Global Optimization Approach for the Ascent Problem of Multi-stage Launchers

Abstract

This paper deals with a trajectory optimization problem for a three-stage launcher with the aim to minimize the consumption of propellant needed to steer the launcher from the Earth to the GEO (geostationary orbit). Here we use a global optimization procedure based on Hamilton-Jacobi-Bellman approach and consider a complete model including the transfer from a GTO (geostationary transfer orbit) to the GEO. This model leads to an optimal control problem involving also some optimization parameters that appear in the flight phases. First, an adequate formulation of the control problem is introduced. Then, we discuss some theoretical results related to the value function and to the reconstruction of an optimal trajectory. Numerical simulations are given to show the relevance of the global optimization approach. This work has been undertaken in the frame of CNES Launchers’ Research and Technology program.

Appendices

Appendix 1 Definition of Orbital Parameters

For the phase 3 of the flight sequence there are two kinds of orbital maneuvers:

• Injection on a GTO orbit

• Orbital transfer from GTO to the GEO.

We define here all necessary parameters to describe these maneuvers.

Fig. 13

GEO and GTO orbits

1.1 Appendix 1.1 GTO Injection Parameters

A GTO orbit is represented on Fig. 13. We denote A and P the perigee and apogee centers of the orbit respectively. The line of nodes is the line of intersection of the orbital plane with the equatorial plane of the Earth. It’s extremal points are respectively ascending (NA) and descending (ND) nodes of the orbit. A GTO orbit can be fully defined with the following set of parameters $(r_a,r_p,\Upsilon ,\omega ,\mathbf {i})$, where:

• $r_a$ and $r_p$ are respectively the distances from the center of the Earth of the perigee and the apogee of the orbit

• $\Upsilon$ is the polar position of the perigee P, measured positively from the ascending node NA.

• $\mathbf{{i}}$ is the inclination of the orbital plane with respect to the equatorial plane of the Earth.

On can deduce from these parameters the semi-major axis a, and the eccentricity e by:

$$\begin{aligned}a=\frac{r_a+r_p}{2}, \quad e=\frac{r_a-r_p}{r_a+r_p}. \end{aligned}$$

A point Q on the orbit is defined by its angular position $\varsigma$, called true anomaly (see Fig. 13). Let $c_0=3.986\cdot 10^{14}\,m^3\,s^{-2}$ be the Earth’s gravitational constant. Given orbital parameters defined below and the true anomaly of a point Q on the orbit, one can deduce the corresponding orbit radius $r_Q(\eta )$ and orbit velocity $v_Q(\varsigma )$ as follows:

$$\begin{aligned} r_Q(\varsigma )=\frac{r_ar_p}{r_a(1+\cos (\varsigma ))+r_p(1-\cos (\varsigma ))}, \quad v_Q(\varsigma )=\sqrt{\frac{c_0}{a}\frac{1+e^2+2e\cos (\varsigma )}{1-e^2}} \end{aligned}$$

(37)

In this study we consider a family of GTOs with a fixed perigee altitude $r_p^0$ and that intersect a given GEO. Recall that a GEO is a circular orbit in the equatorial plane of the Earth. It is fully defined by the data of its radius $r_{GEO}$. If a GTO intersects a GEO of radius $r_{GEO}$, this fixes the orbit radius of its ascending node NA. Note that by definition (see Fig. 13) the true anomaly of the point NA is $- \Upsilon$. Then we have the following condition:

$$ r_{NA}=r_Q(- \Upsilon )=r_{GEO} $$

from which we deduce the apogee altitude:

$$ r_a=\frac{r_{GEO}\, r_p (1-\cos ( \Upsilon ))}{2r_p - r_{GEO}(1+\cos ( \Upsilon ))}. $$

So, if we fix the perigee altitude $r_p$ and the GEO radius $r_{GEO}$ the parameter $r_a$ is also fixed. Then the set of considered GTO orbits is defined by the inclination $\mathbf {i}\in [i_{min},i_{max}]$ The perigee orientation $\Upsilon$ is fixed. For each GTO orbit we define a segment of injection $\mathcal {C}_\mathbf{{i}}$ as a neighborhood of the perigee.

1.2 Appendix 1.2 Orbital Transfer Parameters

In our study, we assume that the GTO- GEO orbital transfer is performed through an impulse boost (to change the velocity’s modulus and direction). The amount of propellant required for the orbital transfer is determined by the choice of the GTO via Tsiolkovsky’s formula. Let $\overrightarrow{V}_{GTO}$ be the speed at the ascending node of the GTO, and $\mathbf {V}_{GEO}$ be the desired GEO speed. Then the differential gear to provide the vehicle is the vector difference:

$$ \varDelta \mathbf {V}=\mathbf {V}_{GEO}-\mathbf {V}_{GTO} $$

with the modulus:

$$\begin{aligned} \varDelta V=\sqrt{V_{GEO}^2+V_{GTO}^2-2V_{GTO}V_{GEO}\cos (\mathbf {i})}. \end{aligned}$$

(38)

Note that the speed $V _{GTO}$ can be expressed using (37) for the ascending node:

$$ V_{GTO}=v_{NA}=v_{Q}(-\omega ) $$

The formula Tsiolkovsky connects the differential speed to provide the required fuel mass

$$\begin{aligned} \varDelta V=g\,I_{sp}\,\ln \left( \frac{m_{init}}{m_{final}}\right) \end{aligned}$$

(39)

where $m_{init}$ and $m_{final}$ are respectively the launcher’s masses before and after the operation. Thus one can calculate the weight of propellant required to the GTO-GEO orbital transfer in a single impulse, knowing the speed at the height of the GTO and its inclination with respect to the GEO.

Table 5 Numerical data

Appendix 2 Numerical Data Used for Simulations

Table 5 gives the list of numerical constants and data functions describing the dynamical properties of the launcher and the atmospheric model used in our computations. Furthermore, Fig. 14 represents the air density (and sound speed) models, the thrust (for boosters and stage E1), and drag function. All these data were provided by CNES (Table 6).

Fig. 14

Tabulated data used in the computations

Table 6 Numerical data corresponding to the CNES reference trajectory