DETAILED CHARACTERIZATION OF MAGNETIC FIELDS SURROUNDING THE WHORL II SPACECRAFT SIMULATOR AND APPLICATION TO ATTITUDE DETERMINATION

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1 DETAILED CHARACTERIZATION OF MAGNETIC FIELDS SURROUNDING THE WHORL II SPACECRAFT SIMULATOR AND APPLICATION TO ATTITUDE DETERMINATION Robbie Robertson Advisor: Dr. Kevin Shinpaugh Department of Aerospace and Ocean Engineering Virginia Polytechnic Institute and State University, Blacksburg, VA, 6, USA Abstract The research presented in this report focuses on the Whorl II spacecraft simulator in Virginia Tech s Space Systems Simulation Laboratory (SSSL). The goal of this research is to improve the reliability and accuracy of the magnetometer aboard Whorl II that is used for attitude determination. Magnetic interference surrounding the simulator causes unacceptable margins of error its perceived attitude. The first step in solving this problem was to characterize the magnetic fields surrounding the simulator. With this information, it was determined that the primary cause of magnetic interference is the ferrous, steel pedestal that supports the simulator. The behavior of the magnetic interference and the role of the magnetometer in attitude determination led to the final solution: to redesign the method of magnetometer calibration. This new method calibrates the raw magnetometer data dynamically, meaning that the calibration method changes in response to the position of the simulator. This significantly improves the accuracy of the magnetometer in attitude determination.. Spacecraft Simulators The Space Systems Simulation Laboratory (SSSL) at Virginia Tech houses the Distributed Spacecraft Attitude Control Systems Simulator, or DSACSS. The primary purpose of the DSACSS is to provide a test bed to aid in the development of distributed control laws for the pointing and spacing of a formation of satellites. DSACSS is actually composed of two separate spacecraft simulators, Whorl I and Whorl II. These simulators are mounted on spherical air bearings to simulate floating in a space environment. Whorl I (Figure ) is in a tabletop configuration that enables full freedom in yaw and ±5 of tilt in pitch and roll. Whorl II (Figure ) is in a dumbbell configuration that enables full freedom in yaw and roll with ± of freedom in pitch. Figure : Whorl I spacecraft simulator with its magnetometer circled at the end of its boom. 7 Roll Axis Yaw Axis Pitch Axis Figure : Whorl II spacecraft simulator with its magnetometer circled at the end of its boom. 7 Yaw, pitch, and roll axes are labeled. These simulators employ most of the same components for sensing and controlling their motion and orientation. For control, the simulators use momentum wheels and cold gas thrusters in each axis. For sensing, they use -axis accelerometers and magnetometers. The accuracy of these sensors is key to the work being done by researchers in the lab. Robertson

2 A -axis accelerometer and -axis magnetometer are used together to determine each simulator s attitude in pitch and roll. The accelerometers are able to provide attitude in pitch and roll by referencing the downward gravitational acceleration vector. In yaw, the orientation of the simulators with respect to this gravitational acceleration vector is unchanged. Therefore, the magnetometer is the only source for yaw angles. Each simulator has a PC- stack flight computer with a Fedora Linux operating system. Users access the flight computers via a wireless bridge connection to a central computer in the lab. Researchers write code for the simulators in C++ through the VI command line text editor.. Magnetic Field Characterization. Motivation Zarrin Chua conducted undergraduate research to implement a magnetometer on Whorl I for attitude determination. She concluded that magnetic interference from the environment and from equipment on the simulators made the data inadequate. Since then, the addition of a long vertical boom to Whorl I has moved the magnetometer away from this magnetic interference and led to sufficiently accurate attitude determination. Researchers added a boom to Whorl II as well; however, its length is limited by the location of the simulator and by its enclosure, which cannot be expanded due to space constraints. The information from the magnetometer is used, but magnetic interference greatly limits its accuracy and effectiveness. Due to this magnetic interference, the perceived attitude jumps significantly, sometimes by - or more with respect to the actual attitude. As previously discussed, in pitch and roll, this jump can be checked by accelerometer readings, but not in yaw. Therefore, these jumps have the greatest effect on the determination of yaw angle. This is especially unfortunate because yaw angles are the easiest to maintain and therefore the most important