CN102768536A - Route planning method based on multi-target glowworm swarm algorithm - Google Patents

Abstract

The invention provides a route planning method based on a multi-target glowworm swarm algorithm and belongs to the technical field of route planning. The method includes: modeling a route planning problem, initializing the multi-target glowworm swarm algorithm, updating the glowworm position, determining a non-inferior solution set, updating an external archived file, judging whether a preset maximum iteration number is achieved or not, and determining a Pareto optimal route. The basic glowworm swarm algorithm is improved based on the Pareto dominant conception, and global searching and parallel computing capacities of the glowworm swarm algorithm are well used. Multiple route performance indexes are considered simultaneously during planning, a group of Pareto optimal solution sets can be obtained by planning once, and high flexibility is achieved. Further, the route panning method is different from traditional route planning methods for a single target and route planning methods of using a weighing method to convert multiple targets into the single target, and can better meet practical requirements of route planning.

Description

A kind of paths planning method based on multiple goal firefly algorithm

Technical field

The invention belongs to the path planning technical field, be specifically related to a kind of paths planning method based on multiple goal firefly algorithm.

Background technology

Path planning is one of core technology of robot realization independent navigation.Mobile robot path planning is exactly to seek the movement locus that a connection source is breasted the tape and can be avoided the mobile robot of barrier in the environment, i.e. optimum or suboptimum active path according to certain mission requirements (path is the shortest, consumption is minimum or service time the shortest etc.).Traditional path planning is only considered the optimization of single weighing criteria usually, optimizes criterion according to certain and in decision space, finds an optimal path that reaches dbjective state.Yet, in practical application, need usually to consider a plurality of factors are optimized simultaneously, such as path, smoothness, robot energy consumption, security etc.At this moment, path planning problem can be regarded as a multi-objective optimization question.Different with the path planning problem under the single goal condition; Because each target all exists conflict usually; One is separated for certain target is preferably; Possibly be relatively poor for other targets, so just cause the multiple goal path planning problem generally not have well-determined optimal path, but the set of one group of optimal path that can't compare.For practical problems, must therefrom select a suitable path to use according to the actual conditions of problem and decision maker's preference.At present; For the multiple goal path planning problem some achievements in research have been arranged; But most of documents are for the simplification problem; Usually adopting weighted method to be combined into a scalar function to a plurality of performance index functions, make it to be converted into the single goal optimization problem and find the solution, is that 200910113086.8 patent adopts weighted method to be combined as an objective function to sub-goal, security sub-goal and stationarity sub-goal immediately to carry out path planning like application number; The weighted method simple, intuitive is separated but operation once can only obtain one, and the problem that exists weight to choose, and requires the priori understanding very strong to having of problem itself, when decision maker's preference changes, the value of corresponding change weight.

Evolutionary computation is a kind of random optimization technology based on colony's search; Through separating the population of forming and realize global search keeping between generations by potential; Can search for a plurality of the separating in the solution space concurrently; And can utilize the similarity between different the separating to improve the efficient that its exploitation is found the solution, so evolutionary computation relatively is fit to find the solution multi-objective optimization question.More existing patent documentations solve multi-objective problem with evolution algorithm, are 200710038988.0 the patent of invention automatic core-adjusting problem with multi-target evolution algorithm solution waveguide-optical fiber such as application number; Application number is that 201110037461.2 patent of invention solves the engineering design optimization problem based on the multi-target evolution algorithm; Application number is that the patent of invention of 200810153138.X is based on fuzzy expert system a kind of motor optimized design method that adopted multi-target particle crowd algorithm design.In addition; Formed a lot of typical multi-target evolution algorithms now, estimated genetic algorithm (VEGA), multi-objective genetic algorithm (MOGA), microhabitat Pareto genetic algorithm (NPGA), non-bad ordering genetic algorithm (NSGA), intensity Pareto evolution algorithm (SPEA), NSGA2, Pareto archives evolution strategy (PAES) such as vector based on genetic algorithm; Multi-target particle crowd algorithm (MOPSO) based on particle cluster algorithm.These algorithms are each has something to recommend him for the treatment effect of different multi-objective optimization question, become the focus of current multiple-objection optimization area research.But, need a plurality of targets of optimizing simultaneously be converted into corresponding fitness value based on the multi-target evolution algorithm of genetic algorithm, could use the natural law of " survival of the fittest " like this and accomplish the process that search is separated, cause the algorithm more complicated, operand is big; And need solve problem how to choose global optimum's particle based on the multi-target particle crowd algorithm of particle cluster algorithm; Objective function in view of multi-objective optimization question has a plurality of needs to optimize simultaneously instructs the population optimizing so be difficult to choose a rational global optimum particle.

