CN105652664B - A kind of explicit forecast Control Algorithm of quadrotor unmanned plane based on dove group's optimization - Google Patents

A kind of explicit forecast Control Algorithm of quadrotor unmanned plane based on dove group's optimization Download PDF

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CN105652664B
CN105652664B CN201610113345.7A CN201610113345A CN105652664B CN 105652664 B CN105652664 B CN 105652664B CN 201610113345 A CN201610113345 A CN 201610113345A CN 105652664 B CN105652664 B CN 105652664B
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段海滨
陈智隆
邓亦敏
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Beihang University
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    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
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    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance

Abstract

A kind of explicit forecast Control Algorithm of quadrotor unmanned plane based on heuristic bionical dove group optimization, includes the following steps：1st, quadrotor unmanned plane model, and given relation between controlled quentity controlled variable and quantity of state are established according to aerodynamics and kinematics.2nd, dove colony optimization algorithm is initialized, the number D of sample point point is required in given linearisation track.3rd, cost function is designed；4th, optimizing is carried out using the map compass factor；5th, optimizing is carried out using dove group's optimization terrestrial reference operator；6th, the sample object point that dove group optimized is obtained, constructs linear interpolation function；7th, the model that the interpolating function after dove group's optimization is applied to be previously obtained is formed：8th, emulated or tested, obtained correlated results and verify.Control component difficulty can be reduced by this method, overcome explicit PREDICTIVE CONTROL to track requirements height, it is difficult to which the problem of applying, avoids on-line optimization process, saved the plenty of time, reduce control cost.

Description

A kind of explicit forecast Control Algorithm of quadrotor unmanned plane based on dove group's optimization
【Technical field】
The present invention is a kind of explicit forecast Control Algorithm of quadrotor unmanned plane based on heuristic bionical dove group optimization, is belonged to Flying vehicles control field.
【Background technology】
Unmanned plane (Unmanned Aerial Vehicle, UAV) plays the part of more and more important angle in current military operation Color, can efficiently accomplish complicated and dangerous investigation and combat duty.Extensively should although UAV has in fields such as military and civilians With, but many key technologies of its own still need further research and application.And quadrotor is as a kind of less expensive reliability Higher, can be with a kind of unmanned plane of VTOL, just by extensive business and Military application, yet with the various dimensions of quadrotor Property, being taken off to hover by four motor controls turns to, therefore it is higher to control rate requirement.
Model Predictive Control can realize the accurate control to controlled device, but some Model Predictive Controls rely on online Optimization process, than relatively time-consuming.For some Linear Model for Prediction control problems, the explicit of its optimum control can be asked for Solution, but for nonlinear problem, it is relatively difficult to ask for explicit solution.Show Nonlinear Model Predictive Control (Explicit Nonlinear Model Predictive Control, ENMPC) Taylor expansion principle is applied, nonlinear model can be asked for The explicit solution of PREDICTIVE CONTROL.Its major control thought is requirement track --- backstepping PREDICTIVE CONTROL amount --- output trajectory.And this Patent application will simplify prediction process by simplifying track, so as to substantially reduce component difficulty.But its own have it is certain Complexity, and can not carry out derivation operation for complicated track and scatterplot track.So we can pass through interpolation method The linear track for asking it to estimate.
Swarm intelligence is an important branch of bionic intelligence, and people are subject to by the observation to nature biotechnology colony The inspiration of biocenose behavior in nature, general increase, its behavior pattern is depicted with the mode of mathematics on this basis Come.On the basis of Swarm Intelligent Model, there has been proposed the concept of Swarm Intelligent Algorithm, with the behavior mould of biocenose Formula carrys out solving-optimizing problem.Heuristic bionical dove colony optimization algorithm (Pigeon Inspired Optimization, PIO) is A kind of new heuristic colony intelligence that HaibinDuan and PeixinQiao was proposed in 2014 optimizes algorithm, which is subject to The inspiration of pigeon group behavior, according to pigeon during target is found, successively according to the row of magnetic field and terrestrial reference as instruction With the characteristics of, it is established that two kinds of algorithm mechanism of map compass and terrestrial reference.
Dove group can carry out Primary Location, then according to terrestrial reference during destination is found with initial reference to the sun and magnetic field It is accurately positioned, according to this characteristic, dove colony optimization algorithm proposes two kinds of corresponding operators, is respectively that map compass is calculated Both operators to simulate this characteristic of dove group, and are combined solution optimization problem by son and terrestrial reference mechanism.
Traditional control mode is mainly by the way that model linearization is controlled, and explicit PREDICTIVE CONTROL can be realized The control of nonlinear model, but complexity is higher, and this method keeps model as former state path line will be required to realize Control, substantially reduces control forecasting model cootrol difficulty, and maintains the equation of model well.
(1) Nonlinear Explicit PREDICTIVE CONTROL
It is by by non-linear partial Taylor expansion, being carried out with look-ahead required amount in Nonlinear Explicit prediction Control, its main thought are as follows：
For a nonlinear model：
Wherein, x represents quantity of state, and y represents output quantity, and f (x, u), h (x) are two functions.
If ρiRepresent and occur the number of i-th of controlled quentity controlled variable in state equation by differential, the sequence number of input is represented with r. Carrying out Taylor expansion respectively to each variable of above formula can obtain
τ is the predicted time in PREDICTIVE CONTROL,It is i-th of estimator, t is current time, and ρ i measure for i-th of control The exponent number of Taylor expansion is needed in present status, r is continuing deployment of opposite exponent number.For the coefficient matrix of Taylor expansion, yi (t) for i-th of controlled quentity controlled variable in the value of t moment, Yi(t) it is by yi(t) and its all-order derivative composition matrix.
