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Passive Magnetic Attitude Control for Cubesat Spacecraft

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Passive Magnetic Attitude Control for Cubesat Spacecraft SSC10-XXXX-X Passive Magnetic Attitude Control for CubeSat Spacecraft David T. Gerhardt University of Colorado, Boulder, CO 80309 Scott E. Palo Advisor, University of Colorado, Boulder, CO 80309 CubeSats are a growing and increasingly valuable asset specifically to the space sciences community. However, due to their small size CubeSats provide limited mass (< 4 kg) and power (typically < 6W insolated) which must be judiciously allocated between bus and instrumentation. There are a class of science missions that have pointing requirements of 10-20 degrees. Passive Magnetic Attitude Control (PMAC) is a wise choice for such a mission class, as it can be used to align a CubeSat within ±10◦ of the earth's magnetic field at a cost of zero power and < 50g mass. One example is the Colorado Student Space Weather Experiment (CSSWE), a 3U CubeSat for space weather investigation. The design of a PMAC system is presented for a general 3U CubeSat with CSSWE as an example. Design aspects considered include: external torques acting on the craft, magnetic parametric resonance for polar orbits, and the effect of hysteresis rod dimensions on dampening supplied by the rod. Next, the development of a PMAC simulation is discussed, including the equations of motion, a model of the earth's magnetic field, and hysteresis rod response. Key steps of the simulation are outlined in sufficient detail to recreate the simulation. Finally, the simulation is used to verify the PMAC system design, finding that CSSWE settles to within 10◦ of magnetic field lines after 6.5 days. Introduction OLUTIONS for satellite attitude control must Sbe weighed by trading system resource allocation against performance. CubeSats are a unique form factor varying from one (1U) to three (3U) stacked cubes of dimensions 10 × 10 × 10 cm.1 For these small satellites, Passive Magnetic Attitude Control (PMAC) is a robust and simple attitude solution, using no electrical or soft- ware components. PMAC is composed of a bar magnet Figure 1. CSSWE CubeSat with magnetic com- to supply restoring torque and hysteresis rods to supply ponents: longitudinal bar magnet (blue arrow), dampening torque. As a passive system, PMAC draws transverse hysteresis rods (red arrows), and mag- no system power and, for microsatellites and smaller, netometer (green arrow). uses less than 50g of mass. The low resource draw and simplicity of PMAC reduces the risk of an orbit sys- tem failure which can be mission ending for single-string Michigan / SRI International 3U CubeSat RAX (ex- CubeSats. However, PMAC is typically limited to atti- pected 2010).4 However, none of these missions doc- ◦ tude pointing of ±10 about the earth's local magnetic ument the clear design and simulation of a CubeSat field. Also, the simplicity of implementation is con- PMAC system. trasted by the difficulty of design and verification, which Figure 1 shows the University of Colorado's 3U Cube- this paper describes for a general 3U CubeSat. Sat, the Colorado Student Space Weather Experiment Early satellites using PMAC include: Transit-1B (CSSWE). In-depth information on CSSWE can be (1960), Transit-2A (1960), Injun-3 (1962), ESRO-1A found in the paper and presentation by Dr. Scott (1968), ESRO-1B (1969), Azur (1969), Exos (1978), and Palo.5 The CSSWE mission involves sensing high-energy Magion (1978).2 Recent uses are Swedish nanosatellite charged particles which spiral around magnetic field lines Munin (2000),2 NASA Ames / Santa Clara Univer- as they reach Low Earth Orbit (LEO). PMAC aligns sity 3U CubeSat GeneSat-1 (2006),3 and University of CSSWE to local magnetic field lines, maximizing par- 1 24th Annual AIAA/USU D. T. Gerhardt Conference on Small Satellites SSC10-XXXX-X ticle flux to the science instrument: the Relativistic Table 1. 3U CubeSat environmental torques at Electron Particle Telescope integrated little experiment 600km altitude. (REPTile). A detailed description of REPTile can be Torque Value found in the paper and presentation by Quintin Schiller [N·m] and Abhishek Mahendrakumar.6 In the case of CSSWE, the high-energy particles are roughly contained within Aerodynamic 8E − 8 earth's outer radiation belt, which intersects LEO at Gravity Gradient 6E − 8 high latitudes (>55◦). Also, particles of interest begin Radiometric 1E − 8 to be absorbed by the atmosphere at altitudes less than RMS Sum 1E − 7 450km. For these reasons, CSSWE requires a 600km or- bit at 55◦ inclination; these values are also used for all calculations and simulations within this paper. The at- magnet dipole moment. titude control system of CSSWE has two performance requirements: PMAC System Design A successful attitude system design begins with an 1. The attitude control system shall have a settling analysis of