(Firefly Algorithm FA) is a kind of new biological heuristic algorithm that was proposed in 2008 by X.S Yang to the firefly algorithm.FA also is a kind of random optimization algorithm based on colony's search; But the information transmission between its individuality and information sharing mechanism and evolution algorithm, particle cluster algorithm are different; In FA; Interact in twos between the firefly individuality, the firefly that brightness is big attracts the little firefly of brightness constantly to move to it, thereby guides whole colony to move to more excellent zone.This distinctive information sharing mechanism makes FA under the prerequisite that guarantees speed of convergence, to be not easy to be absorbed in local optimum.The successful Application of FA in the single goal optimization problem explained the validity of FA, but FA can not directly apply to multi-objective optimization question.Human FA such as Theofanis solve this multi-objective optimization question of economic Emission Load Dispatch (Application of the Firefly Algorithm for Solving the Economic Emissions Load Dispatch Problem in the electric system; 2010); The solution of the document is to be converted into the single goal optimization problem to multi-objective optimization question with weighted method earlier; Adopt FA that the single target function is optimized again, separate but the operation of this method once can only obtain one, and the problem that exists weight to choose; The priori understanding that requirement is very strong to having of problem itself; When decision maker's preference changes, the value of corresponding change weight, do not meet the essence of multi-objective optimization question.Application number is that 201110257951.3 patent of invention has designed a kind of naval vessel paths planning method based on the firefly algorithm, but this method only considers the path total length is optimized, and does not consider other performance index in the path planning problem.

Summary of the invention

To the problem that exists in the prior art, the present invention improves basic firefly algorithm, proposes a kind of paths planning method based on multiple goal firefly algorithm.The notable feature that the method that provides among the present invention is different from existing method is: one of which; Notion based on the Pareto domination; Information sharing mechanism in conjunction with FA; Basic firefly algorithm is improved, make it can directly solve multi-objective optimization question, utilized the global search and the computation capability of firefly algorithm well; Its two, the present invention is directed to the multiple goal path planning problem, in planning, consider a plurality of path performance indexs simultaneously, once planning just can access one group of Pareto optimal solution set, have very big dirigibility.This paths planning method differs from traditional paths planning method to simple target and adopts weighted method to be converted into the paths planning method of single goal to multiple goal, can satisfy the actual needs of path planning better.

A kind of paths planning method based on multiple goal firefly algorithm is characterized in that: specifically comprise following step:

Step 1: path planning problem is carried out mathematical modeling:

(1) environment to path planning carries out mathematical modeling:

In two dimensional surface, carry out path planning, S is the starting point of robot, and G is a terminal point, in the path planning scope, sets up global coordinate system O-XY, if n path point formed a path, then path representation is P={S, p ₁, p ₂..., p _n, G}, wherein (p ₁, p ₂..., p _n) be the sequence of the path point in the global map, be the target of path planning;

In global coordinate system O-XY; The coordinate of path point sequence is a two dimension; Be the length that reduces to encode; Setting up a coordinate system S-X ' Y ', is coordinate origin with starting point S, makes X ' axle with

; The ray of order perpendicular to X ' and through S is as Y ' axle, and the coordinate transform of correspondence is:

$\left[\begin{array}{c}{x}^{\′}\\ y^{\′}\end{array}\right]=\left[\begin{array}{cc}\cos \mathrm{\θ}& -\sin \mathrm{\θ}\\ \sin \mathrm{\θ}& \cos \mathrm{\θ}\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}{x}_s\\ y_s\end{array}\right]$

Wherein (x, y), (x ', y ') be respectively certain some coordinate under coordinate system O-XY and S-X ' Y ' in the map, θ is the angle between coordinate axis X and the coordinate axis X ', (x _s, y _s) be the coordinate of S point under coordinate system O-XY;