By multiple variables and in a matrix, then above formula, which arranges, is：
WhereinFor the Taylor expansion coefficient of i-th of amount.
There to be controlled quentity controlled variable and without controlled quentity controlled variable state variable to separate, rearrange matrix, following formula can be obtained：
Wherein symbol definition is as follows：
And whereinOccur the part of controlled quentity controlled variable after more subdifferentials, can be opened up Open as follows：
WhereinFor to i-th of ρ i subdifferential without input control quantity, A (x) is measured in order to control arrives quantity of state Function, u inputs in order to control.I=1,2 ..., n；
Above formula is subjected to differential to the time, can be obtained as follows：
Wherein p1(x, u) is the input of non-linear partial.
It is as follows that differential is repeated with this method, until there is controlled quentity controlled variable：
Same method, can be predicted output quantity ω (t), and by the portion containing controlled quentity controlled variable amount and without controlled quentity controlled variable Separation,The matrix formed for reality output amount：
Similarly, controlled quentity controlled variable u is also subjected to Taylor expansion, can then obtains the predicted value of controlled quentity controlled variable u
Can evaluated error, wherein J be with the following method error criterion, Q is certainly close to w to make output y The matrix of fixed each quantity of state Error weight,To estimate output valve, w is real output value：
In theory when being optimal, condition is：
Then the predicted value of controlled quentity controlled variable can be obtained
Error requirements matrix Q is determined wherein K again, Mρ,MiBy defined below：
WhereinThe matrix formed for estimation output quantity.
(2) heuristic bionical dove colony optimization algorithm
The first step：Pass through map compass operator optimizing
Given dove group quantity Np, maximum iteration t1max, and map compass operator R, evolvement method can be by such as Under provide：
Wherein xgIt is the dove that cost function fitness is maximum or minimum in current iteration for current globally optimal solution Group position, vi(t) it is speed of i-th of pigeon in the t times iteration, xi(t) it is position of i-th of pigeon in the t times iteration Put, rand is the random number between one 0 to 1.
With such a method iteration, the speed of the dove group after iteration t1imax is walked, and part and global and local are obtained Optimal solution.
Second step：Pass through terrestrial reference operator optimizing
Since pigeon is finding the later stage of destination, what is relied primarily on is terrestrial reference to carry out the guiding of target, for this basis Its behavioral trait proposes terrestrial reference operator.Change so result of the top Jing Guo map compass operator optimizing is continued through and changes method Generation.The operator provides that dove group's number per a generation halves, and in order to arrive at faster, remaining pigeon directly flies to mesh Ground.Specific replacement criteria is shown below：
Xi(t)=Xi(t-1)+rand·(Xc(t)-Xi(t-1)) (20)
In above formula, NpFor the number of dove group, fitness is the cost function of pigeon positional information, in order to try to achieve cost Functional minimum value, can take fminAs object function, XcIt is the weighting place-centric of dove group.By this kind of method, due to a Exponentially type declines and chooses an optimal part every time group's quantity, it is possible to quickly find optimal value.
Dove colony optimization algorithm schematic diagram is as shown in figure 3, the speed of next iteration is closed by present speed and object velocity vector Into.
【The content of the invention】
1st, goal of the invention：
The present invention proposes a kind of explicit forecast Control Algorithm of quadrotor unmanned plane based on heuristic bionical dove group optimization, The purpose is to provide one kind to have higher reliability quadrotor unmanned aerial vehicle (UAV) control method.
This method using dove group's algorithm to track into row interpolation, and be applied to explicit non-linear Model Predictive Control, can be with Overcome the problem of Nonlinear Model Predictive Control is high to track requirements, and complicated track is difficult to application, it is pre- to avoid nonlinear model The problem of multiderivative is sought in observing and controlling system, saves control cost.
2nd, technical solution：
The step of quadrotor unmanned plane based on heuristic bionical dove group optimization explicit forecast Control Algorithm, is as follows：
Step 1：Quadrotor unmanned plane model, and given controlled quentity controlled variable and state are established according to aerodynamics and kinematics Relation between amount.And to the control track of provisioning request.It is straight to the quantity of state progress differential of Controlling model (quadrotor unmanned plane) To there is the part of controlled quentity controlled variable, and controlled quentity controlled variable is solved according to that first differential for occurring controlled quentity controlled variable in state equation is counter, obtain correlation The module of backstepping amount.The input of micro component and quantity of state as anti-solution controlled quentity controlled variable module is predicted in desired track, The anti-controlled quentity controlled variable solved is applied to master mould, establishes and shows predictive control model.Formed
Predict micro component --- the anti-controlled quentity controlled variable that solves --- controlled quentity controlled variable is applied to master mould --- model of result.
Step 2：Beginningization dove colony optimization algorithm is given to linearize the number D that sample point point is required in track, the number of point The dimension of pigeon position as in dove colony optimization algorithm.
Determine dove group quantity Np, maximum iteration t1max, and map compass operator R.
Step 3：Design cost function
Cost function it is definite be intelligent optimization algorithm core, determine the accuracy of target detection.In this method, use Deviation accumulation method, that is, require the summation of functional value absolute difference between track and the multiple spot of error locus.
