BACKGROUND

The current push towards the commercialization of space has expanded the utilization of satellites deployed in low Earth orbit (LEO). As more satellites are deployed in this space, accurate orbital forecasting is becoming increasingly important. In particular, the position and orbit of each satellite should be tracked and/or predicted with greater accuracy in order to ensure safe usage of this space, especially as commercial constellations of hundreds or thousands of small satellites in LEO become more common. The accuracy of orbital forecasting, however, is dependent, in part, on accurate predictions of LEO perturbations. Typically, one of the largest contributors to orbital propagation error in low orbits is atmospheric drag, which can vary significantly with altitude, latitude, and solar activity.

SUMMARY

The Inventors have recognized and appreciated accurate and real-time orbital forecasting can enable safer deployment of satellites, especially in LEO, where commercial constellations of hundreds or thousands of satellites are becoming more common. However, the Inventors have also recognized that conventional approaches are typically unable to simultaneously provide accurate and real-time estimates of the atmospheric density. Furthermore, conventional approaches often rely upon specialized high-precision instrumentation or ground-based systems, which increase system complexity, the cost, and the payload size.

The present disclosure is thus directed to various inventive apparatuses and methods for estimating the atmospheric density. In one example, a satellite (also referred to herein as the “Satellite Producing Aerodynamic Torque to Understand LEO Atmosphere (SPATULA)”) includes a bus and a drag flap coupled to the bus where the drag flap is configured to deploy during an orbit to subject the satellite to a drag torque about a first axis passing through the bus of the satellite. The drag flap may include a moment arm and a drag panel. The drag flap may have a proximal end coupled to the bus and a distal end coupled to the drag panel. The drag flap is configured to deploy during orbit to subject the satellite to a drag torque about a first axis of the satellite. The bus may include a reaction wheel to measure a reaction wheel angular velocity output, a star tracker to measure a star tracker output, a gyroscope to measure a gyroscope output, and a global position system (GPS) receiver to measure a position and a velocity of the satellite. The satellite may also include a processor configured to execute a method of estimating the atmospheric density described below.

In some implementations, the satellite is not coupled to a tracking system or a processor on the ground. In some implementations, the satellite does not include a magnetic torquer or a thruster to desaturate the reaction wheel.

An exemplary method of estimating the atmospheric density using a satellite includes the following steps: (A) operating the satellite during the orbit to subject the satellite to a significant external torque, (B) determining the external torque to which the satellite is subjected during the orbit, and (C) estimating the atmospheric density proximate to the satellite based at least in part on the external torque determined in (B).

In some implementations, the satellite may include a bus and a drag flap coupled to the bus. In some implementations, step (A) includes (A1) deploying the drag flap during the orbit to subject the satellite to a drag torque about a first axis passing through the bus of the satellite, the significant external torque comprising the drag torque. The drag flap may include a moment arm having a proximal end coupled to the bus and a distal end and a drag panel coupled to the distal end of the moment arm, where in (A1), the moment arm has a length such that a significant angular acceleration of the satellite is generated about the first axis per unit drag force applied to the flap and the moment arm has a natural frequency greater than or equal to about 1 min−1. The drag flap may have a proximal end coupled to the bus and a distal end where the proximal end and the distal end of the drag flap define a longitudinal axis of the drag flap and (A1) includes deploying the drag flap such that the longitudinal axis of the drag flap is substantially parallel to a gravitational gradient.

The satellite may also include a reaction wheel such that (A) includes after (A1), (A2) applying a first torque to the satellite about the first axis, via the reaction wheel, to counteract the external torque to which the satellite is subjected so as to substantially mitigate rotation of the satellite about the first axis. In some implementations, the step (A) may further include the following steps: (A1) deploying the drag flap such that the longitudinal axis of the drag flap is in a plane perpendicular to the first axis and is positioned at a first angle of rotation about the first axis, during (A2), a first angular momentum of the reaction wheel about the first axis increases until reaching a first saturation point, (A3) upon the first angular momentum of the reaction wheel about the first axis reaching the saturation point, allowing the satellite to rotate about the first axis to reposition the drag flap in the plane perpendicular to the first axis at a second angle of rotation about the first axis so as to desaturate the first angular momentum of the reaction wheel about the first axis, (A4) applying a second torque to the satellite about the first axis, via the reaction wheel, to counteract the external torque to which the satellite is subjected so as to substantially mitigate rotation of the satellite about the first axis, wherein during (A4), a second angular momentum of the reaction wheel about the first axis increases until reaching a second saturation point, (A5) upon the second angular momentum of the reaction wheel about the first axis reaching the second saturation point, allowing the satellite to rotate about the first axis to reposition the drag flap in the plane perpendicular to the first axis at the first angle of rotation about the first axis so as to desaturate the second angular momentum of the reaction wheel about the first axis, and (A6) repeating step (A2).