piece of attitude information. When interpreting data from the simulator, a jump usually hints at something significant about the dynamics of the simulation. The jumps caused by magnetometer errors make picking out these significant jumps very challenging. Therefore, eliminating the effect of these regions of interference would make it significantly easier to extract good information from simulations. Zarrin Chua characterized the magnetic fields in the lab. However, to compensate for the magnetic interference that she referred to and that researchers in the lab have recognized, a significantly more detailed characterization is necessary. This will show what causes the magnetic interference and how its influence varies over the space. This improved understanding of the problem is necessary to design the most appropriate solution.. Method.. Inertial Frame and Data Points To characterize the magnetic fields surrounding Whorl II, it is first necessary to define an appropriate inertial frame in which to locate the position of each magnetic field reading. The square enclosure surrounding the simulator is the basis for this reference frame with the x, y, and z-axes oriented as shown in Figure. The square enclosure is what limits the length of the boom on Whorl II, so any readings beyond this enclosure would be unnecessary. Shelves z y x Column z` Figure : Scaffold for collecting with inertial frame and magnetometer axes. Note the positioning of Whorl II to avoid interfering with the measurement. Figure : Magnetometer and level placed on the scaffold for a measurement. The previous magnetic reference field in the lab was determined using readings taken at 7 data points around Whorl I and at a fixed height above the floor of the lab. It is clear that many more points are necessary to both locate the regions of magnetic interference around Whorl II and understand what is causing them. I chose a six-inch separation because it provides points at both ends of the enclosure in the x y` Crosspiece x` Magnetometer Robertson

3 and y-directions and leads to over total data points... Scaffold For Collecting Data Since the enclosure is the basis for the reference frame, it is appropriate to utilize it for collecting the data. The scaffold used to collect the data is made of pressed paper and wood, non-ferrous materials that will not influence the magnetic fields. Figure shows the scaffold with the magnetometer in place. The two columns of pressed paper corner stock have wood shelves attached every six inches. These columns are mounted to slots in the enclosure s aluminum frame, enabling the columns to slide to y- positions marked along the frame. A crosspiece of pressed paper corner stock sits on the wood shelves. The crosspiece has areas marked off for placement of the magnetometer and numbered with x-coordinates. Movement of the magnetometer along the crosspiece, sliding of the columns along the enclosure s frame, and movement of the crosspiece among the shelves allows for accurate positioning of the magnetometer in the x, y, and z directions, respectively... Magnetometer For consistency, we take magnetic field readings with Whorl II s magnetometer. This magnetometer is a Honeywell HMR Smart Digital Magnetometer (Figure ) which is capable of measurements in the range of ± nanoteslas with an uncertainty of 6.7 nanoteslas. 5 Data is collected with the x`, y`, and z`-directions of the magnetometer aligned with the x, y, and z-directions of the inertial reference frame as shown in Figure. To prevent magnetic interference from the electronic equipment aboard Whorl II from influencing the readings, we power off the simulator and disconnect battery-charging cables before collecting data. The other simulator, Whorl I, connects to the magnetometer with a long serial cable and runs the data acquisition program... Data Acquisition Program The role of the data acquisition program is to interface with the magnetometer and store the data in MATLAB files for analysis. The speed and efficiency of the data collection process and quality of the data depend on this program. The program achieves speed and efficiency by minimizing the amount of input required. The program assumes the fastest order for moving the magnetometer through the region on the scaffold. By assuming this order, the user only needs to input a starting location and then follow a set procedure, pressing enter for each measurement. Data sets of samples each are taken repeatedly at each position until the standard deviation of the set falls below 5 nanoteslas in each direction. Considering that magnetic field readings are on the order of at least nanoteslas, this requirement is sufficient. The magnetic field measurement is the average of the accepted sample set. Position, magnetic field, and standard deviation values in each direction are