AERODYNAMIC ATTITUDE STABILIZATION FOR A RAM-FACING CUBESAT

Transcription

1 AAS AERODYNAMIC ATTITUDE STABILIZATION FOR A RAM-FACING CUBESAT Samir Rawashdeh, * David Jones, Daniel Erb, * Anthony Karam, and James E. Lumpp, Jr. This paper describes the design, modeling, and analysis of an attitude control system for a ram-facing pico-class satellite in Low Earth Orbit (LEO). A 3-U (30x10x10 cm 3 ) CubeSat is designed to maintain one 10x10 cm 2 face aligned with the velocity vector throughout the orbit. The solution presented implements deployable drag fins and resembles a shuttlecock design which is shown to be capable of providing passive stabilization for orbits below 500 km. A simplified Direct Simulation Monte Carlo (DSMC) method is used to model the rarefied atmosphere and its interaction with the spacecraft body for a range of fin geometries. An attitude propagator is developed to observe the satellite s dynamic response and steady-state behavior considering perturbing torques due to gravity gradient and solar pressure. Stability characteristics and pointing errors are shown for altitudes ranging from 300 to 450 km with fin lengths from 2 to 30 cm at angles from 0 to 90 degrees. I TRODUCTIO Small spacecraft technology has been shown to reduce cost and development time and to maximize science return. The CubeSat Standard (10x10x10 cm 3 with mass 1 kg) was developed by Stanford University and California Polytechnic University (CalPoly) as a means to standardize pico-satellite buses, structures, and subsystems. 1 The current CubeSat standard allows two or three cubes to be stacked to construct larger 2-U and 3-U CubeSats. CalPoly has also developed a standardized Launch Vehicle Interface (LVI) to accommodate CubeSats known as the Poly-Picosatellite Orbital Deployer (P-POD) which can launch up to 3-U s in several configurations (one 3-U, three 1-U s, etc). This system has opened space exploration to smaller organizations, in particular university student teams, that would not otherwise have the opportunity to build, launch, and operate spacecraft. Figure 1 is an example of a 1-U CubeSat. * Graduate Research Assistant, Electrical and Computer Engineering, University of Kentucky, Lexington, KY Graduate Research Assistant, Mechanical Engineering, University of Kentucky, Lexington, KY Undergraduate Research Assistant, Mechanical Engineering, University of Kentucky, Lexington, KY Associate Professor, Electrical and Computer Engineering, University of Kentucky, Lexington, KY

2 The P-POD and the CubeSat Standard have enjoyed much success since the first CubeSat launch in * The P-POD has flown on four different launch vehicles: the Rockot operated by Eurockot, the Dnepr operated by ISC Kosmotras, the Minotaur from Orbital Sciences, and the Falcon-1 from SpaceX. Six P-PODs have been successfully deployed to date containing twelve CubeSats. There have also been several other international CubeSat launches utilizing other LVI s. CubeSats are designed for high risk, low cost access to space; however, the small size of the CubeSat imposes substantial mass, volume, and power constraints. Therefore novel spacecraft designs can be investigated and are often necessary to meet the constraints of the standard. In particular, attitude control for CubeSats remains a fairly open problem. Experiments have been conducted using actuators such as reaction wheels, magnetic torque coils, and microthrusters. Passive methods such as gravity gradients and aerostabilization are robust, require little to no power, and are an attractive option for several applications. Figure 1. KySat-1 is a 1-U CubeSat designed by Kentucky Space. 2 This paper discusses the design, modeling, and performance of satellite attitude stability obtained using aerodynamic drag fins to maintain a ram-facing steady state in Low Earth Orbits (LEO) below 500 km for one 10x10 cm 2 face of a 3-U CubeSat. The objective is to carry an atmospheric sensor on the front face which requires its aperture to track the velocity vector. The satellite is in a 3-U CubeSat configuration that measures 10x10x30 cm 3 before deployment and weighs less than 5 kg with the center of mass at the geometric center during launch. The satellite is required to recover from the initial tumble after launch then achieve and maintain a ram-facing steady-state attitude. In this configuration, the satellite will perform one revolution about the pitch axis per orbit. The design and simulations presented are based on a 3-U CubeSat with deployable side panels resembling a badminton shuttlecock. Stability is achieved by placing the center of drag pressure behind the center of mass. Figure 2 shows the configuration of the satellite where side panels are deployed to a narrow angle measured from the negative velocity vector (See Figure 4). Aerodynamic pitch-torque profiles were generated by simulation based on the Direct Simulation Monte Carlo (DSMC) method. The panel deployment angle, the length of the deployable side panels, and the orbit altitude were varied and simulated to analyze the effect of these variables on the steady-state behavior of the satellite. The next section discusses related research and results from previous experiments for geometries similar to the one considered here. Next, the design of an aerostabilized CubeSat is discussed, followed by details on an attitude propagator designed to perform orbit simulations. Fi- * CubeSat Program ( NASA Small Satellite Missions ( 2

