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Citation: Jin, X.; Liu, X.; Guo, J.;
Zhou, M.; Wu, K. A Novel
All-Weather Method to Determine
Deflection of the Vertical by
Combining 3D Laser Tracking
Free-Fall and Multi-GNSS Baselines.
Remote Sens. 2022,14, 4156.
https://doi.org/10.3390/rs14174156
Academic Editor: Dong Liu
Received: 3 July 2022
Accepted: 22 August 2022
Published: 24 August 2022
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remote sensing
Article
A Novel All-Weather Method to Determine Deflection of the
Vertical by Combining 3D Laser Tracking Free-Fall and
Multi-GNSS Baselines
Xin Jin 1, Xin Liu 1, *, Jinyun Guo 1, Maosheng Zhou 2and Kezhi Wu 1
1
College of Geodesy and Geomatics, Shandong University of Science and Technology, Qingdao 266590, China
2Institute of Oceanographic Instrument, Qilu University of Technology (Shandong Academy of Sciences),
Qingdao 266100, China
*Correspondence: skd994268@sdust.edu.cn; Tel.: +86-0532-8605-7276
Abstract:
The bright stars in the clear night sky with weak background lights should be observed in
the traditional deflection of the vertical (DOV) measurement so that the DOV cannot be observed
under all-weather conditions, which limits its wide applications. An all-weather DOV measurement
method combining three-dimensional (3D) laser tracking free-fall and multi-GNSS baselines is
proposed in this paper. In a vacuum environment, the 3D laser tracking technique is used to
continuously track and observe the motion of free-fall with high frequency and precision for obtaining
3D coordinate series. The plumb line vector equation is established to solve the gravity direction
vector in the coordinate system of the laser tracker at the measuring point using least squares fitting
coordinate series. Multi-GNSS observations are solved for obtaining the precise geodetic cartesian
coordinates of the measuring point and GNSS baseline information. A direction transformation
method based on the baseline information proposed in this paper is used to convert the gravitational
direction vector in the laser tracker coordinate system into the geodetic cartesian coordinate system.
The geodetic cartesian coordinates of the measuring point are used to calculate the ellipsoid normal
vector, and the angle between this and the gravity direction vector in the geodetic cartesian coordinate
system is estimated to obtain the astrogeodetic DOV. The DOV is projected to the meridian and prime
vertical planes to obtain the meridian and prime vertical components of the DOV, respectively. The
astronomical latitude and longitude of the measuring point are calculated from these two components.
The simulation experiments were carried out using the proposed method, and it was found that
the theoretical precision of the DOV measured by the method could reach 0.2”, which could realise
all-weather observation.
Keywords: deflection of the vertical; free fall; 3D laser tracking technique; GNSS
1. Introduction
The deflection of the vertical (DOV) is the angle between the plumb line and the
normal to the reference ellipsoid, indicating the inclination of the geoid with respect to
the reference ellipsoid, and characterising the spatial and temporal inhomogeneity of
the mass distribution within the earth [
1
,
2
]. The DOV is indeed a basic observation of
geodesy [
3
–
5
], which determines the gravity direction and contains rich high-frequency
information about the gravity field [
6
]. It can be used to infer the size and shape of
the mean earth ellipsoid [
7
,
8
], refine the geoid model, convert astrogeodetic data [
9
,
10
],
and correct precision engineering measurements [
11
]. The time-varying study of DOV
verifies that there is a very close relationship between its change and earthquakes, which
provides a new means of monitoring large earthquakes [
12
,
13
]. In conclusion, the DOV can
provide rich detailed information about the earth’s gravity field and the geoid, which is
of great significance in the fields of geodetic applications, geophysical inversion, resource
exploration, and seismic and volcano monitoring [2,14–16].
Remote Sens. 2022,14, 4156. https://doi.org/10.3390/rs14174156 https://www.mdpi.com/journal/remotesensing
Remote Sens. 2022,14, 4156 2 of 15
The common methods for measuring DOV include astronomical geodesy, gravimetry,
astronomical gravimetry, and GNSS levelling [
1
,
17
]. Astronomical geodesy generally uses
high-precision astronomical theodolites or digital zenith camera systems [
18
–
20
] to observe
low-magnitude stars for achieving high-precision DOV [
21
,
22
]. However, such instruments
are cumbersome and inefficient in measurement, and can only observe lower-magnitude
stars on clear nights, not all-weather. The gravimetry essentially uses methods such as the
Stokes formula or the Vening Meinesz formula to integrate gravity anomalies on geoids to
calculate the DOV [
23
,
24
]. However, the method assumes no disturbances outside the geoid
and global gravity anomalies are known, neither of which can be achieved, so it is rarely
used independently. The astronomical gravimetry method is a combination of astronomical
geodesy and gravimetry to determine DOV. Therefore, this method, like astronomical
geodesy, is seriously constrained by the measurement environment. The GNSS-levelling
method uses height anomaly to calculate DOV [
1
,
25
], requiring the survey area to be
flat. The measurement method is relatively low in accuracy and efficiency, which has
some limitations. In general, these traditional methods to determine DOV have problems
such as difficulty in measurement, being time consuming, susceptibility to environmental
conditions, and an inability to observe under all-weather conditions. It seriously limits the
application of DOV. Therefore, all-weather and high-precision DOV measurement is still a
pressing challenge in the world [2,15,22].
The three-dimensional (3D) laser tracking technique is a polar coordinate-measuring
method with high measuring speed and precision. With characteristics such as a data
sampling rate of up to 1000~3000 Hz, the laser tracker measurement system with sub-
micrometre level precision can track the target automatically and realise accurate and
dynamic three-dimensional tracking measurement [
26
,
27
]. Based on this, an all-weather
DOV measurement method combining 3D laser tracking free-fall and multi-GNSS baselines
is proposed in this paper.
