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Terrestrial GPS Time-Differenced Carrier-Phase
Positioning of Lunar Surface Users
Keidai Iiyama
Aeronautics and Astronautics
William F.Durand Building
496 Lomita Mall
Stanford, CA 94305
kiiyama@stanford.edu
Sriramya Bhamidipati
Aeronautics and Astronautics
William F.Durand Building
496 Lomita Mall
Stanford, CA 94305
sriramya@stanford.edu
Grace Gao
Aeronautics and Astronautics
William F.Durand Building
496 Lomita Mall
Stanford, CA 94305
gracegao@stanford.edu
Abstract— To support the increasing interest in lunar explo-
ration, the future lunar users will require precise position,
velocity, and timing (PVT) services on the Moon. Active ef-
forts are being invested in utilizing terrestrial GPS signals at
lunar distances to achieve precise lunar surface localization.
Compared to conventional pseudoranges of terrestrial GPS
satellites, carrier phase measurements are 1000 times more
accurate when integer ambiguities (unknown number of cycles)
are accurately resolved. We propose a precise localization
technique for lunar users, wherein we design a tightly-coupled
Kalman filter framework that utilizes the terrestrial GPS time-
differenced carrier phase (TDCP) measurements and inertial
measurement unit (IMU)-based motion dynamics. To account
for the time-correlation across TDCP measurements, we design
an augmented state vector that incorporates consecutive rover
states and perform fixed point smoothing before each measure-
ment update. Through high-fidelity Monte-Carlo simulations,
we validate that our framework not only isolates the TDCP
measurements suffering from cycle slips but also demonstrates
precise localization on the lunar surface.
TABLE O F CONTENTS
1. INTRODUCTION......................................1
2. TERRESTRIAL GPS TDCP FORMULATION.........2
3. PRO PO SED TDCP/IMU KA LM AN FILTER ..........2
4. EXPERIMENT SETUP AND RES ULTS .................4
5. CON CLU SI O NS .......................................7
ACKNOWLEDGMENTS ..................................7
REFERENCES ...........................................8
BIOGRAPHY ............................................8
1. INTRODUCTION
To support the increasing interest for human and robotic ex-
ploration, the future lunar users (e.g., rovers, astronauts) will
require position, velocity, and timing (PVT) services on the
Moon. A precise PVT knowledge is crucial for various lunar
applications, including mission activity planning, search-and-
rescue, and geotagging of scientific payload samples. The
Global Exploration community has expressed the need to
achieve a stringent positioning accuracy of less than 50 m on
the Moon within the next 10 years [1].
Given these stringent requirements, it is worthwhile to inves-
tigate if we can harness the signals already broadcast by the
legacy terrestrial GPS, to provide precise lunar surface local-
ization. The transmit antennas of terrestrial GPS satellites
point towards Earth, thus, only weak satellite signals can be
received at lunar users (from the side lobes and the main lobe
not occluded by Earth and the Moon).
978-1-6654-9032-0/23/$31.00 ©2023 IEEE
Prior works on navigation of lunar satellites [2] and landers
[3] successfully developed receiver algorithms using pseu-
dorange and pseudorange rate measurements from terrestrial
GPS signals. Regarding lunar surface navigation, previous
works [4, 5] have investigated the use of additional infras-
tructure, including lunar base stations on the Moon and lunar
satellites in orbit, for aiding terrestrial GPS pseudoranges to
achieve precise positioning. However, these approaches re-
quire the setup and maintenance of lunar infrastructure, which
incurs additional costs. In addition, navigation methods that
rely on lunar base stations could not be used when communi-
cation links to the base stations could not be secured, which
could reduce operation flexibility when exploring through
intricate terrains. Thus, to improve positioning accuracy
without using additional infrastructures, it would be benefi-
cial to explore leveraging carrier phase measurements that are
1000 times more accurate than pseudoranges when integer
ambiguities (unknown cycle ambiguities) are correctly fixed.
