MARCO REFERENCIAL

This section describes the lunar landing phases. The mission timeline is based on the NLL mission introduced in Section 2.1 and detailed in [21]. The targeted landing site is close to the Moon South Pole, i.e. at the Malapert’s peak(−86.024 deg latitude and 2.6133 deg longitude). The landing site altitude is 5076.7 m above the mean surface of the Moon.

The mission starts when the spacecraft is placed into a Low-Lunar Orbit (LLO). This orbit is circular at an altitude of 100 km. At this phase of the mission, the spacecraft position and velocity are estimated using ground-tracking techniques. This estimate is used to initialize the on-board navigation algorithm. The initial state accuracy is shown in Table 4.1. The communication delay and its limited accuracy prevent the utilization of the ground tracking localization during the next phases of the mission. When the descent is engaged, the on-board guidance and control software needs an accurate and real-time estimation of the spacecraft states to successfully complete the mission.

Table 4.1: State Error Covariance at DOI

States State Error Covariance (1σ)

Along Track Position 300 m

Cross Track Position 200 m

Vertical Position 25 m

Along Track Velocity 0.025 m/s

Cross Track Velocity 0.2 m/s

Vertical Velocity 0.25 m/s

At a downrange of about 6750 km, the Descent Orbit Insertion (DOI) is commanded from ground. After this point, the mission cannot be aborted. The thrusters are fired to modify the trajectory of the spacecraft, so it reaches an orbit with a periapsis of 10 km and an apoapsis of 100 km. During the maneuvers, thrust is augmented gradually to settle the propellant in the tanks. The thrust ramp up lasts about 100 s. When the estimated Δ𝑣 matches 19.57 m/s, the thrusters are turned off and the Descent Orbit (DO) phase starts. The burn Δ𝑣 is computed using Hohmann transfer techniques [136]. At a downrange of 1000 km, the Powered Descent (PD) phase is initiated. This phase consists in reducing the velocity of the spacecraft from orbital speed (1692 m/s) to velocities compatible with soft landing (1 m/s). At the beginning of the PD, the spacecraft follows a predefined optimal trajectory computed using numerical optimisation. The optimization problem is formulated following the Pontryagin’s maximum/minimum principle (minimum fuel to reach the landing site) [137]. The optimization algorithm finds the optimal trajectory of the vehicle from a downrange of 1000 km to

the touch down. This optimal trajectory shows that from 1000 km to 799.24 km of downrange, the vehicle must continue its coasting. It is only after this point that the thrusters are fired. Again, a thrust ramp up is used to settle the propellant. The maximum thrust is reached within 30 seconds. At 80 km of downrange, the Powered Explicit Guidance (PEG) algorithm is enabled [138]. This algorithm has been originally developed for the Space Shuttle. It aims at computing the thrust direction and magnitude to reach a desired terminal position 𝒓𝑑 and velocity 𝒗𝑑in a minimum time. In the 𝑥𝑧 plane

of the frame ℑ𝐿, the motion of the spacecraft is given by:

𝒓̈𝐿₌ 𝑓

𝑚𝝀𝑓𝐿+ 𝒈𝐿 (4.3)

where the rotation of the Moon is neglected, 𝒓̈𝐿 is the radial and the downrange acceleration of the spacecraft, 𝑚 is the spacecraft mass, 𝒈𝐿 _{is the gravitational acceleration, 𝑓 is the thrust magnitude}

and 𝝀_𝑓𝐿 is the thrust vector. The time history of the vector 𝝀_𝑓𝐿 is defined by:

𝝀𝑓𝐿=

𝝀0+ 𝝀̇(𝑡 − 𝑡0) ‖𝝀0+ 𝝀̇(𝑡 − 𝑡0)‖

(4.4)

where 𝝀₀ is a vector in the direction of the velocity-to-be gained by thrust, 𝝀̇ is a vector perpendicular to 𝝀0 representing the rate of change of 𝝀𝑓𝐿 and 𝑡0 is the reference time. It is noted that the

superscript designating in which frame 𝝀0 and 𝝀̇ are expressed is omitted to simplify the notation.

The parameters 𝝀0, 𝝀̇ and 𝑡0 are also referred to as steering parameters. At each guidance cycle, the

algorithm re-computes the steering parameters and the thrust magnitude using the current position and velocity of the vehicle. This is done using an efficient iterative approach summarized below: 1. Set the thrust magnitude 𝑓 at its value computed at the previous guidance cycle (or at its initial

value if the algorithm is run for the first time).

2. Assuming that 𝑓 is constant during the burn, solve for 𝝀0, 𝝀̇ and 𝑡0 (steering parameters) such

the spacecraft reaches a desired altitude ℎ𝑑 at a desired velocity 𝒗𝑑 in a minimum time (the

terminal conditions of the PEG for the particular problem of the lunar is shown in Table 4.2). 3. Adjust the thrust magnitude so the desired terminal conditions are reached in a desired

downrange 𝑠_𝑑.

The second step of the previous algorithm is not trivial. At each iteration, the steering parameters is computed using the following approach:

1. Set the altitude and the velocity to be gained by thrust, denoted respectively ℎ𝑔𝑜 and 𝒗𝑔𝑜, to

their value obtained at the previous guidance cycle (or a their initial value for the first run). 2. Compute the time to reach the desired terminal condition, denoted 𝑡𝑔𝑜 from the 𝒗𝑔𝑜 using the

rocket equation [138].

3. Compute the steering parameters from 𝑡𝑔𝑜, ℎ𝑔𝑜 and 𝒗𝑔𝑜. In order to do so, the thrust direction

is approximated by:

𝝀𝑓𝐿≈ 𝝀0+ 𝝀̇(𝑡 − 𝑡0) (4.5)

so Eq. (2.1) can be integrated analytically and solved for 𝝀0, 𝝀̇ and 𝑡0.

4. Predict the terminal altitude and velocity from the steering parameters computed at the previous step. This is done using a numerical integration of Eq. (4.3) and Eq. (4.4) or an approximated analytical method as presented in [138].

5. Use the difference between the predicted terminal spacecraft altitude and velocity and the desired ones to correct ℎ𝑔𝑜 and 𝒗𝑔𝑜 using a user-defined relaxation parameter.

At an altitude of 55 m, a downrange of 25 m, a vertical velocity of −10 m/s and a horizontal velocity of 12 m/s, the Gravity Turn (GT) segment is engaged. This guidance law lines up the engines of the vehicle to fire in the opposite direction of its current surface velocity vector. As the vehicle loses horizontal velocity, the gravity of the Moon pulls the spacecraft trajectory closer to a vertical descent. At 10 m of altitude or when the spacecraft reaches a vertical velocity of 1 m/s, the Terminal Descent (TD) phase starts. During this phase, the spacecraft descends at a constant velocity toward the ground. Finally, when the landing legs touch the ground, the main engines are cut-off. This phase is referred to as Main Engine Cut-Off (MECO). The mission phases are summarized in the following table:

Table 4.2: Mission Phases

Mission Phase Triggering

Principle

Low-Lunar Orbit Initial mode

Descent Orbit Insertion (DOI)

Propellant Settling Commanded from ground (downrange about _{6750 km)}

DOI Burn Time-based (100 s)

Descent Orbit Acceleration-based _{(estimated Δ𝑣 matches 19.57 m/s)}

Power Descent (PD)

Coasting Downrange-based (1000 km)

Propellant Settling Downrange-based (799.24 km)

Optimal Braking Time-based (30 s)

Powered Explicit Guidance (PEG) Segment

Downrange-based (80 km) and PEG convergence verified

Gravity-Turn (GT)Segment

PEG segment completed (altitude of 55 m, downrange of 25 m, vertical velocity of −10 m/s and horizontal velocity of 12 m/s)

Terminal Descent (TD) Target altitude(10 m) or descent rate reached (1

m/s)

Main Engine Cut-Off (MECO) Contact with ground