mars atmospheric entry integrated navigation with partial...

Research ArticleMars Atmospheric Entry Integrated Navigation with PartialIntermittent Measurements

Tai-shan Lou,1 Yan Wang,1 Guo-qiang Ding,1 and Liang-yu Zhao2

1School of Electrical and Information Engineering, Zhengzhou University of Light Industry, Zhengzhou 45002, China2School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China

Correspondence should be addressed to Tai-shan Lou; [email protected]

Received 12 June 2017; Accepted 13 August 2017; Published 24 September 2017

Academic Editor: Christopher J. Damaren

Copyright © 2017 Tai-shan Lou et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Signal degradation suffered by the vehicle is a combination brownout and blackout during Mars atmospheric entry. Thecommunications brownout means that signal fades and blackout means that the signal is lost completely. The communicationsbrownout and blackout periods are analyzed and predicted with an altitude and velocity profiles. In the brownout period, therange measurements between the vehicle and the orbiters are modeled as intermittent measurements with the radio signalarrival probabilities, which are distributed as a Rayleigh distribution of the electron number density around the entry vehicle. Anew integrated navigation strategy during the Mars atmospheric entry phase is proposed to consider the probabilities of theradio measurements in the communications brownout and blackout periods under the IMU/beacon scenario based on theinformation filter with intermittent measurements. Numerical navigation simulations are designed to show the performance ofthe proposed navigation strategy under the integrated navigation scenario.

1. Introduction

On 5 August 2012, the Mars Science Laboratory’s (MSL)Curiosity rover successfully landed on Mars after sufferingthe Seven Minutes of Terror, in which the MSL entry,descent, and landing (EDL) system brings the velocity ofthe vehicle from about 5.9 km/s to 0.75m/s. Especially,during Mars entry phase, the vehicle experienced the mostrugged aerodynamic environment, which includes hightemperature and pressure, peak heating (peak temperaturesof up to 2090°C), and peak deceleration (out at 15g), andthe signal between the vehicle and the orbiters can fade orsuffer from total loss to make the ultrahigh frequency(UHF) relay links from the MSL to the orbiters suffer aperiod of about 70 s of degradation [1, 2]. The Mercury,Gemini, and Apollo spacecraft entering the Earth’s atmo-sphere and the Mars Pathfinder entering Mars’ atmosphereall endured communications blackouts, and the lasting timeis from 30 seconds to several minutes [3]. During communi-cations blackout period, it is difficult to perform real-timecommunication between the vehicle and the orbiter to

provide the range measurements in time for the navigationsystem. So, it is necessary to predict trends or likelihood ofthe signal degradation and design appropriate navigationtechnology and strategy.

The communications blackout is caused by reflecting orabsorbing the electromagnetic at some frequencies by thesheath of ionized atmospheric gases around the spacecraftwhen the spacecraft enters into a planetary atmosphere witha hypersonic velocity. Whether or not the communicationsblackout occurs depends on the electron number density(END), the critical electron number density (CEND) at thelink frequency, such as UHF. If the electron number densityexceeds the corresponding critical electron number density,the communications blackout may happen under some fre-quencies. In order to analyze and forecast any possible black-out problem, two aerothermodynamic analysis programs,which include the Jet Propulsion Laboratory (JPL) NormalShock and Chemical Equilibrium Program (also called theHorton program) [4] and the Langley AerothermodynamicUpwind Relaxation Algorithm (LAURA) program [5], havebeen used in Mars entry vehicles as they fly through Mars

HindawiInternational Journal of Aerospace EngineeringVolume 2017, Article ID 4378640, 12 pageshttps://doi.org/10.1155/2017/4378640

https://doi.org/10.1155/2017/4378640

atmospheric environment [6]. The JPL Horton program wasdeveloped in the 1960s [4] and then used to predict any pos-sible communications blackout problem [2, 6]. These toolsoutput the velocity of the shock layer and the electron num-ber density at the stagnation point by using the composition,pressure, temperature, and density of the Mars atmosphere[6]. Morabito analyses the communications blackoutproblem of the Mars Pathfinder at X-band based on thewake-region electron density estimates produced by the JPLHorton program and the LAURA program and comparesthe predicted results of the two programs [6]. Morabito andEdquist predicted the communications blackout at UHF fordifferent MSL entry cases by using the above two tools [3].After the MSL successfully landed on Mars, Morabito andEdquist analyzed the UHF communications brownout andblackout experienced by the MSL during Mars atmosphericentry, which coincided with the predicted signal degradationfrom preflight analyses in literature [3]. Schratz et al. dis-cussed the performance of the MSL telecommunications forUHF during the EDL phase and analyzed the signal strengthand the plasma blackout during Mars entry [7].

