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Citation: Deng , H.; Wang, H.; Liu, Y.;
Jin, Z. Short-Arc Horizon-Based
Optical Navigation by Total
Least-Squares Estimation. Aerospace
2023,10, 371. https://doi.org/
10.3390/aerospace10040371
Academic Editor: Xiaofeng Wu
Received: 7 March 2023
Revised: 7 April 2023
Accepted: 11 April 2023
Published: 13 April 2023
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aerospace
Article
Short-Arc Horizon-Based Optical Navigation by Total
Least-Squares Estimation
Huajian Deng 1,2,† , Hao Wang 1,2,∗,† , Yang Liu 1,2 and Zhonghe Jin 1,2
1Micro-Satellite Research Center, School of Aeronautics and Astronautics, Zhejiang University,
Hangzhou 310027, China; denghuajian@zju.edu.cn (H.D.)
2Zhejiang Key Laboratory of Micro-Nano Satellite Research, Hangzhou 310027, China
*Correspondence: roger@zju.edu.cn; Tel.: +86-137-7741-8785
† These authors contributed equally to this work.
Abstract:
Horizon-based optical navigation (OPNAV) is an attractive solution for deep space explo-
ration missions, with strong autonomy and high accuracy. In some scenarios, especially those with
large variations in spacecraft distance from celestial bodies, the visible horizon arc could be very
short. In this case, the traditional Christian–Robinson algorithm with least-squares (LS) estimation is
inappropriate and would introduce a large mean residual that can be even larger than the standard
deviation (STD). To solve this problem, a simplified measurement covariance model was proposed
by analyzing the propagation of measurement errors. Then, an unbiased solution with the element-
wise total least-squares (EW-TLS) algorithm was developed in which the measurement equation
and the covariance of each measurement are fully considered. To further simplify this problem, an
approximate generalized total least-squares algorithm (AG-TLS) was then proposed, which achieves
a non-iterative solution by using approximate measurement covariances. The covariance analysis
and numerical simulations show that the proposed algorithms have impressive advantages in the
short-arc horizon scenario, for the mean residuals are always close to zero. Compared with the
EW-TLS algorithm, the AG-TLS algorithm trades a negligible accuracy loss for a huge reduction in
execution time and achieves a computing speed comparable to the traditional algorithm. Further-
more, a simulated navigation scenario reveals that a short-arc horizon can provide reliable position
estimates for planetary exploration missions.
Keywords: optical navigation; horizon sensor; total least-squares estimation
1. Introduction
Autonomous navigation is being widely valued with the bloom of deep space explo-
ration missions [
1
,
2
]. It is an effective supplement and backup for ground-based tracking
to reduce ground maintenance costs, overcome communication delays, and prevent com-
munication loss [
3
]. Optical navigation is an image-based solution with strong autonomy,
high accuracy, and good real-time performance [
4
]. It has been successfully applied to
interplanetary exploration missions such as Apollo [
5
], Deep Space 1 [
6
], and SMART-1 [
7
]
and will soon be applied to Orion missions [8].
OPNAV comes in many forms, including extracting the line-of-sight direction of
natural or artificial targets [
9
–
11
], using the horizon of a celestial body [
12
–
15
], and tracking
the landmarks on the body’s surface [
16
–
19
]. The most appealing solution for a planetary
exploration mission is the horizon-based OPNAV, which guarantees a good balance of
accuracy and complexity. The position of the spacecraft can be determined by using the
planet’s centroid and apparent diameter in the image [
12
]. In most applications, the OPNAV
camera’s field of view (FOV) is carefully designed to exactly cover the entire celestial body
for better navigation accuracy. However, in some cases, the visible horizon arc could
be very short, for example, due to the influence of lighting conditions, deviation of the
Aerospace 2023,10, 371. https://doi.org/10.3390/aerospace10040371 https://www.mdpi.com/journal/aerospace
Aerospace 2023,10, 371 2 of 22
pointing angle, and, most notably, change in altitude. In a typical Mars exploration mission,
the distance
between the spacecraft and Mars varies from a few hundred kilometers to
tens of thousands of kilometers [
1
]. To ensure full-stage navigation accuracy, the FOV
of the navigation camera is generally small such that only a small part of the horizon or
scenery can be captured at the perigee. Although landmark-based OPNAV techniques can
provide higher accuracy, they imply additional modes of navigation sensors and higher
computational requirements. Thus, OPNAV, which only relies on a short-arc horizon, is
essential.
The perspective projection of an ellipsoid planet or moon is an approximate ellipse,
so many
ellipse fitting algorithms have been used for the problem of horizon-based OP-
NAV
[12,20].
In some applications, the celestial body with low ellipticity is directly ap-
proximated as a sphere, resulting in a simple and robust solution at the expense of some
accuracy [
21
,
22
]. The traditional method of direct ellipse fitting in the image plane will
incorrectly converge and introduce large errors in the case of a short-arc horizon. To
solve this problem, Hu et al.
[23]
takes ephemeris and the spacecraft’s position as prior
information and simplifies the ellipsoid to a sphere, thus obtaining the size and shape
information of the ellipse in the image plane beforehand. The fitting accuracy of a short-arc
horizon is greatly improved because there are only two unknown pieces of information,
a scalar of the ellipse’s rotation and a two-dimensional vector of the ellipse’s translation
on the image plane, to be estimated. This approach is inappropriate for our application
due to the lack of position information, but it still reveals the difficulty of short-arc ellipse
fitting and the importance of using prior information. The state-of-the-art solution is the
Christian–Robinson algorithm [
13
,
24
,
25
], in which the ellipsoid celestial body is transferred
into a sphere through Cholesky factorization, thus leading to a non-iterative solution.
This is undoubtedly a novel and elegant solution that makes full use of prior information
and greatly simplifies the problem. It has been employed in evaluating various OPNAV
missions and has shown excellent performance [21,26].
In the Christian–Robinson algorithm, it was mentioned that the measurement equa-
tions should be solved using either the least-square (LS) estimation or the total least-square
(TLS) estimation, but no detailed analysis was given [
25
], and LS estimation has widely
been used for its simplicity [
21
,
24
,
26
]. The ordinary LS estimation assumes that the basis
function
H
is error-free. However, in the measurement equation of the Christian–Robinson
algorithm, the measurement errors are entirely embedded in the basis function
H
. TLS
estimation addresses errors in the
H
matrix, thus obtaining higher estimation accuracy and
smaller estimation bias than the LS estimation [
27
]. TLS estimation is an asymptotic unbi-
ased estimator whose error covariance achieves the Cramér–Rao lower bound [
28
,
29
]. It has
been widely used in various measurement and estimation applications and demonstrates
excellent performance [
30
–
34
]. However, compared with LS estimation, TLS estimation
is more complex to apply and brings a non-negligible time consumption. In addition,
TLS estimation is a general term for a class of algorithms with many variants [
35
–
37
].
It is
necessary to further analyze why and when TLS estimation is needed and how it
is implemented.
In this paper, we concentrate on improving the performance of horizon-based OPNAV
based on a short-arc horizon. The main contributions of this paper lie in the following as-
pects: (1) Theoretical analysis reveals that LS estimation applied to the traditional Christian–
Robinson algorithm is inappropriate and may introduce estimation bias.