We propose a precise positioning technique for lunar users,
wherein the lunar rovers will harness the intermittently avail-
able terrestrial GPS time-differenced carrier phase (TDCP)
values and the motion dynamics based on inertial measure-
ment unit (IMU) measurements. In theory, the TDCP for-
mulation is beneficial in removing the need for estimating
unknown integer ambiguities, which can be a computation-
ally expensive process. Furthermore, the TDCP implemen-
tation also eliminates the slow-varying signal errors caused
by terrestrial GPS satellite clock and relativistic effects. To
the best of our knowledge, this is the first work to apply
TDCP techniques for lunar surface positioning. Our proposed
technique provides a standalone and absolute PVT solution.
Our key contributions are listed as follows:
1. To account for time-correlation in TDCP measurements
of an terrestrial GPS satellite, we propose a tightly-coupled
TDCP/IMU Kalman filter framework with an augemented
state vector that encompasses two consecutive rover states.
2. We further address time-dependency across TDCP mea-
surements by performing a fixed point smoothing of the
lagged rover states using future IMU and terrestrial GPS
measurements. We also account for the different sampling
rate between the IMU and the terrestrial GPS signals.
3. We perform cycle slip detection via analysis of the post-
measurement TDCP residuals along with their covariances.
4. Through Monte-Carlo simulations, we validate that our
method demonstrates precise lunar localization, which meets
the accuracy target set by the Global Exploration community.
The rest of the paper is organized as follows. Section 2 de-
scribes our terrestrial GPS TDCP formulation and Section 3
discusses our proposed framework. Section 4 explains our
simulation setup and results. Section 5 provides conclusions.
1
2. TERRESTRIAL GPS TDCP FORMULATION
While the carrier phase measurements could provide more
precise measurements compared to pseudorange (i.e. ϵi
ϕ<
ϵi
ρ), the drawback is that the unknown integer ambiguity,
Nihas to be accurately fixed to use them. Resolving inte-
ger ambiguity is often done using Least-squares AMBiguity
Decorrelation Adjustment (LAMBDA) method [6] or via
an Extended Kalman Filter. However, integer ambiguity
resolution are known to be difficult in a non-differential mode
and poor GPS geometry scenario, as considered in this paper.
However, under the condition that Nis a constant over time,
which is true except when cycle slip occurs, Ncould be
eliminated by differencing the carrier phase measurements
between two timesteps as shown in Figure.1.
Ter r es t ri a l G PS
Satellite (i)
Lunar Rover
!!"# $ %&' !!"#'
%"()
#
)!("# $ %&' )!("#'
)
#("#( $ %&' )
#("#'
Figure 1: TDCP measurement formulation associated with
any terrestrial GPS satellite iover sample interval ∆t.
Considering ∆tto be the sample interval, the TDCP mea-
surement ∆Tϕiusing carrier phase values at timesteps kand
(k−∆t), i.e., ϕi
kand ϕi
(k−∆t), respectively, is given by
∆Tϕi=ϕi
k−ϕi
(k−∆t)(from [7])
=
rik−ru,k·
eiu,k
−
ri(k−∆t)−ru,(k−∆t)·
eiu,(k−∆t)
+ ∆T(cτu)−∆T(cτi)+∆T˜ϵi
ϕ
(1)
where, eliminating subscripts kand (k−∆t)for sake of
simplicity, ruis the user’s position in the local East-North-
Up (ENU) frame with origin as the Moon’s South Pole, τu
is the receiver’s clock bias, and cis the speed of light. For
any terrestrial GPS satellite i,
riis the satellite position,
ei
uis the unit line-of-sight (LOS) vector from the user to
the GPS satellite iand τiis the satellite clock error. Also,
∆Tis sufficiently small such that the time difference in
the tropopheric, ionospheric, and multipath related terms in
standard carrier phase equation [7] are considered negligible,
and thereby, absorbed into the residual noise term ∆T˜ϵi
ρ.
Additionally, we utilize terrestrial GPS-based pseudorange
and pseudorange rate measurements in our proposed filter,
whose standard equations are taken from [7].
3. PRO POSE D TDCP/IMU KALMA N FILTER
The high-level architecture of the proposed tightly-coupled
filter framework is shown in Fig. 2, wherein we execute the
time update step based on IMU, and measurement update step
via terrestrial GPS-based pseudorange, pseudorange rate, and
TDCP measurements. We also perform cycle slip detection in
carrier phase measurements during the measurement update
step. The details of each process will be explained in the
following subsections.
