Time-optimal Attitude Reorientation Research of a Rigid Spacecraft

Xi-Jing Wang

, An Wang

, Tao Hong

and Min Tian

Xi’an Satellites Control Center, Xi’an, China

Engineering University of CAPF, Xi’an, China

{wanganmm, tmtw}@163.com

Keywords: The Reorientation of a Non-inertial Asymmetrical Rigid-body Spacecraft, Optimization, Genetic Algorithm.

Abstract: The modified switching time optimization algorithm is used to solve minimum-time problem for the rest-to-

rest reorientation of a non-inertial asymmetrical rigid-body spacecraft. The essential conditions for solving

the problem is inducted with the minimum value principle. Based on the idea of homotopy algorithm, the

relaxation time factor introduced into the Genetic algorithm, which is optimized to determine the switch

features without the gyroscopic coupling. The improved Simulated Annealing method (Simulated Annealing

SA) to determine the optimal trajectory of the switch point. Computer simulation results show its

practicability.

1 INTRODUCTION

Space missions have higher request for spacecraft

attitude, of which the accurate posture keeping ability

is a basic need. In order to perform scheduled tasks,

aircraft must have a certain attitude. There are some

articles about the shortest time attitude adjustment

(Li, and Bainum, 1990). The spacecraft can be

classified as three kinds according its structure, which

are perfectly rigid body, rigid body with some flexible

parts and gyroscopic system. It is meaningful to

research the shortest time three dimensional

maneuvering of rigid spacecraft (Bilimoria and Wie,

1993). On the one hand, there is potential for space

applications. Future spacecraft requires quick

reorientation in the limited thrust to solve a variety of

tasks, for example the rescue or defense and dodge

something. On the other hand it has the academic

value, and it is a basic problem in attitude dynamics

(Bai and Junkins, 2009). The rigid body is the

simplest model of spacecraft, and it is the base of

complex multi-body, flexible and charging model of

t-he spacecraft.

2 THE BASIC PRINCIPLE OF

RIGID BODY MOVEMENT

AROUND THE FIXED POINT

Before we discuss the problem of rigid body

movement around the fixed point, we must choose

attitude parameter, and give the attitude dynamics

equations and attitude motion equation. Quaternion

overcomes the shortcomings of the singularity of

Euler angle method, and the attitude dynamics

equations expressed by Quaternion is linear

equations. So the Quaternion solution is adopted in

this article.

MωIωωI 









)(



(1)



















2103

3102

3201

3210

qqqq

xyz

xzy

yzx

zyx







(2)

Where I is central inertia tensor with a 3×3 moment

of inertia, and



is the rotating angular velocity of the

stars in space. M is the resultant moment of force

relative to the center of mass.



、



、



are

projections of



in local coordinate system

respectively.

3 THE OPTIMAL CONTROL

PROCESS

The least time maneuvering problem can be solved

481

Wang N., Wang X., Tian M. and Hong T.

Time-optimal Attitude Reorientation Research of a Rigid Spacecraft.

DOI: 10.5220/0006029004810485

In Proceedings of the Information Science and Management Engineering III (ISME 2015), pages 481-485

ISBN: 978-989-758-163-2

481

based on the minimum theory, which can deduce the

necessary conditions of solution. These conditions

form two-point boundary value problem, solved by

numerical algorithm. The optimal control process can

be confirmed after getting the solution meet the

necessary conditions.

3.1 The Necessary Conditions of the

Least Time Reorientation

When the moment of inertia is zero, the formula (1)

can be transformed as











32133

23122

13211

uωωDω





(3)

When the original state and the termination state are

both stationary, the least time reorientation problem

can be described as: search the way to control

321



uuu , and make the dynamic system

decided by formula (2) and (3) change from the

original state

)(

3,2,1,0 t

qqqq to the termination state

qqqq )(

3,2,1,0

, getting the least value objective

function











At the same time, the control constraint should be

meet

)3,2,1(

max,max,

 iuuu

iii

(5)

Where

max,i

u is the greatest of

u .

According the minimum value principle, the

select optimum control parameters should make the

Hamilton function the least. Firstly we define

function Si related with the control process.

