Angular resolution of cosmic rays

Research Group: Matthew Lochridge, Azizullah Moltafet, and Behzod Tojiev

Launch: Whitworth Spring 2019

''This project focused of analyzing coincident collisions on two Geiger counters based on the relative orientations of the pod during its flight. The goal was to find any patterns in coincident CPM based on the directions from which they collide with the pod. The circuit used an inertial measurement unit containing a gyroscope and magnetometer to record instantaneous orientations of the pod in Euler angles. The data showed that two streaks of cosmic rays were coincident on the Geiger counters within the pod. In the future, groups researching related topics may want to focus on optimizing the precision of the IMU's data collection as well as comparing orientation data with altitude data.''

Background


High-energy cosmic rays traveling through space at nearly the speed of light collide with particles in Earth's atmosphere, losing energy and changing direction (as shown in Figure 0 ). It is difficult to track the paths of these rays since their courses are altered more by collisions and magnetic fields the further they travel into the atmosphere; there is little known yet of the origins of cosmic rays, although some experts predict that they are generated in supernovas. Last spring's cosmic ray orientation experiment explored a correlation between coincident Geiger counter readings and light intensity, hypothesizing that the amount of coincident hits detected depends on the pod's orientation relative to the Sun. This experiment focuses on the precise measurement of the spatial orientation of the pod so as to interpret graphically from which directions cosmic rays collide with the pod during flight.

Theory
Using an inertial measurement unit (IMU) containing a gyroscope that measures 3-dimensional rotation in Euler angles, we can measure the pod's orientation in space at any point. We can also use two Geiger counters fixed in the same orientation and along the same axis to measure cosmic ray hits coincident on both counters; the angle of coincident reception between the two increases with distance from one another. Combining the angle data collected from the IMU and coincident collision counts per minute (CPM) collected from the Geiger counters, we can create a "heat map" of coincident CPM at given points on a graph of heading angles plotted against the square root of the sum of the squares of roll and pitch angles.

Pod Layout
As shown in Figures 1 and 2, the flight computer is secured to the "front" of the pod (relative to the direction of the IMU, to be discussed) with a power switch & LED assembly wired out to the right side of the pod and a cardboard sheet fitted between it and the inner pod wall, so as to insulate any conduction through the pod. The flight computer is wired to the inertial measurement unit, secured atop the bottom Geiger counter at the back right of the pod. Two Geiger counters are secured with bolts in that back right corner, one at the top of the pod and one at the bottom. There is a hole in the bottom-back-left of the pod for camera placement, made by the group that constructed this pod in 2018.

Inertial Measurement Unit
As shown in Figure 2, an Adafruit BNO055 Absolute Orientation Sensor is secured atop the bottom Geiger counter oriented toward the launch computer (hence the launch computer is at the front). The IMU measures and reads out Euler angles of heading (the measure of rotation about the vertical axis), roll (the measure of rotation about the minor horizontal axis), and pitch (the measure of rotation about the major horizontal axis). IMU had to be secured along the same vertical axis and in the same orientation as the Geiger counters so that the angles collected from the IMU correspond to the actual orientation of the Geiger counters while they are collecting data.

Geiger Counters
As shown in Figure 3, two Aware RM-60 Micro-Roentgen Radiation Monitors are secured along the same vertical axis and in the same orientation as each other. They were placed at the top and bottom of the pod so as to maximize the angle of coincident ray detection between the two devices. Due to the size of the flight computer and the already-cut camera hole in the bottom of the pod, we had to secure them at the back right corner; this resulted in a coincident angle range of approximately 31.6 degrees.

Flight Computer


As shown in Figure 4, the Whitworth University Flight Computer V.1 is secured to the front of the pod. Wired into the flight computer is an NXP mbed LPC1768 Development Board, which stores and executes programs. Through the flight computer, the mbed is wired to the IMU, both Geiger counters, the LED/switch assembly on the right outside of the pod, and a microSD Transflash2 for storing data mid-flight. The mbed can read Euler angle values from the IMU on command, store coincident CPM from the Geiger counters, activate diagnostic LEDs, and write all data to a microSD card with timestamps.

Code
Written in C++ in the mbed browser interface, this program writes data to a tab-separated values (TSV) formatted .txt file for easy analysis via Excel or Mathematica.