stored in a MATLAB file.. Error Analysis In this process, there are four potential sources of error. The first potential source of error is incorrect placement of the magnetometer. The marked and numbered areas on the scaffold crosspiece, and the use of a level as shown in Figure, minimize this error enough to make it negligible for this characterization. The second potential source of error is movement of the magnetometer during measurement. The data acquisition program takes care of this. Assuming an approximately Gaussian distribution of values about the mean, the standard deviation requirement indicates that 99% of the magnetometer readings in a given measurement vary from the mean by less than 8.8 nanoteslas. More importantly, the standard deviation requirement indicates that the maximum probable error of each reading is, at most,.75 nanoteslas. The magnitudes of these values relative to magnitudes of the fields being measured, which are on the order of nanoteslas, indicate that the standard deviation requirement sufficiently minimizes errors due to magnetometer movement. The fourth potential source of error is temporary disturbances in the magnetic field. Taking samples for each measurement effectively reduces these errors. This ensures that a temporary disturbance is out-weighed by the other samples in the measurement, and if the disturbance is significant enough, the standard deviation requirement will not be met and the point will be retaken automatically.. Results Figure 5 shows the complete set of magnetic field data. Each arrow represents the magnitude and direction of the magnetic field at the position of its tail end. Shaded planes represent the floor and boundaries of the enclosure. The graphical representation of the air bearing pedestal begins at the floor and protrudes partially into the center of the vector field. Because of the amount of data involved, it is difficult to make any thorough conclusions about the behavior of the Robertson

4 magnetic fields from this plot. However, there is an apparent overall disturbance around the pedestal. z distance (ft) y distance (ft) Figure 5: All magnetic field vectors with air bearing pedestal, floor, and enclosure boundaries. Isolating individual planes of data in each direction enables a more thorough analysis by simplifying the identification of patterns. For clarity, it is necessary to normalize the magnetic field vectors in the following plots. When two planes of the same orientation are shown (i.e. x-y planes at two different z- positions), the data is normalized so that in both plots the scaling of the vectors is equivalent. The scaffold cannot be set up at the y-position of feet, because it is blocked by the air bearing pedestal. As a result, points with a y-position of feet are missing. It is also necessary to leave out points close to the pedestal. Their large relative magnitudes would result in MATLAB scaling down the other vectors, greatly reducing the clarity of the plots... x-y Planes Figures 6 and 7 show the magnetic fields as viewed from above the simulator. Figure 7 shows an x- y plane above the pedestal and Figure 6 shows a lower x-y plane which the pedestal passes through. The magnetic field in this lower plane seems to be drawn into the pedestal. There is a significant change in magnitude of the field in this plane as well. The magnetic field seems to exit the pedestal with less magnitude. These trends are apparent, but far less significant, in Figure 7, farther from the pedestal. This indicates that the pedestal is the only significant source x distance (ft) 5 of magnetic interference, and is not only affecting the directions of the field but also causing a variation in magnetic field strength in the x-y planes. This is especially significant because these values in the x-y planes are the only source of yaw angles for the simulator, which, as previously stated, are the most important piece of attitude information. y distance (ft) x distance (ft) Figure 6: Magnetic field vectors in the x-y plane of z= ft. The circle at the center represents the pedestal. y distance (ft) x distance (ft) Figure 7: Magnetic field vectors in the x-y plane of z=.5 ft. The circle at the center represents the pedestal... Vertical Planes (x-z and y-z) Figures 8 and 9 and Figures and show the magnetic fields as viewed from the sides of the simulator in the y and x directions, respectively. While the values shown from this perspective are less directly associated with yaw angle calculation, they are important in further understanding the magnetic interference. On Whorl II, the position of the magnetometer in these planes varies with the pitch of the simulator. Figures and show the same magnetic interference behavior observed in the Figures 6 and 7. Robertson