(2) A simplified
measurement covariance model is proposed by further analyzing the propagation of
measurement errors. The covariance model is no longer limited by the narrow FOV ap-
proximation and can be adapted to large FOV cameras. (3) An unbiased solution with
the EW-TLS algorithm is developed. By considering the measurement equation and the
covariance of each measurement, mean residuals close to zero can be achieved. (4) To
further simplify this problem, an AG-TLS algorithm is proposed, in which a non-iterative
solution is achieved by using the covariance of any one measurement as an approxima-
tion of all measurement covariances. Compared with the EW-TLS algorithm, the AG-TLS
Aerospace 2023,10, 371 3 of 22
algorithm trades a negligible accuracy loss for a huge reduction in execution time. Its
computing speed is comparable to the traditional algorithm and is suitable for onboard
applications.
(5) Covariance
analysis and numerical results prove that the covariances of
the LS estimation and the TLS estimation in this problem are identical, and the difference
between them lies in the possible estimation bias. Due to the drawbacks of LS estimation,
the Christian–Robinson method with ordinary LS estimation introduces large mean residu-
als under a short-arc horizon scenario, while the EW-TLS algorithm can achieve near zero
mean residuals.
The remainder of this paper is organized as follows. Section 2reviews the horizon-
based OPNAV problem. Section 3analyzes the problem in detail at the theoretical level
and describes the proposed algorithms using TLS estimation. Section 4presents the results
of numerical simulations, and the main conclusions are summarized in Section 5.
2. Brief Review of Horizon-Based Optical Navigation
2.1. Geometry Fundamentals
The geometry of horizon-based OPNAV is shown in Figure 1. When the navigation
camera takes a picture of an ellipsoid planet, the vectors from the camera’s optical center to
the planet’s apparent horizon will form an approximate conical surface that tightly bounds
the planet. Here is a brief review of the geometry fundamentals behind this.
y
YC
ZC
x
XC
OC
YP
ZP
OP
XP
Os
f
Figure 1. The geometry of the triaxial ellipsoid planet’s projection process.
To clarify the statement, several reference frames are introduced, which are defined
as follows. The planetary frame (P) is established with the center of the planet (OP) as the
origin. The X-axis(XP), Y-axis(YP), and Z-axis(ZP) of frame Pare directions of the planet’s
three principal axes, respectively. The origin of the camera frame (
C
) is the optical center
of the camera (
Oc
). The X-axis (
Xc
) and Y-axis (
Yc
) of frame
C
are directions of the row
and column of the image sensor, respectively. The Z-axis (
Zc
) of frame
C
is normal to the
Xc−Yc
plane. The image frame takes the imaging center of the image sensor (
Os
) as the
origin. The row and column of the image sensor are taken as the X-axis (
x
) and Y-axis (
y
)
of the
image frame, and the unit is a pixel. The image plane is right in the image frame,
while its size is limited by the image sensor.
Note that the following analysis is performed in the camera frame
C
unless otherwise
indicated. For an ideal navigation camera that perfectly follows the perspective imaging
relationship [
38
], a detected planetary horizon point
ψi=[xi,yi]T
in the image frame
corresponding to the following horizon vector si:
si=xiyifT(1)
Aerospace 2023,10, 371 4 of 22
where
f
is the focal length of the navigation camera in pixels. The shape of most regular
planets can be approximately described by a triaxial ellipsoid, then all the apparent horizons
form a conical surface tightly bounding the triaxial ellipsoid during the imaging process [
12
].
For any horizon vector
si
belonging to the conical surface, it obeys the following constraints:
siTMsi=0 (2)
where Mis a full rank symmetric 3 ×3 matrix. The expression of Mis as follows:
M=ArrTA−rTAr −1A(3)
where
r
is the position of camera center
Oc
relative to planet center
OP
, and
A
describes the
shape and orientation of a triaxial ellipsoid. The expression of Ais as follows:
A=TP
C
1/a20 0
0 1/b20
0 0 1/c2
TC
P(4)
where
{a,b,c}
represents the length of the planet’s three principal axis, and
TP
C
represents
the rotation matrix from the planetary frame
P
to the camera frame
C
. With the development
of high-precision ephemeris and star trackers, a high-precision
TP
C
can be easily obtained,
which will serve as prior information for the entire problem.
2.2. The Christian–Robinson Algorithm
The Christian–Robinson algorithm makes full use of prior ellipsoid shape and camera
attitude information. The algorithm transforms the triaxial ellipsoid into a unit sphere
through Cholesky factorization, thereby fully simplifying the triaxial ellipsoid projection
constraint. Here is a brief review of the algorithm.
(1)
Convert the camera’s horizon measurement sito ¯siin Cholesky factorization space:
¯si=Usi(5)
where Ucan be obtained from Equation (4) as:
U=QTC
P(6)
and the expression of Qis as follows:
Q=
1/a0 0
0 1/b0
0 0 1/c
(7)
(2)
Normalize ¯sito hi:
hi=¯si
k¯sik(8)
(3)
The problem is converted to solving for nin the following measurement equations:
Hn =y(9)
where:
H=
hT
1
hT
2
.
.
.
hT
n
(10)
y=1n×1,nis the total number of horizon measurement points.
Aerospace 2023,10, 371 5 of 22
(4)
The solution of rcould be found by:
r=−nTn−1−(1/2)U−1n(11)
3. Total Least-Squares Estimation
3.1. Estimation Bias Caused by Least-Squares Estimation
With the help of the Christian–Robinson algorithm, the problem has been greatly
simplified to solving for
n
in Equation
(9)
. Although it was mentioned in [
25
] that the
measurement equations should be solved using either the LS estimation or the TLS es-
timation, no detailed analysis was given. The LS estimation seems to bring the same
results with much lower computational consumption in most cases, so it is more widely
used [21,24,26].
However, the limitation of the LS estimation seriously affects the perfor-
mance of the algorithm in short-arc horizon-based OPNAV. This will be demonstrated in
the following derivations.
Let us revisit the problem described in Equation
(9)
. If the problem is treated as a LS
problem, then its loss function is:
J(ˆnls) = 1
2˜y −˜
H ˆnlsT˜y −˜
H ˆnls(12)
where
˜y
is the measurement of
y
,
˜
H
is the measurement of
H
, and
ˆnls
is the LS solution.
As shown in Section 2.2,
˜
H
is converted from the camera’s horizon measurement by
Equations
(5)
and
(8)
,
˜y
is set as
1n×1
directly from Equation
(9)
, and the expression
of ˆnls is:
ˆnls = ( ˜
HT˜
H)−1˜
HT˜y (13)
If we treat
ˆnls
as a function of the measurements, then the expectation of the LS
solution is:
E(ˆnls) = E(( ˜
HT˜
H)−1˜
HT˜y)(14)
According to Equation (9), we have:
˜y =1n×1=y=Hn (15)
thus ˜y can be rewritten as:
˜y=Hn = ( ˜
H+H−˜
H)n(16)
Then, substituting Equation (16) into Equation (14) yields:
E(ˆnls) = E(( ˜
HT˜
H)−1˜
HT˜y)
=E(( ˜
HT˜
H)−1˜
HT(˜
H+H−˜
H)n)
=E(( ˜
HT˜
H)−1˜
HT˜
Hn) + E(( ˜
HT˜
H)−1˜
HT(H−˜
H)n)
=n+E(( ˜
HT˜
H)−1˜
HT(H−˜
H)n)
(17)
It can thereby be concluded that LS estimation is a biased estimation for this
problem [27],
and the estimation bias is:
E(ˆnls)−n=E(( ˜
HT˜
H)−1˜
HT(H−˜
H)n)(18)
Estimation bias is unavoidable in LS estimation due to the error contained in
˜
H
,
and its
existence is verified in subsequent simulations. The estimation bias will increase
with the lack of measurement information due to the shortening of the horizon arc length.