Validation and Calibration
The Geiger counters were first calibrated by measuring ambient single and coincident hits in Eric Johnston Science Center. Consistent with last year's iteration, ambient hits ranged from 15 to 25 single CPM for both Geiger counters; the counters picked up no ambient coincident hits. Using a weak radioactive source, strontium-90, single hits were measured between 110 and 125 CPM for both Geiger counters, again consistent with last year's iteration, and coincident hits between 0 and 2 CPM. As expected, we measured virtually no coincident hits due to the low level of radiation emitted from strontium-90.

Already-published code was used to configure the BNO055 to measure and read out three Euler angles: heading, the measure of rotation about the vertical axis; roll, the measure of rotation about the minor horizontal axis; and pitch, the measure of rotation about the major horizontal axis. Testing the ranges of these measurements by rotating the pod in various directions yielded ranges of 0 to 360 degrees for heading, -90 to 90 degrees for roll, and -180 to 180 degrees for pitch.

Composite Testing
Initially the mbed was programmed to read IMU data every second and write all data to the microSD card every minute. This affected the Geiger counter measurements when they were tested in combination; the mbed could not continuously store three CPM measurements while reading the IMU every second. We decided to sacrifice measurement precision to amend this by programming the mbed to read the IMU every minute and write all data to the microSD card every five minutes. This resulted in a latency while reading the IMU of about 4 milliseconds every minute instead of 4 milliseconds every second. After more testing, the mbed successfully recorded all CPM and angle data within the established respective ranges simultaneously.