Future Mars explorations need the capability of pinpointlanding on the preselected high scientifically valuable regionsto meet the scientists’ requirements. It is difficult to achievethis target if the only available measurement sensor is theon-board default inertial measurement unit (IMU) to pro-vide measurement information for the navigation, guidance,and control system during Mars atmospheric entry phase[8, 9]. An innovative integrated navigation scheme duringMars atmospheric entry was proposed, which is adding therange measurements between the vehicle and the surfacebeacons or orbiters by using the UHF radio communication[10, 11]. The key technology of the above navigation schemeis that the UHF radio communication is not blocked by theplasma sheath around the vehicle. National Aeronauticsand Space Administration (NASA) develops an advancedMars Network’s Electra UHF transceiver, which is a proto-type, embedded, real-time navigation system and can provideradio communication information between the vehicle andan orbiter or a surface beacon [10, 12]. Levesque and deLafontaine researched four integrated navigation scenariosand their observability based on radio ranging by using theunscented Kalman filter during the Mars atmospheric entry[11]. Li et al. did a lot of works based on a six-dimensiondynamic model by using radio communication under theIMU/beacon integrated navigation schemes [9, 13, 14]. Yuet al. primarily analyzed the optimization of the Mars entrynavigation beacon configuration based on the system observ-ability by using radiometric measurements [15, 16]. Lou et al.considered the uncertain model parameters and the mea-surement biases by using consider Kalman filter (also calledSchmidt-Kalman filter) [17–19] and analyzed the sensitivityof the parameter uncertainties based on the robust desensi-tized extended Kalman filter [20] under the radio beaconnavigation schemes during Mars atmospheric entry.

Morabito et al. [2] pointed that the UHF radio signalsfade or break off in the communications brownout andblackout periods for the MSL during Mars atmosphericentry. This signal degradation makes the two-way range

measurements for the navigation system cannot reach (orat a certain probability) the MSL. Li and Peng primarilyanalyzed the effect of the communications blackout underthe radio beacons/IMU integrated navigation and pointedthat as the communications blackout occurs, the naviga-tion errors will gradually increase [9]. Wang and Xia com-pared the performance of the unscented Kalman filterunder the surface beacons/IMU integrated navigation andthe orbiters/IMU integrated navigation when the communi-cations blackout happens [21]. The above researches consid-ered that the vehicle does not receive any radio signal in thecommunications brownout period during Mars atmosphericentry and the probabilities of the arrival of the range mea-surements are not analyzed [2]. Adding the potential rangemeasurements, which arrives at a certain probability, intothe navigation system undoubtedly improves the navigationaccuracy in the communications brownout period.

Recently, the intermittent measurements, which are alsocalled missing measurements and modeled by a Bernoullidistributed sequence [22–24], were introduced into a seriesof filtering algorithms to deal with these intermittent dataphenomena, such as the Kalman filtering [24, 25], unscentedKalman filter [25, 26], and robust filtering [23, 27]. These fil-ters are committed to obtain the optimal estimates when thearrival of the measurement is not in time; in other words, themeasurement is lost at a certain probability. This filteringproblem with intermittent measurements has motivatedresearcher’s interest, because these missing measurementphenomena are more popular in practical applications, suchas the navigation of a moving vehicle [28], and networkedcontrol systems [29]. Subsequently, the intermittent mea-surement problem with different loss probabilities, whichmeans that the measurement loss might be partial, has beeninvestigated, and some solutions are proposed [30, 31].

The aim of this paper is to develop an integratednavigation scenario with the probabilities of the radiomeasurements in the communications brownout and black-out periods during the Mars atmospheric entry phase byusing the information filter with intermittent measurements(IFIM). The communications brownout and blackoutperiods are analyzed and predicted with an altitude andvelocity profiles during Mars atmospheric entry. The signalarrival probabilities of the UHF radio signals are modeledas the Rayleigh distribution of the electron number densityaround the entry vehicle. The range measurements betweenthe vehicle and the orbiters are described as a Bernoulli dis-tributed sequence with different probabilities in the commu-nications brownout period. Then, the IFIM algorithm isderived and designed for the navigation filter. The numericalnavigation simulations are designed to show the performanceof the proposed navigation strategy.