In practice,
estimation biases are not easily detected without additional correction methods.
Furthermore, it is undoubtedly catastrophic for subsequent Kalman filter or other estima-
tion frameworks if the estimation bias is large enough compared with the
estimation error.
Aerospace 2023,10, 371 6 of 22
3.2. Simplified Measurement Covariance Model
As shown in the last section, the key to the estimation bias is the coupling of the
measurement error with the measurement model. To solve this problem, we need to
find out the covariance of measurement errors beforehand, which means studying the
propagation of the measurement error in the transition of horizon measurements in the
image to
˜
H
in the measurement model. The narrow FOV approximation was used in
previous covariance derivations, which greatly limited its applicability [
20
,
24
,
39
,
40
]. In this
paper, a more general and simple measurement covariance model is given by reducing the
normalization step of the measurement vector.
At first, a detected horizon point
ψi=[xi,yi]T
in the image frame describes the
horizon point’s line-of-sight direction. The measured error should be uniformly Gaussian
distributed in the image plane, then the covariance of ψican be expressed as:
Rψi=σ20
0σ2(19)
where σis the STD of the horizon point location error, and the unit is one pixel.
Then, the covariance changes during the projection of the point
ψi
in the image frame
to the vector
si=xiyifT
in the camera frame
C
. The narrow FOV approximation is
commonly used to estimate the covariances of
si
when the camera has a small
FOV [20,24]
.
A small FOV indicates that the focal length
f
is much greater than the size of the image
sensor, allowing for the assumption that the covariance of any normalized
si
within the
narrow FOV remains constant [
40
]. However, we do not use the narrow FOV approximation
in our covariance model because it would restrict the model from being applied to cameras
with larger FOVs. Instead, we apply the normalization step only once at a later stage. This
approach further simplifies the covariance calculation and increases the model’s versatility.
The projection corresponds to the Jacobi matrix as follows:
∂si
∂ψi
=
1 0
0 1
0 0
(20)
Thus the covariance of siis:
Rsi=∂si
∂ψi
Rψi∂si
∂ψiT
=
σ20 0
0σ20
0 0 0
(21)
Due to the characteristics of camera measurements, the covariance of
si
is not uniformly
distributed in three directions, and the errors mainly accumulate in
XC
and
YC
directions.
Note that a more general theory of camera measurement covariance could be found in [
39
],
while the one used here is an adequate simplification.
Next, consider the transformation corresponding to Equation
(5)
, in which
si
in the
camera frame
P
is transferred into
¯si
in the Cholesky factorization space. The transformation
corresponds to the Jacobi matrix as follows:
∂¯si
∂si
=U(22)
In addition, as shown in Equation (8), the vector ¯sineeds to be normalized to hi. The
normalization corresponds to the Jacobi matrix as follows:
∂hi
∂¯si
=1
k¯sikI3×3−hihT
i(23)
Aerospace 2023,10, 371 7 of 22
Defining the 3 ×3 matrix Φi=∂hi
∂¯sithus obtains:
∂hi
∂si
=ΦiU(24)
Since the magnitude of every measurement is different, the covariance of each mea-
surement varies after normalization.
Finally, the covariance of hiis found to be:
Rhhi=ΦiURsiUTΦT
i(25)
3.3. Element-Wise Weighted Total Least-Squares Algorithm
As shown in the previous section, the measurement error propagation leads to different
covariance for each measurement. In addition, the measurements
H
act as a model in the
measurement equation, which means that this is an entirely error-in-variables problem. To
better address this problem, the TLS estimation is adopted, which takes into account the
error in the
H
matrix and weights the measurement equations based on the covariance of
the measurements.
For this problem, define the following matrix:
D=H 1n×1(26)
z=nT−1T(27)
The constraint condition Equation (9) can then be rewritten as:
Dz =0(28)
TLS solution ˆ
ntls for this problem is the optimal estimation of nthat minimizes[28]:
J(ˆntls) = 1
2vecT˜
DT−ˆ
DTR−1vec˜
DT−ˆ
DT,
subject to ˆ
Dˆz =0(29)
where
ˆz =ˆnT
tls −1T
denotes the estimation of
z
,
˜
D=˜
H 1n×1
denotes the mea-
surement of
D
,
ˆ
D=ˆ
H 1n×1
denotes the estimation of
D
,
ˆ
H
denotes the estimation of
H
,
R
is the covariance matrix of
˜
D
, and vec denotes a vector formed by stacking the con-
secutive columns of the associated matrix. Ris given by following block diagonal matrix:
R=
R104×4· · · 04×4
04×4R2· · · 04×4
.
.
..
.
.....
.
.
04×404×4· · · Rn
(30)
where Riis covariance matrix of hi, and it is given by:
Ri="RhhiRhyi
RT
hyiRyyi#(31)
As defined in Equation (15), there is:
Rhyi=03×1and Ryyi=01×1(32)
In this case, since the errors are element-wise correlated and non-stationary, each
measurement element needs to be weighted according to its covariance. It implies that
this is an element-wise weighted TLS (EW-TLS) problem and can be solved by using an
Aerospace 2023,10, 371 8 of 22
iterative procedure [
27
,
36
]. In addition, it is necessary to impose strict time limits on the
algorithm to ensure implementation onboard. The specific steps of the EW-TLS algorithm
are as follows:
(1)
Calculate the covariance Rhhiof each set of measurements with Equation (25).
(2)
Take the LS estimation ˆnls as the initial estimate ˆn(1)with Equation (13).
(3)
The estimate ˆn(j+1)at the (j+1)th iteration is given by [36]:
ˆn(j+1)=
n
∑
i=1e
hie
hT
i
γiˆn(j)−e2
iˆn(j)Rhhi
γ2
iˆn(j)
−1"n
∑
i=1
˜
yie
hi
γiˆn(j)#(33)
where:
γiˆn(j)=ˆn(j)TRhhiˆn(j)(34)
eiˆn(j)=e
hT
iˆn(j)−˜yi(35)
(4)
The iteration is stopped when the final change is less than the tolerance value, or
when the number of iterations is greater than the maximum number of iterations
(MaxIterations). The iteration stopping conditions are specified as:
ˆn(j+1)−ˆn(j)
2≤tolerance value or j>MaxIterations (36)
The tolerance value is set according to the calculation precision, and it is set to 10
−10
in
this paper. MaxIterations is limited to five to avoid excessive time consumption. Although
it is certainly satisfactory that the algorithm usually converges in only one to three iteration
cycles, it is definitely safer to impose a limit on the number of iterations.
3.4. Approximate Generalized Total Least-Squares Algorithm
Although the accuracy of the EW-TLS algorithm is satisfactory, its computation time
is still too long. Considering the limitations of onboard computing capabilities, it is usu-
ally worthwhile to trade some precision loss for faster computation. To further simplify
this problem, an AG-TLS algorithm is proposed in this paper, which uses the covari-
ance of any one measurement as an approximation of all covariances, thus achieving a
non-iterative solution.