The structure of this paper is as follows: Section 2introduces the principles for the
all-weather method to determine DOV by combining 3D laser tracking free-fall and multi-
GNSS baselines. Section 3conducts simulation experiments to analyse the theoretical
precision of the proposed method, and separately discusses the influence of laser tracking
and GNSS errors on DOV. Section 4contains the conclusion.
2. Principle and Methodology
The schematic diagram and technical route of DOV measurement using the proposed
method are shown in Figures 1and 2. The method involves two kinds of coordinate systems:
one is the spatial cartesian coordinate system established with the centre of the laser tracker
as the origin, called laser tracker coordinate system (LTCS), and the other is the geodetic
cartesian coordinate system (GCCS). Firstly, the central coordinates of the GNSS antennas
are measured using the laser tracker, and they are differentiated to obtain the baseline
information in LTCS. The laser tracker is used to dynamically track and measure the 3D
coordinates of the free-fall target. Then, the gravity direction vector in LTCS is calculated
using least squares (LS) fitting 3D coordinate series. With the sub-micrometre precision
distance constraint of baseline, the GNSS data are processed to obtain high-precision
geodetic coordinates (or geodetic cartesian coordinates) of the measuring point P and the
GNSS antennas’ baseline vectors in GCCS. According to the baseline information in LTCS,
the direction transformation parameters between LTCS and GCCS are calculated to convert
the gravity direction vector in LTCS into GCCS. According to the ellipsoid normal vector
determined by the geodetic coordinates of the point P, the astrogeodetic DOV is estimated
and projected onto the meridian and prime vertical planes to obtain the meridional and
prime vertical components of the DOV, respectively. Finally, the astronomical coordinates
are obtained from the two components and the geodesic coordinates of the point P.
Remote Sens. 2022,14, 4156 3 of 15
RemoteSens.2022,14,xFORPEERREVIEW3of16
ridionalandprimeverticalcomponentsoftheDOV,respectively.Finally,theastronomi‐
calcoordinatesareobtainedfromthetwocomponentsandthegeodesiccoordinatesof
thepointP.
Figure1.SketchofDOVmeasurement.1:Base,2:measuringcylinder,3:lasertracker,4:GNSSan‐
tenna,5:free‐falltarget.
Laser tracker and
device leveling
Dynamic tracking the free-
fall t arget using laser trac ker
for 3D coordinates
Measuring GNSS
antenna center by
laser tracker
GNSS data acquisition
GNSS data processing
(
Distance constrai nt of
baseline
)
High-precision
GNSS baseline
information
High-precision
measuring point P
geodetic coordinates
Gross error elimination
Using a new method of direction
transformation
to obtain the direction
transformation
parameters between
LTCS and GCCS
Obtaining gravity direction
vector in LTCS using least-
square s fitting 3D
coordinates
Calculating the
normal vector of
ellip soid at the
measuring point P
Normal vector of the
meridian and prime
vertical planes
Gravity direction vector
in GCCS
Astrogeodetic DOV
Meridian andprime
vertical components of
DOV
Astronomical coordinates
Distance constraint
Direction conversion
Projection
of baseline
Figure2.TechnicalrouteofDOVmeasurement.
Figure 1.
Sketch of DOV measurement. 1: Base, 2: measuring cylinder, 3: laser tracker, 4: GNSS
antenna, 5: free-fall target.
RemoteSens.2022,14,xFORPEERREVIEW3of16
ridionalandprimeverticalcomponentsoftheDOV,respectively.Finally,theastronomi‐
calcoordinatesareobtainedfromthetwocomponentsandthegeodesiccoordinatesof
thepointP.
Figure1.SketchofDOVmeasurement.1:Base,2:measuringcylinder,3:lasertracker,4:GNSSan‐
tenna,5:free‐falltarget.
Laser tracker and
device leveling
Dynamic tracking the free-
fall t arget using laser trac ker
for 3D coordinates
Measuring GNSS
antenna center by
laser tracker
GNSS data acquisition
GNSS data processing
(
Distance constrai nt of
baseline
)
High-precision
GNSS baseline
information
High-precision
measuring point P
geodetic coordinates
Gross error elimination
Using a new method of direction
transformation
to obtain the direction
transformation
parameters between
LTCS and GCCS
Obtaining gravity direction
vector in LTCS using least-
square s fitting 3D
coordinates
Calculating the
normal vector of
ellip soid at the
measuring point P
Normal vector of the
meridian and prime
vertical planes
Gravity direction vector
in GCCS
Astrogeodetic DOV
Meridian andprime
vertical components of
DOV
Astronomical coordinates
Distance constraint
Direction conversion
Projection
of baseline
Figure2.TechnicalrouteofDOVmeasurement.
Figure 2. Technical route of DOV measurement.
2.1. Extracting Gravity Direction Vector
In a vacuum environment, the laser tracker is used to track the free-falling target
with high precision and dynamics for obtaining its three-dimensional coordinate series
in LTCS. According to the coordinate series, the LS and total least squares (TLS) fitting
methods [28–30] is used to calculate the gravity direction vector in LTCS, respectively.
Remote Sens. 2022,14, 4156 4 of 15
2.1.1. Spatial Linear Fitting Using LS
Assume that the coordinates of the free-falling target in LTCS are
Li(xi,yi,zi)i=1, 2, · · · ,k. The equation of the fitted space line is:
x−x0
l=y−y0
m=z−z0
n(1)
where
(l,m,n)
is the direction vector of the space straight line, and
(x0,y0,z0)
is the coordi-
nate of a point on the space straight line. Equation (1) is converted into a spatial straight
line projective equation:
x=l
n(z−z0) + x0=alz+bl
y=m
n(z−z0) + y0=clz+dl
(2)
where al=l
n,bl=x0−l
nz0,cl=m
n,dl=y0−m
nz0.
The observation equations are constructed to satisfy:
Qx=k
∑
i=1[xi−(alzi+bl)]2=min
Qy=k
∑
i=1[yi−(clzi+dl)]2=min
(3)
Based on the data of
k
3D coordinates, the values of
al
,
bl
,
cl
,
dl
are estimated by solving
the equations with the LS, and the gravity direction vector
gLTCS
of the free-falling trajectory
in LTCS is obtained as alcl1.
