SDRE Guidance and Navigation for Spacecraft Relative Motion

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Universit`

a di Pisa

Dipartimento di Ingegneria dell’ Informazione

Laurea Magistrale in

Ingegneria Robotica e dell’ Automazione

SDRE Guidance and Navigation for

Spacecraft Relative Motion

Candidate: Michele Galullo

Supervisors: Prof. Mario Innocenti Ing. Giovanni Franzini

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This report presents the SDRE guidance and navigational technique with state constraints that is capable of driving a spacecraft approaching a target vehicle on Near Rectilinear Orbit in the Earth-Moon system. Nonlinear varying-time relative equations were used to design the guidance and navi-gation system. Constraints and weight functions parameters are maintained as generic as possible, in a way to delineate a different requirements for the different missions.

In order to fully validate the proposed solution, the Monte Carlo simulation has been performed over a range of 5 - 20 km for the relative position, in which the chaser can begin its rendezvous maneuvers even outside the cone. The results show that the performances, evaluated in terms of maneuvers time and fuel consumption, are comparable with real mission.

1. Introduction

The rendezvous and docking (RVD) process consists of a series of orbital manoeuvres and controlled trajectories, which successively bring the active vehicle (chaser ) close and eventually into contact with the passive vehicle (target ). The complexity of the rendezvous approach results from the multitude of conditions and constraints which must be fulfilled. The target station may impose safety zones, approach-trajectory corridors and hold points along the way to check out the chaser conditions. Any dynamic state (position and velocities, attitude and angular rates) of the chaser vehicle outside the nominal limits of the approach trajectory could lead to collision with the target, a situation dangerous for crew and vehicle integrity [7].

Traditionally the rendezvous and proximity maneuver have been performed using open-loop maneuver planning techniques ed ad hoc error correction. Examples of constrained maneuvers include the thrust magnitude constraints, constraints on the approaching spacecraft to maintain its position within a Line-of-Sight cone emanating from the docking port on the target platform, and constraints on the terminal translational velocity for soft-docking are proposed in [5, 22]. However, since the 1960s, the problem of control of rendezvous in Earth’s orbit has been studied [2, 9, 11] and real missions are practiced continuously. Now another step forward in the exploration of space must be done. In fact, one of the objectives of the main national space agencies is to build a modular space station close to the moon [23]. The optimal location for this space infrastructure can be a Halo orbit in L1 or L2 (Lagrangian point) [6]. In this region the classical RVD

control strategies based on two-body dynamics fail.

Several studies aimed at the analysis on the station access by the incoming vehicles performing logistic flights, crew transportation missions, or samples return from the Moon surface are currently ongoing [17, 1].

Some of studies using the terminal sliding mode control which enables a time-fixed process with the flight prescribed a priori [15]; a fixed-time glideslope guidance algorithm on a quasi-periodic halo orbit [4].

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This report presents the SDRE technique for the rendezvous scenario formulated as a control with state constraints: the LOS cone has been designed as a constraint for the relative position and a weight function that encourages the chaser to follow the axis of the approaching cone. This new method was tested, by means of Monte Carlo simulation, on relative motion in Near Rectilinear orbit, a particular Halo orbit.

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2. Phase of rendez-vous

The purpose of this chapter is to give a short overview of the different phases of a rendezvous approach and to describe the major issues of these phases. A rendezvous mission can be divided in this simple phases:

• launch; • phasing;

• far range rendezvous; • close range rendezvous; • mating.

During these phases, the kinematic and dynamic conditions that will eventually allow the connection of the chaser to the target spacecraft are successively established.

2.1. Launch phase

Owing to the rotation of the Earth, each point on its surface passes twice per day through any orbit plane. Since at most launch sites have a limited sector of launch directions that can be used (e.g. toward the sea), and so there is only one opportunity per day to launch a spacecraft into a particular orbit plane. During every minute the launch site move ≈ 0,25 deg w.r.t. the orbital plane so the size of the launch window, i.e. the margin around the time when the launch site passes through the orbital plane, will mainly be determined by the correction capabilities of the launcher. At the end of the launch phase, the chaser vehicle has been brought by the launcher into a stable orbit in the target orbital plane. After separation from the launcher, the spacecraft has to deploy its solar arrays and antennas and must initialise all its subsystems.

2.2. Phasing

After separation from the launcher the chaser vehicle is on a lower orbit and may be at an arbitrary phase angle behind the target. The objective of this first orbital phase of a rendezvous mission is to reduce the phase angle between the chaser and target spacecraft, by making use of the fact that a lower orbit has a shorter orbital period.

Phasing ends with the acquisition of either an ”initial aim point”, or with the achievement of a set of margins for position and velocity values at a certain range, called the ”trajectory gate” or ”entry gate”. The margins of the ”aim point” or the ”gate” must be achieved to make the final part of the approach possible. The aim point or “gate” will be on the target orbit, or very close to it, and from this position the far range relative rendezvous operations can commence.

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2.3. Far range Rendez-vous

In many publications this phase is called “homing”, by analogy to the navigation term used for aircraft when approaching an airport. The major objective of the far range rendezvous phase is the reduction of trajectory dispersions, i.e. the achievement of position, velocity and angular rate conditions which are necessary for the initiation of the close range rendezvous operations. Major tasks of this phase are the acquisition of the target orbit, the reduction of approach velocity and the synchronisation of the mission timeline. Far range rendezvous can start when relative navigation between chaser and target is available. The end point of this phase is usually a point from which standard rendezvous operations on standard trajectories at a fixed timeline can commence, a feature which is particularly desirable for an automatic rendezvous process.

2.4. Close range Rendez-vous

The close range rendezvous phase is usually divided into two subphases:

• preparatory phase leading to the final approach corridor, often called ”closing”; • final approach phase leading to the mating conditions.

There are, of course, cases where no distinction can be made between a closing and a final approach subphase. This may be the case, e.g., for a V-bar approach, where the direction of motion remains the same and where no change of sensor type occurs. The proximity to the target makes all operations safety-critical, requiring particular safety features for trajectory and onboard system design and continuous monitoring and interaction possibility by operators on ground and in the target station.

2.4.1. Closing phase

The objectives of the closing phase are the reduction of the range to the target and the achievement of conditions allowing the acquisition of the final approach corridor. This means that at the end of this phase the chaser is, concerning position, velocities, attitude and angular rates, ready to start the final approach on the proper approach axis within the constraints of the safety corridor.

