CN110487280A - Wind disturbs UAV Landing bootstrap technique under environment - Google Patents
Wind disturbs UAV Landing bootstrap technique under environment Download PDFInfo
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
- G01C21/00—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00
- G01C21/20—Instruments for performing navigational calculations
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- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
- G05D1/00—Control of position, course or altitude of land, water, air, or space vehicles, e.g. automatic pilot
- G05D1/08—Control of attitude, i.e. control of roll, pitch, or yaw
- G05D1/0808—Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for aircraft
- G05D1/0816—Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for aircraft to ensure stability
- G05D1/0825—Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for aircraft to ensure stability using mathematical models
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- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
- G05D1/00—Control of position, course or altitude of land, water, air, or space vehicles, e.g. automatic pilot
- G05D1/10—Simultaneous control of position or course in three dimensions
- G05D1/101—Simultaneous control of position or course in three dimensions specially adapted for aircraft
Abstract
Wind disturbs UAV Landing bootstrap technique under environment, solves the problems, such as to disturb UAV Landing bootstrap technique under environment by wind that there are guidance accuracies is not high, belongs to UAV Landing control field.The present invention include: S1, control constraints condition and it is calm under conditions of, solve auxiliary optimal control problem by unmanned plane from arbitrary initial position transfer to final position, the guiding movement track of unmanned plane be calculated；And give specific control constraints condition；The unmanned plane motion profile and guiding movement track maximum approximation problem of S2, solution under the conditions of the random component of given wind, determine that unmanned aerial vehicle (UAV) control is restrained；The random component of the wind speed is the random function with given statistical property, 0≤σW≤|σW|M, σWIndicate the random component root mean square of wind speed, | σW|MIndicate σWMaximum permissible value；S3, it is restrained using the unmanned aerial vehicle (UAV) control and realizes that UAV Landing guides.Unmanned plane under environment (UAV) is disturbed applied to wind to land on small-sized movable platform.
Description
Technical field
The present invention relates to a kind of UAV Landing bootstrap technique, in particular to a kind of wind is disturbed UAV Landing under environment and is guided
Method belongs to UAV Landing control field.
Background technique
As the warships such as aircraft carrier, battleship, destroyer, escort vessel and amphibious warship equipment Shipborne UAV increasingly increases
Add, unmanned plane plays an important role in " non-contact fighting " that intelligent weapon and smart weapon are leading.Along with
Modern naval battle carries unmanned plane to three-dimensional, multi-level development, using the small naval vessel of displacement, reaches certain special operations
Region executes the hot missions such as battle reconnaissance, antisubmarine anti-ship, amphibious assault, air-borne early warning, grasps the system sea in future war
Power, control of the air, to enhance national military capability of the country, and the major issue of international common concern.
Wind disturbs unmanned plane under environment (UAV) when landing on small-sized movable platform, needs in unmanned plane and landing gear pair
Connect boundary (terminal) condition for meeting given state vector constantly.Existing wind disturbs UAV Landing bootstrap technique under environment
There is a problem of that guidance accuracy is not high.
Summary of the invention
Disturbing UAV Landing bootstrap technique under environment for existing wind has that guidance accuracy is not high, and the present invention mentions
UAV Landing bootstrap technique under environment is disturbed for a kind of wind.
Wind of the invention disturbs UAV Landing bootstrap technique under environment, which comprises
S1, control constraints condition and it is calm under conditions of, solve unmanned plane from arbitrary initial position transfer to final
The landing motion profile of unmanned plane is calculated, it may be assumed that guiding movement track in the auxiliary optimal control problem of position；
The control constraints condition are as follows:0 < ρ≤1, α (t) indicates the angle of attack of unmanned plane, αMIt indicates
The maximum permissible value of α (t), R (t) indicate the electric propeller pulling force of unmanned plane, RMIndicate the maximum permissible value of R (t)；
In the random component allowed band of wind speed, under given wind speed constant value component, ρ keeps the mathematic expectaion J of criterion J small
In the permissible value ε of setting；
The criterion J are as follows:
V, θ, x and y respectively indicate the flying speed of unmanned plane, the inclination angle of flight velocity vector, rectangular coordinates system
ox0y0In unmanned plane mass center x-axis coordinate and y-axis coordinate,WithIt is represented to deckle dividing value, tFIndicate terminal
At the time of position corresponds to；
The unmanned plane motion profile and guiding movement track maximum of S2, solution under the conditions of the random component of given wind approach
Problem determines that unmanned aerial vehicle (UAV) control is restrained；
The random component of the wind speed is the random function with given statistical property, 0≤σW≤|σW|M, σWIndicate wind
The random component root mean square of speed, | σW|MIndicate σWMaximum permissible value；
S3, it is restrained using the unmanned aerial vehicle (UAV) control and realizes that UAV Landing guides.
Preferably, the S1 includes:
In the case where calm and constraint conditionUnder, guarantee in t=tFMoment unmanned plane is along given
The direction unit vector l=[0 0 sin ξ cos ξ] slave arbitrary initial position to the displacement of the position of given edge-restraint condition
Maximum determines guide track；ξ indicates vector l and ox0The angle of between centers, when determining guide track, utilize maximum principle
The direction of unit vector l is solved, and then makes criterion J2=-lTz(t0) it is maximum value or criterion J3=lTz(t0)=y (t0)sinξ
+x(t0) cos ξ be minimum value；
Z (t)=[V, θ, y, x]T, V, θ, x and y respectively indicate the flying speed of unmanned plane, flight velocity vector inclines
Angle, rectangular coordinates system ox0y0In unmanned plane mass center x-axis coordinate and y-axis coordinate, tFWhen indicating that terminal location is corresponding
It carves, t0At the time of indicating initial position；
Calculate the phase vectors of guide track are as follows:
wT(t)=[wV(t) wθ(t) wy(t) wx(t)]
wV(t)、wθ(t)、wy(t) and wx(t) the corresponding phase vectors of V, θ, x and y are respectively indicated.Preferably, described
The representation of unmanned aerial vehicle (UAV) control rule in S2 are as follows:
Wherein,R(t),α(t),XW(t),YW(t)) unmanned plane is vertical under the conditions of, expression wind is disturbed
Movement in plane；XW(t) frontal drag of unmanned plane, Y are indicatedW(t) lift of unmanned plane is indicated.
Preferably, the representation that the unmanned aerial vehicle (UAV) control in the S2 is restrained are as follows:
Wherein,R(t),α(t),XW(t),YW(t)) unmanned plane is vertical under the conditions of, expression wind is disturbed
Movement in plane；XW(t) frontal drag of unmanned plane, Y are indicatedW(t) lift of unmanned plane is indicated；
t*According toIt is calculated, wy(t*) indicate that height is recently on guiding movement track
Point；H indicates to determine the search time interval of height closest approach.
Preferably, in the S1, the random component allowed band of wind speed and the acquisition methods of η are as follows:
Meeting 0≤σW≤|σW|M, 0 < ρ≤1 while, single change σWOr η, it recalculates, obtainLess than setting
All σ when fixed permissible value εWWith the corresponding relationship of η, the random component allowed band of wind speed is determined according to corresponding relationship.
Beneficial effects of the present invention, the present invention are directed to there are in the case where wind disturbance, are meeting given end conswtraint
Under the premise of, unmanned plane is within a specified time guided to the problem of boat-carrying landing gear and studied, proposes a kind of wind-force
The bootstrap technique that the unmanned plane with end conswtraint under bumpy weather lands on small-sized movable platform, wherein the statistics of wind
Characteristic is unknown, but is limited.Regard the problem as confrontation differential game problem, is solved using guiding control methods
The problem.Simulation results show the error of method of the invention is very small, the precision of guidance is improved.
Detailed description of the invention
Fig. 1 is flow diagram of the invention.
Specific embodiment
Following will be combined with the drawings in the embodiments of the present invention, and technical solution in the embodiment of the present invention carries out clear, complete
Site preparation description, it is clear that described embodiments are only a part of the embodiments of the present invention, instead of all the embodiments.It is based on
Embodiment in the present invention, those of ordinary skill in the art are obtained all without creative labor
Other embodiments shall fall within the protection scope of the present invention.
