Aas 19-725 Improved Atmospheric Estimation for Aerocapture Guidance

AAS 19-725

IMPROVED ATMOSPHERIC ESTIMATION FOR AEROCAPTURE GUIDANCE

Evan Roelke,∗ Phil D. Hattis,† and R.D. Braun‡

Increased interest in Lunar or Mars-sample return missions encourages considera- tion of innovative orbital operations such as aerocapture, which generally provides significant mass-savings for orbital insertion at Earth or Mars. Drag modulation architectures offer a straightforward approach to orbital apoapsis targeting by enabling ballistic entry, among other benefits. A shortcoming of these architectures is the poor estimation of atmospheric density resulting in target apoapsis altitude errors. This research seeks to assess and improve upon current atmospheric density estimation techniques in order to support the flight viability of discrete event drag modulated aerocapture. Three different estimation techniques are assessed in terms of estimation error and apoapsis altitude error: a static density factor, a density array interpolator, and an ensemble correlation filter. The density interpolator achieves a 5% improvement in median apoapsis altitude over the density factor when entering at −5.9◦ and targeting a 2000km apoapsis altitude, while the ensemble correlation filter achieves a 7% improvement under identical simulation conditions. The ensemble correlation filter was found to improve with decreasing density search tolerance, achieving a 4.6% improvement in median apoapsis altitude for a tolerance of 1% over 5%. These improvements are dependent on entry and vehicle parameters and improve as the entry angle becomes more shallow or the target apoapsis is reduced. Errors in the density factor measurements are main contributors to the error in estimated versus true density profiles.

INTRODUCTION The revitalized interest in Lunar missions as well as the drive for Mars sample-return missions in recent years encourages innovative solutions to orbital operations. One such innovation is aerocapture, which falls under the realm of aeroassist maneuvers. Aerocapture is a re-entry maneuver in which a spacecraft flies through the upper atmosphere of a planetary body from an inbound hyperbolic trajectory in order to capture into a closed orbit about the central body by means of atmospheric deceleration.1,2 Although it has yet to be flight-proven, aerocapture has been shown to provide significant mass savings for orbital insertion. A cost-benefit analysis of aerocapture mission sets shows that this maneuver can provide mass savings of up to 15% for low-altitude orbit missions to Mars.3

Aerocapture Guidance In order to mitigate error from uncertainty in the atmospheric density, entry state, and other parameters, an active GNC system is required. Lift modulation architectures have typically been ∗Draper Laboratory Fellow, The Charles Stark Draper Laboratory, Inc. Graduate Research Assistant, Entry systems Design Lab, Colorado Center for Astrodynamics Research, 1111 Engineering Drive, Boulder, CO 80309 †The Charles Stark Draper Laboratory Inc., 555 Technology Square, Cambridge, MA 02139 ‡Smead Professor of Space Technology, Dean of Engineering and Applied Science, 1111 Engineering Drive, Boulder, CO 80309

1 studied due to their flight heritage.2,4,5 These systems utilize a reaction control system (RCS) thruster framework to alter the vehicle’s bank angle, and consequently lift vector direction, to steer the vehicle through the atmosphere for range control.4,6 While flight-proven, these systems require ballast mass to enforce an off-centered center of gravity, allowing the vehicle to fly at an angle of attack (AOA) and, subsequently, with a non-zero lift force. In addition, the active bank modulation GNC algorithm is computationally intensive and prone to actuator latency.7 Drag modulation architectures offer a more straightforward re-entry architecture to lift modulation systems. Such systems negate the need for ballast mass, enabling the vehicle to fly in a ballistic configuration, or at an AOA of zero. These systems instead provide a control event by modifying the vehicle’s ballistic coefficient, defined in Equation1. This modification is typically performed by jettisoning a rigid drag skirt or ballute to modify the vehicle mass and reference area mid-flight.8–10

m β = (1) cDAref

Changing the vehicle’s ballistic coefficient ultimately affects its energy loss rate due to atmospheric drag, enabling these systems to target a specific orbital energy, or geometry, based on the time and location of the jettison event during the atmospheric trajectory.9, 11 Such architectures are ideal candidates for innovative re-entry systems; the removal of propellant, propellant tanks, and ballast mass simplify the packaging in addition to providing mass savings beyond the benefit of aerocapture itself.7 Additionally, the aforementioned reduction in GNC complexity by requiring only discrete control events as opposed to continuous control reduces the risk of failure. Finally, reduction in uncertainty of flight parameters and flow field interaction are reduced with a ballistic configuration.7,8 This research is concerned with the performance of a discrete, rigid drag-skirt jettison event, detailed in Figure1, at Earth. The control authority allotted to the spacecraft in this event is measured by the ratio of β2 , where β refers to the ballistic coefficient of the vehicle after separation, and β β1 2 1 refers to the value before separation.

Figure 1: Diagram of Discrete-Event, Drag Modulation Aerocapture with a Rigid Drag Skirt Jettison Event9

The control authority of the spacecraft affects how well the spacecraft can mitigate against atmospheric uncertainties. If the entry conditions are off nominal and/or the atmospheric density significantly perturbed, the GNC system can adjust its jettison time to reflect these changes in order to alter the total energy lost throughout the trajectory. In doing so, the spacecraft can target a specific orbit, typically by means of an orbital apoapsis altitude. Of course, errors in the jettison time due to guidance system biasing, atmospheric uncertainties, or other uncertainties can result in large apoapsis errors, surface impacts, or failure to enclose the resulting orbit in extreme cases.9

2 Atmospheric Variations There are several sources of atmospheric uncertainty in the density, temperature, pressure, and wind profiles. Solar heating causes numerous secondary perturbing effects such as geomagnetic storms, atmospheric tides by surface heating, and gravity waves resulting from jet stream shear.12, 13 The periodicity of these atmospheric disturbances ranges between short scale, primarily due to atmospheric turbulence, and seasonal up to quasi-biennial scales.14 Fortunately, the time scale of an aerocapture trajectory is small, meaning that the longer time scale perturbations are effectively held constant throughout the aerocapture trajectory. The primary sources of atmospheric uncertainty then lie with diurnal (daily) and semi-diurnal perturbations, including solar heating, geomagnetic fluctuations, gravity waves, planetary waves, and atmospheric tides. In this investigation, we will focus on accurate estimation of atmospheric density. Solar Heating. Solar heating directly affects parameters such as the geomagnetic index, and heats up water vapor in the troposphere and stratosphere, resulting in density perturbations through atmospheric tides. Many of the perturbations resulting from solar heating can be characterized by a diurnal or semi-diurnal period, making them significant to re-entry guidance atmospheric models.12 The correlation between solar heating and density perturbations can also be recognized as a correlation between atmospheric temperature and density. Geomagnetism. Geomagnetism typically has low impact on the atmospheric density, particularly below the Thermosphere. However, geomagnetic storms due to increased solar wind pressure may influence the atmospheric density distribution, especially at lower latitudes.15 A Disturbance Storm Time (Dst) index is used to quantify the severity of the geomagnetic storm. Dst > −75nT causes minimal density disturbances, while a severe geomagnetic storm can result in a Dst of less than −200nT .15 Again, these disturbances primarily affect the Thermosphere density distribution, limiting the overall perturbations on an aerocapture trajectory. Planetary Waves. Wave patterns in the atmosphere characterized mainly as gravity waves, atmospheric tides, and planetary waves can add both small and large-scale fluctuations in the atmospheric density (and other parameters).16 Planetary waves, also referred to as Rossby waves, are caused by planetary rotation, having periods of several days and are modeled as a cosine function in EarthGRAM.12 On the other hand, atmospheric tides and gravity waves produce shorter time scale perturbations due to jet stream shear or surface heating due to solar radiation. Atmospheric tide variations have also been observed to intensify with altitude, resulting in larger variations at aerocapture altitude/density ranges.12 The small-scale perturbations such as gravity waves and general atmospheric turbulence are highly irregular in nature, and must be modeled with the stochastic approach of Equation 2b.

Atmospheric Modeling Earth Global Reference Atmosphere Model (EarthGRAM) is used to simulate atmospheric parameters as functions of altitude. Developed at NASA Marshall Spaceflight Center, EarthGRAM pulls data from numerous data sources and empirical models over various altitude ranges such as the Marshall Thermosphere Model (MET), Jacchia Model, and others.17–19 A nominal, or average, atmospheric profile is generated based on the input latitude, longitude, day-of-flight, and other parameters. The perturbation model is then assumed to be a linear combi- nation of functions corresponding to each perturbation source, such as in Equation 2a, where F (z) is some atmospheric parameter at some altitude, z, F0 is the time invariant profile, S(z, t) is a long

3 period perturbation function (such as seasonal or annual variations), P (z, t) is the planetary wave perturbation function, T (z, t) is the tidal forcing function, G(z, t) is the gravity wave perturbation function, and E(z, t) contributes to the turbulence modeling.14

F (z) = F0 + S(z, t) + P (z, t) + T (z, t) + G(z, t) + E(z, t) (2a) p µ(x’) = rµ(x) + (1 − r2)q(x) (2b)

The perturbations themselves are assumed to be correlated based on Equation 2b, where µ(x) is some normalized variable, µ(x’) is the normalized variable value at the next position step, q(x) is a random, standard Gaussian variable with a mean of 0 and variance of 1, and r is the auto- correlation between successive values of µ(x) and µ(x’).20 There are other factors that contribute to the perturbation model, but for this investigation’s purpose it is sufﬁcient to understand that the perturbations are correlated with previous values as well as an inherent correlation with other parameters such as temperature and pressure.

Figure 2: Earth Atmospheric Density Proﬁles and ±3σ Bounds, EarthGRAM 2016

EarthGRAM enables Monte Carlo sampling to produce a variety of atmospheric parameter profiles over the specified range of coordinates. The density variations of 1000 Monte Carlo generated profiles are shown in Figure2. Atmospheric variations are observed to reach up to ±0.4 of the average density value, highlighting the importance of both active GNC systems as well as accurate estimation of atmospheric perturbations for precise orbit targeting.

MOTIVATION The need for accurate atmospheric estimation is a prevalent issue for all re-entry problems. Tra- jectory reconstruction efforts attempt to validate deceleration measurements against the models on- board the vehicle.21, 22 Dwyer et. al. used Mars Global Surveyor (MGS) data to develop a Monte Carlo atmospheric model of the Martian atmosphere for future aerobraking missions.23 Tolson and Prince investigated atmospheric density modeling and altitude-based data tracking for autonomous aerobraking.24 Unfortunately, these atmospheric estimation techniques are only applicable when a full set of trajectory data is present. Aerocapture trajectories must use the assumed day-of-ﬂight density proﬁle and descending deceleration data to estimate density in a geographic location the vehicle hasn’t yet

4 flown through. This is difficult due to the many sources of atmospheric uncertainty, resulting in a non-trivial structure of the atmospheric density profile. Accurate atmospheric density prediction aids in preventing a biased guidance system. The discrete, drag modulation architecture considered in this research, while offering a straightforward solution to aerocapture control, lacks sufficient control authority to mitigate post-jettison atmospheric uncertainties. Unexpected density variations downrange of the jettison event typically result in apoapsis altitude errors.9 The more proficient the guidance system is at predicting downrange density values, the more accurate the jettison event timing will be to the optimal value, ideally resulting in tighter apoapsis altitude error bounds. This is because the on-board, estimated vehicle dynamics will better match the resulting trajectory through reliable predictions of atmospheric drag. There have been several studies concerning atmospheric estimation for aerocapture in the past couple decades. Perot and Rousseau estimate the atmospheric scale height from a linear least- squares solution using deceleration data to fit the exponential atmospheric model.25 Masciarelli et. al. also estimates atmospheric density from deceleration data applied to an exponential density model, but keeping the scale height constant.5 More recently, several studies have utilized the Hybrid Predictor-corrector Aerocapture Scheme (HYPAS) algorithm to modify a nominal density profile by some density scale factor, also using vehicle deceleration data.26–28 Wagner et. al. investigated machine learning approaches for atmospheric forecasting, expanding upon the static density factor used in previous studies. Putnam and Braun investigate density estimation for drag modulation systems with the same density scale factor applied to a nominal GRAM density profile.8 The majority of other drag modulation aerocapture studies utilize similar atmospheric estimation techniques.9, 10, 29, 30 This research seeks to investigate improvements to the typical density scale factor estimation technique for discrete-event, drag modulation aerocapture trajectories.

METHODOLOGY Numerical Integration A 4th order Runge-Kutta integrator is written in Matlab to integrate the vehicle dynamics through the atmosphere. The Matlab code is compiled into C++ using the Mex compiler for computational speed. The vehicle’s atmospheric exit state is used to calculate Keplerian orbital elements and, subsequently, the apoapsis altitude error. The simulation models Earth an oblate spheroid, including J2 perturbations. Both nominal and dispere atmospheric profiles were generated with EarthGRAM and subsequently parsed into Matlab tables for simple integration with the dynamics simulation. The simulation linearly interpolates density values between altitudes not explicitly present in the table. For Monte Carlo analysis, the nominal, or expected, atmospheric profile is loaded into the guidance system structure, while a disperse atmospheric profile is loaded into the main input structure for actual dynamics calculations.

Base Simulation A base set of simulation parameters are defined to aid in quantifying the relative performance of each density estimation technique. Table1 details the entry conditions, vehicle parameters, and guidance parameters for the comparison simulation. The 3σ uncertainty values are based off of Austin et. al.31 The drag coefficient is assumed to be constant across the jettison event, which is itself assumed to be instantaneous. The second stage aerodynamic properties are calculated to achieve a nominal ballistic coefficient ratio of 10. The uncertainty of each ballistic coefficient is

5 calculated as the sum of the uncertainties in mass and drag coefﬁcient (assuming perfect backshell radius knowledge), multiplied by 2 to calculate the uncertainty in the ballistic coefﬁcient ratio. This large uncertainty in the vehicle’s measure of control authority is another leading cause of apoapsis altitude targeting error.

Parameter Nominal 3σ Uncertainty v0 12km/s 0.5m/s h0 125km 150m ◦ ◦ γ0 -5.9 0.2 ◦ −3 lat0 0 1.5e ◦ −3 long0 0 1.5e m1 100 0.75kg m2 54.4 0.75kg cD 1.05 .0525 β2/β1 10 1.6 ha,target 2000km N/A GNC Rate 0.5Hz N/A Traj Rate 250Hz N/A Table 1: Nominal Values and Uncertainties for Estimation Method Comparison Simulations

GUIDANCE ALGORITHM This research employs a numerical predictor-corrector (NPC) guidance algorithm equipped with a Newton-Raphson root-ﬁnding method to target the optimal drag-skirt jettison time. The current vehicle state, xi, time, ti, atmospheric estimation history, Π = [h, ρest, p, T ], and current jettison curr time estimate, tjett , are fed into the guidance function for numerical integration. The guidance predictor phase integrates the vehicle dynamics up to atmospheric exit, using expected values for all vehicle coefﬁcients and atmospheric properties. Keplerian orbital elements are then calculated from the inertial exit state to determine the estimate apoapsis error for the given drag skirt jettison time. A small error is then applied to this jettison time, and the predictor is re-run to numerically calculate the change in orbital apoapsis error as a function of jettison time. The guid- curr ance corrector then uses the Newton-Raphson method to calculate a new estimate of tjett that is saved inside the guidance computer until the next guidance call.

Atmospheric Data Storage The atmospheric estimation history, Π, is initialized with a set of exponentially-spaced altitudes from atmospheric interface, assumed to be 125km for Earth, to the simulation’s minimum termination altitude. The minimum termination altitude is set to 50km to reduce computation time for cases that would impact the surface. An exponentially-spaced altitude set is chosen to improve density clarity at lower altitudes, where density uncertainties would result in the largest errors. As the vehicle descends through the atmosphere, the navigation computer saves atmospheric estimation parameters upon reaching the pre-determined altitude values, adjusting the initialized altitudes as needed to match the vehicle’s current altitude. The density estimate is calculated with the density factor described by Equation3, where m is the vehicle mass, a is the acceleration, v 5,8 is the planet-relative velocity, CD is the drag coefﬁcient, and Aref is the vehicle reference area.

