You are using a rotary encoder to determine the distance a robot has traveled. the rotary encoder is directly connected to one of the wheels that has a diameter of 10cm. assume the encoder has 16 lines/revolution and that both the rising and falling edges (2x mode) are detected. what is the smallest detectable change of the rover traveling distance? if you want the rover to travel 2 meters before making a turn, how many encoder counts (rise and fall) are needed before sending a turn command? if you want to upgrade the encoder so that the smallest detectable distance change is ±0.20cm, what would be the required lines/revolution?

The smallest angle is 0.196 rad or 11.25º ,  

102 signals

Explanation:

As the decoder sends a signal every change of intensity, 2 signals are sent per line, in total we have 2 * 16 = 32 signals / rev

The angle covered for each signal is

     Δθ = 1 2π / 32 = 0.196 rad

     Δθ = 0.196 rad (180º / π rad) = 11.25º

The smallest angle is 0.196 rad or 11.25º

Let's track the signals send to have 2 meters

Angular and linear variables are related.

     θ = S / r

     S = R θ

     S = 0.1  0.196

     S = 0.0196 m

     N = 1 2 / 0.0196 = 102.04 signals

    102 signals should be read

Change resolution to ΔS = 0.20 m

The angle for this distance is

    Δθ = ΔS / R

    Δθ = ΔS / R

    Δθ = 0.20 / 0.1

    Δθ = 2 rad

If we have one signal every 2 rad how many signals for 2π rad

    N = 2π 1/2 = π = 3.14

As each line sent signals 2 lines are needed only

a.   mass number from atomic number