State Representation
As defined in Eq. (2), the rover state ˜xk∈R12 includes
position ru∈R3, velocity vu∈R3, attitude ψu∈R,
accelerometer bias ba∈R2, gyroscope bias bω∈R, receiver
clock bias τu∈Rand clock drift ˙τu∈R. We consider
user to navigate on the lunar surface, thus, our attitude state
comprises only rover heading. The position ruand velocity
vuare in the local ENU frame.
˜xk= [ru,k vu,k ψkτu,k ˙τu,k ba
kbw
k]⊤(2)
To maintain the Markov property under the existence of time-
correlated TDCP measurements, the following augmented
state is estimated in the filter
xk=˜xk
˜xtGP S (k)∈ R24×1(3)
where kis the timestep when IMU is received while tGPS (k)
is when the last terrestrial GPS measurement was taken.
Prediction Model and Time Update
The linearized dynamics of the augmented state xkis
xk+1 =˜xk
˜xtGP S(k)=Akxk−1+ϵk(6)
E[ϵkϵ⊤
k] = Qk(7)
where Ak=˜
Ak012×12
012×12 I12 , the state transition ma-
trix ˜
Akis defined in Eq. (4), an n×nidentity matrix is In, an
n×marray of zeros is 0n×m,Qk=˜
Qk012×12
012×12 012×12,
and the process noise covariance matrix ˜
Qkin Eq. (5).
Measurement Update
Ter re st r ia l
GPS
Receiver
Augmented State and Covariance
Re-initialization
Cycle Slip Detection
IMU Time Update
Accelerator
&Gyro
Measurements
State%and%Covariance%Estimate 𝑥
2!"#, 𝑃
4!"#
Pseudoran ge,
Pseudoran ge-rate,
TDCP
Predicted State and Covariance
GPS available ?
State and Covariance%
Estimate
𝑥2!, 𝑃
4!
Yes
No
To
Next
Time
Step
Figure 2: Flowchart of our tightly-coupled filter architecture,
wherein time update is performed using IMU, while measure-
ment update is executed via the use of terrestrial GPS-based
measurements (if at least one satellite is visible).
2
˜
Ak=
I2∆tIM U I2
1 ∆tIM U
I2
dCE
B
dt ∆tIM U (ˆaB−ba)−∆tI MU CE
B
11−∆tIM U
1 ∆tIM U
1
exp −∆tIM U
TaI2
exp −∆tIM U
Tw
(4)
˜
Qk=
07×7
Qclk QaI2
Qω
+BQbaI2
QbωB⊤, B =
03×203×1
∆tIM U CE
B02×1
01×20
01×2∆tIM U
05×205×1
∈ R12×3(5)
In Eq. (4), ˆaBis the IMU-measured acceleration of the body
frame relative to the inertial frame, Ta, Tωis the correlation
time of the first order Gauss Markov processes of the ac-
celerometer and gyro random bias, respectively, ∆tIM U is
the IMU sampling interval, and CE
B=cos ψ−sin ψ
sin ψcos ψis
the transformation matrix from body frame to the ENU frame.
In Eq. (5), Qaand Qωdenote the covariance of IMU-
measured acceleration and angular velocity of the body frame
relative to the inertial frame, while the covariances of the
accelerometer and gyroscope biases [8], are modeled as
Qba =2B2
alog 2
π(0.4365)2Ta
∆tIM U
Qbω =2B2
wlog 2
π(0.4365)2Tw
∆tIM U
(8)
where Baand Bwis the bias instability of the accelerometer
and the gyroscope, respectively.
Based on the dynamics model above, we perform the time
update step using standard equations of Kalman filter []. We
denote ¯xkand ¯
Pkas the predicted augmented state vector
and its covariance at time step k, respectively, while ˆxk−1
and ˆ
Pk−1is the estimated augmented state vector and its
covariance at time step k−1, respectively.
Measurement Model and the Measurement Update
Our proposed filter processes three types of measurements,
which include pseudorange ρi, pseudorange rate ˙ρi, and
TDCP ∆Tϕi. Let zkthe stacked measurements obtained at
time step kas follows:
zk=ρ1
k,· · · , ρm
k,˙ρ1
k,· · · ,˙ρm
k,∆Tϕ1
k,· · · ,∆Tϕm′
k⊤
(9)
where mand m′is the number of available GPS signals and
time-difference carrier phase signals, respectively.