Line segment SG is carried out the n+1 five equilibrium, make vertical line, obtain (the l of parallel lines family at each Along ent ₁, l ₂..., l _n), the intersection point of parallel lines family and path P to be determined is destination path point sequence (p ₁, p ₂..., p _n); Definition S is initial path point p ₀, G is for stopping path point p _N+1, a path candidate is expressed as the set P=(p of a series of available path points ₀, p ₁, p ₂..., p _n, p _N+1), the purpose of path planning is n the path point (p that finds outside starting point and the terminating point ₁, p ₂..., p _n);

Because (the l of parallel lines family ₁, l ₂..., l _n) distance between the adjacent straight line is identical; So confirm its horizontal ordinate in S-X ' Y ' coordinate system according to the sequence number of these path points in the set of path point; Ordinate is initialized as the random number in the perform region, is part to be optimized, therefore; For a certain sequence number is the path point of i, its horizontal ordinate in S-X ' Y ' coordinate system

And ordinate

Be expressed as respectively:

$\left\{\begin{array}{c}{x}_i^{\′}=i\·\frac{L_{\mathrm{SG}}}{n+1}\\ y_i^{\′}=\mathrm{rand}(Y_{\min }^{\′},Y_{\max }^{\′})\end{array}\right.$

Wherein,

Represent that respectively sequence number is abscissa value and the ordinate value of path point in S-X ' Y ' coordinate system of i; L _SGBe the distance between starting point S and the impact point G,

With

Be respectively the minimum value and the maximal value of ordinate,

Be illustrated in

The equally distributed random number of interior obedience.

(2) confirm three evaluation functions in path, weigh path, path smoothness and path security respectively:

If any feasible path is P=(p ₀, p ₁, p ₂..., p _n, p _N+1), then 3 of the multiple goal path planning problem performance index definitions are following:

(1) path f ₁(P)

For a paths P=(p ₀, p ₁, p ₂..., p _n, p _N+1), to form by the n+1 path segments, the length in this path is the length sum of n+1 path segments;

$f_1\left(P\right)=\underset{i=0}{\overset{n}{\mathrm{\Σ}}}\left|p_i{p}_{i+1}\right|=\left|p_0{p}_1\right|+\underset{i=1}{\overset{n-1}{\mathrm{\Σ}}}\sqrt{{(y_{i+1}^{\′}-y_i^{\′})}^2+{(x_{i+1}^{\′}-x_i^{\′})}^2}+\left|p_n{p}_{n+1}\right|$

Wherein i representes the sequence number of path point; | p _ip _I+1| expression path point p _iWith path point p _I+1Between the length of route segment;

With

Represent that respectively sequence number is abscissa value and the ordinate value of path point in S-X ' Y ' coordinate system of i+1;

With

Represent that respectively sequence number is abscissa value and the ordinate value of path point in S-X ' Y ' coordinate system of i,

With Represent that respectively sequence number is abscissa value and the ordinate value of path point in S-X ' Y ' coordinate system of i+1; | p ₀p ₁| expression path point p ₀With path point p ₁Between the length of route segment.

(2) path smoothness f ₂(P)

Regard each route segment as a vector, calculate the angle of itself and X ' axle, be the path direction angle, calculate the difference of the deflection of adjacent two sections route segments, obtain deflection angle α according to slope meter _i, the level and smooth degree in path is described with deflection angle size:

$f_2\left(P\right)=\frac{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{\mathrm{\α}}_i+k\×\mathrm{\π}/2}{n}$

f ₂(P) be the average corner value of path P, α _i(i=1,2 ... n) expression two vectorial p _I-1p _iAnd p _ip _I+1Angle (0≤α _i＜π); N is the number of angle between the adjacent path vector paragraph in the n+1 path segments; K is α _iIn more than or equal to the number of pi/2, when a certain angle during, desired value is punished more than or equal to pi/2;

(3) path safe distance f ₃(P)

Degree of safety is meant the distance size between robot and the barrier, if mobile robot's size is bigger, then can not be regarded as a particle, bumps in order to prevent its a certain position and barrier, makes itself and barrier keep certain path safe distance f ₃(P):

$f_3\left(P\right)=\frac{1}{d}$

Wherein, d representes the bee-line of path P apart from all barriers;

