Sliding mode attitude control using thru

will be inserted by the editorSliding Mode Attitude Control using Thrusters and Pulse Modulation for the ASTER Mission Bruno Victorino Sarli · Andr´e Lu´ıs da Silva · Pedro Paglione Rece

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Sliding Mode Attitude Control using Thrusters and Pulse Modulation for the ASTER Mission

Bruno Victorino Sarli · Andr´e Lu´ıs da Silva · Pedro Paglione

Received: date / Accepted: date

Abstract ASTER, the first Brazilian mission to the

deep space, targets the exploration of the triple

aster-oid system known as 2001 SN263 The mission requires

an attitude controller robust and capable of coping with

the non-linearities and the uncertainties present during

the exploration phase For such requirements, this

pa-per studies the applicability of two controllers, designed

based on the sliding mode control (SMC) technique,

one of the controllers include an adaptive law used to

compensate for the spacecraft’s inertia variation One

application is performed where gain scheduling is used

for controlling two different phases: exit from tumbling

and track a dynamic reference The actuators of the

at-titude control loops are impulsive thrusters They are

activated by pulse width modulation (PWM) or pulse

width pulse frequency modulation (PWPFM)

Simula-tion studies, performed in realistic scenarios, show that

the SMC can maintain stability and performance when

these modulation techniques are used to approximate

B.V Sarli

Department of Space and Astronautical Science, The

Grad-uate University for Advanced Studies

Sagamihara, 252-5210, Japan

E-mail: sarli@ac.jaxa.jp

A L da Silva

Universidade Federal do ABC, Centro de Engenharia,

Mod-elagem e Ciˆ encias Sociais Aplicadas

S˜ ao Paulo, Brazil

Tel.: +55-11-23206338

E-mail: andreluis.silva@ufabc.edu.br,taurarm@gmail.com

P Paglione

Aeronautics Engineering Division, Aeronautical Institute of

Technology

P¸ c Marechal Eduardo Gomes 50, CEP 12.228-900, S˜ ao Jos´ e

dos Campos, S˜ ao Paulo, Brazil

E-mail: paglione@ita.br

the continuous commands It is also shown that PWM can provide better performance, but at a higher control cost In this sense, PWPFM is more appropriate with respect to the fuel consumption and activation times Keywords asteroid mission · nonlinear control · attitude · ASTER · sliding mode · PWM · PWPFM

1 Introduction ASTER, the first deep space Brazilian mission, tar-gets the group of asteroids in the system know as 2001 SN263, which is composed by a central body named Alpha and two satellites, the largest one orbiting fur-ther from the system’s center is named Beta, while the smaller asteroid orbiting closer to Alpha is named Gamma The objective of this mission is to explore the system, taking measurements and pictures from all three asteroids and, if possible, finalize the mission by touching down Alpha, [3, 7]

One of the many elements for the success of the mis-sion is the ability of correct pointing and stabilization

of the spacecraft, which is useful for the orientation of instruments and also for the navigation, which is based

on a fixed ion-thruster technology; therefore, in order to change the trajectory for translation within the system, the attitude of the spacecraft needs to be changed

A triple asteroid system is highly non-linear due

to the nature of the gravity environment generated by the three bodies, allied with the solar radiation pres-sure Therefore, a control law based on a linear approx-imation is not suited, neither for the orbit around the system nor the navigation inside it, because the short maneuver time and the long engine operation Further-more, it is required from the spacecraft to perform large

angle maneuvers during its operation within the

as-teroid system, such tracking cannot be accurately

per-formed by linear controllers, as demonstrated on figure

1 taken from [6], where the track of a large amplitude

angle is attempt, resulting in an oscillatory behaviour

with high amplitude By the way, some of the tests (in

the case of using a reaction wheel) do not generate a

steady response in the observed time horizon

Fig 1: Closed loop spacecraft’s response to step inputs

of high amplitudes, [6]

