CN105279586A - One-to-many on-orbit refueling task planning method of GEO satellite group - Google Patents

Abstract

The invention provides a one-to-many on-orbit refueling task planning method of a GEO satellite group. One service satellite and one oil storage station are adopted for refueling the GEO satellite group, the task planning problem is shown through a refueling sequence X and decision variables S, and a corresponding two-layer optimization model is established; a genetic algorithm is adopted to solve the upper-layer optimization problem, a random search method is adopted for low-layer optimization, the decision variables S in the number of Q are randomly generated, a solution with the minimum optimization index is selected to serve as an optimal solution, the encoding mode and the global optimization capacity of the method are very suitable for solving the path planning problem, and the global optimal scheme can be fast obtained. The method has the advantages that the GEO target group on-orbit refueling capacity is high, and fuel consumption is low.

Description

A kind of " one-to-many " of GEO satellite group annotates mission planning method in-orbit

Technical field

The present invention relates to On-orbit servicing field, particularly one mission planning method that multiple geostationary orbit (GEO) satellite is annotated in-orbit.

Background technology

GEO is the very special track of a class, and orbit altitude is 35786km, and the orbital period is identical with earth rotation period.The satellite being positioned at this track keeps synchronous with the earth all the time, and area coverage is wide, plays an important role in civil and military fields such as early warning, communication, data relay, electronic reconnaissances.The key factor affecting the GEO lifetime of satellite is the fuel mass that it carries, one is due to earth aspherical, lunisolar attraction, the effect of the perturbation such as solar light pressure, the speed increment that GEO satellite need apply 52m/s approximately every year carries out the maintenance of thing and north-south position, and two is GEO satellites is cover the orbit maneuver carried out in other region to need consume fuel.The fuel of GEO satellite will approach exhaustion time, common way to leave the right or normal track operation to it, again launches an alternative satellite.Due to the high cost of space launch and the manufacturing cost of satellite itself, this is very expensive.Therefore, each spacefaring nation life-span of proposing one after another to improve GEO satellite by annotating in-orbit and in-orbit maneuverability in recent years.

To the dosing method in-orbit of GEO satellite group, and corresponding mission planning is the technical matters needing solution badly.Existing mode of annotating in-orbit to multiple satellite mainly contains " a service star mode ", " multiple service star mode ", and " distributed (P2P) ".When adopting one to serve star, the fuel carried due to it is limited, and the ability of annotating in-orbit is very limited.When adopting multiple service star, ability of annotating in-orbit can only be improved to a certain extent.When adopting distributed, each satellite both can be used as service star also can be used as target satellite, but current most satellite does not possess the ability as service star.Therefore, to the filling in-orbit of GEO satellite group, existing method obviously can not meet the demands.

Summary of the invention

For problems of the prior art, " one-to-many " that the invention provides a kind of GEO satellite group to be annotated in-orbit mission planning method, is namely adopted the mode that a service star and a tank farm are annotated to GEO satellite group.

Task scene of the present invention is: multiple GEO satellite with different orbit inclination, right ascension of ascending node and phase place proposes demand of annotating in-orbit, a service star and a tank farm are emitted on GEO, and preliminary orbit parameter is identical.Service star realizes the transport of fuel by travelling to and fro between GEO target and tank farm, thus completes the filling task in-orbit to multiple GEO target.Optimization aim is that the fuel that service star Orbit Transformation consumes is minimum.Constraint condition is that service star carries the limited in one's ability of fuel, and each GEO target can only be once serviced, and after task completes, service star is transferred to preliminary orbit.Annotate in-orbit in task process, service star has following five kinds of states: 1, be docked on initial GEO; 2, GEO target is transferred to; 3, for GEO target is annotated; 4, tank farm is transferred to; 5, supply is obtained from tank farm.Tank farm has following two states: 1, be docked on initial GEO; 2, for service star is annotated.

The preliminary orbit of known service star and tank farm, the mission planning problem that multiple GEO target is annotated in-orbit needs to solve following three subproblems: one is the filling order of Optimized Service star to GEO target; Two is determine that whether getting back to tank farm after service star has been annotated to GEO target carries out supply; Three be determine to serve star orbit maneuver impulse speed and each time from the fuel mass carried during tank farm.