3 nally, the design space is discussed along with the simulation results for various configurations and altitudes. RELATED WORK Passive attitude stabilization can be achieved using a gravity gradient, magnetically, and aerodynamically in low orbit altitudes. Aerodynamic stability acts in pitch and yaw to maintain a ramfacing attitude while leaving roll uncontrolled. Stability of light-weight satellites using drag fins can be achieved for altitudes below 500 km, while active attitude control methods can be employed to complement the passive designs to achieve better controllability and steady-state stability. 3 Aerostabilization in LEO was flight tested as an experiment on the shuttle Endeavour in ,5 The Passive Aerodynamically Stabilized Magnetically-damped Satellite (PAMS) experiment demonstrated the feasibility of aerostabilization with magnetic hysteresis material for damping. The PAMS satellite was designed as a cylindrical stove pipe having a significantly thicker shell on one end to shift the center of mass of the satellite and produce an aerodynamically stable design for altitudes from 250 to 325 km. The simulations were based on free-molecular aerodynamics and incorporated variations in atmospheric density, global winds, and solar radiation. It also simulated the behavior of hysteresis material cycling in a model of the earth s magnetic field, and showed damping within 1 day, and a worst-case cone angle of 9 degrees. The dimensions of PAMS are similar to those of CubeSats; however the CubeSat Standard does not allow such an offset in the center of mass unless a shift is performed post-deployment. In the design studied here, a shuttlecock design is used as an effective way to shift the center of drag pressure behind the center of mass after orbit insertion while still conforming to the CubeSat standard. Psiaki proposes a shuttlecock design to obtain aerodynamic stability. 6 The system uses four deployable feathers that resemble retractable tape measures extending from a 1-U CubeSat. It also incorporates active magnetic torque coils for damping and was shown through simulation to achieve stability for all altitudes below 500 km. The design was evaluated by studying and comparing a simplified stiffness model with a model based on free-molecular aerodynamics. The narrow one-meter-long feathers were deployed at 12 degrees. The design was shown to stop tumbling within 1 hour, and achieved a steady-state pointing error of 2 degrees within 15 hours. AEROSTABILIZED CUBESAT DESIG Figure 2. Aerodynamically Stable Cube- Sat Design Concept. This section describes the issues and considerations in designing an aerodynamically stable CubeSat. The modeling of the aerodynamics in orbital altitudes, satellite geometric constraints, and the expected conditions of operation are highlighted. Atmosphere modeling. At altitudes near the Kármán line (100 km), the Knudsen number typically begins to exceed 1 indicating that the atmosphere more accurately corresponds to a rarefied, free molecular flow regime than a continuum flow regime. 7 Therefore, continuum Computational 3