Step 4：Optimizing is carried out using the map compass factor
Using the group position and speed of initialization, global optimum position is chosen according to initial individual cost function value Xg.The position X of each individual of formula renewal in formula (17)i, the cost function value of newly-generated pigeon is calculated, if newly The cost function value of pigeon is lower than the cost function value of global optimum position, then newly-generated pigeon position is defined as new Global optimum position Xg.Optimizing is carried out using map compass operator repeatedly, until operation algebraically is more than map compass operator maximum Stop during algebraically t1max.
Step 5：Optimizing is carried out using dove group's optimization terrestrial reference operator
By the use of map compass operator optimizing result as terrestrial reference operator initial population, according in formula (18)~(20) The each individual of formula renewal speed ViWith position Xi, the cost function value of newly-generated pigeon is calculated, if the cost of new pigeon Functional value is lower than the cost function value of global optimum position, then newly-generated pigeon position is defined as new global optimum position Put Xg.The Population of new population is calculated according to formula (18), cost letter in colony is given up according to the result that formula (18) calculates Number small portion is individual, and preferably colony carries out next round optimizing as colony is retained in selection current group, answers repeatedly Optimizing is carried out with terrestrial reference operator, is stopped when running algebraically and being more than terrestrial reference operator maximum algebraically t2max.
Step 6：The sample object point that dove group optimized is obtained, constructs linear interpolation function.
Step 7：The model that interpolating function after dove group's optimization is applied to be previously obtained is formed：
Track --- prediction locus --- backstepping amount --- being input to model --- result after dove group's optimization
Such structural model
Step 8：Emulated or tested, obtained correlated results and verify
3rd, advantage and effect：
The present invention proposes a kind of explicit forecast Control Algorithm of quadrotor unmanned plane based on heuristic bionical dove group optimization. It by heuristic bionical dove colony optimization algorithm will require that through row linearisation control component difficulty can be reduced, overcome explicit prediction The problem of control is high to track requirements, and complicated track is difficult to application, avoids on-line optimization process, has saved the plenty of time, reduces Control cost, can be very good to be applied to quadrotor or more rotor control fields.
【Brief description of the drawings】
Fig. 1 is quadrotor unmanned plane schematic diagram.
Fig. 2 is explicit PREDICTIVE CONTROL schematic diagram.
Fig. 3 is dove colony optimization algorithm schematic diagram.
Fig. 4 shows for experiment quadrotor simulates schematic diagram with PREDICTIVE CONTROL.
Fig. 5 is dove group's Optimal Fitting curve.
Fig. 6 is conventional equidistant interpolation method matched curve.
Fig. 7 is error convergence curve.
Fig. 8 is flow chart of the present invention.
Fig. 9 contrasts for final reality output track and track after dove group's optimization.
Figure 10 is final reality output track with initially requiring track to contrast.
Figure label and symbol description are as follows：
F-lift；φ, θ, Ψ-unmanned plane angular speed；Feedback matrix in M-explicit PREDICTIVE CONTROL；
Proportionality coefficient in K-explicit PREDICTIVE CONTROL, is determined by error requirements；U-controlled quentity controlled variable；
-A(x)-1- pass through the matrix of required amount backstepping amount；
【Embodiment】
Having for design method proposed by the invention is verified below by a specific unmanned plane target detection example Effect property.This experimental calculation machine is configured to i7-4710MQ processors, and 2.50Ghz dominant frequency, 4G memories, software is MATLAB 2014a Version.
The specific implementation step of this example is as follows：
Step 1：Quadrotor unmanned plane model and controlled quentity controlled variable are established, gives provisioning request track, and establishes display PREDICTIVE CONTROL mould Type.Quadrotor unmanned plane simplification figure such as attached drawing 1, body coordinate system xyz and earth axes XYZ defined in figure, Eulerian angles definition Such as attached drawing 2.
Wherein Eulerian angles are defined as follows：
Yaw angle Ψ：Ox is projected and the angle of X-axis in plane OXY
Pitching angle theta：Oz is projected and the angle of Z axis in plane OXZ
Roll angle φ：Oy is projected and the angle of Y-axis in plane OYZ
It is hereby achieved that transfer matrix of the body coordinate system to earth axes：
Rx, Ry, RzRespectively with respect to x, the rotational transform of y, z.
For simplified model, hypothesis below is done：
1. quadrotor unmanned plane is a rigid body symmetrically.
2. initial body origin is at unmanned plane center.
3. air drag is not with unmanned plane height change.
Therefore equation below can be established according to Newton's second law：
Wherein F is unmanned plane institute stress, and m is unmanned plane quality, and V is unmanned plane speed, and t is the time, and M rotates for unmanned plane Inertia, H are the moment of momentum suffered by unmanned plane.
The equation of Specific amounts can further be obtained：
G is gravity, and g is acceleration of gravity, DiFor the resistance of each rotor, TiFor the lift of each rotor.DdHindered for air Force coefficient, DtFor air lift coefficient.ωiFor engine speed.
Above-mentioned all formulas are arranged, the state equation of position and the state equation of angle can be obtained：
Wherein Ix,Iy,IzFor the rotary inertia of three axis, x, y, z is coordinate position,For three directional accelerations, P, q, r are three directional angular velocities, Fx,Fy,FzFor x, y, the bonding force in z directions, ktFor lift coefficient, k1,k2,k3For resistance system Number Mx,My,MzIt is able to can be obtained by following formula：
For the ease of control, controlled quentity controlled variable is defined as follows：
Wherein U1For altitude channel controlled quentity controlled variable, U2For roll channel controlled quentity controlled variable, U3For pitch channel controlled quentity controlled variable, U4For yaw Passage controlled quentity controlled variable.ωiRepresent motor rotary speed, FiFor the lift of each engine.