)3,2,1( 



 i





The optimal control logic has relations following:

max,





sifuu 

(7)

max,





sifuu

(8)

0, 



sifuu

(9)

where

max,max, iii

uuu







, and

u is the

singular controls corresponding with S

ways being 0.

In the non-singular segment, each component of

the control vector uses the boundary value to

constitute a maximum torque control, which is named

as Bang-Bang control. However, the optimal

parameter can’t be defined during the singular

segment. It must be noted that, the case of non-

singular means neither the nonexistence of optimal

time control, nor the undefinable of the optimal time

control, but it means only that the exact relationship

between



and





and t can’t be deduced using

the control equations. Based on the system physical

properties, the greater of the control moment mode,

the faster the revolution, and the time may be shorter.

So during the singular segment maximum torque

control is adopted.

3.2 The Solution of the Least Time

Reorientation Problem

In the process of the optimal control, the control

parameters are in a state of extreme value. So the key

is to decide the switching point. And the solution of

the primary question is changed to search a time when

control torque change from one situation to another.

In this paper, the basic idea is to using modified

switching time optimization algorithm to solve static

- static shortest time problem of non-inertial

asymmetrical aircraft, ignoring singular segment. The

exact approach is firstly to use genetic algorithms to

determine the switch characteristics of no coupling

term, and the Simulated Annealing algorithm is used

to determine the switching point, using relaxation

time factor.

3.2.1 Relaxation Process

The homotopy method to search the zero point for

nonlinear algebraic problem is based on topology

theory of continuous. Assuming nonlinear equation

)(



, f can be decomposed as

fff





(10)

is the simple solution, while

is difficult

solution.

Then we can define an auxiliary function

),( xg



)()(),( xfxfxg







(11)

Where

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482

)()()(),1( xfxfxfxg



(12)

The homotopy method is, searching the known

solution from original value when α=0 to 1. When

α=1, the solution is the answer off.

The homotopy method needs to using relaxation

parameter



)10(0

321









Which make formula (3) to











32133

23122

13211







ωDω

ωωDω



(13)

It is a special situation when

0



, and its solution

is relatively easy to obtain as for the inertial body

symmetry spherical rigid object. Relaxation process

begins from the existed boundary conditions and

solution, to a series of solution of different dynamic

equations, began to linear dynamics equation, finally

reached for nonlinear dynamic equations of the

problem. By appropriate control of the relaxation

ratio, each iteration can get a solution close to the

initial iteration. That is to say, in the process of

gradual change of relaxation parameters, the

objective function must be ensured to achieve a

certain requirements.

3.2.2 Seek Switch Features

Multiaxial time optimal control problem is very

complicated since the existence of dynamic coupling,

so the analytical solution can’t be obtained even for

the symmetry spherical rigid object. The only way to

solve it is using numerical method.

The step-by-step method is used to solve the

multiaxial time optimal control problem. Firstly to

make clear the switching property using genetic

algorithm the when

0



. Calculation steps are as

follows.

a) Randomly generate initial population. The initial

state of the controlling variety is positive in the

value when BZ=1, and the state changes once

coming across a state switching point; when

BZ=0, The initial state of the controlling variety

is negative.

b) Calculate the adaptability

||||

0303

202101

303202

101000

ttcc

qqcqqc

qqcqqcp

ftff

fqfq



















(14)

Where



are weight coefficient,

t is reference

value,

q and



is optimization value. Given

control function u, using fixed step method to

integrate the state equation, we can get the value of p.

c) Census to choose, exchange and variate to make

new population. Calculate the optimal

adaptability to make sure if it meets the index

requirements. If it is, turn to Step d), or turn to step

b) if it is smaller than M. Turn to Step a) if it is

greater than M.

d) Export the result of calculation. The program is

over.

The final step to make sure the optimal trajectory

switching point of every



is calculated using

simulated annealing algorithm. During the

calculation process, the relaxation parameter is small,

so the optimal trajectory change slowly. And the

selection of the original temperature

T and the

persistent time can be a minor value to decrease the

calculation.

