- Email: [email protected]

Corso di Laurea in Ingegneria Spaziale

ROBUST CONTROL FOR PLANETARY LANDING MANEUVERS

Relatore: Prof. Michèle LAVAGNA Correlatore: Ing. Roberto ARMELLIN

Tesi di laurea di: Paolo LUNGHI Matr. 734407

Anno Accademico 2011 - 2012

A Papà.

Abstract This work focuses on an adaptive guidance algorithm for planetary landing that updates the trajectory to the surface by means of a minimum fuel optimal control problem solving. A semi-analytical approach is proposed. The trajectory is expressed in a polynomial form of minimum order to satisfy a set of 17 boundary constraints: 12 constraints on initial and ﬁnal state and 5 control constraints, added in order to include attitude requirements. By imposing boundary conditions, a fully determined guidance proﬁle is obtained, function of only two parameters: time-of-ﬂight and initial thrust magnitude. The optimal guidance computation is reduced to the determination of these parameters, according to additional path constraints due to the actual lander architecture: available thrust and control torques, visibility of the landing site, and other additional constraint not implicitly satisﬁed by the polynomial formulation. Solution is achieved with a simple two-stage compass search algorithm: the algorithm ﬁrstly ﬁnds a feasible solution; whenever detected, it keeps solving for the optimum; nonlinear constraints are evaluated numerically, by pseudospectral methods. Results on diﬀerent scenarios for a Moon landing mission are shown and discussed to highlight the eﬀectiveness of the proposed algorithm and its sensitivity to the navigation errors. Keywords: pinpoint landing, adaptive guidance, retargeting, hazard avoidance

Sommario Il presente lavoro è focalizzato sulla formulazione e la veriﬁca di un algoritmo di guida adattiva per l’atterraggio planetario, che in seguito alla modiﬁca del sito di atterraggio riformuli la traiettoria risolvendo un problema di ottimizzazione del carburante. Viene qui proposto un approccio semi-analitico. La traiettoria viene rappresentata con una forma polinomiale, del minimo ordine necessario per soddisfare un set di 17 condizioni al contorno, di cui 12 per gli stati iniziale e ﬁnale del sistema e 5 riguardanti le variabili di controllo, imposte dall’assetto iniziale e ﬁnale desiderato. Imponendo queste condizioni al contorno, separatamente su ogni asse, si ottiene un completo proﬁlo di guida, funzione di soli due parametri, tempo di volo e spinta iniziale. Il problema è quindi ridotto all’individuazione dei valori di questi parametri, tali da ottimizzare il consumo di propellente soddisfacendo al tempo stesso i vincoli addizionali posti dall’architettura del lander: valori minimi e massimi erogabili di spinta e coppie di controllo, requisiti di visibilità sul sito di atterraggio e tutti gli eventuali altri vincoli non implicitamente soddisfatti dalla formulazione polinomiale. La soluzione del problema di ottimo viene ottenuta mediante un semplice algoritmo di compass search in due stadi: da principio l’algoritmo procede alla ricerca di una soluzione che non violi i vincoli; una volta individuata, prosegue alla ricerca dell’ottimo. I vincoli non lineari vengono valutati discretamente, mediante metodi pseudospettrali. L’algoritmo viene quindi applicato a diﬀerenti scenari di un atterraggio lunare, in modo da stimarne ﬂessibilità di applicazione e sensitività agli errori di navigazione. Parole chiave: atterraggio di precisione, guida adattiva, retargeting, hazard avoidance

Ringraziamenti

Un ringraziamento alla professoressa Lavagna, che mi ha dato l’opportunità di svolgere questa tesi e nei cui corsi sono rimasto coinvolto nel problema dell’atterraggio. Grazie all’Ing. Roberto Armellin che, con i suoi consigli al momento giusto, è stato determinante per la buona riuscita del lavoro. Grazie alla mia famiglia, a mia madre, alle mie sorelle Mari, Claudia e Daniela, a Simone, Angelo e Alberto, che mi hanno sempre sostenuto in ogni momento e in ogni situazione, e sono sempre stati per me il migliore degli esempi. Grazie a tutti i miei nipotini, Michele, Milena, Andrea, Giulia, Arianna, Marta e Edoardo: distraendo lo zio Paolo che lavorava, gli hanno consentito di riposare i neuroni e ripartire ogni volta più scattante di prima. Grazie al Ricky, al Dimpo, al Manu, a Stephane a Michel e al Mike, senza i quali l’università sarebbe stata davvero più pesante. Grazie al Rosso, al Toga e al Betto, che han sempre sopportato le mie sparizioni prolungate. Grazie infine a Pao, pungolo costante a dare il meglio di me e che in me ha sempre creduto.

Contents

Nomenclature

xvii

1 Introduction 1.1 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Motivations and Goal . . . . . . . . . . . . . . . . . . . . . . . 1.3 Dissertation overview . . . . . . . . . . . . . . . . . . . . . . . 2 Nominal Landing Trajectory 2.1 Landing phases . . . . . . . . . . . . . . . . 2.2 Nominal Landing Simulation . . . . . . . . . 2.2.1 Discretization of Control Variables . 2.2.2 Nominal Landing Simulation Results 3 Retargeting Algorithm 3.1 Landing Model . . . . . . . . . 3.1.1 Reference systems . . . . 3.1.2 Equation of motions . . 3.2 Trajectory Design . . . . . . . . 3.2.1 Polynomial Formulation 3.2.2 Additional Constraints . 3.3 Optimization Algorithm . . . . 3.4 Algorithm Performances . . . . 3.4.1 Order of approximation 3.4.2 Divert Capability . . . . 3.4.3 Optimality . . . . . . . .

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4 Retargeting Simulation 45 4.1 Modeling criteria . . . . . . . . . . . . . . . . . . . . . . . . . 46 xi

4.2

4.1.1 Lander Dynamics . . . . . . . . 4.1.2 Navigation Errors Model . . . . 4.1.3 Guidance . . . . . . . . . . . . 4.1.4 Control and Actuation . . . . . Results . . . . . . . . . . . . . . . . . . 4.2.1 Single Landing Case . . . . . . 4.2.2 Sensitivity to TLS . . . . . . . 4.2.3 Sensitivity to Initial Dispersion 4.2.4 Cameras Field of View . . . . .

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46 50 52 54 58 58 63 65 67

5 Conclusions 71 5.1 Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.2 Future Developments . . . . . . . . . . . . . . . . . . . . . . . 72 Bibliography

xii

75

List of Figures

1.1 Historical Perspective over the Landing Accuracy on Mars . .

2

2.1 2.2 2.3 2.4 2.5 2.6 2.7

Landing phases. . . . . . . . . . . . . . . . . . . . . . Global landing reference system. . . . . . . . . . . . . Nominal landing - control proﬁles . . . . . . . . . . . Nominal landing path, Vertical and horizontal speed . Nominal landing path, altitude and mass . . . . . . . Nominal landing - view angle onto NLS . . . . . . . . Nominal landing - Trajectory . . . . . . . . . . . . .

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3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16

Body-Fixed, Flight and Ground reference systems. Body-Fixed reference system and euler angles. . . Glide-slope constraint. . . . . . . . . . . . . . . . Considered RCS thrusters scheme. . . . . . . . . . ESA Lunar Lander dimensions. . . . . . . . . . . Mean Computation time vs N . . . . . . . . . . . Position precision vs N . . . . . . . . . . . . . . . Speed precision vs N . . . . . . . . . . . . . . . . Feasibility performances. . . . . . . . . . . . . . . Feasibility performances (2) . . . . . . . . . . . . Computational performances. . . . . . . . . . . . Fuel consumption performances. . . . . . . . . . . Optimality comparison, CASE 1. . . . . . . . . . Optimality comparison, CASE 2. . . . . . . . . . Optimality comparison, CASE 3. . . . . . . . . . Optimality comparison, CASE 3. . . . . . . . . .

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4.1 Complete landing system logical scheme . . . . . . . . . . . . 45 4.2 Simpliﬁed Landing System . . . . . . . . . . . . . . . . . . . . 46 xiii

4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19

xiv

Parametric lander inertial model . . . . . . . . . . . . . Guidance logic diagram . . . . . . . . . . . . . . . . . . Control logic diagram . . . . . . . . . . . . . . . . . . . PWPF Modulation . . . . . . . . . . . . . . . . . . . . Single Retargeting, Position . . . . . . . . . . . . . . . Single Retargeting, Speed . . . . . . . . . . . . . . . . Single Retargeting, target and actual attitude . . . . . Single retargeting. Angular velocities . . . . . . . . . . Single retargeting. Mass trend . . . . . . . . . . . . . . Single retargeting. ACS Thrusters ﬁrings . . . . . . . . Single retargeting. Main thrust magnitude trend . . . . Landing Position Sensitivity to the TLS . . . . . . . . Landing Velocity Sensitivity to the TLS . . . . . . . . Landing Position Sensitivity to Navigation dispersions . Landing Velocity Sensitivity to Navigation dispersions . Camera pointing during retargeting . . . . . . . . . . . Sightline-TLS angle of view . . . . . . . . . . . . . . .

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48 53 54 57 58 59 60 61 61 62 62 63 64 65 66 67 68

List of Tables

2.1 Nominal landing simulation assumptions. . . . . . . . . . . . . 15 2.2 Nominal landing simulation - Additional constraints values. . . 16 3.1 Lander architecture assumptions. . . . . . . . . . . . . . . . . 35 4.1 4.2 4.3 4.4 4.5 4.6

Components of lander inertial model. . . . . . Sensors performance properties. . . . . . . . . Navigation errors covariance at 2000 m. . . . . Navigation errors covariance at ground (0 m). PID gains and equivalent dynamics properties. PWPF Modulator parameters. . . . . . . . . .

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xv

Nomenclature

∆

Compass search mesh size

Φ

Compass search Modiﬁed cost function

β

Angle of thrust

β

Angle of thrust pseudospectral representation vector

φ, θ, ψ Euler angles: Roll, Pitch and Yaw γ

Glide-slope angle

µ

Moon standard gravitational parameter

ω

Angular speed vector

ωM

Measured angular speed vector

ωT

Target angular speed vector

ωe

Error angular speed vector

ψ

Yaw angle pseudospectral representation vector

ρ

Margin of safety weight on maximum control torques

τ

Pseudospectral computational time

τ∆

Compass search minimum mesh size

τk

Chebyshev-Gauss-Lobatto nodes

τm

PWPFM ﬁlter time constant xvii

θM

Azimuth in polar coordinates, Moon centered frame

θ

Pitch angle pseudospectral representation vector

ADC

Attitude director cosine matrix

AGF

Director cosine matrix coordinates transformations between Flight and Ground frames

B

Gyro bias error

DN

Chebyshev diﬀerentiation matrix

˜N D

Modiﬁed Chebyshev diﬀerentiation matrix

F

Feasibility function

gM

Gravitational acceleration on Moon’s surface

I

Matrix of inertia

Imax

Maximum moment of inertia

Isp

Speciﬁc impulse

Kd

PID derivative gain

Ki

PID integral gain

Km

PWPFM ﬁlter gain

Kp

PID proportional gain

M

Gyro misalignment error

MC

Control torque

MC

Control torques vector

˜C M

Theoretical control torques vector

MD

Disturbance torques vector

Mmax Maximum available control torque N

Order of pseudospectral approximation

P

Thrust-to-mass ratio

xviii

P

Cartesian Thrust-to-mass ratio vector

RM

Moon mean radius

S

Gyro scale factor error

T

Thrust magnitude

T

Cartesian thrust vector

TT

Target thrust magnitude

Tmag Thrust magnitude pseudospectral representation vector a

Semi-major axis of orbit

a

Cartesian acceleration vector

b

Thrust point of application vector

cl , cU Non-linear constraints Lower and Upper bounds c

Optimization non-linear constraints vector

˜ c

Generalized non-linear constraints vector

ea

Euler angles vector

f

Optimization cost function

g0

Standard gravity acceleration

g

Cartesian gravity acceleration vector

h

Altitude

itF max Maximum feasibility iterations limit itOmax Maximum optimality iterations limit m

Mass of the spacecraft

mM

Measured spacecraft mass

mPDI Spacecraft mass at Powered Descent Initiation mdry Dry mass of the spacecraft mfuel Total fuel mass on the spacecraft xix

n

Gyro sensor noise

p, p

Generic function of time, and vector of its discrete pseudospectral representation

q

Quaternions vector

qM

Measured quaternions vector

qT

Target quaternions vector

qe

Error quaternions vector

r

Radial distance from origin in polar coordinates, Moon centered frame

r

Cartesian position vector

rM

Measured position vector

t

Time

t0 , tf Initial and ﬁnal time tToF

Time-of-ﬂight

u

Tangential speed in polar coordinates, Moon centred frame

umax PWPFM Schmitt-trigger output uoff

PWPFM Schmitt-trigger oﬀ-value

uon

PWPFM Schmitt-trigger on-value

v

Radial speed in polar coordinates, Moon centered frame

v

Cartesian speed vector

vM

Measured speed vector

wC

Clenshaw-Curtis quadrature scheme weights

wF

Generalized non-linear constraints weights

xL , xU Lower and Upper bounds of optimization variables x

Optimization variables vector

˜ x

Normalized optimization variables vector

xx

B

Body-ﬁxed reference frame

F

Flight axes reference frame

G

Ground axes reference frame

ACS Attitude Control System AG

Approach Gate

CGL Cebyshev-Gauss-Lobatto CoM Center of Mass DOI Descent Orbit Initiation FoV

Field of View

HDA Hazard Detection and Avoidance HG

High Gate

IMU Inertial Measurement Unit LG

Low Gate

LLO Low Lunar Orbit MB

Main Brake

NLS Nominal Landing Site PDI

Powered Descent Initiation

TD

Terminal Descent

TLS Target Landing Site

xxi

Chapter

1

Introduction In last years, a renewed interest in planetary exploration has brought to the realization of several missions, especially towards Mars, culminated with the recent landing of the rover Curiosity in August 2012. Together with Mars, the Moon is a main destination for exploration. The European Space Agency has conducted several studies concerning a possible unmanned lunar lander, while NASA is planning to send humans back to the Moon. ESA will supply the Orion/MPCV Service Module (SM) for the 2017 unmanned Exploration1 Mission, including ground and ﬂight operation support. Provisions for the construction and delivery of a second SM have been taken. Recently ESA and the Russian federal space agency, Roscosmos, have signed a formal agreement to work in partnership on the ExoMars programme towards the launch of two missions in 2016 and 2018, in order to bring a rover on Mars surface. In all these missions, the Entry Descent and Landing phase fulﬁls a critical role. During last decades, several improvements in automatic landing precision have been done (see Figure 1.1), but the relative uncertainty over the ﬁnal achieved position still imposes strict requirements onto the landing site choice. This is why an autonomous, precise and safe landing capability is a key feature for the next space systems generation. A dynamical landing site selection could allow the reaching of more scientiﬁcally relevant places, avoiding eventual hazardous terrain items. The short duration of the terminal descent phase and the delay of communications at large distances make a direct ground control impossible, and impose the develop of a fully autonomous system. After the landing site selection, the system needs to recalculate a pinpoint feasible trajectory toward the target. The pinpoint landing problem can be deﬁned as guiding a lander spacecraft to a given target on the planet’s surface 1

Chapter 1 with an accuracy of fewer than several hundred meters [2]. This implies the resolution of a minimum fuel optimal control problem.

Figure 1.1: This image illustrates how spacecraft landings on Mars have become more and more precise over the years. Since NASA’s first Mars landing of Viking in 1976, the targeted landing regions, or ellipses, have shrunk. Image credit: NASA/JPL-Caltech/ESA.

1.1

State of the art

A trajectory based on a quartic polynomial in time was used during the Apollo missions [19]. This allowed to perform in 1969 the ﬁrst recognized pinpoint landing (also if manually aided) during the Apollo 12 mission [4], whose lunar module landed at only 600 m from the target, the Surveyor 3 probe landing site. A derivative of the Apollo lunar descent guidance was still considered in recent years, for the Mars Science Laboratory (MSL) [35]. Various other approaches to obtain both numerical and approximate solutions of the pinpoint landing terminal guidance problem have been described over the last few years. In [32] the ﬁrst-order necessary conditions for the problem are developed, and it is shown that the optimal thrust proﬁle has a maximum-minimum-maximum structure. 2

Introduction Direct numerical methods for trajectory optimization have been widely investigated, not requiring the explicit consideration of the necessary conditions [5]. These methods have been used together with Chebyshev pseudospectral techniques, in order to reduce the number of the optimization variables [10]. Also convex programming approach has been proposed, in order to guarantee the convergence of the optimization [2]. Direct collocation methods has showed that the size of the region of feasible initial states, for which there exist feasible trajectories, can be increased drastically (more than twice) compared to the traditional polynomial-based guidance approaches, but at the price of a higher computational cost [2].