At low speeds, air drag can be ignored, and assumed：
Then nonlinear model and the backstepping mode that unmanned plane can be obtained are as follows：
I in this examplex=Iy=0.8kg.m2Iz=1kg.m2L=0.15mm=1kg.
Data are substituted into obtain unmanned plane model and inverse equation：
Since controlled quentity controlled variable is appeared in second dervative, so needing prediction second dervative is counter to solve controlled quentity controlled variable, thus may be used Prediction model is shown to be established according to display forecast Control Algorithm, and completes to build in simulink.And due to method in Since track is linear track after optimization, second dervative input is 0.Block mold attached drawing, Fig. 2 show for explicit PREDICTIVE CONTROL It is intended to, Fig. 4 schemes for overall realize.Only to control angle in this exampleExemplified by.
Step 2：Dove colony optimization algorithm is initialized, gives the number of linearity requirements point, the number of point is dove group's optimization The dimension of pigeon position in algorithm.
(1) Optimal Parameters dimension D is initialized
Eight optimum points (non-endpoint) will be found in this example through row linear interpolation, so D is 8.
(2) population quantity Np is initialized
The population quantity Np of colony intelligence optimization algorithm influences effect of optimization very big, one appropriate population quantity of selection, Ensure the accuracy and rapidity of algorithm.It is 100 that this method, which removes Np,.
(4) population position and speed are initialized
The position upper limit P for not setting colony in search spacemax=100 and position lower limit Pmin=0, to each in population Particle all initializes an initial position XiWith initial speed Vi.The upper limit is the section upper limit of function in this example, and lower limit is For the interval limit of function.Wherein rand is 0 to 1 random number.
Vi(t)=rand
xi(t)=rand (100-0) (29)
(5) algorithmic algebra is set
Dove colony optimization algorithm has two operators, is map compass operator and terrestrial reference operator respectively, is needed before algorithm computing point Not She Ding the operation of two algorithms maximum algebraically NC1=50 and NC2=8.
Step 3：Design cost function
Cost function it is definite be intelligent optimization algorithm core, determine the accuracy of target detection.In this method, use Deviation accumulation method, that is, require the summation of functional value absolute difference between track and the multiple spot of error locus.
Because explicit PREDICTIVE CONTROL is higher to pattern function requirement, therefore considers to linearize track, on function segment D point is taken, respectively linearizes track on every section.
A section [a, b] is given, x will be selectediAs sample point, wherein xi≠a,b.(i=1,2,3 ..N)
It will select section x ∈ [xi,xi+1] or x ∈ [a, x1] or x ∈ [xn, b], and define estimate such as Under：
Draw is taken into n error sample point afterwards, distance d is defined as follows between error sampling point：
Then each error sampling point can be obtained so：XXi=a+id
Then accumulated error can be obtained：
Error sampling points are 100 in this example, so n=100.
And cumulative errors E is cost function fitness.
Step 4：Optimizing is carried out using the map compass factor
Using the group position and speed of initialization, global optimum position is chosen according to initial individual cost function value Xg.The position X of each individual of formula renewal in formula (17)i, the cost function value of newly-generated pigeon is calculated, if newly The cost function value of pigeon is lower than the cost function value of global optimum position, then newly-generated pigeon position is defined as new Global optimum position Xg.Optimizing is carried out using map compass operator repeatedly, until operation algebraically is more than map compass operator maximum Stop during algebraically t1max.
Iterative formula is as follows：R=0.2 in this example, brings into as follows
vi(t)=vi(t-1)·e-Rt+rand·(xg-xi(t-1)) (30)
xi(t)=xi(t-1)+vi(t)
R=0.2 in this example, brings into as follows
Vi(t)=Vi(t-1)·e-0.2*t+rand·(xg-xi(t-1)) (31)
xi(t)=xi(t-1)+Vi(t)
Step 5：Optimizing is carried out using dove group's terrestrial reference operator
By the use of map compass operator optimizing result as terrestrial reference operator initial population, according in formula (18)~(20) The each individual of formula renewal speed ViWith position Xi, the cost function value of newly-generated pigeon is calculated, if the cost of new pigeon Functional value is lower than the cost function value of global optimum position, then newly-generated pigeon position is defined as new global optimum position Put Xg.The Population of new population is calculated according to formula (18), cost letter in colony is given up according to the result that formula (18) calculates Number small portion is individual, and preferably colony carries out next round optimizing as colony is retained in selection current group, answers repeatedly Optimizing is carried out with terrestrial reference operator, is stopped when running algebraically and being more than terrestrial reference operator maximum algebraically t2max.
Step 6：D optimal interpolation sample point is obtained, and linear interpolation is established according to the functional value of these points, this is inserted Value is the track applied to explicit PREDICTIVE CONTROL.
Step 7：The linear track that dove group optimized is used to show PREDICTIVE CONTROL, is constructed
Dove group's optimization optimum results --- prediction locus --- backstepping amount --- result
Model, such as attached drawing 4.