Particularly in this work, only the case where the

spacecraft orbits the center of the system will be

con-sidered That is, a circular orbit around Alpha The

attitude control actuators are small thrusters that can

provide an anti-symmetrical setting, which means that

the attitude can be controlled without inducing

trans-lation The design of a feedback control law is

per-formed using continuous control techniques After, the

control thrusters are activated by pulse width

lation (PWM) or pulse width pulse frequency

modu-lation (PWPFM), which allows for an approximation

of the continuous thrust calculated by the controller

A realistic modelling is performed taking into account

thrust levels, activation time (on an off times) and time

response of practical control thrusters of small

space-craft

The control design technique chosen in this study

is the sliding mode control (SMC) This formulation

is very appealing specially by its invariance to

distur-bances and model uncertainties, [11, 2, 10] This

charac-teristic is highly desirable for a mission such as ASTER,

where many uncertainties in the model of the asteroid

system are present The chosen SM formulation has

an-other attractive feature for the mission: an adaptive law

that is of great importance when using ion-thrusters,

which consumes propellant for long periods of time,

making the inertia matrix vary, [13] The focus of this

study is to develop the control algorithm; the precise

de-termination of the moment of inertias’ values was not

the objective Therefore, the parameters used in the inertia matrix are hypothetical Each controller makes use of two sets of gains (gain schedule) that are tuned for a specific purpose, the first set is applied to take the spacecraft from tumbling and bring it to an equilibrium position, once the equilibrium is achieved, the second set is used and the controller starts to track a time vary-ing attitude angles The determination of a unique set

of parameters for different tasks may be very hard and could not determine a suitable performance for both cases

The main contribution of the paper, is to show the feasibility of using continuous nonlinear control tech-niques to activate the control thrusters, via the PWM

or PWPFM in the scenario in question For this pur-pose, the adoption of a robust feedback control is im-portant, in order to cope with the inherent time delays caused by the modulation, and also uncertainties of the thrusters modelling and inertia matrix Particularly, re-garding this aspect of perturbations and uncertainties, this paper shows that the gravity gradient torque per-turbations generated by the triple asteroid system, and also the solar radiation pressure, are irrelevant when compared with the former effects

As it follows, section 2 presents the equations of motion of the spacecraft Section 3, based in the work

of [13], presents the formulation of the non-linear con-troller for the attitude pointing, discussing the sliding surface and control law addressing the problem of chat-tering, followed by the formulation of the SM adaptive controller, featuring the same important points as in the previous controller Section 4 outlines the imple-mentation of the designed controllers using PWM and PWPFM The arrangement of the control thruster of this work is shown, real design data obtained from simi-lar spacecraft are also discussed Section 5 presents the simulation results, the continuous and pulsed control actions are evaluated and analysed with a detailed set

of performance measures Finally, section 6 presents the conclusions

2 Equations of motion Particularly for a circular orbit, or long orbits, the rota-tional and translarota-tional equations of motion of a space-craft can be treated independently Specifically for the

2001 SN263 system, the main forces acting there are the gravity of the three asteroids and the solar radiation pressure; the gravity of the Sun, due to the distance, has

a very small intensity, which can be considered negligi-ble Figure 2 presents an initial evaluation of the mag-nitude of the accelerations acting on the spacecraft at

Alpha’s equatorial plane (fixed frame at Alpha’s

cen-ter of mass (cm) corrected for a central system, [4])

For this work, the rotational motion will not affect the

20 40 60 80 100 120 140 160 180

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−7

Distance from Alpha [km]

2 ]

Alpha Beta Gamma Sun SRP

Fig 2: Acceleration induced by the disturbances acting

on the spacecraft

translational motion, because the gravity force of

Al-pha around the spacecraft will be constant around the

circular orbit and the gravity of Beta, Gamma and the

solar radiation pressure will be treated as perturbations

that are corrected by the orbit control system That is,

the perturbing torque or disturbance considered here

will come from the gravity gradient torque generated

by the three asteroids and the solar radiation pressure

Their perturbing torque, denoted by D, can be

calcu-lated as:

where, β denotes the body frame, gc is the gravity

gra-dient, FSRP is the force due to the solar radiation

pres-sure and h is the vector from the spacecraft’s optical

pressure center to its geometrical cross-section center

The evaluation of the vector from Alpha’s reference into

body frame can be made by using the classical

quater-nion based rotational matrix, 1-3-2 rotation sequence,

with q4 being the scalar quaternion:

R0b=





1 − 2(q2+ q2) 2(q1q2+ q3q4) 2(q1q3− q2q4)

2(q1q2− q3q4) 1 − 2(q2+ q2) 2(q2q3+ q1q4)

2(q1q3+ q2q4) 2(q2q3− q1q4) 1 − 2(q2+ q2)



 (2) The perturbing torque due to the gravity gradient can

be calculated as, [6]:

