Robust control of asteroids' bearing rate during mineral extraction

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School of Engineering

Department of Information Engineering

Master's Degree Course in Robotics and Automation Engineering

Master's Degree Thesis

Robust control of asteroids' bearing rate

during mineral extraction

Contestant:

Ruben Agnesi

Advisors:

Prof. Andrea Caiti

Prof. Mario Innocenti

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Abstract

The European Space Agency (ESA) is studying the possibility of extracting minerals from asteroids.

Here we will deal with the control of an asteroid's bearing rate, which is per-turbed by the extraction operations. Indeed, due to the extraction, we modify the system's mass distribution, namely its density, and consequently its inertia tensor. Therefore, due to the angular momentum conservation law, also the angular velocity of the system is modied.

The purpose of this project is to design a robust controller which keeps the motion of the asteroid unperturbed, as the excavation never has occurred, despite the measurement errors and the uncertain knowledge of the asteroid's parameters.

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Preface

In this work we are interested in the study of the excavation's eects on the asteroids' orbit and rotation in order to design a controller which was able to reject these eects. Indeed our purpose is to keep the asteroid's motions unperturbed by the human extraction. Our work is limited at the extraction phase, therefore at the time window from when the spacecraft is already landed on the asteroid and it is starting the process of excavation, till when the minerals are collected and the excavation is over. Through these operations we change the asteroid mass distribution, because we move its material from a location to other inside it. The inertia tensor of a system depends from the entity of the mass and from how this mass is located inside it, therefore a modication of the system's mass distribution implies a change in the inertia tensor. The angular momentum of a system is determined by its inertia tensor and velocity. Because of the angular momentum is a conserved quantity, a modication of the inertia tensor leads to a change of the angular velocity, in order to keep the angular momentum constant.

The purpose of this work is to delete the eects of the excavation on the asteroid motions, in order to keep the orbit and the rotation unperturbed from the human activ-ities. The request has to be satised despite the presence of errors in the measurements and uncertainties in the asteroid's model, due to system dynamics's approximations and an inaccurate knowledge of its parameters. Therefore it is necessary a robust controller which, for every system whose uncertainties are inside the set of the possible ones, is able to guarantee the system's stability and a minimum performance, namely a maximum error on the desired velocity never surpassed.

In order to do this one, we will use an H∞ lopp-shaping controller and we will

design it by the Glover-McFarlane procedure [9], which lets to easily set the system's performance and it provides a closed form of the controller's expression based on the maximum achievable stability margin. the technique does not let to reach the maximum margin, but one as near to it as we desire.

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Notations

Here we indicate the usual notations used in this text. Whether in some sections we need to introduce new notations or we does not use these ones, we will say in such sections.

v by lowercase block letters we represent constants, scalar functions, scalar variables and

vectors' norms.

v by bold lowercase block letters we represent vectors. Only exception to this rule is the

angular momentum vector for which we use the bold uppercase block letters L

V by bold uppercase blocks letters we represent matrices. Only exception to this rule is

the angular momentum vector for which we use the symbol L. The same symbol is not used for any matrix in this text.

f

pV

e

i the position of superscripts and subscript in this text has a specic meaning:

• the superscript e after variable is always an exponent, so it is always a number

and it has the usual mathematical meaning.

• the superscript f before the variable represents the reference frame in which

the variable is expressed. Often it is omitted, but in these case is specied at the beginning of the analysis in which frame we are working.

• the subscript p before the variable is used for specifying the spatial point in

relation to which are calculated some particular variables, like angular momen-tum and inertia tensor. Therefore it is used only for the variables which need it and sometimes it is omitted, but in these case is specied at the beginning of the analysis which point we are using for dening the variable.

• the subscript i after the variable is used for add an information, extend the

name of the variable or for indicating the time in which that variable is eval-uated or measured. Usually the dependence by the time is omitted despite of in the case when this omission may cause confusion, as when we are deal with a variable quantity and its evaluation in a specied time. In these case, if a change of notation is needed, we specify at the beginning of the analysis.

I,Id Very often we deal with inertia tensor and identity matrix together. Unfortunately

historically the same symbol is used for both quantities, that is a capital "I". For this work we must change this notation for avoid misunderstanding, so we will call

identity matrix Id and inertia tensor I. When we refer to the inertia tensor at the

time 0 we use the notation I0.

Sv represent the skew-matrix of a generic vector v, namely given a vector v = [vx, vy, vz]T

its skew-matrix is dened as

Sv=   0 −vz vx vz 0 −vy −vx vy 0  

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0

R1 represents the rotation matrix for expressing the coordinates of a vector from the

reference frame 1 to the reference frame 0 ˜

v all the estimated quantities (function, constants, vectors and matrices) are represented

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Chapter 1 Model

1.1 Asteroid's model

Our purpose in this chapter is to nd a state-space representation of the asteroid, as in (1.1), which can describe its motion and the modications to it due to the mineral

excavation. ₍

˙

x = Ax + Bu

y = Cx + Du (1.1)

It is probable that the excavation will be executed on near Earth objects (NEO), due to their proximity to the Earth extremely reduces costs and mission's duration. We will use this assumption for do some approximations on the asteroid behaviour, in order to design a reliable model, but not more complex than the necessary.

1. Circular orbit

Due to we are studying a NEO, its orbit probably has a low eccentricity, namely is almost circular. Moreover the duration of the mission will be probably short enough to not noticing signicant dierence between the real orbit and the approximated circular one. This hypothesis let us to consider the linear orbital velocity of the asteroid barycenter only composed by a component tangential to the orbital plane. 2. Punctual gravitational sources

If we exclude particular cases, the greatest gravitational sources for a NEO are the Sun, the Earth, and the Moon. For the purposes of this work we consider them as punctual gravitational sources located in the barycenter of these objects, which act on the asteroid's barycenter. We have done this choice due to the great distances from these objects and the asteroid, the small size of it (from few meters to hundreds) in relation to the sources, and the duration of the mission, which is too short for revealing signicant eects caused by these irregularities in the gravitational eld.

Before starting with the designing of the model it is useful dening some characteristic points and frames of our system (see also gures 1.1 and 1.2). The description of the asteroids' behaviour is possible through the following three points.

• C is the center of the approximated asteroid's circular orbit, namely the

instanta-neous rotational centre of the real orbit considered at a specied time

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Figure 1.1: Asteroid's orbit and its circular approximation. They are underlined the reference frames 0 and 1, the Sun's barycenter S, the asteroid's barycenter B and the center of the approximated circular orbit C

Figure 1.2: Zoom of the asteroid's orbit. They are underlined the reference frames 00

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• D is the the origin of the body reference frame.

• Pi is the location of a generic point or particle inside the asteroid.

Other than these points, for our treatise we need to dene these four reference frames:

• one inertial frame located in the orbital centre of the asteroid, which we call by

number 0.

• a twin of the previous inertial frame, but located in the asteroid's barycenter. We

refer to it by number 00_.

• a non-inertial frame, located in the orbital centre, which rotates at the same velocity

of the orbital motion, namely it follows the asteroid's barycenter during its orbital movement. We use number 1 as reference to it.

• a non-inertial frame, located in the asteroid's barycenter and constrained to it,

namely it rotates with it at the same rotational velocity, so all the particles of asteroid are xed in relation to this frame. We refer to this frame by number 2.

1.1.1 Basic model

In this section, given two points P1 and P2, we use the notation 0−P1P2−−→ to refer to the

vector position of the point P2related to the point P1, expressed in the reference frame

0.

We can imagine the asteroid as it was composed by a nite number of small particles.