2.1.2. Spatial Linear Fitting Using TLS
The spatial straight line projective Equation (2) is transformed into matrix form [30]:
x
y=z100
0 0 z1
al
bl
cl
d
(4)
According to Equation (4), the error equation is constructed:
Vl=z100
0 0 z1
ˆ
al
ˆ
bl
ˆ
cl
ˆ
dl
−x
y(5)
Let
Bl=z100
0 0 z1
,
Ll=x
y
,
ˆ
Xl=ˆ
alˆ
blˆ
clˆ
dlT
, and then Equation (5) is
simplified to:
Vl=Blˆ
Xl−Ll(6)
Based on the 3D observations
Li
of the spatial line, the error Equation (6) is solved
using the TLS iterative method to meet the adjustment criteria:
k
∑
i=1ˆ
Li−Li2+
j=t,i=k
∑
j=1,i=1ˆ
Bij
l−Bij
l2=min (7)
By substituting Equation (6) and taking derivatives of each element, the iterative
equation is obtained:
(ˆ
BT
lˆ
Xl=ˆ
BT
lLl
Nbˆ
BT
l=BT
l+ˆ
XlLT
l
(8)
where
Nb=E+ˆ
Xlˆ
XT
l
. The algorithm flow of the TLS iterative method is shown in Figure 3.
Remote Sens. 2022,14, 4156 5 of 15
RemoteSens.2022,14,xFORPEERREVIEW5of16
ˆ
llll
VΒXL
(6)
Basedonthe3Dobservationsi
Lofthespatialline,theerrorEquation(6)issolved
usingtheTLSiterativemethodtomeettheadjustmentcriteria:
,
22
11,1
ˆˆmin
jtik
k
ij ij
ii l l
iji
LL B B (7)
BysubstitutingEquation(6)andtakingderivativesofeachelement,theiterative
equationisobtained:
ˆˆ ˆ
ˆˆ
TT
ll ll
TT T
bl l ll
BX BL
NB B XL (8)
whereˆˆ
T
bll
NEXX
.ThealgorithmflowoftheTLSiterativemethodisshowninFigure3.
Input the i nitial values of 4
parameters f or the space line
Calculate the initial values of the
coefficient matrix based on the
observed information
Obtaining the adjustment values of
4-para meter for the space line
Obtaining iterative values of
4-para meter for the space line
Yes
No
1
ˆˆ
ii
ll
XX
ˆ
l
B
Figure3.FlowchartoftheTLSiterativemethod.
2.1.3.ComparativeAnalysisofLSandTLS
Toillustratetheaccuracyofthetwomethodsforextractinggravitydirectionvectors,
wecarryoutsimulationexperiments.Assumingthatthepointprecisionofthelaser
trackeris10μm,thefallingdistanceofthefree‐falltargetisabout1m,andthelasertrack‐
ingmeasurementfrequencycanreach2000Hz.Thetheoreticalaccuracyofthetwometh‐
odsforextractinggravitydirectionvectorsiscalculatedby1000free‐fallingsimulation
experiments.Theaverageof30observationswasrecordedasoneobservation,andthe
resultsareshowninFigure4andTable1.
FromFigure4andTable1,itcanbeseenthattheSTDandRMSofLSandTLSare
thesame,bothbetterthan0.15,indicatingthattheaccuracyofthetwomethodsiscon‐
sistent.However,sincetheTLSinvolvesaniterativecalculation,whichismorecompli‐
catedandtime‐consumingcomparedtoLS,theLSischosenforextractinggravitydirec‐
tionvectorinthispaper.
Figure 3. Flowchart of the TLS iterative method.
2.1.3. Comparative Analysis of LS and TLS
To illustrate the accuracy of the two methods for extracting gravity direction vectors,
we carry out simulation experiments. Assuming that the point precision of the laser tracker
is 10
µ
m, the falling distance of the free-fall target is about 1 m, and the laser tracking
measurement frequency can reach 2000 Hz. The theoretical accuracy of the two methods for
extracting gravity direction vectors is calculated by 1000 free-falling simulation experiments.
The average of 30 observations was recorded as one observation, and the results are shown
in Figure 4and Table 1.
RemoteSens.2022,14,xFORPEERREVIEW6of16
Figure4.ComparisonofgravitydirectionvectorsofLSandTLS.
Table1.StatisticsofgravitydirectionvectorsofLSandTLS(″).
MAXMINMEANSTDRMS
LS0.1920.1020.1460.0140.146
TLS0.1920.1020.1460.0140.146
2.2.SpaceDirectionVectorTransformation
IntheproposedDOVmeasurementmethod,itisnecessarytoconvertthegravity
directionvectorinLTCStoGCCS,butthistransformationinvolvestherotationofthe
largerotationangle,andthetraditionalBursamodelisnolongerapplicable.Basedonthe
nonlinear13‐parametertransformationmodelwithalargerotationangle[31,32],thispa‐
perproposesadirectiontransformationmethodusingbaselineinformationtorealisethe
transformationofthegravitydirectionvectorinLTCStoGCCS.
SeveralGNSSantennasarefixedontothemeasuringdevice,andthecoordinatesof
theantennaphasecentreinLTCSaremeasuredusingthelasertracker,andthedistance
betweeneachantennaiscalculatedtoobtainthebaselineinformation.Becausethemeas‐
urementprecisionofthelasertrackerreachesthesub‐micrometrelevel,thesub‐microme‐
treprecisiondistanceconstraintisaddedtoGNSSbaselinesolutiontoobtainhigh‐preci‐
sionGNSSbaselines.Basedonseveralbaselines,thedirectiontransformationparameters
aresolvedasshownbelow.