This paper is organized as follows: firstly, the Mars entrydynamic model and measurement model based on the IMUand three orbiters are introduced; secondly, the communica-tions brownout and blackout phenomena are predicted by anaerodynamical tool during Mars atmospheric entry; thirdly,the information filter with intermittent measurements isderived; and lastly, the results of the navigation simulationare discussed.

2 International Journal of Aerospace Engineering

2. Dynamics of the Navigation System

2.1. Dynamic Equations for Mars Entry. During Mars atmo-spheric entry, the dynamic model of the vehicle is modeledas six state equations of a six-dimensional state, whichincludes the altitude r (radial distance from the center ofthe vehicle mass to the Mars center), radial velocity v, flightpath angle (FPA) γ, longitude θ, latitude λ, and azimuthangle ψ (a clockwise rotation angle starting at due north),in the Mars-centered Mars-fixed coordinate system. Someassumptions are that the planet is nonrotating, its atmo-sphere is stationary and quiet, and winds in the atmosphereand the centripetal and Coriolis effects are neglected [17, 32].Then, the six entry dynamic equations of the vehicle aredescribed by [11, 17, 19, 32]

r = v sin γ,v = −D − g sin γ,

γ =1v

v2

r− g cos γ + L cos ϕ ,

θ =v cos γ sin ψ

r cos λ,

λ =v cos γ cos ψ

r,

ψ =1

v cos γv2

rsin ψ cos2γ tan λ + L sin ϕ ,

1

where ϕ is the bank angle, g = μ/r2 is the simplified gravita-tional acceleration, and μ is the Mars gravitational constant(μ = 42 828 29 × 109m3/s2). D and L are, respectively, theaerodynamic drag and lift accelerations given as follows:

D =12ρv2

CDSm

,

L =12ρv2

CLSm

,2

where CD and CL are, respectively, the vehicle drag coefficientand lift coefficient, S is the reference surface area of vehicle,m

is the mass of the vehicle. ρ represents the Mars atmosphericdensity with an exponential model is given by [18, 33]

ρ = ρ0expr0 − rhs

, 3

where ρ0 is the nominal reference density on the Marssurface, r0 denotes the nominal reference radial positionr0 = 3 437 2 km , and hs denotes the constant scale heighths = 7 5 km . For convenience’s sake in this work, the lift-to-drag ratio is defined by L/D = CL/CD, and the ballisticcoefficient is defined as B = CDS/m.

2.2. Measurement Models. To improve the navigation accu-racy based on the IMU information, a potential integratednavigation scenario with both IMU and UHF radio commu-nication is proposed to support pinpoint landing accuracyfor future Mars missions. The IMU provides three compo-nents of acceleration for the navigation system. The UHFradio provides communication between the entry vehicleand the orbiting satellites or surface beacons (e.g., a presetfixed beacon or a previous Mars lander) and measures thedistance between the vehicle and an orbiter or a surfacebeacon for the navigation system [12, 18, 19, 34]. At pres-ent, there are only three available Mars orbiters to provideradio communications, which are Mars ReconnaissanceOrbiter (MRO), Mars Odyssey (MO), and Mars Express(MEX), and no surface radio beacons are preset on Mars.In this work, three orbiting radio satellites are consideredto provide the two-way range measurements by using theUHF radio communications. The diagram of the threeorbiter navigation scenarios and the position of the UHFantenna, the stagnation point and the wake-region, whichneed to estimate the electron number density, is shownin Figure 1.

2.2.1. IMU Measurements. The IMU is used to measure thespecific force components along three orthogonal axes andthe angular rates about the axes by using the accelerometers

OrbiterOrbiter

Mars

VehicleTwo-way radio range

Wake region

Two-way radio range

Stagnation point

Mars atmosphere

UHF antenna

Orbiter

Figure 1: Diagram of three orbiter navigation scenarios, stagnation point, and wake region.

3International Journal of Aerospace Engineering

and gyroscopes. Three components of acceleration a aregiven by

a = a + ba + ηa, 4

where a is the true linear acceleration along body axes, ba isthe drift-rate biases of the accelerometer, and ηa is assumedas zero-mean white Gaussian noise. In this work, the acceler-ometer measurement model is defined by

a = −D − Lsin ϕ Lcos ϕ T 5

2.2.2. Range Measurements. The UHF radio communicationsprovide the two-way range measurements between the vehi-cle and the orbiters within sight. The two-way range mea-surement R in the Mars-centered inertial coordinate systemis reconstructed by

R = r − ro T r − ro + ξR, 6

where r is the position vector of the entry vehicle, which iscalculated by

rl = r cos λcos θ, cos λsin θ, sin λ T , 7

ro is the position vector of the orbiter, and ξR is the rangemeasurement noise. In this work, three orbiters within sightare utilized to provide the range measurements for the navi-gation system, and the corresponding parameters of the threeorbiters are listed in the following section.