5 The magnetic field lines flow into and then out of the pedestal shortened in the direction perpendicular to the sides of the pedestal. These effects are absent from the planes furthest from the simulator shown in Figures 9 and, reinforcing the assumption that the pedestal is the only significant source of magnetic interference. z distance (ft) z distance (ft) 5 x distance (ft) Figure 8: Magnetic field vectors in the x-z plane of y= ft. The rectangle at the center represents the pedestal. z distance (ft) 5 6 y distance (ft) Figure 9: Magnetic field vectors in the x-z plane of y=7 ft. The rectangle at the center represents the pedestal. 5 6 y distance (ft) Figure : Magnetic field vectors in the x-z plane of x= ft. The rectangle at the center represents the pedestal. z distance (ft) 5 x distance (ft) Figure : Magnetic field vectors in the y-z plane of x=6 ft. The rectangle at the center represents the pedestal... Magnetic Interference and Position In order to design a solution, it is important to identify how the magnetic interference varies with position. Figure is a plot of the magnetic interference as a function of y-position. The value used to quantify magnetic interference is standard deviation from the reference field. The value plotted in Figure is the average of the standard deviations in the x and y- directions. Magnetic interference in the z-direction is disregarded because it has no effect on the determination of yaw angles. Robertson 5

6 Average Standard Deviation, σ (nt) av Position, y (ft) Figure : Average standard deviation from the reference field as a function of y-position. The vertical lines represent the pedestal. The curve fit is a third order polynomial. According to Figure, the magnetometer will experience the most magnetic interference when it is closest to the pedestal. The minimum distance from the pedestal occurs when the magnetometer end of the simulator is pitched downwards at the maximum pitch angle. The geometry that yields this minimum distance is shown in Figure. Figure : Whorl II illustrated at maximum pitch angle to show the minimum distance from magnetometer to pedestal With the magnetometer at its minimum distance from the pedestal,.58 feet, it is very close to a y-position of 5 feet. For this plane of y data points, the average deviations in the x and y-directions are 6.% and 7.7% of the reference field, respectively. The influence of the pedestal on the magnetic fields surrounding Whorl II is clearly significant, especially at higher pitch angles... Pedestal Properties The pedestal s magnetic properties are a result of its shape and material. The pedestal is a hollow cylinder made from 8 grade steel. 5 This grade corresponds to a composition of only about.8 percent carbon by weight, making it a mild steel. Mild steel is a magnetically soft, meaning it has relatively high permeability. As a result, mild steel tends to attract and distort magnetic fields. The field is attracted to the pedestal because its permeability is higher than that of the surrounding air and is therefore the path of least resistance. The field exits the pedestal with reduced magnitude in the direction that it entered but with the same overall magnitude. This indicates that the pedestal is redirecting the field upward, in the z- direction.. Application to Attitude Determination. Magnetometer Calibration On Whorl II, magnetic field vectors used in attitude determination are calibrated to compensate for hard iron and soft iron effects. Only the x and y- components of the magnetic field vectors are calibrated because these are the components important to the determination of yaw angle. Ideally, if the simulator were rotated through a series of yaw angles, the resulting magnetometer measurements would trace out a circle centered at the origin when plotted y-component versus x-component. The effects of hard iron lead to a constant distortion of Earth s magnetic field, offsetting the center of this circle. The effects of soft iron cause a distortion of Earth s magnetic field which varies with the position of the magnetometer, distorting the ideal circle into an ellipse. Determining the constants necessary for calibration requires the collection of a set of measurements like the one mentioned above. This is done by rotating the simulator through various yaw angles with the magnetometer at constant pitch and roll. An ellipse is then fit to this data set using a least squares fit to the following conic equation for an ellipse: = () where A, B, C, D, E, and F are constants describing the resulting ellipse and x and y are the Cartesian coordinates. From these coefficients, the calibration constants a (semi-major axis), b (semi-minor axis), (tilt angle), x off (x-offset), and y off (y-offset) are calculated. If it is determined that the ellipse is significantly rotated (if the B coefficient is sufficiently Robertson 6