Basedonthenonlinear13‐parametertransformationmodel,therelationshipbetween
thecommonpointi
A
inGCCSandLTCScanbeexpressedas:
ii
ii
ii
AA
AA
AA
Xx
Tx
Y
y
T
y
Tz
Zz
R1, 2, ,in(9)
where
,,
iii
AAA
x
yz isthecoordinateofthecommonpointi
A
inLTCS,
,,
ii i
AA A
X
YZ
isthecoordinateofthecommonpointi
A
inGCCS,
123
123
123
=
aaa
bb b
ccc
R,
123 123 123
,, ,, ,,aaa bbb ccc，， arethedirectioncosinesofthex,y,andzaxesinGCCS,
isthescaleparameter,and
,,Tx Ty Tz arethetranslationparameters.
SincetherotationmatrixRisanorthogonalmatrixandthecorrespondingcoordi‐
natetransformationisanorthogonaltransformation,theconditionalequationisobtained:
Figure 4. Comparison of gravity direction vectors of LS and TLS.
Table 1. Statistics of gravity direction vectors of LS and TLS (”).
MAX MIN MEAN STD RMS
LS 0.192 0.102 0.146 0.014 0.146
TLS 0.192 0.102 0.146 0.014 0.146
From Figure 4and Table 1, it can be seen that the STD and RMS of LS and TLS are the
same, both better than 0.15, indicating that the accuracy of the two methods is consistent.
However, since the TLS involves an iterative calculation, which is more complicated and
Remote Sens. 2022,14, 4156 6 of 15
time-consuming compared to LS, the LS is chosen for extracting gravity direction vector in
this paper.
2.2. Space Direction Vector Transformation
In the proposed DOV measurement method, it is necessary to convert the gravity
direction vector in LTCS to GCCS, but this transformation involves the rotation of the
large rotation angle, and the traditional Bursa model is no longer applicable. Based on
the nonlinear 13-parameter transformation model with a large rotation angle [
31
,
32
], this
paper proposes a direction transformation method using baseline information to realise the
transformation of the gravity direction vector in LTCS to GCCS.
Several GNSS antennas are fixed onto the measuring device, and the coordinates of the
antenna phase centre in LTCS are measured using the laser tracker, and the distance between
each antenna is calculated to obtain the baseline information. Because the measurement
precision of the laser tracker reaches the sub-micrometre level, the sub-micrometre precision
distance constraint is added to GNSS baseline solution to obtain high-precision GNSS
baselines. Based on several baselines, the direction transformation parameters are solved
as shown below.
Based on the nonlinear 13-parameter transformation model, the relationship between
the common point Aiin GCCS and LTCS can be expressed as:
XAi
YAi
ZAi
=ψR
xAi
yAi
zAi
+
Tx
Ty
Tz
i=1, 2, · · · ,n(9)
where
xAi,yAi,zAi
is the coordinate of the common point
Ai
in LTCS,
XAi,YAi,ZAi
is the
coordinate of the common point
Ai
in GCCS,
R=
a1a2a3
b1b2b3
c1c2c3
,
(a1,a2,a3)
,
(b1,b2,b3)
,
(c1,c2,c3)
are the direction cosines of the x, y, and z axes in GCCS,
ψ
is the scale parameter, and (Tx,Ty,Tz)are the translation parameters.
Since the rotation matrix
R
is an orthogonal matrix and the corresponding coordinate
transformation is an orthogonal transformation, the conditional equation is obtained:
a12+a22+a32=1
b12+b22+b32=1
c12+c22+c32=1
a1a2+b1b2+c1c2=0
a1a3+b1b3+c1c3=0
a2a3+b2b3+c2c3=0
(10)
From Equation (10), it can be seen that there are only three independent parameters in
the rotation matrix
R
, taking
a2
,
a3
,
b3
as the independent parameters. The remaining six
parameters are expressed as:
a1=p1−a22−a32
c3=p1−a32−b32
b1=−a1a3b3−a2c3
1−a32
b2=p1−b12−b32
c1=a2b3−a3b2
c2=a3b1−a1b3
(11)
According to Equation (9), the common points
Ai
and
Aj
respectively construct equa-
tions. The two equations are differenced to eliminate the translation parameters. The scale
parameters will be added to each baseline separately:
∆XAij
∆YAij
∆ZAij
=ψij R
∆xAij
∆yAij
∆zAij
(12)
Remote Sens. 2022,14, 4156 7 of 15
where
ψij
is the scale parameter of the baseline
Aij
,
∆XAij ∆YAij ∆ZAi j
are the GNSS
baselines of common points
Ai
and
Aj
, and
∆xAij ∆yAi j ∆zAij
are the common point
baselines in LTCS.
According to more than three common baselines, the rotation matrix
R
can be obtained
by solving Equation (12) using the LS method. However, the rotation matrix
R
only has
three independent parameters, and the remaining six parameters are its nonlinear functions.
It is very complicated to directly solve Equation (12), so the first-order expansion of Taylor
series is used to solve it.
Assume that the unknowns are nine direction cosine parameters,
N
scale parameters,
and
N
is the number of common baselines. Then, Equation (12) is expanded by Taylor
series to obtain:
∆XAij
∆YAij
∆ZAij
=ψ0
ij
a0
1a0
2a0
3
b0
1b0
2b0
3
c0
1c0
2c0
3
∆xAij
∆yAij
∆zAij
i
+
a0
1∆xAij +a0
2∆yAij +a0
3∆zAij
b0
1∆xAij +b0
2∆yAij +b0
3∆zAij
c0
1∆xAij +c0
2∆yAij +c0
3∆zAij
dψij +
ψ0
ij ∆xAij ψ0
ij ∆yAij ψ0
ij ∆zAij 0 0 0 0 0 0
0 0 0 ψ0
ij ∆xAij ψ0
ij ∆yAij ψ0
ij ∆zAij 0 0 0
0 0 0 0 0 0 ψ0
ij ∆xAij ψ0
ij ∆yAij ψ0
ij ∆zAij
da1da2da3db1db2db3dc1dc2dc3T
(13)
where the variables with superscript 0 represent the approximate value of the variable,
and
dψij
,
da1
,
da2
,
da3
,
db1
,
db2
,
db3
,
dc1
,
dc2
,
dc3
are the corrections. Equation (13) is converted
into the form of the error equation to obtain:
Vdt =Adt Xdt −Ldt (14)
where
Vij
dt =hV∆XAij V∆YAi j V∆ZAij iT
denotes the corrections of each GNSS baseline
Aij
in the X, Y, and Z directions, respectively, and
Xdt =dψij da1da2da3db1db2db3dc1dc2dc3T.