3. Communications Blackout Estimations andSignal Loss Probabilities during Mars Entry

The UHF radio communications between the vehicle and anorbiter or a surface beacon during Mars entry can becomedisrupted, when the electron number density of the ionizedparticles around the hypersonic vehicle increases sufficientlyhighly. The intervening charged particles can reflect orabsorb the radio signal energy to degrade the quality of thecommunications, such as brownout or blackout. The com-munications brownout/blackout problem can decrease thesources of the navigation measurement and make the mea-surement model with range information mismatch [6, 22].The navigation accuracy during Mars entry will greatlydegrade or even make unexpected results. So, it is necessaryto analyze the degree of the signal degradation and the effectof the different signal attenuations for the communication.

3.1. Effects of Charged Particles on Communications withUHF. The communication signal between the entry vehicleand the preselected radio beacons will be attenuated, becausethe surrounding plasma on the vehicle during Mars atmo-spheric entry will absorb and reflect the signal if the plasmadensity is sufficiently great. Spencer points that the mainmatters to reflect waves in a plasma gas are electrons, andthe principle factors to determine the signal propagationcondition are the transmission frequency and the plasmafrequency of the charged particle sheath [35, 36].

During Mars atmospheric entry, the heating of atmo-spheric gases or an ablating heat shield generates greatcharged particles, and when the density of the charged parti-cles surpasses the plasma density at the frequency, the plasmasheath will black out the communication signal [6]. Theplasma frequency f p is defined as

f p =12π

q2neε0me

, 8

where q is the electron charge q = 1 6 × 10−19 coulombs ,ne is the electron number density per cubic meter (particles/m3), me is the electron mass me = 9 1 × 10−31 kg , and ε0 isthe permittivity ε0 = 8 85 × 10−12 F/m2 . So, when the UHFcommunication frequency, f UHF = 401MHz, is greater thanthe plasma frequency f p, the signal attenuation usually isnegligible; otherwise, the signal attenuation should be con-sidered. The signal attenuation is described by the amountattenuation per unit length, which is defined by [6]:

dBzp

=54 6λ

f 2p − f 2UHF

f 2UHF, 9

where dB is the total attenuation, zp is the thickness of theplasma sheath, and λ is the wavelength of the signal.Figure 2 shows the attenuation amount per unit length forUHF versus the electron number density. It can be seen thatthe signal strength decreases rapidly when the real electronnumber density reaches and exceeds the critical electronnumber density of UHF.

3.2. Communications Blackout Estimations. In this paper, theJPL Horton program numerically simulates the chemicalequilibrium process. We assume that the Martian atmo-sphere model with a surface pressure of about 6.1mb andan argon-free simplification proportion, in which the pro-portion of CO2 is 96.5% and N2 is 3.5% in the JPL Hortonprogram, and other assumptions are used as in the literatures[4, 6, 21]. To predict the communications blackout duringMars entry, the atmospheric relative velocity and altitudeare shown in Figure 3, and the atmospheric density is shownin Figure 4, because the combination of the atmosphericrelative velocity and atmospheric density during Mars entry

109 1010 1011 1012 1013 1014 1015100

101

102

103

104

105

Electron number density (m−3)

Atte

nuat

ion

per u

nit l

engh

t (dB

/m)

Figure 2: Attenuation per unit length versus electron numberdensity for UHF.


is the main factor to affect the received signal degradationdue to charged particles. Figure 3 shows the wake-regionelectron number density, the stagnation point electron num-ber density, the blackout critical electron number density(blackout CEND), and the brownout critical electron number

density (brownout CEND) for the 401MHz UHF. Theblackout CEND for UHF is calculated by [6]

nblackout,crit =f 2UHF

80 64 × 10610

150

150 200 250

100A

ltitu

de (k

m)

100

50

50

150Times (s)

200 25010050

00

6000

4000

Vel

ocity

(m/s

)

2000

0

Figure 3: Atmospheric relative velocity and altitude of the entry vehicle.

