MA 240  Discrete Mathematics                                    NAME: ________________________________ 

Fall 2007               Instructor:  Jeffrey Horn                                                                    

HOMEWORK 3:  Combinatorial (Propositional) Logic as Stimulus-Response Brains (Digital Braitenberg Vehicles)
 

Handed out/Assigned:	Wednesday, September 19, 2007
Due:	Wednesday, September 26, 2007

(1)  DIGITAL Braitenberg Vehicle 1:

Assume two inputs and two outputs.  Input is from photo detectors.  Assume detector geometry in the above, right diagram.

So a "01" input vector would mean a light source detected to the right, but nothing center or left, while "11" would mean a light source in the center .  Also assume a finite detector range, defined by an arc (not shown above) with its center at the robot's center.   We have two binary outputs, one for each motor.  LM is left motor, RM is right.  A "0" output means the motor stops, while a "1" means go forward.  (so an output of 01, which is 0 to the left and 1 to the right motor, would result in an arcing turn to the left, while 10 would be a right turn)
Here is an example truth table (ie., a "brain") for a strategy to TURN AWAY FROM lights (hopefully!):

INPUTS			OUTPUTS
LS	RS		LM	RM
0	0		1	1
0	1		0	1
1	0		1	0
1	1		0	1

For each of the following parts of question 1, fill out the truth table (thus determining the indicated strategy), then write out the corresponding boolean expressions for the two outputs, simplify as much as possible (extra credit for any simplification), and finally draw the circuit for each output (i.e., for each motor).  See above example.  

Boolean expressions:                               LM                                                         RM

                            

Simplified (extra credit):

 

CIRCUITS:                  LM                                                                                                RM

 

  

 

 


(1A)  Fill in the following blank truth table for a strategy to ORBIT lights CLOCKWISE (that is, it tries to keep a light source on its right hand side):    (you can try out your strategy in the on-line simulator)  

INPUTS			OUTPUTS
LS	RS		LM	RM
0	0
0	1
1	0
1	1

 

Boolean expressions:                                                LM                                                            RM

 

Simplified (extra credit):

 

CIRCUITS:                  LM                                                                                                RM

 

 

(2)  DIGITAL Braitenberg Vehicle 2:

This is the same as vehicle 1, except that we add one more sensor bit and one more output bit.   Actually, the extra bits are really the same:  M.  We now have a "sensor mode indicator " M.  Our two sensors are now dual mode:  they can detect light AND obstacles, but not at the same time!  If M is 0 then the sensors are in light detection mode.  If M is 1, then the sensors are in obstacle detection mode (keep these straight!).    In our output, we can set the sensor mode M.  (M sets, and thus indicates, the mode of BOTH sensors.  Assume the time it takes to switch modes insignificant, or at least less than the time between "firings" of the truth table.  So if in time step t your output changes M from 0 to 1, then by the next time step t+1, when the truth table is read again, the value of M will be 1.  You may seek and destroy light or seek and worship/absorb light.  But please indicate which, in words!)

THE STRATEGY:  Design a vehicle that seeks out light sources, BUT avoids obstacles.  (If you hit an obstacle, you take damage!)

(2 A)  Now fill out the 3-input truth table for the above strategy:

 

(2 B)  Explain in plain English what strategy your truth table above implements:

 

 

 

 

(2 C)  Now give the boolean expressions, which do NOT have to be minimized, FOR EACH OF THE THREE OUTPUTS above.

 

 

(2 D)  Now try minimizing the boolean expressions above.

 

 

 

 

(2 E)  Now try creating the circuits (or giving the boolean expressions, which do NOT have to be minimized) FOR EACH OF THE THREEE OUTPUTS above.
 

      LM:

 

 

 

 

     RM:

 

 

 

 

 

     M: