Environmental models .1 Trajectory generator

A trajectory generator has been developed with the help of a student so as to compute the position of the centre of gravity of a civil aviation aircraft along an RNAV/GNSS approach. An example of RNAV/GNSS procedure is shown in Figure 17.

Figure 17: RNAV/GNSS procedure for LILLE Lesquin airport [SIA,2010]

This type of procedures has standardized shapes which are “Y” and “T” approaches. The generator is currently only able to generate “T” approaches. The trajectory is expressed in the north-east-down coordinate frame centred at the runway threshold. The characteristics are fully customizable so as to allow representing a wide variety of airports. An example trajectory generated with this module is represented in Figure 18 and Figure 19.

The trajectory generated takes into account the velocity profile of the aircraft along the approach.

Below can be found an example of generated trajectory:

Figure 18: Lateral simulated trajectory profile

Figure 19: Vertical simulated trajectory profile

Aircraft Behaviour:

The behaviour of the aircraft along its trajectory has been taken into account. It means that we modelled the attitude of the aircraft that is to say pitch, roll and yaw angles. Raised cosine functions were used so as to generate continuous evolutions of these angles. An example is shown in Figure 20.

-122.55 -122.5 -122.45 -122.4 -122.35 -122.3

-0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01

LONGITUDE in DEG

LATITUDE in DEG

GROUND PROFILE OF TRAJECTORY

0 50 100 150 200 250 300

0 100 200 300 400 500 600

METERS

SECONDS VERTICAL PROFILE OF TRAJECTORY

Figure 20: Raised Cosine for attitude angles simulation

Knowing the position of the centre of gravity of the aircraft, the attitude of the aircraft and the relative position of the receiver antenna with respect to the gravity centre, we are able to compute the position of the GNSS receiver antenna along the RNAV GNSS approach. This trajectory is used as the reference position in our simulations.

One additional feature of the trajectory generator is that it is also capable to compute the position of the wings of the aircraft. This information can be used to detect signal outages due to wings during turns for example.

4.3.3.2 Satellite Position

Satellites position and velocities are generated on the basis of GALILEO and GPS almanacs. As presented in [SPILKER,1996], almanacs provide a reduced precision set of the ephemeris parameters used to compute position of GNSS satellites. For simulation purpose, it is sufficient to reproduce realistic positions and dynamics of the space vehicles that will generate the observation of the most significant geometry configurations. We remind here the algorithm used to compute GPS and GALILEO satellites position using almanacs.

The algorithm used in the case of almanacs is based on the user algorithm for ephemeris determination published in [ARINC, 2004]. Some of the parameters necessary for positioning using ephemeris are considered null in the case of almanacs thus resulting in far simpler equations and therefore an easier implementation.

Different formats of almanacs exist but in this PhD we used YUMA almanacs which have the following structure:

Phase 1 Phase 2 Phase 3 Maximum

Amplitude

Initial Amplitude

𝑀₀ Mean Anomaly at Reference Time

∆𝒏 Mean Motion Difference From Computed Value 𝒑 Eccentricity

√𝑨 Square Root of the Semi-Major Axis

𝛀𝟎 Longitude of Ascending Node of Orbit Plane at Weekly Epoch 𝐢_𝟎 Inclination Angle at Reference Time

𝝎 Argument of Perigee 𝛀̇ Rate of Right Ascension IDOT Rate of Inclination Angle

𝐂_𝒖𝒄 Amplitude of the Cosine Harmonics Correction Term to the Argument of Latitude

𝐂_𝒖𝒔 Amplitude of the Sine Harmonics Correction Term to the Argument of Latitude 𝐂_𝒑𝒄 Amplitude of the Cosine Harmonics Correction Term to the Orbit Radius 𝐂𝒑𝒔 Amplitude of the Sine Harmonics Correction Term to the Orbit Radius

𝐂_𝒊𝒄 Amplitude of the Cosine Harmonics Correction Term to the Angle of Inclination 𝐂_𝒊𝒔 Amplitude of the Sine Harmonics Correction Term to the Angle of Inclination 𝐭_𝒑𝒑 Reference Time Ephemeris

IODE Issue of Data

Table 14: YUMA almanacs parameters

To compute SVs positions we have to use the ephemeris algorithm with the following additional assumptions:

- Sinusoidal corrections are null i.e. : 𝛿𝑢_𝑘 = 𝛿𝑟_𝑘 = 𝛿𝑖_𝑘 = 0

- For the inclination angle, a nominal value of 0.30 semicircles is implicit, only the correction to the inclination 𝛿₁ is transmitted.

