Winter Interim
Contents
1. This program was run using initial values of O for everything It also assumed that there was no velocity or angular rates This was done just to see what would happen to the Euler angles The plots of these angles can be seen below Roll Time in seconds 100 0 1 2 Pitch time in Seconds 8 0 1 2 3 Yaw Time in S2conds 7 E 3 Figure 1 Plots of Roll Pitch and Yaw in seconds This program was run at a rate of 24 seconds 3600 times This is equal to one day It can be seen in figure 1 that the yaw and pitch stayed at zero The roll however went from 0 to 90 90 to 90 and 90 to 0 or a 360 degree rotation This is due to the rotation of the earth To show the accuracy of the equations this is the data for one iteration or 24 seconds The earth rotates about 15 041067 hour In 24 seconds the earth rotates 0 10027378 When running the program the roll after 24 seconds 0 100273676798553 Error eoretical Experimental ny 0 000103 Theoretical When running the program 3600 times the roll after 1 day 360 985236474792547 This also is an error of 0 000103 When this is done at an increment of 0 001 seconds and starting at 0 the error is 0 015041067 0 015041066876065 0 015041067 This error is very minimal over 3600 iterations This is also the same value over 36000 iterations a step of 0 0001 seconds This was tried with different time step values and different2. Winter Interim 12 19 02 12 21 02 All individual functions for the attitude computer are written Now the program for the attitude computer can be written First here is the code for the initial Cp It looks a little messed up because there are a lot of sine and cosine equations involved in this matrix To see the equation for the directional cosine matrix see equation 9 function cbn cbninit yaw pitch roll CBNINIT yaw pitch roll Derive the initial directional cosine matrix Written By Brian Bleeker Rob MacMillan yaw yaw p1 180 pitch pitch pi 180 roll roll pi 180 cbn cos pitch cos yaw cos roll sin yaw sin roll sin pitch cos yaw sin roll sin yaw cos roll sin pitch cos yaw cos pitch sin yaw cos roll cos yaw sin roll sin pitch sin yaw sin roll cos yaw cos roll sin pitch sin yaw sin pitch sin roll cos pitch cos roll cos pitch Just for reference of the new Euler angles a function was written to derive the Euler angles from the directional cosine matrix The equation for this was found on page 50 Note the function has not been written yet for the provision that approaches 90 This will have to be done in the future because the following equations will approach 0 0 Here is what has been written so far function euler cbn pos euler cbn position Derive the euler angles from cbn Written By Brian Bleeker Rob MacMillan global a
3. he IMU Upon startup the software recognizes when the IMU is connected to the computer and power supply properly Data Collection The IMU has three modes with which data can be collected Voltage Mode The analog sensors are sampled and converted to digital data with 1 mV resolution The data is 12 bit unsigned The data is just voltages which would need to be converted to numerical angular rates and accelerations before we could analyze the data Scaled Sensor Mode The analog sensors are sampled converted to digital data temperature compensated and scaled to engineering units The digital data represents the actual value of the quantities measured The acceleration data is measured in g s and therefore must be converted to force to be useful data The angular rate data is measured in degrees per second Angle Mode This mode acts the same as the scaled sensor mode however it outputs the yaw pitch and rolls angles with the other data We will be using the scaled sensor mode The data is outputted to a text file in a spreadsheet format which can be edited nicely in Matlab We didn t use the angle mode because it gives the angles of the yaw pitch and roll which is data we don t need IMU Coordinate System X axis From face with connector through the IMU Y axis Along the face with connector from left to right Z axis Along the face with the connector from top to bottom We didn t have much time but were able to connec