50 100 150 200 25010‒9

10‒8

10‒7

10‒6

10‒5

10‒4

10‒3

10‒2

10‒1

Times (s)

Atm

osph

eric

den

sity

(kg/

m3 )

Figure 4: Atmospheric density during Mars entry.


The value of the blackout CEND for UHF is nblackout,crit =1 9941 × 109 particles/m3. The brownout CEND for UHF isestimated as nbrownout,crit = 3 08 × 108 particles/cm3, whichcomes from Figure 5 in the literature [2], and is just less thanan order of magnitude compared with the blackout CEND.

From Figure 6, it can be seen that the peak predictedstagnation point electron number density at 64 s past entryis about 3 69 × 1012 particles/cm3, and the peak predictedwake-region electron number density at this time is about3 70 × 1010 particles/cm3. The likely worst case vehicle

20 40 60 80 100 120

10‒8

10‒6

10‒4

10‒2

100

Brownout BrownoutBlackout

Times (s)

Sign

al ar

rival

pro

babi

lity

Figure 5: Probabilities of the received signal during Mars atmospheric entry.

0 20 40 60 80 100 120104

106

108

1010

1012

Times (s)

Elec

tron

num

ber d

ensit

y (m

‒3 )

UHF blackout CEND

UHF brownout CEND

Blackout

Stagnation endWake end

Brownout Brownout

Figure 6: Predicted electron number density profiles (brownout: blue 30 s~41 s, 81 s~87 s; red 6 s~15 s, 91 s~98 s; blackout: blue 41 s~81 s;red 15 s ~91s).


degradation period at UHF spans from 6 s to 98 s for the stag-nation and from 30 s to 87 s for the wake region. The signalfades between 6 s~15 s and 91 s~98 s for the stagnation andbetween 30 s~41 s and 81 s~87 s for the wake region, andthe signal is lost completely between 15 s~91 s for the stagna-tion and between 41 s~ 81 s for the wake region, similarly tothe signal link data the period around plasma degradationin literature [2].

3.3. Signal Arrival Probabilities.When the real electron num-ber density reaches and exceeds the critical electron numberdensity of UHF, the signal strength will decrease rapidly, andthen the communication quality between the entry vehicleand the preselected radio beacons will seriously fade or evenblack out. Morabito et al. pointed that the UHF relay linksfromMSL to the MRO and to the MEX suffered degradation,which is a combination of brownout and blackout [2], andthe communication outage is approximately 70 s duringMars atmospheric entry. The received carrier-to-noise ratio(CNR) is recorded and processed by using the signal datafrom a bandwidth of ±30Hz, which its undisturbed level isabout 42 dB, and its outage threshold level is 10 dB, duringMars atmospheric entry. For the MSL-to-MRO signal linkin the degradation period, the calculated CNR shows thatthe brownout period is between roughly 30 s and 40 s, wherethe CNR decreases from about 42 dB to below 10dB and82 s~100 s, where the CNR increases from below 10 dB to42 dB, and the blackout period is between 40 s and 80 s, wherethe CNR drops below 10dB as expected for a small periodabout 70 s [2].

For the brownout period, the electron number densityis above this brownout CEND and below the blackoutCEND for UHF, and at the same time, the CNR is alsoabove its outage threshold level and below its undisturbedlevel, but possibly, communications blackout is deemeduncertain. So, the uncertain received navigation signalsbetween the vehicle and the radio beacons with UHF willmake the navigation data cannot arrive to its destinationin time to be used for navigation data process. Here, theuncertain navigation measurement information is describedas an intermittent measurement, in which the arrival of mea-surement is a Bernoulli process with parameter 0 ≤ β ≤ 1 [2].Next, we will define the probability of the arrival of navi-gation measurement information by using the electronnumber density.

The communication signal energy can be absorbed,reflected, refracted, and diffracted by the sheath of thecharged particles around the hypersonic vehicle duringMars atmospheric entry. The magnitude of the signal willfade in otroposheric and ionospheric signal propagation,and it is approximately modeled as Rayleigh fading, whichdistributes to the Rayleigh distribution, and its probabilitydensity function and cumulative distribution function aregiven by [37].

f x =xτ2

e−x2/2τ2 , x ≥ 0,

F x = 1 − e−x2/2τ2 , x ≥ 0,

11

where τ is the scale parameter of the distribution. Here,we assume that the signal strength in the brownoutperiod during Mars atmospheric entry obeys the Rayleighdistribution, in which random variable is the electronnumber density. So, the probability of the arrival of naviga-tion measurement information is calculated by the aboveRayleigh distribution.

4. Information Filter with IntermittentMeasurement Design

In this section, the intermittent measurement problem withmultiple arrival probabilities is introduced into the tradi-tional information filter (IF) [38, 39]. When the arrival prob-abilities of the measurements coming from navigationsensors are different, the information filter has a great advan-tage to deal with these intermittent measurement problemscompared with the Kalman filter.

4.1. Problem Formulation. Consider the linear discrete-time process and measurement models with intermittentmeasurements:

xk =Φk/k−1xk−1 +wk−1, 12

zk =ΘkHkxk + vk, 13

where xk ∈Rn is a state vector and Φk/k−1 is the state transi-

tion matrix. zk = z1kT , z2k

T ,…, zNkT T

∈Rm is a mea-surement vector, and Hk is the state measurement matrix.For every sensor, the measurement equation is

zi,k = λi,kHi,kxk + vi,k 14

Θk is defined as Θk = diag λ1,k, λ2,k,…, λN ,k , in whichrandom variable λi,k ∈R is a Bernoulli distributed sequencetaking on values of 0 and 1 with

Prob λi,k = 1 = βi,k, 15

Prob λi,k = 0 = 1 − βi,k, 16

where βi,k is a known time-varying positive scalar satisfying0 ≤ βi,k ≤ 1. If t ≠ s, λt,k is independent of λs,k, and λi,k isassumed to be independent of x0, wk, and vi,k. wk ∈R

n isindependent zero-mean Gaussian noise processes withcovarianceQk. Themeasurement noise vi,k ∈Rm is defined as

p vi,k λi,k =Ν 0, Ri,k , λi,k = 1,Ν 0, σ2i I , λi,k = 0,

17

where σ2i I is the variance of the measurement at time k. If λi,kis 0 for σ2i I, and λi,k is 1 for Ri,k. When the real measurementdoes not arrive, the given variance of the measurement istaken as the limiting case of σi →∞ [22].

Here, the aforementioned measurement (13) repre-sents intermittent measurements, missing measurements,or uncertain observations, in which the arrival of a measure-ment is the above Bernoulli distribution [23, 40, 41].


4.2. Traditional Information Filter. The information filter isthe information form of the Kalman filter. The informationmatrix, P−1, is defined as the inverse of the covariance matrixP in the Kalman filter. In the information filter, some propa-gation equations must be rederived by using the Sherman-Morrison-Woodbury matrix inversion lemma [39]

A + BCD −1 = A−1 − A−1B DA−1B + C−1 −1DA−1 18

Here, the probability βi,k of the random variable λi,k in(15) for all i and k is assumed as βi,k = 1, and this implies thatthe measurement arrives to its destination in time to be usedfor filtering. So, the measurement equation of the informa-tion filter is given by

zk =Hkxk + vk 19

Based on the process and measurement models in (12)and (19), the information is concluded as follows [22, 39].The time update equations are

x̂k/k−1 =Φk/k−1x̂k−1,P−1k/k−1 =Ak−1 −Ak−1 Ak−1 +Q−1

k−1−1Ak−1,

20

and the measurement update equations are

P−1k = P−1

k/k−1 + 〠N

i=1HT

i,kR−1i,kHi,k, 21

Ki,k = PkHTi,kR−1

i,k , 22

x̂k = x̂k/k−1 + 〠N

i=1Ki,k zk −Hi,kx̂k/k−1 , 23

where Ak−1 =Φ−1k/k−1P−1

k−1Φ−Tk/k−1

4.3. Information Filter with Intermittent MeasurementAlgorithm. As the Kalman filter with intermittent measure-ments in [22], the variance of the measurement can bereconstructed as

R∗i,k = λi,kRi,k + 1 − λi,k σ2

i I 24

Substituting Ri,k in (21) and (22) by R∗i,k gives

P−1k = P−1

k/k−1 + 〠N

i=1HT

i,k λi,kRi,k + 1 − λi,k σ2i I−1Hi,k, 25

K∗i,k = PkHT

i,k λi,kRi,k + 1 − λi,k σ2i I−1 26

When λi,k = 0, σi is taken as the limit of σi →∞. Then(25) and (26) become

P−1k =

P−1k/k−1 + 〠

N

i=1λi,kHT

i,kR−1i,kHi,k, λi,k = 1,

P−1k/k−1, λi,k = 0,

K∗i,k =

PkHTi,kR−1

i,k , λi,k = 1,0, λi,k = 0

27

Comparing with (21), (22), and (23) in the traditionalinformation filter, some formulas of the information filterwith intermittent measurements (IFIM) are reconstructedas follows:

P−1k = P−1

k/k−1 + 〠N

i=1λi,kHT

i,kR−1i,kHi,k, 28

x̂k = x̂k/k−1 + 〠N

i=1λi,kKi,k zk −Hi,kx̂k/k−1 29

Note that the gain Ki,k in (28) and (29) has the same gaincompared with the classical information filter. The IFIMalgorithm is summarized in Table 1.