The residual of the measurement ˆykcould be calculated as
ˆyk=zk−h(¯xk)(10)
where h(¯xk)is the expected measurement for predicted
augmented state ¯xk, defined as a stack of the measure-
ment equations related to pseudorange, pseudorange rate and
TDCP (derived in Eq. (1)), respectively.
We perform measurement update step utilizing the standard
equations of Kalman filter [9], wherein the linearized mea-
surement model Hkand the measurement noise covariance
matrix Rkare given in Eq. (11) and Eq. (12), respectively. We
now explain how each term, which is related to measurement
model and measurement noise covariance matrix.
Pseudorange measurements—The Jacobian Hi
ρfor any ter-
restrial GPS satellite i(shown in Eq. (11)) with respect to the
augmented state is written as
Hi
ρ=h−
eiu⊤01×4101×16i.(14)
We also model the pseudorange residual measurement vari-
ance σ2
ρ,ki(shown in Eq. (12)) based on the receiver delay
lock loop (DLL) tracking error σ2
DLLki. More details
regarding this covariance modeling can be found in [10–12].
Pseudorange rate measurement — The Jacobian Hi
˙ρ,k with
respect to augmented state (shown in Eq. (11)) is given by
Hi
˙ρ,k =hHvr,k −
eiu⊤01×5101×15i(15)
where
Hvr,k =−
viu,k −
viu,k ·
rik−ru,k·
rik−ru,k⊤
∥
rik−ru,k∥3
(16)
We model the uncertainty σ2
˙ρ,kiin pseudorange rate residu-
als of the terrestrial GPS satellite i(shown in Eq. (12)) based
on the variance in the receiver frequency lock loop (FLL)
tracking loop σ2
FLLki. For more details, refer to [10–12].
3
Hk=H1
ρ,k,· · · , H m
ρ,k, H 1
˙ρ,k,· · · , H m
˙ρ,k, H 1
∆Tϕ,k,· · · , H m′
∆Tϕ,k⊤
(11)
Rk=diag σ2
ρ,k1,· · · ,σ2
ρ,km,σ2
˙ρ,k1,· · · ,σ2
˙ρ,km,σ2
∆Tϕ,k1,· · · ,σ2
∆Tϕ,km′(12)
Hi
∆Tϕ,k =h−(
eiu,k)⊤01×4101×4(
eiu,tGP S (k))⊤01×4−101×4i(13)
TDCP measurement—The Jacobian Hi
∆Tϕ,k with respect to
the augmented state (shown in Eq. (11)) is defined in Eq. (13).
From [11], we model the uncertainty σ2
∆Tϕ,kiin the TDCP
residuals of the terrestrial GPS satellite i(shown in Eq. (12))
based on the variance in PLL tracking loop (σ2
PLLkias
σ2
∆Tϕ,ki=E[∆T˜ϵi
ϕ,k∆T˜ϵi
ϕ,k⊤]
= 2 c2
4π2f2
L1 σ2
PLLki(17)
A multiplicative factor 2indicates that the error is doubled to
consider the variance of time-differenced measurement. It is
assumed that the two consecutive carrier phase measurements
are not correlated with each other.
Filter Re-Initialization
We re-initialize the augmented state vector and covariance
after the measurement update to process the next TDCP
measurements. Let the augmented state and covariance after
the measurement update at time kbe
ˆxk=ˆ
˜xk
ˆ
˜xsand ˆ
Pk=ˆ
Pkk ˆ
Pks
ˆ
Psk ˆ
Pss ,(18)
where ˆ
Pkk ,ˆ
Pks,ˆ
Psk,ˆ
Pss ∈R12×12 is the previous timestep
where GPS measurement was sampled (s=tGP S (k)< k).
The re-initialized augmented state to process the next TDCP
measurement between timestep s′(s′> k)and kis
ˆxk,new =ˆ
˜xk
ˆ
˜xk,and ˆ
Pk,new =ˆ
Pkk ˆ
Pkk
ˆ
Pkk ˆ
Pkk (19)
at time step kafter the measurement update. The augmented
state ˆxk,new ,ˆ
Pk,new is propagated via time update steps until
the next terrestrial GPS measurement at timestep s′. As
pointed in reference [13], this procedure could be viewed as a
fixed point smoothing of the lagged state xkusing the future
IMU and GPS measurements at timestep k+ 1, . . . , s′.