Every particle i has mass mi and it is located in the position Pi. First we dene the

kinematic equations of a generic particle Pi inside the asteroid, namely its position and

velocity related to the orbital center C. For the basic model the acceleration's equation is unnecessary. 0_−→ CPi=0R1 1₋₋_→ CB +0R₀00 0 R2 2₋₋_→ BPi (1.2)

where the rst addend represents the position of the asteroid's barycenter B, related to

the orbital center C. Instead the second addend represents the position of Pi related

to the barycenter B. Notice that the frames 00 _{and 0 have the same bearing, so the}

rotation matrix0

R00 is the identity one. Therefore, in order to not complicate uselessly

the notation, we delete from the expression (1.2) the matrix 0

R00 and we substitute in

00_R2 _{the reference 0}0 _{by 0. This assumption does not involve any error, because given}

a vector, its coordinates in a particular reference frame only depends from the mutual bearing between the vector and the frame and not from the location of the frame in the space. Therefore we can rewrite (1.2) in the following simpler form

0_−→

CPi=0R1

1₋₋_→

CB +0R22−BP−→i (1.3)

Now we do the derivative of (1.3) for nding the expression of the velocity. 0 ˙ −→ CPi=0Sω1/0 0_R 1 1₋₋_→ CB +0Sω2/0 0_R 2 2₋₋_→ BPi (1.4) where

ω1/0 is the angular velocity of the reference frame 1 as it rotated around the reference

frame 0

ω2/0 is the angular velocity of the reference frame 2 as it rotated around the reference

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Notice we have applied the assumption 1 of section 1.1; indeed we have derived the orbital

velocity of the point Pi as it was composed by only a tangential component, dependent

on the angular velocity 0_{ω1/0. Also for the rotational velocity we have considered only}

the tangential component, but this is not an approximation, indeed we are dealing with a rigid body, namely every asteroid's particles is xed to each other and no reciprocal translations between them are possible.

We remind that the vector product is anticommutative, namely taken two vectors v1

and v2 it is veried the following property

v1∧ v2= −v2∧ v1

Obviously this property is kept also in the skew-matrix representation of the cross prod-uct. Therefore through this property we can rewrite equation (1.4) in the following form

0_−→_˙ CPi= −0R11S−→_CB0RT1 0_ω 1/0− 0 R22S−→ BPi 0 RT₂0ω2/0

Now we calculate the total angular momentum of the asteroid in relation to the orbital centre C and in the coordinates of inertial frame 0

0 CL = X mi 0_−→ CPi∧ 0 ˙ −→ CPi= X mi0S−→_CP i 0 ˙ −→ CPi = −Xmi 0R11S−→_CB0RT₁+0R22S−→_BP i 0_RT 2 0 R11S−→ CB 0 RT10ω1/0+ 0 R22S−→ BPi 0 RT20ω2/0 = −Xmi 0R11S2−→ CB 0 RT10ω1/0+ 0 R22S2−→ BPi 0 RT20ω2/0+ +0R11S−→_CB0RT₁0R22S−→_BP i 0_RT 2 0_ω 2/0+ +0R22S−→_BP i 0_RT 2 0_R11_S−→ CB 0_RT 1 0_ω 1/0 = −0R1 M01S2−→ CB 0_RT 1 0_ω 1/0−0R2B2I 0_RT 2 0_ω 2/0+ −Xmi 0S−→_CB0S−→ BPi 0_ω 2/0+0S−→_BP i 0_S−→ CB 0_ω 1/0 (1.5) where

M0 is the total mass of the asteroid

2

BI is the inertia tensor of the asteroid related to its barycenter B and expressed in the

reference frame 2

We can further simplify equation (1.5), because both the summations of the last line are null. In order to demonstrate it, we analyze separately the two addends. We begin with the rst addend −Xmi0S−→ CB 0 S−→ BPi 0_ω2/0 ₌X mi0S−→ CB 0 Sω2/0 0₋₋_→ BPi =0S−→_CB0Sω2/0 X mi 0₋₋_→ BPi=0S−→_CB0Sω2/0 M0 0₋₋_→ BB = 0 (1.6)

Indeed the summation (P mi0−BP−→i) gives the mass of the asteroids multiplied for the

distance of the barycenter from itself, so a null distance. The same result is obtained if we analyze the second addend of the last summation in (1.5).

−Xmi0S−→_BP i 0_S −→ CB 0_ω 1/0= X mi 0S−→_CB0Sω_1/0−0_Sω 1/0 0_S−→ CB 0−−→ BPi = 0S−→ CB 0_S ω1/0− 0_S ω1/0 0_S−→ CB X mi 0₋₋_→ BPi = 0S−→ CB 0 Sω_1/0−0Sω_1/00S−→ CB M0 0₋₋_→ BB = 0 (1.7)

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where in the rst equivalence we have used the Jacobi identity, which is a property of

the cross product, such that, given three vectors v1, v2 and v3 it is veried that

v1∧ (v2∧ v3) + v2∧ (v3∧ v1) + v3∧ (v1∧ v2) = 0

As in (1.6), also in (1.7) we have obtained the vector (M00−BB)−→ , which is null. Therefore,

by the results obtained in (1.6) and (1.7), we can rewrite (1.5) as 0 CL = − 0 R1 M01S2−→ CB 0 RT₁0ω1/0− 0 R2_B2I0RT₂0ω2/0 (1.8)

Now we calculate the derivative of (1.8) 0 CL = −˙ 0_S ω1/0 0_R 1 M01S 2 −→ CB 0_RT 1 0_ω 1/0−0R1 M01S 2 −→ CB 0_RT 1 0_ω_˙ 1/0 −0Sω2/0 0_R 2B2I 0_RT 2 0_ω 2/0−0R2 2 BI˙ 0_RT 2 0_ω 2/0−0R2B2I 0_RT 2 0_ω_˙ 2/0 (1.9) Now, in order to nd a relation between the angular velocity and the torque, we apply the law of the angular momentum conservation. Indeed for an inertial reference frame, as the 0 one, we can write

0

CL =˙

0

Cτext (1.10)

Where 0

Cτextrepresents the summation of all external torques which eect on the system

is expressed by an equivalent torque related to the point C.

We now investigate the possible natural external torques. In this environment the principal forces which act on the asteroid are gravitational and, as revealed at the begin-ning of this chapter, the principal gravitational sources of our system are the Sun, the Earth and the Moon. As we said in the second assumption of section 1.1, we consider them as punctual source, located in the barycenter of the bodies. We use the letters S,

E, M for indicating respectively the barycenters of the Sun, the Earth and the Moon.

We begin with the calculation of the force generated by the Sun on an asteroid's particle

Pi, the calculation of the Earth and Moon forces is analogue.

0_f i= − GMmi SP3 i 0_−→ SPi' − GMmi SB3 0_−→ SB (1.11)

where G is the universal gravitational constant and M is the Sun mass.

Notice we have considered as if every point had the same distance from the Sun and we have considered this distance as if it was the distance between the Sun and the asteroid's barycenter. This approximation is derived by the second assumption of the previous section 1.1, however, by equation 1.11, the reasons of this approximation

becomes clearer. Indeed the approximation−→SPi'

−→

SBis valid, due to the great distance

between the Sun and the asteroid and the small sizes of this last one, related to that distance. Obviously the same approximation is valid in the case of the Earth and Moon forces.

Now we calculate the total torque applied by the Sun on the asteroid, by the sum-mation of the applied torques on every particle, namely the sumsum-mation of the forces' momentum related to the orbital centre C.

0 CτSun = X ₀ Cτi= X0−→ CPi∧0fi= X0−→ CPi∧ −GMmi SB3 0_−→ SB = −GM SB3 X0−→ CPi∧ mi 0_−→ SB=GM SB3 0_−→ SB ∧Xmi0−→CPi =GM SB3 0_−→ SB ∧ M0 0₋₋_→ CB = GMM0 SB3 0_−→ SB ∧ 0₋₋_→ CB (1.12)

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We nd similar expressions for the Earth and Moon torques. 0 CτEarth= GM⊕M0 EB3 0_−→ EB ∧0−CB−→ (1.13) 0 CτM oon= GM_$M0 M B3 0_−−→ M B ∧0−CB−→ (1.14)

We observe that the gravitational attraction of these objects inuences the asteroid's orbit, but it is not clear if it changes also its rotation. If this one was the case, we should nd also a torque applied around the asteroid's barycenter. We try to calculate the torque exerted by the Sun around the asteroid's barycenter B. The results in the case of the Earth and Moon torques is analogue.

0 BτSun = X ₀ Bτi= X0−−→ BPi∧0fi= X0−−→ BPi∧ −GMmi SB3 0_−→ SB = −GM SB3 X0−−→ BPi∧ mi 0_−→ SB =GM SB3 0_−→ SB ∧X mi 0₋₋_→ BPi =GM SB3 0_−→ SB ∧ M00−BB = 0−→

We have just demonstrated that, to a rst approximation, every gravitational torque does not modify the asteroid's rotation. However this result depends from the approximation in (1.11), but also from the omission of other non-gravitational eects (as the YORP eect) which in this work are not considered. Therefore in general cases the gravitational forces exerted by these objects (and any other one suciently near to the asteroid), together with other phenomena, inuence the asteroid's rotational motion; but in this work we have considered these eects negligible, because of they are usually relevant on much longer period than the duration of the mission, namely, within the mission's time window, their eect on the rotational motion is very much smaller than that one produced by the excavation.

Now we substitute the founded torques in (1.12), (1.13) and (1.14), together with the angular momentum derivative in (1.9), inside the angular momentum conservation law in (1.10). −0_S ω1/0 0_R1M01_S2_−→ CB 0_RT 1 0_ω 1/0−0R1M01S 2 −→ CB 0_RT 1 0 ˙ ω1/0 −0_S ω2/0 2 BI 0_RT 2 0_ω 2/0−0R2 2 BI˙ 0_RT 2 0_ω 2/0−0R2B2I 0_RT 2 0 ˙ ω2/0 =GMM0 SB3 0_−→ SB ∧0−CB +−→ GM⊕M0 EB3 0_−→ EB ∧0−CB +−→ GM$M0 M B3 0_−−→ M B ∧0−CB−→ (1.15)

As we said the gravitational torques only inuence the orbital velocity, moreover in (1.9) we notice that the eects of the orbital velocity and the rotational one, on the angular momentum rate, are separated. Therefore we can split expression (1.15) in an orbital component and in a rotational one.

0 CLorb˙ = − 0 Sω_1/00R1M01S2−→ CB 0 RT1 0 ω1/0−0R1M01S2−→ CB 0 RT1 0 ˙ ω1/0 =GM0 M SB3 0_−→ SB ∧ 0₋₋_→ CB + M⊕ EB3 0_−→ EB ∧ 0₋₋_→ CB + M$ M B3 0_−−→ M B ∧ 0₋₋_→ CB ! (1.16) 0 CL˙rot= 0 BL˙rot= −0Sω2/0 0_R 2B2I 0_RT 2 0_ω 2/0−0R2 2 BI˙ 0_RT 2 0_ω 2/0+ −0R2_B2I 0R₂T0ω2/0˙ =_B0τ (1.17) where 0

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Notice in (1.17), by the rst equivalence 0

CLrot˙ =

0

BLrot˙

we have armed that the rotational angular momentum rate, calculated in relation to the orbital centre, is equal to that one calculated in relation to the asteroid's barycen-ter. Indeed in (1.17) there is not any references to the orbital centre C, namely for its calculation the knowledge of the orbital centre location is unnecessary. We can see this fact as another proof of the independence between the rotational motion and the orbital one (in the limits of our approximations).

The same separation of (1.16) and (1.17), between the orbital motion and the rota-tional one, is applicable also on the expression of the angular momentum in (1.8), indeed we can easily split this last one in the following two components.

0 CLorb= − 0_R1M01_S_−→ CB 20 RT₁0ω1/0 (1.18) 0 CLrot=B0Lrot= −0R2B2I 0_RT 2 0_ω 2/0 (1.19)

Again in (1.19), for the same reasons of the equation (1.17), we have declared that 0

CLrot=

0

BLrot

Due to the excavation only acts on the rotational angular momentum and its deriva-tive, from now we will deal with these two quantities and we will omit the orbital ones.

We suppose that all the measurements are taken directly by the lander. Maybe in a real mission some measurements could be taken also from the Earth, but we think that these last ones would be only supplementary or redundant to those one gotten directly by the lander on the asteroid. Indeed the measurement accomplished by the lander have the advantages to be immediately available (there is not a delay of transmission) and probably, about the rotational angular velocity, they would be more accurate.

In general it is dened a non-inertial reference frame, called body, located on the lander and integral to it. For not complicating useless the calculations and the model, we have considered as if frame 2 and the body one were the same frame, or at least two aligned ones, so related one to each other by a constant identity rotational matrix. Indeed both the frames are non-inertial and integral to the asteroid. Frame 2 is integral to it for denition, instead the body one is integral to the lander, which, during the excavation phase, is stopped on the asteroid and so integral to it. Notice that this choice keeps the treatise general, indeed if the two frame were not aligned we would use a rotational

matrix 2_Rbody _{for passing from reference 2 to the body one. This matrix would be}

constant due to both, the frame 2 and the body frame, are integral to the asteroid, so

2 _˙

Rbody= 0

and no other element would be introduced to the expression of the angular momentum derivative. Anyway it would be introduced to the expressions a constant rotational matrix, which does not change the theoretical treatise.

Therefore, in order to express the angular momentum and its conversation law in the reference frame 2, we pre-multiply left and right members of (1.19) and (1.17) for the

rotation matrix0_RT

2. In this way we derive the following expressions.

2 BLrot= − 2 BI 2_ω2/0 _(1.20) 2 BL˙rot= −0R2B2I 2_ω 2/0− 2 BI˙ 2_ω 2/0+ − 2 BI 2 ˙ ω2/0=B2τ (1.21)

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Now the equations are almost ready to be used for representing the system in the form of equation (1.1); we have only to choose the outputs, inputs and status vectors.

The output is the quantity which we want control, therefore the angular velocity

2_ω_{2/0. The input is the only quantity we can produce in order to change the rotational}

motion, namely the control torque 2

Bτ, as it is showed also in (1.21). The status vector

is usually the most complex to choose and when we have no ideas about it, a feasible way is to use the output as status. Initially we have followed this way, but it leaded to numerical instability, during Matlab simulations. At later time we have notice that the use of the angular momentum as status vector solve the computational instability and moreover it is more natural, clear and it reduces the complexity of the system. However, by the equations (1.20) and (1.21) which we have just derived, it is not very clear the role of the angular momentum inside the system and so this form of the equations is not suitable for our state-space representation. In order to obtain this purpose, it is sucient express equation (1.21) in a more implicit way, where the direct dependence between the inertia tensor, angular velocity and angular momentum is hide. Therefore we substitute (1.20) in (1.21) and we obtain the state-space representation of the system which we were searching for. (2 BL˙rot= −2Sω2/0 2 BLrot+B2τ 2_ω2/0_{= −}2 BI −1 2 BLrot (1.22) Finally, as we have said, the measurements and estimations will be done by the lander, so we have also to consider to refer all the quantities to the origin of the body reference frame instead of the barycenter, which has the disadvantage of being a mobile point which introduces a lot of estimation and measurements errors than a xed point as the body frame origin. In order to convert equation (1.22), it is helpful use the generalized Huygens-Steiner theorem as expressed in (A.3) from which we saw it exist a simple relation between the angular momentum calculate from the barycenter and that one calculated in another point. We notice that the size of the inertia tensor is changed, and consequently also that one of the angular momentum. Moreover also the torque is dierent, because it acts on another point. However we are not interested in nding an explicit conversion for our model, due to we suppose that the inertia tensor is estimated directly by the lander and the control torque calculated directly for this application point and not for the barycenter. Therefore we write the nal expression of the basic model

(2 DLrot˙ = − 2 Sω_2/0_D2Lrot+_D2τ 2_ω 2/0= −D2I −1 2 DLrot (1.23) where A = −2Sω_2/0 (1.24) B =Id (1.25) C = −_D2I−1 (1.26) D =0 (1.27)

1.1.2 Extended model

It can be useful to dene a model with a second order dynamic, in order to have more degree of freedom in the control and therefore obtaining behaviours more complex. In order to do this, we have to increase the dimensions of the status vector and computing the derivative of the angular momentum conservation law in (1.21). From this operation

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we obtain the following expression of the second derivative of the angular momentum. 2 BLrot¨ = − 2_S ˙ ω2/0 2 BLrot− S2ω2/0 2 BLrot˙ + 2 Bτ˙ (1.28) where 2

Bτ˙ is the rotatum, namely the derivative of the torque.

For the computational purposes, it is useful also to dene an explicit form of the second derivative of the angular momentum, not dependent from the angular momentum and its derivative. In order to nd it we substitute in (1.20) and (1.21) in (1.28) and we obtain 2 BLrot¨ = 2 Sω˙2/0 2 BI 2_ω 2/0+ S2_ω 2/0 0 R2_B2I 2ω2/0+ 2 BI˙ 2_ω 2/0+ 2 BI 2 ˙ ω2/0+ 2 Bτ˙ (1.29)

Also the dimensions of the output have to be increased, so we need to derive an

implicit expression of the angular acceleration 2

˙

ω2/0, dened in the reference frame 2.

This variable appears in the expression (1.21), so it is enough to move the member inside the equation for obtaining the expression of the rotational angular acceleration.

2_ω_˙ 2/0=B2I −1₋2 BL˙rot−2Sω2/0 2 BI 2_ω 2/0− 2 BI˙ 2_ω 2/0 (1.30) The expression is in an explicit form, whereas we need an implicit one in which the angular velocity does not appear explicitly. Indeed we have to show the relation of the angular acceleration with the status, in order to build the output equation of the state-space representation. We solve the problem by the angular momentum equation in

(1.20), which denes a direct relation between the rotational angular momentum 2

BLrot

and the rotational angular velocity2_{ω2/0. Therefore we substitute (1.20) in (1.30) and}

we derive the nal expression of2

˙ ω2/0 2_ω_˙ 2/0 =B2I −1h₋2 BL˙rot+ ₂ Sω2/0+ 2 BI˙ 2 BI −12 BLrot i (1.31) Now we can derive the state-space representation of the extend model. We use the rotational angular momentum and its derivative as state vector, the derivative of the torque (rotatum) as input and the rotational angular velocity and the angular acceleration as outputs. Therefore by (1.28) and (1.31) we derive the following representation of the system                " 2 BL˙rot 2 BLrot¨ # = " 0 Id −2_S ˙ ω2/0 − 2_S ω2/0 # " ₂ BLrot 2 BLrot˙ # + " 0 Id # 2 Bτ˙ " 2_ω 2/0 2 ˙ ω2/0 # =   −2 BI −1 ₀ 2 BI −12_S ω2/0+ 2 BI˙ 2 BI −1 ₋2 BI −1   " ₂ BLrot 2 BLrot˙ # (1.32)

which if moved to the body reference origin it becomes                " 2 DL˙rot 2 DLrot¨ # = " 0 Id −2_S ˙ ω2/0 − 2_Sω 2/0 # " ₂ DLrot 2 DLrot˙ # + " 0 Id # 2 Dτ˙ " 2_ω2/0 2_ω_˙ 2/0 # =   −_D2I−1 0 2 DI −12_Sω 2/0+ 2 DI˙ 2 DI −1 ₋2 DI −1   " ₂ DLrot 2 DL˙rot # (1.33)

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where A = " 0 Id −2Sω˙2/0 − 2_S ω2/0 # (1.34) B = 0 Id (1.35) C =   −2 DI −1 ₀ 2 DI−1 ₂ Sω_2/0+_D2I˙_D2I−1 −_D2I−1   (1.36) D =0 (1.37)

From the next chapter we will alway refer to the rotational quantities expressed in the body frame coordinates and related to the origin of the body frame D, so the complex notation used until now is useless. Therefore from now we will adopt the following simplied notation. L=4_D2Lrot (1.38) ˙ L=4_D2Lrot˙ (1.39) ¨ L=4_D2Lrot¨ (1.40) ω=42ω2/0 (1.41) ˙ ω=42ω2/0˙ (1.42) ¨ ω=42ω¨2/0 (1.43) I=4_D2I (1.44) ˙ I=4_D2I˙ (1.45) ¨ I=4_D2I¨ (1.46)

1.1.3 Observations about the relation between orbital and

rota-tional motions

Inuence of the excavation

During the excavation we change the asteroid's mass distribution and so its inertia tensor, but not its total mass; this means we observe a change only in the rotational angular momentum but not in the orbital one, because of this last one does not depend by the inertia tensor, but only by the total mass and the distance between the orbital center C and the asteroid's barycenter B. Despite the total mass does not change, it could seem that during the excavation the mass center is moving inside the asteroid shape and so also its distance from the orbital center is changing. The rst thought can be that the movement of the barycenter is very small compared to the distance from orbital center, so we can neglect it. If the center of mass changed is position, this approximation would be surely right, but actually we are not doing any approximation. Indeed from the point of view of our lander is natural and more useful consider that the barycenter B is moving inside the asteroid's shape, but actually, if we see the system from the point of view of the orbital center C, we observe the point B at a constant distance and the asteroid shape which is changing its position and bearing around it (see gure 1.3).

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Figure 1.3: During the extraction, despite we are moving the asteroid mass, due to the total mass keeps the same, the orbit does not change. Therefore from an inertial reference frame an observer would see the shape of the asteroid moving around its barycenter. Inuence of the actuators

From what we have said in section 1.1.1 and 1.1.2, we are interested in controlling only the asteroid's rotation, because this is the only movement which is inuenced by the excavation. Actually the rotation and orbital control are two complex problems, not totally separable, due to there are many secondary eects, more or less relevant, which act on the system and on the actuators and make these two motions not completely independents in the control. For example in this work we do not consider the kind of propulsion we use for the bearing control. If we use a propellant engine (e.g. hydrazine) the control is actuated by the expulsion of propellant from the thrusters, so the total system, composed by the asteroid plus the lander, loses mass. Another problem could be caused by the manner in which we apply the control torque. Indeed for the rotational control we have to apply a torque around the asteroid barycenter B, whereas for the orbital control we need to exert a torque around the orbital centre C, so a force toward the barycenter B. If the bearing control is realized by thrusters, located on the lander, in order to apply only a pure torque to the barycenter, they always should exert a force orthogonal to the position vector which links the barycenter B to the application point of the force. This one is in general false in the reality, because it is not possible to realize it with an absolute precision, so we always have an orbital disturbance due to the rotational actuation. Anyway it does not exist an a priori general rule for dealing with these eects. Therefore every time the designers have to analyze the particular system and decide if these eects and many other ones are negligible or decide how to deal with them.

This last argument is only for showing how much it is extended the problem and for saying that we are neglecting many of these eects, because of in many cases, depending by the order of precision we want, they are negligible. However a general rule is not denable, because of they depend from the particular mission and consequently they are hardly generalizable. Finally this is a general work which wants to be useful as a start point for other detailed studies applied to particular missions.

1.2 Extraction's model

We consider that the excavation occurs by a drill, so the shape of the space left void by the extracted material is the same of the drill, namely almost a cylinder. The irregularities in the cylinder's shape are located on the lateral surface, which is rough because of the drill activity, and on the bases areas, which are not completely at. Indeed the base area,

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where the lander digs, is not perfectly at and on the other side, on the drill's bottom, due to the detachment action, the material is not separated from the rest of the asteroid by a perfect plane. Moreover some material can detach from the lateral surface of the cylindrical hole and falling on the bottom of it. However in this work we will neglect all these irregularities and we will consider a perfect cylinder.

After we have dened the shape of the hole, we must understand the way in which the material is moved from its original location to the inner of the lander. In order to do this, we have to decide what kind of drills we will use for the excavation. Through a great simplication, we can divide the kinds of drills into two general groups:

• Helicoidal drill

This kind of drill is used when we are interested in collecting material for a following manufacturing of it. Through its shape, the helicoidal drill, transforms its rotational motion in a translational one. Indeed, by the rotation, it destroys the material in contact with it and advances inside the ground. When the excavation is over the material is crumbled and so detached from its original environment, therefore it is sucient to stop the rotation and translating the drill outside the hole with the crumbled material inside its hollows.