Recall the analysis of measurement error propagation in Section 3.2. For a planet
with a small oblateness, its three major axes
a
,
b
,
c
are very close, and Equation
(6)
can be
approximated as:
U=QTC
P≈1
Re
I3×3TC
P=1
Re
TC
P(37)
where Reis the mean radius of the target planet. Thus, ¯simay be rewritten as:
¯si=1
Re
TC
Psi(38)
The attitude transformation matrix
TC
P
does not change the norm of
si
, which lead to:
k¯sik=qx2
i+y2
i+f2
Re(39)
Then, substituting Equations (38) and (39) into Equation (24) yields:
∂hi
∂si
≈qx2
i+y2
i+f2
R2
eI3×3−hihT
iTC
P(40)
Aerospace 2023,10, 371 9 of 22
For a narrow FOV camera, the focal length
f
is much larger than
qx2
i+y2
i
and is
approximately: qx2
i+y2
i+f2≈f(41)
Further substituting Equations (40) and (41) into Equation (25) yields:
Rhhi≈f2
R4
eI3×3−hihT
iTC
PRsiTP
CI3×3−hihT
iT(42)
As shown in Equation
(21)
,
Rsi
is constant and mainly accumulated in
XC
and
YC
directions. Then, as shown in Equation
(42)
, when horizon measurements
si
in the image
transitions to
hi
in the measurement model,
Rhhi
are redistributed to three directions with
I3×3−hihT
iTC
P
. The variation of
Rhhi
is dominated by the variation of
hihT
i
, while
hihT
i
changes relatively little since all measurements
hi
are gathered in a small area for a narrow
FOV camera.
According to the above analysis, it can be concluded that when both camera’s FOV and
planetary oblateness are small,
Rhhi
are relatively close for all measurements
hi
. Therefore,
the problem can be further simplified with a small loss of precision by taking the covariance
of any one measurement as an approximation of all covariances. Considering the error
distribution and correlation among the elements, the simplified problem is a generalized
TLS estimation problem, which has a closed-form solution [
27
,
35
,
36
]. The specific steps of
the AG-TLS algorithm are as follows:
(1)
Calculate the covariance
Rhhi
of any one measurement
hi
with Equation
(25)
, then
construct Riwith Equation (31).
(2)
Prepare for Cholesky factorization of matrix
Ri
. According to Equation
(32)
,
Ryyi
should be equal to zero, while this will lead to a rank deficient of
Ri
. In addition,
the covariance matrix
Rhhi
is a positive semi-definite matrix. With the change in
spacecraft position and attitude, and considering the machine computational accuracy,
Rhhi
will sometimes have an eigenvalue close to zero or even negative. These factors
block the Cholesky decomposition of the matrix Riand affect the implementation of
the algorithm. Thus, the approximate covariance matrix
¯
R
is obtained by making a
simple but effective adjustment to Ri.
¯
R=Ri+εI4×4(43)
where
ε
is a very small value, e.g., 10
−15
, and is much smaller than every element in
the matrix
Rhhi
. Such an approximation hardly affects the distribution of covariances
and ensures that the matrix ¯
Ris positive definite.
(3)
Perform a Cholesky factorization on the matrix ¯
R:
¯
R=CTC(44)
Then partition the inverse of Cas:
C−1=C11 c
01×3c22 (45)
where C11 is an 3 ×3 matrix, cis an 3 ×1 vector , and c22 is a scalar.
(4)
Perform a singular value decomposition on the following matrix:
ˆ
DC−1=NSVT(46)
Then partition Vas:
V=V11 v
wTv22 (47)
Aerospace 2023,10, 371 10 of 22
where
N
,
S
and
V
are 4
×
4 matrixes,
V11
is an 3
×
3 matrix,
v22
is a scalar, and
w
and
vare both 3 ×1 vectors.
(5)
The final solution is given by [27]:
ˆnagtls =1
c22 −1
v22
C11v−c(48)
3.5. Analytic Solution Covariance
In this section, the analytic solution covariance of the TLS estimation and the LS
estimation are evaluated to compare their performance under this problem.
According to [28], the analytic solution covariance of nwith TLS estimation is:
tlsPn= (
n
∑
i=1
hihT
i
zTRiz)−1(49)
According to Equation (32), it can be simplified as:
tlsPn= (
n
∑
i=1
hihT
i
nTRhhin)−1(50)
Take the variation of Equation (11) with n:
∂r
∂n=F(51)
where:
F=−nTn−1−(1/2)TP
CQ−1I3×3−nnT
nTn−1(52)
So the analytic solution covariance of ris:
tlsPr=Ftls PnFT(53)
The analytic covariance of nwith LS estimation is:
lsPn= (
n
∑
i=1
hihT
i
lsRyyi
)−1(54)
Since hiis assumed to be unperturbed in the ordinary LS estimation, there are:
lsRyyi=nTRhhin(55)
Therefore, it can be obtained:
lsPn= (
n
∑
i=1
hihT
i
nTRhhin)−1=tlsPn(56)
That is to say:
lsPr=Fls PnFT=tlsPr(57)
It can be seen that the analytic covariances of the LS estimation and the TLS estimation
are completely consistent, then the difference between the two kinds of estimations lies in
the possible estimation bias. As demonstrated in [
28
], the TLS estimation is asymptotically
unbiased, which is the main advantage over the ordinary LS estimation in this problem,
so it is certainly attractive to adopt the TLS estimation without any loss of precision. In
the actual performance of the EW-TLS algorithm, the analytic solution covariance will be
gradually approximated as the iteration number grows, while in the AG-TLS algorithm,
there may be a gap between the actual performance and the analytics solution covariance
Aerospace 2023,10, 371 11 of 22
due to the presence of approximations. The actual performance of the proposed algorithms
is verified in the following numerical simulations.
4. Numerical Results
4.1. Performance under a Short-Arc Horizon
The simulated navigation camera assumed a FOV of 8x8 degrees and a resolution
of 1024
×
1024 pixels. The exact numerical solution of the apparent horizon in the im-
age plane could be found using Equations
(1)
–
(3)
. One horizon point is sampled from
each pixel where the horizon appears, and Gaussian distributed errors with
σ=
0.3 pixel
are added to the horizon points, thus getting closer to the actual horizon detection situa-
tion. The above simulation conditions of navigation camera are applied to all subsequent
simulations.
The simulated
navigation camera is pointing to the center of Mars at a range
of 65,000 km. At this point, the horizon of Mars occupies almost the entire FOV of the
navigation camera. However, due to the possible deviation of the pointing angle and
the influence of lighting conditions, only a small part of the horizon may be detected.
Simulated images of the horizon ellipse with different arc lengths in the image frame are
shown in Figure 2.
− 6 0 0 − 4 0 0 − 2 0 0
0 200 400 600
− 4 0 0
− 2 0 0
0
200
400
A c t u a l M e a s u r e d
(a) 15 degrees.
− 6 0 0 − 4 0 0 − 2 0 0
0 200 400 600
− 4 0 0
− 2 0 0
0
200
400
A c t u a l M e a s u r e d
(b) 60 degrees.