4 Fluid Dynamics (CFD) methodologies cannot be used to study satellites in LEO. Instead, an approach based on free molecular aerodynamics or direct simulation of individual atmospheric particles on the satellite is necessary. Simplified aerodynamic stiffness and torque models based on free-molecular aerodynamics have been developed. 6 Another approach is the direct simulation of rarefied gas flows using the DSMC method which is convenient for spacecraft with complex geometries that are difficult to describe analytically. 8 For a 3-U CubeSat, with fins that are 10 cm-wide and relatively short compared to the satellite, the shading effect of the satellite body plays a significant role in the aerodynamics and can easily be taken into account using DSMC. The DSMC method can be simplified by assuming a uniform atmospheric density, negligible thermal motion, and modeling collisions as elastic impacts without reflections. Satellite Geometry. The P-POD launch requirements state that the center of mass of a 3-U CubeSat lie at most 2 cm from the geometric center prior to deployment. In general, the inability to customize mass distribution of CubeSats poses limitations on the ability to control the distance to the center of aerodynamic pressure. The deployable side-panel configuration provides the possibility for a centered mass distribution at launch and an aerodynamically stable geometry once deployed. The angle of deployment and length of panels determine the dynamic behavior. Expected Behavior in Orbit. A ram-facing satellite in a 90-minute circular orbit, performing one rotation per orbit around the pitch axis, would ideally have a steady angular pitch rate of deg/s. However, local oscillations were observed to be on the order of 4 cycles per orbit with body-frame angular rates ranging between 0.01 deg/s and 0.12 deg/s. These small rates pose a rate estimation challenge in attitude determination. Energy dissipation caused by aerodynamics is negligible and raises a need to include a method of angular rate damping by design. 4,5 The amount of dissipation torque caused by particles impacting a moving surface due to the satellite s angular rate is approximately four orders of magnitude smaller than the torque component caused by the translation of the satellite in the atmosphere and can be ignored. 9 Therefore, passive or active angular rate damping methods are required. Passively, magnetic hysteresis material when cycling through a magnetic field follows a hysteresis pattern that introduces heat losses which then dampens angular rotations. 10 Active reaction wheels and torque coils can complement or replace passive methods to improve steady-state behavior under external perturbations. Some of these techniques have been developed and tested. 4,6 ATTITUDE PROPAGATOR Introduction. In order to estimate the satellite s response to initial tumble conditions and observe the steady-state behavior, a simulation that takes into account atmospheric torques, gravity gradient, and solar pressure effects is implemented. In the 1-DOF (degree of freedom) implementation of the satellite rotations in pitch, aerodynamic torque profiles for several geometric combinations of panel length and deployment angle are employed to study the design space over the range of panel lengths and deployment angles. The attitude propagator also includes a gravity gradient torque profile, a simplified model for solar pressure, a random disturbance torque factor, and a damping coefficient to introduce a viscosity factor. Table 1 shows the parameters for the disturbance torques and the damping coefficient used in the propagator. Aerodynamic Torque Profile. To reduce simulation time over many orbits, it is advantageous to calculate the aerodynamic torque profiles over the attitude angles. This allows the attitude 4