In this case, toward the end of the closing phase, the acquisition conditions for the new sensor type have to be met. The rule of thumb is that the measurement accuracy must be of the order of 1% of range or better.

2.4.2. Final approach phase

In this phase the objective is to achieve docking or berthing capture conditions in terms of positions and velocities and of relative attitude and angular rates. The attempted end condition is the delivery of chaser docking or capture interfaces into the reception range of the target docking mechanism or of the capture tool of the manipulator in the case of berthing.

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2.5. Maiting

The mating phase starts when the GNC system of the chaser has delivered the capture interfaces of the chaser into the reception range of those of the target vehicle. This must be achieved within the constraints of the interface conditions, concerning:

• for docking:

- approach velocity; - lateral alignment; - angular alignment; - lateral and angular rates; • for berthing:

- position accuracy; - attitude accuracy; - residual linear rate; - angular rate.

More of this paraghaph is ispired by [7].

Figure 1: This figure shows the rendez-vous phases. The labels are the nomenclature used for all hold and intermediate way points. The shaded area is the Keep Out Zone, which is defined for safety reasons.

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Part I.

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3. Two-Body Problem

The Two-Body Problem is the dynamic problem of two point masses under the influence of their mutual gravitational attraction according to Newton’s Law. Consider two particles of masses m1 and m2 with position vectors R1 and R2, as you can see in fig. 2, the

equations of motion for the two bodies can be written using the Newton’s Second Law: m1 _¨ R1 I = G m1m2 r3 r m2 _¨ R2 I = −G m1m2 r3 r (1)

where G is the universal gravitational constant and r = R2− R1. So the relative

acceleration of two bodies is ¨r_I =_¨ R2 I− _¨ R1 I = −G m1+ m2 r3 r (2)

In general, when we study the motion of satellite (or spacecraft) its mass is neglegible so the equation eq. (2) begins:

¨r_I = −µp

r

r3 (3)

The gravitational parameter µp = GM is introduced and M is the mass of primary.

The eq. (3) describe the motion of small body (satellite or spacecraft) around a primary body (planet). This type of motion is usually called Keplerian Motion.

Figure 2: Two Body Problem.

The Keplerian motion takes place on a plane, as matter of fact the angular momentum of small mass is constant:

h = r × v = r × d dtr =

d

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The solution of eq. (3) can be rapresented by the conic polar equation

r = p

1 + e cos f (4)

where e is the eccentricity, p is the semilatus rectum (p = h_µ2

p) and f is the true anomaly.

The eccentricity define the orbit shape: • e = 0 circumference;

• 0 < e < 1 ellipse; • e = 1 parabola; • e > 1 hyperbola

The primary body occupies one of the conic focus. The closest point on the ellipse to the primary focus is called periapse, whereas the furthest point is the apoapse.

Other important quantities are the orbital period

T = 2π s

a3

µp

(a is the conic semi-major axis) and the mean motion n = 2π

T =

r µp

a3

The true anomaly is measured from periapse and its time derivative can be computed as ˙

f = h r2

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4. Three-Body Problem

In general, the classical Three-Body Problem regard the motion of infinitesimal body, m3,

in the gravity field of two primary massive bodies, m1 and m2. Suppose the positions of

the three masses are located through the vectors R1, R3 and R3 relative to their center

of mass (fig. 4), and assume the external forces are neglected, we see that the motion of the center of mass is a constant velocity straight-line, so we can adopt the center of mass as an inertial origin, then the three equations of motion are given by:

m1 _¨ R1 I = G m1m2 r3 12 r12+ G m1m3 r3 13 r13 m2 _¨ R2 I = −G m1m2 r3₁₂ r12+ G m2m3 r₂₃3 r23 m3 _¨ R3 I = −G m1m3 r3₁₃ r13− G m2m3 r₁₂3 r23 (5)

where r13 = R3− R1, r23 = R3− R2 and r12 = R2− R1 and G is the universal

gravitational constant. Relative positions norms are indicated with r13, r23 and r12. The

notation R¨i

I denotes the acceleration of the body as seen from the inertial frame.

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The equation of motion of m3 with respect to m1 and m2 are given by: ¨r13 I = _¨ R3 I− _¨ R1 I = −G (m1+ m3) r3 13 r13− Gm2 r23 r3 23 +r12 r3 12 ¨r23 I = _¨ R3 I− _¨ R2 I = −G (m2+ m3) r3 23 r23− Gm1 r13 r3 13 −r12 r3 12 (6)

If the mass m3 is negligible with respect to the primaries masses, i.e. m3 m1 and

m3 m2, we consider the restricted three-body problem. Under this assumption the m3

doesn’t affect the motion of two primaries, so the eq. (6) becomes: ¨r13 I = −µ1 r13 r3₁₃ − µ2 r23 r₂₃3 + r12 r₁₂3 ¨r₂₃ I = −µ2 r23 r3₂₃ − µ1 r13 r₁₃3 − r12 r₁₂3 (7)

where µ1 = Gm1 and µ2 = Gm2 are the gravitational parameter for m1 and m2

rispectively.

4.1. Elliptical Restricted Three-Body Problem

If the motion of the two primary bodies is constrained to elliptic orbits about their barycenter, the Elliptic Restricted Three-Body Problem (ER3BP), the distance between the two primary bodies varies periodically.

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By the following choice of units, the equation of motion can be written in non-dimensional form

• the unit of mass is taken to be m1+ m2;

• the unit of length is chosen to be the constant separation between m1 and m2;

• the unit of time is chosen such that the orbital period of m1 and m2 about their

center of mass is 2π.