Step 8：Final result is obtained, is verified, final result is as shown in Figure 10
By the contrast of attached drawing 5 and attached drawing 6, it will be seen that compared with common equidistant interpolation, by dove, group optimizes The non-linear interpolation of progress can preferably restore original function, and the accumulated error with smaller.And it can be seen that dove from attached drawing 7 Group optimization there is good convergence property, can be quickly obtain optimal value and make error very little.
From attached drawing 10, in this way, our last output trajectories can be very good to track original complicated track, And this method avoids track fluctuating seriously by simplifying track, and reduce control cost.By dove, group optimizes into line The method of property interpolation can also be applied to scatterplot track well, so as to fulfill the control to discrete point track requirements, Ke Yiyong In avoidance, path planning, avoids the application of radar monitoring etc..

Claims (5)

  1. A kind of 1. explicit forecast Control Algorithm of quadrotor unmanned plane based on dove group's optimization, it is characterised in that：The control method Step is as follows：
    Step 1：Quadrotor unmanned plane model is established according to aerodynamics and kinematics, and given controlled quentity controlled variable and quantity of state it Between relation；And to the control track of provisioning request；Differential is carried out until there is controlled quentity controlled variable to the quantity of state of quadrotor unmanned plane Part, and controlled quentity controlled variable is solved according to that first differential for occurring controlled quentity controlled variable in state equation is counter, obtain the mould of related backstepping amount Block；Using the prediction micro component and quantity of state of desired track as the anti-input for solving controlled quentity controlled variable module, the anti-controlled quentity controlled variable solved Applied to master mould, explicit prediction quadrotor unmanned plane is established；Form prediction --- counter the to solve controlled quentity controlled variable --- controlled quentity controlled variable of micro component Applied to master mould --- the model of result；
    Step 2：Dove colony optimization algorithm is initialized, the number D of sample point point is required in given linearisation track, the number of point is For the dimension of pigeon position in dove colony optimization algorithm；Given dove group quantity Np, maximum iteration t1max, and map compass Operator R；
    Step 3：Design cost function
    Cost function it is definite be intelligent optimization algorithm core, determine the accuracy of target detection；Using deviation accumulation method, i.e., It is required that between track and the multiple spot of error locus functional value absolute difference summation；
    Step 4：Optimizing is carried out using the map compass factor
    Using the group position and speed of initialization, global optimum position X is chosen according to initial individual cost function valueg；Root According to the position X of each individual of formula renewal in formula (1)i, the cost function value of newly-generated pigeon is calculated, if new pigeon Cost function value is lower than the cost function value of global optimum position, then newly-generated pigeon position is defined as new global optimum Position Xg；Optimizing is carried out using map compass operator repeatedly, until operation algebraically is more than map compass operator maximum algebraically t1max When stop；
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    Wherein xgIt is the dove group position that cost function fitness is maximum or minimum in current iteration for current globally optimal solution Put, vi(t) it is speed of i-th of pigeon in the t times iteration, xi(t) it is position of i-th of pigeon in the t times iteration, Rand is the random number between one 0 to 1；
    Step 5：Optimizing is carried out using dove group's optimization terrestrial reference operator
    By the use of map compass operator optimizing result as terrestrial reference operator initial population, according to the formula in formula (2)~(4) The speed V of each individual of renewaliWith position Xi, the cost function value of newly-generated pigeon is calculated, if the cost function value of new pigeon It is lower than the cost function value of global optimum position, then newly-generated pigeon position is defined as new global optimum position Xg； The Population of new population is calculated according to formula (2), small one of cost function in colony is given up according to the result that formula (2) calculates Some individuals, select preferably colony's conduct reservation colony progress next round optimizing in current group, repeatedly using terrestrial reference operator Optimizing is carried out, is stopped when running algebraically and being more than terrestrial reference operator maximum algebraically t2max；
    <mrow> <msub> <mi>N</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>N</mi> <mi>p</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <msub> <mi>X</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>&amp;Sigma;X</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>f</mi> <mi>i</mi> <mi>t</mi> <mi>n</mi> <mi>e</mi> <mi>s</mi> <mi>s</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>N</mi> <mi>p</mi> </msub> <mo>&amp;CenterDot;</mo> <mi>&amp;Sigma;</mi> <mi>f</mi> <mi>i</mi> <mi>t</mi> <mi>n</mi> <mi>e</mi> <mi>s</mi> <mi>s</mi> <mrow> <mo>(</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
    Xi(t)=Xi(t-1)+rand·(Xc(t)-Xi(t-1)) (4)
    In above formula, NpFor the number of dove group, fitness is the cost function of pigeon positional information, in order to try to achieve cost function Minimum value, takes fminAs object function, XcIt is the weighting place-centric of dove group；Due to a group of quantity exponentially type decline and Choose an optimal part every time, thus quickly find optimal value；
    Step 6：The sample object point that dove group optimized is obtained, constructs linear interpolation function；
    Step 7：The model that interpolating function after dove group's optimization is applied to be previously obtained is formed：
    Track --- prediction locus --- backstepping amount --- being input to model --- result after dove group's optimization；
    Step 8：Emulated or tested, obtained correlated results and verify；
    For simplified model, hypothesis below is done：