5 propagator to simply interpolate on the torque profile to determine the torque at any pitch, rather than calculate the torque during the simulation itself. A table that contains a direct correlation between the satellite s pitch and the resulting torque is generated by running the simulation over a sufficiently high resolution of pitch angles. Simulation Assumptions: - Rotation in pitch is the single degree of freedom. - The orbit is perfectly circular. - Atmospheric density at a given altitude is constant. All results correlate to the mean value of the atmospheric density at that altitude. - Impacts by atmospheric molecules are modeled as elastic without reflection. 9 - Random thermal motion is negligible compared to the satellite s velocity. The collective effect of the atmospheric particles is simulated using a simplified DSMC method by spawning a sufficiently large number of particles in front of the satellite. The sum of the masses of the particles in the simulation is equal to the mass of all the particles which would be present in the volume of atmosphere travelled during the simulation step: m p = ( V * ρ ) / n p (1) where m p is the mass of each simulation particle in kg, V is the volume of space traveled during the discretized time in m 3, ρ is the density of the atmosphere in kg/m 3, and n p is the number of atmospheric particles created to impact the satellite to calculate the effective torque. Particles are distributed uniformly in V since a uniform atmospheric distribution is assumed. The rigid body mechanics equation for the torque applied by a single particle in an elastic impact with thermal reflection is: Τ p = m p ( U p ) (u v s p ) (2) t It was verified that the torque caused by a particle impact using the above equation is equivalent to the generally accepted fluid mechanics equation for atmospheric torque on a satellite surface discretized and integrated over the area that is not shadowed and is facing the particle stream: where: Τ aero = ½ρU 2 C d A(u v s cp ) (3) - Τ p is the torque applied by the single particle in N.m - U p is the change in velocity of the particle in m/s - m p is the mass of the particle in kg - t is the length of the discretized period of time in s - Τ aero is the aerodynamic torque in N.m - ρ is the density of the atmosphere in kg/m 3 - U is the velocity of the satellite in m/s - A is the surface area in m 2 - u v is a unit vector corresponding to the velocity vector - s cp is a vector from the center of mass to the center of pressure - s p is a vector from the point of impact to the center of mass The collective torque affecting the satellite at a given attitude is obtained by summing the individual torques due to all discrete particles that impact the satellite. Reactions after impacts are 5

6 neglected as the particles are assumed to reflect thermally at a velocity much smaller than the velocity of impact. Figure 3 overviews the satellite/particle interaction. Figure 3. Particle Impact. Gravity Gradient Torque Profile. The gravity gradient torque for an earth orbiting satellite is caused by differences in the distance to earth across the satellite body; mass that is closer to Earth experiences higher gravitational attraction. For a given satellite geometry the torque profile due to the gravity gradient is a function of attitude. For the 1-DOF implementation, in which the mass can be estimated to fall on the centerline of the satellite, the relationship resolves to a sinusoid. Namely, at zero degrees to the velocity vector, which is earth tangential for a circular orbit, the satellite feels negligible gravity gradient torque, at 45 degrees the gravity gradient is at a maximum, and at 90 degrees it is zero again. If the distribution of mass in the satellite is modeled as a row of mass points on the centerline of the satellite, the total torque on the satellite about the pitch axis as a result of gravity gradient can be calculated by summing the torques applied by each point mass. Τ p,gg = Σ(F mp s mp ) (4) where Τ p, gg is the torque about the satellite's pitch axis as a result of the gravity gradient in N.m, F mp is the gravitational force of attraction between the point mass and the Earth in N, and s mp is the distance vector from the satellite center of mass to the point mass in m. The calculation is performed over the range of pitch angles to produce the gravity gradient torque profile, which correlates the angle to the velocity vector with the torque that the satellite experiences. Damping Factor. A damping factor is necessary to achieve steady state because, as discussed earlier, the aerodynamic dissipation torque is insufficient. A damping factor of N.m per angular rate of 1 radian/second was added to the attitude propagator. The torques required to achieve our set damping factor are on the order of 3.5 x 10-9 N.m. Damping can be realized in the satellite using passive magnetic hysteresis material countering angular rotation. modeling and simulations are required to predict its behavior. 4,10 Solar pressure. A solar model was implemented to account for torques due to solar pressure. The torque due to solar pressure can be calculated by: 3 T solar = c F s A s (1+q) (c ps c g ) cos(i) (5) 6