7 large), then the x and y values are rotated by the tilt angle,, using the following equations: = arctan ( ) () = cos( ) sin ( ) () = sin( )+ cos ( ) () = ( ) ( ) () = ( )+ ( ) () This rotation eliminates the coefficient in the conic equation. Equations 5-8 determine the remaining calibration constants from the A, B, C, and D constants of the resulting ellipse. = (5) = (6) = = (7) (8) Equations 9-5 calibrate each raw magnetometer measurement using these constants. The figures shown with the equations (Figures -8) illustrate how each calibration step is manipulating the magnetometer data. Figure 6: Step, Equations and rotate the data points to align the major axis of the ellipse with the x- axis. = () Figure 7: Step, Equation scales the x values so that the fit ellipse is a circle. = cos( )+ sin ( ) () Figure : Example of magnetometer data plotted y- component versus x-component with initial fit ellipse. = sin( )+ cos ( ) (5) = (9) = () Figure 8: Final step, Equations and 5 reverse step by rotating the data points back by the tilt angle of the ellipse,. Figure 5: Step, Equations 9 and shift the data points to center the fit ellipse at the origin. This method of calibration is quite effective at compensating for the magnetic interference that it is designed for. However, the magnetic field characterization and subsequent analysis have shown that the magnetic interference around Whorl II varies Robertson 7

8 significantly in the z-direction. Regardless, a single set of constants currently calibrates all magnetometer data. We will call this method static magnetometer calibration. Potential solutions that would reduce the magnetic interference or avoid it, such as implementation of multiple magnetometers and installation of magnetic shielding, were thoroughly researched and considered. However, the most effective and practical solution is to change the way that the raw magnetometer data is calibrated.. Dynamic Magnetometer Calibration Section shows that magnetic interference experienced at higher and lower pitch angles vary significantly from the magnetic interference experienced at zero pitch. Dynamic magnetometer calibration considers this and uses the current pitch angle of the simulator as input, which the accelerometers can accurately provide. This pitch angle is used to choose an appropriate set of constants for calibrating the magnetometer data... Implementation This method requires that the full range of pitch angles (- to ) be split up into some number of regions, and that a set of calibration constants be determined for each one. As a starting point to evaluate the effectiveness of this method, the range of pitch angles is divided into three regions: - to -.5, -.5 to.5, and.5 to. The method outlined in Section., using data collected at pitch angles of -,, and, is used to determine calibration constants for the lower, middle and upper regions, respectively. A program on Whorl II collects this data. Given a desired pitch angle from the user, the data collection program tells the user where to place the simulator in yaw. The setup shown in Figure 9 enables accurate simulator positioning. Markings on the floor enable the user to accurately place the telescoping guide stand, which leads straight up to the magnetometer. The program continues to display pitch and roll angles from the accelerometers until the simulator is close enough to the desired pitch, zero roll, and zero angular rates. When these requirements are satisfied, the program stores the x and y-components of the magnetic field vector in a MATLAB file. A MATLAB program then uses this data to produce a set of calibration constants. Magnetometer Guide Stand Yaw Angles Figure 9: Setup for collecting calibration data. Yaw angles are marked on the floor. The level taped to the magnetometer eases positioning in roll. Repeating this process three times at each of the three pitch angles is necessary to determine uncertainties. The final constants (shown in Table ), are the averages of the calibration constants resulting from each of these trials. The following equation determines the uncertainty associated with each constant: = (6) where is the uncertainty, is the standard deviation, and is the number of samples. Table : Calibration constants: semi-major axis, a, semi-minor axis, b, x-offset, x off, y-offset, y off, and tilt angle, ϕ, determined from data collected at each of the three pitch angles, ϴ. Note that some of the uncertainties are relatively large. Section 5. addresses this issue. ϴ a b xoff yoff ϕ (deg) (nt) (nt) (nt) (nt) (deg) + ± ±. -9 ± ± ±.8 88 ±.6 98 ± ± -8. ± ±. - 9 ± ± ± ± ±.9 Robertson 8

