Flight results of a low-cost attitude determination system

Acta Astronautica 99 (2014) 201–214

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Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro

Flight results of a low-cost attitude determination system John C. Springmann n, James W. Cutler University of Michigan, 1320 Beal Avenue, Ann Arbor, MI 48109, United States

a r t i c l e i n f o

abstract

Article history: Received 4 September 2013 Received in revised form 10 February 2014 Accepted 27 February 2014 Available online 11 March 2014

This paper presents flight results of the attitude determination system (ADS) flown on the Radio Aurora Explorer (RAX) satellites, RAX-1 and RAX-2, which are CubeSats developed to study space weather. The ADS sensors include commercial-off-the-shelf magnetometers, coarse sun sensors (photodiodes), and a MEMs rate gyroscope. A multiplicative extended Kalman filter is used for attitude estimation. On-orbit calibration was developed and applied to compensate for sensor and alignment errors, and attitude determination accuracies of 0.51 1–s have been demonstrated on-orbit. The approach of using low-cost sensors in conjunction with on-orbit calibration, which mitigates the need for pre-flight calibration and high-tolerance alignment during spacecraft assembly, reduces the time and cost associated with the subsystem development, and provides a low-cost solution for modest attitude determination requirements. Although the flight results presented in this paper are from a specific mission, the methods used and lessons learned can be used to maximize the performance of the ADS of any vehicle while minimizing the pre-flight calibration and alignment requirements. & 2014 IAA. Published by Elsevier Ltd. All rights reserved.

Keywords: Attitude determination CubeSat Nanosatellite Sensor calibration On-orbit calibration

1. Introduction This paper presents flight results and lessons learned from the attitude determination system (ADS) flown on the Radio Aurora Explorer (RAX) satellites, RAX-1 and RAX-2. This follows a previously published paper on the ADS design [30] by analyzing and assessing flight system performance. The ADS includes coarse sun sensors (photodiodes), three-axis magnetometers, and a three-axis rate gyroscope, and each of these is a commercial-off-the-shelf sensor designed for terrestrial use. The system, developed primarily by students and under an initial time constraint of one year between the start of the project and delivery of the flight satellite, was developed to be a low-cost solution to the 51 (1–s) attitude determination requirements of the RAX mission [9,30]. Attitude determination accuracies of

n

Corresponding author. E-mail address: [email protected] (J.C. Springmann).

http://dx.doi.org/10.1016/j.actaastro.2014.02.026 0094-5765 & 2014 IAA. Published by Elsevier Ltd. All rights reserved.

approximately 21 are typical of coarse sun sensor and magnetometer-based systems [1,5,30,31], but after the development and application of on-orbit calibration methods to estimate and compensate for various sensor errors, accuracies better than 0.51 1–s have been demonstrated with the RAX flight data. The methods and lessons learned from the RAX data presented in this paper can be extended to any ADS to maximize the angular accuracy while minimizing the pre-flight development and calibration requirements. The RAX mission was developed to study magnetic field-aligned irregularities (FAI) of electron density in Earth's ionosphere, an aspect of space weather. These irregularities are known to scatter radio signals, potentially interfering with space-based resources such as GPS and global communication, but their formation is not well understood. The scientific payload is an ultra-high frequency radar receiver that works in conjunction with ground-based transmitters in a bistatic configuration: a ground-based transmitter illuminates the FAI while the

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satellite-based receiver passes overhead, recording both the direct signals and those scattered from the irregularities. This data, combined with other measurements of ionospheric conditions provided by the ground-based radar, provides information on the formation and structure of the FAI. The ultimate goal of the mission is to enable the development of short-term forecast models for the ionospheric irregularities. There are currently two satellites in the RAX mission; RAX-1 launched in November 2010 and RAX-2 launched in October 2011. Both satellites are 3U CubeSats, a standardized nanosatellite form factor. RAX-1 successfully completed science experiments (radar measurements), demonstrating the payload and bus capabilities, but the mission ended after three months of operations due to a flaw in the solar panel design [10]. RAX-2 corrected the solar panel issue and operated for 1.5 years on-orbit, successfully completing its planned one year mission and providing unprecedented measurements of the ionospheric irregularities [2]. More information on the scientific mission and satellite development can be found in Refs. [3,9]. A wide variety of methods – both in terms of sensor types and estimation algorithms – exist for spacecraft attitude determination. Common types of attitude determination sensors for spacecraft in low-Earth orbit include magnetometers, sun sensors, Earth horizon sensors, and star trackers. Magnetometers and sun sensors are the most common types of sensors because of their relative ease of implementation, simplicity, and low cost [33]. For nanosatellite attitude determination, magnetometers and sun sensors have been used almost exclusively, but as miniature star trackers and horizon sensors have become commercially available over the last few years, these sensors are also becoming more prevalent on small spacecraft. Still, magnetometer and sun sensors are lower-cost solutions that meet the requirements of many missions, including the requirements of RAX, and thus are the focus of this paper. Magnetometers and sun sensors provide a measurement of the local magnetic vector and line-of-sight vector to the sun in the satellite body-fixed frame. A MEMs gyroscope is also used to measure the spacecraft angular velocity. Many methods exist for attitude estimation with these types of sensors, including single-point methods such as QUEST and TRIAD, and recursive methods such as extended Kalman filters [8]. Of the available methods, the multiplicative extended Kalman filter (MEKF) has become a default approach to attitude estimation [8,19,22], and it is one of the methods utilized in this paper. The MEKF supplements the additive state update of the traditional Kalman filter with a multiplicative update, which maintains the unity norm constraint of attitude quaternions (or equivalently, the attitude matrix orthogonality constraint). Additionally, the filter utilizes a three-axis rate gyroscope instead of a dynamic model to propagate attitude, which alleviates the difficulties in spacecraft dynamics modeling [19]. More details on the estimation algorithms used in this work are presented in Sections 4 and 5. The unique contribution of this paper is the dissemination of the methods and results from actual flight data. This follows a previously published paper in which the