5. Navigation Results

Numerical simulations are carried out to analyze the per-formance of the proposed IFIM navigation algorithm dur-ing Mars atmospheric entry, when the arrivals of partialmeasurements between the orbiters and the vehicle byusing UHF signal in the brownout period is described by aBernoulli distributed sequence with a continuous probability.The state variables of the vehicle during Mars atmosphericentry are defined as x = r θ λ v γ ψ T , and themeasurement information includes the acceleration a andthree ranges R1, R2, and R3.

In this work, the Mars entry initial conditions andparameters are listed in Table 2. Some state variable errorsare considered, which include an altitude error of 1 km, avelocity error of 20m/s, a longitude and latitude angle errorof 0.02 deg, and azimuth and FPA errors of 1 deg. Further-more, the reference atmospheric density, the ballistic coeffi-cient, and the L/D ratio are set as an arbitrary 10% modelparameter error. The bank angle ϕ is assumed to be zero.The IMU accelerometer bias ba used in the simulation isset as ba = 0 003, 0 003, 0 003 m/s2. We assumed thatthree orbiting beacons are available for the UHF signalcommunication during Mars atmospheric entry. The initialpositions and velocities of three orbiters in the Mars-centered Mars-fixed coordinate system are given in Table 3[13, 19]. The numerical simulation is terminated afterapproximately 250 s.

Table 1: Information filter with intermittent measurements.

State propagation x̂k/k−1 =Φk/k−1x̂k−1Covariance propagation P−1

k/k−1 =Ak−1 −Ak−1 Ak−1 +Q−1k−1

−1Ak−1

Covariance update P−1k = P−1

k/k−1 + 〠N

i=1λi,kHT

i,kR−1i,kHi,k

Gain Ki,k = PkHTi,kR−1

i,k

State update x̂k = x̂k/k−1 + 〠N

i=1λi,kKi,k zk −Hi,kx̂k/k−1


In this simulation, the communications blackout prob-lem between the entry vehicle and the orbiters is consideredas two parts, which are the brownout period and blackoutperiod. Since the antennas used for UHF communicationare preinstalled on the back shell of the entry vehicle asin Figure 1, the communication quality is affected by theelectron number density in the wake-region. In Section3.3, the probabilities of the arrival of navigation measure-ments are introduced by using the Rayleigh distributionwith the electron number density for the wake region inthe brownout period. The scale parameter τ of the Ray-leigh distribution is set as τ = 2 5 in the simulation tocharacterize the received signal probabilities of the entryvehicle, and the probabilities of the received signal inbrownout period during Mars atmospheric entry are shownin Figure 5. In addition, the probabilities of the receivedsignal in blackout period are set as 10−8, which is a verysmall probability to receive signals in the blackout periodas in literature [2].

Considering the communications brownout and black-out problem during Mars atmospheric entry, the numericalsimulations are run for three navigation schemes. One isthe default navigation scenario based on IMU measurementalone, because the IMU is the only available conventionalsensor and other sensors are impaired by the vehicle heatshield during Mars atmospheric entry. Another one is theinformation filter integrated navigation scenario based onthe IMU and range measurements without considering theloss of the range measurements when the communicationssuffered signal degradation during Mars atmospheric entry.

For this navigation scenario, the loss of the range measure-ment with a big order of magnitudes makes the measurementmodel seriously unmatch to generate large navigation errorsor even diverge. For this reason, the navigation errors may bediverge in the numerical simulation for this second naviga-tion scenario, when first communication signal cannot bereceived by the UHF antenna mounted on the vehicle backshell. So, the navigation results of the second scenarioare not shown in this paper. The last one is the IFIM inte-grated navigation scenario based on the IMU and rangemeasurements with arrival probabilities. When the rangemeasurements between the vehicle and the orbiters cannotarrive to navigation compute center in time, the rangemeasurement noise is set as infinity, which implies thatthe range measurement is not adopted and only informationprovided by the IMU is used.

