Sense and Move

Question:

Start Quiz:

#Given the list motions=[1,1] which means the robot 
#moves right and then right again, compute the posterior 
#distribution if the robot first senses red, then moves 
#right one, then senses green, then moves right again, 
#starting with a uniform prior distribution.

p=[0.2, 0.2, 0.2, 0.2, 0.2]
world=['green', 'red', 'red', 'green', 'green']
measurements = ['red', 'green']
motions = [1,1]
pHit = 0.6
pMiss = 0.2
pExact = 0.8
pOvershoot = 0.1
pUndershoot = 0.1

def sense(p, Z):
    q=[]
    for i in range(len(p)):
        hit = (Z == world[i])
        q.append(p[i] * (hit * pHit + (1-hit) * pMiss))
    s = sum(q)
    for i in range(len(q)):
        q[i] = q[i] / s
    return q

def move(p, U):
    q = []
    for i in range(len(p)):
        s = pExact * p[(i-U) % len(p)]
        s = s + pOvershoot * p[(i-U-1) % len(p)]
        s = s + pUndershoot * p[(i-U+1) % len(p)]
        q.append(s)
    return q
#
# ADD CODE HERE
#
print p

User's Answer:

(Note: The answer done by the user is not guaranteed to be correct)

studentMain.py

#Given the list motions=[1,1] which means the robot 
#moves right and then right again, compute the posterior 
#distribution if the robot first senses red, then moves 
#right one, then senses green, then moves right again, 
#starting with a uniform prior distribution.

p=[0.2, 0.2, 0.2, 0.2, 0.2]
world=['green', 'red', 'red', 'green', 'green']
measurements = ['red', 'green']
motions = [1,1]
pHit = 0.6
pMiss = 0.2
pExact = 0.8
pOvershoot = 0.1
pUndershoot = 0.1

def sense(p, Z):
    q=[]
    for i in range(len(p)):
        hit = (Z == world[i])
        q.append(p[i] * (hit * pHit + (1-hit) * pMiss))
    s = sum(q)
    for i in range(len(q)):
        q[i] = q[i] / s
    return q

def move(p, U):
    q = []
    for i in range(len(p)):
        s = pExact * p[(i-U) % len(p)]
        s = s + pOvershoot * p[(i-U-1) % len(p)]
        s = s + pUndershoot * p[(i-U+1) % len(p)]
        q.append(s)
    return q
#
# ADD CODE HERE
#
for i in range(len(measurements)):
    p = sense(p, measurements[i])
    p = move(p, motions[i])

print p

Solution:

Clarification Regarding Entropy

The video mentions that entropy will decrease after the motion update step and that entropy will increase after measurement step. What is meant is that that entropy will decrease after the measurement update (sense) step and that entropy will increase after the movement step (move).

In general, entropy represents the amount of uncertainty in a system. Since the measurement update step decreases uncertainty, entropy will decrease. The movement step increases uncertainty, so entropy will increase after this step.

Let's look at our current example where the robot could be at one of five different positions. The maximum uncertainty occurs when all positions have equal probabilities [0.2, 0.2, 0.2, 0.2, 0.2]

Following the formula Entropy = \Sigma (-p \times log(p)), we get -5 \times (.2)\times log(0.2) = 0.699.

Taking a measurement will decrease uncertainty and entropy. Let's say after taking a measurement, the probabilities become [0.05, 0.05, 0.05, 0.8, 0.05]. Now we have a more certain guess as to where the robot is located and our entropy has decreased to 0.338.

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