6 Kgain is the gain value associated with the low-pass smoothing ﬁlter, and is typically set to 0.2. The pressure and temperature histories are assumed to be obtained from a sensor suite such as the Mars Science Laboratory Entry, Descent, and Landing Instrumentation (MEDLI).32

2m|a| ρest = 2 |v| CDAref ρest (3) Kcurr = ρexpected i+1 i Kρ = (1 − Kgain)Kcurr + KgainKρ

ATMOSPHERIC ESTIMATORS Density Factor The density factor, a static multiplier on average atmospheric density profiles described in Equa- tion3, has been the primary density estimation technique for drag modulation aerocapture studies, as well as most lift-modulation re-entry analyses. Thus, the density factor is set to the baseline performance level for performance quantification. The density factor allows the vehicle to estimate atmospheric density based on its deceleration and presumed drag coefficient and reference area. The gain value, Kgain, can be tuned to emphasize previous density measurements over current measurements in an attempt to smooth the density history. Too large of a gain value can destabilize the measurements, resulting in over- or under- prediction of the true density value.

Density Interpolator This estimation technique capitalizes on the atmospheric density history, operating under the assumption that, because the time scale of the perturbations in atmospheric properties is typically larger than that of an aerocapture trajectory, the atmospheric density profile exiting the atmosphere is the same as that when entering the atmosphere. It also assumes that the latitude and longitude dependence on atmospheric density is negligible again due to the short time scale of an aerocapture trajectory. At altitudes within range of those the vehicle has flown through on its descent, the predictor linearly interpolates the presumed density value from the data storage table. At all other altitudes, the density factor is used to estimate the density from the expected density profile due to lack of knowledge of density values at lower altitudes. Figure3 showcases the guidance predictor’s accuracy using the density interpolator over the density factor. While the density factor accurately estimates the density at the current vehicle state (when the guidance predictor is initiated), this value fails to match the majority of other density values. Of course, this also means that the density interpolator is only as effective as the density scale factor because the multiplier is used at altitudes outside of the current density array. This implies that the density interpolator’s performance is inherently tied to the jettison timing of the drag skirt; a later jettison time typically means that the vehicle will fly lower into the atmosphere and thus be able to interpolate between acceleration data at more points, as shown in Figure 3b. When the vehicle is at the orbit periapsis, or lowest altitude of its atmospheric trajectory, the guidance computer is assuming near-perfect knowledge of the atmospheric density forward in time

7 (a) Altitude vs. Predictor Density Variations (b) Altitude vs. Predictor Density Variations Initiated at 100km Initiated at 85km

Figure 3: Density Interpolator and Density Factor Variations vs. Altitude

(ignoring changes in density as a function of latitude and longitude). If the density estimation history during the descent portion of the trajectory is inaccurate, the density interpolator may perform worse than a more accurate density factor.

Ensemble Correlation Filter The third atmospheric estimator developed for this research involves an ensemble approach to estimating downrange density from the vehicle’s deceleration measurements. This method, detailed in Figure4, samples density values within a given tolerance of the current density estimate to match the atmospheric estimation proﬁle to on-board models generated from EarthGRAM Monte Carlo dispersion analysis. It operates under the assumption that the overall structure of the density proﬁle is deterministic due to the sinusoidal nature of the larger-scale perturbations and correlation of the small-scale perturbations from Equation 2b.

Figure 4: Ensemble Correlation Filter Guidance Algorithm Block Diagram

After searching for density values within the speciﬁed tolerance, the residual sum of squares

8 (RSS) error is calculated for each density, pressure, and temperature profile for each of the corresponding density indices. The profile that has the minimum RSS error of the respective atmospheric property histories is then multiplied by the current density factor and loaded into the guidance predictor for integration. This multiplication factor is used so as not to presume a finite number of feasible atmospheric profiles. In addition, the vehicle has no knowledge about the true atmospheric density value. In other words, implementing this multiplicative factor reduces the presumptive nature of the method’s assumptions.