− 6 0 0 − 4 0 0 − 2 0 0
0 200 400 600
− 4 0 0
− 2 0 0
0
200
400
A c t u a l M e a s u r e d
(c) 120 degrees.
Figure 2. Simulated images of the horizon ellipse with different arc lengths.
As an example scenario, assume that a 15-degree of horizon arc is detected, and the ac-
tual number of valid horizon points is about 100. For convenience, the Christian–Robinson
algorithm with LS estimation is called the LS algorithm, while the algorithms proposed in
this paper are called the EW-TLS algorithm and AG-TLS algorithm. A total of
5000 Monte
Carlo simulations are performed with three algorithms. Various horizon measurements
are produced by adding randomly generated Gaussian noise to the same scenario, and
the results are shown in Figure 3. Errors of OPNAV are presented in the camera frame
C
.
Gray points represent measurement residuals of Monte Carlo simulations, the black cross
represents mean residuals of simulations, the red dashed line ellipse represents 3
σ
bounds
of analytic covariance, and the black solid line ellipse represents 3
σ
bounds of numeric
covariance from simulations. Note that the covariance in Figure 3is superimposed on the
mean residuals. In the simulations of all three algorithms, the analytical covariance and the
numeric covariance are in good agreement, and the correctness of the analytic covariance is
fully verified.
The detailed results of Monte Carlo simulations are listed in Table 1. The magnitude
of the solution covariances is the same for all three algorithms. However, as shown in
Figure 3a, the mean residuals of the LS algorithm are clearly not zero. The mean to
the standard deviation ratio (MSTDR) is introduced here to evaluate the influence of
mean residuals. MSTDR is set to be a positive value. MSTDR of the LS algorithm in
XC
,
YC
,
and ZC
directions are 311.63%, 301.23%, and 311.67%, respectively. This means
that the mean residuals are much larger than the STD, which is unacceptable in practical
applications. In contrast, as shown in Figure 3b,c, the MSTDR of the EW-TLS algorithm and
AG-TLS algorithm are all near zero. The root mean squared error (RMSE) is introduced to
evaluate the estimation effectiveness of all three algorithms with respect to mean residuals,
and two TLS algorithms both show a significant accuracy advantage of three times over
the LS algorithm. The mean residuals of the AG-TLS algorithm are slightly larger than the
Aerospace 2023,10, 371 12 of 22
average error of the EW-TLS algorithm. From the perspective of MSTDR, the performance
difference between the two algorithms is within 2%, while it is basically no different from
the perspective of RMSE, which indicates that the approximation and simplification of the
AG-TLS algorithm are effective under a short-arc horizon.
− 6 0 0 − 4 0 0 − 2 0 0
0
− 8 0
− 6 0
− 4 0
− 2 0
0
− 8 0 − 6 0 − 4 0 − 2 0
0
0
3,000
6,000
9,000
12,000
− 6 0 0 − 4 0 0 − 2 0 0
0
0
3,000
6,000
9,000
12,000
M e a s . R e s . M e a n R e s .
(a) LS algorithm.
− 4 0 0 − 2 0 0
0 200 400
− 4 0
− 2 0
0
2 0
4 0
− 4 0 − 2 0
0 2 0 4 0
− 6 , 0 0 0
− 3 , 0 0 0
0
3,000
6,000
− 4 0 0 − 2 0 0
0 200 400
− 6 , 0 0 0
− 3 , 0 0 0
0
3,000
6,000
M e a s . R e s . M e a n R e s .
(b) EW-TLS algorithm.
− 4 0 0 − 2 0 0
0 200 400
− 4 0
− 2 0
0
2 0
4 0
− 4 0 − 2 0
0 2 0 4 0
− 6 , 0 0 0
− 3 , 0 0 0
0
3,000
6,000
− 4 0 0 − 2 0 0
0 200 400
− 6 , 0 0 0
− 3 , 0 0 0
0
3,000
6,000
M e a s . R e s . M e a n R e s .
(c) AG-TLS algorithm.
Figure 3. Monte Carlo results with a 15-degree arc horizon.
Table 1. Monte Carlo results with a 15-degree arc horizon.
Algorithm Direction Mean
Residuals, km STD, km MSTDR RMSE, km
LS
XCaxis −296.82 95.25 311.63% 311.73
YCaxis −39.71 13.18 301.23% 41.84
ZCaxis 5717.96 1834.61 311.67% 6005.01
EW−TLS
XCaxis 0.84 95.77 0.88% 95.76
YCaxis 0.04 13.22 0.34% 13.21
ZCaxis −16.24 1845.42 0.88% 1845.30
AG−TLS
XCaxis −1.89 95.87 1.97% 95.88
YCaxis −0.37 13.23 2.78% 13.24
ZCaxis 36.54 1847.53 1.97% 1847.71
Aerospace 2023,10, 371 13 of 22
Although there is no circle fitting or ellipse fitting of the projected horizon in the
image frame in the Christian–Robinson algorithm, it is still more intuitive to use the
projected horizon in the image frame to show the solution results. Substituting the true
value of
r
, LS algorithm’s estimation of
r
, EW-TLS algorithm’s estimation of
r
, and AG-TLS
algorithm’s estimation of
r
into Equations
(1)
–
(3)
yields the actual projected horizon ellipses,
LS-projected horizon ellipses, EW-TLS-projected horizons, and AG-TLS-projected horizon
ellipses in the image frame, respectively. These ellipses are shown in Figure 4together with
the measured horizon arc. Compared with the actual projected horizon ellipse, LS-projected
horizon ellipses are generally offset and relatively small, which means that the distance
from the camera to the planet estimated by the LS algorithm is larger than the actual value,
and there will be a direction error. On the contrary, EW-TLS-projected horizon ellipses
and AG-TLS-projected horizon ellipses can both basically fit the actual projected horizon
ellipse. This is consistent with our previous analysis that the LS estimation introduces a
non-negligible bias, while the TLS estimation eliminates the effect of the bias.
− 6 0 0 − 4 0 0 − 2 0 0
0 200 400 600
− 4 0 0
− 2 0 0
0
200
400
L S A c t u a l M e a s u r e d
(a) LS algorithm.
− 6 0 0 − 4 0 0 − 2 0 0
0 200 400 600
− 4 0 0
− 2 0 0
0
200
400
E W - T L S A c t u a l M e a s u r e d
(b) EW-TLS algorithm.
− 6 0 0 − 4 0 0 − 2 0 0
0 200 400 600
− 4 0 0
− 2 0 0
0
200
400
A G - T L S A c t u a l M e a s u r e d
(c) AG-TLS algorithm.
Figure 4.
Projected horizon ellipse in the image frame. For clarity’s sake, only 30 out of 5000 Monte
Carlo sample histories are reported.
4.2. Performance under Different Horizon Arc Lengths
More Monte Carlo simulations are conducted to evaluate the algorithms’ performance
under different lengths of detected horizon arc. The actual number of valid horizon
points increases linearly with increasing arc length. A total of 5000 sets of independent
simulations are conducted for each experimental condition. Various horizon measurements
are produced by adding randomly generated Gaussian noise to the same scenario, and
the results shown are the average values of these measurements. Errors of OPNAV are
presented in the camera frame
C
. RMSE and MSTDR are used to evaluate the performance
of three algorithms.