• Core drill

This kind of excavation is used when it is important keeping the original ground structure, so usually for scientic studies (e.g. geological ones). In this case the drill is a hollow cylinder, with no base on the excavation's side; there, on the edge of this open base, they are located the tips for the excavation. Through its rotational motion, the drill only removes a thin layer of material from the cylinder's lateral surface. The dust and the grains generated during the excavation are removed by a uid circuit. When the drill is lled by the core, only the core's bottom side follows to be connected with its original surrounding environment. Dierent mechanisms exist in order to complete the probe detachment; for example, in the ice excavation, it is located on the drill head a core catcher (core dog), which is a series of particular tips, which laterally harpoons the core. During the excavation the catcher is open, so it does not cut the core, but when the excavation is over and we begin to lift the drill, the catcher shuts and rotates, causing the detachment of the core's bottom. Despite the purpose of these missions is to collect minerals for manufacturing them on the Earth, therefore for commercial purposes, in this work we will use the core drill, because it implies less uncertainties about what happens to the mass distribution inside the drill during the excavation and the extraction. Indeed, except for the thin layer under the drill, during the excavation phase and till we start the extraction, the density of the asteroid keeps almost unperturbed and so we can follow to use the initial estimate of it. The hypothesis of the unperturbed mass distribution is an important approximation of how it acts the core drill. Indeed during the excavation we move the material from the thin layer under the drill edge to some place outside the hole, and the removed material in the hole is substituted by the drill. Therefore a small change in the mass distribution happens also in this kind of excavation. In our model we neglect this change, namely we consider an innitesimal thin drill, with no mass, which exactly passes between the molecules of the core and those ones of the rest of the asteroid and it exactly detaches the two parts without any loss of mass. This assumption further simplies the analysis of the density modications, due to this means that during the drill penetration no changes in the system's mass distribution happens. Therefore in our model all the modications in the mass distribution occur only during the core extraction and so we have to focus our analysis only on that phase.

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Figure 1.4: Overview of the main position vectors

In summary our core is an ideal one, with an exact cylinder shape, as large as the hole, in which we neglect the modication on mass distribution caused by the penetration and detachment phase, and where we consider only the changes produced by the extraction phase.

1.3 Barycenter's kinematic

In order to know what it happens to the system's barycenter we must understand how the density changes in the space occupied by the core during the extraction. In this location the mass is translated towards the lander and there it is stored somehow. The space left free from the core instantly decreases its density to zero, because of, how we have explained in section 1.2, there are no empty layers between the lateral cylinder's surface and the rest of the asteroid, so no uids can enter inside the hole. Anyway the hole has zero density also in the real system, because of there is not atmosphere around the asteroid.

In order to understand how the barycenter changes its position, we analyze the cti-tious case in which the extracted material disappears as it comes out from the excavation's hole. We apply the barycenter's denition, which imposes to divide the system's static moment by the total mass.

df =R 1 V_A0δ0(p) dV − R V_cr0δ0(p) dV + R Vcrδ(p) dV + Ml Z V0 δ0(p)p dV + − Z V_cr0 δ0(p)p dV + Z Vcr δ(p)p dV + Mldl (1.47) where

df is the barycenter's position in the ctitious case in which the mass disappears after

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Figure 1.5: Overview of the local reference frames, cylindrical and cartesian coordinates R

V represents a triple integral calculated on the three spacial coordinates. In order to

simplify the notation to use this symbol instead of the devoted onet_V

δ_{0(·) : R}3

→ R is the spacial function of the initial mass distribution, namely the initial asteroid density function. It is related to the spacial coordinates.

δ(·) : R3→ R is the current mass distribution which changes during the excavation, so

it is related to the time, as well as the spacial coordinates

VA₀ is the space occupied by the asteroid at the initial time, before the excavation starts

Vcr is the current space occupied by the core

Vcr₀ is the initial space occupied by the core

dV is the volume of an innitesimal point inside the integration space

p is the position of the innitesimal point whose volume is dV

Ml is the lander's mass

dl is the position of the lander's barycenter

The initial mass is composed by the asteroid's mass plus the lander's one, so from the

sum _Z

V_A0

δ0(p) dV + Ml (1.48)

In the ctitious model, while we extract the core, the total mass decreases of the quantities which comes out from the hole. The easiest way to establish the material mass which does not come out to the hole, namely the mass which still belongs to the ctitious system, is to subtract, from the total initial system's mass, all the initial core's mass and

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adding the mass still located inside the hole. Therefore we add, to the initial system's mass (1.48), the following quantity, composed by the initial core mass and the current one. − Z V_cr0 δ0(p) dV + Z Vcr δ(p) dV (1.49)

About the static moment, namely the numerator of equation (1.47), the reasoning is the same; from the initial static moment of the system we subtract the initial static moment of the core and we add the current one. Now we try to dene in a more suitable way the elements of the equation (1.47).

Initial asteroid mass and static moment We dene the initial mass of the asteroid

MA₀ =

Z

V_A0

δ0(p)dV (1.50)

Now we call dA0 the initial asteroid's barycenter, which is dened through the division

of the initial asteroid's static moment by its initial mass, so

dA0= 1 MA0 Z V_A0 δ0(p)p dV (1.51)

therefore, from (1.51) we derive the initial asteroid's static moment by Z

V_A0

δ0(p)p dV = MA0dA0 (1.52)

Initial core mass and static moment

Similarly to the asteroid's mass, we dene the initial core's mass

Mcr₀ =

Z

V_cr0

δ0(p)dV (1.53)

About the initial core static moment we could nd a denition similar to the asteroid static moment one in (1.52). Although in this case we are dealing with a cylinder, so, severally from the asteroid shape, an object with a known and particular geometry, which allows to nd a more explicit solution of the problem. Indeed, due to its particular geometry, it is easier to relate the particles' position to the core's center, whose location is known because it coincides with that one of the drill's center. Therefore we introduce the vector p∆, namely the particles' position referred to the centre of the core, dened in a reference frame aligned with the body one.

p∆=   xp_∆ yp_∆ zp_∆  =   r cos θ r sin θ zp_∆   (1.54)

where (see also gure 1.4)

xp_∆, yp_∆, zp_∆ are the cartesian coordinates in the initial cylinder local frame

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θ is the anticlockwise angle from the cartesian coordinate xp_∆ to the projection of p∆

on the (xp∆, yp∆)plane.

In order to calculate the static moment we need the cartesian coordinates, but in a cylinder it is easier to measure the cylindrical ones, for this reason in (1.54) the cartesian coordinates are related to the cylindrical ones.

In order to obtain the position of dV related to the body reference frame, we add to

p∆ the position of the cylinder centre 1₂dcr0, referred to the body reference frame. Due

to the two vectors are expressed in two constantly aligned reference frames, any rotation matrix is unnecessary (see gure 1.4).

p = 1

2dcr0+ p∆ (1.55)

Notice that in (1.55) the vector dcr0, which is the position of the initial core bottom, is

equal to dcr0 =   0 0 zcr₀  

where the rst two coordinates are null, due to they are always aligned to the body reference frame and the last one is implicitly negative by how we have chosen the body reference frame (see gures 1.4 and 1.5). Finally we have to consider that the vector p∆ represents the position of the particle in the initial core, referred to the cylinder's centre, so its range of integration is symmetric in all the directions, namely we are working in a new integration space, which now we dene (see also gure 1.6). In order to do this, it is useful to split the integration variable dV in its three cylindrical coordinates and explicitly showing the triple integral and its integration variables. Therefore we rst dene the integration variable dV by the three cylindrical variables.

dV = (rdθ)drdz where

dθ is the integration variable along the angular coordinate

dr is the integration variable along the radial coordinate

dh is the integration variable along the height

Therefore the new integration space Vcr_sm0 is dened by

Z V_crsm0 dV = y V_crsm0 dV = Z 2π 0 dθ Z r 0 dr Z −zcr0₂ zcr0 2 r dz (1.56)

whereas the original space Vcr0 was dened through

Z V_cr0 dV = y V_cr0 dV = Z 2π 0 dθ Z r 0 dr Z 0 z_cr0 r dz (1.57)

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Therefore we can nally rewrite the initial core static moment as Z V_cr0 δ0(p)p dV = Z V_cr0 δ0(p) 1 2dcr0+ p∆ dV =1 2 Z V_cr0 δ0(p)dcr0dV + Z V_crsm0 δ0(p)p∆dV =1 2 Z V_cr0 δ0(p) dV dcr0+ Z V_crsm0 δ0(p)p∆dV =1 2Mcr0dcr0+ Z V_crsm0 δ0(p)p∆dV (1.58)

Notice that in (1.58), the density of both addends follows to be calculated in the position

p, namely in the position referred to the body reference frame, due to δ0(·)is dened in

this frame.