Raw Data
-- Time	Face	Roll	Pitch	Coincidence	Single 1	Single 2 1.000 	 359.81 	 0.62 	 -0.62 	 2 	 19 	 18 2.000 	 359.81 	 0.62 	 -0.62  	 0 	 16 	 39 3.000 	 359.81 	 0.62 	 -0.62  	 0 	 21 	 59 4.000 	 359.81 	 0.62 	 -0.62  	 0 	 15 	 80 5.000 	 359.81 	 0.62 	 -0.62  	 0 	 15 	 94 6.000 	 359.44 	 0.38 	 -0.50  	 0 	 17 	 111 7.000 	 357.56 	 -4.12 	 8.81  	 0 	 18 	 130 8.000 	 105.94 	 -3.75 	 20.12  	 0 	 14 	 143 9.000 	 117.31 	 -9.31 	 3.94  	 0 	 16 	 161 10.000 	 73.31 	 22.12 	 78.44  	 0 	 12 	 172 11.000 	 312.94 	 -3.38 	 59.06  	 0 	 13 	 188 12.000 	 132.94 	 0.00 	 87.25  	 0 	 9 	 204 13.000 	 249.69 	 0.88 	 86.12  	 0 	 11 	 215 14.000 	 126.44 	 -0.19 	 85.31  	 0 	 11 	 228 15.000 	 31.88 	 -0.31 	 88.44  	 0 	 10 	 243 16.000 	 250.31 	 -1.69 	 87.25  	 0 	 16 	 259 17.000 	 359.56 	 0.56 	 86.88  	 0 	 19 	 274 18.000 	 249.50 	 -0.38 	 88.06  	 0 	 12 	 292 19.000 	 159.50 	 -2.44 	 87.06  	 0 	 15 	 317 20.000 	 3.31 	 -0.62 	 87.81  	 0 	 22 	 337 21.000 	 257.62 	 1.81 	 87.06  	 0 	 25 	 358 22.000 	 119.75 	 -1.56 	 85.69  	 0 	 22 	 387 23.000 	 13.75 	 0.00 	 87.56  	 0 	 28 	 417 24.000 	 250.44 	 -1.25 	 87.56  	 0 	 30 	 451 25.000 	 251.50 	 2.56 	 87.88  	 0 	 36 	 484 26.000 	 194.81 	 2.25 	 89.12  	 0 	 55 	 525 27.000 	 340.06 	 -1.12 	 86.38  	 0 	 50 	 559 28.000 	 162.88 	 -0.06 	 85.81  	 0 	 60 	 616 29.000 	 180.00 	 0.69 	 87.00  	 0 	 70 	 682 30.000 	 265.31 	 0.00 	 88.94  	 0 	 67 	 753 31.000 	 298.38 	 -0.44 	 88.94  	 0 	 71 	 826 32.000 	 180.81 	 1.06 	 89.00  	 0 	 61 	 918 33.000 	 205.75 	 -0.88 	 88.19  	 0 	 105 	 1025 34.000 	 359.19 	 0.75 	 87.75  	 0 	 140 	 1152 35.000 	 167.06 	 0.25 	 87.56  	 4 	 158 	 1294 36.000 	 59.50 	 0.88 	 85.25  	 0 	 145 	 1451 37.000 	 159.31 	 0.50 	 89.94  	 0 	 157 	 1614 38.000 	 290.88 	 0.12 	 91.06  	 0 	 175 	 1793 39.000 	 214.75 	 -0.31 	 89.81  	 0 	 209 	 2005 40.000 	 44.31 	 0.19 	 88.19  	 0 	 221 	 2240 41.000 	 268.19 	 -1.44 	 87.44  	 2 	 225 	 2473 42.000 	 170.50 	 1.25 	 88.38  	 2 	 235 	 2755 43.000 	 346.50 	 -0.88 	 85.81  	 2 	 244 	 3009 44.000 	 330.00 	 1.62 	 87.50  	 2 	 301 	 3286 45.000 	 232.19 	 -1.88 	 88.31  	 2 	 295 	 3545 46.000 	 308.31 	 -1.62 	 87.19  	 2 	 335 	 3853 47.000 	 286.56 	 1.12 	 87.38  	 2 	 366 	 4179 48.000 	 324.75 	 -0.56 	 87.06  	 3 	 372 	 4532 49.000 	 281.25 	 1.25 	 89.50  	 1 	 366 	 4937 50.000 	 60.44 	 0.25 	 85.19  	 4 	 428 	 5336 51.000 	 287.00 	 -0.88 	 89.75  	 6 	 473 	 5754 52.000 	 116.00 	 -1.06 	 88.56  	 0 	 446 	 6217 53.000 	 207.06 	 -1.94 	 85.50  	 2 	 504 	 6687 54.000 	 116.44 	 -1.56 	 91.12  	 5 	 477 	 7202 55.000 	 231.75 	 0.44 	 90.44  	 2 	 518 	 7679 56.000 	 352.19 	 1.31 	 88.50  	 2 	 548 	 8246 57.000 	 101.75 	 -0.94 	 87.06  	 8 	 589 	 8818 58.000 	 118.25 	 2.38 	 88.50  	 6 	 629 	 9406 59.000 	 42.50 	 0.62 	 87.44  	 6 	 634 	 10036 60.000 	 228.75 	 -0.19 	 86.69  	 7 	 687 	 10678 61.000 	 237.81 	 0.94 	 88.50  	 6 	 660 	 11349 62.000 	 113.56 	 -5.56 	 82.62  	 5 	 687 	 12028 63.000 	 149.00 	 1.62 	 88.44  	 4 	 715 	 12745 64.000 	 