    (a) quadrotors unmanned plane is a rigid body symmetrically；
    (b) body origin initial is at unmanned plane center；
    (c) air drag is not with unmanned plane height change；
    Therefore equation below is established according to Newton's second law：
    <mrow> <mtable> <mtr> <mtd> <mrow> <mi>F</mi> <mo>=</mo> <mi>m</mi> <mfrac> <mrow> <mi>d</mi> <mi>V</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow> <mi>M</mi> <mo>=</mo> <mfrac> <mrow> <mi>d</mi> <mi>H</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
    Wherein, F is unmanned plane institute stress, and m is unmanned plane quality, and V is unmanned plane speed, and t is the time, and M rotates used for unmanned plane Amount, H is the moment of momentum suffered by unmanned plane；
    Further obtain the equation of Specific amounts：
    <mrow> <mtable> <mtr> <mtd> <mrow> <mi>G</mi> <mo>=</mo> <mi>m</mi> <mi>g</mi> </mrow> </mtd> <mtd> <mrow> <msub> <mi>D</mi> <mi>i</mi> </msub> <mo>=</mo> <msub> <mi>C</mi> <mi>d</mi> </msub> <msubsup> <mi>&amp;omega;</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mo>/</mo> <mn>2</mn> <mo>=</mo> <msub> <mi>D</mi> <mi>d</mi> </msub> <msubsup> <mi>&amp;omega;</mi> <mi>i</mi> <mn>2</mn> </msubsup> </mrow> </mtd> <mtd> <mrow> <msub> <mi>T</mi> <mi>i</mi> </msub> <mo>=</mo> <msub> <mi>&amp;rho;C</mi> <mi>t</mi> </msub> <msubsup> <mi>&amp;omega;</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mo>/</mo> <mn>2</mn> <mo>=</mo> <msub> <mi>D</mi> <mi>t</mi> </msub> <msubsup> <mi>&amp;omega;</mi> <mi>i</mi> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
    G is gravity, and g is acceleration of gravity, DiFor the resistance of each rotor, TiFor the lift of each rotor；DdFor air drag system Number, DtFor air lift coefficient；ωiFor engine speed；
    Arrange the state equation for obtaining position and the state equation of angle：
    <mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>F</mi> <mi>x</mi> </msub> <mo>-</mo> <msub> <mi>K</mi> <mn>1</mn> </msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>/</mo> <mi>m</mi> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mi>t</mi> </msub> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>4</mn> </munderover> <msubsup> <mi>&amp;omega;</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mo>(</mo> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;psi;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;theta;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;phi;</mi> <mo>+</mo> <mi>sin</mi> <mi>&amp;psi;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;phi;</mi> </mrow> <mo>)</mo> <mo>-</mo> <msub> <mi>K</mi> <mn>1</mn> </msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>/</mo> <mi>m</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>y</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>F</mi> <mi>y</mi> </msub> <mo>-</mo> <msub> <mi>K</mi> <mn>2</mn> </msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>/</mo> <mi>m</mi> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mi>t</mi> </msub> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>4</mn> </munderover> <msubsup> <mi>&amp;omega;</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mo>(</mo> <mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;psi;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;phi;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;phi;</mi> <mo>+</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;psi;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;phi;</mi> </mrow> <mo>)</mo> <mo>-</mo> <msub> <mi>K</mi> <mn>2</mn> </msub> <mover> <mi>y</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>/</mo> <mi>m</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>z</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>F</mi> <mi>z</mi> </msub> <mo>-</mo> <msub> <mi>K</mi> <mn>3</mn> </msub> <mover> <mi>z</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>-</mo> <mi>m</mi> <mi>g</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>m</mi> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mi>t</mi> </msub> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>4</mn> </munderover> <msubsup> <mi>&amp;omega;</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mo>(</mo> <mrow> <mi>cos</mi> <mi>&amp;phi;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;theta;</mi> </mrow> <mo>)</mo> <mo>-</mo> <msub> <mi>K</mi> <mn>3</mn> </msub> <mover> <mi>z</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>/</mo> <mi>m</mi> <mo>-</mo> <mi>g</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mover> <mi>p</mi> <mo>&amp;CenterDot;</mo> </mover> </mtd> </mtr> <mtr> <mtd> <mover> <mi>q</mi> <mo>&amp;CenterDot;</mo> </mover> </mtd> </mtr> <mtr> <mtd> <mover> <mi>r</mi> <mo>&amp;CenterDot;</mo> </mover> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mi>x</mi> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mi>x</mi> </msub> <mo>-</mo> <msub> <mi>I</mi> <mi>z</mi> </msub> </mrow> <mo>)</mo> </mrow> <mi>q</mi> <mi>r</mi> <mo>/</mo> <msub> <mi>I</mi> <mi>x</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mi>y</mi> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mi>z</mi> </msub> <mo>-</mo> <msub> <mi>I</mi> <mi>x</mi> </msub> </mrow> <mo>)</mo> </mrow> <mi>r</mi> <mi>p</mi> <mo>/</mo> <msub> <mi>I</mi> <mi>y</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>M</mi> <mi>z</mi> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mi>x</mi> </msub> <mo>-</mo> <msub> <mi>I</mi> <mi>y</mi> </msub> </mrow> <mo>)</mo> </mrow> <mi>p</mi> <mi>q</mi> <mo>/</mo> <msub> <mi>I</mi> <mi>z</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
    Wherein Ix,Iy,IzFor the rotary inertia of three axis, x, y, z is coordinate position,For three directional accelerations, p, q, r For three directional angular velocities, Fx,Fy,FzFor x, y, the bonding force in z directions, ktFor lift coefficient, k1,k2,k3For resistance coefficient Mx, My,MzObtained by following formula：
    <mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>M</mi> <mi>x</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>M</mi> <mi>y</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>M</mi> <mi>z</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>F</mi> <mn>4</mn> </msub> <mo>-</mo> <msub> <mi>F</mi> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> <mi>l</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>F</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>F</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> </mrow> <mi>l</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>F</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>F</mi> <mn>4</mn> </msub> <mo>-</mo> <msub> <mi>F</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>F</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> </mrow> <mi>l</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>U</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>U</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>U</mi> <mn>3</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mi>l</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
