Approach to Spacecraft Functional Stability in Changes in Moments of Inertia

Approach to Spacecraft Functional Stability in Changes in Moments of Inertia Procedia Computer Science 103 ( 2017 ) 549 – 555 Available online at www sciencedirect com 1877 0509 © 2017 The Authors Pub[.]

Procedia Computer Science 103 ( 2017 ) 549 – 555

1877-0509 © 2017 The Authors Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license

( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

Peer-review under responsibility of the scientific committee of the XIIth International Symposium “Intelligent Systems”

doi: 10.1016/j.procs.2017.01.056

ScienceDirect

XIIth International Symposium «Intelligent Systems», INTELS’16, 5-7 October 2016, Moscow,

Russia

Approach to spacecraft functional stability in changes in moments

of inertia Vuong Anh Trung*, Nguyen Van Thinh

Vietnam Air defence-Air force Academy, Vietnam

Abstract

The paper considers comprehensive solution providing functional stability of the spacecraft with changes in its moments of inertia There was developed the mathematical description of the spacecraft angular position stabilization system (APSS) in the linear and discrete forms There was shown the process of obtaining spacecraft APSS diagnostic models for the particular kind and class of failure There was proved possible to diagnose and restore functionality spacecraft APSS at the systemic level, using the regulator coefficients parametric adjustment

© 2017 The Authors Published by Elsevier B.V

Peer-review under responsibility ofthe scientific committee of the XIIth International Symposium «Intelligent Systems»

Keywords: spacecraft; angular position stabilization system; fault tolerance; diagnosis; failure; restore; parametric adjustment

1 Introduction

Analysis of the progress in space exploration indicates an increase in the requirements for weight and size characteristics, lifetime and functionality of small spacecraft These requirements lead to the complication of the spacecraft angular position stabilization system (APSS) Complexity makes the system more sensitive to failure Failure changes the behavior of the control object (spacecraft) so that the system no longer satisfies the goal Failures

in the system can occur due to aging ingredients and depreciation or due to errors in configuring and operating the system Also, the failure may occur due to changes in environmental conditions In any case, the failure is a major

* Corresponding author

E-mail address: vuonganhtrung@gmail.com

© 2017 The Authors Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license

( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

Peer-review under responsibility of the scientific committee of the XIIth International Symposium “Intelligent Systems”

cause of changes in the system structure or its parameters, which leads to poor performance or even loss of the system functionality

Using the traditional methods associated with the construction of the majority circuit parry of failures resulting in significant increase in power, mass and dimensional, cost parameters of spacecraft Therefore, research related to the construction of systems with active fault tolerance, recently received the development They are based on the principles of diagnosis and recovering of the functional system state Currently, various methods and models of fault tolerance of separate functional blocks spacecraft APSS was developed1,2 However, the study is not considered APSS fail as a whole, at the systemic level Therefore, the actual problem is a systematic approach to the development of an APSS active fault tolerant

In this paper, we consider a systematic approach to a place of failure associated with the change in the spacecraft moment of inertia One cause of this failure is incomplete disclosure spacecraft solar panels Incomplete disclosure,

as presented in the various reports, may be due to some structural defect in the mechanism of disclosure, strain solar surface, mounting damaging one of the solar panels on startup, etc.3

2 Mathematical description spacecraft apss

Let us consider equations of motion of the spacecraft relative to the center of mass One of the most characteristic features of the spacecraft as the control object is the impact on its angular position internal moment’s movements resulting from the relative motions of the device parts There is possible following two cases:

x internal moments are absent or compared with the external moments They are so small that their effect on the angular motion can be ignored;

x internal moments are commensurate with the external or they are the main cause of the angular motion of the vehicle, so that they can not be ignored

The second case is often used to control the angular position of the spacecraft

The principle of using the reaction wheel as the device for creating control moments based on the law of angular momentum conservation If the flywheel rotated by the motor in one direction, the spacecraft will rotate in the opposite direction

Next, consider spacecraft with control executive bodies in form of contactless torque DC motors There rotor with permanent magnets located at the maximum diameter and performs the function of centrifugal mass (reaction wheel)

As the installation diagrams of four reaction wheels (RW) will take the circuit shown in Fig.1

X

Y

Z

E

D

0

2

M

3

M

4

M

2 RW

3 RW

4 RW

1 RW

Fig 1 Installation diagram of RW on the spacecraft M i , i=1…4 – angular moments of RW; RW i , i=1 4 – reaction wheels

In addition, we assume that the angular speed of rotation is much greater than the angular velocity of angular motion RW rotation does not change position the spacecraft center of mass and the total moment of inertia

Dynamics equations for this object obtained on the basis the Lagrange equations of the second kind4 Let us consider linearized equation for the closed angular position stabilization system (APSS) of spacecraft

Fig 2 Functional diagram of the closed-loop system

Linearized equations for the APSS can be written as:



I a t a a t t a k t a k t

a k t t a k G t a k G t a k G t

a k t dt a k t dt a k t dt

(1)

where V is an output signal of the regulator, Кг, K2are constant coefficients, Vel = Ve = (Fi -V2) is an error signal of

the force reflection channel of a system, V = V = (F — Zcp ap) is an error signal of a torque and position

integrating system, i=1,2,3, I is a spacecraft tensor of inertia, I M i is a RW tensor of Mi, i=1,2,3,4, direction cosine

matrix characterizing the position of the i-th RW in the coupled system of coordinates; ωx, ωy, ωz – spacecraft

angular speed relatively corresponding axes; ωRWi, i=1 4 – i-th RW angular speed; kj5, j= 1…10 – coefficients that

are functions of Kp, Kd – regulator coefficients, Kdus – AVS coefficient and Kum, TM – motor coefficients; Gx, Gy, Gz