The universal constant of gravitation then becomes G = 1, the mean motion of the primaries is also n = 1 (or ω = 1) and the mass parameter of the system is

µ = m2

m1+ m2

The position vector R3 can be expressed in non-dimensional component

R3 =x~i +e y~ej +ez~k

so the equations of motion, under assumption that the primaries revolve around their common barycenter in elliptic orbits are:

           ◦◦ e x −2ω_e ◦ e y − ◦ e ωy −_e ω_e2 e x = −µx+ ee R2 e r3 23 − (1 − µ)ex− eR1 e r3 13 + 1 e r2 12 ◦◦ e y +2ω_e ◦ e x + ◦ e ωx −_e ω_e2y = −µ_e ey e r3 23 − (1 − µ) ye e r3 13 ◦◦ e z = −µ ez e r3 23 − (1 − µ) ez e r3 13 (8)

where ◦ denote the time derivative in non-dimentional ander

2 13= (ex − eR1) 2₊ e y2+ez 2_, e r2 23= (x + ee R2) 2₊ e y2₊ e

z2_{, as we can see in fig. 5}

4.2. Circular Restricted Three-Body Problem

This section considers the Circular Restricted Three-Body Problem (CR3BP) in which the motion of the two primary bodies is constrained to circular orbits about their barycenter. The CR3BP was formulated in the first time by Euler (1772) for the Sun–Earth–Moon system to study the motion of the Moon about the Earth but perturbed by the Sun. By means of the Lagrangian formulation, as given in [24], we obtain the following set of equations:          ◦◦ e x −2 ◦ e y= ∂U (x,eey,z)e ∂xe ◦◦ e y +2 ◦ e x= ∂U (ex,ey,ez) ∂ey ◦◦ e z = ∂U (ex,y,eez) ∂ze (9)

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where ◦ denote the time derivative in non-dimentional variables and in the right hand of equations there are partial derivative of gravitational potential

U (_ex,y,_e z) =_e 1 2(ex 2₊ e y2) +1 − µ e r13 + µ e r23 and e r2₁₃= (x + µ)e 2₊ e y2+ez 2 e r2₂₃= (x − 1 + µ)e 2₊ e y2+ez 2

Note that eq. (8) becames eq. (9) ifω = 1 and_e

◦

e ω= 0. 4.2.1. Equilibrium or Lagrangian Points

Setting all derivatives in eq. (9) to zero, we can find equilibrium points of the m1-m2

system. In every equilibrium points, the gravitational forces and the centrifugal force acting on the m3 are balanced. Five equilibrium points of the Circular Restricted

Three-Body Problem exist, as illustrated in fig. 6.The first three are on the line connecting the two large bodies and the last two, L4 and L5, each form an equilateral triangle with

the two large bodies. The L1, L2, and L3 points are nominally unstable. In [20, p. 38]

explain how compute the x values, γi, of the collinear points.

Although the Lagrange point is just a point in empty space, its peculiar characteristic is that it can be orbited and we can find more kind of orbit.

Figure 6: Lagrangian Points in Circular Restricted Three-Body Problem. The same considerations and the same resutls can get for ER3BP.

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5. Orbit around collinear point

Robert Farquhar [18] discovered trajectories around L2 in the Earth-Moons system. He

named such a trajectory a halo orbit as it appeared from the Earth to be a halo encircling the Moon. Halo orbits are the result of a complicated interaction between the gravitational pull of the two planetary bodies and the Coriolis and centrifugal accelerations on object with mass m3.

Farquhar used analytical expressions to represent halo orbits, but Kathleen Howell [13] showed that more precise trajectories could be computed numerically.

The studies on the orbits near libration point discovered other particular kind of orbit: • planar and vertical families of Lyapunov periodic orbits;

• three-dimensional quasi-periodic Lissajous orbits; • periodic halo orbits;

• quasi-halo orbits.

5.1. Halo Orbit

The first halo orbit mission ISEE-3 was designed using methodology described by Richard-son [19].

The halo orbit was constructed using the third-order analytical solution obtained by an application of successive approximations using the Lindstedt–Poincar’e method.

Some computational advantages can be obtained if the eq. (9) are developed using Legendre polynomials Pnto expand the nonlinear terms 1−µ_r₁₃ +_rµ₂₃ in the gravitational

potential. The following formula is used: 1 p(x − A)_e 2_{+ (} e y − B)2_{+ (} e z − C)2 = 1 D ∞ X n=0 ρ D n Pn A_ex + By + C_e _ez Dρ (10) where D2 = A2+ B2+ C2 and ρ2 =xe 2₊ e y2+ze 2_.

After that we can write the eq. (9) as:                      ◦◦ e x −2 ◦ e y −(1 + 2c2)ex = ∂ ∂ex P n≥3cnρnPn e x ρ ◦◦ e y +2 ◦ e x +(c2− 1)y =e ∂ ∂ey P n≥3cnρnPn e y ρ ◦◦ e z +c2z =e ∂ ∂ze P n≥3cnρnPn e z ρ (11)

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5.1.1. Linearized Motion

Considering the linear part of eq. (11):          ◦◦ e x −2 ◦ e y −(1 + 2c2)ex = 0 ◦◦ e y +2 ◦ e x +(c2− 1)y = 0e ◦◦ e z +c2ez = 0 (12)

we can be deduced the existence of periodic solution. As we know, the solution of the characteristic equation of eq. (12) has two real and four imaginary roots (±λ, ±iωp, ±iωv).

If we choose the initial conditions adequately so that only the non-divergent mode is allowed, the xy-solution will be bounded. In this case, the linearized equations have solutions of the form

     e x = −Axcos(ωpet + φ) e y = κAxsin(ωpet + φ) e z = Azsin(ωvet + ψ) (13) where κ = ω 2 p+1+2c2

2ωp . Note that the linearized motion will become quasi-periodic if the

in-plane and out-of-plane frequencies are such that their ratio is irrational.

If the amplitudes Axand Azare constrained by a certain non-linear algebraic relationship:

l1A2x+ l2A2z+ ∆ = 0 (14)

that it was found as result of the applicaiton of the perturbation method, the eigenfre-quencies of the eq. (12) are equal (for ∆, l1, l2 see [20, p. 156], [19]).

Another importat condition there is between the phases φ and ψ that are related in linear fashion: ψ − φ = pπ 2, p = 1, 3 (15) Note that if Ax > 2 q ∆

l1 we have a bifurcation that manifests itself through the condition

espressed in eq. (15) so any halo orbit can be characterized completely by specifying a particular out-of-ecliptic plane amplitude Az:

• p = 1 and Az is positive, we have the northern halo whose maximum out-of-plane

component is above the xy-plane;

• p = 3 and Az is negative, we have the southern halo whose maximum out-of-plane

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Northern and southern halo orbits with the same Az amplitude are specular images

across the xy-plane fig. 7.