7 Where F s is the solar constant, c is the speed of light, A s is the surface area, c ps is the location of the center of solar pressure, c g is the center of gravity, q is the reflectance coefficient, and i is the angle of incidence. The torque calculated for a design with full 30cm panels perpendicular to the incident solar pressure was found to be 4.5 x 10-8 N.m. The maximum value for torque corresponds to when the maximum area faces the Sun. In the attitude propagator, the maximum value is modulated by a sinusoid with a period of an orbit length to simulate the satellite in orbit. The solar torque is set to zero when the satellite is in eclipse. Random oise. A zero-mean random torque value is also placed in the model with maximum torques lower than the other major effects. It is used to simulate torques due to other factors such as transient magnetic torques induced by onboard circuitry, Earth albedo, and changes in atmospheric density (e.g., when coming out of eclipse). It introduces randomness into the system and reveals the system response to transients. The random torque is updated four times per orbit, with a standard deviation of 10-9 N.m, in order to simulate orbit harmonics of the system. Eclipse and atmosphere changes due to the Sun occur once per orbit, while the magnetic field cycles twice per orbit. The noise torque was designed to include single and double torque cycles to model their effect on the attitude. Table 1. Simulation Parameters. Peak Solar Torque Mean 4 x 10-8 N.m Peak Solar Torque Variance 5 x N 2.m 2 Peak Gravity Gradient Torque 1 x 10-7 N.m Zero-mean Noise Torque Variance 1 x N 2.m 2 Angular Rate Damping Factor -1 x 10-6 (N.m)/(rad/s) DESIG SPACE Figure 4 illustrates the geometric variables and the attitude to the velocity vector defined by φ. Figure 4. Geometry of satellite design. 7

8 The main body dimensions α and ß are constant across the simulations at 10 cm and 30 cm respectively. The deployable panel length (λ) and deployment angle (θ) are the parameters varied to optimize performance. An exhaustive search through the panel deployment angle, panel length, and orbit altitude was performed to determine the optimal deployment angle and panel length. Pitch Torque Profiles. Figure 5 shows a set of torque profiles for three designs with a panel length λ = 20 cm at different deployment angles θ = 10, 30 and 50 at an altitude of 400 km as a function of its attitude to the velocity vector (φ). Negative sloped zero-crossings indicate stable points at which the satellite will settle temporarily or permanently; a positive error angle produces negative torque to realign the satellite to the stable point, and vice-versa. At shallow deployment angles the shadowing of the drag panels by the satellite main body affects the linearity of the torque profile through the 0 degree pitch angle. At deployment angles greater than 75 degrees, secondary stable points begin to appear near ±90 degrees pitch, where the projected drag area of the fins perpendicular to the flow begins to diminish and the torque affecting them balances out with the atmospheric drag on the satellite main body. In general the panel length was found to mainly scale the torque profile in amplitude for panel lengths greater than 10 cm. Likewise, evaluating the torque profiles at lower altitudes with higher atmospheric density increases the torque experienced and is manifested as a scaling in the torque profile. Figure 5. Pitch Torque Profiles showing torque experienced as a function of the angle to velocity vector (φ) 8

9 Stiffness. The main performance parameter considered was the amount of stiffness through the ram-facing angle. Stiffness is defined as the amount of correcting torque the satellite experiences for every 1 degree of error, which is calculated as the negative of the derivative of the pitch torque relative to the pitch attitude angle evaluated at the zero degrees pitch angle (φ = 0). Simulations showed that satellites with greater stiffness resulted in smaller steady state errors and higher oscillation frequencies. Varying the deployment angle yields an optimal value at which stiffness is greatest for a given panel length. Figure 6 illustrates stiffness curves over variable deployment angles for several panel lengths. The most efficient deployment angle is around θ = 50 degrees. The drag area by the satellite with deployed panels has a direct effect on orbit life, orbit dwell times were calculated to be below 1 year for a wide range of design combinations at altitudes of 400km and below. Therefore, the optimal design for a specific mission is a trade study between the stiffness (pointing accuracy) and orbit life Figure 6. Aerodynamic stiffness at 400 km altitude for a range of panel lengths (λ). Effect on Steady-state Error. Figure 7 shows equal-stiffness curves over the geometric design variables the panel length (λ) and the deployment angle (θ). Each curve represents length and angle combinations that have equal stiffness and provide the same steady state performance. The orbit propagator was run on a range of ideal constant aerodynamic stiffness values to correlate them to the resulting steady-state error values. This ideal approximation is valid when the slope of the torque profile spans linearly beyond the range of expected worst case steady-state error. The ideal stiffness values in Figure 7 translated to steady state errors of ± 2.5 to ± 31. It is not recommended to use values of the deployment angle θ < 20 where the linearity of the stiffness slope does not span beyond φ = ± 7 from the main stable point. 9