It should be noted that in the absence of conflict, the feature in embodiment and embodiment in the present invention can phase
Mutually combination.
The present invention will be further explained below with reference to the attached drawings and specific examples, but not as the limitation of the invention.
The movement of unmanned plane in vertical plane can be expressed as following vector differential equation form under the conditions of wind is disturbed:
Wherein, z=[V, θ, y, x]T；The flying speed of V expression unmanned plane；The inclination angle of θ expression flight velocity vector；X, y
Indicate rectangular coordinates system ox0y0In unmanned plane center-of-mass coordinate；R indicates electricity
Dynamic propeller pulling force；M indicates unmanned plane quality；α indicates the angle of attack；G indicates acceleration of gravity；Indicate frontal drag；Indicate lift；Indicate unmanned plane air speed；WxAnd WyIndicate projection of the wind vector in velocity coordinate system；Cx
(α+△αW)=Cx0+A(α+△αW)2Indicate frontal drag coefficient；Indicate lift
Coefficient；Cx0, A,Indicate nondimensional aerodynamic coefficient；△αWIndicate angle of attack added value caused by wind disturbance,ρ indicates atmospheric density, ignores it with the variation of height； S
Indicate maximum secting area.
In the present embodiment, a control amount of unmanned plane is the angle of attack, constraint condition are as follows:
|α(t)|≤αM (2)
Another control amount of unmanned plane is electric propeller pulling force, constraint condition are as follows:
0≤R(t)≤RM (3)
Wind vector is in earth axes ox0y0Corresponding axis on projection can indicate are as follows:Wherein,WithIndicate the Unit Vector of corresponding axis
Amount； WithIndicate projection of the wind speed constant value component on corresponding axis；ζx(x0,y0) and ζy(x0,
y0) indicate projection of the wind speed random component on corresponding axis.
Projection W of the wind vector on the corresponding axis of velocity coordinate systemxAnd WyPass through relational expression Wx=Wx0cosθ+
Wy0Sin θ and Wy=Wx0sinθ+Wy0Cos θ and wind vector are in earth axes ox0y0Corresponding axis on projection it is related
Connection.
In the landing mission of unmanned plane, the constant value component of wind speed has almost no change, thus, it is supposed that wind speed constant value component
Projection be constant value.Using shaping filter equation by the random component ζ of wind speedx=ζx(x0,y0), ζy=ζy(x0,y0) indicate
For the random function form with given statistical property.
The random component of wind speed is unknown, but bounded.In simulation process, the random component of wind speed is arranged to
σW(root mean square of turbulent velocity), variation range are as follows:
0≤σW≤|σW|M (4)
The motor-driven initial time t of UAV Landing0With completion moment tFIt is known.Item in control constraints (2) and (3)
Under part, the influence of wind disturbance is considered, stationary component is it is known that random component meets constraint condition (4)) design unmanned plane
Control law guarantees that criterion (5) are minimized:
Wherein,Indicate given boundary (terminal) condition.
Due to the presence of the random component of wind, it is studied the problem of contain uncertain factor.In the present embodiment,
Regard the problem of required solution as there are two the antagonism differential game problem that player participates in.Wherein, first player according to
Our interests are acted, and seek the minimum value of criterion (5), the interests of second player are on the contrary, find the maximum of criterion (5)
Value.
The domination set U of first player is made of all feasible control rates for meeting constraint condition (2) and (3), by its point
At two subset U1And U2.Subset U1For solving the auxiliary optimal control problem in calm situation, the landing that will be calculated
Motion profile is known as guide track.Subset U2For compensating the deviation between actual motion track and guide track.