Figure 7: Northern halo orbit (above) and Southern halo orbit (below)

5.1.2. Third-Order Approximation

Recall the eq. (11), Richardson stopped the Legendre polynomial at third-order so the equations of motion begin:

                   ◦◦ e x −2 ◦ e y −(1 + 2c2)ex = 3 2c3(2xe 2₋ e y2−_ez2) + 2c4ex(2ex 2_{− 3} e y2− 3z_e2) + O(4) ◦◦ e y +2 ◦ e x +(c2− 1)y = −3ce 3exy −e 3 2c4ey(4xe 2₋ e y2−z_e2) + O(4) ◦◦ e z +c2ez = −3c3xeez − 3 2c4ez(4ex 2₋ e y2−z_e2) + O(4) (16)

By forcing the linearized z equation with conditions in eqs. (14) and (15) and replacing c2 with ωp in eq. (16), so it becomes necessary to rewrite the left-hand side of the z

equation and to introduce a correction term ∆ = ω_p2− c₂ = ω_p2− ω2

v. In addition, to help

remove secular terms, Richardson used a new independent variable τ = νet,

where

ν = 1 +X

n≥1

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It is important to note that νn are assumed to be O(Anz), so Az 1 [20, p. 155].

This is the Lindstedt–Poincar’e Method and the equations of motion are written in terms of new independent variable:

                   ν2 ◦◦ e x −2ν ◦ e y −(1 + 2c2)x =e 3 2c3(2xe 2₋ e y2−z_e2) + 2c4x(2e xe 2_{− 3} e y2− 3_ez2) + O(4) ν2 ◦◦ e y +2ν ◦ e x +(c2− 1)ey = −3c3xeey − 3 2c4y(4e xe 2₋ e y2−_ez2) + O(4) ν2 ◦◦ e z +ω_p2z = −3c_e 3xeez − 3 2c4z(4e ex 2₋ e y2−z_e2) + ∆z + O(4)_e (17)

Most of the secular terms can be removed by using that:

ν1 = 0, ν2 = s1A2x+ s2A2z (18)

After that, the halo orbit period is T = _ω2π

pν.

5.2. Near Rectilinear Orbit

Analytical approximations must be combined with numerical techniques to generate a halo orbit accurate enough. We need the 6 × 6 State Transition Matrix (STM), Φ(et, 0) of _∂X(0)∂X(et) where X =            e x e y e z ◦ e x ◦ e y ◦ e z           

is the column vector of the states of eq. (9). We started from Φ(0, 0) = I6×6, and then

we integrate simultaneously the following ODEs: d

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with A(et) =   03×3 I3×3 UXX 2Ω   Ω =   0 1 0 −1 0 0 0 0 0  

UXX is the symmetric matrix of second partial derivatives of U (x,e y,e ez) with respect to _ex,_ey,z, evaluated along the orbit. The stability of the orbit is determined by the_e eigenvalues of Φ(TF, 0), where TF is the period of orbit.Two of the eigenvalues are always

1 and the other four are in reciprocal pairs (λi,_λ1_i).

As we can see in [12], the point with the maximumx and maximum_e z value is unique_e for each orbit, so we can choose the xmax value to identify each member of the family.

It can be seen in figs. 8 and 9, the family is tending toward orbits to have a rectilinear shape, which have a large out-of-plane component z and increasingly smaller in-plane components.

Figure 8: Example of family of halo orbits in L3 xz -plane.

Figure 9: Example of family of halo orbits in L3 yz -plane.

5.2.1. Modelling as a Perturbated Keplerian Orbit

When the orbits are very close to the larger mass (if we consider the L3 point) or the

smaller mass (if we consider the L1 or L2), the distance from mi is given by r, and we

can modell their as a perturbed Kepler orbits.

Let the eccentricity tend to unity, the ellipse becomes more and more elongated, with the perihelion distance a(1 − e) tending to zero: this is called a rectilinear ellipse. In the [21, p. 88] we can find the following equation, that relating time, position and velocity when the motion is rectilinear as in the two-body problem:

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M = n(t − τ ) (20)

M = E − sin(E) (21)

r = a(1 − cos(E)) (22)

r ˙r = a1/2µ1/2sin(E) (23)

They may also be used for near-rectilinear motion. In the elliptic case when e ≈ 1 Kepler’s equation begin:

M = E − e · sin(E)

= E − sin(E) + (1 − e) · sin(E) = E − sin(E) + · sin(E)

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≈ 0, so we have the eq. (21). After that, as we can see in [13], the presence of the farthermost mass of primary bodys not only disturb the position of m3 from that in a

2-body Kepler orbit, it also make a time correction necessary. The equation of perturbed motion turn into:

¨ r = −n

2_a2

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6. Relative Motion

6.1. Relative Motion in Two-Body Problem

The relative motion involves two or more spacecraft/bodies in orbit around a primary body: a target and a chaser in the case of a rendezvous mission; a chief, or formation leader, and one ore more deputies, or followers, if we are considering a formation.

Figure 10: Rendezvous scenario. Figure 11: LVLH frame.

In the Cartesian formulation of relative motion dynamic is developed in the local -vertical local - horizon frame (LVLH), centered on the target (or chief) center of mass. For two-Body problem, in this report, the unit vectors are given by:

ˆi = ˆj × ˆk, ˆj = −ht/e

h_t/e, ˆk = − rt

rt

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ρ = xˆi + yˆj + z ˆk ρ = ˙˙ xˆi + ˙yˆj + ˙z ˆk

and ω and ˙ω denote the angular velocity and acceleration of the LVLH frame w.t.r. the ECI frame

ω = ωxˆi + ωyˆj + ωzkˆ ω = ˙˙ ωxˆi + ˙ωyˆj + ˙ωzkˆ

where the dot operator indicates the derivation w.r.t. time in LVLH frame.

We can define the absolute position of chaser, expressed in LVLH frame, in this manner: rc= rt+ ρ = xˆi + yˆj + (z − rt)ˆk (27)

Chaser’s absolute velocity can be obtained deriving w.r.t. time eq. (27), d dtrc= d dtrt+ d dtρ = d dtrt+ ˙ρ + ω × ρ (28)

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Note that the derivation in the inertial frame is denoted with _dtd.

