Coordinate frames

In order to provide navigation and therefore the kinematic state of a vehicle, a specific point of the vehicle must first be selected. This may be the center of mass, the geometrical center or another convenient point; for radio positioning, the phase center of the antenna is typically adopted. Second, if the kinematic state includes the orientation and the angular motion, a set of three axes must be selected as well. Typically, the axes are mutually perpendicular, with one axis corresponding to the direction of motion.

Furthermore, some form of reference frame is needed, relative to which the kinematic state of the vehicle is expressed. This reference frame is defined by an origin and a set of axes.

The origin, of either an object or a reference, together with a set of axes define a coordinate frame.

Any navigation problem then involves at least two coordinate frames: the object frame, which describes the body whose kinematic state is desired, and the reference frame, which describes a known body (e.g. the Earth), relative to which the object’s kinematic state is desired.

2.5.1.1 Earth-Centered Inertial frame

By definition, in physics, an inertial frame is any coordinate frame that does not accelerate or rotate with respect to the rest of the Universe; thus an Earth-centered frame is not strictly an inertial frame since the Earth actually accelerates in its heliocentric orbit, its spin axis slowly moves and, additionally, the galaxy rotates. However, these effects are negligible if compared to the navigation sensors’ noise. For this reason, an Earth-centered inertial (ECI) coordinate frame is practically treated as a true inertial frame [25]. The origin of the ECI coordinate system is the geocenter. According to [24], “the xy-plane is taken to coincide with the Earth’s equatorial plane, the +x-axis is permanently fixed in a particular direction relative to the celestial sphere, the +z-axis is taken normal to the xy-plane in the direction of the north pole, and the +y- axis is chosen so as to form a right-handed coordinate system”. In order to fully define the ECI frame, it is necessary to provide temporal information. Indeed, the Earth’s shape is oblate, and due mainly to the gravitational effects of Sun and Moon, the Earth’s equatorial plane moves with respect to the celestial sphere (and, as already mentioned, the spin axis slowly moves). Since the x-axis is defined with respect to the celestial sphere and the z-axis is defined with respect to the equatorial plane, the ECI frame as defined earlier would not to be truly inertial. The solution to this problem is to define the orientation of the axes at a particular instant in time, or epoch.

In this thesis, as in the GPS ECI coordinate system, the z-axis of this system is perpendicular to the Earth’s mean equator at epoch J2000, the x-axis points to the vernal equinox of the Earth’s mean orbit at epoch J2000, while the y-axis is perpendicular to the yz-plane in such a way to define a right-handed xyz coordinate system. The epoch J2000 corresponds to the Julian Date 2451545.0, equivalent to January 1, 2000, 12 hours Terrestrial Time (TT) [48]. Since the orientation of the axes remains fixed over time, the ECI coordinate system defined in this way can be considered inertial for GNSS purposes [24].

Figure 2:13 Earth mean equator and equinox of the J2000 coordinate system [48].

2.5.1.2 Radial, In-track, Cross-track

Another frame used in this thesis, is the radial, in-track, cross-track (RIC) frame, identified by the unit vectors <, => and ?<.

In this frame < lies along the instantaneous radius vector, => lies in the orbit plane normal to < and in the direction of motion of the spacecraft, and ?< is normal to the orbit plane and lies along the angular momentum vector. Even though <and => rotate together with the radius vector of the spacecraft, the frame is fixed at each instant in time. This means that these unit vectors do not have to be differentiated when transforming velocity to this frame, which thus has the same magnitude in this frame as in the ECI frame [49]. This coordinate frame is very useful to show the difference between two orbits in the radial, in-track, and cross-track directions. A good quantitative description of such a coordinate frame is provided in [49] and it is here summarized as follows.

In order to compute the transformation matrix from ECI to RIC, let us assume to have position @ and velocity @A vectors in the ECI frame, expressed as:

@ = BĈ + EF̂ + GH< (2:4)

@A = BAĈ + EAF̂ + GAH< (2:5)

Where Ĉ, F̂ and H< are respectively the unit vectors along ECI x-, y-, and z- axes. We can define:

< ≡ @ = B Ĉ + E F̂ + G H< (2:6) ?< ≡J_{ℎ =‖@ K @A‖}@ K @A (2:7)

Where

J = @ K @A (2:9)

Where K denotes the vectorial product. In matrices this is equivalent to:

M<=> ?<N = O PQ PR PS 9Q 9R 9S Q R S T MĈF̂ H<N (2:10) Where UVW+:+W≡ O PQ PR PS 9Q 9R 9S Q R S T (2:11)

where the elements of U_VW+:+Ware the direction cosines of the RIC unit vectors with respect to the ECI frame, given in equations (2:6), (2:7) and (2:8). Thus,

MP9 N = UVW+:+WM X Y ZN and M X Y ZN = UVW+ :+W[_MP_{9 N} (2:12)

In order to compute the difference between two orbits using the <, => , ?< directions, first one orbit has to be chosen as the reference, where @ and @A are the position and velocity of the orbit. Then, we use equations (2:6), (2:7) and (2:8) to compute the unit vectors <, => , ? <and equation (2:11) to compute U_VW+:+Wusing @ and @A. We compute the position and velocity difference in the ECI frame,

M∆X∆Y

∆ZN = ∆@, M ∆XA ∆YA

∆ZAN = ∆@A (2:13)

Finally, we use equation (2:12) for the differences in the radial, in-track and cross-track directions.

M∆P∆9 ∆ N = UVW+ :+W_M∆X_∆Y ∆ZN, M ∆PA ∆9A ∆ AN = UVW+ :+W_M∆XA_∆YA ∆ZAN (2:14)