Cycle Slip Detection
To correctly process TDCP measurements, cycle slips should
be detected and discarded. For each TDCP measurement
i∈1,...m′, we perform the cycle slip detection by
comparing the normalized residual metric, which is formu-
lated as the ratio of the squared post-measurement resid-
ual ˆyi
∆Tϕ,k2(based on Eq. (10)) and its expected covari-
ance S∆Tϕ,ki,i ∈Rm′×m′. We discard the measurement i
if the normalized residual metric is greater than the pre-
defined threshold γ2
cs. The selection of γcs depends on the
desired balance between sensitivity and false alarm rate.
4. EXPERIMENT SETUP AND RESU LTS
We validate our proposed TDCP navigation framework using
a MATLAB simulation of lunar rover equipped with a terres-
trial GPS receiver, an onboard clock, and an IMU.
Experiment Setup
In the baseline scenario, we simulate the navigation perfor-
mance of the rover traveling around the Rima Bode region,
located at 12.0◦N and 3.5◦W. The rover follows a circular
trajectory with a radius of 1200 m, at a constant velocity of
1m/s. The starting epoch of the simulation was set to 2022
August 01, 01:00:00, and the simulation time is set to 2 hours.
To model the transmit antenna onboard the terrestrial GPS
satellites in our simulation setup, we utilize the transmit
power and antenna gain patterns of the L1 C/A signals
from the NASA Antenna Characterization Experiment (ACE)
study [14] for Block II-F, and the original data by Lockheed
Martin For Block IIR and IIR-M satellites [15]. For the
GPS receiver, we consider a high-gain steering antenna with
14 dBi at 0◦off-boresight angle, and a 3dB beamwidth
of 12.2◦, representative of the current design of the Pretty
CubeSat mission [16].
The visibility of the GPS signal is calculated considering the
blockage from the Earth and Moon, receiver C/N0threshold
of 15 dB-Hz, and constant elevation mask of 5 ◦. Figure 3
shows the number of terrestrial GPS signals that can be
tracked, their skyplots as seen from the rover moving at
1m/s, and the HDOP, VDOP, TDOP, GDOP values, which
are respectively Horizontal, Vertical, Time, and Geometric
Dilution of Precision. The satellites are clustered in a similar
direction seen from the rover, which worsens the GDOP and
creates a challenging scenario for the rover, given the need
for precision navigation.
We sample the position and velocity of the terrestrial GPS
satellites at a rate of 10 Hz, and the IMU measurements at a
rate of 100 Hz. Correspondingly, we design our filter with an
update interval of ∆t= 0.01 s. Parameters of the Microsemi
CSAC [17] is used for the on-board clock modeling and
parameters of the LN-200S, which is a tactical-grade IMU
for space applications is used for the IMU modeling.
To evaluate the navigation performance of the filter, the root-
mean-square (RMS) and the 68, 95 percentile value of the
position, velocity, and clock bias estimate error were calcu-
lated. We assess the mean performance over 24 Monte-Carlo
simulations with different clock motion and state estimate
initializations. The initial state estimation error was set to
100 mfor the horizontal position, 10 mfor the altitude,
0.1 mfor the velocity, 100 mfor the clock bias (converted
to meters by multiplying light speed), and 0.1 m/sfor the
clock drift. Each performance metric was calculated using
the last 1/4th duration of all the Monte-Carlo runs to evaluate
4
20 40 60 80 100 120
T [min]
2
3
4
5
6
7
8
9
10
Visible Sats
(a) The number of visible GPS satellites
0
30
60
90
120
150
180
210
240
270
300
330
90
75
60
45
30
15
0
(b) Skyplot of the GPS satellites
20 40 60 80 100 120
T [min]
0
500
1000
1500
2000
2500
3000
3500
4000
DOP
HDOP
VDOP
TDOP
GDOP
(c) The DOP values of the satellites
Figure 3: Satellite visibility from the rover moving around a circular trajectory around Shackleton crater at 1 m/s; (a) the
number of tracked terrestrial GPS satellites (b) the skyplot of the visible terrestrial GPS satellites (c) the Dilution of Precision
(DOP) values of the satellites. Note that the DOP values are not calculated when less than 4 satellites are visible
the navigation performance after the convergence.