design of the ADS was described [30]. The techniques used for attitude estimation are a combination of both newly developed and previously existing methods. The demonstrated determination accuracy of 0.51 1–s surpasses the expected performance of this class of sensors, and to the authors' knowledge, surpasses the accuracy of flight results with this class of sensors previously reported in the literature. The remainder of this paper is organized as follows. Details on the RAX ADS, including the performance specifications of the individual sensors, are given in Section 2, and a description of the flight data used in this paper is given in Section 3. Before using the sensors measurements for attitude determination, the sensors are calibrated using flight data, which is discussed in Section 4. Attitude estimation algorithms and results are then presented in Section 5. A summary and discussion of the approach to RAX attitude determination is then given in Section 6, and conclusions are given in Section 7. 2. The RAX attitude determination system The RAX-1 and RAX-2 flight satellites are shown in Fig. 1. Both spacecraft are 3U CubeSats with physical dimensions of 34
10
10 cm3 and masses under 3 kg. The ADS of each satellite is identical with the exception of their photodiode configurations. The design of the RAX-1 ADS is described in Ref. [30], which includes details on the requirements, sensor selection, and sensor integration that are beyond the scope of this paper. In this section, the sensors and their performance parameters are reviewed in Section 2.1, and the RAX-1 and RAX-2 photodiode configurations are then discussed in Section 2.2. 2.1. Attitude determination sensors The attitude determination sensors include two threeaxis magnetometers, a three-axis rate gyroscope, and photodiodes for sun sensing. The specific sensors and their parameters are given in Table 1, and they are discussed in the following subsections. 2.1.1. Magnetometers Two three-axis magnetometers are included in RAX-1 and RAX-2: a PNI MicroMag3 and an Analog Devices ADIS16405, which is an inertial measurement unit that includes a magnetometer. All sensors are affected by errors that cause sensor scale factors, bias, and nonorthogonality. While these errors can be estimated and compensated through calibration, the sensor noise is the limiting factor in magnetometer performance. The noise floor of both magnetometers, given in Table 1, was estimated by computing the standard deviation of measurements taken by the stand-alone, de-integrated magnetometers in a magnetically shielded environment to reduce noise from nearby objects. When integrated in the satellite, magnetometer accuracy is also affected by the surrounding electronics, which increases the noise environment as well as causes additional scale factor, bias, and time-varying errors. On-orbit calibration is used to

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203

Fig. 1. The flight-ready RAX satellites with photodiode locations circled. Photodiodes are located on all six sides of the spacecraft, but only the locations visible in the photo are indicated. (a) RAX-1 and (b) RAX-2.

Table 1 Attitude determination sensors and parameters. Type

Model

Parameters

Three-axis magnetometer Three-axis magnetometer Three-axis rate gyroscope

PNI MicroMag 3

128 nT resolution o 128 nT noise floor 50 nT resolution 111 nT noise floor

Photodiode

Analog Devices ADIS16405 Analog Devices ADIS16405

Oram SFH2430

2:8
10  2 1=s1=2 ARW 1:8
10  4 1=s3=2 RRW 3–4 V max output 0.02 V standard deviation 1401 field of view

estimate and compensate for these errors; this is discussed in Section 4. 2.1.2. Rate gyroscopes One three-axis MEMs rate gyroscope, part of the Analog Devices ADIS16405, is included in the ADS. Like all inertial sensors, rate gyroscopes are subject to drifting bias. A widely used model of a single-axis of a rate gyroscope with no scale factor error is given in Eq. (1), where ω~ is the measured angular rate, ω is the true angular rate, β is the bias, and ηv and ηu are the independent mean-zero Gaussian white noise processes

with Efηv ðtÞηv ðτÞg ¼ s2v δðt  τÞ and Efηu ðtÞηu ðτÞg ¼ s2u δðt  τÞ [6,12]. In practice, the power spectral densities s2v and s2u are typically called angle random walk (ARW) and rate random walk (RRW), respectively. While gyros are also subject to scale factor errors and non-orthogonality, ARW and RRW define the gyro performance, whereas scale factors and alignment errors can be compensated with calibration. Typically, ARW is given on manufacturerprovided specification sheets, but RRW is not. Both ARW and RRW can be characterized experimentally [11,34]. The ARW and RRW listed in Table 1 were measured and provided in Ref. [32]; these parameters were not measured by the RAX team: ~ ωðtÞ ¼ ωðtÞ þβðtÞ þηv ðtÞ

ð1aÞ

_ ¼ ηu ðtÞ βðtÞ

ð1bÞ

2.1.3. Photodiodes Osram SFH430 photodiodes are used for sun sensing. Photodiodes generate current as a function of incoming light, and when individual photodiodes1 are used for sun sensing, they are sometimes referred to as coarse sun sensors. The photodiode measurement model is given in 1 Multiple photodiodes can be combined with a mask to form different types of sun sensors, but in their implementation in the RAX ADS, the stand-alone photodiodes are used, with each photodiode providing a measurement of a single sun vector component.

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1

2

3

60

60

30

30

0 −30 −60 −90 −180

1

90

Elevation,deg

Elevation,deg

90

2

3

0 −30 −60

−120

−60

0

60

120

180

−90 −180

−120

−60

0

60

120

180

Azimuth, deg

Azimuth, deg

Fig. 2. Photodiode coverage over the body-fixed frame for RAX-1 and RAX-2. Each symbol on the plot is a direction in the body-fixed frame, and the symbols have a uniform angular distribution of approximately 41. The symbols are shaped and color-coded by the number of sun vector components measured when the sun is in the direction of the symbol. On RAX-1 the three component availability corresponds to exactly three photodiodes being illuminated, but on RAX-2, up to seven photodiodes are illuminated for portions of the body frame. (a) RAX-1 photodiode coverage and (b) RAX-2 photodiode coverage. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

Eq. (2), where I~ is the measured current output, C is a dimensional scale factor that corresponds to the maximum current output when illuminated by direct sunlight, n^ is the 3
1 unit vector that is normal to the photosensitive plane, s^ is the 3
1 unit sun vector, Ea is the irradiance of Earth albedo, which is attitude- and position-dependent, EAM0 is the irradiance of direct sunlight, and η is zero mean Gaussian measurement noise [4,28]. In their implementation in the RAX ADS, the voltage across a resistor is measured rather than current, which is directly proportional to the current output of Eq. (2). The dimensional scale factor C is a function of the photodiode and the surrounding circuitry. As indicated in Table 1, the scale factor for RAX is between 3 and 4 V, and the standard deviation of η is approximately 0.02 V. The standard deviation was calculated from measurements taken under constant irradiance. The photodiode model is generally not valid over a full 1801 field of view; the Osram SFH2430s have a half-angle field of view of approximately 701. In practice, the 0.02 V standard deviation is optimistic metric of photodiode accuracy because it does not account for uncertainty in Earth albedo or other deviations from the T cosine dependence (n^ s^ ¼ cos ðθÞ) of Eq. (2): Ea T þη I~ ¼ C n^ s^ þC EAM0