9 .. Testing and Results Recall that the data collected for determining calibration constants was collected at specified yaw angles marked on the floor below the simulator (Figure 9). This was not necessary for determining the calibration constants. However, it is necessary for testing because the actual yaw angles can be referred to for error analysis. To test the static and dynamic calibration methods, the magnetometer data is calibrated using each method and then compared to the reference field in the lab to determine yaw angles. Table presents the results of applying each of the two methods to sets of magnetometer data at high, medium, and low pitch angles. Table : Comparison of dynamic and static calibration methods. σ is the standard deviation of the perceived yaw with respect to the actual yaw angles. ɛ is the maximum error in yaw angle. ϴ is the pitch angle at which the magnetometer measurements are taken. ϴ = ϴ = ϴ = - Calibration σ ɛ σ ɛ σ ɛ (deg) (deg) (deg) (deg) (deg) (deg) Static Dynamic Figure : Plot of perceived yaw angle versus actual yaw angle with the simulator at negative degrees pitch (ϴ=- ). Note the significant improvement of the dynamic calibration in this plot and in Table. The significant improvement resulting from dynamic calibration is apparent in Figures and. The yaw angles from this method are consistently closer to actual yaw angles. Table shows a large reduction in maximum errors and standard deviations from dynamic as opposed to static calibration. Figure : Plot of perceived yaw angle versus actual yaw angle with the simulator at positive degrees pitch (ϴ= ). Note the significant improvement of the dynamic calibration in this plot and in Table. Robertson 9

10 . Conclusions ) The magnetic interference around the Whorl II spacecraft simulator is significant and varies considerably throughout the space. This magnetic interference is caused by the magnetic properties and shape of the 8 steel air bearing pedestal. ) Static magnetometer calibration is not sufficient because it does not consider the significant variation of magnetic interference with simulator pitch angle. ) Dynamic magnetometer calibration will significantly improve the quality of attitude determination on the Whorl II spacecraft simulator. This will improve the capabilities of Whorl II and DSACSS as a whole for developing distributed control laws and performing control simulations. 5. Further Work 5. Improvements to Dynamic Calibration As previously mentioned, the dynamic magnetometer calibration has been implemented only to evaluate its effectiveness. The uncertainties presented in Table and errors in Table are not acceptable. Collecting orders of magnitude more data points for the calculation of each set of calibration constants and repeating the process more than three times will reduce these errors. Increasing the number of regions of pitch angles and selecting their boundaries more carefully will also improve the calibration. Further testing will utilize the new dynamic calibration function written into the software for Whorl II. Altering existing control simulations to utilize this new calibration function would thoroughly test its effectiveness. The success of these tests will determine the degree to which the improvements are applied. 5. Magnetic Coils Another objective of this research is the development of a magnetic coil system to surround Whorl II. These coils will produce static magnetic fields to overpower Earths field for a new, more consistent reference field. The design of such a system is underway with the help of Todd Bonalsky at NASA Goddard s Spacecraft Magnetic Test Facility. VSGC funding has paid for the components of a single axis system. This single axis system will consist of two coils oriented in the z-direction, the axis most aligned with the current reference field. 6. Acknowledgements The author thanks the Virginia Space Grant Consortium for proving funding for this project, Dr. Shinpaugh for his advice and guidance throughout the project, Dr. Hall for the opportunity to work in his lab, Brian Williams for his continuing technical assistance, Todd Bonalsky for his assistance with magnetic coil design, and Jin Share, Daniel Martin, and Anthony Rinaldi for their help in the SSSL. 7. Literature Cited ) Bevington, Philip R. Data reduction and error analysis for the physical sciences. st ed. New York: McGraw-Hill, 969. Print. ) Chua, Zarrin K. Implementation of a Three-Axis Magentometer On-Board an Attitude Simulator. Tech. ) Furlani, Edward P.. Permanent Magnet & Electromechanical Devices: Materials, Analysis, and Applications (Electromagnetism). st ed. Toronto: Academic Press,. Print. ) Konvalin, Christopher. Compensating for Tilt, Hard Iron and Soft Iron Effects. Memsense, 6 August 8. Web. 9 March. < 8_._Magnetometer_Calibration.pdf>. 5) Kunsa, Tim. Phone interview. Dec. 9. 6) "Magnetometer RS W/Case - HMR-D-."Digi-Key Corporation. N.p., n.d. Web. 9 Dec. 9. < magnetometer-rs-w-case-hmr-d-.html>. 7) "Simulators. Space Systems Simulation Laboratory. N.p., n.d. Web. 9 Dec. 9. < Robertson