Moreover derivation of eq. (28) leads to the chaser’s vectorial equation of motion. Intro-ducing chaser’s and target’s equations

d dtrc= − µ rc3 rc+ u (29) d dtrt= − µ r3 t rt (30)

the general vectorial expression for the relative motion is given by ¨ ρ + 2ω × ˙ρ + ˙ω × ρ + ω × (ω × ρ) − µ r3_trt+ µ rc3 (rt+ ρ) = u (31)

In absence of perturbation the angular velocity ω becomes ω = ˙fˆj = h

r2_tˆj and the angular acceleration in Keplerian orbits is:

˙ ω = −2h ˙rt r_t3ˆj = −2 ˙f ˙rt rt ˆj

Now we have all necessary to write the Nonlinear Equation of Relative Motion (NERM) along the LVLH unit vectors separately:

¨ x = 2 ˙f ˙rt rt z − ˙z + ˙f2x − µ rc3 x + ux ¨ y = − µ rc3 y + uy ¨ z = −2 ˙f ˙rt rt x − ˙x + ˙f2z + µ r2 t − µ rc3 (z − rt) + uz (32)

Note that rt, ˙rt and ˙f are time - varing terms; x =x y z x˙ y˙ z˙

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6.2. Relative Motion in Three-Body Probelm

Consider a target and a chaser spacecraft, orbiting around the second primaries, and subject to both first and second primaries gravitational influence. This section describe the motion of the chaser relative to the target, in a frame centered on the latter. To this end, the local-vertical local-horizon (LVLH) frame L :Rt;î,î,î is introduced, with unit

vectors defined as follow

ˆi = ˆj × ˆk, ˆj = −ht/m

ht/m

, ˆk = −rmt rmt

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where rmt is the target position w.r.t. the Moon, rmt= ||rmt||, ht/m = rmt× ˙rmt

M

is the target angular momentum w.r.t. the Moon and ht/m= ||ht/m||. The unit vectors

ˆi,ˆj, ˆk in the rendezvous literature are generally referred to as V-bar, H-bar and R-bar, respectively [10].

Figure 12: Target and Chaser in Three-Body system.

Figure 13: LVLH reference frame.

The equation of relative motion in the LVLH frame was derived for the first time in [10] and are: ¨ρ_L+ 2ωl/i× ˙ρ L+ ˙ωl/i L× ρ + ωl/i× ωl/i× ρ = = µm rmt r3_mt − rmc r3 mc + µe ret r_et3 − rec r3 ec ₍₃₄₎

where, as you can see in fig. 12,

rmc = rmt+ ρ, ret= rmt+ rem, rec = ret+ ρ

The second primary is not negligible in this case, a second gravitational term is introduced.

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Part II.

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7. SDRE Nonlinear Control

The goal of this work is to find a controller that permit a rendezvous between the target and the chaser using the SDRE technique, an extension of LQR theory on nonlinear problems known as State-Dependent Riccati Equation (SDRE) nonlinear regulation [3].The SDRE strategy provides an effective algorithm for synthesizing nonlinear feedback controls by allowing nonlinearities in the system states.

7.1. Problem Formulation

Consider a nonlinear dynamic system affine in the control:

˙x = f (x) + g(x)u (35)

where x ∈ Rnis the state vector, u ∈ Rmis the input vector, f : Rn→ Rn_{and g(x) 6= 0}

∀x ∈ Rn _{. As the performance index to be minimized, the following infinite-time integral}

is considered: J (x, u) = 1 2 Z ∞ 0 xTQ(x)x + uTR(x)u dt (36)

where Q(x) ≥ 0 and R(x) > 0 are weighing matrices of the state and the control vectors respectively. To solve the optimal control problem using SDRE method. If the nonlinear dynamic equation eq. (35) can be written into a pseudo-linear structure using State Dependent Coefficient (SDC) parameterization, like this:

˙x = A(x)x + B(x)u (37)

the SDRE technique can be applied. However, there are some requirements on the SDC parametrization to guarantee that eq. (37) has a solution: the pair A(x); B(x) must be pointwise controllable in the linear sense.

The control technique consists of the following two phases for each time step: 1. Solve the state-dependent Riccati equation:

AT(x)P(x) + P(x)A(x) − P(x)B(x)R−1(x)BT(x)P(x) + Q(x) (38) 2. Derive the feedback controller

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7.2. SDC Parametrization

SDC parametrization is a mathematical factorization of nonlinear system into a linearlike structure.

f (x) = A(x)x

The folowing statements, under conditions expressed in [3], garantee the parametrization existence and some linear system property:

• Let f : Ω → Rn _{be such that f (0) = 0 and f (·) ∈ C}k_{(Ω), k > 1. Then, for all x ∈ Ω,}

a SDC parametrization for f (x) always exists for Ck−1 _{matrix-valued function}

A : Ω → Rn×n.

• The SDC parametrization is a stabilizable (or controllable) parametrization of a nonlinear system [eq. (37)] in region Ω if the pair {A(x), B(x)} is pointwise stabilizable (or controllable) in the linear sense for all x ∈ Ω.

• The SDC parametrization is a detectable (or observable) parametrization of a non-linear system [eq. (37)] in region Ω if the pair {C(x), A(x)} is pointwise detectable (or observable) in the linear sense for all x ∈ Ω.

Note that, in multivariable case, the parametrization is not unique. This allows to achieve better performance or to satisfy some project’s requirement.

7.2.1. Parametrization in Two-Body Problem Case

Nonlinear equation of relative motion can be written in affine form like eq. (35), as follow:

A(x) =          0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ˙ f2−_rµ c3 0 2 ˙f ˙ rt rt 0 0 2 ˙f 0 −_rµ c3 0 0 0 0 −2 ˙fr˙t rt + γx γy ˙ f2 −2 ˙f + γ(z − rt) −_rµ3 c 0 0          , B(x) =03×3 I3×3 (40) The Earth nonlinear gravitational term can be written in linearlike structure as follow:

µ rt r_t3 − rc r3 c =    −µ r3 c 0 0 0 −_rµ3 c 0 γx γy γ(z − rt) −_rµ3 c   ρ = Ag(x)ρ (41) where rc= rt+ ρ, ρ =x y z T

, f is the target true anomalie, and γ = −µ(r

2

c+ rtrc+ rt2)

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This SDC representation make system controllable. For more details see [8]. 7.2.2. SDC Parametrization for Relative Motion around Moon

Nonlinear equation of relative motion, eq. (34) can be easily rewritten in affine form like eq. (35) and all conditions expressed in [3] are met, so a possible SDC parametrization exists. Note that the nonlinearities are concentrated in gravitational terms.