(4) infeasible path is punished

Whether can collision obstacle based on the path, the path is divided into feasible path and infeasible path, judge whether a paths is feasible path; A given paths is judged the crossing information of it and environment; Barrier is set at polygon, is described out by one group of apex coordinate, therefore; Calculate the crossing information on every path segments and every limit of barrier, draw the crossing information of whole piece path and barrier;

For the target function value that guarantees every infeasible path is bigger than the fitness value of all feasible paths, when calculating the infeasible path target function value, add a penalty value, for infeasible path, more than the calculating of three target function values following:

f _i(P)＝W _i+m×C _i,i＝1,2,3

Wherein, W _iFor all feasible paths at objective function f _iOn worst-case value; M is the number of infeasible path section among the infeasible path P; C _iBe penalty factor;

Step 2: initialization multiple goal firefly algorithm:

The parameter of at first, initialization multiple goal firefly algorithm: population size N, outside files size N _aWith maximum iteration time T _MaxThe position of initialization firefly, every firefly is represented an alternative path, and through the simplification of step 1 to path code, each dimension component of firefly position vector is represented the ordinate of each path point on the alternative path successively.The initial position of random initializtion firefly in the search volume, if confirm n path point, then the position vector of firefly is n-dimensional vector;

Step 3: upgrade the position of firefly, and definite noninferior solution collection:

In FA, firefly adopts brightness to distinguish the quality of separating of firefly representative through luminous realization information sharing; The firefly that brightness is big attracts the little firefly of brightness to move to it; Thereby make whole population move towards better zone, be defined as its brightness to the target function value of firefly position, the notion that adopts Pareto to arrange is distinguished the brightness size of the firefly among the FA; The information sharing mechanism that combines FA simultaneously, firefly is constantly moved towards better zone in guiding;

The Pareto domination:

In the multi-objective problem of asking the objective function minimum value, establish the set of feasible solution that X is a multi-objective optimization question,

Be design variable,

Be objective function to be optimized, then $F\left(x\right)=(f_1\left(\stackrel{\&RightArrow;}{x}\right),f_2\left(\stackrel{\&RightArrow;}{x}\right),...,f_m\left(\stackrel{\&RightArrow;}{x}\right))$ Be object vector, $\&ForAll;{\stackrel{\&RightArrow;}{x}}_k\&Element;X,{\stackrel{\&RightArrow;}{x}}_l\&Element;X.$ And if only if $\&ForAll;i\&Element;\{\mathrm{1,2},...,m\}:f_i\left({\stackrel{\&RightArrow;}{x}}_k\right)\&le;f_i\left({\stackrel{\&RightArrow;}{x}}_l\right),$ And $\&Exists;j\&Element;\{\mathrm{1,2},...,m\}:f_j\left({\stackrel{\&RightArrow;}{x}}_k\right)<f_j\left({\stackrel{\&RightArrow;}{x}}_l\right)$ The time, claim

Domination

The Pareto optimum solution:

If in the whole set of feasible solution of multi-objective optimization question, do not exist any other to separate x ' Pareto domination x, claim that then x is the Pareto optimum solution of this problem;

The Pareto optimal solution set:

The set of all Pareto optimum solutions of multi-objective optimization question has constituted the Pareto optimal solution set of this problem;

The concrete grammar that upgrades firefly position and definite noninferior solution collection is:

At first, be updated to these three objective functions of path, path smoothness and path security to the position vector of each firefly successively, and judge whether the path is feasible, and infeasible path is punished, obtain the corresponding objective function vector of every firefly;

To any two fireflies in the population,, judge the Pareto dominance relation between the firefly based on the notion of Pareto domination; If certain firefly iPareto domination firefly j; The path of then representing the i representative is more excellent, and j can be attracted and the position of renewal oneself by i, and its position renewal formula is:

${\stackrel{\&RightArrow;}{x}}_j(t+1)={\stackrel{\&RightArrow;}{x}}_j\left(t\right)+{\mathrm{\&beta;}}_{\mathrm{ij}}\left(r_{\mathrm{ij}}\right)({\stackrel{\&RightArrow;}{x}}_i\left(t\right)-{\stackrel{\&RightArrow;}{x}}_j\left(t\right))+\mathrm{\&alpha;}{\stackrel{\&RightArrow;}{\mathrm{\&epsiv;}}}_i$