    For the ease of control, controlled quentity controlled variable is defined as follows：
    <mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>U</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>U</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>U</mi> <mn>3</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>U</mi> <mn>4</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>F</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>F</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>F</mi> <mn>3</mn> </msub> <mo>+</mo> <msub> <mi>F</mi> <mn>4</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>F</mi> <mn>4</mn> </msub> <mo>-</mo> <msub> <mi>F</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>F</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>F</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>F</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>F</mi> <mn>4</mn> </msub> <mo>-</mo> <msub> <mi>F</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>F</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mi>t</mi> </msub> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>4</mn> </munderover> <msubsup> <mi>&amp;omega;</mi> <mi>i</mi> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mi>t</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mi>&amp;omega;</mi> <mn>4</mn> <mn>2</mn> </msubsup> <mo>-</mo> <msubsup> <mi>&amp;omega;</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mi>t</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mi>&amp;omega;</mi> <mn>3</mn> <mn>2</mn> </msubsup> <mo>-</mo> <msubsup> <mi>&amp;omega;</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mi>t</mi> </msub> <mrow> <mo>(</mo> <msubsup> <mi>&amp;omega;</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>-</mo> <msubsup> <mi>&amp;omega;</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;omega;</mi> <mn>3</mn> <mn>2</mn> </msubsup> <mo>-</mo> <msubsup> <mi>&amp;omega;</mi> <mn>4</mn> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
    Wherein U1For altitude channel controlled quentity controlled variable, U2For roll channel controlled quentity controlled variable, U3For pitch channel controlled quentity controlled variable, U4For jaw channel control Amount processed；ωiRepresent motor rotary speed, FiFor the lift of each engine；
    At low speeds, ignore air drag, and assume：
    <mrow> <mover> <mi>&amp;phi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mi>p</mi> <mover> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mi>q</mi> <mover> <mi>&amp;psi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mi>r</mi> </mrow>
    Then nonlinear model and the backstepping mode for obtaining unmanned plane are as follows：
    <mrow> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>=</mo> <mo>(</mo> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;psi;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;theta;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;phi;</mi> <mo>+</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;psi;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;phi;</mi> </mrow> <mo>)</mo> <msub> <mi>U</mi> <mn>1</mn> </msub> <mo>)</mo> <mo>/</mo> <mi>m</mi> </mrow>
    <mrow> <mover> <mi>y</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;psi;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;phi;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;phi;</mi> <mo>+</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;psi;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;phi;</mi> <mo>)</mo> </mrow> <msub> <mi>U</mi> <mn>1</mn> </msub> <mo>/</mo> <mi>m</mi> </mrow>
    <mrow> <mover> <mi>z</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;phi;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;theta;</mi> <mo>)</mo> </mrow> <msub> <mi>U</mi> <mn>1</mn> </msub> <mo>/</mo> <mi>m</mi> <mo>-</mo> <mi>g</mi> </mrow>
    <mrow> <mover> <mi>&amp;phi;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>=</mo> <mo>&amp;lsqb;</mo> <msub> <mi>lU</mi> <mn>2</mn> </msub> <mo>+</mo> <mover> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mover> <mi>&amp;psi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mi>y</mi> </msub> <mo>-</mo> <msub> <mi>I</mi> <mi>z</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>/</mo> <msub> <mi>I</mi> <mi>z</mi> </msub> </mrow>
    <mrow> <mover> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>=</mo> <mo>&amp;lsqb;</mo> <msub> <mi>lU</mi> <mn>3</mn> </msub> <mo>+</mo> <mover> <mi>&amp;phi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mover> <mi>&amp;psi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mi>z</mi> </msub> <mo>-</mo> <msub> <mi>I</mi> <mi>x</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>/</mo> <msub> <mi>I</mi> <mi>y</mi> </msub> </mrow>
    <mrow> <mover> <mi>&amp;psi;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>=</mo> <mo>&amp;lsqb;</mo> <msub> <mi>lU</mi> <mn>4</mn> </msub> <mo>+</mo> <mover> <mi>&amp;phi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mover> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mi>x</mi> </msub> <mo>-</mo> <msub> <mi>I</mi> <mi>y</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>/</mo> <msub> <mi>I</mi> <mi>z</mi> </msub> </mrow>
    Ix=Iy=0.8kg.m2Iz=1kg.m2L=0.15m m=1kg；
    Data are substituted into obtain unmanned plane model and inverse equation：
    <mrow> <mover> <mi>&amp;phi;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>=</mo> <mo>&amp;lsqb;</mo> <mn>0.15</mn> <mo>*</mo> <msub> <mi>U</mi> <mn>2</mn> </msub> <mo>+</mo> <mover> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mover> <mi>&amp;psi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mn>0.8</mn> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>/</mo> <mn>0.8</mn> </mrow>
    <mrow> <mover> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>=</mo> <mo>&amp;lsqb;</mo> <mn>0.15</mn> <mo>*</mo> <msub> <mi>U</mi> <mn>3</mn> </msub> <mo>+</mo> <mover> <mi>&amp;phi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mover> <mi>&amp;psi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mn>0.8</mn> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>/</mo> <mn>0.8</mn> </mrow>