We remind here the user algorithm for ephemeris determination which gives the space vehicle antenna phase centre position in WGS-84 Earth-centred, Earth fixed reference frame [ARINC, 2004]:

𝜇 = 30986005. 10¹⁴ 𝑚³/𝑖² WGS-84 value of the Earth’s

universal gravitational parameter Ω̇_e= 7.2921151467. 10⁻⁵ 𝑟𝑎𝑟/𝑖 WGS-84 value of the Earth’s rotation

rate

𝐴 = �√𝐴�² Semi major axis

𝑛₀= �𝜇 𝐴³

Computed mean motion – rad/s

𝑡_𝑘= 𝑡 − 𝑡_𝑜𝑣 Time from ephemeris reference

epoch

𝑛 = 𝑛₀+ ∆𝑛 Corrected mean motion

𝑀_𝑘 = 𝑀₀+ 𝑛. 𝑡_𝑘 Mean anomaly

𝜋 = 3.1415926535898 GPS standard value for 𝜋

𝑀_𝑘 = 𝐸_𝑘− 𝑒. sin 𝐸_𝑘 Kepler’s equation for the eccentric

anomaly 𝐸_𝑘 – rad

𝜈_𝑘 = 𝑡𝑎𝑛⁻¹�_{cos 𝜈}^{sin 𝜈𝑘}

𝑘�=𝑡𝑎𝑛⁻¹�^√1−𝑣_{(𝑐𝑜𝑠 𝐸}^{2sin 𝐸𝑘�(1−𝑣.𝑐𝑜𝑠 𝐸𝑘)}

𝑘−𝑣) (1−𝑣.𝑐𝑜𝑠 𝐸⁄ _𝑘) � True anomaly 𝜈_𝑘 as a function of the eccentric anomaly

𝐸_𝑘 = 𝑐𝑜𝑖⁻¹� 𝑒 + cos 𝜈_𝑘

1 + e. cos 𝜈𝑘� Eccentric anomaly

Φk= 𝜈𝑘+ 𝜔 Argument of Latitude

𝛿𝑢_𝑘 = 𝐶_𝑢𝑠sin 2Φ_k+ 𝐶_𝑢𝑐cos 2Φ_k Argument of latitude correction 𝛿𝑟𝑘 = 𝐶𝑣𝑠sin 2Φk+ 𝐶𝑣𝑐cos 2Φk Radius correction

𝛿𝑖_𝑘 = 𝐶_𝑖𝑠sin 2Φ_k+ 𝐶_𝑖𝑐cos 2Φ_k Inclination correction

𝑢_𝑘 = Φ_k+ 𝛿𝑢_𝑘 Corrected argument of latitude

𝑟_𝑘 = 𝐴(1 − 𝑒. cos 𝐸_𝑘) + 𝛿𝑟𝑘 Corrected radius

𝑖_𝑘 = 𝑖₀+ 𝛿𝑖_𝑘 + (𝐼𝐷𝑂𝑇). 𝑡_𝑘 Corrected inclination

𝑥𝑘′= 𝑟𝑘. cos 𝑢𝑘

Satellite position in orbital plane 𝑦_𝑘^′ = 𝑟_𝑘. sin 𝑢_𝑘

Ω_k = Ω₀+ �Ω̇ − Ω̇e�𝑡_𝑘− Ω̇_et_𝑜𝑣 Corrected longitude of ascending node

𝑥_𝑘 =𝑥_𝑘^′. cosΩ_k−𝑦_𝑘^′. cos 𝑖_𝑘. sin Ω_k 𝑦𝑘 = 𝑥𝑘′. sin Ωk+ 𝑦𝑘′. cos 𝑖𝑘. cos Ωk

𝑧_𝑘 = 𝑦_𝑘^′. sin 𝑖_𝑘

Satellite position in Earth-centred, Earth-fixed coordinates

Table 15: Ephemeris equations [ARINC, 2004]

To compute an estimate of the space vehicles velocities we derivate the expressions presented previously. Here are the equations allowing computing the Space Vehicle velocities in the WGS-84 coordinate frame.