For solving the problems of the technologies described above, " one-to-many " of a kind of GEO satellite group of the present invention annotates mission planning method in-orbit, and the technical scheme of employing is:

Step one: input initial parameter: the orbital tracking and the demand for fuel that comprise n GEO target, serves the orbital tracking of star, architecture quality and fuel carrying capacity, the orbital tracking of tank farm;

Step 2: determine dosing method: service star realizes the spacecrafts rendezvous with tank farm by orbit maneuver, obtains the supply of fuel; After GEO target spacecrafts rendezvous, for it is annotated in-orbit; Service star is travelled to and fro between GEO target and tank farm, transports fuel.During orbit maneuver, first service star adjusts orbital plane, then carries out the adjustment of phase place in orbital plane, thus realizes the spacecrafts rendezvous with tank farm or GEO target.Carry out phase modulation by applying twice velocity pulse, phase modulation speed increment and phase modulation time claim inverse ratio.Analysis shows, when the phase modulation time more than one month time, phase modulation speed increment with adjustment orbital plane compared with can ignore, therefore ignore the fuel needed for phase modulation in the present invention.

For task process of annotating in-orbit is described visually, Fig. 1 gives service star and the tank farm process of annotate to GEO target: serve star and first obtain supply in tank farm, annotate in-orbit with No. 8 target spacecrafts rendezvous after orbit maneuver, then No. 1 target is annotated in-orbit, get back to after tank farm obtains supply again, and No. 2 targets are annotated in-orbit, until complete the filling task in-orbit to all selected GEO targets.

Step 3: mission planning of annotating in-orbit:

S3.1 design optimization variable

Definition optimized variable

X＝[x ₁,x ₂,…,x _n],S＝[s ₁,s ₂,…,s _n]

Wherein, X represents the order of filling in-orbit to n GEO target, the decision variable that S is made up of 0 and 1, and 0 expression is directly transferred to next GEO target, and 1 represents that getting back to tank farm carries out supply, is then transferred to next target.Such as, X=[2,4,5,1,3], S=[00101] represents and has 5 GEO targets, and the service in-orbit filling order of star to target is 2-4-5-1-3, and after No. 5 targets being completed to filling, service star is got back to tank farm and carried out supply, then No. 1 target is annotated, get back to tank farm to after No. 3 target filling.

The speed increment of S3.2 calculation services star orbit maneuver, determines optimizing index simultaneously

(1) serve star when adjusting orbital plane, impulse speed need change ascending node of orbit right ascension Ω and orbit inclination I simultaneously; If service star will from a GEO satellite (I ₁, Ω ₁) motor-driven to another satellite (I ₂, Ω ₂), the speed increment of needs is

$\mathrm{\Δ}v=2vsin\frac{\mathrm{\γ}}{2}$

Wherein v is the travelling speed of GEO satellite, and γ is obtained by following formula.

cosγ＝sinI ₁sinI ₂cos(Ω ₁-Ω ₂)+cosI ₁cosI ₂

The fuel mass of orbit maneuver consumption is

$\mathrm{\Δ}m=m_0(1-e^{-\mathrm{\Δ}v/I_{sp}{g}_0})$

Wherein m ₀be the initial mass of service star, comprise architecture quality and fuel mass; I _spengine/motor specific impulse, g ₀it is normal gravity.

Given X and S, can calculate the speed increment of each orbit maneuver of service star and the fuel consumption of orbit maneuver.

(2) optimizing index is defined as total fuel mass that service star obtains from tank farm

$M_{fuel}=\underset{i=1}{\overset{N}{\Σ}}{m}_{i,fuel}$

Wherein, N is service star obtains supply number of times in tank farm, m _{i, fuel}it is the fuel mass obtained for i-th time.

Provide optimizing index M below _fuelcomputing method.