Variable core mass and static moment

The core mass still inside the asteroid is time-variant, indeed it decreases as we extract it. From (1.47) we can derive its denition

Mcr=

Z

Vcr

δ(p) dV (1.59)

From (1.59) we observe that the core mass is dependent from two time-variant variables: the density function δ(·) and the integration space Vcr. The density function is time-variant, because during the extraction we move the core from a location to other, so we change the total mass distribution, as we have explained in section ??. Instead the integration space, that is the space lled by the core's portion still inside the hole, decreases during the extraction. Therefore we could dene two time-variant functions which describe these modications, but it is not a trivial and low cost computational procedure, especially for the update of the density function. Therefore we handle the problem in a dierent way. We imagine to deal with a time-invariant density function, so equal to the initial one, as the core was xed in the initial position. In this scenario, in order to obtain a behaviour equivalent to that one of the real system, we take an integration space xed on the bottom of the initial core and which reduces its height

from the top; we call this integration space VcrM. This space is dened only inside the

initial core shapes and it works only for the mass calculation. Indeed in order to compute the mass, it is not important where the material is really located, but rather its quantity. Similarly to what we have done in (??) and (??) to deal with a cylinder, we express the cartesian coordinates by the cylindrical ones. Moreover we introduce the new integration

space VcrM related to the constant density δ0(·), so the equation (1.59) becomes

Mcr= Z Vcr δ(p) dV = y Vcr δ(p) (rdθ)drdz = y V_crM δ0(p) (rdθ)drdz = Z 2π 0 dθ Z r 0 dr Z zcr0−zcr z_cr0 δ0(p) r dz (1.60) where

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zcr₀ is the body frame z coordinate of the core bottom at the initial time

zcr is the body frame z coordinate of the core bottom at the current time

The space VcrM, used for the Mcr computation, is not applicable to the static moment,

which, dierently from the mass, depends also from the particle's position. However we organize its expression in a more suitable way for the computation. Similarly to the previous subsection we imagine the position vector composed by the sum of the core's centre position and the particle's position referred to the current cylinder's centre.

p = 1

2dcr+ q∆ (1.61)

Also in this case we notice that the vector dcr, namely the position of the current core bottom, is dened by dcr=   0 0 zcr  

where the rst two terms follow to be null, due to the alignment with the body reference frame, and the last coordinate is implicitly negative. Therefore, similarly to the initial

core, we dene the new space Vcrsm in which the static moment of q∆ is integrated.

Z V_crsm dV = y Vcrsm dV = Z 2π 0 dθ Z r 0 dr Z −zcr₂ zcr 2 r dz (1.62)

which is dierent from the original space Vcr, namely Z Vcr dV = y V_cr0 dV = Z 2π 0 dθ Z r 0 dr Z 0 zcr r dz (1.63)

Therefore we can rewrite the core static moment in (1.47) as Z Vcr δ(p)p dV = Z Vcr δ(p) 1 2dcr+ q∆ dV =1 2 Z Vcr δ(p) dV dcr+ Z Vcrsm δ(p)q∆dV =1 2Mcrdcr+ Z V_crsm δ(p)q∆dV (1.64)

Similarly to the mass in (1.60), we want to deal with the initial density function

δ0(·), in order to simplify the calculation. For obtaining the right values of the density

we should integrate it in the space VcrM, whereas we are working in the Vcrsm space.

In order to obtain an equivalent result, we have to translate the density argument of a

quantity (dcr0 −

1

2dcr). Indeed in this way, due to the integration range of the height

is [zcr

2 ; −

zcr

2 ], it is like we are integrating the density argument in (1.61) in the range

[zcr₀; zcr₀− zcr].

Now we substitute the δ0function in (1.64) and we obtain

Z Vcr δ(p)p dV =1 2Mcrdcr+ Z 2π 0 dθ Z rdr 0 dr Z −zcr₂ zcr 2 δ0(p + dcr0− 1 2dcr)q∆r dz =1 2Mcrdcr+ Z 2π 0 dθ Z rdr 0 dr Z −zcr₂ zcr 2 δ0(q)q∆r dz (1.65) where q = p + dcr0− 1 2dcr

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Figure 1.6: The core with the several integration spaces used in this section Fictitious barycenter

Now we substitute (1.50), (1.52),(1.53), (1.58), (1.59) and (1.65) in (1.47) and we obtain the nal expression of the ctitious barycenter

df = 1 MA0− Mcr0+ Mcr+ Ml MA0dA0− 1 2Mcr0dcr0− Z V_crsm0 δ0(p)p∆dV + +1 2Mcrdcr+ Z V_crsm δ0(q)p∆dV + Mldl (1.66)

Kinematics of the real barycenter

We have still to consider that the mass does not disappear once it come out from the hole (as in (1.66)), but rather it is stored inside the lander. Moreover, in order to obtain a more realistic system, we suppose that not all the extracted material is useful for the mission purposes and so a part is discarded. For simulating this behaviour we set a parameter α, ranged of 0 to 1, which indicates the percentage of the extracted material which is stored in the lander. Through it we obtain the following denitions of the stored

mass Ms and the discarded one Md

Ms= Z αd dt(Mcr0− Mcr) dt = − Z α ˙Mcrdt (1.67) Md= Z (1 − α)d dt(Mcr0− Mcr) dt = − Z (1 − α) ˙Mcrdt (1.68)

We consider that Ms is gathered in one ideal point in the position ds. We take this

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mass; indeed it changes depending of the mission purposes and the kind of lander. In the same way we consider the discarded material grouped in an ideal point, located in a position dd, outside of the lander.

After these last assumptions we derive the position of the system's barycenter, in

which, compared to df in (1.66), there are also the static moments of the stored and

discarded masses and where the total mass is constant, due to the system does not lose material. d = 1 MA0+ Ml MA0dA0− 1 2Mcr0dcr0− Z V_crsm0 δ0(p)p∆dV + +1 2Mcrdcr+ Z V_crsm δ0(q)p∆dV + Mldl+ Msds+ Mddd = 1 MA₀+ Ml MA₀dA₀−1 2Mcr0dcr0− Z V_crsm0 δ0(p)p∆dV + +1 2Mcrdcr+ Z Vcrsm δ0(q)p∆dV + Mldl− Z α ˙Mcrdtds+ − − Z (1 − α) ˙Mcrdtdd (1.69)

Now we derive the expression of the barycenter velocity, by the derivative of (1.69)

v = ˙d = 1 MA₀+ Ml 1 2 ˙ Mcrdcr+1 2Mcr ˙ dcr+ d dt Z Vcrsm δ0(q)q∆dV − α ˙Mcrds+ − (1 − α) ˙Mcrdd (1.70) where we have showed explicitly the expression of the derivative of the stored and dis-carded mass. Finally we compute the derivative of (1.70) for obtaining the barycenter acceleration a = ˙v = 1 MA₀+ Ml 1 2 ¨ Mcrdcr+ ˙Mcrdcr˙ +1 2Mcr ¨ dcr+d 2 dt Z Vcrsm δ0(q)q∆dV + − ˙α ˙Mcrds− α ¨Mcrds+ ˙α ˙Mcrdd− (1 − α) ¨Mcrdd (1.71)

1.4 Inertia tensor's dynamic

From (1.22) and (1.32), we need to calculate the inertia tensor related to the body reference frame origin and expressed in the coordinates of this last one. Even if it was requested to calculate the inertia tensor in another reference frame, the computation of it in the body frame coordinates would be a logic rst step, because of it is the more natural frame in which computing the inertia. Indeed the inertia tensor only depends from the position and the mass of the system's particles, which are constant such reference frame, due to it is integral to the asteroid and we are dealing with a rigid body. Therefore also the inertia tensor will be constant, except for the eects of the extraction activities which perturb the distribution of the particles. Moreover, during the mission, all the measurements are accomplished in relation to the body reference frame located in the lander.