Wherein t is an iterations;

is firefly i and j location in space; α is a constant; Get α ∈ [0; 1],

is the random number vector;

β _Ij(r _Ij) be the attractive force of firefly i to firefly j, be defined as:

${\mathrm{\&beta;}}_{\mathrm{ij}}\left(r_{\mathrm{ij}}\right)={\mathrm{\&beta;}}_0{e}^{-\mathrm{\&gamma;}{r_{\mathrm{ij}}}^2},$ $r_{\mathrm{ij}}=\left|\right|{\stackrel{\&RightArrow;}{x}}_i-{\stackrel{\&RightArrow;}{x}}_j\left|\right|=\sqrt{\underset{k=1}{\overset{n}{\mathrm{\&Sigma;}}}{(x_{i,k}-x_{j,k})}^2}$

β wherein ₀Be greatest attraction forces, γ is the absorption coefficient of light, r _IjBe the distance of firefly i to firefly j, n is a firefly position vector

Dimension, x _{I, k}Be firefly i position vector

K dimension component, x _{J, k}Be firefly j position vector

K dimension component.State in realization in the process of iteration, preserving the firefly that does not receive any other firefly domination is the noninferior solution collection of this iteration;

Step 4: upgrade outside files.

The all properties that adopts outside files to be used for being kept to obtain in the iterative process is path preferably all; Initial outside files are empty; Along with the carrying out of iteration, with each upgrades outside files for the noninferior solution collection that produces in the step 3, the update strategy of files is: for each concentrated noninferior solution of noninferior solution; The member arranges if noninferior solution receives archives, then refuses noninferior solution and adds in the archives; If noninferior solution has been arranged part archives member, then remove those ridden members, simultaneously noninferior solution is added in the archives; If all members insubjection each other in noninferior solution and the archives then directly adds noninferior solution in the archives;

Limit the size of files,, have the criterion of part noninferior solution in the file that deletes files, surpass the maximum-norm N of setting when the size of files when the upper limit that the scale of files is set in advance _aThe time, the method for the noninferior solution that has more in the file that deletes files is: calculate the density of the individual neighborhood of all files member, and ordering from small to large, keep the wherein minimum N of neighborhood density _aIndividual archives member, other members delete from files;

Definition for individual neighborhood density; PAES algorithm use adaptive mesh method defines the density of individual neighborhood; Concrete is: the search volume is divided into plurality of grids, and individual neighborhood density is defined as with it and is in all the individual numbers in the same grid, and the division of grid is adaptively modifying along with outside files member's variation; Individuality in being inserted into archives is positioned at outside the existing boundary of grid, then repartitions grid;

Step 5: judge whether to reach predefined maximum iteration time:

If reached the maximum iteration time of setting in the step 2, then forward step 6 to; Otherwise, forward step 3 to;

Step 6: confirm the Pareto optimal path, path planning finishes:

Export the noninferior solution collection in the outside files, then obtain the set of one group of Pareto optimal path, need, therefrom select the result of a Pareto optimal path as path planning based on practical problem.

The invention has the advantages that:

The first, what the present invention proposed carries out paths planning method based on multiple goal firefly algorithm, and the firefly algorithm is improved, and proposes a kind of multiple goal firefly algorithm.This algorithm can solve multi-objective optimization question, and is more more simple than multi-target evolution algorithm, the multi-target particle crowd algorithm of classics, has versatility.

Second; What the present invention proposed carries out paths planning method based on multiple goal firefly algorithm; Adopt multiple goal firefly algorithm to solve the path planning problem of considering a plurality of performance index simultaneously, can rational Pareto optimum solution be provided, meet the needs of practical problems for the decision maker.

Description of drawings

Fig. 1: what the present invention proposed carries out the process flow diagram of paths planning method based on multiple goal firefly algorithm.

Fig. 2: among the present invention to planning regional environment modeling figure.

Fig. 3: multiple goal firefly algorithm flow chart among the present invention.

Embodiment

Below in conjunction with accompanying drawing the present invention is elaborated.