The Moon gravitational attraction term can be written as:

µm rmt r3 mt −rmc r3 mc =    −µm r3 mc 0 0 0 −µm r3 mc 0 γmx γmy γm(z − rmt) −_rµ3m mc   ρ = Am(x)ρ (42)

The Earth gravitational attraction term can be written as:

µe rmt+ rem ||rmt+ rem||3 − rmt+ rem+ ρ ||rmt+ rem+ ρ||3 = Ae(x)ρ =    γerxem(2remx + x) − rµ3e ec γer x em(2r y em+ y) γeremx (2(remz − rmt) + z) γeryem(2rxem+ x) γeremy (2ryem+ y) − _rµ3e ec γer y em(2(remz − rmt) + z) γe(rzem− rmt)(2rxem+ x) γe(remz − rmt)(2remy + y) γe(rzem− rmt)(2(rzem− rmt) + z) − _rµ3e ec   ρ (43) where ρ =x y zT, rmt=0 0 −rmt T , rmc=x y z − rmt T , rem=remx r y em rzem T , rec = rem+ rmt+ ρ and γm = −µm (r_mc2 + rmtrmc+ r2mt) (rmc+ rmt)(rmc3 rmt2 ) , γe= µe (r_ec2 + retrec+ r2et) (rec+ ret)(r3ecret3)

The SDC parametrization of relative motion, eq. (34), now can be written as follow:

A(x) = " 03×3 I3×3 −_˙ Ω_l/i_L− Ω2 l/i+ Am(x) + Ae(x) −2Ωl/i # , B(x) =03×3 I3×3 (44)

x =ρ ρ˙T is the state of dynamical system,Ω˙l/i

Land Ωl/idenote the skew-symmetric

matrix associated to ˙ωl/i

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7.3. Constraints Formulation

The rendezvous maneuver can be faced imposing constraint on relative spacecraft position and velocity while approaching the docking port on a target platform.

Considering an input-affine system as eq. (35), with x(0) = x0 ∈ Ω and the set of

allowable states defined by

Ω =x : l(x) ≤ 0, l(x) ∈ Rp, l(·) ∈ C1 (45) it is possible to design a state feedback controller such that the closed-loop system is stable and x does not cross ∂Ω, the boudary of Ω, defined as:

∂Ω =x : l(x) = 0, l(x) ∈ Rp, l(·) ∈ C1 (46) The sufficient condition for x to remain in Ω is ˙l(x) = 0:

∇l(x) ˙x = ∇l(x)f (x) + g(x)u = 0 (47)

It is possible to express the condition above in SDC form and add a fictitious output z to SDRE problem

z = ∇l(x)A(x)x + B(x)u

= C(x)x + D(x)u (48)

A controller that satisfies the condition on eq. (47) forces the closed-loop trajectories to follow the level set of Ω.

So we need an augmented cost function: J (x, u) = J0(x, u) + JΩ(x, u) = = 1 2 Z ∞ 0 xTQ(x)x + uTR(x)u dt +1 2 Z ∞ 0 zTWz(x)z dt (49)

where Wz is a p × p diagonal matrix, such that its i-th element is large when x is close

to the boundary of i th constraint and large otherwise. It increases JΩ(x, u) far above

the performance index J0(x, u) when the chaser hold not constraints.

In this case the SDRE problem become: ¯ AT(x) ¯P(x) + ¯P(x) ¯A(x) − ¯P(x) ¯B(x) ¯R−1(x) ¯BT(x) ¯P(x) + ¯Q(x) = 0, ¯ R(x) = R(x) + DT(x)Wz(x)D(x), ¯ Q(x) = Q(x) + CT(x)Wz(x)I − DT(x) ¯R−1(x)D(x) Wz(x)C(x) ¯ A(x) = A(x) − B(x) ¯R−1(x)DT(x)Wz(x)C(x). (50)

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The resulting control law is composed: u = −K(x)x = −K0(x) + KΩ(x) x (51) where: K0(x) = ¯R−1(x)B(x) ¯P(x) KΩ(x) = ¯R−1(x)DT(x)Wz(x)C(x) (52)

K0(x) is used for stabilization/performance and KΩ(x) is used to satisfy the constraints

[16].

7.3.1. Constraint design

The constraints of LOS require the chaser to remain within the LOS-cone. The approach-ing cone is mathematically defined by:

     − sin(β) · x + cos(β) · z < 0 − sin(β) · x − cos(β) · z < 0 x < 0     

a(x, z): − sin(β) · x + cos(β) · z = 0 b(x, z): − sin(β) · x − cos(β) · z = 0

c(x, z): z = 0

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The Wzweight function is similar to Artificial Potential Function (APF) that describes

the attractive area (fig. 15) and the avoidance region (fig. 16). A suitable choice can be to create a shell around the target with a preferential region, like LOS cone, on the docking port. To this it is preferable to add another function that encourage the chaser to come following the cone axis.

Figure 15: Weight functions on thirth fictious output, f2.

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The mathematical expression of functions are: fcone(x, z) = [sign(a(x, z)) + sign(b(x, z)) + 2] · S(

√ x2_{+ z}2_{) · S(r} w− √ x2_{+ z}2₎ faxis(x, z) = z − mx √ 1+m2 (53)

where S(·) is logistic function, sign(·) is the signum function, m is the slope of line and rw is the blast radius of LOS cone.

However the constraints are softly imposed by the minimization of cost function, so it not guarantee that the states reach the desired values if the weight matrices are not suitable.

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Part III.

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8. Relative Motion Control around Earth

This section presents the preliminary part of project. Here we investigate the feasibility of SDRE algorithm in presence of constraints in rendezvous scenario, around the Earth. Here we used the assumption of Two-Body Problem. The parametrization proposed in the previous sections, eq. (40), were used to develop SDRE controllers. Their performance were compared setting up a terminal rendezvous mission scenario. Simulations were developed in Simulink. Dorman-Price integration algorithm was used. SDRE controllers weight matrices coefficients were tuned in order to achieve a rendezvous condition in an acceptable time and passable propellent consumption.