{gc}β=

n

X

j=1

3GMj

R3

j

where, c3is the third column of the transformation

ma-trix from the inertial frame to the body-fixed frame

with hc3xi being its skew-symmetric matrix hc3xi =





0 −c33 c32

c33 0 −c31

−c32 c31 0



, Rjis the distance between the space-craft’s cm and the asteroid’s cm, J is the inertia matrix,

j represents the number of the body: Alpha is 1, Beta

2 and Gamma 3, G is the gravitational constant and

Mj is the mass of each asteroid The torque due to the solar radiation can be calculated as:

where, PSRP is the local solar radiation pressure given

rSunis the position vector Sun-spacecraft in astronom-ical units, PSRP 0= 9.15N/km2 is the value at 1 AU (Astronomical Unit), CR is the reflectivity coefficient considered fixed at 1.14 [12], and A is the cross-section

of the spacecraft, assumed to be equal to 5m2

It is important to point out that the evolution of Beta and Gamma around Alpha were studied in [1] However, no official model has been derived yet There-fore, the ephemeris model used in this work was calcu-lated using the initial conditions from Fang’s work and propagated using an Adams integrator for 20 days from June 1st of 2019 until June 20th of the same year, [7] The rotational equations of motion for the space-craft as a rigid body can be derived by Euler’s formu-lation, having this form:

J ˙Ω = − ˙J Ω − Ω × (J Ω) + Tb+ D (5) where, Ω is the angular velocity, Tb is the moment due

to the actuators and D is the disturbance torque The ASTER spacecraft will have a large ion-thrust for trajectory control and a cluster of small thrusters for attitude control These small thrusters are arranged

in such a way to generate three independent control torques Tx, Ty and Tz around the x, y and z axes

of the body, respectively, without inducing translation; then, Tb = [Tx Ty Tz]T The geometrical arrangement

of thrusters and technological data are discussed in sec-tion 4.1

It is convenient to evaluate the rotational kinematics

in quaternions, rather than in Euler angles, in order to avoid singularities The quaternion formulation is:

Q = q1q2q3q4T

=Q¯T q4T

¯

QT =





q1

q2

q3



= U sin Φ2
, q4= cos Φ2
(6)

where Q is the quaternion, ¯Q is its vectorial part and

q4 is its scalar part, as mentioned before U is the ro-tation vector and Φ is the roro-tation angle, that describe

a rigid body rotation, according to the Euler’s rotation

theorem

The derivative of the quaternion is:

˙¯

Q = 1

2

¯

Qx Ω +1

2q4Ω,

¯

Qx =





0 −q3 q2

q3 0 −q1

−q2 q1 0





˙

q4= −1

2Ω

With Eq 5 and Eq 7, the final form of the

space-craft’s dynamic model can be obtained, where e means

an error and d means a desired behaviour,

˙

Qe=1

2

¯

Qex Ωe+1

2qe4Ωe, ˙qe4= −

1

2Ω

T

J ˙Ω = − ˙J Ω − Ω × (J Ω) + Tb+ D (8)

where Ωe = Ω − Ωd is an error angular speed with

respect to a desired angular speed Ωd and:

Qe=







qe1

qe2

qe3

qe4







=







qd4 qd3 −qd2−qd1

−qd3 qd4 qd1 −qd2

qd2 −qd1 qd4 −qd3

qd1 qd2 qd3 qd4













q1

q2

q3

q4





 (9)

Equation 9 defines an incremental rotation with

re-spect to a desired attitude Qd The desired angular

speed and attitude will be considered in the next

sec-tion, where the problem of attitude tracking control is

developed

3 Non linear attitude controller

The SMC design adopted in this work came from

ref-erence [13] It was derived for the attitude control of

a spacecraft with thrusters The approach concerns a

nonlinear controller robust with respect to external

dis-turbances and parametric uncertainties A variation of

the robust controller is also developed, with an

addi-tional adaptive law to estimate the uncertain inertia

matrix The complete evaluation of the control scheme

can be found in the reference In the following, a

syn-thesis of this approach is presented This particular

dis-cussion gives special attention to the meaning of each

design parameter on the controls By explaining the role

of each element in the control law, the designer can

un-derstand the impact of such parameters when tuning a

specific control law for some application One

applica-tion of such concepts is presented in secapplica-tion 5.1, where

the continuous control law is determined in order to

generate control actions below the physical limits of the

thrusters, while still delivering reasonable performance

measures

3.1 Sliding mode attitude controller design

In the development of a SMC, one needs to define a sliding surface This surface is determined in order to obtain a sliding mode (SM) that satisfy the require-ments of some application, in this case, the tracking of

a desired attitude After the determination of a sliding surface, a reaching condition shall be described, from which the sliding surface can be reached This condi-tion determines the control law that can make the SM possible and, consequently, the desired behaviour, [11, 2]