Given input: filling order X, decision variable S;

1) by M _fuelbe initialized as 0;

2) finding out all elements in S is the position of 1, represents with s, and is expanded s=[0, s];

3)fori＝1toN

i) $Q=s(i+1)-s\left(i\right),m=m_{dry}\×e^{{\mathrm{\Δv}}_{i,Q+1}/\left(I_{sp}{g}_0\right)};$

ii)forj＝Qto1

$m=(m+{\mathrm{\δm}}_{i,j})\×e^{{\mathrm{\Δv}}_{i,j}/\left(I_{sp}{g}_0\right)}$

Iii) m _{i, fuel}=m-m _dryif, m _{i, fuel}< C, then M _fuel=M _fuel+ m _{i, fuel}, otherwise S is not a feasible solution, makes M _fuel=10 ⁵, terminate to calculate;

Wherein, m _dryfor service star architecture quality, Δ v _{i, Q+1}represent that service star i-th time is after tank farm, gets back to the speed increment needed for tank farm, Δ v _i,jbe service star i-th time from tank farm, with the speed increment needed for a jth target intersection, δ m _i,jfor the demand for fuel of target, C carries fuel capability, if m for serving star _{i, fuel}> C, then illustrate it is not a feasible program.

S3.3 design optimization model

Set up a service star and tank farm to annotate in-orbit to multiple GEO target the bi-level optimization model of mission planning, it is optimization to filling order X that upper strata is optimized, and it is optimization when given X to decision variable S that lower floor optimizes, specific as follows:

Upper strata is optimized: find optimum X=[x ₁, x ₂..., x _n], make optimizing index minimum, and meet following condition: (1) x _i≤ n, x _i∈ 1,2 ..., n}; (2) x _i≠ x _j, i, j ∈ (1,2 ... n), i ≠ j;

Lower floor optimizes: in given X situation, find optimum S, and satisfy condition: (a) s _i=0/1, i ∈ (1,2 ... n-1); (b) s _n=1; (c) m _{i, fuel}< C;

S3.4 utilizes genetic algorithm and Monte Carlo analysis to be optimized

The present invention adopts the genetic algorithm with efficient ability of searching optimum to carry out upper strata optimization, and genetic algorithm is a kind of conventional intelligent optimization method, and its each step is all the art common practise;

Lower floor optimizes employing stochastic search methods, a stochastic generation Q S, and the solution selecting optimizing index minimum is as optimum solution.Fig. 2 gives the optimizing process of genetic algorithm and Monte Carlo analysis.First stochastic generation is annotated the initial population of order X in-orbit, and makes genetic algebra Gen be 0.For the X that each is given, in lower floor optimizes, utilize Monte Carlo analysis to be optimized decision variable S, and obtain the adaptive value of X.If genetic algebra does not reach the maximum algebraically of setting, carry out genetic manipulation to X, comprise selection, intersection, variation, genetic algebra Gen adds 1.When genetic algebra reaches maximum algebraically, algorithm stops.

Different from basic genetic algorithmic, selection operation in genetic algorithm of the present invention have employed elitism strategy, specific as follows: to find out the individuality that adaptive value in colony is higher and lower, destroyed by ensureing that good individuality is not operated by crossover and mutation, directly enter colony of future generation, individuality high for adaptive value replaced the low individuality of adaptive value, the population then newly obtained carries out selecting, intersect, mutation genetic operation; The advantage of elitism strategy to ensure that the optimized individual adaptive value of every generation does not reduce;

Variable X adopts sequence crossover: 1) Stochastic choice two point of contact k ₁and k ₂, 2) and exchange center section, 3) from k ₂rear first gene rises lists former order, and removes existing gene, 4) from k ₂rear first position is risen, and by inserting without duplicate factor order of obtaining, shown below is an example:

P ₁:24|357|16→C ₁:37|514|62

P ₂:63|514|27→C ₂:14|357|26

k ₁k ₂k ₁k ₂

The variation mode of variable X is as follows: 1) Stochastic choice two point of contact k ₁and k ₂, 2) and center section rearranges from small to large, shown below is an example:

37|514|62→37|145|62

k ₁k ₂k ₁k ₂

S3.5 exports optimal case:

Optimal case comprises service star to filling order X, the decision variable S of GEO target, the speed increment of service star orbit maneuver and from the fuel mass carried during tank farm.

Compared with prior art, the present invention has the following advantages:

1, the present invention proposes employing one service star and the tank farm mode of annotating in-orbit to GEO target, has GEO target complex ability of annotating in-orbit strong, the features such as fuel consumption is few.

2, the present invention proposes to utilize filling order X and decision variable S to represent mission planning problem, and sets up corresponding bi-level optimization model.

3, the present invention adopts genetic algorithm to solve upper strata optimization problem, and its coded system and global optimization ability are very applicable to path planning problem, can obtain the scheme of global optimum fast.