A kind of paths planning method that the present invention proposes based on multiple goal firefly algorithm, as shown in Figure 1, specifically comprise following step:

Step 1: path planning problem is carried out mathematical modeling.

(1) environment to path planning carries out mathematical modeling.

As shown in Figure 2, in two dimensional surface, carry out path planning, S is the starting point of robot, G is a terminal point.In the path planning scope, set up global coordinate system O-XY, represent barrier with the object that solid black is filled.The path planning of robot is the set of seeking a path point, supposes path of n path point composition, and then the path can be expressed as P={S, p ₁, p ₂..., p _n, G}, wherein (p ₁, p ₂..., p _n) be the sequence of the path point in the global map, i.e. the target of path planning.Requirement to path point is, path point is non-barrier point, and with the line of adjacent path point on do not have barrier point.

In global coordinate system O-XY, the coordinate of path point sequence is two-dimentional, for the length that reduces to encode, sets up a new coordinate system S-X ' Y '.With starting point S is coordinate origin; Make X ' axle with

, the ray of ordering perpendicular to X ' and through S is as Y ' axle.Corresponding coordinate transform is:

$\left[\begin{array}{c}{x}^{\&prime;}\\ y^{\&prime;}\end{array}\right]=\left[\begin{array}{cc}\cos \mathrm{\&theta;}& -\sin \mathrm{\&theta;}\\ \sin \mathrm{\&theta;}& \cos \mathrm{\&theta;}\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}{x}_s\\ y_s\end{array}\right]$

Wherein: (x, y), (x ', y ') be respectively certain some coordinate under coordinate system O-XY and S-X ' Y ' in the map, θ is the angle between coordinate axis X and the coordinate axis X ', (x _s, y _s) be the coordinate of S point under coordinate system O-XY.

Line segment SG is carried out the n+1 five equilibrium, make vertical line, obtain (the l of parallel lines family at each Along ent ₁, l ₂..., l _n), the intersection point of they and path P to be determined is destination path point sequence (p ₁, p ₂..., p _n).Definition S is initial path point p ₀, G is for stopping path point p _N+1, such path candidate just is expressed as the set P=(p of a series of available path points ₀, p ₁, p ₂..., p _n, p _N+1).The purpose of path planning is exactly n the path point (p that finds outside starting point and the terminating point ₁, p ₂..., p _n).

Because (the l of parallel lines family ₁, l ₂..., l _n) distance between the adjacent straight line is identical, so just can confirm its horizontal ordinate in S-X ' Y ' coordinate system according to the sequence number of these path points in the set of path point, ordinate can be initialized as the random number in the perform region, is part to be optimized.Therefore; For a certain sequence number is the path point of i, and its horizontal ordinate

and ordinate

in S-X ' Y ' coordinate system can be expressed as respectively:

$\left\{\begin{array}{c}{x}_i^{\&prime;}=i\&CenterDot;\frac{L_{\mathrm{SG}}}{n+1}\\ y_i^{\&prime;}=\mathrm{rand}(Y_{\min }^{\&prime;},Y_{\max }^{\&prime;})\end{array}\right.$

So just the two-dimensional encoded one dimension that is reduced to of path planning problem.

Wherein, Represent that respectively sequence number is abscissa value and the ordinate value of path point in S-X ' Y ' coordinate system of i; L _SGBe the distance between starting point S and the impact point G,

With Be respectively the minimum value and the maximal value of ordinate,

Be illustrated in

The equally distributed random number of interior obedience.

(2) confirm three evaluation functions in path, weigh path, path smoothness and path security respectively.

If any feasible path is P=(p ₀, p ₁, p ₂..., p _n, p _N+1), then 3 of the multiple goal path planning problem performance index definitions are following:

(1) path f ₁(P)

For a paths P=(p ₀, p ₁, p ₂..., p _n, p _N+1), to form by the n+1 path segments, the length in this path is the length sum of n+1 path segments.