Theoretically, be difficult from it is studied the problem of in isolate one for compensate wind disturbance act on subset
U2.For this purpose, the wind of the invention of present embodiment disturbs UAV Landing bootstrap technique under environment, comprising:
S1, control constraints condition and it is calm under conditions of, solve unmanned plane from arbitrary initial position transfer to final
The landing motion profile of unmanned plane is calculated, it may be assumed that guiding movement track in the auxiliary optimal control problem of position；
Present embodiment introduces the coefficient η changed from 0 to 1, that is, 0 η≤1 <, and by unmanned aerial vehicle (UAV) control constraint set U1Table
It is shown as:
α (t) indicates the angle of attack of unmanned plane, αMIndicate that the maximum permissible value of α (t), R (t) indicate the electric screw of unmanned plane
Paddle pulling force, RMIndicate the maximum permissible value of R (t)；
In the random component allowed band of wind speed, under given wind speed constant value component, η makes the mathematic expectaion of criterion J
Less than the permissible value ε of setting；
The criterion J are as follows:
V, θ, x and y respectively indicate the flying speed of unmanned plane, the inclination angle of flight velocity vector, rectangular coordinates system
ox0y0In unmanned plane mass center x-axis coordinate and y-axis coordinate,WithIt is represented to deckle dividing value, tFIndicate terminal
At the time of position corresponds to；
The unmanned plane motion profile and guiding movement track maximum of S2, solution under the conditions of the random component of given wind approach
Problem determines that unmanned aerial vehicle (UAV) control is restrained；
The random component of the wind speed is the random function with given statistical property, 0≤σW≤|σW|M, σWIndicate wind
The random component root mean square of speed, | σW|MIndicate σWMaximum permissible value；
S3, it is restrained using the unmanned aerial vehicle (UAV) control and realizes that UAV Landing guides.
Present embodiment is directed to there are in the case where wind disturbance, under the premise of meeting given end conswtraint, by nobody
Machine is within a specified time guided to the problem of boat-carrying landing gear and is studied, wherein the statistical property of wind be it is unknown,
But it is limited.Regard the problem as confrontation differential game problem, introduces η using guiding control methods to solve the problems, such as this.It is imitative
The error that true result demonstrates method of the invention is very small, improves the precision of guidance.
Present embodiment introduces the coefficient η changed from 0 to 1, that is, 0 η≤1 <, and by unmanned aerial vehicle (UAV) control constraint set U1Table
It is shown as:
Present embodiment solves auxiliary optimal control problem, determines that calculation obtains the landing motion profile of unmanned plane, preferably in fact
It applies in example, the S1 of present embodiment includes:
In the case where calm, guiding movement is determined according to vector equation (1), control amount meets constraint condition (6).It is given
The initial time t of controlled motion0With end time tFAnd t=tFThe boundary condition at momentIt protects
Card is in t=tFMoment along the direction given unit vector l=[0 0 sin ξ cos ξ] slave arbitrary initial position (t0Moment) it arrives
Final position (tFMoment) displacement it is maximum, wherein ξ indicates vector l and ox0The angle of between centers, that is, need to find criterion J1=
lT[z(tF)-z(t0)] maximum value.Due to z (t hereinF) it is known, so only needing to find criterion J2=-lTz
(t0) maximum value or criterion J3=lTz(t0)=y (t0)sinξ+x(t0) cos ξ minimum value.
According to initial time (t=t0) unmanned plane flying height setting vector l direction.It is maximum by Pang Te lia king
The necessary condition of value principle solves the directionality problem of vector l, and is solved using Krylov-Chernousko successive approximation method
The border issue.
Calculate the phase vectors of guide track are as follows:
wT(t)=[wV(t) wθ(t) wy(t) wx(t)] (7)
wV(t)、wθ(t)、wy(t) and wx(t) the corresponding phase vectors of V, θ, x and y are respectively indicated.
Present embodiment gives the constant value component of wind speed and the root mean square σ of turbulent velocityW；
The S2 of present embodiment solves the random component (σ in given windW) under the conditions of unmanned plane motion profile and guiding
Motion profile maximum approximation problem.The constant value component of wind is constant.The control amount of unmanned plane meet constraint condition (2) and
(3), that is, use complete control ability.Unmanned plane is in t0The primary condition at moment corresponds to the pass solution auxiliary optimum control
The t that problem obtains0The value of the guide track phase vectors at moment.
In preferred embodiment, the representation of the unmanned aerial vehicle (UAV) control rule in the S2 of present embodiment are as follows:
Wherein,R(t),α(t),XW(t),YW(t)) unmanned plane is vertical under the conditions of, expression wind is disturbed
Movement in plane；XW(t) frontal drag of unmanned plane, Y are indicatedW(t) lift of unmanned plane is indicated.