208.44 	 -2.56 	 90.06  	 6 	 714 	 13460 65.000 	 239.81 	 -0.06 	 88.50  	 9 	 760 	 14183 66.000 	 198.75 	 0.81 	 86.25  	 7 	 711 	 14944 67.000 	 291.62 	 -1.38 	 89.69  	 12 	 801 	 15737 68.000 	 252.88 	 -4.44 	 89.50  	 9 	 803 	 16559 69.000 	 69.31 	 -1.94 	 90.06  	 17 	 821 	 17374 70.000 	 95.12 	 -0.69 	 90.56  	 7 	 817 	 18172 71.000 	 67.94 	 -1.38 	 88.94  	 11 	 854 	 18956 72.000 	 153.19 	 -0.38 	 88.19  	 7 	 823 	 19890 73.000 	 352.62 	 1.06 	 84.19  	 4 	 850 	 20723 74.000 	 7.06 	 0.19 	 87.06  	 2 	 849 	 21596 75.000 	 0.75 	 0.25 	 87.69  	 6 	 825 	 22507 76.000 	 198.62 	 -0.25 	 88.75  	 17 	 867 	 23363 77.000 	 339.31 	 -2.75 	 83.69  	 5 	 910 	 24240 78.000 	 338.56 	 3.88 	 86.19  	 10 	 879 	 25126 79.000 	 146.81 	 -1.12 	 92.25  	 8 	 906 	 26030 80.000 	 270.81 	 -0.38 	 87.81  	 15 	 922 	 26961 81.000 	 166.19 	 -1.56 	 88.62  	 15 	 923 	 27862 82.000 	 119.75 	 -3.88 	 90.06  	 14 	 856 	 28788 83.000 	 349.44 	 0.69 	 90.25  	 21 	 885 	 29745 84.000 	 274.75 	 1.00 	 89.31  	 11 	 960 	 30637 85.000 	 197.88 	 2.69 	 89.06  	 7 	 905 	 31575 86.000 	 262.62 	 0.69 	 88.75  	 15 	 910 	 32457 87.000 	 24.31 	 0.06 	 89.38  	 15 	 928 	 33410 88.000 	 193.56 	 -1.19 	 87.31  	 5 	 926 	 34329 89.000 	 195.31 	 2.38 	 87.25  	 9 	 936 	 35352 90.000 	 195.94 	 0.25 	 89.94  	 8 	 939 	 36283 91.000 	 276.56 	 0.44 	 90.81  	 8 	 956 	 37225 92.000 	 147.31 	 5.38 	 90.06  	 19 	 948 	 38172 93.000 	 123.44 	 0.12 	 88.75  	 10 	 948 	 39120 94.000 	 211.06 	 -1.75 	 89.00  	 12 	 896 	 40006 95.000 	 318.06 	 -1.44 	 90.06  	 20 	 945 	 40985 96.000 	 83.50 	 -2.81 	 87.12  	 14 	 922 	 41937 97.000 	 334.69 	 -2.81 	 88.19  	 24 	 936 	 42922 98.000 	 55.56 	 0.06 	 87.25  	 9 	 857 	 43803 99.000 	 50.56 	 1.56 	 88.88  	 16 	 904 	 44717 100.000 	 119.06 	 -0.75 	 90.75  	 31 	 924 	 45629 101.000 	 120.00 	 2.94 	 83.69  	 23 	 937 	 46602 102.000 	 36.38 	 0.81 	 87.44  	 13 	 892 	 47504 103.000 	 230.25 	 -0.31 	 86.62  	 22 	 899 	 48380 104.000 	 180.69 	 0.00 	 86.69  	 10 	 873 	 49244 105.000 	 230.19 	 1.50 	 88.50  	 21 	 949 	 50091 106.000 	 103.75 	 0.19 	 91.25  	 16 	 894 	 50999 107.000 	 81.31 	 -2.75 	 91.75  	 17 	 849 	 51834 108.000 	 64.12 	 -0.19 	 87.75  	 15 	 884 	 52668 109.000 	 80.06 	 0.56 	 93.50  	 21 	 878 	 53582 110.000 	 99.50 	 -0.31 	 82.44  	 10 	 863 	 54458 111.000 	 299.81 	 2.88 	 83.94  	 14 	 899 	 55353 112.000 	 348.31 	 -2.19 	 91.44  	 26 	 865 	 56207 113.000 	 325.00 	 0.88 	 91.81  	 12 	 869 	 57052 114.000 	 337.00 	 -0.50 	 93.25  	 26 	 816 	 57906 115.000 	 320.75 	 -0.31 	 87.31  	 15 	 882 	 58732 116.000 	 111.38 	 0.31 	 90.31  	 14 	 856 	 59600 117.000 	 330.44 	 -2.00 	 87.25  	 8 	 804 	 60471 118.000 	 44.50 	 -0.12 	 83.38  	 16 	 809 	 61316 119.000 	 263.81 	 -1.25 	 84.69  	 9 	 867 	 62122 120.000 	 332.06 	 4.88 	 86.62  	 22 	 831 	 62917 121.000 	 262.94 	 -0.12 	 85.50  	 19 	 773 	 63743 122.000 	 135.75 	 -3.38 	 90.69  	 21 	 830 	 64555 123.000 	 170.06 	 3.56 	 88.94  	 18 	 862 	 65360 124.000 	 123.75 	 1.06 	 85.50  	 14 	 