4. ngles angles y pos atan cbn 2 1 cbn 1 1 180 pi angles p pos asin cbn 3 1 180 pi angles r pos atan cbn 3 2 cbn 3 3 180 pi These equations also look a little messy in the Matlab code so here they are written out It can be seen from equation 9 how these equations are derived It can now be seen why new equations have to be written in the case that the denominator is 0 This will be done later Roll arctan eq 1 Cs Pitch arcsin C eq 2 Yaw Q arctan eq 3 C 1 It was found that the equation for ny was incorrect When relating the turn rate of the navigation frame with respect to the earth in the navigation frame and the turn rate of the earth with respect to the inertial frame in the navigation frame C should be used instead of Cp Cp relates the body frame to the navigation frame Since the turn rate in the body frame is desired C should be used Not much had to be done to the overall attitude computer Deriving Cn is done by transposing Cp This was found on page 44 C C C G eq 4 ET O EN Ob eq 5 cosy sny 0 Yaw Rotation about the z axis C siny cosy 0 eq 6 0 0 1 cos8 0 sim0 Pitch Rotation about the y axis C 0 1 0 eq 7 sn 0 cos8 1 0 0 Roll Rotation F about the x axis C 0 cos sind eq 8 0 sind cosd cos8 cosp coso sin sin sin cos sin 6sin cos sin 8 cos C cosOsin cosdco
5. number of iterations Every time it was done there was very minimal error x 100 8 2394x10 1 28 03 Testing Platform The original plan for today was to finish the development of the IMU testing platform We haven t been able to find a gearbox for the rotating portion of the testing platform y et therefore we are restricted from building anything but the base of the platform We need to know the dimensions of the gearbox to be able to construct the uprights that hold the gearbox and rotating rod We did purchase the wood for the base and uprights and the casters We will have to wait two lab periods for Dave Miller lab shop supervisor to return from vacation He will be constructing the rotating rod made of metal that needs to be flattened where it will be attached to the metal plate that will hold the IMU IMU User s Manual We picked up the IMU that was sent to Bradley over winter interim We read the DMU User s Manual which is the generic manual from Crossbow for their inertial sensors We had to read over it carefully so that we don t damage the 3 000 00 product Most importantly for now the power supply must be connected properly Red Black GND The Gyro View software that came with the IMU was installed on the laboratory computer Set up for the IMU is pretty simple Just connect the power supply as explained above and connect the 9 pin connector to the serial port of the computer and the 15 pin connector to t
6. s sin sinOsin sin p cos cos sin O sin eq 9 sin 8 sin 4 cos cosd cos This is the newest attitude computer written thus far SATTITUDE COMPUTER oo SGLOBAL VARIABL global angles E un Get IMU data from file oo Enter the roll pitch yaw yaw 0 pitch 0 roll 0 lat 0 delt 24 h 0 i 1 time 0 len 3600 earth earth_param prevcbn cbninit yaw pitch roll matrix cnb prevcbn body for wnb Sobtain earth data Sinital directional cosine lt get cnb to relate NED to euler prevcbn i Sobtain the euler angles time i 0 for i 1 len win trEARTH_IF_NED lat Sturn rate of the earth w r t inertial frame in the NED frame wen trLGF_earth v_e_n lat h Sturn rate of the NED frame w r t earth in the NED frame wnb trBODY_NED_BODY win wen wib cnb Sturn rate of the body w r t NED in the body frame skewwnb skew wnb cnb update_chbn cnb delt skewwnb cosine matrix cbn cnb euler cbn 1 1 time itl time i delt end lt skew wnb Supdate the directional sobtain the euler angles figure 1 subplot 311 plot time angles r grid title Roll Time in seconds subplot 312 plot time angles p grid title Pitch Time in seconds subplot 313 plot time angles y grid title Yaw Time in seconds
7. t the IMU to the computer and gather some data The default sample rate was 207 samples per second Gyro View has a real time graph which displays all 6 measured values 3 angular rates and 3 accelerations We tested moving the IMU in all directions and were able to see the lines move as expected We took about a couple minutes of data with the IMU just sitting on the table to see what kind of errors we would get Each trial requires a lot of data to be stored in the text file Here is a sample of some data Time s Roll rate Pitch rate Yaw rate X Accel Y Accel ZAccel Temperat Timer deg s deg s deg s g g g ure C counts 406 436 0 42114 0 88348 0 109863 0 00467 0 00458 0 997284 27 721 42381 406 4409 0 37537 0 86975 0 247192 0 00559 0 00339 0 998016 27 721 36281 406 4457 0 521851 0 40283 0 32959 0 00522 0 00403 0 997284 27 721 30185 406 4506 0 06409 0 27008 0 672913 0 00559 0 00485 0 997284 27 721 24086 406 4554 0 13733 0 0412 0 169373 0 00559 0 00339 0 997375 27 721 17987 We ran three trials and looked at the average roll pitch and yaw rates See data below Trial01 txt Roll Pitch Yaw Data Points 25 47457 1307 31 254 0772 3208 Avg Error 0 007941 0 40756 0 079201 Trial02 txt Roll Pitch Yaw Data Points 28 299 592 328 58 75848 1437 Avg Error 0 01969 0 4122 0 040889 Trial03 txt Roll Pitch Yaw Data Points 320 8099 3854 71 1460 88 15150 Avg Error 0 021176 0 25444 0 096428
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