$f_1\left(P\right)=\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}\left|p_i{p}_{i+1}\right|=\left|p_0{p}_1\right|+\underset{i=1}{\overset{n-1}{\mathrm{\&Sigma;}}}\sqrt{{(y_{i+1}^{\&prime;}-y_i^{\&prime;})}^2+{(x_{i+1}^{\&prime;}-x_i^{\&prime;})}^2}+\left|p_n{p}_{n+1}\right|$

Wherein i representes the sequence number of path point; | p _ip _I+1| expression path point p _iWith path point p _I+1Between the length of route segment;

With

Represent that respectively sequence number is abscissa value and the ordinate value of path point in S-X ' Y ' coordinate system of i+1;

With Represent that respectively sequence number is abscissa value and the ordinate value of path point in S-X ' Y ' coordinate system of i,

With

Represent that respectively sequence number is abscissa value and the ordinate value of path point in S-X ' Y ' coordinate system of i+1; | p ₀p ₁| expression path point p ₀With path point p ₁Between the length of route segment.

(2) path smoothness f ₂(P)

Because mobile robot's geometric shape has certain influence to its kinetic characteristic, so the path should be mild and smooth, i.e. angle of deflection between route segment and the route segment _iShould be as much as possible little.

If regard each route segment as a vector, can calculate the angle of itself and X ' axle according to slope, i.e. the difference of the deflection of adjacent two sections route segments is calculated at path direction angle, just can obtain deflection angle α _iThe level and smooth degree in path can be described with the deflection angle size.

$f_2\left(P\right)=\frac{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_i+k\&times;\mathrm{\&pi;}/2}{n}$

f ₂(P) be the average corner value of path P.In the following formula, α _i(i=1,2 ... n) expression two vectorial p _I-1p _iAnd p _ip _I+1Angle (0≤α _i＜π); N is the number of angle between the adjacent path vector paragraph in the n+1 path segments; K is α _iIn more than or equal to the number of pi/2, promptly, desired value is punished when a certain angle during more than or equal to pi/2.

(3) path safe distance f ₃(P)

Degree of safety is meant the distance size between robot and the barrier, if mobile robot's size is bigger, then can not be regarded as a particle.At this moment, bump, should make itself and barrier keep certain path safe distance f as far as possible in order to prevent its a certain position and barrier ₃(P).

$f_3\left(P\right)=\frac{1}{d}$

Wherein, d representes the bee-line of path P apart from all barriers.

(4) infeasible path is punished

Whether can collision obstacle based on the path, can be divided into feasible path and infeasible path to the path.Judge whether a paths is feasible path, a promptly given paths is judged the crossing information of it and environment.Barrier is set at polygon, is described out by one group of apex coordinate.Therefore, as long as calculate the crossing information of every path segments and every limit of barrier (being the adjacent vertex line), can draw the crossing information of whole piece path and barrier.

Because the path that the define objective functional value is more little is good more,, when calculating the infeasible path target function value, add a penalty value for the target function value that guarantees every infeasible path is bigger than the fitness value of all feasible paths.For infeasible path, more than the calculating of three target function values following:

f _i(P)＝W _i+m×C _i,i＝1，2,3

In the formula, W _iFor all feasible paths at objective function f _iOn worst-case value; M is the number of infeasible path section among the infeasible path P; C _iBe penalty factor.

Step 2: initialization multiple goal firefly algorithm.

As shown in Figure 3, at first, the parameter of initialization multiple goal firefly algorithm: population size N, outside files size N _aWith maximum iteration time T _Max, the size of this tittle will be confirmed according to problem to be solved.Population size N can get 20～40; Outside files size N _aConfirmed the size of the set of paths of final acquisition, N _aValue big more, then final alternative number of path is many more; Maximum iteration time can get 300～600.

Need the initialized position that also has firefly simultaneously.Every firefly is represented an alternative path.Through the simplification of step 1 to path code, each dimension component of firefly position vector is represented the ordinate (in S-X ' Y ' coordinate system) of each path point on the alternative path successively.The initial position of random initializtion firefly in the search volume.As stated, if confirm n path point, then the position vector of firefly is n-dimensional vector.

Step 3: upgrade the position of firefly, and definite noninferior solution collection.

