Using the RANDOM NUMBERS sequenced you withdraw from EXCEL, for which you may use the RANDOM TESTING radio button in CyberRiskSolver for this random sequence by hand; perform the MARS-lander robot problem. If it lands on MARS where will it be {what coordinates if it lands initially at (0,0) } i.e. X axis=0, Y axis=0 on the center of MARS) a certain time say 1000 simulation runs later? Start moving now. Generate random distances from uniform density with mean U(a=-100 feet, b=100 feet) for both X and Y coordinates. With 30% chance it will fall and not get up. After n=5 runs (must do), doing this by hand; where is the MARS-lander robot landed by the MARSshooter now in terms of X and Y coordinates with respect to (0,0) center of Mars. Then find the location and angle (0, 360 counter clockwise) as seen from the center of MARS (0,0) by using arctangent? Use your MARS landing map scale to mark your progress by a ruler from (0,0), to the end when it stops ay n=1000.
(20 pts) Hints (Your solution will be different) :
A) Given, fall probability = 30%; X & Y axis movement is uniform distribution with a = -100ft, b = 100ft. Assume robot landed safely at (0, 0). Rounding to nearest integers, Step 1
X1 = a + (b - a) u21 = -100 + (100 + 100) (0.0785) = -84 Y1 = c + (d - c) u22 = -100 + (100 + 100) (0.1467) = -71 Robot falls? u23 < 0.3 = 0.388 < 0.3 = No Continuing to step 2 X2 = X1 + a + (b - a) u24 = -84 -100 + (100 + 100) (0.5274) = -79 Y2 = Y1 + a + (b - a) u25 = -71 -100 + (100 + 100) (0.8723) = 3 Robot falls? u26 < 0.3 = 0.7517 < 0.3 = No Continuing to step 3 X3 = X2 + a + (b - a) u27 = -79 -100 + (100 + 100) (0.9905) = 19 Y3 = Y2 + a + (b - a) u28 = 3 -100 + (100 + 100) (0.8904) = 81 Robot falls? u29 < 0.3 = 0.8177 < 0.3 = No Continuing to step 4 X4 = X3 + a + (b - a) u30 = 19 -100 + (100 + 100) (0.666) = 52 Y4 = Y3 + a + (b - a) u31 = 81 -100 + (100 + 100) (0.1158) = 4 Robot falls ? u32 < 0.3 = 0.6635 < 0.3 = No Continuing to step 5 X5 = X4 + a + (b - a) u33 = 52 -100 + (100 + 100) (0.4992) = 52 Y5 = Y4 + a + (b - a) u34 = 4 -100 + (100 + 100) (0.907) = 85 Robot falls ? u35 < 0.3 = 0.2975 < 0.3 = Yes After 5 moves, robot falls at (X5, Y5) = (52, 85). Angle = arc tan (Y/X) = 58.5 degrees. NOTE: Carry on with this hand calculation routine until you complete 3 such falls.
B) Repeat A with EXCEL for n=1000 runs, now use 5% for falling and getting up straight, so the MARSLANDER project is not doomed! When no more action, where is it (coordinates and angle w.r.t. starting point: 0,0) so that we can locate it and send repair crews from the base or space station? C) Now redo this problem using EXCEL using 1000 SIM RUNS with +100ft or -100ft values changed to +50ft or -50ft ft. and 20% of not getting back up.
D) Now redo this problem using EXCEL using 1000 SIM RUNS USING NORMAL DISTRIBUTION WITH MEAN=0 FT AND STD DEV = 100FT
Solution Hints (30pts) Taking X & Y as uniform random variables using X= a + (b - a)rand() & Y = c + (d – c)rand(). For a given iteration, robot is at point (Sum X, Sum Y). Fall probability as z = rand(). If z < 0.05, robot falls. At this point angle is calculated using tan-1(Y/X). Use the ARCTAN calculator as in class.
Sample Solution