Accompanying drawing explanation

Fig. 1 is the schematic diagram that the present invention one service star and a tank farm are annotated in-orbit to multiple GEO target;

Fig. 2 is the process flow diagram of bilevel optimization of the present invention;

Fig. 3 is genetic algorithm optimization process of the present invention;

Fig. 4 is that the present invention annotates the optimal case of task in-orbit;

Fig. 5 is the impulse speed size that the present invention serves star orbit maneuver;

Fig. 6 is the change that the present invention serves star quality.

In figure, symbol description is as follows:

X annotates order in-orbit;

S decision variable;

Gen genetic algebra;

M _fueloptimizing index;

H _xthe component of angular momentum in J2000 coordinate system x-axis of satellite;

H _ythe component of angular momentum in J2000 coordinate system y-axis of satellite;

| Δ v| serves the size of star impulse speed;

Δ m serves the change of star quality.

Embodiment

Below with reference to specific embodiment and Figure of description, the present invention is described in further details.

The present embodiment is to the filling in-orbit of 15 GEO targets, the invention will be further described, and its concrete steps are as follows:

Step one: input initial parameter: the right ascension of ascending node of 15 the GEO targets chosen, orbit inclination and demand for fuel are in table 1, service star and tank farm are deployed on GEO, there is identical right ascension of ascending node and orbit inclination, be 0 degree, the architecture quality of service star is 500kg, fuel carrying capacity is 1000kg, I _sp=320s, g ₀=10m/s ²;

Table 1GEO target component

Step 2: determine dosing method: service star realizes, with the spacecrafts rendezvous of tank farm, obtaining the supply of fuel by orbit maneuver; After GEO target spacecrafts rendezvous, for it is annotated in-orbit; Service star is travelled to and fro between GEO target and tank farm, transports fuel.

Step 3: mission planning of annotating in-orbit:

1, design optimization variable: definition optimized variable is

X＝[x ₁,x ₂,…,x _n],S＝[s ₁,s ₂,…,s _n]

Wherein, n=15, X represent the order of filling in-orbit to 15 GEO targets, the decision variable that S is made up of 0 and 1, and 0 expression is directly transferred to next target, and 1 represents that getting back to tank farm carries out supply, is then transferred to next target.

2, the calculating of optimizing index: optimizing index is defined as total fuel mass that service star obtains from tank farm

$M_{fuel}=\underset{i=1}{\overset{N}{\&Sigma;}}{m}_{i,fuel}$

Wherein, N is service star obtains supply number of times in tank farm, m _{i, fuel}it is the fuel mass obtained for i-th time.Provide optimizing index M below _fuelcomputing method.

Wherein, find (S==1) represents that in S, element is the position of 1, m _dryfor service star architecture quality, Δ v _i,jbe service star i-th time from tank farm, with the speed increment needed for a jth target intersection, δ m _i,jfor the demand for fuel of target, C carries fuel capability, if m for serving star _{i, fuel}> C, then illustrate it is not a feasible program

3, design optimization model:

4, set up a service star and tank farm to annotate in-orbit to multiple GEO target the bi-level optimization model of mission planning, it is optimization to filling order X that upper strata is optimized, and it is optimization when given X to decision variable S that lower floor optimizes, and Optimized model is as follows

FindX＝[x ₁,x ₂,…,x _n]

$\begin{array}{cc}Min& M_{fuel}=\underset{i=1}{\overset{N}{\&Sigma;}}{m}_{i,fuel}=F(X,S)\end{array}$

$Subjectto\left\{\begin{array}{c}\begin{array}{cc}{x}_i\&le;n,& x_i\&Element;\{1,2,\mathrm{...},n\}\\ x_i\&NotEqual;x_j,& i,j\&Element;(1,2,\mathrm{...}n),i\&NotEqual;j\end{array}\\ finds=\left[\begin{array}{cccc}{s}_1,& s_2,& \mathrm{...}& s_n\end{array}\right],Min{M}_{fuel}=\underset{i=1}{\overset{N}{\&Sigma;}}{m}_{i,fuel}=F(X,S)\\ Subjectto\left\{\begin{array}{c}{s}_i=0/1,i\&Element;(1,2,\mathrm{...}n-1)\\ s_n=1\\ m_{i,fuel}<C\end{array}\right.\end{array}\right.$

5, genetic algorithm and Monte Carlo analysis is utilized to be optimized

Employing genetic algorithm is optimized on upper strata, and parameter is as shown in table 2.Lower floor optimizes employing Monte Carlo analysis, and parameter Q is taken as 100 and 500, and Optimizing Flow as shown in Figure 2.