    <mrow> <mover> <mi>&amp;psi;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>=</mo> <mo>&amp;lsqb;</mo> <mn>0.15</mn> <mo>*</mo> <msub> <mi>U</mi> <mn>4</mn> </msub> <mo>+</mo> <mover> <mi>&amp;phi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mover> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mn>0.8</mn> <mo>-</mo> <mn>0.8</mn> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>/</mo> <mn>1</mn> </mrow>
    <mrow> <msub> <mi>U</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mover> <mi>&amp;phi;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>*</mo> <mn>0.8</mn> <mo>-</mo> <mover> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mover> <mi>&amp;psi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mn>0.8</mn> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> <mn>0.15</mn> </mfrac> </mrow>
    <mrow> <msub> <mi>U</mi> <mn>3</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mover> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>*</mo> <mn>0.8</mn> <mo>-</mo> <mover> <mi>&amp;phi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mover> <mi>&amp;psi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mn>0.8</mn> <mo>)</mo> </mrow> </mrow> <mn>0.15</mn> </mfrac> </mrow>
    <mrow> <msub> <mi>U</mi> <mn>4</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mover> <mi>&amp;psi;</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>*</mo> <mn>1</mn> <mo>-</mo> <mover> <mi>&amp;theta;</mi> <mo>&amp;CenterDot;</mo> </mover> <mover> <mi>&amp;phi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mn>0.8</mn> <mo>-</mo> <mn>0.8</mn> <mo>)</mo> </mrow> </mrow> <mn>0.15</mn> </mfrac> </mrow>
    Since controlled quentity controlled variable is appeared in second dervative, so needing prediction second dervative is counter to solve controlled quentity controlled variable, thus according to aobvious Formula forecast Control Algorithm establishes explicit prediction model, and completes to build in simulink；
    Since track is linear track after optimization, second dervative input is 0.
  2. 2. a kind of explicit forecast Control Algorithm of quadrotor unmanned plane based on dove group's optimization according to claim 1, it is special Sign is：In step 1, quadrotor unmanned plane body coordinate system xyz and earth axes XYZ is defined；Eulerian angles define such as Under：
    Yaw angle ψ：Ox is projected and the angle of X-axis in plane OXY；
    Pitching angle theta：Oz is projected and the angle of Z axis in plane OXZ；
    Roll angle φ：Oy is projected and the angle of Y-axis in plane OYZ；
    Body coordinate system is obtained to the transfer matrix of earth axes：
    <mrow> <mi>R</mi> <mo>=</mo> <msub> <mi>R</mi> <mi>x</mi> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>R</mi> <mi>y</mi> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>R</mi> <mi>z</mi> </msub> <mo>=</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;psi;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;phi;</mi> </mrow> </mtd> <mtd> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;psi;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;theta;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;phi;</mi> </mrow> </mtd> <mtd> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;psi;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;theta;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;phi;</mi> <mo>+</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;psi;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;phi;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>sin</mi> <mi>&amp;psi;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;theta;</mi> </mrow> </mtd> <mtd> <mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;psi;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;theta;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;phi;</mi> </mrow> </mtd> <mtd> <mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;psi;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;theta;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;phi;</mi> <mo>-</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;phi;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;psi;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mi>sin</mi> <mi>&amp;theta;</mi> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi>&amp;theta;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;phi;</mi> </mrow> </mtd> <mtd> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;theta;</mi> <mi>cos</mi> <mi>&amp;phi;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
    Rx, Ry, RzRespectively with respect to x, the rotational transform of y, z.
  3. 3. a kind of explicit forecast Control Algorithm of quadrotor unmanned plane based on dove group's optimization according to claim 1, it is special Sign is：In step 2, it is 8 to set D；Np is 100；Set the position upper limit P of populationmax=100 and position lower limit Pmin= 0, initialize an initial position X to each particle in populationiWith initial speed Vi；The upper limit is the section of function The upper limit, lower limit are the interval limit of function；
    Wherein rand is 0 to 1 random number；
    Vi(t)=rand
    xi(t)=rand (100-0) (13).
  4. 4. a kind of explicit forecast Control Algorithm of quadrotor unmanned plane based on dove group's optimization according to claim 1, it is special Sign is：In step 3, track is linearized, D point is taken on function segment, respectively linearizes track on every section；
    A section [a, b] is given, x will be selectediAs sample point, wherein xi≠a,b；I=1,2,3 ..N；
    It will select section x ∈ [xi,xi+1] or x ∈ [a, x1] or x ∈ [xn, b], and it is as follows to define estimate：
    Draw is taken into n error sample point afterwards, distance d is defined as follows between error sampling point：
    <mrow> <mi>d</mi> <mo>=</mo> <mfrac> <mrow> <mi>b</mi> <mo>-</mo> <mi>a</mi> </mrow> <mi>n</mi> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>
    Then each error sampling point so obtains：XXi=a+id
    Then accumulated error is obtained：
    <mrow> <mi>E</mi> <mo>=</mo> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mo>|</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>XX</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mover> <mi>f</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>XX</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>|</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>
    N=100；E is cost function fitness.
  5. 5. according to a kind of explicit forecast Control Algorithm of quadrotor unmanned plane based on dove group's optimization described in claim 1, its feature It is：In step 4, R=0.2.
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