The results of the simulations are shown in Figures 5and 6. The RMSE of the three
algorithms gradually decreased as the arc length was increased, which is consistent with
the previous study of horizon-based OPNAV covariance [
41
]. At the same time, the gap
between the RMSE and MSTDR between the LS algorithm and the TLS algorithms also
decreased. As seen in Figure 5, both the EW-TLS algorithm and AG-TLS algorithm show
great advantages over the LS algorithm in terms of RMSE when the horizon arc is less
than 50 degrees, while the advantage of the two TLS algorithms is no longer obvious
as the horizon arc length grows further. As shown in Figure 6, the mean residuals of
the LS algorithm are larger than the STD when the horizon arc is less than 35 degrees.
Even when the horizon arc is 95 degrees, the MSTDR of the LS algorithm is still larger
than 20%. This will cause problems for the subsequent Kalman filter or other estimation
frameworks, since they all assume that the measurements are zero mean error. In contrast,
the two TLS algorithms maintain almost zero mean residuals as the horizon arc degree
is changed. When the horizon arc length is long enough, the LS algorithm has almost no
performance loss when compared with the two TLS algorithms. This indicates that as the
measurement information is increased, the bias introduced by the LS algorithm becomes
progressively smaller. As long as the captured horizon arc is longer than a certain value
(e.g., 120 degrees), the LS algorithm can be used to obtain a faster calculation speed without
Aerospace 2023,10, 371 14 of 22
any loss of accuracy; otherwise, it is recommended that the TLS algorithms be used to avoid
mean residuals. As the horizon arc length increases, the MSTDR of the EW-TLS algorithm
never exceeds 4%, while the MSTDR of the AG-TLS algorithm never exceeds 9%. From
the perspective of RMSE, there is almost no significant difference in the performance of the
two algorithms, and such a performance gap is negligible in practice.
30 60 90 120 150 180
0
100
200
300
400
R M S E , k m
H o r i z o n a r c l e n g t h , d e g
3 0 6 0 9 0
0
1 0
2 0
3 0
(a) XCaxis.
30 60 90 120 150 180
0
1 0
2 0
3 0
4 0
5 0
R M S E , k m
H o r i z o n a r c l e n g t h , d e g
3 0 6 0 9 0
0
2
4
6
8
(b) YCaxis.
30 60 90 120 150 180
0
2,000
4,000
6,000
8,000
R M S E , k m
H o r i z o n a r c l e n g t h , d e g
L S
E W - T L S
A G - T L S
3 0 6 0 9 0
0
200
400
600
(c) ZCaxis.
Figure 5. RMSE with different horizon arc lengths.
30 60 90 120 150 180
0 %
200%
400%
M S T D R
H o r i z o n a r c l e n g t h , d e g
50 100 150
0 %
20%
40%
60%
(a) XCaxis.
30 60 90 120 150 180
0 %
200%
400%
M S T D R
H o r i z o n a r c l e n g t h , d e g
50 100 150
0 %
20%
40%
60%
(b) YCaxis.
30 60 90 120 150 180
0 %
200%
400%
M S T D R
H o r i z o n a r c l e n g t h , d e g
L S
E W - T L S
A G - T L S
50 100 150
0 %
20%
40%
60%
(c) ZCaxis.
Figure 6. MSTDR with different horizon arc lengths.
Additionally, it is necessary to evaluate the execution times of the algorithms. Simula-
tions are running multiple times at different horizon arcs and the average value is obtained
as execution times. They are running in a computer with an intel
r
i7-10700 CPU and
MATLAB
r
2022a. The execution times include only the navigation algorithms, not the
time for image processing, etc. The number of horizon points gradually increases with the
horizon arc, and the relationship between the number of horizon points and execution time
is shown in Figure 7. It can be seen that the number of measurement points dominates the
execution time of the algorithms. The execution time of the EW-TLS algorithm increases
from 1.3 to 8.1 ms with the increase of horizon points. Most execution time of the EW-TLS
algorithm is consumed during the iterations, and a small portion is used to calculate the
covariance of each measurement. The execution time of the LS algorithm never exceeds
0.23 ms, while the execution time of the AG-TLS algorithm never exceeds 0.43 ms, and such
extra computation consumption is often acceptable. Compared with the EW-TLS algorithm,
the AG-TLS algorithm brings a huge speed improvement with a slight loss of precision,
which is very suitable for onboard applications with limited computing resources.
Aerospace 2023,10, 371 15 of 22
200 400 600 800 1,000 1,200 1,400
0
2
4
6
8
1 0
E x e c u t i o n t i m e , m s
N u m b e r o f h o r i z o n p o i n t s
L S E W - T L S A G - T L S
Figure 7. Execution time with different number of horizon points.
4.3. Opnav Performance in a Highly Elliptical Orbit
To fully evaluate the performance of the short-arc horizon-based OPNAV, a navigation
scenario in a highly elliptical orbit is tested. The scientific exploration orbit is usually
a highly elliptical orbit to meet the requirements of communication and observation at
the same time. Elements of the orbit around Mars are listed in Table 2. The test starts
from the epoch 1 January 2022 07:49:00 (UTCG). In the frame
P
, the spacecraft’s initial
position is (1.3212
×
10
3
, 2.4107
×
10
3
,
−
2.4834
×
10
3
) km, and its initial velocity is (1.0999,
2.6389, 3.2311) km/s. The test covers 500 min, with perigee at the 0 and 470th minutes
and the apogee at the 236th minute. The sampling interval for OPNAV is one minute.
The parameters of the navigation camera are the same as those introduced earlier. The
optical axis of the navigation camera is always aligned with the edge of Mars during the
simulation, and the Mars horizon is on the diagonal of the image plane to ensure the best
navigation performance. The influence of lighting conditions is not considered here to
show the ideal situation, which may also be excluded by using an infrared camera, and the
visible limb portion is always the entire one fitting within the camera’s FOV. The actual
number of valid horizon points is always between 1700 and 1950. A total of 5000 sets
of independent simulations are performed for each position, and the results shown are
average values. Errors of OPNAV are presented in the camera frame C.
Table 2. Elements of a highly elliptical orbit around Mars.
Parameter Value
Semi-major axis 9500 km
Eccentricity 0.61
Inclination 87◦
Argument of perige 320◦
RAAN 64◦
True anomaly 0◦
The Monte Carlo results of the LS algorithm, EW-TLS algorithm, and AG-TLS algo-
rithm are shown in Figure 8. Gray points represent measurement residuals of Monte Carlo
simulations, the blue solid line represents mean residuals of simulations, the red dash
line represents 3
σ
bounds of analytic covariance, and the black solid line represents 3
σ
bounds of numeric covariance from simulations. Note that the covariance in Figure 8is
superimposed on the mean residuals. The three algorithms’ performances with the change
in orbital altitude during the simulation are shown in Figure 9. The orbital altitude is raised
from
300
to 12,000 km from perigee to apogee. The visible horizon arc length in the image
plane gradually increases from 2 to 18 degrees as the orbital altitude is increased while, at
the same time, the accuracy of each edge point gradually decreases.
The horizon
arc length
and the edge point accuracy both affect the navigation accuracy and have
opposite effects.