According to Equation (11), the conditional equation is listed to obtain:
A0
dtXdt +Wdt =0 (15)
where Xdt means the same as above, and A0
dt and Wdt are:
A0
dt =
0 2a0
12a0
12a0
10 0 0 0 0 0
0 0 0 0 2b0
12b0
22b0
3000
0 0 0 0 0 0 0 2c0
12c0
22c0
3
0a0
2a0
10b0
2b0
10c0
2c0
10
0a0
30a0
1b0
30b0
1c0
30c0
1
0 0 a0
3a0
20b0
3b0
20c0
3c0
2
;
Wdt =
a0
1+a0
2+a0
3−1
b0
1+b0
2+b0
3−1
z0
1+z0
2+z0
3−1
a0
1a0
2+b0
1b0
2+c0
1c0
2
a0
1a0
3+b0
1b0
3+c0
1c0
3
a0
2a0
3+b0
2b0
3+c0
2c0
3
.
To improve the calculation efficiency, the conditional equation is transformed into a
pseudo-observation equation, and then a new error equation is formed with Equation (14):
Vdt
V0
dt=Adt
A0
dtXdt +Ldt
Wdt(16)
The parameters
Xdt
are obtained by solving Equation (11) using the indirect adjustment.
Remote Sens. 2022,14, 4156 8 of 15
The iterative computation is performed to obtain the rotation matrix
R
, and the steps
are as follows:
(i)
The approximate value is generally desirable:
ψij =1, R=
100
010
001
. (17)
(ii) The error equation is composed according to Equation (11), and if there are
N
baselines,
3N+6 error equations can be composed.
(iii)
The corrections of N+9 unknowns are solved by 3N+6 equations.
(iv)
The latest values of unknowns are calculated.
(v)
According to the corrections, judge whether the convergence requirement is satisfied.
If not, repeat steps (ii) to (v) until the convergence is satisfied.
(vi)
According to the rotation matrix RTobtained iteratively, the gravity direction vector
gLTCS in LTCS is converted to GCCS:
gGCCS =RTgLTCS (18)
where gLTCS is the gravity direction vector in GCCS.
2.3. Calculation of Astrogeodetic DOV
The astrogeodetic DOV, also known as the relative DOV, has relative significance
because the normals of the measured points under different reference ellipsoids are different,
and the astrogeodetic DOV of the point is also different. Users can obtain the DOV of
the corresponding ellipsoid through different ellipsoid transformations. The WGS1984
ellipsoid is selected as an example to illustrate the principle of the proposed method in
the paper.
The semi-major axis of the reference ellipsoid corresponding to GCCS is
a
. The earth’s
ellipticity is
f
, and the semi-minor axis of the ellipsoid is
b=a−a∗f
. The reference
ellipsoid can be expressed as:
X2
a2+Y2
a2+Z2
b2=1 (19)
Let
F(X,Y,Z)=X2
a2+Y2
a2+Z2
b2−
1, and for X, Y, and Z, the first order derivatives are:
FX(X,Y,Z)=2
a2·X
FY(X,Y,Z)=2
a2·Y
FZ(X,Y,Z)=2
b2·Z
(20)
Assuming that the geodetic cartesian coordinates of the measuring point P measured
by GNSS are (X0,Y0,Z0), the ellipsoid normal vector through point P is:
p=2
a2·X0,2
a2·Y0,2
b2·Z0(21)
In GCCS, the meridian plane through point P can be determined by the space vectors
(
0, 0, 1
)
and
(X0
,
Y0
,
Z0)
. Assuming that the normal direction vector of the meridian plane
is (Xme,Yme,Zme), there are:
Zme =0
X0Xme +Y0Yme +Z0Zme =0(22)
Let
Xme=1
, and the normal direction vector of the meridian plane through point P is
pme1, −X0
Y0, 0.
Remote Sens. 2022,14, 4156 9 of 15
The prime vertical plane through point P can be determined by the space vectors
p2
a2·X0,2
a2·Y0,2
b2·Z0
and
pme1, −X0
Y0, 0
. Assuming that the normal direction vector
of the prime vertical plane is Xpr,Ypr,Zpr , there are:
(2
a2X0Xpr +2
a2Y0Ypr +2
b2Z0Zpr =0
Xpr −X0
Y0Ypr =0(23)
Let
Xpr=1
, and the normal direction vector of the prime vertical plane through point
P is ppr1, Y0
X0,−b2
a2·X02+Y02
X0Z0.
According to the gravity direction vector
gGCCS
and ellipsoid normal vector
p
, the
astrogeodetic DOV uis calculated as:
u=arccos gGCCS ·p
|gGCCS ||p|(24)
The gravity direction vector
gGCCS
is projected onto the meridian and prime vertical
planes, respectively:
gme =gGCCS −pme
|pme |·gGCCS ·pme
|pme |
gpr =gGCCS −ppr
ppr
·gGCCS ·pp r
ppr
(25)
where
gme
is the meridian component of
gGCCS
, and
gpr
is the prime vertical component of
gGCCS .
The meridian and prime vertical components of astrogeodetic DOV are calculated as:
ξ=arccos gme·p
|gme ||p|
η=arccos gpr·p
gpr|p|
(26)
where
ξ
is the meridian component of astrogeodetic DOV, and
η
is the prime vertical
component of astrogeodetic DOV.