The search tolerance should be chosen to reduce the risk of biasing the algorithm towards a specific density profile. Uncertainties in the vehicle state and aerodynamics can cause errors in the measured density estimate, potentially preventing the true density value from being within the search region for small error percentages. This problem may also be worsened by the linear in- terpolation required to calculate the RSS error of each density value at the altitudes specified in the on-board atmospheric history table. Thus, several search tolerances are simulated in order to observe their effects on the apoapsis altitude error distributions.

Figure5 highlights the risk of the ensemble filter. Lower tolerances on the density search percent- age, shown in Figure 5a, can cause the estimated atmospheric density to vary considerably from the true density profile, likely resulting in targeting errors, particularly because each predictor function call will likely be initialized with a different density profile estimate, depending on how small the search tolerance is. However, if the search tolerance is high enough, as shown in Figure 5b, the estimated density profile will be a much better match to the true density structure.

(a) Altitude vs. Predictor Density Variations, 1% (b) Altitude vs. Predictor Density Variations, Search Tolerance 10% Search Tolerance

Figure 5: Ensemble Correlation Filter and Density Factor Variations vs. Altitude

Unsurprisingly, the accuracy of this estimation method is then heavily reliant on the accuracy of the density estimate. Figure 5b details how even a 10% error between the density estimate and truth value can skew the density proﬁle considerably from the actual density variation proﬁle. This constant skew will always result in either over-predicting or under-predicting the energy loss experienced by the spacecraft, the degree of which depends on the accuracy of Equation3.

9 RESULTS Density Interpolator Figure6 details the 3σ error of the density interpolator against the density factor and average density proﬁles at each simulation time step. The density interpolator error exactly matches the density factor up until the periapsis of the vehicle’s orbit (lowest altitude of the trajectory). After this point, the density interpolator’s error is a mirror image of the error on the descent, except it is stretched out slightly due to the lower vehicle speed. The difference in error at each time step between the density factor and the density interpolator is small; however, Figure6 depicts the error of the estimated value at each simulation time step, rather than the error of the predictor’s estimation method. Instead, Figures 7a and 7b showcase the improvement in apoapsis targeting over the density factor.

Figure 6: Comparing 3σ Errors for the Density Interpolator and Density Factor Estimation Methods

Figure 7a plots the capture rate (percent out of 1000 Monte Carlo cases) against varying EFPAs for a constant target apoapsis of 2000km, while Figure 7b details the capture rate against target apoapsis altitudes between 400km and 40000km for a constant EFPA of −5.9◦. Note that these trade studies employ non-optimal EFPAs to showcase the resilience to off-nominal conditions. Each curve depicts the rate of capture within a specified tolerance of the target value. The cases with low capture rates are caused by the static EFPA and target apoapsis in which the allotted control authority is insufficient to correct for the particular degree of off-nominal conditions. The cases where the control authority is sufficient to mitigate off-nominal conditions and uncertainties show definitive targeting improvements. For example, Figure 7b shows a 52% capture rate within 25km for a 400km apoapsis altitude target using a density factor, and a 98% capture rate using the density interpolator. In general, the percent improvement between density interpolator and density factor diminishes with increasing apoapsis tolerance as well as increasing target apoapsis altitude. In addition, analyzing Figure 7a indicates that the improvement of the density interpolator over the density factor increases with shallower entry angles. This is because the optimal jettison time

10 (a) Density Interpolator Capture Rates vs. Entry (b) Density Interpolator Capture Rates vs. Flight Path Angle Target Apoapsis

Figure 7: Comparison of Orbital Capture Rates for Density Interpolator and Density Factor Estimation Methods for shallower entry ﬂight path angles occurs at lower altitudes relative to periapsis, allowing the guidance system to gain more information on the assumed density proﬁle.

Ensemble Correlation Filter At a given simulation time step, the accuracy of the ensemble correlation filter is only as good as the density factor because it searches for a range of new density values and profiles at each guidance function call. Therefore, the RSS error of the estimated density profile vs. simulation time is shown in Figure 8a to detail the method’s uncertainty throughout the flight compared to the density factor. As discussed previously, this estimation technique works best with sufficient density history data; therefore, the guidance algorithm is not run until several tens of seconds into the atmosphere before searching for densities and calculating RSS errors. This explains the equal RSS error between the density factor and correlation filter at the beginning of the trajectory.