ð2Þ

2.2. Photodiode configuration RAX-1 and RAX-2 utilize different photodiode configurations, where the improvement of the RAX-2 system was motivated by the desire for improved sun sensor coverage over the body-fixed frame. The RAX-1 photodiodes are mounted parallel to the surfaces of the six sides of the satellite, a configuration that is typical of CubeSats. Referring to the coordinate system in Fig. 1(a), there is one photodiode each on the x/y surfaces and multiple photodiodes on the z surfaces (two on  z and three on þz), with the redundant sensors motivated by the potential for shadowing by the antennas. Since the FOV of each photodiode is less than 1801, this configuration does not provide

three-component sun sensor coverage over the entire body-fixed frame. Although three-component coverage is not required for use in the attitude estimator, threecomponent coverage is required for a 3
1 sun vector measurement. A full 3
1 vector measurement, as opposed to measurements of one or two components, is useful because it can be used to verify sensor parameters and performance through scalar checking2 [20]. The RAX-1 sun sensor coverage over the body-fixed frame is shown in Fig. 2(a). The symbols in the figure represent directions on the attitude sphere; they have a uniform angular distribution in the body-fixed frame and represent directions in the body frame. The symbols are shaped and color-coded by how many measured sun vector components are available when the sun is in the direction of the symbol. Three non-parallel and noncoplanar photodiodes must be simultaneously illuminated for a three-component sun vector measurement. RAX-2 includes additional photodiodes mounted at angles on the side panels to increase to three-component coverage to nearly the entire body-frame. The orientation of the photodiodes is given in Table 2. This configuration was chosen because it provides three-component sun sensor coverage over 96% of the body frame while adhering to the CubeSat form factor specifications that limit the height of objects above each CubeSat surface. Since the RAX-2 design, a design method for optimizing the photodiode configuration was developed and is presented in Ref. [29]. The methods of Ref. [29] can also be used to map the uncertainty of the sun vector measurement over the attitude sphere for a given photodiode configuration. An additional change to the photodiodes of RAX-2 compared to RAX-1 is the added use of coverglass. On RAX-1, the photodiodes were mounted on the spacecraft

2 Various scalar metrics, such as the magnitude of the measured vector or the angle between two vectors, can be used for fault detection, sensor calibration, and verification of sensor calibration parameters. For example, a scalar objective function, the magnitude of the measured vector, is used for the magnetometer calibration discussed in Section 4.1, and the magnitude of the sun vector measurement is one metric for verification of the photodiode calibration discussed in Section 4.3.

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Table 2 The intended azimuth and elevation angles of the photodiode normal directions on RAX-2. The side panel is the surface to which the sensors are mounted (coordinate system shown in Fig. 1(b)). Photodiode #

1

2

3

4

5

6

7

8

9

10

11

12

13–15

16-17

Side panel Azimuth (degree) Elevation (degree)

þx 17  10

0 20

 17  10

x  162  10

180 20

162  10

þy 72 10

107 10

90  20

y  107 10

 72 10

 90  20

þz 0 90

z 0  90

as-is, with no additional shielding. On RAX-2, solar cell coverglass was adhered to each photodiode. This was done to prevent photodiode degradation due to (primarily ultraviolet) radiation. On RAX-1, the output of the photodiodes degraded over time [26], and no such degradation is evident on RAX-2 with the addition of the coverglass.

well as the ability to sample the gyroscope and directional sensors at different frequencies, but the 1 Hz gyro sampling frequency is sufficient for the steady-state portion of the flight, so the higher frequency sampling capabilities were not utilized during the mission. 4. On-orbit calibration

3. Flight data description In this section, we provide an overview of the flight data that will be utilized throughout the rest of the paper. RAX-1 launched into a 721 inclination, 650 km circular orbit, and RAX-2 launched into a 1021 inclination, 400
820 km orbit. Neither RAX-1 nor RAX-2 have real-time attitude estimation requirements; the sensor measurements are downloaded and processed off-line. Nonetheless, an MEKF, which is a recursive method that can be implemented in real-time, is one of the methods used for attitude estimation (this will be discussed further in Section 5). Even though the MEKF can be implemented real-time, it is used for batch processing so that the results are comparable to those that could be achieved in realtime implementation. The RAX satellites utilize a passive magnetic attitude control system, which is a common scheme on nanosatellites due to its simplicity (no moving parts and no processing required) and small volume and mass requirements [15]. This system consists of permanent magnets to align the spacecraft with Earth's magnetic field, and soft magnetic material to dissipate rotational kinetic energy. After deployment from the launch vehicle, the passive magnetic system gradually dissipates the kinetic energy and the spacecraft settles to steady-state dynamics about the magnetic field. These steady-state dynamics consist of oscillations of approximately 7201 about the geomagnetic field with 1–21/s spin around the magnetic field. The portion of the mission following deployment from the launch vehicle and before reaching steady-state dynamics is referred to as tumbling. Data from both the tumbling and steady-state portions of the flight are utilized in this paper. The initial RAX-1 and RAX-2 angular velocity was approximately 201/s, and it took both satellites approximately three weeks to reach steady-state dynamics. Throughout the remainder of this paper, various sets of 1 Hz measurements are used. These data sets were collected for initial spacecraft checkout and sensor calibration, as well as during science experiments and other times of interest. The same sampling frequency was used for all sensors, and these measurements were recorded and downloaded from the spacecraft. The RAX ADS has the capability to sample the sensors at higher frequencies, as

Sensor calibration is critical for accurate attitude determination. In general, sensor measurements are corrupted by errors such as scale factors, bias, and angular misalignments. Sensor calibration using only in-flight data is referred to as on-orbit calibration. On-orbit calibration can be advantageous over pre-flight calibration because it accounts for any changes to sensor parameters due to the launch or orbit environments, and it reduces the need for complex in-lab calibration equipment. Some pre-flight calibration was performed on the integrated RAX ADS [30], but all pre-flight calibration was ultimately replaced by on-orbit calibration, completely mitigating the need for pre-flight calibration. New methods were developed for on-orbit calibration of the RAX magnetometers and photodiodes, and these techniques have general applicability to these sensors on any platform. The on-orbit calibration for each attitude sensor is described in the following sections. 4.1. Magnetometer calibration Magnetometers are corrupted by both constant and time-varying errors. The constant error sources include hard iron errors, null shift errors, soft iron errors, scale factors, and non-orthogonality [13,14]. The combined effect of these errors is quantified by constant scale factors, offsets, and angular mis-alignments in the measurement model. Time-varying errors are caused by nearby electronics: current-carrying wires generate magnetic fields, resulting in a time-varying magnetometer bias [27]. Traditionally, this time-varying bias is mitigated by either using a boom to physically separate the magnetometer from the spacecraft, or by using design and manufacturing techniques to minimize the influence of the electronic components on the magnetometers. Both of these methods add cost and complexity to the satellite design, which is not conducive to low-cost ADS. The RAX magnetometers are embedded within the satellite and subject to the magnetic fields of the surrounding electronics. To compensate for the resulting errors, a new calibration technique was developed to compensate for the effect of the nearby electronics. The magnetometer calibration method is described in Ref. [27] and summarized here for completeness before

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describing the results of its application to RAX-1 and RAX-2. The magnetometer measurement model is given in Eq. (3), where B~ j is the measured magnetic field in each sensor axis (jA fx; y; zg); Bj are the true magnetic field components; a, b and c are scale factors; x0, y0, and z0 are constant bias; ρ, λ, and ϕ define the axis nonorthogonality using the definition shown in Fig. 3; I~j are measured currents in the various satellite subsystems; si;x~ , si;y~ , and si;z~ are scale factors that map the i-th current measurement to magnetometer bias; N is the total number of current measurements used; and ηj is mean zero Gaussian noise. This model accounts for the time-varying effect of nearby electronics by including actual measurements of the nearby currents in the measurement model, and mapping them to the corresponding magnetometer error by constant scale factors. There are 9 þ 3N calibration parameters that are estimated and used to correct the magnetometer measurements: a, b, c, x0, y0, z0, ρ, λ, ϕ, si;x~ , si;y~ , and si;z~ , with i ¼ 1; …; N. N