Aerospace 2023,10, 371 16 of 22
As shown in Figures 8b,c and 9, there is a significant improvement in navigation
accuracy with the increase of orbital altitude in the vicinity of the perigee. This shows that
the increase in horizon arc length plays a dominant role in navigation accuracy at a low
altitude. The navigation accuracy slowly decreases as the orbit altitude is further increased,
which indicates that the decrease in the edge point accuracy at this time will affect the
navigation accuracy to a greater extent. The STD of the navigation error ranges from 0.040
to
0.054 km
in the
XC
and
YC
axes and from 2.374 to 3.188 km in the
ZC
axis.
The phase
angle of the spacecraft does not show much effect in our simulation, which should be
related to the small oblateness of Mars. As seen in Figures 8a and 9, the LS algorithm
shows a significant mean residual in the navigation results for all three axes at low orbital
altitudes. The mean residuals of all three axes gradually decreases as the orbital altitude is
further increased, and the influence in the
XC
and
YC
axes can be neglected. However, a
certain mean residual has been maintained in the
ZC
axis,
and even
at the apogee, there
is still a mean residual of 1.16 km. While the mean residuals appear to be concentrated
primarily along the
ZC
axis, it is important to note that the impact of residuals becomes
more intricate when analyzed within the planetary frame. The
ZC
axis, which corresponds
to the camera’s optical axis, is aligned with the edge of Mars.
The attitude
of the camera
within the planetary frame varies as the spacecraft’s position changes. Consequently, the
direction of the mean residuals will also be variable within the planetary frame. This means
that the effect of the mean residuals should not be overlooked. On the contrary, the EW-TLS
algorithm and AG-TLS algorithm both showed a better performance with near-zero mean
residuals in different orbital altitudes. In general, the short-arc horizon-based OPNAV
exhibits good accuracy throughout the entire orbital period, and it provides unbiased
position estimation with the help of the TLS estimation.
0 100 200 300 400 500
− 0 .2
0 . 0
0 . 2
M e a s . R e s . M e a n R e s .
a x i s , k m
T i m e , m i n u t e s
0 100 200 300 400 500
− 0 .2
0 . 0
0 . 2
a x i s , k m
T i m e , m i n u t e s
0 100 200 300 400 500
− 2 0
− 1 0
0
1 0
2 0
a x i s , k m
T i m e , m i n u t e s
(a) LS algorithm.
0 100 200 300 400 500
− 0 .2
0 . 0
0 . 2
M e a s . R e s . M e a n R e s .
a x i s , k m
T i m e , m i n u t e s
0 100 200 300 400 500
− 0 .2
0 . 0
0 . 2
a x i s , k m
T i m e , m i n u t e s
0 100 200 300 400 500
− 2 0
− 1 0
0
1 0
2 0
a x i s , k m
T i m e , m i n u t e s
(b) EW-TLS algorithm.
0 100 200 300 400 500
− 0 .2
0 . 0
0 . 2
M e a s . R e s . M e a n R e s .
a x i s , k m
T i m e , m i n u t e s
0 100 200 300 400 500
− 0 .2
0 . 0
0 . 2
a x i s , k m
T i m e , m i n u t e s
0 100 200 300 400 500
− 2 0
− 1 0
0
1 0
2 0
a x i s , k m
T i m e , m i n u t e s
(c) AG-TLS algorithm.
Figure 8.
Monte Carlo results of OPNAV performance. For clarity’s sake, only 30 out of 5000 Monte
Carlo sample histories are reported (gray points).
0 4,000 8,000 12,000
0 . 0 0
0 . 0 2
0 . 0 4
0 . 0 6
0 . 0 8
0 . 1 0 L S E W - T L S A G - T L S
R M S E , k m
A l t i t u d e , k m
(a) XCaxis.
0 4,000 8,000 12,000
0 . 0 0
0 . 0 2
0 . 0 4
0 . 0 6
0 . 0 8
0 . 1 0 L S E W - T L S A G - T L S
R M S E , k m
A l t i t u d e , k m
(b) YCaxis.
0 4,000 8,000 12,000
0
2
4
6
8
1 0
R M S E , k m
A l t i t u d e , k m
L S E W - T L S A G - T L S
(c) ZCaxis.
Figure 9. OPNAV performance with different orbital altitude.
Aerospace 2023,10, 371 17 of 22
In addition to orbital altitude, the performance of short-arc horizon-based OPNAV is
affected by pointing errors. Ideally, the optical axis of the navigation camera should point
to Mars’ horizon, and the horizon should remain on the diagonal of the image plane to
ensure the best navigation performance, while it is difficult to achieve such precise pointing
in practice. Taking the perigee as an example scenario, the performance of horizon-based
OPNAV with different pointing errors on the
XC
,
YC
, and
ZC
axes of the navigation camera
are shown in Figure 10, Figure 11, and Figure 12, respectively.
As the pointing error increases, the horizon arc taken by the navigation camera be-
comes shorter and the OPNAV performance gradually decreases. The performance degra-
dation of the LS algorithm is particularly significant compared to the EW-TLS algorithm
and AG-TLS algorithm. When the pointing error on
XC
or
YC
axis is -4 degrees, the LS
algorithm’s positioning accuracy is worse than 10 km for both
XC
and
YC
axes and 200km
for
ZC
axis, while the EW-TLS and AG-TLS algorithms are better than 1 km for both
XC
and
YC
axes and better than 25 km for
ZC
axis. With the pointing error around the
ZC
axis, the horizon shifts from the diagonal of the image plane to the horizontal or vertical
direction. For the EW-TLS and AG-TLS algorithms, the positioning accuracy of
XC
and
YC
axes decreases from 0.054 km to the worst, 0.108 km, and the positioning accuracy of
ZC
axis decreases from 3.188 km to the worst, 6.000 km. Meanwhile, for the LS algorithm, the
positioning accuracy of
XC
and
YC
axes ranges from 0.050 to 0.272 km, and the positioning
accuracy of the
ZC
axis ranges from 10.214 to 25.052 km. The pointing error in the
ZC
axis
has less impact on the optical navigation performance than the other two axes because the
reduction of the horizon arc is smaller.
Pointing errors lead to the degradation of horizon-based OPNAV performance, and
the EW-TLS algorithm and AG-TLS algorithm demonstrate a higher tolerance for pointing
errors. The huge performance gap between the LS algorithm and TLS algorithms reveals
the necessity of TLS estimation. Higher requirements should be placed in the pointing
accuracy on XCand YCaxes than the ZCaxis in the horizon-based OPNAV.
− 4 − 3 − 2 − 1
01234
0
2
4
6
8
1 0 L S E W - T L S A G - T L S
R M S E , k m
(a) XCaxis.
− 4 − 3 − 2 − 1
01234
0
2
4
6
8
1 0 L S E W - T L S A G - T L S
R M S E , k m
(b) YCaxis.
− 4 − 3 − 2 − 1
01234
0
5 0
100
150
200
R M S E , k m
L S E W - T L S A G - T L S
(c) ZCaxis.
Figure 10. OPNAV performance with different pointing error in the XCaxis.
− 4 − 3 − 2 − 1
01234
0
2
4
6
8
1 0 L S E W - T L S A G - T L S
R M S E , k m
(a) XCaxis.
− 4 − 3 − 2 − 1
01234
0
2
4
6
8
1 0 L S E W - T L S A G - T L S
R M S E , k m
(b) YCaxis.
− 4 − 3 − 2 − 1
01234
0
5 0
100
150
200
R M S E , k m
L S E W - T L S A G - T L S
(c) ZCaxis.