10 Altitude. The atmospheric density varies exponentially with altitude. Figure 8 gives insight into the effect of orbit altitude on the achievable steady state. The plot shows the steady-state error as a function of the panel length for panels deployed at θ = 50 over several altitudes. These plots were obtained using the actual torque profiles with non-ideal stiffness. Damping. It was found that increasing the damping factor improves the performance of a certain configuration and reduces the pointing error while increasing tracking lag. Stability at 500 km was achievable with an error of ± 35 by assuming a damping factor to N.m.sec/rad Figure 7. Constant stiffness curves at 400km altitude. Panel length (λ) and deployment angle (θ) combinations to obtain equal steady-state performance. Figure 8. Effect of varying the Panel Length (λ) at different altitudes for panels deployed at θ = 50 deg, computed using actual calculated torque profiles. 10

11 RESULTS This section highlights detailed results for two sample runs, including an Orbit Life estimate that correlates to the ram-facing area. The selected configurations were simulated at 400km, and were chosen to highlight the tradeoff between the degree of stability and orbit life. Table 2 contains results from two simulations along with a description of the satellite simulated. Figure 9 shows the time responses of the two runs in Table 2. Table 2. Sample Runs at 400km and Simulation Results. Type Description Sample Run A Sample Run B Unit Initial Conditions Pitch (φ) Tumbling rate 5 5 /s Satellite Description Moment of Inertia kg.m 2 Drag Panel Length (λ) Cm Panel Deployment Angle (θ) Ram-Facing Area m 2 Results Settling Time Orbits Steady State Error +/- 5 +/- 2.5 Pitch Angular Rates (bodyframe) Orbit Life (No panels: 3 years) /s Months Orbit life was calculated using the Orbit Dwell Time tool of the NASA Debris Assessment Software (DAS 2.0) for a polar orbit. * The launch year was chosen to be 2018 and the calculation was based on a satellite mass of 3.5 kg and a drag cross-sectional area of a perfectly tracking 3-U CubeSat configured as described in the sample runs. The results are dependent on the year of launch due to solar cycles. The results obtained are based on a series of approximations, the most significant of which is the 1-DOF implementation of the satellite rotations in the orbit plane and the effects of environmental forces on only the pitch rotations of the satellite. In practice, inducing a roll may be desirable to avoid strong temperature gradients. The satellite in our calculations is assumed to be symmetric around the roll-axis with no stored momentum. The attitude propagator was implemented in Simulink. The tool includes several differential equation solvers. Issues were encountered where the results varied across the different solvers; these were overcome by reducing the simulation time step and tolerances to increase the accuracy of the simulations while sacrificing simulation run time. Results that were consistent across solvers were sought. * NASA Orbital Debris Program Office ( 11