835 	 66165 125.000 	 126.38 	 -0.25 	 90.00  	 20 	 797 	 66987 126.000 	 195.12 	 -1.94 	 89.94  	 19 	 844 	 67717 127.000 	 265.75 	 1.69 	 86.00  	 13 	 822 	 68505 128.000 	 64.94 	 1.75 	 84.94  	 12 	 790 	 69244 129.000 	 158.75 	 -1.50 	 89.38  	 19 	 780 	 70011 130.000 	 38.31 	 -5.44 	 86.00  	 14 	 772 	 70796 131.000 	 64.81 	 -9.44 	 85.94  	 17 	 870 	 71603 132.000 	 174.19 	 -1.81 	 85.00  	 21 	 895 	 72514 133.000 	 280.75 	 13.81 	 78.69  	 12 	 960 	 73389 134.000 	 245.81 	 -13.94 	 63.62  	 16 	 893 	 74296 135.000 	 329.81 	 8.75 	 99.06  	 17 	 955 	 75194 136.000 	 252.94 	 -34.06 	 116.75  	 5 	 903 	 76071 137.000 	 230.31 	 -12.88 	 68.19  	 18 	 860 	 76840 138.000 	 30.81 	 -3.81 	 116.31  	 9 	 806 	 77585 139.000 	 343.38 	 32.44 	 111.12  	 8 	 713 	 78284 140.000 	 217.25 	 -23.38 	 17.31  	 6 	 667 	 79009 141.000 	 37.62 	 5.44 	 112.75  	 8 	 595 	 79561 142.000 	 241.88 	 31.06 	 72.94  	 4 	 463 	 79996 143.000 	 257.06 	 2.56 	 92.56  	 6 	 411 	 80393 144.000 	 229.56 	 21.88 	 111.81  	 3 	 351 	 80755 145.000 	 97.38 	 -35.06 	 119.75  	 0 	 288 	 81050 146.000 	 124.88 	 -41.88 	 82.12  	 0 	 249 	 81295 147.000 	 355.12 	 31.38 	 103.31  	 6 	 211 	 81513 148.000 	 272.06 	 22.75 	 118.94  	 2 	 176 	 81668 149.000 	 38.06 	 18.12 	 71.00  	 0 	 128 	 81802 150.000 	 7.44 	 -35.50 	 80.56  	 2 	 120 	 81910 151.000 	 255.25 	 4.56 	 91.44  	 0 	 80 	 81989 152.000 	 39.56 	 37.19 	 84.19  	 0 	 67 	 82045 153.000 	 270.69 	 12.06 	 69.25  	 0 	 55 	 82097 154.000 	 70.94 	 14.81 	 69.75  	 0 	 43 	 82148 155.000 	 189.19 	 -13.50 	 55.31  	 0 	 32 	 82179 156.000 	 358.69 	 25.19 	 65.44  	 0 	 42 	 82201 157.000 	 42.88 	 -37.81 	 94.00  	 0 	 27 	 82214 158.000 	 228.62 	 16.06 	 59.31  	 0 	 19 	 82229 159.000 	 188.06 	 20.62 	 58.38  	 0 	 21 	 82251 160.000 	 233.50 	 10.69 	 92.81  	 0 	 11 	 82272 161.000 	 220.19 	 10.94 	 91.25  	 0 	 10 	 82282 162.000 	 163.62 	 -26.75 	 74.94  	 0 	 15 	 82295 163.006 	 1.62 	 1.75 	 -1.88  	 1 	 8 	 82302 164.006 	 1.62 	 1.75 	 -1.88  	 0 	 0 	 82302 165.006 	 -1934.38 	 -1934.25 	 1934.12  	 0 	 0 	 82302 166.006 	 1.62 	 1.75 	 -1.88  	 0 	 0 	 82302 167.006 	 1.62 	 1.75 	 -1.88  	 0 	 0 	 82302 168.006 	 1.62 	 1.75 	 -1.88  	 0 	 0 	 82302 169.006 	 1.62 	 1.75 	 -1.88  	 0 	 0 	 82302 170.007 	 1.62 	 1.75 	 -1.88  	 0 	 0 	 82302 171.006 	 1.62 	 1.75 	 -1.88  	 0 	 0 	 82302 172.006 	 1.62 	 1.75 	 -1.88  	 0 	 0 	 82302 173.006 	 1.62 	 1.75 	 -1.88  	 0 	 0 	 82302 174.006 	 1.62 	 1.75 	 -1.88  	 0 	 0 	 82302 175.007 	 1.62 	 1.75 	 -1.88  	 0 	 0 	 82302 176.006 	 1.62 	 1.75 	 -1.88  	 0 	 0 	 82302 177.006 	 1.62 	 1.75 	 -1.88  	 0 	 0 	 82302 178.006 	 1.62 	 1.75 	 -1.88  	 0 	 0 	 82302 179.000 	 -70.00 	 0.00 	 0.00  	 15 	 14 	 82338 180.007 	 1.62 	 1.75 	 -1.88  	 1 	 2 	 82339 181.006 	 1.62 	 1.75 	 -1.88  	 0 	 0 	 82339 182.006 	 -1934.38 	 -1934.25 	 1934.12  	 0 	 0 	 82339 183.006 	 1.62 	 1.75 	 -1.88  	 0 	 0 	 82339 184.006 	 1.62 	 1.75 	 -1.88  	 2036 	 2531 	 86044 185.006 	 1.62 	 1.75 	 -1.88  	 8949 	 10195 	 98195