– control voltages on the input of system

xx yy zz

I

I

;

M

M

I

I

,

cos cos sin sin sin cos cos cos sin

i

We define the state variables, control vectors and output signals for further transition from the equations

describing the motion of the functional elements to the system of equations relating the spacecraft input and output

signals:

x

( ) [G (t) ( ) ( )]

T

T

u t G t G t

x t t dt t dt t dt

³ ³ ³

Measurement and production control actions are performed with a time interval T0 because the system contains

ADC and DAC, and all procedures of processing, transformation and analysis are carried out in microcontroller We

use the representation of derivatives in the form of finite differences when describing the behavior of the continuous

part of the automatic control object at time kT0, k = 0,1 Then we can obtain the system of equations describing the

behavior of nominal APSS in discrete form

3 Diagnostic support of spacecraft apss

We will make parameterization of the kinds of faults for the control object spacecraft Under the kind of

functional element with fault, we mean any event associated with the deviation of its static and dynamic

characteristics from its nominal value

Spacecraft set of kinds of faults includes the following elements: dsp1, dsp2 – respectively, the kinds of faults,

characterized by a decrease and increase of the moment of inertia Ixx; dsp3, dsp4 – respectively, the kinds of faults,

characterized by a decrease and increase of the moment of inertia Iyy; dsp5, dsp6 – respectively, the kinds of faults,

characterized by a decrease and increase of the moment of inertiaIzz

Many types of spacecraft faults can be represented by three classes: αsp1 – class of spacecraft faults, including the

kinds of faults dsp1, dsp2 and characterized by the parameter I ~xx; αsp1 – class of spacecraft faults, including the

kinds of faults dsp3, dsp4 and characterized by the parameterI ~yy

; αsp3– class of spacecraft faults, including the kinds of faults dsp5, dsp6 and characterized by the parameter I~zz

We associate spacecraft direct diagnostic features with deviations of ARS output signals while compiling diagnostic

functional models (DFM) for the classes of spacecraft faults DFM links direct diagnostic features for classes of spacecraft

faults αsp1 and αsp2 with their deviations from their estimations of main CRS output signals established along axes of the

spacecraft associated coordinate system, have the form:

α :

3

1

j

r

'

¦

¦

10

3

1

( , , , , ) )

j

r

'

¦

¦

10

3

1

( , , , , ) )

j

r

¦

αsp2:

10

3

1

( , , , , ) )

j

r

'

¦

¦

10

3

1

( , , , , ) )

j

r

'

¦

¦

10

3

1

( , , , , ) )

j

r

'

¦

αsp3:

10

3

1

( , , , , ) )

j

r

'

¦

¦

10

3

1

( , , , , ) )

j

r

'

¦

¦

10

3

1

( , , , , ) )

j

r

'

¦

Let us analyze these DFM The evidence is reviewing by using the criteria of the DFM structure and signal diagnosability It is not difficult to show that is provided the condition of structural diagnosability, and the unique determination of direct diagnostic characteristics of failures classes is provided with non-zero control and angular

momentum generated by RW, as well as with non-zero spacecraft speeds Consequently, these models are possible

to use for the purpose of deep system diagnosing, the result of which is to determine the failures characteristics Failures

characteristics, such as the detection time, place, class and kind, obtained as a result APSS diagnosis They are the

foundation for the next phase for active fault tolerance - parrying failures by using the available in the object

structural, functional and information redundancy5

4 Development of procedures restore functionality of spacecraft APSS

According to the systems approach to the active fault tolerance, the development of parrying failures procedures

is carried out APSS research, which includes the following steps: identification for each failure mode every possible

variants of its parrying through available excess resources; the allocation from the whole set of options the parrying

subsets that characterize element and system level fault tolerance; establishing for each kind of failure the relative

priority, necessary for the formation procedures for the effective use of appropriate redundancies

According to (2, 3), the DFM for this kind of failure includes transfer coefficients PD controller If we allow

these coefficients change according to the change of parameters I , I , Ixx yy zz, there is possible to parry these kinds

of failures at the system level by adjusting the regulator coefficients Thus, there is the possibility of parametric

adjustment which is to change the parameters of system components (transmission coefficients, voltages, etc.) in

order to parry kinds of failure detected in the object

The figure shows the connection diagram of the parametric adjustment block

Fig 3 Scheme of parametric adjustment realization

For example, a parameter adjustment of the regulator matrix transfer coefficient ΔKp(k) guarantees asymptotic

stability of the spacecraft motion; it is based on the following expressions:

where ΔK*(k) - component to compensate the influence of parametric perturbations; ΔKp.st - component that

provides the asymptotic stability of the object perturbed motion

Expression for calculating ΔK*(k)6:

1

0

ˆ

ˆ

p

K k B B n k C x k k k

n k F f k B K k C x k B K g k b

O

where B – control matrix of spacecraft APSS, ˆ ( )

i

B ' O

' , ˆ ( )

i

F ' O

' , ˆ ( )

b ' O , - matrix and vectors describing

spacecraft dynamics of change in the deviation relative to the nominal; C * - matrix consisting of the transmission

coefficients of sensors

The equations for calculating values ΔKp.st(k) are determined based on the discrete analog of the Lyapunov second method:

. 0;

p st

k

) ( ) ( )) ( ˆ ( 2 )

5 Conclusion

The research resulted in received mathematical models of spacecraft APSS They are used as a basis for formation diagnostic functional models for the failure place related with the change of spacecraft moments of inertia The impact of the controller parameters on the spacecraft APSS parameters was shown Possible recovery procedures of the APSS functional state are developed on the basis of the system (2, 3) at the system level It is shown that by using the equations (5, 6) is possible to carry out a parametric adjustment of regulator coefficients that provide the asymptotic stability of the spacecraft angular motion

References

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