In some implementations, the satellite may include a star tracker, a gyroscope, and a reaction wheel and step (B) includes the following: (B1) estimating a moment of inertia of the satellite and a moment of inertia of the reaction wheel, (B2) applying a Kalman filter to the moment of inertia of the satellite from (B1), the moment of inertia of the reaction wheel from (B1), and a measurement vector to determine an attitude state of the satellite where the measurement vector comprises a star tracker output, a gyroscope output, and a reaction wheel angular velocity output and the attitude state of the satellite comprises an angular position of the satellite, an angular velocity of the satellite, a reaction wheel angular velocity, and a reaction wheel torque of the reaction wheel, (B3) determining an angular acceleration of the satellite based on the angular velocity of the satellite determined in (B2), (B4) determining a net torque on the satellite as the product of the angular acceleration determined in (B2) and the moment of inertia of the satellite determined in (B1), and (B5) determining the external torque to which the satellite is subjected during the orbit by subtracting the reaction wheel torque determined in (B2) from the net torque determined in (B4).

In some implementations, the method described may further include between (B1) and (B2), the following: applying a first low-pass filter transfer function to the star tracker output to obtain a filtered star tracker output, applying a second low-pass filter transfer function to the gyroscope output to obtain a filtered gyroscope output, determining a time derivative of the filtered gyroscope output, subtracting the filtered star tracker output from the time derivative of the filtered gyroscope output to obtain a bias estimate, subtracting the bias estimate from the gyroscope output to obtain a corrected gyroscope output, and using the corrected gyroscope output in the measurement vector in place of the gyroscope output.

In some implementations, the step of estimating the moment of inertia of the satellite and the moment of inertia of the reaction wheel in (B1) includes: (B1 a) applying a first torque to the satellite, (B1b) measuring the gyroscope output as a function of time during (B1a) to determine a first angular acceleration of the satellite, (B1c) measuring the reaction wheel angular velocity output as a function of time during (B1a) to determine a first angular acceleration of the reaction wheel, (B1d) applying a second torque to the satellite, the second torque being substantially similar in magnitude with the first torque and in a direction substantially opposite to the first torque, (B1e) measuring the gyroscope output as a function of time during (B1d) to determine a second angular acceleration of the satellite, (B1f) measuring the reaction wheel angular velocity output as a function of time during (B1d) to determine a second angular acceleration of the reaction wheel, (B1g) dividing an average of the respective magnitudes of the first torque and the second torque applied in (B1a) and (B1d), respectively, by an average of the first angular acceleration determined of the satellite in (B1b) and the second angular acceleration of the satellite determined in (B1e) to estimate the moment of inertia of the satellite, (B1h) dividing an average of the respective magnitudes of the first torque and the second torque applied in (B1a) and (B1d), respectively, by an average of the first angular acceleration determined of the reaction wheel in (B1c) and the second angular acceleration of the reaction wheel determined in (B1f) to estimate the moment of inertia of the satellite.

In some implementations, the step (C) includes the following: (C1) measuring a position and a velocity of the satellite, (C2) determining a solar radiation pressure torque and a gravity gradient torque from the position and the velocity measured in (C1) and the attitude state in (B2), and (C3) determining a drag torque applied to the satellite by subtracting the solar radiation pressure torque and the gravity gradient torque determined in (C2) from the external torque measured in (B).

In some implementations the satellite may be a cubesat. In some implementations, the estimated atmospheric density has a root mean square (RMS) error ranging between about 1×10−13kg/m3to about 2×10−12kg/m3. In some implementations, the estimated atmospheric density has a bandwidth ranging between about 0.03 min−1to about 1 min−1.

It should be appreciated that all combinations of the foregoing concepts and additional concepts discussed in greater detail below (provided such concepts are not mutually inconsistent) are contemplated as being part of the inventive subject matter disclosed herein. In particular, all combinations of claimed subject matter appearing at the end of this disclosure are contemplated as being part of the inventive subject matter disclosed herein. It should also be appreciated that terminology explicitly employed herein that also may appear in any disclosure incorporated by reference should be accorded a meaning most consistent with the particular concepts disclosed herein.