Internationale State Space (ISS) orbit was chosen as target’ s orbit with the following orbital elements [14]:

a = 6.7644 · 103 km; e = 0.00051; i = 51.6391◦; Ω = 270.8069◦; ω = 116.0974◦; f (t0) = 0◦;

Figure 17: International Space Station orbit.

8.1. Guidance and Navigation

The position is assumed as available measurement, so H(x) =I3×3 03×3. The error

that affect the measuremets is considered as purely random with a Gaussian distribution and standard deviation σ = 1/3 × 10−2 m [16].

In this subsection the SDRE filter and control were tested in the same way as before. The control weight matrices were set as follow:

Q0 = 10−3_{· I} 3×3 03×3 03×3 102· I3×3 R0= 105· I3×3 (54)

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Also the constraint matrix is set as follow, section 7.3.1: Wz =   50 · fcone(x) 0 0 0 50 · fcone(x) 0 0 0 5 · faxis(x)   (55)

The measurements noise covariance matrix is set as: Rf = 10−5· I3×3

the process noise covariance was chosen as: Qf =

03×3 03×3

03×3 10−12· I3×3

the initial condition for error covariance matrix is: P0_f =10

−4_{· I}

3×3 03×3

03×3 10−5· I3×3

the filter start from state:

x0_f = x0+ξp ξv

where x0 is the real relative position and velocity of chaser, ξp and ξv are 3 × 1 vectors

of uniformly distributed random number, rispectively in the interval (0,10−4) [km] and (0,10−5) [km/s].

Simulations results for this first set of tests are shown section 8.1.

Figure 18: Simulation with 20 points for 5, 8, 11, 14, 17, 20 km of relative distance between target and chaser.

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Figure 19: The average time needed to reach the rendezvous conditions around Earth.

Figure 20: Propellant consumption is the integral over time of ||u|| for rendezvous around Earth. Our goal is to determine the feasibility of algorithm and the results show a good behaviour in our region of interest: Line-of-Sight cone while approaching the docking port on target platform. The time of flight is also comparable with the time of flight of real mission [7], so we investigated the feasibility of this algorithm in another rendezvous scenario: rendezvous around Moon in Near-Rectilinear Orbit where the relative motion is influenced by both Earth and Moon gravitational fields, see section 6.2.

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9. Motion Control in Earth-Moon system

Due to the nonlinearity of the gravitational acceleration and the presence of several time-varying parameters, the equations of relative motions may be difficult to use, specially in the navigation system where the filter needs a lot of information such as: position, velocity acceleration from target, and position, velocity, angular velocity, angular acceleration, angular jerk of Moon. Under the assumption of primaries revolving in circular orbits, the number of time-varying parameters reduces and the filter requires only position and velocity from target [10]. In terms of maneuver time and fuel consumption the comparison between elliptical and circular assumption model was done. In this work we suppose that the rendezvous maneuvers take place close periselene, the point in orbit closest to the Moon.

The parametrization expressed in eq. (44) was used in SDRE controllers. Simulations were developed in Simulink. Dorman-Price integration algorithm was used. The guidance and navigation system run at 1 Hz. The distances were normalized in units of Moon orbit semi-major axis a, time in units of the inverse of mean angular motion n and the masses such that Me+ Mm= 1 [20].

9.1. Guidance

The SDRE controller was tested by means of Montecarlo simulation for six different ρ = 5, 8, 11, 14, 17, 20 [km]. For each ρ is chosen 20 random uniformly distributed points. The weight matrices coefficients used are:

Q0 = Qp 03×3 03×3 I3×3 Qp = 4 ·   105 0 0 0 106 ₀ 0 0 105   (56) R0= 5 · 10−8· I3×3 (57)

and the constraint matrix is set as follow:

Wz =   20 · fcone(x) 0 0 0 20 · fcone(x) 0 0 0 106· f_axis(x)   (58)

The terminal condition are ρ ≤ 1 m (relative distance) and ˙ρ ≤ 0.03 m/s (relative velocity). Note that ESA’s ATV docking, the deputy had to converge and to keep the following conditions: ρ < 20 m, ˙ρ < 0.01 m/s [7].

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Figure 21: Simulation with 20 points for 5, 8, 11, 14, 17, 20 km of relative distance between target and chaser.

In the fig. 21 we can see how the chaser moves close to the target. In the simulation for ρ = 17 km, fig. 22 it is more evident how the controller moves the chaser in the corridor of approach, avoiding keep-out zone that surrounds the target.

Figure 22: Simulation with 20 points for 17, 20 km of relative distance between target and chaser.

As we can see in fig. 23 and fig. 24, the Time of Flight (tof) and the total Control Usage

Index (δv) increase with increasing the relative distance between chaser and target as we

expected. Another important aspect is the standard deviation on tof: when the chaser

initial condition isn’t in LOS cone, and the relative distance is less than 15 km the chaser moves very slow as long as reaches the approach corridor. It needs more time for moving

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Figure 23: The average time needed to reach the rendezvous conditions.

Figure 24: Propellant consumption is the integral over time of ||u||.

close target. However the standard deviation can be reduced if increase the number of test for each given distance. Analogous consideration can be done for δv standard

deviation.

The same test is done with Circular Restricted 3 Body problem (CR3BP) assumption. The results are very similar to elliptical case as can be seen from table 1, so the circular case was chosen for the controller and filter design.

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ρ [km] 5 8 11 14 17 20 tc_of − te of [min] 10 −13 ₀ ₀ ₀ ₀ _0.2842 ₀ δ_vc− δe v [m/s] 10−5 0.3908 0.3018 0.2264 0.0980 0.5358 0.5853

Table 1: Flight of Time mean difference and Control Usage Index mean difference between circular and elliptical case.

In the fig. 25 we can see the behaviour of the guidance system when the initial velocities of chaser point outward of the approach cone. The flight of time and fuel consumption are similar to the previous case.

Figure 25: Simulation with 20 points for 5, 8, 11, 14, 17, 20 km of relative distance between target and chaser with relative velocities that point outward of the approach cone.

9.2. Guidance and Navigation

The position is assumed as available measurement, so H(x) =I3×3 03×3. The error

that affect the measurements is considered as purely random with Gaussian distribution, 0 mean and standard deviation σ = 1/3 × 10−2 m [16].