B~ x ¼ aBx þx0 þ ∑ si;x~ I~i þ ηx ;

ð3aÞ

i¼1

N

B~ y ¼ bðBy cos ðρÞ þ Bx sin ðρÞÞ þ y0 þ ∑ si;y~ I~i þ ηy ;

ð3bÞ

i¼1

B~ z ¼ cðBx sin ðλÞ þBy sin ðϕÞ cos ðλÞ þBz cos ðϕÞ cos ðλÞÞ N

þ z0 þ ∑ si;z~ I~i þ ηz :

ð3cÞ

i¼1

The calibration parameters are estimated by minimizing the difference between the measured magnetic field ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi magnitude, B2x þ B2y þ B2z , and the expected field magnitude as a function of the calibration parameters. The expected magnetic field magnitude is given by a geomagnetic field model, such as the International Geomagnetic Reference Field (IGRF), coupled with spacecraft position. The minimization is carried out numerically using nonlinear least-squares [27]. This is a batch method that was carried out in ground-processing of the data; real-time magnetometer correction can be implemented by uploading the calibration parameters to the spacecraft, and realtime estimation of the calibration parameters using a recursive method is left for future work. The measurement

Fig. 3. The convention used to define the sensor non-orthogonality by ~ y, ~ and z~ represent the actual sensor axes, and x, y, the angles ρ, ϕ, and λ. x, and z denote the axes of the orthogonal frame. The x-axis of the true and orthogonal frames are coincident, and the y-axis is in the x~  y~ plane.

model is general in that any number of current measurements can be included in the calibration, and the currents that impact the magnetometer will depend on the satellite design. Current monitoring is typically already part of satellite health monitoring, so this calibration does not necessarily require additional current sensors, but the ADS would need access to the current measurements for realtime magnetometer correction, if desired. This should be accounted for in the satellite design. The calibration was applied to flight data from RAX-1 and RAX-2. The data used in the calibration was taken early in the mission while the spacecraft was still tumbling following deployment from the launch vehicle. The calibration parameter estimates are most accurate when using measurements throughout the sensor field of view [27]. Since RAX is passively magnetically stabilized, the magnetometer measurements are predominately along a single direction after reaching steady-state dynamics, whereas the tumbling phase provides excellent data for calibration. A sample of raw and calibrated RAX-2 magnetometer data is shown in Fig. 4. This data set is from November 4, 2011 18:29:45 UT, and is the data during the tumbling portion of the flight that has been used for calibration. The magnitude of the raw, uncalibrated measurements from the PNI MicroMag3 magnetometer are shown in Fig. 4(a), and the magnitude of the calibrated measurements is shown in Fig. 4(b). The root mean square error (RMSE) of the calibrated magnitude, where error is the difference between the measured and IGRF magnitudes, is 306 nT. The histogram of this error is shown in Fig. 4(c). The accuracy and consistency of the estimated calibration parameters are demonstrated by using the parameters estimated from this data set to correct magnetometer measurements from other data sets. Using this process to correct other RAX-2 data sets, the RMSE of the calibration remains between 300 and 400 nT. This corresponds to an angular accuracy of approximately 0.71 in an ambient magnetic field of 30 μT. When using the same calibration method with RAX-1 data, the RAX-1 calibration resulted in RMSE of 200 nT. We hypothesize that the RAX-2 residual magnetometer error is higher than RAX-1 because of the difference in the solar panels. RAX-1 had a design flaw in the solar panels which limited the power generation and resulted in a power failure approximately two months after launch [9,10]. RAX-2 flew corrected solar panels which resulted in more electrical activity on the spacecraft. Compared to RAX-2, RAX-1 did not have as much electrical activity in each solar cell, corresponding to a reduction in the current-generated magnetic fields compared to RAX-2. In application to both satellites, the current measurements used in the calibration were from sensors included in the design for satellite health monitoring; no current sensors were added specifically for magnetometer calibration. Additional current sensors could potentially be used to further improve the results of the calibration, but this has not been tested. After calibration, which resolves the angular axis nonorthogonality in the magnetometer-fixed frame, the relative alignment of the two magnetometers can be estimated by applying any single-point attitude estimation

60

45

50

40

40

35

µT

µT

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30

30 25

20

IGRF

Measured 10

207

0

20

40

IGRF

Measured 60

80

100

20

120

0

20

Minutes Elapsed

40

60

80

100

120

Minutes Elapsed

# of Occurances

1000 800 600 400 200 0 −600 −400 −200

0

200

400

600

Error, nT Fig. 4. Magnetometer calibration results for RAX-2. (a) and (b) The measured magnetic field magnitude before and after calibration overlaid with the magnitude predicted by the IGRF model is shown. (c) Histogram of the difference between the calibrated measurements and IGRF magnitudes shown in (b). (a) Raw measurements, (b) Calibrated measurements and (c) Histogram of the residual error after calibration.

method, such as the q-method [8], to a batch of measurements from both magnetometers. In application of the attitude estimation method, one magnetometer-fixed frame is treated as the reference frame, and the other is treated as the measurement frame. The attitude is then the orientation of the measurement frame relative to the reference frame. Using this method, the estimated attitude of the PNI magnetometer relative to the ADIS16405 magnetometer on RAX-2 is 2 3 0:9996 0:0090 0:0280 6 7 A^ PNI=IMU ¼ 4 0:0083 0:9997 0:0249 5; ð4Þ 0:0282 0:0247 0:9993 which corresponds to rotation angles of 0.51, 1.61, and 1.41 about the x-, y-, and z-axes, respectively. For comparison, the estimated non-orthogonality errors of the PNI magnetometer are 0.51,  4.01, and 2.21, and the estimated non-orthogonality errors of the ADIS16405 magnetometer are  0.61,  2.41, and  1.71. From this analysis, we see that angular errors of up to 41 were present in the RAX-2 magnetometers, due to both sensor non-orthogonality and relative mis-alignments. These angular errors have been estimated and corrected using on-orbit calibration. 4.2. Rate gyroscope calibration In addition to the drifting bias discussed in Section 2.1.2, a tri-axial rate gyroscope is also subject to scaling and non-orthogonality errors. Various on-orbit rate gyroscope calibration methods exist to estimate and compensate for these errors [16,17,23], but no on-orbit gyroscope calibration was applied to the RAX data. Pre-flight

calibration was used to estimate the gyroscope scale factors, and no scale factor errors were detected [30]. Additionally, since the magnetometer and gyroscope are packaged together in the ADIS16405, we assume that the non-orthogonality error of the gyroscope is the same as the magnetometer, which was discussed in Section 4.1. This is a reasonable assumption because within the ADIS16405, the channels of the two sensors are mounted to the same structure. After application of the MEKF for attitude estimation, which will be discussed in Section 5, no additional gyroscope errors are evident, and any unmodeled non-orthogonality or scale factor errors are captured in the process noise. 4.3. Photodiode calibration Photodiodes provide a measurement of the sun vector component along the direction normal to the photosensitive plane, herein referred to as the photodiode normal T direction. This component is equivalent to n^ s^ , where n^ is the normal direction and s^ is the sun vector. As seen in the photodiode measurement model of Eq. (2), the measurements are subject to uncertainty in the dimensional scale factor C, uncertainty in the normal direction, and Earth albedo. Thorough pre-flight calibration of photodiodes to determine the scale factor requires a light source that matches the characteristics of sunlight in orbit, which can be a barrier to pre-flight calibration. Additionally, accurate knowledge of the photodiode normal direction may require high-tolerance integration procedures, which negates the low-cost benefit of using photodiodes. This is true particularly when photodiodes are mounted at angles from the spacecraft surfaces, as is the case on RAX-2. Therefore, we developed on-orbit calibration techniques to

