1 (distance of the object from the robot body)
  	L*sin(theta3) + L* sin(theta4) = D1 (distance of the object from the robot body)
  	L*cos(theta1) + L* cos(theta2) + L*cos(theta3) + L* cos(theta4) = D2 (distance between the arms)
  Thus if we fix one angle out of the four, the others are governed by these equations, which means only one of the four angles can be varied freely.
  Hence, the dimensionality is 1 as observed form the graph.