12 Figure 9. Time response of attitude (φ) of chosen scenarios in Table 1 at 400km. The plot duration of 150 orbits corresponds to approximately 10 days. Damping of the angular rotations was assumed to be proportional to the angular rate. There are several challenges to achieving this in practice in a CubeSat, both in accurately measuring small angular rates, and in producing the appropriate response. Magnetic torquers and hysteresis dampers are limited to producing torques normal to the ambient magnetic field. However, the use of hysteresis material has been shown to be effective and is often used in passive magnetic stabilization designs. 4 High-fidelity models of the effects hysteresis dampers will improve the analysis. 10 System resonances occurred where a certain stiffness value produced a steady state oscillation frequency that was a harmonic of the solar torques that affected the satellite resulting in notably larger steady-state error. Because the simulation was a 1-DOF implementation, these resonances were believed to be an artifact of the model because satellite roll could reduce these effects. However, parametric and classical resonances may occur and should be considered in a more comprehensive simulation. 5 CO CLUSIO A direct simulation of the rarefied atmosphere was implemented using the DSMC method to compute atmospheric torque profiles across pitch angles. The components of an attitude propagator that was used to simulate the time response of different designs were also described. For a 3- U CubeSat design with deployable side panels, it was found that the most efficient deployment angle lays around 50. Stability was achieved for altitudes below 450 km and worst-case pointing errors of 2.5 degrees at 400 km. Orbit life was shown to be a concern and is recommended to be taken as a limiting factor in the design process. The next step in modeling an aerodynamically stabilized CubeSat is the implementation of a 3-DOF simulation that incorporates variations in atmospheric density beyond the random torque 12

13 used here. It is also desirable to more precisely model the hysteresis material for angular rate damping. Finally, higher fidelity solar, atmospheric, and magnetic field models will further increase the accuracy of the results. ACK OWLEDGME TS This work was supported by the Kentucky Science and Technology Corporation (KSTC), the Kentucky Space Grant Consortium (KSGC), and the University of Kentucky. REFERE CES 1 H. Heidt, J. Puig-Suari, A.S. Moore, S. Nakasuka, R.J. Heidt, CubeSat: A new Generation of Picosatellite for Education and Industry Low-Cost Space Experimentation 14th Annual/USU Conference on Small Satellites. August G. D. Chandler, D. T. McClure, S. F. Hishmeh, J. E. Lumpp, Jr., J. B. Carter, B. K. Malphrus, D. M. Erb, W. C. Hutchison, III, G. R. Strickler, J. W. Cutler, R. J. Twiggs, Development of an Off-the-Shelf Bus for Small Satellites, IEEEAC paper #1365, IEEE Aerospace Conference, Big Sky, Montana. March J. R. Wertz, W. J. Larson, Space Mission Analysis and Design. 3rd ed., R. R. Kumar, D. D. Mazanek, M. L. Heck, "Simulation and Shuttle Hitchhiker Validation of Passive Satellite Aerostabilization" Journal of Spacecraft and Rockets. Vol. 32, No. 5, 1995, pp R. R. Kumar, D. D. Mazanek, M. L. Heck, "Parametric and Classical Resonance in Passive Satellite Aerostabilization" Journal of Spacecraft and Rockets. Vol. 33, No. 2, 1996, pp M. L. Psiaki, "Nanosatellite Attitude Stabilization Using Passive Aerodynamics and Active Magnetic Torquing" Journal of Guidance, Control, and Dynamicss. Vol. 27, No. 3, 2004, pp F. J. Regan, S. M. Anandakrishnan, Dynamics of Atmospheric Re-entry. 1993, pp G. A. Bird, Molecular Gas Dynamics. Oxford Univ. Press, J. R. Wertz, Spacecraft Attitude Determination and Control. Vol. 73, J. Levesque, Passive Magnetic Attitude Stabilization using Hysteresis Materials 17th AIAA/USU Conference on Small Satellites, August S. K. Shrivastava, V. J. Modi "Satellite Attitude Dynamics and Control in the Presence of Environmental Torques A Brief Survey" Journal of Guidance, Control, and Dynamics. Vol. 6, No. 6, 1983, pp R. Ravindran, P. C. Hughes "Optimal Aerodynamic Attitude Stabilization of Near-Earth Satellites" Journal of Spacecraft and Rockets. Vol. 9, No. 7, 1972, pp T. Bak, R. Wisniewski "Passive Aerodynamic Stabilization of a Low Earth Orbit Satellite" European Space Agency - Publications- ESA SP. Vol. 381, 1997, pp