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24 EKF estimate

UF estimate

EKF 3−σ

UF 3−σ

UF estimate

EKF 3−σ

UF 3−σ

20

3.6 3.4 60

EKF estimate

22

3.8

deg

V

4

18 70

80

90

100

110

16 60

120

70

80

Minutes Elapsed

deg

−4

90

100

110

120

Minutes Elapsed

EKF estimate

UF estimate

EKF 3−σ

UF 3−σ

−6 −8 −10 60

70

80

90

100

110

120

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estimate the scale factors and photodiode normal directions without pre-flight characterization. The on-orbit photodiode method is derived and demonstrated through both simulation and application to RAX flight data in Ref. [28], and the method is summarized again here for completeness. Unlike the attitudeindependent magnetometer calibration, the photodiode calibration is a recursive method that simultaneously estimates spacecraft attitude and the photodiode calibration parameters. There are three calibration parameters for each photodiode: the scale factor and two angles, azimuth and elevation in the body frame, that parametrize the photodiode normal direction. The recursive, attitudedependent formulation enables the inclusion of an attitude-dependent Earth albedo model, which is used to compensate for the albedo contribution in the photodiode measurements. Additionally, it enables the inclusion of any number of photodiodes in the calibration, as opposed to an attitude-independent method that would require a full sun vector measurement and therefore at least three simultaneously illuminated photodiodes. The photodiode calibration has been implemented with both an extended Kalman filter (EKF) and unscented filter (UF) [28]. Like the MEKF [19,22], the calibration uses biascorrected rate gyroscope measurements directly in the attitude propagation, and thus requires a three-axis rate gyroscope. The traditional six states of the MEKF, the three-state gyro bias vector and three-state attitude error vector, are expanded to include the photodiode calibration parameters. The EKF- and UF-based calibrations provide the same estimation accuracy if the initial state estimates are sufficiently close to the true states, but the UF provides better performance under a wider range of initial conditions. In application to RAX-1 and RAX-2, we have found that the initial conditions derived from the flight data3 are sufficiently accurate to use the EKF for photodiode

calibration [28]. Given the lower computational requirements of the EKF, this facilitates real-time implementation, but the UF should be used when possible, such as during ground-based processing of the flight data, since it is robust to a broader range of initial conditions. The photodiode calibration parameter state estimates are shown over time for one of the RAX-2 photodiodes in Fig. 5, and the behavior is representative of all photodiodes. This is the same portion of flight data that was discussed in Section 4.1 and shown in Fig. 4, but only the portion of the orbit that is illuminated by the sun is used for the calibration. The initial estimate of the scale factor estimate was taken to be the maximum measured output of the photodiode. The dimensional scale factor maps the sun vector component to the measured current output, and it is equivalent to the current output under direct sunlight, so the maximum current output measured when the spacecraft is tumbling provides a reasonable initial state estimate. The initial estimate of the photodiode normal direction was the intended mounting orientation on the spacecraft. In application of the calibration to the RAX-2 data, the improvement in scale factor estimate relative to its initial condition ranged from 0.17 to 0.46 V, which is equivalent to 4.4–12.7% of the calibration scale factor. The improvement in azimuth and elevation estimates ranged from 01 to 91. For RAX-1, the respective improvements were 0.09–0.29 V, equivalent to 3.4–10.2%, and 0–41. A histogram of the angular improvement in the sun vector estimate resulting from the RAX-2 on-orbit calibration, where the improvement is defined as the difference between the sun vector estimate using the initial parameters and those resulting from on-orbit calibration, is shown in Fig. 6. The sun vector is calculated from the photodiode measurements using a linear least-squares formulation [29]. The mean improvement in angular

3 The initial conditions for the scale factors are the maximum measured output of each photodiode while the spacecraft tumbling, determined from inspection of the flight data, which is a reasonable estimate since the scale factor corresponds to the sensor output caused

(footnote continued) by direct sunlight. The initial conditions for the photodiode orientation are the intended mounting angles.

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accuracy of the sun vector resulting from the on-orbit calibration is 9.11. The analogous improvement in RAX-1 measurements is 5.61. These angular improvements are significant given that the angular accuracy of photodiodes is on the order of degrees, and as seen in Fig. 5, the 3–s accuracy for the normal direction estimates is less than 11.

5. Attitude estimation methods and results With on-orbit sensor calibration complete, the rate gyroscope, magnetometer, and photodiode measurements are used to estimate satellite attitude. Two methods have been used for attitude estimation. The first method is the photodiode calibration filter of Section 4.3, which estimates the photodiode calibration parameters and spacecraft attitude simultaneously, and the second is the conventional 6-state MEKF [19,22]. Of the many methods available for attitude estimation, the MEKF has become a widely used approach [8]. The MEKF estimates six states: three components of attitude and three components of rate gyroscope bias. This filter alleviates the difficulties of modeling spacecraft dynamics by using bias-corrected rate gyroscope measurements directly in the attitude propagation, and it maintains attitude orthogonality constraints through a multiplicative quaternion update. Although attitude estimation for RAX is done off-line, the MEKF is well suited for real-time implementation [7]. In this section, we discuss results of attitude estimation using both the MEKF and the photodiode calibration filter. In general, non-linear recursive estimators require initial conditions for the state estimates. The attitude is initialized to the estimate provided by a single-point estimator, such as QUEST or TRIAD, using the magnetometer and photodiode measurements. Since both directional measurements are required, the estimator is initialized at a time when the spacecraft is in the sun so the sun vector measurement is available. The gyro bias states are initialized to zero, and the initialization of the photodiode calibration states was discussed in Section 4.3.