𝑀𝑘̇ = 𝑛 𝐸𝑘̇ = 𝑀_𝑘̇

(1 − 𝑒. cos 𝐸𝑘)

𝜈_𝑘̇ = sin 𝐸𝑘 . 𝐸_𝑘̇ 1 + 𝑒. cos 𝜈_𝑘 sin 𝜈_𝑘. (1 − 𝑒. cos 𝐸_𝑘) 𝑢_𝑘̇ =𝜈_𝑘̇ + 2(𝐶_𝑢𝑠. cos 2𝑢_𝑘− 𝐶_𝑢𝑐. sin 2𝑢_𝑘). 𝜈_𝑘̇ 𝑟_𝑘̇ = 𝐴. 𝑒. 𝑛. sin 𝐸𝑘

1 − 𝑒. cos 𝐸𝑘 + 2(𝐶_𝑣𝑠. cos 2𝑢_𝑘− 𝐶_𝑣𝑐. sin 2𝑢_𝑘). 𝜈_𝑘̇ 𝚤_𝑘̇ = 𝐼𝐷𝑂𝑇 + 2(𝐶_𝑖𝑠. cos 2𝑢_𝑘− 𝐶_𝑖𝑐. sin 2𝑢_𝑘). 𝜈_𝑘̇

(𝑥𝑘̇ =′) 𝑟𝑘̇ cos 𝑢𝑘−𝑦_𝑘^′. 𝑢𝑘̇

(𝑦𝑘̇ = 𝑟′) 𝑘̇ sin 𝑢𝑘+𝑥_𝑘^′.𝑢𝑘̇ Ω_k̇ = �Ω̇ − Ω̇e�

Satellites velocity in Earth-centred, Earth-fixed coordinates:

𝑥𝑘̇ = �(^𝑥𝑘̇′)₋𝑦𝑘′cos(𝑖𝑘) Ωk̇ �^{. cos}Ωk−�𝑥_𝑘^′.Ωk̇ + (𝑦𝑘̇ cos𝑖′) 𝑘− 𝑦𝑘′sin(𝑖𝑘) 𝚤𝑘̇ �^sinΩk

𝑦𝑘̇ = �(^𝑥𝑘̇′)₋𝑦𝑘′cos(𝑖𝑘) Ωk̇ �^{. sin}Ωk+�𝑥_𝑘^′.Ωk̇ + (𝑦𝑘̇ cos(𝑖′) 𝑘) − 𝑦𝑘′sin(𝑖𝑘) 𝚤𝑘̇ �^cosΩk

𝑧_𝑘̇ = (𝑦_𝑘̇ sin𝑖^′) 𝑘+ 𝑦_𝑘^′cos(𝑖_𝑘) 𝚤𝑘̇

Table 16: Ephemeris Velocity Equations

4.3.3.2.1 Distance computation

The different models gathered in this section are used so as to build a realistic true distance between an aircraft receiver and GNSS satellites. The true geometrical distance can be written as:

𝜌_𝑖(𝑘) = �(𝑋𝑠𝑖 (𝑘) − 𝑋𝑣(𝑘))²+ (𝑌_𝑠^𝑖(𝑘) − 𝑌𝑣(𝑘))²+ (𝑍_𝑠^𝑖(𝑘) − 𝑍𝑣(𝑘))²

(4-57)

Where:

- �𝑋𝑠𝑖 , 𝑌𝑠𝑖 , 𝑍𝑠𝑖 � is the position of satellite 𝑖 - (𝑋𝑣, 𝑌_𝑣, 𝑍_𝑣) is the receiver position

We have already presented the two modules which provide the satellites positions and the user position. The next step is to provide the obtained distance to the IQ generator.