Performance of the Baseline Scenario
The result of the baseline scenario is summarized in Table 1,
and the transition of the estimation error and covariance of the
rover position and velocity are shown in Fig. 4. From Fig. 4,
we could observe that the estimation error in the clock bias
and the altitude (RMS: 33.7 m) is larger than the horizontal
positioning error because of the unfavorable geometry, as rep-
resented in the VDOP and TDOP values in Fig.3. Therefore,
as a future direction adding an altitude constraint using a
lunar Digital Elevation Model (DEM) [10] could be effective
in mitigating unfavorable satellite geometry and improving
navigation accuracy.
Cycle Slip Detection
The cycle slip detector was tested by randomly adding cycle
slips to 30%of the simulated carrier phase signals. Each
cycle slip was simulated as an instantaneous change in the
integer ambiguity term Nfrom the previous timestep. The
number of cycles that changes from the previous time step
when a cycle slip occurs was limited to either -1 or 1.
The navigation performance for the scenario with simulated
cycle slips is shown in Table 2. The estimation accuracy
is significantly degraded if the cycle slip detector was not
incorporated into the filter. In contrast, our proposed cycle
slip detector is able to avoid degradation by successfully
removing the outliers.
Measurement Combinations
Different measurement combinations are tested to investigate
the contribution of each to the navigation accuracy, as fol-
lows: 1) Pseudorange only; 2) Pseudorange + Pseudorange
rate; 3) TDCP only; and 4) Pseudorange + Pseudorange rate
+ TDCP.
The results are summarized in Table 3. As expected, adding
pseudorange rate measurement and/or TDCP measurements
to the pseudorange measurement improves the velocity esti-
mates. Especially, compared to only pseudorange + pseudo-
range rate solution, the addition of the TDCP measurement
reduces the 95 percentile value of the positioning error by
21.1% and the velocity estimation error by 40.0%. However,
when TDCP is used alone without the aid of the pseudorange
measurement, the position estimate could not be bounded,
resulting in a large positioning error estimate despite the good
velocity estimate achieved.
Performance at Other Landing Sites on the Moon
The achievable positioning accuracy using the terrestrial GPS
signals depends on the geometry of the terrestrial GPS satel-
lites seen from the rover. To investigate the achievable po-
sitioning performance by the proposed algorithm at different
locations on the Moon with different satellite visibility, we
conducted navigation simulations on several potential landing
site options across the surface of the Moon. Four landing
sites are selected from the high-priority landing site options
proposed in reference [18], which would meet the scientific
goal in a wide variety of areas. Fig.5 and Table.4 shows the
four selected landing site options.
Shackleton
Rima Bode
Marius Hills
Gruithuisen domes
Figure 5: Four target landing sites where the navigation
performance was investigated
For each simulation at each location, parameters are set the
same as the baseline scenario. In particular, the rover is
equipped with CSAC and a tactical grade IMU and follows
a circular trajectory with a radius of 1200 mwith a constant
velocity of 1 m/s. The initial epoch of the simulation is set
to August 01, 01:00:00. Cycle slips and their detection are
ignored in each simulation.
The number of visible terrestrial GPS satellites and the HDOP
values at each landing site are shown in Fig.6. The HDOP
5
Table 1: Baseline settings and results
Landing Site IMU Clock Rover Horizontal Position [m] 3D Velocity [m/s] Clock Bias [m]
velocity RMS 68 %95 %RMS 68 %95 %RMS 68 %95 %
Rima Bode Tactical CSAC 1.0 m/s9.4 8.2 17.2 0.07 0.07 0.12 14.7 16.0 26.0
Figure 4: For one Monte-Carlo run of the baseline scenario at the Rima Bode region: The estimation error (red line) and 3σ
covariance of rover position and velocity.
Table 2: Navigation results (24 Monte-Carlo runs) when cycle slips are added to the simulator. The estimation accuracy is
significantly degraded if the cycle slip detector is not incorporated in the filter, while our proposed cycle slip detector is able to
avoid the degradation by successfully removing the outliers.