Figure 7 shows the state estimate errors and its 3σ boundsof the information filter for the default IMU navigationscenario. Figure 8 shows the state estimate errors and the3σ bounds of the IFIM for the IMU/orbiter integrated navi-gation scenario. It is shown that the state errors of thealtitude, velocity, FPA and latitude of the information filterare not captured by their 3σ covariance bounds. Figure 8shows that the 3σ covariance bounds of all the states canalmost capture the corresponding errors. When the com-munication signals between the vehicle and the orbitersbegin to degrade, and the range measurements from theUHF communications cannot arrive in time, the state esti-mation error covariance continues to increase, becauseonly used IMU information is used. The covariance ofthe IFIM navigation algorithm reconverges again and stayssmall for the reminder of the numerical simulation sometime after reacquisition of the range measurements whenthe communications blackout disappears.

The state estimate root mean square error (RMSE) withlogarithmic scales in 1000 time simulations is introduced tocompare the performance of both the traditional informationfilter for default IMU navigation scenario and the IFIM forIMU/orbiter integrated navigation scenario. The RMSEs ofboth navigations are plotted in Figure 9. The all state RMSEsof the IFIM for IMU/orbiter integrated navigation scenarioare obviously smaller than those of the traditional informa-tion filter for the default IMU navigation scenario; the stateRMSEs of altitude and velocity using the traditional informa-tion filter are too large to meet the requirements of the futureMars pinpoint landing. For IMU/orbiter integrated naviga-tion scenario with the intermittent range measurements inMars communications brownout and blackout periods, theIFIM algorithm deals with the measurement partial lossproblem very well, and the final navigation accuracy canmeet future Mars missions.

6. Conclusions

An integrated navigation scenario with the probabilities ofthe radio range measurements in the communicationsbrownout period during the Mars atmospheric entry byusing the proposed IFIM is designed. The UHF signalattenuation per unit length versus electron number density

Table 2: Initial conditions and parameters during Mars entry.

Initial parameters Values Standard deviation

Altitude r 3522.20 km 1 km

Velocity v 5900m/s 20m/s

FPA γ −16 deg 0.5 deg

Longitude θ 0.00 deg 0.02 deg

Latitude λ 0.00 deg 0.02 deg

Azimuth ψ 5.156 deg 0.5 deg

Reference density ρ0 2.0× 10−4 kg/m3 2× 10−5 kg/m3

L/D ratio 0.156 0.016

Ballistic coefficient B 0.016m2/kg 0.0016m2/kg

Table 3: Positions and velocities of orbiters.

Beacon Initial positions (km) Initial velocities (m/s)

Orbitingbeacon 1

(780.44, 13276.13, 1267.20) (−317.7, −50, 1697.2)

Orbitingbeacon 2

(3492.70, 3659.20, 215.11) (−605.1, 97, 3232.8)

Orbitingbeacon 3

(3927.85, −230.90, 374.91) (93.6, 3120.8, −584.1)


is analyzed. The vehicle will suffer about 17 s brownoutperiod and about 40 s blackout period with an altitudeand velocity profiles. In the brownout period, the intermit-tent range measurements are modeled by a Bernoulli dis-tributed sequence with different arrival probabilities,which distributed to Rayleigh distribution of the electronnumber density around the entry vehicle for the UHF. To

deal with the potential missing range measurements andthe IMU information, the information filter with multipleintermittent measurements is derived. In fact, the measure-ments are partial intermittent in simulation, because theIMU measurements always arrive in time and only therange measurements are lost at a certain probability. Thenumerical simulations under the integrated navigation

0 50 100 150 200 250‒500

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Figure 8: State estimate errors and 3σ bounds of IFIM for IMU/orbiter integrated navigation scenario.

IF IMU3�휎 bounds

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Figure 7: State estimate errors and 3σ bounds for the default IMU navigation scenario.


scheme based on IMU and three orbiters are carried out.The results show that the IFIM navigation filtering has agood performance to deal with partial intermittent rangemeasurements in the communications brownout period.

Conflicts of Interest

The authors declare no conflict of interest.

Acknowledgments

The work described in this paper was supported by theNational Nature Science Foundation of China (Grantnos. 61603346 and 61603347), the Fundamental ResearchPrograms for the Provincial Universities (Grant no.15KYYWF01), the key research projects of Henan highereducation institutions (Grant no. 18A413003), and theDoctor Startup Funds Foundation of Zhengzhou Universityof Light Industry (Grant no. 2015BSJJ026). The authors fullyappreciate the financial support.

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