The random component allowed band of wind speed and the acquisition methods of ρ in present embodiment are as follows:
Meeting 0≤σW≤|σW|M, 0 < ρ≤1 while, single change σWOr ρ, it recalculates, obtainLess than setting
All σ when fixed permissible value εWWith the corresponding relationship of ρ, the random component allowed band of wind speed is determined according to corresponding relationship.Tool
Body includes:
Step 1 is solved in the random component (σ for giving windW) under the conditions of unmanned plane motion profile and guiding movement track
Maximum approximation problem；
Step 2, clearance solve the problem, have found the mathematic expectaion of criterion (5)
If the mathematic expectaion of step 3, criterion (5)Less than the permissible value ε of setting, then in unmanned plane and the dress that lands
Random component (the σ of wind can be completely eliminated by setting the docking momentW) influence.By σWValue increase Δ σW, then go to step 1；
If the mathematic expectaion of step 4, criterion (5)Greater than the permissible value ε of setting, then for given ρ value, nothing
Man-machine random component (the σ that will be unable to the given wind of compensationW) and stationary component influence.
Change ρ value, and calculates the maximum permissible value σ for corresponding to the wind speed random component mean square deviation of each ρ valueW, find wind
Random component allowed band, in the range, under given wind speed constant value component, the mathematic expectaion of criterion (5)It is less than
The permissible value ε of setting.
Simulating, verifying:
End conswtraint condition are as follows: tF=60s；
The parameter of unmanned plane are as follows: cx0=0.34；A=0.01；M=20kg；S=0.0357m2； αM=
0.4°；RM=98N.
Under conditions of coefficient ρ=0.8, the problem of studied is solved.
Under conditions of control constraints (6), by solving position (t=t of the unmanned plane from arbitrary initial0) be transferred to it is given
Position (the t=t of end conswtraint conditionF) subsidiary equation determine guide track.When determining guide track, unit vector l
Direction according to t=t0Moment is y (t in height0)=y0Unmanned plane locality condition at=250m determines.
Optimum angle of attack is constant value, and the α of α (t)=0.8M=0.32 °.Turbulent velocity root mean square σMVariation range 0.1m/s-
30m/s。
It when designing unmanned aerial vehicle (UAV) control rule, does not use equation (8), and uses (x (t)-w (t*)) calculate unmanned plane and guiding
Deviation between track, that is, use following equation:
In equation (9), t*It is calculated according to equation (10).
Wherein, wy(t*) indicate the point that height is nearest on guiding movement track；When H indicates to determine the search of height closest approach
Between be spaced.In simulation process, H=5s is chosen.
In this case, when selection unmanned aerial vehicle (UAV) control sampling period Δ t=0.01s and calm condition under, unmanned plane with
Landing gear docks the unmanned plane at moment as the deviation of coordinate and given terminal condition is minimum.Wherein, flying speed is inclined
Difference-
When studying the influence of wind speed random component, change the value σ of turbulent velocity root mean squareM, the stationary component of wind speed is
Zero.
When being given in Table 1 coefficient ρ=0.8 and ρ=0.6 and ρ=0.4, the mathematic expectaion of Optimal condition (5).
The data provided in table 1It is to be calculated according to 20 the simulation experiment results.
The mathematic expectaion of 1 criterion of table (5)
If can be seen that the absolute value of the mathematic expectaion permissible value of criterion (5) less than 2 from the experimental data in table 1,
So the value of wind speed random component root mean square should not be more than 1m/s, wind under conditions of ρ=0.6 under conditions of ρ=0.8
The value of fast random component root mean square should not be more than 5m/s, and the value of wind speed random component root mean square is not under conditions of ρ=0.4
It should be more than 10m/s.
The control algolithm based on guide track that simulation result shows that present embodiment proposes can be used for that unmanned plane is guided to exist
It is recycled on the landing gear of small-sized motion platform, and can be used for assessing wind disturbance to unmanned plane and landing receiving apparatus pair
The influence of the guidance accuracy connect.