Cycle Slip Rate Cycle Slip Detector Horizontal Position [m] 3D Velocity [m/s] Clock Bias [m]
RMS 68 %95 %RMS 68 %95 %RMS 68 %95 %
0%N/A 9.4 8.2 17.2 0.07 0.07 0.12 17.1 18.9 29.1
30%OFF 33.9 34.7 61.5 0.36 0.36 0.67 68.8 65.6 129.9
30%ON 10.2 9.2 20.2 0.08 0.08 0.15 18.3 20.2 31.0
Table 3: Navigation results (24 Monte-Carlo runs) for different measurement combinations. Other parameters were aligned
with the baseline scenario. Adding pseudorange rate and TDCP measurements improved the velocity estimates. Compared to
the pseudorange + pseudorange rate solution, the addition of the TDCP measurement reduced the 95 %positioning error by
21.1% and the 95 %velocity estimation error by 40.0%. However, when TDCP is used alone without the aid of the pseudorange
measurement, the position estimate could not be bounded, resulting in poor positioning accuracy.
Measurement Combinations Horizontal Position [m] 3D Velocity [m/s] Clock Bias [m]
RMS 68 %95 %RMS 68 %95 %RMS 68 %95 %
Pseudorange only 15.3 13.6 35.7 0.18 0.18 0.34 17.1 18.9 29.1
Pseudorange + Pseudorange rate 11.0 10.7 21.8 0.11 0.12 0.20 14.8 15.8 26.2
TDCP only 45.8 51.5 81.9 0.08 0.8 0.13 133.3 116.4 311.7
Pseudorange + Pseudorange rate + TDCP 9.4 8.2 17.2 0.07 0.07 0.12 14.7 16.0 26.0
Table 4: Four target landing sites on Moon where the navigation performance was investigated
Name Latitude Longitude Potential scientific targets
Shackleton Crater 89.9◦S 0.0◦E Water Ice
Rima Bode 12.0◦N 3.5◦W Extensive, high-Ti pyroclatic deposit
Marius Hills 14.1◦N 56.7◦W Lava tubes
Gruithuisen Domes 36.5◦N 40.2◦W Highly silicic deposits
value becomes smaller as the landing site becomes closer
to the equator and the prime meridian because the elevation
of the terrestrial GPS satellites seen from the rover becomes
higher.
The result of the simulation is summarized in Table.5. The
horizontal positioning error is the smallest at Rima Bode, and
the largest at Shackleton, which aligns with the comparison
of the HDOP values among the landing sites. Regarding the
velocity, estimation errors below 0.1 m/sin RMS is obtained
at all four observation sites, thanks to the highly accurate
TDCP measurements.
6
Table 5: Navigation results (24 Monte-Carlo runs) for each target landing site. ’PR’ in the measurement column stands for
the pseudorange measurement. For each case, the rover operates at the constant velocity of 1 m/s, and is equipped with a
CSAC and tactical grade IMU. The best horizontal positioning performance was achieved at Rima Bode, which is closet to the
equator and the prime meridian. Velocity estimates under 1 m/scould be obtained for all four landing sites thanks to the highly
accurate TDCP measurements.
Landing Site Measurement Horizontal Position [m] 3D Velocity [m/s] Clock Bias [m] Mean
RMS 68 %95 %RMS 68 %95 %RMS 68 %95 %GPS No.
Shackleton PR only 67.3 66.1 135.2 0.18 0.18 0.32 66.7 65.4 135.2 6.40
PR + TDCP 64.8 61.3 129.8 0.07 0.07 0.12 64.9 61.1 130.4
Rima Bode PR only 15.3 13.6 35.7 0.18 0.18 0.34 17.1 18.9 29.1 7.52
PR + TDCP 9.4 8.2 17.2 0.07 0.07 0.12 14.7 16.0 26.0
Marius Hills PR only 16.9 18.3 28.8 0.19 0.20 0.36 14.1 15.3 25.6 7.50
PR + TDCP 12.0 12.8 21.6 0.09 0.08 0.19 12.3 13.3 22.9
Gruithuisen Domes PR only 50.4 53.7 63.3 0.18 0.17 0.35 60.1 63.9 75.7 7.51
PR + TDCP 47.6 50.9 61.2 0.07 0.07 0.13 54.3 58.1 69.9
0 20 40 60 80 100 120
Time [min]
0
1
2
3
4
5
6
7
8
9
10
Visible Sat No.