The switching function is defined as:

where, P is a positive diagonal matrix

When S = 0, the switching surface is obtained The path of the system constrained to this surface is the SM This SM shall represent the desired behaviour, which,

in this case, is the tracking of the desired attitude So, during the SM, one should expect that the errors in the quaternion and angular velocity Qeand Ωetend to zero

The reference [13] shows that by using Lyapunov function of the form Ve Qe
= kQTeQe, where k is a positive number, the following property is obtained:

−ckQTeP Qe≥ ˙Ve Qe
≥ −kQTeP Qe (11)

so, with c > 0 the Lyapunov stability theory proves that the SM is stable, in such a way that the origin of the error dynamics is a stable equilibrium point Thus, the tracking error converges to zero during the sliding mode:

Qe, Ωe
→ (03×1, 03×1) , t → 0 (12) This shows that the sliding surface is capable of sat-isfy the requirement of attitude tracking In this sense, note the meaning of the positive definite matrix P : from

Eq 11 the decaying rate of the Lyapunov function de-pends on the magnitude of P , so, by increasing the magnitude of the elements of this matrix, one can de-crease the convergence time to the origin of the error dynamics

Once a stable sliding surface is obtained, it is neces-sary to establish a control law in order to guarantee that the sliding surface is reached, from any initial condition, such that the SM exists That is, the control shall be determined in order to satisfy a reaching condition, [2] The reaching condition can be specified in a variety of ways The Lyapunov function method is chosen by [13] Using a function of the form Vs= 1STJ S, it is shown

that exponential stability and robust performance can

be obtained with the control torque:

Tb= − KsS + ˙J0Ω − 1

2

˙

J0S + Ω × (J0Ω) +

−1

2J0P

¯

Qex Ωe+ qe4Ωe
− J0Ω˙d+ Λs (13) where element KsS is a feedback of the switching

func-tion, Ks= diag ks1ks2ks3 is a 3 × 3 positive definite

matrix, Λsis a discontinuous control action, responsible

for the generation of the SM:

Λs=





λs1

λs2

λs3



, λsi= −csi



Q, Ω, Qd, ˙Qd, ¨Qd

 sgn(si) (14) the terms csi



Q, Ω, Qd, ˙Qd, ¨Qd

 are control amplitudes, sgn(si) is the sign function:

sgn(si) =







1, si> 0,

0, si= 0,

−1, si< 0,

and S = s1s2s3T

is the sliding surface The remain-ing elements in the control torque of Eq 13 comprehend

the equivalent control, it is responsible for guaranteeing

that the SM trajectories are tangent to the sliding

sur-face, [2] Note that it is a continuous nonlinear function

that depends on the dynamics of the plant

Special attention shall be paid for the meaning of

the control gains Ks= diag ks1ks2ks3 and the

con-trol amplitudes csiQ, Ω, Qd, ˙Qd, ¨Qd The amplitude

of the gains ksiare related to the rate of convergence of

the reaching mode, in other words, increasing the

mag-nitude of these gains, the time of convergence to the

sliding surface is decreased In other way, the control

amplitudes csi



Q, Ω, Qd, ˙Qd, ¨Qd

 shall be determined

in order to guarantee that the reaching condition is

sat-isfied in the presence of the external disturbance and

parametric uncertainties There are a variety of ways

for satisfy this, one possibility is consider the worst case

scenario with a subsequent substitution of the maximal

expected value of each variable, [9]

Finally, the chattering problem is originated due to

the existence of physical limitations in the

implemen-tation of the sign function sgn(si), some problems are

delays, dead zones, hysteresis In order to improve the

solution and avoid the problem caused by chattering,

the sign function can be replaced by a saturation

func-tion; in this way, forcing the system to stay within the

limits of the boundary layer |Si| < ε (ε represents a

positive small scalar value) and no longer exactly on

the sliding surface Such change in the control law will

be, of course, followed by a reduction in the accuracy of the desired performance, [10] The saturation function

is defined as follows:

sat(si, ε) =







1, si> ε

ε |si| < ε

−1, si< ε

(16)

where ε is a constant that shall be chosen as small as possible

3.2 Sliding mode adaptive attitude controller design The variation spacecraft’s inertia cannot be neglected

if a particular long phase consumes fuel continuously Because the calculation of the inertia variation is too complex, a very useful solution is to design an adap-tive controller that can compensate for the effect of this variation and respective disturbances

In short, the adaptive control problem consists in generating a control law and a parameter vector esti-mation law, such that the tracking error tends expo-nentially to zero:

1 Given the desired attitude, quaternion Qd, and an-gular velocity, Ωd, with some of all the parameters unknown of J ;

2 Derive a control law for the thrusters torques and

an estimation law for the unknown parameters, such that Q and Ω precisely track Qd and Ωd after the initial adaptation process;

Let A be a constant 6 × 1 vector containing the un-known parameters and ˆA (the time-varying parameter vector estimate); with the error ˜A = ˆA − A Reference [13] shows that, when maintaining the sliding mode sur-face defined in Eq 10 and using a new Lyapunov func-tion candidate of the form V = 12STJ S + 12A˜TΓ−1A˜ for the reaching condition, where Γ is a positive diago-nal matrix, the global stability of the attitude tracking system is guaranteed, provided by choosing a control torque:

Tb=Ω × ˆJ Ω

−1 2 ˆ

J P Q¯ex Ωe+ qe4Ωe
+ ˆJ ˙Ωd+

where, Ka= diag ka1 ka2 ka3 is a 3 × 3 positive defi-nite matrix and Λ = λ1λ2λ3

T

is a vector of discon-tinuous functions These terms are analogous to that in

Eq 13, but note that the nominal matrix J was changed

by the estimated matrix ˆJ

In [13], it is shown that this controller also provide the convergence of the states ¯Qe and Ωe and parame-ter estimation error ˜A The demonstration involves the

Lyapunov stability theory and the Barbalat’s lemma.

This proves that the adaptive controller can also solve

the attitude tracking problem

Again, special attention shall be paid for the

mean-ing of the control gains Ka = diag ka1 ka2 ka3, the

control amplitudes ki and the elements of Γ The

am-plitude of the gains kaiare related to the rate of

conver-gence of the reaching mode, in other words, by

increas-ing the magnitude of these gains, the time of

conver-gence to the sliding surface is decreased In other way,

the control amplitudes ki shall be determined in order

to guarantee that the reaching condition is satisfied in

the presence of the disturbance D, which is related to

the external perturbations and the uncertainties in the

inertia matrix Finally, one shall note that the elements

of Γ are related with the convergence rate of the

pa-rameter estimation law

Regarding chattering, in the same way of the

stan-dard controller, the sign function should be changed by

a saturation function in order to mitigate its effects

4 Control Implementation with Thrusters

4.1 Thrusters arrangement and technology

The control design of section 3 assumes that a pure

torque can be generated by the attitude control thrusters

without inducing translation, this effect can be obtained

by using a configuration of anti-symmetric thrusters

Figure 3 illustrates one of the many possible

configura-tions that generate the desired torque with no

transla-tion Naturally, this scheme shall be repeated for all the

3 axis of rotation, resulting in 12 control thrusters Note

that a thruster, in fact, generate a force, the torque

re-sults from the respective arm until the spacecraft’s cm

In practice, the control torques can assume only

three levels: zero, minimum negative and maximum

pos-itive The force, in fact, can be only zero or positive

along the thruster axis; for a specific control axis,

posi-tive torque is generated by choosing one pair of thrusters,

negative torque is generated by the other pair

On the other hand, a feedback control law assumes

continuous torques, but they can be approximated by

pulse modulation as described in the section 5.2,

ac-cording a suggestion made by [5]

In order to obtain an appropriate result, the

magni-tude of the torques are required to be within the limits

of the force that can be delivered by typical actuator

used in small and light spacecraft like ASTER’s Some

examples are described at table 1 Based on nominal

thrust values presented on table 1 It is assumed that

each thruster can provide 0.5 [N] and their distance

from the cm is 0.5 [m] which results in a maximum torque of 0.5[m] × 0.5[N ] + 0.5[m] × 0.5[N ] = 0.5[N m] Another critical issue of pulse modulation is the minimal “on” and “off” times of the thrusters These times strongly influence the quality of the pulsed ap-proximation Theoretically, smaller modulation times generate finer controls However, control hardware have limitations as described in the table 1

Fig 3: Illustration of control thrusters arrangement This scheme repeats in the other 4 sides

Table 1: Attitude control thrusters for small spacecrafts

Gas

(N2) Nominal

thrust [N]

0.32 ∼ 0.95 0.22 ∼ 1.02 0.1 ∼ 1.0 Pressure

[MPa]

0.69 ∼ 2.75 0.62 ∼ 2.76 1.51 ∼ 1.80

Min ON Time [ms]

Min OFF Time [ms]

-Response time [ms]

-Min Im-pulse [Ns]

-Total Impulse [Ns]