In this subsection the SDRE filter and control were tested in the same way as before. The control weight matrices were set as in section 9.1. The process noise covariance and the measurements noise covariance matrix were set as:

Qf =

03×3 03×3

03×3 10−12· I3×3

Rf = 0.6768 · 10−21· I3×3

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P0_f =2.6015 · 10

−10_{· I}

3×3 03×3

03×3 0.9820 · 10−5· I3×3

the initial condition of filter is:

x0_f = x0+ξp ξv

where x0 is the real relative position and velocity of chaser, ξp and ξv are 3 × 1 vectors of

uniformly distributed random number, respectively in the interval (0, 10) [cm] and (0, 1) [cm/s].

Simulations results are shown in fig. 26.

Figure 26: Simulation with 20 points for 5, 8, 11, 14, 17, 20 km of relative distance between target and chaser with assumption of CR3BP for Guidance and Navigation system.

ρ [km] 5 8 11 14 17 20

Ieρ 5.0322 8.0935 11.1210 14.2286 17.6183 20.5973

Ieρ˙ 0.0158 0.0202 0.0194 0.0210 0.0245 0.0240

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The indexes were evaluated as follow: Ieρ= Z tof 0 ||eρ(t)|| dt Ieρ˙= Z tof 0 ||eρ˙(t)|| dt

where eρ(t) and eρ˙(t) are the error vectors respectively between real position and

estimated position and real velocity and estimated velocity.

Figure 27: The average time needed to reach the rendezvous conditions with assumption of CR3BP for Guidance and Navigation system.

Figure 28: Propellant consumption is the integral over time of ||u|| with assumption of CR3BP for Guidance and Navigation system.

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10. Conclusion

This thesis work present a guidance and navigation system based on SDRE technique with state constraint in rendezvous missions with cis-lunar space station. Constraints and weight functions parameters were maintained as generic as possible. One possible SDC parametrization for relative motion in Earth-Moon system was found. Comparison between Elliptical and Circular assumption for relative motion equations, evaluated on time of flight and fuel consumption, was done. The results show that the performances are very close, so the Circular assumption was used to design the guidance and navigation system, taking the advantage that navigation system need only position and velocity measurements from target. Simulations demonstrate feasibility of the proposed control.

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References

[1] Simone Flavio Rafano Carn´a Andrea Colagrossi Mich´ele Lavagna. ‘Dynamical analysis of rendezvous and docking with very large space infrastructures in non -keplerian orbits.’ In: CEAS Space Journal (2017).

[2] Finn Ankersen. ‘Guidance, Navigation, Control and Relative Dynamics for Space-craft Proximity Maneuvers’. PhD thesis. Aalborg University, 2010.

[3] T. Cimen. ‘Survey of State-Dependent Riccati Equation in Nonlinear Optimal Feedback Control Synthesis’. In: Journal of Guidance, Control and Dynamics 35 (2012).

[4] ‘Constant-thrust glideslope guidance algorithm for time-fixed rendezvous in real halo orbit’. In: Acta Astronautica (2012).

[5] A. G. Richards E. N. Hartley P. A. Trodden and J.M Maciejowski. ‘Model predictive control system design and implementation for spacecraft rendezvous’. In: Control Engineering Practice (2012).

[6] Kathleen C. Howell Emily M. Zimovan and Diane C. Davis. ‘Near Rectilinear Halo Orbits and their application in cis-lunar space.’ In: IAA Conference on Dynamics and Control of Space Systems. (2017).

[7] W. Fehse. Automated Rendezvous and Docking of Spacecraft. Cambridge University press, 2003.

[8] Giovanni Franzini. ‘Nonlinear Control of Relative Motion in Space using Extend Linearization Technique’. MA thesis. University of Pisa, 2013/2014.

[9] Giovanni Franzini and Mario Innocenti. ‘Nonlinear H-infinity Control of Relative Motion in Space via the State-Dependent Riccati Equations.’ In: Annual Conference on Decision and Control (CDC) (2015).

[10] Giovanni Franzini and Mario Innocenti. ‘Relative Motion Equations in the Local-Vertical Local-Horizon Frame for Rendezvous in Lunar orbits.’ In: AAS/AIAA Astrodynamics Specialist Conference (2017).

[11] Lorenzo Pollini Giovanni Franzini and Mario Innocenti. ‘H-infinity Controller Design for Spacecraft Terminal Rendezvous on Elliptic Orbits using Differential Game Theory.’ In: American Control Conference (ACC) (2016).

[12] K. C. Howell. ‘Almost Rectilinear Halo orbits’. In: Celestial Mechanics 32 (1984). [13] K. C. Howell. ‘Three-Dimensional, periodic, halo orbits’. In: Celestial Mechanics

32 (1984).

[14] International Space Station orbital elements. url: https://go.nasa.gov/2jx11D2. [15] Yijun Lian and Guojian Tang. ‘Libration point orbit rendezvous using PWPF modulated terminal sliding mode control’. In: Elsevier, Advances in Space Reserch (2013).

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[16] F. Bernelli-Zazzera M. Massari. ‘Nonlinear Control of Formation Flying with State Constraints’. In: Journal of Guidance, Control and Dynamics 35, No 6 (2012). [17] ‘Practical Rendezvous Scenario for Transportation Missions to Cis-Lunar Station

in Earth?Moon L2 Halo orbits.’ In: Proceedings of ISSFD (2015).

[18] A. A. Kamel R. W. Farquar. ‘Quasi-Periodic Orbits About the Translunar Libration Point’. In: Celestial Mechanics 7 (1973).

[19] D. L. Richardson. ‘Halo Orbit Formulation for the ISEE-3 Mission’. In: J Guidance and Control 3, No.6 (1980).

[20] Wang Sang Koon Martin W. Lo Jerrold E. Marsden Shane D. Ross. Dynamical Systems, the Three-Body Problem and Space Mission Design. 2011.

[21] A. E. Roy. Orbital Motion. Institute of Physics Publishing, 2005.

[22] H. Park S. Di Cairano and I. Kolmanovsky. ‘Model Predictive Control approach for guidance of spacecraft rendezvous and proximity maneuvering’. In: International Journal of Robust and Nonlinear Control (2012).

[23] The global exploration roadmap. 2013. url: www.nasa.gov/sites/default/files/ files/GER-2013_Small.pdf.

[24] Bong Wie. Space Vehicle Dynamics and Control. American Institute of Aeronautics and Astronautics, Inc., 2008.