Table 2 genetic algorithm parameter

6, optimal case is exported:

Optimal case comprises service star to filling order X, the decision variable S of GEO target, the speed increment of service star orbit maneuver and from the fuel mass carried during tank farm.

Study the mission planning of the filling in-orbit problem of 15 GEO targets in a service star and tank farm his-and-hers watches 1.The parameter Q that lower floor optimizes gets 100 and 500, carries out 30 independent experiments respectively, the results are shown in Table 3, and done overstriking to optimum solution.

The result of table 330 suboptimization

The M that optimal case is corresponding _fuel=4265.5kg, the fuel consumption for orbit maneuver is 1265.5kg.The filling order X of optimal case is 9-11-12-1-4-13-3-8-5-7-15-10-14-6-2, and decision variable S is the optimizing process that 0-0-1-0-0-1-1-0-0-1-0-1-0-0-1, Fig. 3 give genetic algorithm, converges to optimum solution greatly after 50 generations.Fig. 4 gives the loading plan in-orbit to 15 GEO targets, and the state of service star and target represents with angular momentum.Fig. 5 give service star apply speed increment, be applied with altogether 21 velocity pulses, wherein 6 times for tank farm intersection.Fig. 6 gives the mass change of service star, and service star is respectively 930.89kg from the fuel carried during tank farm, 864.12kg at every turn, 220.43kg, 874.96kg, 507.52kg and 867.52kg, these fuel parts are for serving the orbit maneuver of star, and remaining is supplied to GEO target.

Claims (5)

1. " one-to-many " of GEO satellite group annotates a mission planning method in-orbit, it is characterized in that, comprises the following steps:

Step one: input initial parameter: the orbital tracking and the demand for fuel that comprise n GEO target, serves the orbital tracking of star, architecture quality and fuel carrying capacity, the orbital tracking of tank farm;

Step 2: determine dosing method: service star realizes the spacecrafts rendezvous with tank farm by orbit maneuver, obtains the supply of fuel; After GEO target spacecrafts rendezvous, for it is annotated in-orbit; Service star is travelled to and fro between GEO target and tank farm, transports fuel;

Step 3: mission planning of annotating in-orbit:

S3.1 design optimization variable

Definition optimized variable

X＝[x ₁,x ₂,…,x _n],S＝[s ₁,s ₂,…,s _n]

Wherein, X represents the order of filling in-orbit to n GEO target, the decision variable that S is made up of 0 and 1, and 0 expression is directly transferred to next GEO target, and 1 represents that getting back to tank farm carries out supply, is then transferred to next target;

The speed increment of S3.2 calculation services star orbit maneuver, determines optimizing index simultaneously

A. serve star when adjusting orbital plane, impulse speed need change ascending node of orbit right ascension Ω and orbit inclination I simultaneously; If service star will from a GEO satellite (I ₁, Ω ₁) motor-driven to another satellite (I ₂, Ω ₂), the speed increment of needs is

$\mathrm{\&Delta;}v=2vsin\frac{\mathrm{\&gamma;}}{2}$

Wherein v is the travelling speed of GEO satellite, and γ is obtained by following formula:

cosγ＝sinI ₁sinI ₂cos(Ω ₁-Ω ₂)+cosI ₁cosI ₂

The fuel mass of orbit maneuver consumption is

$\mathrm{\&Delta;}m=m_0(1-e^{-\mathrm{\&Delta;}v/I_{sp}{g}_0})$

Wherein m ₀be the initial mass of service star, comprise architecture quality and fuel mass; I _spengine/motor specific impulse, g ₀it is normal gravity;

Given X and S, can calculate the speed increment of each orbit maneuver of service star and the fuel consumption of orbit maneuver;

B. optimizing index is defined as total fuel mass that service star obtains from tank farm

$M_{fuel}=\underset{i=1}{\overset{N}{\&Sigma;}}{m}_{i,fuel}$

Wherein, N is service star obtains supply number of times in tank farm, m _{i, fuel}it is the fuel mass obtained for i-th time;