MonoH is mono-propellant: hydrazine (N2H4)

4.2 Thrusters activation

According to figure 4, first, a continuous control is

de-signed according to the method of section 3; after, a

pulse modulation technique is used to approximate the

continuous control by time activation of control thrusters

Spacecraft Dynamics

u

Sliding Mode

Control

Continuous

Control action

Spacecraft Dynamics Sliding Mode

Control

Modulation

Modulation

to activate the control thrusters

u

Pulsed

Control action

Fig 4: Insertion of the pulse modulation in the control

loop

Two techniques are evaluated in this paper,

accord-ing to the suggestions in the reference [8]: PWM and

PWPFM Diagrams that illustrate these techniques are

shown in figures 5 and 6, respectively The PWM

algo-rithm operates in discrete time, the sampling and hold

time is Ts The basic idea is to generate an activation

time that is proportional to a constant thrust level:

ton= Tcom

Tmax

where ton is the activation time, Tcom is the required

thrust level (generated by the control law) and Tmax

is the thrust level generated by the thruster In order

to simplify the analysis, the algorithm is developed to

calculate torque levels instead of thrust levels (in this

case, the torque arm is tacked in consideration)

T com

T s

T max

sawtooth generator

T s

T s T s

zero order

holder - zoh

u

abs

sign

quantizer

+

compare

to zero

X

prod

T max

Gain

Sample and hold

the signal at each

time interval Ts.

Take the absolute

value in order

to cope with both

negative and

positive torques.

Quantize a signal in order

to limit the minimal shooting time.

Generate the signal that controls the size of the shooting time.

At each interval

Ts, the output

is equal to 1 during the time (|Tcom|/Tmax)Ts.

Choose the thrusters for generating positive or negative torque.

During the simulation, the gain represents the thruster itself, with its maximum torque and respective sign.

T out

PWM generator

Fig 5: Illustration of the PWM algorithm

The activation time generated by the PWM does not take into account whether the pulse is negative or positive, for this purpose, the “sgn” function is used to store the signal of the demanded torque In a practical implementation, the minus or the plus signal indicate which pair of control thrusters shall be activated for the specific control axis

In figure 5, one shall note the presence of a quantizer block This block is used to avoid short activation times

If Ndis the number of quantization levels, the minimum activation time will be:

tminon = Ts

T com

u

abs

sign

prod

Take the absolute value in order

to cope with both negative and positive torques.

Choose the thrusters for generating positive or negative torque.

The thrusters activation time

is defined by the relay state During the simulation, the amplitude of the relay's output already defines the value of the thruster's torque.

T out

PWPFM generator

relay

U m

U off U on

km

1+τs

low pass filter LPF The relay function

opens the output

at U on and closes

at U off The last defines a deadzone.

The feedback of the relay output switches the input of the LPF, generating the "on" and "off"

times of the modulator.

The convergence of the LPF's output

defines to U on or U off switches the relay state.

Fig 6: Illustration of the PWPFM algorithm

On the other way, the PWPFM shown in figure 6 modulates both pulse width and pulse frequency Un-like the PWM, the PWPFM is a continuous time tech-nique, it has two main blocks: a low pass filter and an ideal hysteresis The last is high nonlinear, so, this tech-nique inserts a nonlinear control action into the system The on and off times depend on several parameters: filter gain km, filter time constant τ , turn on voltage

Uon, turn off voltage Uof f, output voltage Um Analyt-ical expressions involving these quantities are shown in the reference [8] Briefly, in order to design the control block, some basic properties can be tacked into account: – The Uof f voltage can be seen as a dead zone, used

to limit the activation time and cope with noises and spurious torque commands;

– The Umis related with the thruster level;

– The Uonvoltage is related with the sensitivity in the thruster activation, small Uon voltage generate fast commutations;

– The time constant τ is related with the order of magnitude of the on and off times;