Shackleton
MariusHills
Rima-Bode
Gruithuisen-Domes
(a) The number of visible terrestrial GPS satellites at each landing sites
0 20 40 60 80 100 120
Time [min]
0
500
1000
1500
2000
2500
3000
HDOP
Shackleton
MariusHills
Rima-Bode
Gruithuisen-Domes
(b) HDOP at each landing sites
Figure 6: The number of terrestrial GPS satellites and Horizontal Dilution of Precision (HDOP) at each landing sites. HDOP
is not calculated when the number of visible satellites falls below 4. Fewer terrestrial GPS satellites could be seen from the
Shackleton Crater compared to other landing site options closer to the equator.
5. CONCL USI O NS
We design a tightly-coupled Kalman filter framework for
lunar surface positioning that utilizes terrestrial GPS TDCP
measurements and motion dynamics from IMU. The devel-
oped filter accounts for the time-correlation across TDCP
measurements by introducing an augmented state vector that
incorporates consecutive rover states and performing fixed
point smoothing before each measurement update. Addi-
tionally, the filter discards any corrupted TDCP measure-
ments (cycle slips) by comparing the post-measurement
residuals and their estimated covariance.
The performance of our TDCP/IMU tightly-coupled Kalman
Filter was validated through high-fidelity Monte-Carlo sim-
ulations. Our work successfully isolated the TDCP mea-
surements suffering from cycle slips, and also demonstrated
horizontal positioning RMS error under 10 mfor the baseline
scenario where a rover equipped with a tactical grade IMU
and CSAC operates at the velocity of 1 m/sat the Rima Bode
region near the equator. The results of the navigation simula-
tion with different measurement combinations indicated that
by using TDCP measurements, we could reduce the 95 % po-
sitioning error by 21.1% and the 95 % velocity error by 40%,
compared to using only pseudorange and pseudorange rate
measurements. In addition, navigation simulations on several
candidate target landing sites were conducted to test the ap-
plicability of the proposed framework over the lunar surface.
The results showed that despite the challenging environment
to perform terrestrial-GPS-based navigation, horizontal RMS
positioning estimates below 70 mcould be achieved at the
Shackleton crater near the south pole.
Overall, simulation results display promising positioning per-
formance to meet the target set by the Global Exploration
community. Moreover, the improvement in the velocity esti-
mate accuracy by using the TDCP measurements is important
for both scientific and engineering applications on the Moon,
such as the estimation of rover slip rate and surface features
of the unstructured lunar terrain.
Future research areas include incorporating terrain effects on
terrestrial GPS visibility analysis and rover motion simula-
tion, adding digital elevation model (DEM)-based altitude
constraints, and leveraging lunar base stations for improved
navigation performance.
ACKNOWLEDGMENTS
We would also like to thank Daniel Neamati for reviewing
this paper.
7
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BIOGRAPHY[
Keidai Iiyama is a Ph.D. student in
the Department of Aeronautics and As-
tronautics at Stanford Univerisity. He
received his M.E. degree in Aerospace
Engineering in 2021 from the University
of Tokyo, where he also received his B.E.
in 2019. His research is on positioning,
navigation, and timing of lunar space-
craft and rovers, and system designs for
lunar navigation systems.
Sriramya Bhamidipati is a postdoc-
toral scholar in the Aeronautics and As-
tronautics Department at Stanford Uni-
versity. She received her Ph.D. in
Aerospace Engineering at the University
of Illinois, Urbana-Champaign in 2021,
where she also received her M.S in 2017.
She obtained her B.Tech. in Aerospace
from the Indian Institute of Technology,
Bombay in 2015. Her research interests
include GPS, power and space systems, artificial intelligence,
computer vision, and unmanned aerial vehicles.
Grace Gao is an assistant professor
in the Department of Aeronautics and
Astronautics at Stanford University. Be-
fore joining Stanford University, she was
an assistant professor at University of
Illinois at Urbana-Champaign. She ob-
tained her Ph.D. degree at Stanford Uni-
versity. Her research is on robust and
secure positioning, navigation, and tim-
ing with applications to manned and un-
manned aerial vehicles, autonomous driving cars, as well as
space robotics.
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