Claims (5)
1. wind disturbs UAV Landing bootstrap technique under environment, which is characterized in that the described method includes:
S1, control constraints condition and it is calm under conditions of, solve by unmanned plane from arbitrary initial position transfer to final position
Auxiliary optimal control problem, the landing motion profile of unmanned plane is calculated, it may be assumed that guiding movement track；
The control constraints condition are as follows:0 < η≤1, α (t) indicates the angle of attack of unmanned plane, αMIndicate α (t)
Maximum permissible value, R (t) indicate the electric propeller pulling force of unmanned plane, RMIndicate the maximum permissible value of R (t)；
In the random component allowed band of wind speed, under given wind speed constant value component, η makes the mathematic expectaion of criterion JLess than setting
Fixed permissible value ε；
The criterion J are as follows:
V, θ, x and y respectively indicate the flying speed of unmanned plane, the inclination angle of flight velocity vector, rectangular coordinates system ox0y0In
Unmanned plane mass center x-axis coordinate and y-axis coordinate,WithIt is represented to deckle dividing value, tFIndicate terminal location
At the time of corresponding；
The unmanned plane motion profile and guiding movement track maximum of S2, solution under the conditions of the random component of given wind are approached and are asked
Topic determines that unmanned aerial vehicle (UAV) control is restrained；
The random component of the wind speed is the random function with given statistical property, 0≤σW≤|σW|M, σWIndicate wind speed with
Machine component root mean square, | σW|MIndicate σWMaximum permissible value；
S3, it is restrained using the unmanned aerial vehicle (UAV) control and realizes that UAV Landing guides.
2. wind according to claim 1 disturbs UAV Landing bootstrap technique under environment, which is characterized in that the S1 includes:
In the case where calm and constraint conditionUnder, guarantee in t=tFMoment unmanned plane is along given unit
The displacement slave arbitrary initial position to the position of given edge-restraint condition in the direction vector l=[0 0 sin ξ cos ξ] is maximum,
Determine guide track；ξ indicates vector l and ox0The angle of between centers, when determining guide track, solved using maximum principle single
The direction of bit vector l, and then make criterion J2=-lTz(t0) it is maximum value or criterion J3=lTz(t0)=y (t0)sinξ+x(t0)
Cos ξ is minimum value；
Z (t)=[V, θ, y, x]T, V, θ, x and y respectively indicate the flying speed of unmanned plane, the inclination angle of flight velocity vector, ground
Rectangular coordinate system ox0y0In unmanned plane mass center x-axis coordinate and y-axis coordinate, tFAt the time of indicating that terminal location corresponds to, t0It indicates
At the time of initial position；
Calculate the phase vectors of guide track are as follows:
wT(t)=[wV(t) wθ(t) wy(t) wx(t)]
wV(t)、wθ(t)、wy(t) and wx(t) the corresponding phase vectors of V, θ, x and y are respectively indicated.
3. wind according to claim 2 disturbs UAV Landing bootstrap technique under environment, which is characterized in that the nothing in the S2
The representation of Human-machine Control rule are as follows:
Wherein,Unmanned plane is vertically being put down under the conditions of expression wind is disturbed
Movement in face；XW(t) frontal drag of unmanned plane, Y are indicatedW(t) lift of unmanned plane is indicated.
4. wind according to claim 2 disturbs UAV Landing bootstrap technique under environment, which is characterized in that the nothing in the S2
The representation of Human-machine Control rule are as follows:
Wherein,Unmanned plane is vertically being put down under the conditions of expression wind is disturbed
Movement in face；XW(t) frontal drag of unmanned plane, Y are indicatedW(t) lift of unmanned plane is indicated；
t*According toIt is calculated, wy(t*) indicate that height is nearest on guiding movement track
Point；H indicates to determine the search time interval of height closest approach.
5. wind according to claim 1 disturbs UAV Landing bootstrap technique under environment, which is characterized in that in the S1, wind
The random component allowed band of speed and the acquisition methods of η are as follows:
Meeting 0≤σW≤|σW|M, 0 < ρ≤1 while, single change σWOr η, it recalculatesIt obtainsLess than setting
All σ when permissible value εWWith the corresponding relationship of η, the random component allowed band of wind speed is determined according to corresponding relationship.
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