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Attitude determination accuracy is quantified by the state error covariance of the tuned filter. Filter tuning is the process of adjusting the process and measurement covariance to produce accurate and consistent results, with consistent results meaning that the state error covariance accurately quantifies the state estimation error. During simulated testing, the true states are known (simulated) so the state error covariance can be compared directly to the true estimation error. When working with flight data, the true error is unknown, and tuning is based on the measurement residuals. The residuals are the difference between the measured magnetic and sun vector components and the corresponding reference components after rotation into the body-fixed frame with the estimated attitude. Given the assumption of Gaussian measurement and process noise, the residuals are expected to be zero mean and accurately bounded by the measurement and process covariance. In application to flight data, the covariances are tuned to meet this criteria. Throughout this section, we discuss accuracy in terms of the 1–s (one standard deviation) bounds on the three attitude components, as well as through an approximation of the “total” attitude uncertainty. The 1–s bounds are taken directly from the diagonal elements of the attitude portion of the state error covariance matrix of the filters and, under the small angle approximation, quantify uncertainty in rotation about each axis of the body-fixed frame. Although no single number adequately represents attitude error (the uncertainty is generally not identical in all directions), a single number representing total attitude error is often desired. The major principal axis of the error ellipsoid provides a conservative approximation of this total attitude error [21]. The total attitude error has a chisquare probability distribution, and thus is discussed in terms of confidence bounds rather then 1–s or 3–s bounds that are typically used to quantify a Gaussian distribution. The 68% and 99% confidence bounds used to provide an approximate total attitude uncertainty are derived from the attitude state error covariance; they are given by 1.872 and 3.368 times the square root of the maximum element of the diagonalized covariance matrix (the major axis of the error ellipsoid), respectively [21]. The attitude estimation accuracy from the earliest available RAX-1 and RAX-2 data sets are shown in Figs. 7 and 8. These results are from application of the simultaneous photodiode calibration and attitude estimation extended Kalman filter. The RAX-1 data was recorded 12 days after launch, and the RAX-2 data was recorded seven days after launch. Both data sets are 1 Hz measurements from the tumbling phase of the mission. Since the extended Kalman filter uses bias-corrected rate gyroscope measurements directly to propagate attitude, an inherent assumption is that the angular velocity of the spacecraft is constant between measurements. The 1 Hz gyro sampling frequency is not sufficient to validate this assumption during the tumbling phase of the mission, so the process covariance is increased to rely more heavily on the directional measurements. The angular velocities of the spacecraft during both data sets, shown in Fig. 9, are comparable such that the same process covariance is used for both data sets. From comparison of Figs. 7(a) and 8(a), we see

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that the uncertainty in estimating the RAX-1 attitude is greater than the uncertainty during the RAX-2 data set. The 1–s accuracy of the RAX-1 data varies between 0.51 and 1.51, whereas the 1–s accuracy of the RAX-2 data varies between 0.51 and 1.01 for the majority of the data set. This difference is due to the increased number of photodiodes in the RAX-2 ADS. In both data sets, the dramatic increase in uncertainty for approximately 35 min is when the spacecraft is in eclipse and the photodiodes are not illuminated. The decrease in accuracy in the RAX-2 data around 95 min corresponds to a position where the magnetic and sun vectors are approximately parallel, minimizing the amount of information provided by the sensors. Attitude measurements taken after the satellite has reached steady-state dynamics about the geomagnetic field demonstrate the full potential of the attitude determination system. During this time, as opposed to the tumbling phase of the mission, the 1 Hz sampling frequency more accurately captures the dynamics and direct gyro-based attitude propagation within the filter can be carried out with a smaller process covariance. This process covariance is dictated largely by the noise characteristics of the gyros, rather than additional inflation due to the sampling frequency. Attitude estimation accuracy for a representative data set during this phase of the mission is shown in Figs. 10–12 using three different methods: the full photodiode calibration filter, the conventional MEKF with an Earth albedo model to correct the photodiode measurements, and the conventional MEKF without an Earth albedo model, each of which is discussed in the following paragraphs. This data was collected on December 9, 2011 16:00:00 UT. Fig. 10 shows the attitude determination accuracy when using the full photodiode calibration and attitude estimation filter. The x- and y-components of the 1–s

attitude error remain below 0.51 when in the sun, and below 0.751 when in eclipse. The 1–s accuracy of the z-component remains below 0.751 in the sun and below 2.11 in eclipse. The uncertainty is higher about the z-axis because it is aligned with the geomagnetic field and is also the predominant spin axis of the spacecraft. Since the spacecraft is spinning about the magnetic field, the magnetometer and gyroscope provide little non-redundant information about that axis, increasing the uncertainty. The attitude accuracy of the same data set using the 6state MEKF with an albedo model (the same albedo model used in the photodiode calibration filter [4,28]) to compensate the photodiode measurements is shown in Fig. 11. The photodiode calibration parameters used in the measurement model are not taken from calibration with this data set; the calibration parameters used are those estimated from the first available data set, discussed previously in this section. The calibration parameters remain constant over time,4 and this highlights the utility of batch calibration used in conjunction with real-time attitude estimation: the on-orbit calibration can be applied during ground processing of an initial set of data collected for spacecraft health assessment and calibration, and the parameters can be uploaded for real-time sensor correction and attitude estimation using existing methods. This can simplify the algorithms running onboard the spacecraft when real-time estimation is needed. Compared to the performance of the photodiode calibration filter, the MEKF converges faster, as seen particularly in the z component accuracy of the two filters. In the photodiode

4 On RAX-2, the scale factor remains constant over time. On RAX-1 the scale factor did decrease over time, and this decrease can be compensated for with period calibration. The decrease was attributed to the effects of ultra-violet radiation, and was mitigated on RAX-2 by utilizing solar cell coverglass to shield the photodiodes.

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calibration filter, the uncertainty continues to decrease throughout nearly the entire initial illuminated portion of the data, whereas with the MEKF, the filter converges almost immediately. This is due to the higher number of state variables in the photodiode calibration filter (6 þ 3N, where N ¼17 is the number of photodiodes) and is also dependent on the accuracy of state estimates used to initialize the filter. In addition to the difference in convergence time, the MEKF provides slightly higher accuracy attitude estimates throughout the data set. The 1–s

accuracy of all three components remains below 0.51 for the majority of the time in the sun, and the 99% bound on total attitude accuracy remains below 21 during the time in the sun. The increase in accuracy of the MEKF compared to the photodiode calibration filter also stems from the reduced number of states: the same amount of information (measurements) are used to estimate fewer state variables, resulting in overall slightly higher accuracy. A third method is shown in Fig. 12, which is again the 6-state MEKF but without inclusion of an Earth albedo