According to the meridian and prime vertical components of astrogeodetic DOV at
the measuring point P, and the geodetic coordinate
(B,L)
, the astronomical longitude and
latitude (ϕ,λ)are calculated using the following relationship:
ϕ=B+ξ
λ=L+ηsec ϕ(27)
3. Simulation Study
3.1. Theoretical Precision of DOV
The main error sources of the proposed method are laser tracker measurement error
and GNSS measurement error. The influence of laser tracker error on the DOV measure-
ment is reflected in two aspects: one is that when tracking the free-falling target, the
measurement errors will make the measured gravity direction vector different from the
theoretical vector. Second, the errors of the laser tracker will affect the common point
coordinates during the direction transformation between the GCCS and LTCS, and then
affect the direction transformation parameters. Similarly, the GNSS baselines are used
to solve the direction transformation parameters between the GCCS and LTCS, and its
solution precision determines the accuracy of the rotation matrix. The GNSS coordinate
precision of the measurement point P also affects the ellipsoidal normal vector, but its effect
on the ellipsoidal normal vector is negligible due to the large lengths of the semi-major and
semi-minor axes of the earth.
The simulation experiments are carried out to analyse the theoretical precision of the
proposed measurement method. Assuming that the point precision of the laser tracker
is 10
µ
m, the falling distance of the free-fall target is about 1 m, and the laser tracking
measurement frequency can reach 2000 Hz, so the observation time of each free-fall mea-
surement is about 0.45 s, and the sampling data reaches about 900. Before observing the
Remote Sens. 2022,14, 4156 10 of 15
free fall, the fixed GNSS antennas were measured 1000 times to determine the coordinates
in LTCS. Suppose the GNSS positioning precision is 1 mm, and there are four GNSS devices
in total. Due to the addition of the distance constraints of sub-micrometre precision during
the GNSS baseline solving, the calculated GNSS baseline accuracy is set to 0.1 mm. The
reference geodetic cartesian coordinates of measuring point P are: X =
−
2,148,744.4656 m,
Y = 4,426,641.1849 m, and Z = 4,044,656.0516 m. The ellipsoid WGS1984 is selected, the
ellipsoid semi-major axes a = 6,378,137 m, and the earth’s ellipticity f= 1/298.257223563.
Firstly, a 3D coordinate series of the falling free-fall target is simulated, and the laser
tracker error is added to the theoretical coordinates to generate the simulated observations
in LTCS. The method using the fixed GNSS antennas to obtain the direction transformation
parameters of GCCS and LTCS is as follows: the theoretical values of the fixed GNSS
antennas in LTCS are known in advance, and the laser tracker error is added to the the-
oretical coordinates to generate simulation observations of the fixed GNSS antennas in
LTCS. The theoretical coordinates of the fixed GNSS antennas in LTCS and the theoretical
transformation parameters between the two coordinate systems are used to obtain the
theoretical coordinates of the fixed GNSS antennas in GCCS, and then the theoretical coor-
dinates of each GNSS antenna are differenced to obtain the theoretical values of the baseline
vectors. According to the solving precision of the GNSS baseline, the error is added to
the GNSS baseline vector to obtain its simulated observations. Based on the simulated
observations of the fixed GNSS antennas in LTCS and the simulated observation baseline
vectors in GCCS, the direction transformation parameters between the two coordinate
systems affected by the errors of the laser tracker and GNSS can be obtained using the
space direction vector transformation method in Section 2.3. The direction transformation
parameters with errors between GCCS and LTCS are used to calculate the gravity direction
vector containing errors in GCCS. Combined with the ellipsoid normal vector determined
by the geodetic coordinates of the measuring point P, the astrogeodetic DOV with errors
is calculated. Finally, the theoretical precision of DOV can be obtained by calculating the
difference between the error-containing DOV and the theoretical DOV.
Four GNSS antennas are evenly distributed in a circle with the radius
r
centred on the
laser tracker. To consider the size of the measuring device, we separately discussed the
precision of the DOV for different radius
r
. The average of 30 observations was recorded
as one observation, and the above experiments were repeated 1000 times. The results are
shown in Figure 5and Table 2.
RemoteSens.2022,14,xFORPEERREVIEW12of16
theprecisionoftheDOVfordifferentradiusr.Theaverageof30observationswasrec‐
ordedasoneobservation,andtheaboveexperimentswererepeated1000times.There‐
sultsareshowninFigure5andTable2.
Table2.PrecisionstatisticsofDOVwithdifferentradiusr(″).
Radiusr2m2.5m3m3.5m
u
MAX0.910.780.630.72
MIN−0.85−0.75−0.72−0.48
MEAN0.00−0.01−0.010.01
STD0.280.240.200.17
RMS0.280.240.200.17
ξ
MAX1.130.910.690.76
MIN−0.99−0.85−0.84−0.65
MEAN−0.01−0.01−0.010.02
STD0.320.280.230.20
RMS0.320.280.230.20
η
MAX1.120.880.830.54
MIN−1.31−0.93−0.65−0.55
MEAN0.000.000.000.00
STD0.330.250.220.19
RMS0.330.250.220.19
ItcanbeseenfromFigure5andTable2thattheproposedmeasurementmethodhas
goodstability,theerrorfluctuatesbetween±1.2″,andtheaveragevalueapproaches0.
WiththeincreaseoftheradiusrofthecircleformedbyfourGNSSantennas,themeas‐
urementprecisionoftheDOVbecomeshigher,whichbasicallymeetstherequirementsof
thefirst‐classastronomicalprecision(0.3″)inChina.Whentheradiusris3.5m,thepreci‐
sioncanreach0.2″.
Figure5.DistributionofDOVindifferentradiusr(″).
3.2.InfluenceofLaserTrackingMeasurementErroronDOV
TodiscusstheinfluenceofthelasertrackingmeasurementerrorontheDOV,itis
assumedthattheGNSSbaselinesolutioniserror‐free,andonlythelasertrackingmeas‐
urementerrorisaddedtothesimulatedtheoreticalvalues,andtheremainingconditions
Figure 5. Distribution of DOV in different radius r(”).
Remote Sens. 2022,14, 4156 11 of 15
Table 2. Precision statistics of DOV with different radius r(”).