– The role of k is similar to that of U

5 Simulations

In this section, the control laws of section 3 and the

pulse modulation techniques of section 4 are evaluated

First, the controllers are designed and tested in order

to provide a suitable response with continuous control

action After, these controllers are used to generate the

modulating signals for pulsed thrusters, which are

ac-tivated by PWM or PWFM described

As the ASTER is still being prepared and final

val-ues for most of the variables are yet to be available, the

values for the variables used in this work were based

on the references [7, 6, 13] The orbital conditions were

chosen in a way to put the spacecraft under the most

severe conditions with respect to the perturbations: the

initial orbit is a circular, on Alpha’s ecliptic plane

be-tween Beta and Gamma, with a radius of 11.9 km

(fig-ure 2) As mentioned before, the controllers make use of

a gain scheduling that is set for two distinct parts: first,

to rescue the spacecraft from tumbling and bring it to

a stable position; second, to track some time varying

attitude angles

5.1 Continuous control

Two simulations were performed, one with the standard

SMC of section 3.1, other with the adaptive variation

of section 3.2 The initial attitude conditions are the

same, they are set in order to represent a tumbling of

high amplitude The Euler angles are [ 45◦45◦ 45◦]T,

which converted to quaternions are

Q(0) = [ 0.4619 0.4619 0.1913 0.7325 ]T, and the

angu-lar velocity is Ω(0) = [ −2◦/s −3◦/s 5◦/s ]T

The tracking stage is set in a way to simulate a

mea-surement on the surface of Alpha For this purpose, the

spacecraft needs to rotate with the same angular

ve-locity as the orbit (maintaining the same face towards

Alpha) with an assumed surface inspection of 30◦

de-grees amplitude Therefore, the desired conditions for

the Euler angles are:

φ = 30 cos(t/ωr) sin(30 cos(t/ωr))

θ = 30 sin(t/ωr) sin(30 cos(t/ωr))

ψ = worbcos(30 cos(t/ωr))

(20)

which are converted into quaternions at each time step

Qd(t); where, worb is the orbital angular velocity given

by the classical formulapRµ3 = 0.0011[◦/s], with µ as

Alpha’s standard gravitational parameter and R the

or-bit radius, and ωris the speed at which the inspection

will take place, set as ωr = 0.5[s/◦] The desired

an-gular velocity and its derivative are arbitrated as zero

for a more stable measurement, Ωd(t) = [ 0 0 0 ]T and

˙

Ω (t) = [ 0 0 0 ]T

The inertia conditions of the spacecraft are:

J0=





19.4 0.1 3 0.1 25.7 0.5

3 0.5 18.4



 kg m2, ˙J0= −J0

1000,

∆J = J0

10, ∆ ˙J =

˙

J0

For the standard controller the design data are: cs1=

cs2= cs3= 0.2 and:

P =





0.5 0 0

0 0.5 0

0 0 0.5



, Ks1=





1/6 0 0

0 1/6 0

0 0 1/6



,

Ks2=





5 0 0

0 5 0

0 0 5



Finally, for the adaptive controller, the respective design data are exactly the same: k1= k2= k3= 0.2,

P =





0.5 0 0

0 0.5 0

0 0 0.5



, Γ =





600 0 0

0 600 0

0 0 600



,

Ka1 =





1/6 0 0

0 1/6 0

0 0 1/6



, Ka2=





5 0 0

0 5 0

0 0 5



The only difference is the insertion of the matrix Γ , related to the dynamics of the adaptive algorithm So, the only distinction between the two controllers is the estimation of the inertia matrix in the adaptive version The above values for both controllers were adjusted

by trial and error methods aiming a compromise be-tween the sliding mode and switching surface’s conver-gence speeds represented by P and K respectively, the value of cs’s and k’s were based on the expected uncer-tainty in the inertia matrix which has its largest value

of 2.57 kg m2 The most important idea underlying the procedure is to ensure that the amplitude of the con-trol is not greater than 0.5 N m, which is assumed to be the torque level of the thrusters, as argued in section 4.1 The values of P , K, cs and k direct influence the amplitude of the controls, however, according the the descriptions on section 3, small values of these param-eters can degrade the performance

The results of both simulations for the full non-linear system can be seen in the next figures, where

7, 8 and 9 show respectively the Euler angles, quater-nions and angular velocity evolution, 10 shows the con-trol used in each direction, 11 shows the disturbance over the spacecraft and 12 shows the estimation error in the adaptive controller; the continuous lines represent the non-adaptive and the dash-doted lines, the adaptive controller

0 20 40 60 80 100 120 140 160 180

0

20

40

t [s]

0

10

20

30

40

t [s]

0

10

20

30

40

t [s]