environmental torques experienced by the time of less than 7 days. spacecraft. Table 1 shows the torques experienced by a 2. Once settled, the CubeSat shall stay within ±15◦ of 3U CubeSat at 600 km. The torque supplied by a mag- the local magnetic field lines. netic dipole moment in a magnetic field is quite simple: The ability to meet these requirements is entirely de- T~ = ~m × B:~ (1) pendent on the PMAC system design. Figure 1 also highlights the magnetic components of CSSWE. The where B~ is the magnetic flux density vector and ~m is bar magnet is shown in the middle-right of the CubeSat the magnetic moment vector for the bar magnet, though (blue arrow), with the hysteresis rods to the far left (red this equation is valid for the torque from a hysteresis rod arrows) and the magnetometer to the far right (green dipole moment as well. At 600 km, jB~ j varies from 0:20− arrow). This layout was chosen to minimize the noise 0:45 Gauss. The author presents a modified version of input at the magnetometer due to the polarity-switching Santoni and Zelli's8 minimum recommended bar magnet hysteresis rods while ensuring the bar magnet does not strength: saturate the hysteresis rods or the magnetometer. TRMS mmin = 10 (2) There are two sources of error present when using a Bmin · sin(βmax) PMAC system. The first is steady state error, in which where TRMS is the root means squared sum of indepen- the hysteresis rod magnetic dipole moment causes the to- dent environmental torques, Bmin is the minimum field tal spacecraft dipole moment to be misaligned with the strength at 600 km (2:3E − 5 Tesla), and βmax is the minor inertia axis of the spacecraft, decreasing pointing desired pointing accuracy (10◦). Although the required accuracy. Because the hysteresis rod polarity switches, alignment with the magnetic field is 15◦, we design the this effect cannot be canceled using a particular hystere- system to 10◦ to ensure there is adequate margin in the sis rod orientation. Steady state error becomes notice- PMAC system design. Santoni and Zelli's8 version of able after settling, when the torque supplied by the bar Equation 2 sets the bar magnet at 30 times the sum of magnet is on par with the hysteresis rod torque (when environmental torques. The author modifies this to 10 the angle between the CubeSat minor inertia axis and times the sum of the torques, decreasing the strength of the local magnetic field is small). The second, oscilla- the bar magnet and decreasing the required hysteresis tory error, occurs because the magnetic field changes as material within the volume-limited CubeSat. the spacecraft travels, causing a delay before alignment with the current field.7 The oscillatory error is directly Bar Magnet Design related to the PMAC settling time, which may be of crit- The bar magnet design is concerned with finding a ical importance for a short mission duration. Usually, as suitable bar magnet dipole moment. The environmental hysteresis material is increased, the oscillatory error and torque analysis shown above gives CSSWE the minimum 2 settling time decrease at the cost of increased steady- dipole moment mmin=0.29 A·m . state error, while decreasing the hysteresis material has A second constraint on the chosen bar magnet the opposite effect. A good PMAC system design will strength is magnetic parametric resonance, which can find the balance between these two errors when deter- occur for polar orbits. For a given inertia matrix, there mining the hysteresis material needed for a given bar exist values of bar magnet dipole moment that create 2 24th Annual AIAA/USU D. T. Gerhardt Conference on Small Satellites SSC10-XXXX-X Table 2. CSSWE bar magnet dipole moments which create parametric resonance in polar or- bits. k η mRES [A·m2] 10 263.60 0.229 11 318.83 0.277 12 379.32 0.330 13 445.07 0.387 14 516.08 0.449 15 592.35 0.515 a resonance which increases the amplitude of oscillation about the magnetic field line. This resonance counters Figure 2. Hysteresis loop diagram showing co- the desired dampening effect and should be avoided. Al- ercive force Hc, remanence Br, and saturation though CSSWE is not expecting a polar orbit, this orbit induction Bs. does meet the science requirements and must be consid- ered. The magnetic moments which resonate are given strength, and is generally characterized by three mag- by the following system of equations:2, 9 netic hysteresis parameters: the coercive force Hc, the I remanence Br, and the saturation induction Bs, shown η =∼ 2:63k2 + 0:49 + 0:51 xx (3) in Figure 2. These parameters are important in describ- I yy ing the shape of the hysteresis loop, the area of which 2 Iyy · n0 · η determines the rotational dampening per cycle per unit mRES = (4) Beq volume. However, the magnetic parameters vary with rod length to diameter ratio (L=D), material, and exter- where k is an integer, Ixx is the minor axis moment of nal field strength.
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