Radius r 2 m 2.5 m 3 m 3.5 m
u
MAX 0.91 0.78 0.63 0.72
MIN −0.85 −0.75 −0.72 −0.48
MEAN 0.00 −0.01 −0.01 0.01
STD 0.28 0.24 0.20 0.17
RMS 0.28 0.24 0.20 0.17
ξ
MAX 1.13 0.91 0.69 0.76
MIN −0.99 −0.85 −0.84 −0.65
MEAN −0.01 −0.01 −0.01 0.02
STD 0.32 0.28 0.23 0.20
RMS 0.32 0.28 0.23 0.20
η
MAX 1.12 0.88 0.83 0.54
MIN −1.31 −0.93 −0.65 −0.55
MEAN 0.00 0.00 0.00 0.00
STD 0.33 0.25 0.22 0.19
RMS 0.33 0.25 0.22 0.19
It can be seen from Figure 5and Table 2that the proposed measurement method has
good stability, the error fluctuates between
±
1.2”, and the average value approaches 0. With
the increase of the radius
r
of the circle formed by four GNSS antennas, the measurement
precision of the DOV becomes higher, which basically meets the requirements of the first-
class astronomical precision (0.3”) in China. When the radius
r
is 3.5 m, the precision can
reach 0.2”.
3.2. Influence of Laser Tracking Measurement Error on DOV
To discuss the influence of the laser tracking measurement error on the DOV, it is
assumed that the GNSS baseline solution is error-free, and only the laser tracking mea-
surement error is added to the simulated theoretical values, and the remaining conditions
are consistent with Section 3.1. The above experiments were repeated 1000 times, and the
results are shown in Figure 6and Table 3.
RemoteSens.2022,14,xFORPEERREVIEW13of16
areconsistentwithSection3.1.Theaboveexperimentswererepeated1000times,andthe
resultsareshowninFigure6andTable3.
Table3.InfluencestatisticsoflasertrackingerroronDOV(″).
Radiusr2m2.5m3m3.5m
u
MAX0.070.070.070.06
MIN−0.07−0.07−0.07−0.06
MEAN0.000.000.000.00
STD0.020.020.020.02
RMS0.020.020.020.02
ξ
MAX0.070.070.070.07
MIN−0.06−0.07−0.07−0.07
MEAN0.000.000.000.00
STD0.020.020.020.02
RMS0.020.020.020.02
η
MAX0.070.070.070.06
MIN−0.08−0.07−0.07−0.07
MEAN0.000.000.000.00
STD0.020.020.020.02
RMS0.020.020.020.02
ItcanbeseenfromFigure6andTable3thattheinfluenceofthelasertrackingmeas‐
urementerrorontheDOVis±0.07″,theSTDoftheDOVis0.02″,andtheeffectremains
consistentforeachdirectionanddoesnotchangeasthecircleradiusrincreases.Ingen‐
eral,theinfluenceoflasertrackingmeasurementontheDOVaccountsforabout2%ofthe
totallevel,whichisrelativelystable.
Figure6.InfluencedistributionoflasertrackingerroronDOVindifferentradiusr(″).
3.3.InfluenceofGNSSErroronDOV
TodiscusstheinfluenceoftheGNSSerroronDOV,itisassumedthatthelasertrack‐
ingmeasurementerroris0,andonlytheGNSSerrorisaddedtothesimulatedtheoretical
valueoftheGNSSbaseline,andtheremainingconditionsareconsistentwithSection3.1.
Theaboveexperimentswererepeated1000times,andtheresultsareshowninFigure7
andTable4.
Figure 6. Influence distribution of laser tracking error on DOV in different radius r(”).
Remote Sens. 2022,14, 4156 12 of 15
Table 3. Influence statistics of laser tracking error on DOV (”).
Radius r 2 m 2.5 m 3 m 3.5 m
u
MAX 0.07 0.07 0.07 0.06
MIN −0.07 −0.07 −0.07 −0.06
MEAN 0.00 0.00 0.00 0.00
STD 0.02 0.02 0.02 0.02
RMS 0.02 0.02 0.02 0.02
ξ
MAX 0.07 0.07 0.07 0.07
MIN −0.06 −0.07 −0.07 −0.07
MEAN 0.00 0.00 0.00 0.00
STD 0.02 0.02 0.02 0.02
RMS 0.02 0.02 0.02 0.02
η
MAX 0.07 0.07 0.07 0.06
MIN −0.08 −0.07 −0.07 −0.07
MEAN 0.00 0.00 0.00 0.00
STD 0.02 0.02 0.02 0.02
RMS 0.02 0.02 0.02 0.02
It can be seen from Figure 6and Table 3that the influence of the laser tracking
measurement error on the DOV is
±
0.07”, the STD of the DOV is 0.02”, and the effect
remains consistent for each direction and does not change as the circle radius
r
increases.
In general, the influence of laser tracking measurement on the DOV accounts for about 2%
of the total level, which is relatively stable.
3.3. Influence of GNSS Error on DOV
To discuss the influence of the GNSS error on DOV, it is assumed that the laser tracking
measurement error is 0, and only the GNSS error is added to the simulated theoretical
value of the GNSS baseline, and the remaining conditions are consistent with Section 3.1.
The above experiments were repeated 1000 times, and the results are shown in Figure 7
and Table 4.
RemoteSens.2022,14,xFORPEERREVIEW14of16
Table4.InfluencestatisticsofGNSSerroronDOV(“).
Radiusr2m2.5m3m3.5m
u
MAX0.810.730.580.50
MIN−0.83−0.69−0.57−0.57
MEAN0.010.000.000.00
STD0.290.250.200.17
RMS0.290.250.200.17
ξ
MAX1.000.860.680.59
MIN−1.04−0.86−0.63−0.68
MEAN0.000.000.000.00
STD0.330.280.230.20
RMS0.330.280.230.20
η
MAX1.050.940.610.53
MIN−0.98−0.94−0.74−0.48
MEAN0.010.000.000.01
STD0.320.250.210.18
RMS0.320.250.210.18
Figure7.InfluencedistributionofGNSSerroronDOVindifferentradiusr(″).