Non−adaptive SMC Adaptive SMC Desired

Fig 7: Attitude Euler angles evolution

0

0.1

0.2

0.3

0.4

t [s]

q1

0 0.1 0.2 0.3 0.4

t [s]

q2

0

0.05

0.1

0.15

0.2

0.25

t [s]

q3

0.75 0.8 0.85 0.9 0.95 1

t [s]

q4

Non−adaptive SMC Adaptive SMC Desired

Fig 8: Attitude quaternion evolution

−3

−2

−1

0

1

t [s]

w1

−3

−2

−1

0

t [s]

w2

−2

0

2

4

t [s]

w3

Fig 9: Attitude angular velocity evolution

Numeric performance values associated to the paths

of figures 7 to 12 are shown in table 2 In this table “ct”

means convergence time, which is the time required for

the Euler angles to converge to the equilibrium, from

the initial condition, with a tolerance of 5% “td” is

the maximal time delay between the commanded and

−0.2 0 0.2

t [s]

u1

−0.2 0 0.2 0.4

t [s]

u2

−0.4

−0.2 0

t [s]

u3

Fig 10: Continuous torque profile

−5 0 5

x 10−10

t [s]

d1

−6

−4

−2 0 2

x 10−6

t [s]

d2

0 5 10 15

x 10 −6

t [s]

d3

Fig 11: Orbit gravitational perturbation on the space-craft

−2 0

t [s]

a1

2 ]

−20 4

t [s]

a2

2 ]

−2 0

t [s]

a3

2 ]

−0.01 0 0.01

t [s]

a4

2 ]

−0.4 0 0.2

t [s]

a5

2 ]

−0.05 0 0.05

t [s]

a6

2 ]

Fig 12: Inertia estimation error

the resulting path in steady state Because the response

is non linear, obviously, this time delay is only an ap-proximation when compared with the pure sinusoidal responses of linear systems Note that the td param-eter is not defined for the ψ angle, since its reference

is not dominated by a periodic function The last

per-formance parameter is “ae”, which stands for amplitude

error, which is the maximum amplitude error, in steady

state, between the command and the resulting response

The performance metrics are very similar for both

controllers In general, no substantial improvement is

noted when using the adaptive version Concerning the

mission accomplishment, the convergence time from the

initial condition seem very adequate, since the orbital

period is 12.5 days Concerning the tracking stage, the

errors in the angular amplitudes and time delays shall

be taken in consideration when evaluating the

preci-sion requirements of the surface measurement Smaller

errors and time delays can be obtained, at the cost of

larger controls, that extrapolate the thruster limits

Due to the physical characteristics of the system,

the disturbances faced by the spacecraft, even in one

of its most demanding places, are relatively low and

the amount of control required to track the reference is

orders of magnitude greater than the amount used to

compensate for the disturbances

Table 2: Performance measures - continuous control

ac-tion

variable standard adaptive

φ ct (s) 25.71 25.02

θ ct (s) 25.71 26.51

ψ ct (s) 24.53 22.84

φ td (s) -5.52 -6.28

θ td (s) -3.6 -3.6

φ ae (deg) 1.34 1.39

θ ae (deg) 1.33 1.97

ψ ae (deg) 0.065 0.069

5.2 Pulsed control with thrusters

The modulation techniques described in section 4.2 where

applied to modulate the control commands generated

by both control laws of section 5.1

The parameters of the PWM algorithms are:

– Sampling period: Ts= 1 s (fs= 1 Hz);

– Discretization levels: Nd= 10 for negative, Nd = 10

for positive torques

From equation 19 and the associated above, the

minimal on time of the PWM is 0.1 s Note that this

value is very conservative when compared with the

min-imal on times on table 1, but is close to the response

time

The values adopted, by trial and error, for the PW-PFM are: km= 1.4, τ = 0.6 s, Uon= 0.25, Uof f= 0.05 The responses generated by the PWM modulator are shown in figures 13 and 15, for both the standard (continuous lines) and the adaptive (dash-doted lines) SMC

By its turn, the responses generated by the PWPFM modulator are shown in figures 16 and 18

0 20 40 60 80 100 120 140 160 180 0

10 20 30 40

t [s]

0 20 40 60 80 100 120 140 160 180 0

10 20 30 40

t [s]

0 20 40 60 80 100 120 140 160 180 0

10 20 30 40

t [s]

Non−adaptive SMC Adaptive SMC Desired

Fig 13: Euler angles evolution with thrusters and PWM

0 20 40 60 80 100 120 140 160 180

−3

−2

−1 0 1

t [s]

w1

0 20 40 60 80 100 120 140 160 180

−3

−2

−1 0

t [s]

w2

0 20 40 60 80 100 120 140 160 180

−2 0 2 4

t [s]

w3

Fig 14: Angular velocity evolution with thrusters and PWM

The table 3 summarize the results, some perfor-mance parameters were already defined in the table 2 The new parameters are: “max.pul.” maximum num-ber of pulses (comparing the three control axes) gener-ated by the modulation; “min.on” minimum on time;