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model to compensate the photodiode measurements. The covariance of the photodiode measurements is significantly increased to account for the albedo contribution while utilizing the same magnetometer and process covariance as the MEKF with albedo compensation. The measurement covariance tuning is done by inspection of the residuals as discussed previously. The measurement covariance of the tuned filter corresponds to an irradiance uncertainty of 395 W/m2, 29% of the direct sunlight. Although accounting for the albedo by simply increasing the measurement covariance is not a mathematically correct treatment of the albedo contribution in an EKF (since it does not have a Gaussian distribution), it is a widely used approach that works well in practice. As seen by comparing Figs. 11 and 12, Earth albedo significantly degrades the attitude determination accuracy. Without albedo compensation, the z-component uncertainty reaches 1.51 in the sun, and the x- and y- components reach 0.751. The 68% confidence bound on total attitude determination accuracy is between 21 and 31 in the sun, whereas it remains below 1.11 with albedo compensation. While these accuracies may still satisfy the requirements of many missions, including the RAX mission, this highlights the impact of Earth albedo on the ADS. 6. Discussion and the approach to low-cost attitude determination We have demonstrated 0.51 1–s attitude accuracy using commercial-off-the-shelf (COTS) magnetometers, photodiodes, and a MEMs gyroscope, and without the use of high tolerance integration/alignment procedures or thorough pre-flight calibration. Although the low-cost COTS sensors typically have inherent scaling, bias, and nonorthogonality errors, and in the case of magnetometers, are also affected by the surrounding spacecraft electronics, on-orbit calibration enables accurate attitude determination with these sensors. This approach of using on-orbit calibration in place of ground-based calibration and alignment enables the rapid development and integration of low-cost attitude determination and control subsystems for small satellite missions with modest determination requirements. This contrasts with the traditional spacecraft integration process that includes high-tolerance sensor alignment during bus-level integration [18]. In the remainder of this section, we summarize the approach taken for RAX attitude determination and its impact on the subsystem design. By utilizing the on-orbit magnetometer calibration [27], the magnetometer can be placed anywhere in the spacecraft without careful consideration of nearby electronics. The calibration method requires measurements of the electric current of nearby components that have a significant impact on the magnetometers, and it estimates constant parameters that map the time-varying current to magnetometer bias. In application to RAX, the current sensors that were included for general spacecraft health monitoring were sufficient for magnetometer calibration; no additional sensors were used. This included monitoring the currents in each of the four solar panels as well as the bus power lines. For real-time magnetometer correction,

the ADS needs access to these current measurements, which should be considered in flight software development. Photodiodes can be used to provide measurements of the sun vector component, and on-orbit photodiode calibration is accomplished with the methods of Ref. [28]. The calibration works with any number photodiodes in an arbitrary configuration. Since photodiodes provide a single component of sun vector information, multiple photodiodes are required for a full sun vector estimate. On RAX-2, photodiodes were mounted at various angles over the body frame to enable three-component sun sensing for nearly all attitudes. Since RAX-2 development, a design optimization technique was developed to determine photodiode configurations that minimize the sun vector angular accuracy, and this can be used to provide photodiode configurations for future missions [29]. Although pre-flight calibration was done to test for rate gyroscope scale factor error, various methods exist for on-orbit rate gyro calibration [16,17,23]. Small gyroscope errors can also be accounted for in the process covariance of the extended Kalman filter. Relative attitude and rate sensor alignment needs to also be considered. The magnetometer calibration resolved non-orthogonality errors in the magnetometer-fixed frame. Relative alignment between two identical three-axis sensors can be accomplished with any single-point attitude determination method, such as the q-method or QUEST [8]. Since the RAX gyroscope has a magnetometer in the same package, this was also used to estimate the relative gyro-magnetometer alignment under the assumption that the axes of the packaged gyro and magnetometer are coincident. The photodiode calibration estimates the alignment of the photodiodes relative to the other sensors, which in the algorithm presented in Ref. [28], are assumed to be already aligned. Additional methods also exist for estimating the relative alignments of attitude and angular rate sensors [16,23–25]. The sensor calibration methods, combined with relative alignment estimation methods, enable the assembly of the ADS without regard to high-tolerance integration or alignment procedures. The RAX mission has no real-time attitude determination requirement, and all sensor processing and attitude determination was done offline after downloading sensor measurements from the spacecraft. For real-time attitude determination with these same methods, correction factors can be uploaded to the spacecraft after initial on-orbit checkout and calibration. This can also be advantageous over purely real-time on-orbit calibration as it allows for a thorough inspection of the data and filter tuning. For realtime on-orbit calibration, the photodiode calibration [29] is well-suited for real time use with the exception of the built-in Earth albedo model. The model should be simplified [1] or pre-calculated and tabulated on-board for realtime use. The magnetometer calibration [27] is a bath method in its current form, and extension to real-time implementation is left for future work. It should also be noted that no sensor data is available to independently verify the reported attitude determination accuracy. The estimate of the accuracy accuracy, which is quantified by the state error covariance matrix of the extended Kalman filter, arises from the consistency of the calibrated sensor measurements. Ideally, an independent

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attitude measurement, such as a star tracker measurement, could be used as an additional verification of attitude determination accuracy. This verification is not possible with the RAX data and is left for future work.

7. Conclusion The RAX satellites utilize commercial-off-the-shelf magnetometers, photodiodes, and a MEMs rate gyroscope for attitude determination. Using these sensors in combination with on-orbit calibration methods and an Earth albedo model, attitude determination accuracies of 0.51 1–s in each component have been demonstrated. The on-orbit calibration methods developed to enable this level of attitude determination with the given sensors are general in that they can be applied to any application of the sensor, including attitude determination on other spacecraft. This approach of replacing thorough pre-flight calibration with on-orbit calibration maximizes the performance of the given hardware without requiring hightolerance integration, alignment, or pre-flight calibration procedures. Although these methods were applied the magnetometers and photodiodes in this paper, the approach can also be used with other directional and attitude sensors. For example, on-orbit calibration could be used to verify the focal length of a star camera, or to estimate the alignment between any types of attitude sensors. These methods enable increasingly complex missions to be accomplished by these small, low-cost spacecraft.

Acknowledgments The authors thank Alex Sloboda, who led the development of the RAX attitude determination subsystem, as well as the other students on the RAX ADCS team throughout its development. RAX was funded by the U.S. National Science Foundation, Grant ATM-0838054. Additional funding for this work was provided by the U.S. Department of Defense through a National Science and Engineering Graduate (NDSEG) Fellowship. References [1] Pontus Appel, Attitude estimation from magnetometer and earthalbedo-corrected coarse sun sensor measurements, Acta Astronaut. 56 (January (1–2)) (2005) 115–126, http://dx.doi.org/10.1016/j. actaastro.2004.09.001. [2] H. Bahcivan, J.W. Cutler, M. Bennett, B. Kempke, J.C. Springmann, J. Buonocore, M. Nicolls, R. Doe, First measurements of radar coherent scatter by the radio aurora explorer cubesat, Geophys. Res. Lett. 39 (2012) L14101, http://dx.doi.org/10.1029/2012GL052249. [3] Hasan Bahcivan, James W. Cutler, Radio aurora explorer: mission science and radar system, Radio Sci. 47 (2012) 22, http://dx.doi.org/ 10.1029/2011RS004817. [4] Dan D.V. Bhanderi, Spacecraft attitude determination with Earth Albedo corrected sun sensor measurements (Ph.D. thesis), Aalborg University, Aalborg, Denmark, August 2005. [5] Gian Paolo Candini, Fabrizio Piergentili, Fabio Santoni, Design, manufacturing and test of a self-contained and autonomous nanospacecraft attitude control system, J. Aerosp. Eng., 2012 (in press), doi: http://dx.doi.org/10.1061/(ASCE)AS.1943-5525.0000291. [6] John L. Crassidis, Sigma-point kalman filtering for integrated gps and inertial navigation, in: AIAA Guidance, Navigation, and Control