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In order to deal with the tensor calculation, it is useful to dene the elements which constitute it. They are the inertia moments along the three axes of the body frame and the three products of inertia. The inertia moments constitute the diagonal elements of the tensor, whereas the inertia products are located in the remaining positions. The products of inertia are caused by the non-alignment of the instantaneous rotational axis with the reference frame and also due to the asymmetry of the mass distribution related to the used reference point, namely when the chosen point is dierent from the barycenter. In order to have always a diagonal tensor, we would refer the inertia tensor to the system barycenter and expressing its coordinate in a frame always parallel to the instantaneous rotational axis.

The rst point is not to complex to obtain, but it requests to refer the inertia tensor to a point which is moving, whose equation is expressed in (1.69). In the reality, where there are also the measurements and estimation errors (see chapter 2), this operation leads more uncertainties in the calculation of the tensor, due to the barycenter position will be an estimate aicted by an error which will be added to the position and mass errors already present. Therefore using a xed point, as the body frame origin, as reference one, it avoids to introduce new error sources.

Instead for solving the second point we could dene a variable rotational matrix, which continuously aligns our reference frame to the principal rotational axes, which is variable. This approach complicates the computation and it introduces other errors in the estimate of the rotation matrix's elements.

Deal with a diagonal matrix can be attractive and seems to simplify the analysis of the problem. Actually from one side it really simplies the calculations, but on other side all the structure necessary to obtain the diagonal matrix increases a lot the complexity of the calculation and in the real case it introduces new and many errors on the measurements and the estimations. After all dealing with a diagonal tensor matrix does not provide any substantial advantage, so in order to avoid all the disadvantages we have enunciated, we use the easier denition introduce the product of inertia inside the inertia tensor.

In (1.72) and (1.73) respectively they are showed the general expression of the inertia

moment Iii and the inertia product Iij. Notice that only in this section, given a vector

v, we dene its cartesian coordinates by v = [v1, v2, v3]T, instead of our usual notation

v = [xv, yv, zv]T. The new notation is more suitable for the analysis in this section.

Iii = Z V δ(p)(p2j+ p2k) dV (1.72) Iij = − Z V δ(p)pipjdV (1.73) where

i, j, k are three scalar coecients such that i 6= j 6= k ∧ i, j, k = 1, 2, 3.

Iii represents the moment of inertia along the axis i

Iij represents the product of inertia in the plan ij

pipjpk are the coordinates of the position vector p in the body reference frame.

About the other variables and symbols, we have used the same notations of (1.47) Notice, from (1.73), that the index order is irrelevant in the inertia products' calcu-lation, namely

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Therefore through (1.72) and (1.73) we derive the following general expression of the inertia tensor. I =   Iii Iij Iik Iji Ijj Ijk Iki Ikj Ikk  =   Iii Iij Iik Iij Ijj Ijk Iik Ijk Ikk   =   R V δ(p)(p 2 j+ p2k) dV − R Vδ(p)pipjdV − R V δ(p)pipkdV −R Vδ(p)pipjdV R V δ(p)(p 2 i + p 2 k) dV − R V δ(p)pjpkdV −R V δ(p)pipkdV − R V δ(p)pjpkdV R V δ(p)(p 2 i + p 2 j) dV   = − Z V δ(p)S_p2dV (1.74)

where, as we have previously said, the inertia moments compose the diagonal elements of the tensor and the inertia products are located in the o-diagonal positions. In (1.74) it is showed that the inertia tensor is a symmetric matrix and moreover it is dependent from the square of the particle position's skew-matrix.

Now, similarly to how we have done in the section 1.3, we dene the general expression of the inertia tensor.

I = − Z V_A0 δ0(p)S_p2dV + Z V_cr0 δ0(p)S2_pdV − Z Vcr δ(p)S_p2dV − MlS2_d_l+ + MsS_d2_s+ MdS_d2_d (1.75)

Similarly to the barycenter's position (see section 1.3), from the initial tensor, composed by the sum between the asteroid's tensor with the lander's one, we have subtracted the initial core's tensor and we have added the current one. Moreover we have added to this expression the tensor of the discarded material and the stored one, which represent the core portion extracted from the asteroid. The used notation in (1.75) is the same of (1.47) and (1.69). Now, as we have done in section 1.3 we analyze the rst three elements of (1.75), in order to obtain a more suitable representation.

Initial asteroid inertia tensor

From (1.75) the initial asteroid inertia tensor is dened by −

Z

V_A0

δ0(p)S_p2dV (1.76)

We can split the position vector p into the position of the initial asteroid barycenter plus the position of the particles related to it.

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Therefore we can rewrite (1.76) as − Z V_A0 δ0(p)S2pdV = − Z V_A0 δ0(p)Sd2_A0+a∆dV = − Z V_A0 δ0(p) Sd2_A0 + Sd_A0Sa∆+ Sa∆SdA0 + S 2 a∆ dV = − Z V_A0 δ0(p)S_d2 A0dV − Z V_A0 δ0(p)Sd_A0Sa_∆dV − Z V_A0 δ0(p)Sa_∆Sd_A0dV − Z V_A0 δ0(p)Sa2∆dV = − Z V_A0 δ0(p) dV Sd2_A0− Sd_A0 Z V_A0 δ0(p)Sa∆dV − Z V_A0 δ0(p)Sa_∆dV Sd_A0 − Z V_A0 δ0(p)S_a2_∆dV = − MA₀Sd2_A0− Z V_A0 δ0(p)Sa2∆dV (1.77)

where, in the last equivalence, we have considered Z

V_A0

δ0(p)Sa∆dV = 0 (1.78)

because of the vector a∆is the position referred to the initial barycenter, therefore (1.78)

is the static moment of the distance of the initial asteroid barycenter to itself, and clearly it is null.

Initial core inertia tensor The initial core tensor is dened by

− Z

V_cr0

δ0(p)Sp2dV (1.79)

where the sign is several from that one in (1.74), because there we are subtracting it from the initial tensor. Similarly to how we have done in section 1.3, we can split the position

pinto two elements

p = 1

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Therefore if we substitute the position expression inside (1.79) we obtain − Z V_cr0 δ0(p)Sp2dV = − Z V_cr0 δ0(p)S21 2dcr0+p∆dV = − Z V_cr0 δ0(p) S21 2dcr0 + S12dcr0Sp∆+ Sp∆S1₂d_cr0 + S 2 p∆ dV = − Z V_cr0 δ0(p)S21 2dcr0dV − Z V_crsm0 δ0(p)S1 2dcr0Sp∆dV − Z V_crsm0 δ0(p)Sp_∆S1 2dcr0dV − Z V_crsm0 δ0(p)S2_p_∆dV = − Z V_cr0 δ0(p) dV S21 2dcr0 − S12dcr0 Z V_crsm0 δ0(p)Sp∆dV − Z V_crsm0 δ0(p)Sp∆dV S1₂d_cr0 − Z V_crsm0 δ0(p)S2p∆dV = −1 4Mcr0S 2 d_cr0 − 1 2Mcr0Sdcr0Sd_bcr0−1 2dcr0 −1 2Mcr0Sd_bcr0−1 2dcr0Sdcr0 − Z V_crsm0 δ0(p)Sp2∆dV =1 4Mcr0S 2 d_cr0 − 1 2Mcr0Sdcr0Sdbcr0 −1 2Mcr0Sdbcr0Sdcr0 − Z V_crsm0 δ0(p)Sp2∆dV (1.80)

where dbcr0 is the position of the initial core barycenter referred to the body reference

frame.