1.2

Motivations and Goal

The aim of this work is the development of a guidance algorithm capable to dynamically recompute and correct the landing trajectory during the descent, allowing the on-board choice of the landing site, required by systems that have to operate in full autonomy. Some principles has been considered as guidelines: • Computational Efficiency A full autonomous retargeting requires algorithms executable on-board, with limited hardware capabilities, and minimal time lag (if not in realtime). Low computational cost is a primary requirement. • Flexibility The attainable landing area must be maximized, in order to have the highest probability to ﬁnd a safe landing site. • Landing Accuracy Landing retargeting and hazard avoidance imply the requirement of a precision pinpoint landing. A semi-analytical approach is proposed, in order to conjugate the low computational cost of polynomial approximation to the larger ﬂexibility of direct optimization methods. Fuel consumption has been used as optimality criterion. In this work the last phases of a lunar landing have been examined. This implies that no aerodynamic forces are considered. However, the obtained results remain still valid also in presence of atmosphere with low density, such as the case of Mars, due to the relatively small velocities involved in this phase. 3

Chapter 1 In all the tests performed the ESA Lunar Lander mission has been taken as reference scenario. The European Lunar Lander is a mission under study within the Human Spaceﬂight and Operations Directorate of the European Space Agency (ESA). Originally planned for launch in 2018 and designed for landing near the Moon’s south pole, the mission’s primary objectives include the demonstration of safe precision landing technology as part of preparations for participation to future human exploration of the Moon [26]. Recently, the project was put on hold at the 2012 ESA Ministerial Council [8]. The technology developed in the context of Lunar Lander phase B1 could be exploited for future cooperations in the area of Lunar Exploration with Russia. The Luna-Resource Lander mission, planned by Roscosmos for 2017, could be a testing platform for European precision landing technology, with the proposed Hazard Detection and Avoidance Experiment and the Visual Absolute/Relative Terrain Navigation Experiment (VNE). The obtained results could be employed to perform an autonomous precision landing in the Lunar Polar Sample Return (LPSR) mission, planned for launch in 2020 [12].

1.3

Dissertation overview

In the ﬁrst part of Chapter 2 diﬀerent phases of a lunar landing, from parking orbit to touchdown, are presented. In the second part, a fuel-optimal solution for a nominal lunar landing trajectory (needed as starting point for retargeting) is obtained through numerical optimization in a two-dimensional frame. Chapter 3 focuses on the retargeting guidance algorithm development. The ﬁrst part is dedicated to illustrate the reference systems considered. The ODE system that describes the lander translational dynamics is derived. Then, a polynomial approach is used to reduce the trajectory computation to the search of optimal values for only two variables: the time of ﬂight and the initial thrust magnitude. A compass search algorithm, modiﬁed to handle nonlinear constraints, is employed in order to ﬁnd the fuel optimal solution of the problem, with a light computational cost. The last part of the chapter is dedicated to the validation of the algorithm: computational performances and divert capabilities at diﬀerent altitudes are estimated by Monte Carlo analysis. Finally, the optimality of the achieved solution is appraised through comparisons with direct collocation schemes. In Chapter 4 a closed-loop simulation of a retargeting during a lunar landing is presented. First, the spacecraft dynamic system, completed by 4

Introduction the rotational dynamics, is presented. In the second part of the chapter, the model structure is expounded; assumptions and approximations made are discussed. The guidance algorithm developed in Chapter 3 is coupled with an all-thruster Attitude Control System. A vision-based navigation system is emulated by introducing stochastic errors in the states inputs of the guidance and control modules. In the last part, obtained results are illustrated. First, a typical case is presented, in order to show distinctive behaviours in the system response. Then, the landing accuracy is evaluated using Monte Carlo analysis. Sensitivity to the diversion magnitude and to navigation dispersion are investigated. Possible sources of dispersion, and their relative impact over the landing accuracy are discussed. Finally, preliminary considerations about the obtained visibility of the landing site are presented. In Chapter 5 obtained results are summarized, and some considerations on requirements imposed by the retargeting strategy are given. An outline of possible future developments and further investigations closes the work.