213

Conference and Exhibit, San Francisco, CA, August 2005, Paper number AIAA 2005-6052. [7] John L. Crassidis, John L. Junkins, Optimal Estimation of Dynamic Systems, Chapman and Hall/CRC, CRC Press, LLC. Boca Raton, FL, United States, 2004 (Chapter 7.2). [8] John L. Crassidis, F. Landis Markley, Yang Cheng, Survey of nonlinear attitude estimation methods, J. Guid., Control, Dyn. 30 (January– February (1)) (2007) 12–28. [9] James W. Cutler, Hasan Bahcivan, The radio aurora explorer - a mission overview. AIAA Journal of Spacecraft and Rockets, 2013. Inprint. doi: http://dx.doi.org/10.2514/1.A32436. [10] James W. Cutler, John C. Springmann, Sara Spangelo, Hasan Bahcivan, Initial flight assessment of the radio aurora explorer, in: Proceedings of the 25th Annual Small Satellite Conference, Logan, Utah, August 2011. [11] Naser El-Sheimy, Haiying Hou, Xiaoji Niu, Analysis and modeling of inertial sensors using allan variance, IEEE Trans. Instrum. Meas. 57 (January (1)) (2008) 140–149, http://dx.doi.org/10.1109/TIM.2007. 908635. [12] R.L. Farrenkopf, Analytic steady-state accuracy solutions for two common spacecraft attitude estimators, J. Guid., Control, Dyn. 1 (4) (1978) 282–284. [13] C.C. Foster, G.H. Elkaim, Extension of a two-step calibration methodology to include nonorthogonal sensor axes, IEEE Trans. Aerosp. Electron. Syst. 44 (July 3) (2008) 1070–1078, http://dx.doi.org/ 10.1109/TAES.2008.4655364. [14] Demoz Gebre-Egziabher, Gabriel H. Elkaim, J. David Powell, Bradford W. Parkinson, Calibration of strapdown magnetometers in the magnetic field domain, ASCE J. Aerosp. Eng. 19 (April 2) (2006) 87–102, http://dx.doi.org/10.1061/(ASCE)0893-1321(2006)19:2(87). [15] David T. Gerhardt, Passive magnetic attitude control for cubesat spacecraft, in: 24th Annual AIAA/USU Conference on Small Satellites, Logan, UT, August 2010. Utah State University, Paper Number SSC10-VIII-5. [16] K.-L. Lai, J.L. Crassidis, R.R. Harman, In-space spacecraft alignment calibration using the unscented filter, in: AIAA Guidance, Navigation, and Control Conference, Austin, TX, August 2003, AIAA Paper Number 2003-5563. [17] K.-L. Lai, J.L. Crassidis, R.R Harman, Real-time attitude-independent gyro calibration from three-axis magnetometer measurements, in: AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Providence, RI, August 2004. Paper Number AIAA-2004-4855. [18] David E. Lee, Space Mission Engineering: The New SMAD, in: J. R. Wertz, D.F. Everett, J.J. Puschell (Eds.), Spacecraft Manufacturing, Integration and Test (Chapter 23.3), Microsm Press, Hawthorne, CA, United States, 2011, pp. 711–718. [19] E.J. Lefferts, F.L. Markley, M.D. Shuster, Kalman filtering for spacecraft attitude estimation, J. Guid., Control, Dyn. 5 (September– October (5)) (1982) 417–429. [20] Gerald M. Lerner, Scalar checking. In: Spacecraft Attitude Determination and Control (Chapter 9.3), Kluwer Academic Publishers, Dordrecht, Holland, 1978, pp. 328–334. [21] Gerald M. Lerner, Covariance analysis. In: Spacecraft Attitude Determination and Control (Chapter 12.3), Kluwer Academic Publishers, Dordrecht, Holland, 1978, pp. 429–434. [22] F. Landis Markley, Attitude error representations for Kalman filtering, J. Guid., Control, Dyn. 26 (March–April (2)) (2003) 311–317. [23] Mark E. Pittelkau, Kalman filtering for spacecraft system alignment calibration, J. Guid., Control, Dyn. 24 (November–December (6)) (2001) 1187–1195. [24] Malcolm D. Shuster, Daniel S. Pitone, Batch estimation of spacecraft sensor alignments. II. Absolute alignment estimation, J. Astronaut. Sci. 39 (October–December (4)) (1991) 547–571. [25] Malcolm D. Shuster, Daniel S. Pitone, Gerald J. Bierman, Batch estimation of spacecraft sensor alignments. I. Relative alignment estimation, J. Astronaut. Sci. 39 (October–December (4)) (1991) 519–546. [26] John C. Springmann, James W. Cutler, Initial attitude analysis of the rax satellite, in: Proceedings of the AIAA/AAS Astrodynamics Specialist Conference, Girdwood, AK, August 2011. [27] John C. Springmann, James W. Cutler, Attitude-independent magnetometer calibration with time-varying bias, J. Guid., Control, Dyn. 35 (June–July (4)) (2012) 1080–1088, http://dx.doi.org/10.2514/1.56726. [28] John C. Springmann, James W. Cutler, On-orbit calibration of photodiodes for attitude determination, AIAA J. Guid., Control, Dyn., 2014 (in preparation). [29] John C. Springmann, James W. Cutler, Optimization of Directional Sensor Orientation with Application to Sun Sensing, Journal of Guidance, Control, and Dynamics (2014), http://dx.doi.org/10.2514/ 1.61468.

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[30] John C. Springmann, Alexander J. Sloboda, Andrew T. Klesh, Matthew W. Bennett, James W. Cutler, The attitude determination system of the RAX satellite, Acta Astronaut. 75 (2012) 120–135, http://dx.doi.org/10.1016/j.actaastro.2012.02.001. [31] Michael Taraba, Christian Rayburn, Albert Tsuda, Charles MacGillivray, Boeing's cubesat testbed 1 attitude determination design and on-orbit experience, in: Proceedings of the AIAA/USU Conference on Small Satellites, Logan, UT, August 2009, Utah State University, Paper Number SSC09-X-6.

[32] Jason Tuthill, Design and simulation of a nano-satellite attitude determination system (Master's thesis), Naval Post-Graduate School, Monterey, CA, December 2009. [33] James R. Wertz (Ed.), Spacecraft Attitude Determination and Control, Kluwer Academic Publishers, Dordrecht, Holland, 1978. [34] Zhiqiang Xing, D. Gebre-Egziabher, Modeling and bounding low cost inertial sensor errors, in: IEEE/ION Position, Location and Navigation Symposium, Monterey, CA, May 2008, pp. 1122–1132.