(a) RSS Error of Assumed Density Proﬁle vs. (b) Ensemble Correlation Filter Performance for Simulation Time Various Search Tolerances and β2/β1 Ratios

Figure 8: Ensemble Correlation Filter Performance using RSS Error and Capture Rate vs. Search Tolerance Comparisons

11 Interestingly, the multiplicative factor on the resulting EarthGRAM density structure causes the RSS error to closely match, and sometimes exceed, the error of the average density profile multiplied by this same factor. If this density factor were removed, the RSS error would drop to zero as soon as the correct density profile structure were found. Of course, as previously discussed, this would incorrectly assume that there are a finite set of density profiles the vehicle may experience. In any case, Figure 8a shows that the average density profile multiplied by a static density factor is sometimes more accurate than a true density profile structure multiplied by that same factor. Figure 8b shows the performance of the filter technique for various search tolerances and ballistic coefficient ratios, keeping the other simulation parameters the same as in Table1. It is observed that the search tolerance has little to no impact on the capture rate for this range of ballistic coefficient ratios. While the ensemble correlation filter may repeatedly fail to encompass the true density value due to a low search tolerance, the simulation guidance rate of 0.5Hz allows the system ample time to identify the correct density profile structure. It is important to note that if the guidance rate were lower, or the jettison time to occur much earlier in the trajectory, the guidance system may fail to identify the correct density structure entirely. A more expected result from Figure 8b is that increasing the ballistic coefficient ratio, effectively increasing the control authority allotted to the vehicle, slightly increase the capture rate of the ensemble filter as well.

(a) Monte Carlo Results of Ensemble Filter for (b) Monte Carlo Results of Ensemble Filter over Various Ballistic Coefﬁcient Ratios Various Density Search Tolerances

Figure 9: Monte Carlo Results using the Ensemble Correlation Filter for Various Ballistic Coefﬁcient Ratios and Search Tolerances

This is better visualized by Figures 9a and 9b. The former shows the apoapsis error distributions for the data acquired in Figure 8b across all search tolerances for each β2/β1 ratio, while the latter depicts the error distributions separated by search tolerance for a single β2/β1 ratio. The tighter error distributions for higher β2/β1 ratios in Figure 9a mimic the results from Figure 8b; however, the results in Figure 9b are somewhat counter-intuitive. A wider error distribution for higher search tolerances may be explained by a guidance biasing; a higher search tolerance may cause the guidance system to envelop too many atmospheric density profiles in which the RSS errors are similarly small for the vehicle’s current density history, even if the profiles do not match well downstream. Previous expectations were that a small search tolerance would prevent the guidance system from accurately targeting a density profile structure; however, it appears that if the guidance computer is given ample time to find the profile (eg. a higher guidance rate or later jettison time), the lower

12 search tolerance guards against density proﬁle biasing, assuming the search tolerance is still large enough to account for errors in Equation3.

Estimation Method Comparison An 1000-case Monte Carlo simulation was run using the settings described in Table1 for each estimation method. The apoapsis altitude distributions are plotted in Figure 10, while a summary of the results can be found in Table2. The marginal improvement in each case can be explained by the similarity in errors found in Figures6 and 8a for the density interpolator and ensemble correlation ﬁlter, respectively.

Figure 10: Apoapsis Altitude Error Distributions for Each Estimation Technique.

Method Median ha(km) ha 1σ(km) Median tjett/tf tjett/tf 1σ Median ∆v(m/s) ∆v 1σ(m/s) Density Factor 2337.1 517.0 0.57 0.21 4139.8 105.89 Density Interpolator 2206.1 567.76 0.65 0.21 4131.7 108.7 Correlation Filter, 1% Tol 2160.0 543.33 0.58 0.19 4141.6 104.9 Correlation Filter, 5% Tol 2281.1 573.5 0.55 0.21 4142.8 110.9 Correlation Filter, 10% Tol 2229.6 598.2 0.58 0.22 4126.6 113.2 Table 2: Summary of Results for each Estimation Technique with Simulation Parameters from Table1