5

Chapter

2

Nominal Landing Trajectory In this chapter all the phases of a lunar landing from a LLO parking orbit is presented. Then, a nominal two-dimensional landing trajectory, needed as starting point for the development of a retargeting algorithm, is obtained through a fuel optimization. Because fuel mass is a major part of the total vehicle mass, minimizing the fuel consumption is the most suitable criterion in determining an eﬃcient landing strategy.

2.1

Landing phases

A nominal Descent and Landing proﬁle, as described in ref. [15] is considered. All the landing phases are depicted in Figure 2.1. The lander is assumed to be delivered on a 100 km Low Lunar Orbit (LLO). The landing procedure is made by two main phases, the Coasting and the Powered Descent. The Powered Descent is, in turn, divided in three subphases: the Main Brake, the Approach and the Terminal Descent. • Coasting Phase At Descent Orbit Initiation (DOI), a de-orbit burn inserts the lander in a Descent Orbit, with a periselenium at 15 km altitude. In this phase, the spacecraft needs to execute a 180◦ turn around the pitch axis, in order to point the main engine nozzle at forward direction at the periselenium. • Main Brake Phase At periselenium, the Powered Descent phase is initiated (PDI), starting with the Main Braking phase: the engines are ignited to provide maximum thrust and to eﬃciently remove as much of the orbit velocity 7

Chapter 2 as possible. At the High Gate (HG) point, the Nominal Landing Site (NLS) comes into the ﬁeld of view of the sensors. The lander starts a pitch maneuver to realize a trajectory compatible with navigation and Hazard Detection and Avoidance (HDA) constraints, in terms of visibility of the NLS and time for image processing. • Approach Phase A few seconds later, at Approach Gate (AG) the Hazard Detection System comes into operation. Once the hazard map of the NLS area is completed, the thrust level is reduced to gain manoeuvrability and between 2000 and 100 m of altitude one or two retargetings may be commanded to divert to a safer Landing Site (Target Landing Site, TLS). • Terminal Descent At Low Gate (LG), the Terminal Descent phase begins. This part starts about 30 m above the surface and achieves a vertical uniform trajectory until Touchdown (TD). The thrust is basically equal to the weight of the lander during this last phase.

Figure 2.1: Lunar landing phases with representative timeframe and altitude values (adopted from ref. [15]).

8

Nominal Landing Trajectory

2.2

Nominal Landing Simulation

The landing trajectory is obtained by ﬁnding a fuel-optimal solution of the Main Brake phase and of the subsequent Approach phase, from PDI to Low Gate, towards the Nominal Landing Site. Searching for a nominal trajectory, no retargeting is for now considered. Every position on a planet can be reached with a planar trajectory, if the parking orbit inclination is equal to the ﬁnal desired latitude. Then, the landing is modelled as two-dimensional. Assuming r and θM as polar coordinates of the position of the lander (see Figure 2.2), u as tangential speed, v as radial speed, m as lander mass, the lander translational dynamics is governed by the system: r˙ = v, u θ˙M = , r u˙ = T (t) sin β(t) − 2 uv , m r (2.1) 2 µ u T (t) cos β(t) − 2 + , v˙ = m r r T (t) m ˙ =− , Isp g0

where the thrust magnitude T (t) and the thrust angle β(t) are control variables; µ is the standard gravitational parameter of the Moon, Isp is the speciﬁc impulse of the lander main engine, g0 is the standard gravity acceleration on Earth. Assuming the altitude h(t) = r(t) − RM , where RM is the

u

T β

M oo n s ur f a ce

v

m r ϑ

Figure 2.2: Global landing reference system.

9

Chapter 2 Moon mean radius, indicating with the subscript p the values at the transfer orbit periselenium, the initial system states are: h(0) = hp , θM (0) = 0, (2.2) u(0) = up , v(0) = 0, m(0) = m , p

together with the additional control constraint: π β(0) = − . 2

(2.3)

An initial mass mp of 1500 kg is expected [15]; hp = 15 000 m is the transfer orbit altitude at periselenium, and up is the corresponding orbital velocity: s 2 1 = 1696.02 m s−2 . up = µ − rp a Since a polar landing is contemplated, no horizontal speed is required at the touchdown. The ﬁnal states lower and upper bounds considered are: m, 20 ≤ h(tf ) ≤ 30 −0.5 ≤ u(tf ) ≤ 0.5 m s−1 , −1.5 ≤ v(tf ) ≤ 0 m s−1 ,

(2.4)

along with the ﬁnal control constraint of v