4 NUMERICAL SIMULATION

Firstly simulate the optimal time control of

homotaxial, calculating it using genetic algorithm.

The results matches the theatrical value.

Since the complexity of the optimal time control

of multiaxial, the spherically symmetric rigid body is

discussed firstly using numerical simulation method.

When we use the genetic algorithm, firstly setting the

original population is 1000, the variation rate and

crossing-over rate is 0.04 and 0.2 respectively. There

are 3 switching points for every controlling process.

When the seed adaptability becomes stable, fix two at

a time till everyone is the same. The example is as

following.

















，

Time-optimal Attitude Reorientation Research of a Rigid Spacecraft

483

Time-optimal Attitude Reorientation Research of a Rigid Spacecraft

483

0)0()0()0(

321





，

1)0(



0)()()(

321



fff

ttt



，

0)()(



tqtq

，

)

sin()(





，

)

cos()(





The rotation angle of θ with eigenvector is named as

reorientation angle, and it is originally equals to 180

degrees. It can be known that from the boundary

conditions, the 3 coordinate axises of local coordinate

system are same as reference coordinate system in the

beginning time. After the adjustment, it is only same

of the

and ZO

axises. Consequentially there

are 5 switching points for controlling parameters, of

which the control moment of the

axis changes

once. The total controlling time is 3.243 seconds, and

it is 8.5% less than rotating with the eigenvector axis,

being the same as reference (Bilimoria and Wie,

1993). Simulations above are focus on the equal main

inertia moments. However, mostly the main inertia

moments are unequal and small. Simulations below

are for this situation.

Table 1: Time to change the switch condition.

Time(s) ε= 0 ε= 1

11.3489 9.30363

17.1272 11.5518

6.3945 8.1374

8.6427 8.8572

10.7400 7.0771

17.1435 15.6462

34.2746 31.5866

Where the rotation time around the characteristic shaft:

=17.7439；t

=35.4878

The parameters are changed as A=150, B=50,

C=100, M=1. In this situation, the results are obvious

different. Table 1 and Fig. 1~3 show the calculation

results, where

,05.0



4000/

th  . Fig.1

shows the trajectory of angular velocity, and Fig.2

shows the trajectory of quaternion. Fig. 3 shows the

trajectory of switch parameter

t during the

relaxation process. At last the optimal control ratio

reduce 10.9% compared with rotating eigenvector

axis.

Figure 1: Trajectory of angular velocity.

Figure 2: Trajectory of quaternion.

Figure 3: Trajectory of switch parameter

5 CONCLUSION

This article propose a method to solve the least time

reorientation of rigid body. Compared with existed

accomplishment, it doesn’t need the original value by

hand, and it has better astringency. Simulations show

that the least time reorientation problem can be solved

with different main inertia moment. Conclusions are:

the optimal control trajectory is not unique for

completely equal main inertia moment; when the

dynamics equations are linear, time control for each

axis is half positive and half negative; when the main

inertia moments are different, the numbers of control

switch are uncertain with same rotation angle, and the

switch characteristics is the function of reorientation

angle; the existence of coupling term make

0



t >

1



. Further research needs to be done including

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484

ISME 2015 - International Conference on Information System and Management Engineering

484

the asymmetric information i.e. the existence of

inertia product and with different switch features.

REFERENCES

Feiyue Li, and Peter M. Bainum, vol. 13, no. 1, 1990.

Numerical Approach for Solving Rigid Spacecraft

Minimum Time Attitude Maneuvers. Journal of

Guidance Control and Dynamics.

Bilimoria, K. D., and Wie, B., Vol.16, No.3, 1993. Time-

Optimal Three-Axis Reorientation of Rigid Spacecraft.

Journal of Guidance Control and Dynamics.

Xiaoli Bai and John L. Junkins, vol. 32, no. 4, 2009. New

Results for Time-Optimal Three-Axis Reorientation of

a Rigid Spacecraft. Journal of Guidance Control and

Dynamics.

Time-optimal Attitude Reorientation Research of a Rigid Spacecraft

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Time-optimal Attitude Reorientation Research of a Rigid Spacecraft

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