Analysis




According to the raw data, there is a positive correlation with time and coincident CPM from about the 41st minute to the 132nd minute of flight followed by a negative correlation between the two during the 133rd minute and the 150th minute of flight. The change in correlation marks the popping of the balloon and the distinction between the ascent data and the descent data; although the pod is ascending from the 6th minute (the pod was powered on 5 minutes before the balloon was released) to the 132nd minute and descending from the 133rd minute to (approximately) the 162nd minute, the pod only detected coincident CPM at higher altitudes.

We say that the duration of the descent lasted until approximately the 162nd minute despite our 185 minutes of flight data because some circuit connections were broken, causing orientation data in the ranges of thousands of degrees.

Our data is best interpreted in the form of a heat map, generally used to represent matrices with colors. Figure 5 shows coincident CPM throughout the ascent as blotches of color on the red canvas; red corresponds to 0 CPM at specific orientations, and then cooler colors correspond to higher CPM. The coordinate system is based on the ranges of both the heading measurement from the IMU, corresponding to the y-axis, and square root of the sum of the squares of the roll and pitch measurements from the IMU, corresponding to the x-axis.

The figure 5 and figure 6 have the angle phi on the x-axis and it was obtained by the use of a special formula we have come up with. The formula has the following general form: phi=arctan⁡((tan(α)-tan⁡(β) )/(√2*sec⁡(β)), for α≥β. The α angle could be either pitch or roll angle measured by the IMU, whichever is bigger. The β angle could be either pitch or roll as well, whichever is smaller. By comparing pitch and roll, α gets to be the bigger angle and β gets to be the smaller one. The formula was derived from the combination of both roll and pitch angles. We have used some trigonometry and Pythagorean Theorem to obtain resultant one angle from the usage of two angles in a space of ¼ of a single hemisphere. We have taken the roll and the pitch to measure in angles from 0 to 180 degrees in the upper hemisphere. The upper hemisphere consists of 4 quadrants and our formula only works when both roll and pitch angles are located in the first quadrant, because we based our formula on that quadrant. When they are located in different quadrants, our general formula has its own specific modification for each quadrant. Our formula is only based on the condition when pitch and roll not exceed 90 degrees. However, when pitch and roll exceed 90 degrees, we used specific trigonometric symmetry (π-α,π-β) in order to transform them to angles between 0 to 90 degrees. The formula has the following specific modifications:

phi=arctan⁡((tan(α)-tan⁡(β) )/(√2*sec⁡(β)), for α∈[0,π/2),β∈[0,π/2) and α≥β (1st quadrant),

phi=arctan⁡((tan⁡(π-α)-tan⁡(β) )/(√2*sec⁡(β)), for α∈(π/2,π],β∈[0,π/2) and π-α≥β (2nd quadrant),

phi=arctan⁡((tan⁡(α)-tan⁡(π-β) )/(√2*sec⁡(π-β)), for α∈(π/2,π],β∈[0,π/2) and β≥π-α (2nd quadrant),

phi=arctan⁡((tan⁡(π-α)-tan⁡(π-β) )/(√2*sec⁡(π-β)), for α∈(π/2,π],β∈(π/2,π] and π-α≥π-β (3rd quadrant),

phi=arctan⁡((tan⁡((π-β))-tan⁡(π-α) )/(√2*sec⁡(π-α)), for α∈(π/2,π],β∈(π/2,π] and π-β≥π-α (3rd quadrant),

phi=arctan⁡((tan⁡((α))-tan⁡(π-β) )/(√2*sec⁡(π-β)), for α∈[0,π/2),β∈(π/2,π] and α≥π-β (4th quadrant),

phi=arctan⁡((tan⁡((π-β))-tan⁡(α) )/(√2*sec⁡(α)), for α∈[0,π/2),β∈(π/2,π] andπ-β ≥α (4th quadrant).