Table2 shows that the median apoapsis altitude is closest to the target of 2000km for the ensemble correlation filter with a 1% search tolerance. This result matches those found in Figure 9b. Interestingly enough, the 1σ apoapsis altitude is smallest when considering the density factor. This means that the density factor tends to be the most consistent estimation technique, even if it has the largest median error. The density interpolator achieves a 5% improvement in median apoapsis altitude over the density factor, while the ensemble correlation filter achieves at best a 7% improvement in median apoapsis altitude over the density factor and a 1.4% improvement over the density interpolator. Finally, the ensemble correlation filter showcases a 4.6% error in median apoapsis altitude between cases with a 5% and 1% search tolerance. It is also noticeable that there is marginal

13 difference in the total ∆v for each estimation technique. These marginal improvements may also be a result of the given simulation parameters, more speciﬁcally the entry conditions, target apoapsis altitude, and ballistic coefﬁcient ratio.

CONCLUSIONS Three different atmospheric estimation techniques were assessed in terms of apoapsis altitude error distribution and variation from the true atmospheric profile. The three techniques observed were a static density factor, a density array interpolator, and an ensemble correlation filter. The density factor was taken as a baseline comparison technique due to its frequent use in the literature. The density interpolator assumes the density profile during the spacecraft ascent is the same as that during the descent due to the short time frame of an aerocapture trajectory relative to the time scale of atmospheric perturbations. The correlation filter searches for expected density values around the current estimated density and finds an expected density profile that minimizes the RSS error from the vehicle’s density history, multiplying this profile by the current density factor. The density interpolator was able to achieve a 46% improvement within 25km of the 400km apoapsis altitude target over the static density factor. This improvement was found to diminish with increasing apoapsis altitude target as well as increasing tolerance on the acceptable targeting error. Shallower EFPAs were also found to increase the level of improvement of the density interpolator over the density factor due to the increased flight time before jettison occurred, increasing the vehicle’s knowledge of the atmosphere before the jettison event. The density interpolator was also able to improve the median apoapsis altitude by 5% when considering the same simulation parameters. The ensemble correlation filter is prohibitively dependent on the density variation estimate, which skews the entire profile by a multiplicative value, meaning the resultant expected vehicle energy loss will never equate to what the vehicle will actually experience, unless the density estimate has negligible error. The ensemble correlation filter was found to improve with smaller density search tolerance, more specifically a 4.6% improvement in median apoapsis altitude between a 5% and 1% search tolerance. As expected, the search tolerance played an overall minimal role in the capture rate, which instead is largely affected by the ballistic coefficient ratio. When considering the same simulation parameters, the ensemble correlation filter was found to improve median apoapsis altitude by 7% over the density factor, and 1.4% over the density factor.

FUTURE WORK Future work should include more in-depth models of the atmosphere. This research used a static latitude/longitude output of EarthGRAM, whereas a true atmospheric model would depend on the latitude, longitude, and altitude of the spacecraft for accurate density measurements. This would also provide a more realistic analysis of the density interpolator, as the assumption of mirrored density profile would not match nearly as well on the ascent segment of the trajectory. In addition, the ensemble correlation filter assumes there are a discrete set of atmospheric struc- tures that the vehicle might experience. However, without assuming some form of the atmospheric density profile, the guidance solution would be under-determined, requiring a solution of density as a function of latitude, longitude, and altitude, as well as iterating on the jettison time estimate. Another estimation technique that may prove useful is non-linear regression; a curve-fit of the deceleration data may be able to accurately estimate the large-scale fluctuations in the atmosphere. Ideally, the small-scale turbulence would be smoothed out from the deceleration data to reduce

14 the complexity of the resulting curve ﬁt equation. A smoother density proﬁle would likely be more robust than one that seeks to capture the small scale turbulence, particularly for estimating downrange density values. In general, further understanding on the structure of both short and long-wave perturbations in the atmosphere will aid in developing a more robust, less presumptuous density estimation technique.

ACKNOWLEDGMENTS This research was supported by the Charles Stark Draper Laboratory Fellows Program.

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