AscanbeseenfromFigure7andTable4,theGNSSbaselinesolutionerroristhe
decisivefactoraffectingtheDOV,andthesingleinfluenceofGNSSisbasicallyconsistent
withthesimulationprecisionoftheproposedmethod.Withtheincreaseofthecirclera‐
diusrformedbythefourGNSSantennas,themeasurementprecisionoftheDOVin‐
creasesgradually,whichisduetotheincreaseofthebaselinelength,resultinginthede‐
creaseoftherelativeinfluenceofGNSSerror.Overall,GNSSerroristhekeyfactorforthe
all‐weatherDOVmeasurementmethodproposedinthepaper.
4.Conclusions
Anovelall‐weathermethodtodeterminetheDOVbycombining3Dlasertracking
free‐fallandmulti‐GNSSbaselineswasproposed.Thelasertrackerwasusedtodynami‐
callyobservethethree‐dimensionalcoordinateseriesofthefree‐fallingtargetatthemeas‐
uringpoint,andthegravitydirectionvectorinthelasertrackercoordinatesystemwas
obtainedbytheleastsquaresfittingcoordinateseries.Inthispaper,adirectiontransfor‐
mationmethodusingbaselineinformationwasproposedtoconvertthegravitydirection
vectorinthelasertrackercoordinatesystemtothegeodeticcartesiancoordinatesystem.
Figure 7. Influence distribution of GNSS error on DOV in different radius r(”).
Remote Sens. 2022,14, 4156 13 of 15
Table 4. Influence statistics of GNSS error on DOV (”).
Radius r 2 m 2.5 m 3 m 3.5 m
u
MAX 0.81 0.73 0.58 0.50
MIN −0.83 −0.69 −0.57 −0.57
MEAN 0.01 0.00 0.00 0.00
STD 0.29 0.25 0.20 0.17
RMS 0.29 0.25 0.20 0.17
ξ
MAX 1.00 0.86 0.68 0.59
MIN −1.04 −0.86 −0.63 −0.68
MEAN 0.00 0.00 0.00 0.00
STD 0.33 0.28 0.23 0.20
RMS 0.33 0.28 0.23 0.20
η
MAX 1.05 0.94 0.61 0.53
MIN −0.98 −0.94 −0.74 −0.48
MEAN 0.01 0.00 0.00 0.01
STD 0.32 0.25 0.21 0.18
RMS 0.32 0.25 0.21 0.18
As can be seen from Figure 7and Table 4, the GNSS baseline solution error is the
decisive factor affecting the DOV, and the single influence of GNSS is basically consistent
with the simulation precision of the proposed method. With the increase of the circle radius
r
formed by the four GNSS antennas, the measurement precision of the DOV increases
gradually, which is due to the increase of the baseline length, resulting in the decrease of the
relative influence of GNSS error. Overall, GNSS error is the key factor for the all-weather
DOV measurement method proposed in the paper.
4. Conclusions
A novel all-weather method to determine the DOV by combining 3D laser tracking free-
fall and multi-GNSS baselines was proposed. The laser tracker was used to dynamically
observe the three-dimensional coordinate series of the free-falling target at the measuring
point, and the gravity direction vector in the laser tracker coordinate system was obtained
by the least squares fitting coordinate series. In this paper, a direction transformation
method using baseline information was proposed to convert the gravity direction vector
in the laser tracker coordinate system to the geodetic cartesian coordinate system. The
ellipsoid normal vector at the measuring point can be calculated by the derived formula
from the geodetic coordinates obtained by GNSS measurement, and then the astrogeodetic
DOV of the measuring point can be estimated. The DOV was projected to the meridian
and prime vertical planes to obtain the meridian and prime vertical components of the
DOV, respectively. The astronomical latitude and longitude can be calculated according
to meridian and prime vertical components of DOV and geodetic coordinates. Through
theoretical analysis, the method has good feasibility.
The main error sources of the proposed DOV measurement method are laser tracking
and GNSS measurement errors. The control variable method was adopted to discuss
its influence on the DOV. The results showed that the contribution of the laser tracking
measurement error on the DOV was only 2%, and remained stable. The GNSS error was
the key factor of the proposed all-weather DOV measurement method. Overall, the pro-
posed method can simultaneously determine the astrogeodetic DOV and the astronomical
longitude and latitude at the measuring point. Compared with the traditional methods
of astronomical geodesy and astronomical gravimetry, the proposed method is hardly
affected by the climate and environment, it can be observed in all weathers, and has a high
theoretical precision of 0.2”.
Author Contributions:
Conceptualization, X.J. and X.L.; methodology, X.J., X.L. and J.G.; software,
X.J. and M.Z.; validation, X.J., X.L., J.G. and M.Z.; formal analysis, X.J.; investigation, X.J. and
Remote Sens. 2022,14, 4156 14 of 15
K.W.; resources, X.L. and J.G.; data curation, X.J. and K.W.; writing—original draft preparation, X.J.;
writing—review and editing, X.L.; visualization, X.J.; supervision, J.G.; project administration, X.L.;
funding acquisition, X.L. and J.G. All authors have read and agreed to the published version of the
manuscript.
Funding:
This research was supported by the National Natural Science Foundation of China, grant
numbers 41774001 and 41704015; the Autonomous and Controllable Special Project for Survey-
ing and Mapping of China, grant number 816-517; and the SDUST Research Fund, grant number
2014TDJH101.
Data Availability Statement: Not applicable.
Acknowledgments:
This research is supported by the National Natural Science Foundation of China
(Grant Nos. 41774001 and 41704015), the Autonomous and Controllable Special Project for Surveying
and Mapping of China (Grant No. 816-517), and the SDUST Research Fund (Grant No. 2014TDJH101).
Conflicts of Interest: The authors declare no conflict of interest.
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