Figure 11. OPNAV performance with different pointing error in the YCaxis.
Aerospace 2023,10, 371 18 of 22
− 4 0 − 2 0
0 2 0 4 0
0 . 0
0 . 1
0 . 2
0 . 3 L S E W - T L S A G - T L S
R M S E , k m
(a) XCaxis.
− 4 0 − 2 0
0 2 0 4 0
0 . 0
0 . 1
0 . 2
0 . 3 L S E W - T L S A G - T L S
R M S E , k m
(b) YCaxis.
− 4 0 − 2 0
0 2 0 4 0
0
1 0
2 0
3 0
R M S E , k m
L S E W - T L S A G - T L S
(c) ZCaxis.
Figure 12. OPNAV performance with different pointing error in the ZCaxis.
4.4. OPNAV Performance with Synthetic Images
To further validate the algorithm, the navigation performance after image processing
is tested. Blender [
42
], an open source 3D computer graphics software program, is used
for the synthesis of navigation images. Blender ’s realism and convenience have made it
popular in the field of OPNAV [
43
,
44
]. The specific steps of image synthesis and processing
are as follows:
(1)
First, the basic simulation environment is built in Blender, including building Mars
using its parameters and surface textures [45], setting up the navigation camera, and
adding a light source to model the Sun. The parameters of the navigation camera are
consistent with the previous section.
(2)
Next, automated rendering is implemented using the Python interface provided by
Blender. Specifically, Python scripts are used to set the position and attitude of the
sun and camera for each simulation scene, and the cycles engine is used to render
photorealistic images. The camera is always pointed with its back to the Sun and
toward the horizon of Mars to achieve good imaging.
(3)
Finally, MATLAB
r
is used for image processing of rendered images. The images
are convoluted by a Gaussian kernel with an STD of 1.5 pixels to simulate the effect
of defocusing, then the algorithm proposed in [
13
] is employed to achieve subpixel
edge localization.
The example synthetic images for OPNAV around Mars are shown in Figure 13.
It can
be seen that as the orbit altitude decreases, the horizon no longer bends and gradually
becomes straighter. OPNAV performance with synthetic images under different altitudes
is shown in Figure 14. Due to the small number of simulations, the performance gap
between the LS algorithm, EW-TLS algorithm, and AG-TLS algorithm is not significant.
The navigation
errors on the
XC
and
YC
axes are between
±
0.1 km, while the navigation
error on the
ZC
axis did not exceed
±
15 km. OPNAV performance with synthetic images at
an orbital altitude of 300 km under different latitudes is shown in Figure 15. The navigation
performance of the three algorithms seems to be similar on the XCand YCaxes. However,
there is a noticeable difference in their performance on the
ZC
axis. The RMSE on the
ZC
axis of the LS algorithm is 10.2 km, while the RMSE on the
ZC
axis of the EW-TLS
and AG-TLS algorithms are both 5.0 km. It can be seen that the LS algorithm showed
certain mean residuals, which are well eliminated by both the EW-TLS algorithm and the
AG-TLS algorithm. The performance of the proposed algorithms with synthetic images is
not as good as in numerical simulation due to the limitation of simulation accuracy and
edge positioning accuracy. However, the proposed algorithms can still bring considerable
improvement to the performance of horizon-based OPNAV under a short-arc horizon
scenario compared with the traditional algorithm.
Aerospace 2023,10, 371 19 of 22
(a) Orbital altitude 300 km.
(b) Orbital altitude 12,000 km.
Figure 13. Example synthetic images for OPNAV around Mars.
0 2,000 4,000 6,000 8,000 10,000 12,000
− 0 . 1 5
− 0 . 1 0
− 0 . 0 5
0 . 0 0
0 . 0 5
0 . 1 0
0 . 1 5 L S E W - T L S A G - T L S
A l t i t u d e , k m
0 2,000 4,000 6,000 8,000 10,000 12,000
− 0 . 1 5
− 0 . 1 0
− 0 . 0 5
0 . 0 0
0 . 0 5
0 . 1 0
0 . 1 5
A l t i t u d e , k m
0 2,000 4,000 6,000 8,000 10,000 12,000
− 2 0
− 1 0
0
1 0
2 0
A l t i t u d e , k m
Figure 14. OPNAV performance with synthetic images under different altitudes.
− 9 0 − 6 0 − 3 0
0 3 0 6 0 9 0
− 0 . 1 5
− 0 . 1 0
− 0 . 0 5
0 . 0 0
0 . 0 5
0 . 1 0
0 . 1 5 L S E W - T L S A G - T L S
L a t i t u d e , d e g
− 9 0 − 6 0 − 3 0
0 3 0 6 0 9 0
− 0 . 1 5
− 0 . 1 0
− 0 . 0 5
0 . 0 0
0 . 0 5
0 . 1 0
0 . 1 5
L a t i t u d e , d e g
− 9 0 − 6 0 − 3 0
0 3 0 6 0 9 0
− 1 0
− 5
0
5
1 0
1 5
2 0
L a t i t u d e , d e g
Figure 15.
OPNAV performance with synthetic images at an orbital altitude of 300 km under
different latitudes.
Aerospace 2023,10, 371 20 of 22
5. Conclusions
To improve the performance of horizon-based OPNAV based on a short-arc horizon,
a simplified
measurement covariance model is proposed by further analyzing the propaga-
tion of measurement errors. Then, an unbiased solution with EW-TLS estimation is given
by considering every element’s covariance of measurement equations. To further simplify
this problem, an AG-TLS algorithm is proposed, which achieves a non-iterative solution
by using the covariance of any one measurement as an approximation of all measurement
covariances. Covariance analysis demonstrates that the covariances of the traditional
Christian–Robinson algorithm with ordinary LS estimation and proposed algorithms are
identical, and the difference between them lies in the possible estimation bias.
The numerical simulations under different horizon arc lengths show that when the
horizon arc length is sufficiently long, both the traditional algorithm and the proposed
algorithms achieve near-zero mean residual with consistent accuracy. With the shortening
of the horizon arc length, the traditional algorithm will introduce a large mean residual that
may even be larger than the STD, while the proposed algorithms both showed impressive
advantages of near-zero mean residuals in the short-arc horizon scenario. Compared with
the EW-TLS algorithm, the AG-TLS algorithm trades a negligible accuracy loss for a huge
reduction in execution time and achieves a computing speed comparable to the traditional
algorithm. Thus, the AG-TLS algorithm is very suitable for onboard applications with
limited computing resources. Furthermore, as verified by the simulated navigation scenario
with a highly elliptical orbit around Mars, a short-arc horizon can provide reliable position
estimates for deep space exploration missions. Synthetic images with image processing
further validate the effectiveness of the OPNAV method. The proposed algorithms extend
the applicability of the horizon-based OPNAV method.
Author Contributions:
Conceptualization, H.D. and H.W.; Methodology, H.D.; Software, H.D.;
Validation, Y.L. and H.W.; Formal Analysis, H.D.; Resources, Z.J.; Draft Preparation, H.D. and
H.W.; Writing—Review and Editing, Y.L. and Z.J.; Visualization, H.D.; Supervision, Z.J.; Project
administration, Z.J.; Funding Acquisition, Z.J. All authors have read and agreed to the published
version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
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