Current core inertia tensor

Finally we calculate the current core inertia tensor

− Z

Vcr

δ(p)S2_pdV (1.81)

Again we split the position into two terms

p = 1

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By the substitution of the new position's denition in (1.81) we obtain − Z Vcr δ(p)S_p2dV = − Z Vcr δ(p)S21 2dcr+q∆dV = − Z Vcr δ(p) S21 2dcr+ S12dcrSq∆ + Sq∆S1₂dcr+ S 2 q∆ dV = − Z Vcr δ(p)S21 2dcrdV − Z V_crsm δ(p)S1 2dcrSq∆dV − Z V_crsm δ(p)Sq∆S1₂dcrdV − Z V_crsm δ(p)S_q2 ∆dV = − Z Vcr δ(p) dV S21 2dcr− S12dcr Z Vcrsm δ(p)Sq∆dV − Z Vcrsm δ(p)Sq_∆dV S1 2dcr− Z Vcrsm δ(p)S_q2_∆dV = −1 4McrS 2 dcr− 1 2McrSdcrSdbcr−12dcr −1 2McrSdbcr−12dcrSdcr− Z Vcrsm δ(p)S_q2_∆dV =1 4McrS 2 dcr− 1 2McrSdcrSdbcr −1 2McrSdbcrSdcr− Z Vcrsm δ(p)S_q2_∆dV (1.82)

where dbcr is the position of the current core barycenter referred to the body reference

frame.

Finally, in order to use the initial density function instead of the current one, we apply the same modication on the density argument we have used in section 1.3, during the computation of the current core barycenter.

− Z V_crsm δ(p)S_q2 ∆dV = − Z V_crsm δ0(q)Sq2∆dV where q = p + dcr₀−1 2dcr

Therefore the inertia tensor of the current core in (1.82) becomes − Z Vcr δ(p)S_p2dV =1 4McrS 2 dcr− 1 2McrSdcrSdbcr −1 2McrSdbcrSdcr− Z V_crsm δ0(q)Sq2∆dV (1.83)

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System's inertia tensor

Through the substitution of (1.77), (1.80) and(1.83) in (1.75) we obtain the nal expres-sion of the system's inertia tensor

I = − MA0S 2 d_A0− Z V_A0 δ0(p)S2p∆dV − 1 4Mcr0S 2 d_cr0 + 1 2Mcr0Sdcr0Sdbcr0 +1 2Mcr0Sdbcr0Sdcr0 + Z V_crsm0 δ0(p)S_p2_∆dV +1 4McrS 2 dcr− 1 2McrSdcrSdbcr −1 2McrSdbcrSdcr− Z Vcrsm δ0(q)S_p2_∆dV + MlS_d2_l+ + MsSd2s + MdS 2 dd (1.84) Finally we calculate the derivative of the inertia tensor in (1.84)

˙ I = 1 4 ˙ McrS2dcr+ 1 4McrSd˙crSdcr+ 1 4McrSdcrSd˙cr− 1 2 ˙ McrSdcrSdbcr −1 2McrSd˙crSdbcr− 1 2McrSdcrSd˙bcr− 1 2 ˙ McrSdbcrSdcr+ −1 2McrSd˙bcrSdcr− 1 2McrSdbcrSd˙cr− d dt Z V_crsm δ0(q)Sp2∆dV + α ˙Mcr S2ds+ (1 − α) ˙Mcr S 2 dd (1.85)

1.5 Model validation

In order to validate the model we have taken a parallelepipedal asteroid composed by three dierent constant density spaces (see gure 1.7. The transition between two ad-jacent spaces is discontinues, namely there is not a smooth transition area, but we pass immediately from a density to other, like in a step function. The extraction point is located in the origin of the body reference frame, which lies on the upper base area. The

xand y axes are orthogonal, lie on the upper base and they are parallel to the side of

the area. The x axis points toward right, the y axis points toward behind, the z axis, in order to produce an orthogonal right-handed triplet, is orthogonal to the upper base area and it points toward the exterior of the parallelepiped. Moreover one of the segments along which all the three density areas meet, lies on the negative extension of the z axis. This choice in our example have been done in order to put the core's symmetry axis along this segment, indeed in this way we have a not trivial behaviour of the update functions, but at the same time it is not too hardly validatable, namely we can verify by dierent algorithms, not too hard to obtain, if it is correct. The extraction point is not adjustable in our model, although the radius of the drill is changeable, but there is not an automatic control for advising us in the case we have overstepped the bounds of the asteroid. In this last case our algorithm follow to work, but it provides a wrong result. The drill is located at two meters from the nearest edge, so we must avoid radii bigger than two meters. The variable α, namely the variable which denes the stored and discarded masses, is a third degree polynomial which reaches the unitary amplitude at the end of the extraction. Instead the position of the core's bottom is represented by a fth degree polynomial such that it and its acceleration, reach zero at the end of the extraction.

The validation is done by the evaluation of the results obtained by Simulink with that ones obtained by a proper MATLAB function, which, by a dierent algorithm (see

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Figure 1.7: The example asteroid, with a view of the upper area, the body reference frame and the density of its three areas

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section D.1), calculates the same quantities. We calculate the absolute error on the barycenter's position, velocity, acceleration, on the inertia tensor and its derivative. This test is applied for two dierent values of the drill radius (0.5 and 1.5 meters), four dierent position of the drill bottom (-25,-22,-18,-6 meters) and two extraction durations (2 and 20 minutes).

The four dierent initial drill bottom positions (-25,-22,-18,-6 meters) are not chosen randomly, but they stay more or less in the middle of the dierent asteroid density areas, except one; indeed at -25 meters the drill bottom is outside of the asteroid. We have tested this value in order to verify the model's "robustness", namely it behaviour in an unusual situation. In this work, for this particular case, we have decided to keep all the quantities constant until the drill's bottom reaches the asteroid's bottom at -24 meters in the z coordinate. This one is coherent with a core drill whose surface has no friction with the asteroid. Obviously it is an ideal behaviour and other choices are possible.

In order to simulate our example, we have derived some necessary initial value of some variables, referred to the body reference frame.

MA₀ =14394 kg dA0=[−3.6269; −1.1594; −11.1178] T_m Mcr₀ =1715.3 kg dcr0=[0.4104; −0.0280; −11.6209] T_m 2 DI0=   2.5210 · 109 _{−6.3412 · 10}7 _{−5.6447 · 10}8 −6.3412 · 107 _{2.7123 · 10}9 _{−1.9790 · 10}8 −5.6447 · 108 _{−1.9790 · 10}8 _{3.7982 · 10}8  Kg · m 2 2 C_r0Icr0=   8.2863 · 104 ₋₁₄₄ ₂₂₀₈ −144 8.2863 · 104 ₋₂₁₆ 2208 −216 1.7153 · 103  Kg · m 2

where Cr0 is the point which lies on the core's geometrical center at the initial time.

Finally we have chosen arbitrarily the remaining initial values, namely

acr₀=[0; 0; 1.5 · 10−4]Tm s2 vcr₀=[0; 0; 10−2]Tm s ds0=[−2; −2; 1] T_m dd₀=[1; 1; 2]Tm Ml0=5000Kg dl₀=[0.5; 0.25; 0.75]m 2 ω2/0=[6.114 · 10−4; 4.442 · 10−4; 4.363 · 10−4]Trad s α0=0.1 ˙ α0=0

In section C.1 are available the validation graphs described above, with also a brief description of the simulation details.

Robust control of asteroids' bearing rate during mineral extraction

School of Engineering

Department of Information Engineering

Master's Degree Course in Robotics and Automation Engineering

Robust control of asteroids' bearing rate

during mineral extraction

Ruben Agnesi

Prof. Andrea Caiti

Prof. Mario Innocenti

Contents

Preface

Notations

Chapter 1

Model

1.1 Asteroid's model

1.1.1 Basic model

1.1.2 Extended model

1.1.3 Observations about the relation between orbital and

rota-tional motions

1.2 Extraction's model

1.3 Barycenter's kinematic

1.4 Inertia tensor's dynamic

1.5 Model validation