Moreover, one challenge we had faced was to convert the angles of pitch and roll measured by the IMU to our respective angles of pitch and roll from our perspective. The pitch and roll have the ranges of 0 to 360 degrees and 0 to 180 degrees respectively in our standards. On the other hand, the IMU measures the angles in a different perspective. The roll has the range of -90 to 90 degrees and pitch has the -180 to 180 degrees. In order to convert the angles measured by the IMU to our standards, we have used special conversion formulas. For the roll, we have used the formula α=α_0+90, in which α is the roll measured in our standards and α_0 is the roll angle measured by the IMU and this formula works for the α_0∈[-π/2,π/2]. For the pitch, the following formula is used β=|β_0-90| in which β is the pitch angle in our standards and β_0 is the pitch by IMU and this formula works for β_0∈[-π/2,π/2].

Also, we had to face another challenge when the pitch angles measured in the ranges [-π,-π/2] and [π/2,π] in the IMU standards. This meant that we had to deal with the lower hemisphere of the circle instead of only having our concentration for the upper hemisphere of the circle. The general formula we have come up with phi=arctan⁡((tan⁡(α)-tan⁡(β) )/(√2*sec⁡(β)), for α≥β only deals with angles located on the upper hemisphere of the circle and it is useless when it has to deal with the pitch angles in the lower hemisphere. For our formula to function properly with such angles, we have come up with special conversion formulas. As the pitch measured by the IMU ranges in the lower part of the sphere, we wanted the other angles, which are the heading and the roll, to be included in the lower hemisphere as well. When we did that, we noticed that the lower hemisphere becomes identical to that of the upper hemisphere if the pitch, roll, and the heading were in its range. Our task was to make the lower hemisphere as if it was the upper hemisphere itself, where our general formula would have worked. We used the following formula β_0=β_n-180 in which β_n is the pitch angle measured by the IMU which is in the range of β_n∈ [π/2,π] and β_0 is the pitch angle by the IMU which has the resultant range after the conversion β_0∈[-π/2,π/2]. The other conversion formula we have used is β_0=β_n+180 in which β_n is the pitch angle measured by the IMU which is in the range of β_n∈ [-π,-π/2] and β_0 is the pitch angle by the IMU which has the resultant range after the conversion β_0∈[-π/2,π/2]. The roll angle is also automatically converted to the lower hemisphere by the use of the following formula α_0=-α_u in which α_u is the roll angle of the IMU which is initially in the upper hemisphere and α_0 is the roll angle of the lower hemisphere. As a result, the resultant range of β_0 and α_0 is acceptable to be used in the previous conversion formulas which convert the angles measured by the IMU to our standards (β=|β_0-90|,α=α_0+90). The heading angle stays the same in the lower hemisphere as it did in the upper hemisphere.

Unfortunately, we have found it difficult to interpret the meaning of this data. It did not fit last year's hypothesis that coincident CPM would be more concentrated in orientations directed toward the Sun; there are broad streaks of coincident CPM detected around the pod during the entire flight. It certainly did not help the precision of our data to collect orientation data every minute, but we had to lengthen the wait between IMU readings or we would not have collected any radiation data from the Geiger counters.

Conclusion
This project has successfully recorded cosmic ray collisions and recorded corresponding orientations of the pod with each collision. We measured increased levels of radiation in the upper atmosphere and mapped collisions with the specific directions from which they came. Unfortunately, our data was not as precise as we would have hoped, but we had to sacrifice precision for sheer data collection. In the future, groups might want to explore ways of optimizing the processing speed of the microcontroller, or perhaps reading the separate data from separate microcontrollers.

We also realize in retrospect that it would have been useful to plot coincident CPM against altitude during the ascent and descent to better understand and interpret the entire flight; however, we did not.