International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B1. Istanbul 2004
where r, (1 =1,2,3; j =1,2,3) are elements of the
orientation matrix RM, which includes the
direction cosines of 3 rotation angles, o,
@ and K. G denotes the geodetic
coordinate of ground points, and E denotes
the geodetic coordinates of exposure
stations. It should be mentioned that Eq. 1 is based on a
frame sensor. If the onboard camera is a linear sensor, Eq. 1
can be modified accordingly. The detailed description can be
found in Zhou er a/. (2000). With Eq. 1, we can simultaneously
determine the six absolute DOF and six relative DOF
separately.
3.1.1 Simultaneous Determination of Six Absolute DOF
Equation 1 is traditionally used to determine absolute position
and attitude of a single sensor/satellite, if the appropriate
calibration (sensor interior elements, offset between GPS
antenna and exposure center) is finished. If we connect many
overlapping single images/satellites into a block, and extend
Eq. 1 for a block situation, we can simultaneously compute the
instantaneous absolute position and attitude (absolute 6 DOF)
of all satellites from a number of GPS-based navigation data
and a number of tie points (conjugate points), which connect
adjacent images. The basic principle of this technique is to tie
overlapping images together without the need for ground
control points (GCPs) in each image stereo-model. The input
to the aerial model includes measured image coordinates of tie
points that appear in as many images as possible and the GPS-
based navigation data (or ground coordinates of GCPs). The
x2 2 2 2 2 2 2 2 2 2 2 2 2 2 T 2 T. 2 T x2
a qudX; taodY. ta.dZ, ta da tado, a di. aj 4X € v a, dY; + ay dz —/
Si
For G, in satellite 4:
xq. 4 T 4 7 4 T p
uem ajdX. t asdY; c asdZ; -]
7G,
y4 — 4 T. 4 gyT 4 T y4
Va 7 aydX t axdY; * axdZ; -l; (4)
For point G,, we have obtained 6 observation equations, 9
unknowns (6 absolute navigation elements, 1 tie point ground
coordinates). Similarly, for point G», we can obtain the 6
observation equations. In total, we have 12 observation
equations, containing 12 unknowns (6 absolute DOF, 2 tie point
coordinates). Combining these observation equations, we can
solve the 6 absolute position and attitude of the satellite 2
through least square method. This principle demonstrates that
absolute navigation parameters (absolute 6 DOF) of multi-
satellites can simultaneously be determined, resulting in high
and symmetric accuracy and high-reliability of multi-satellites.
Moreover, not all satellites in the formation flying are required
to mount a GPS receiver.
3.1.2 Simultaneous Determination of Six Relative DOF
The relative states between the satellites are of much greater
interest for formation flying. The goal of the relative
navigation is to estimate the relative state of the formation, i.e.,
where the satellites are located with respect to each other. To
this end, a specific GPS antenna on one of the satellites is
typically selected as a formation reference point. The satellite
associated with this reference point will be referred to as the
“master” or “reference” satellite, with the rest called
system outputs the absolute position and attitude (absolute 6
DOF) of all the satellites (imaging sensor) as well as the ground
coordinates of the tie points. Theoretically, this computational
model can link several hundred satellites (imaging sensors)
together.
The satellite absolute state includes the position, attitude, and
velocity, which are expressed in the ground-frame. All
calculations and integration are also performed in the ground-
frame. Support that the ground point G, and G, are imaged
into gi, g; and g! as well as ul. gi and a in the image
plane 1, 2, and 4 are acquired by the satellites 1, 2, and 4,
respectively. Also support that the absolute position and
attitudes for the satellite 1 and 4 are provided by GPS-based
navigation system. (We also assume that the calibration
between the EO sensor exposure center and the GPS antenna
are implemented.) The absolute position and attitude (6 DOF)
of satellite 2 (and all satellites) can be determined as follows:
For G, in satellite 1:
Nl | 7 1 T | T xl
v= a, dX + Ad Ya t adZo —!6,
v^ = ai dx + Doll + a -HIq
G,
For G; in satellite 2:
a
Vv'7 = a5, dX} +andY} + aidZ, + ay dw, + ade; + adi; + az,dX + a dy + a dz — 7 (3)
"followers". For relative state estimation, the selection of the
reference is arbitrary (it can be any satellite in the formation).
The determination of relative state (position and orientation)
can be conducted by coplanarity condition. For example,
suppose that there is a ground point G, the imaged points in the
“master” image plane acquired by the “master” satellite
(satellite 1 in Figure 1) and the second image plane acquired by
the follower (satellite 2 in Figure 1) is g, and g,. The
coplanarity states that the “master” and “follower” exposure
stations (E, and E;), the object point (G), and the “master” and
"follower" imaged points (g; and g;), all lie in a common plane.
The mathematical model of coplanarity in an established
auxiliary coordinate system, E,-UVW is given by,
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