S3.3 design optimization model s _n=1

Set up a service star and tank farm to annotate in-orbit to multiple GEO target the bi-level optimization model of mission planning, it is optimization to filling order X that upper strata is optimized, and it is optimization when given X to decision variable S that lower floor optimizes, and Optimized model is as follows:

Upper strata is optimized: find optimum X=[x ₁, x ₂..., x _n], make optimizing index minimum, and meet following condition: (1) x _i≤ n, x _i∈ 1,2 ..., n}; (2) x _i≠ x _j, i, j ∈ (1,2 ... n), i ≠ j;

Lower floor optimizes: in given X situation, find optimum S, and satisfy condition: (a) s _i=0/1, i ∈ (1,2 ... n-1); (b); (c) m _{i, fuel}< C;

S3.4 utilizes genetic algorithm and Monte Carlo analysis to be optimized

Employing genetic algorithm is optimized on upper strata, and lower floor optimizes employing Monte Carlo analysis, a stochastic generation Q S, and the solution selecting optimizing index minimum is as optimum solution;

S3.5 exports optimal case:

Optimal case comprises service star to filling order X, the decision variable S of GEO target, the speed increment of service star orbit maneuver and from the fuel mass carried during tank farm.

2. " one-to-many " of GEO satellite group according to claim 1 annotates mission planning method in-orbit, it is characterized in that, in S3.3, and optimizing index M _fuelcircular be:

Given input: filling order X, decision variable S;

1) by M _fuelbe initialized as 0;

2) finding out all elements in S is the position of 1, represents with s, and is expanded s=[0, s];

3)fori＝1toN

i)Q＝s(i+1)-s(i)， $m=m_{dry}\&times;e^{{\mathrm{\&Delta;v}}_{i,Q+1}/\left(I_{sp}{g}_0\right)};$

ii)forj＝Qto1

$m=(m+{\mathrm{\&delta;m}}_{i,j})\&times;e^{{\mathrm{\&Delta;v}}_{i,j}/\left(I_{sp}{g}_0\right)}$

Iii) m _{i, fuel}=m-m _dryif, m _{i, fuel}< C, then M _fuel=M _fuel+ m _{i, fuel}, otherwise S is not a feasible solution, makes M _fuel=10 ⁵, algorithm terminates;

Wherein, m _dryfor service star architecture quality, Δ v _{i, Q+1}represent that service star i-th time is after tank farm, gets back to the speed increment needed for tank farm, Δ v _i,jbe service star i-th time from tank farm, with the speed increment needed for a jth target intersection, δ m _i,jfor the demand for fuel of target, C carries fuel capability, if m for serving star _{i, fuel}> C, then illustrate it is not a feasible program.

3. " one-to-many " of GEO satellite group according to claim 1 annotates mission planning method in-orbit, it is characterized in that, in S3.4, first stochastic generation is annotated the initial population of order X in-orbit, and makes genetic algebra Gen be 0; For the X that each is given, in lower floor optimizes, utilize Monte Carlo analysis to be optimized decision variable S, and obtain the adaptive value of X; If genetic algebra does not reach the maximum algebraically of setting, carry out genetic manipulation to X, comprise selection, intersection, variation, genetic algebra Gen adds 1; When genetic algebra reaches maximum algebraically, algorithm stops.

4. " one-to-many " of GEO satellite group according to claim 3 annotates mission planning method in-orbit, it is characterized in that, in S3.4, selection operation in genetic algorithm have employed elitism strategy, specific as follows: to find out the individuality that adaptive value in colony is higher and lower, destroyed by ensureing that good individuality is not operated by crossover and mutation, directly enter colony of future generation, individuality high for adaptive value replaced the low individuality of adaptive value, the population then newly obtained carries out selecting, intersect, mutation genetic operation.

5. " one-to-many " of GEO satellite group according to claim 3 annotates mission planning method in-orbit, it is characterized in that, in S3.4, variable X adopts sequence crossover: 1) Stochastic choice two point of contact k ₁and k ₂, 2) and exchange center section, 3) from k ₂rear first gene rises lists former order, and removes existing gene, 4) from k ₂rear first position is risen, by inserting without duplicate factor order of obtaining;

The variation mode of variable X is as follows: 1) Stochastic choice two point of contact k ₁and k ₂, 2) and center section rearranges from small to large.