In FA, firefly adopts brightness to distinguish the quality of separating of firefly representative through luminous realization information sharing, and the firefly that brightness is big attracts the little firefly of brightness to move to it, thereby makes whole population move towards better zone.When adopting FA to solve the single goal optimization problem, objective function to be optimized has only one, therefore, can be defined as its brightness to the target function value of firefly position.Yet,,, can't directly adopt the value of some objective functions to weigh the quality of separating owing to will consider a plurality of conflicting objective functions are optimized simultaneously for multi-objective optimization question.The notion of Pareto domination can be estimated the quality of separating in the multiple-objection optimization effectively; The notion that therefore can adopt Pareto to arrange is distinguished the brightness size of the firefly among the FA; The information sharing mechanism that combines FA simultaneously, firefly is constantly moved towards better zone in guiding.

In order to explain this process better, at first provide the related notion of Pareto domination.

Definition 1:Pareto domination

In the multi-objective problem of asking the objective function minimum value, establish the set of feasible solution that X is a multi-objective optimization question, Be design variable,

Be objective function to be optimized, then $F\left(x\right)=(f_1\left(\stackrel{\&RightArrow;}{x}\right),f_2\left(\stackrel{\&RightArrow;}{x}\right),...,f_m\left(\stackrel{\&RightArrow;}{x}\right))$ Be object vector, $\&ForAll;{\stackrel{\&RightArrow;}{x}}_k\&Element;X,{\stackrel{\&RightArrow;}{x}}_l\&Element;X.$ And if only if $\&ForAll;i\&Element;\{\mathrm{1,2},...,m\}:f_i\left({\stackrel{\&RightArrow;}{x}}_k\right)\&le;f_i\left({\stackrel{\&RightArrow;}{x}}_l\right),$ And $\&Exists;j\&Element;\{\mathrm{1,2},...,m\}:f_j\left({\stackrel{\&RightArrow;}{x}}_k\right)<f_j\left({\stackrel{\&RightArrow;}{x}}_l\right)$ The time, claim

Domination

Definition 2:Pareto optimum solution

If in the whole set of feasible solution of multi-objective optimization question, do not exist any other to separate x ' Pareto domination x, claim that then x is the Pareto optimum solution of this problem, is also referred to as noninferior solution.

Definition 3:Pareto optimal solution set

The set of all Pareto optimum solutions of multi-objective optimization question has constituted the Pareto optimal solution set of this problem, or is called the noninferior solution collection.

Because path planning problem will be considered path, path smoothness and three performance index of path security simultaneously; And these performance index are conflicting often; Such as, path smoothness and path security that path is short maybe be relatively poor, therefore; Path planning problem is a multi-objective optimization question in essence, can adopt above-mentioned multiple goal firefly algorithm to address this problem.

The concrete grammar that upgrades firefly position and definite noninferior solution collection is:

At first, be updated to these three objective functions of path, path smoothness and path security to the position vector of each firefly successively, and judge whether the path is feasible, and infeasible path is punished, obtain the corresponding objective function vector of every firefly.

To any two fireflies in the population,, judge the Pareto dominance relation between the firefly based on the notion of Pareto domination.If certain firefly iPareto domination firefly j representes that then the path of i representative is more excellent, j can be attracted and the position of renewal oneself by i, and its position renewal formula is:

${\stackrel{\&RightArrow;}{x}}_j(t+1)={\stackrel{\&RightArrow;}{x}}_j\left(t\right)+{\mathrm{\&beta;}}_{\mathrm{ij}}\left(r_{\mathrm{ij}}\right)({\stackrel{\&RightArrow;}{x}}_i\left(t\right)-{\stackrel{\&RightArrow;}{x}}_j\left(t\right))+\mathrm{\&alpha;}{\stackrel{\&RightArrow;}{\mathrm{\&epsiv;}}}_i$

Wherein: t is an iterations;

is firefly i and j location in space; α is a constant; Can get α ∈ [0; 1],

is the random number vector that is obtained by Gaussian distribution, even perhaps other distributions that distribute.

β _Ij(r _Ij) be the attractive force of firefly i to firefly j, be defined as:

${\mathrm{\&beta;}}_{\mathrm{ij}}\left(r_{\mathrm{ij}}\right)={\mathrm{\&beta;}}_0{e}^{-\mathrm{\&gamma;}{r_{\mathrm{ij}}}^2},$ $r_{\mathrm{ij}}=\left|\right|{\stackrel{\&RightArrow;}{x}}_i-{\stackrel{\&RightArrow;}{x}}_j\left|\right|=\sqrt{\underset{k=1}{\overset{n}{\mathrm{\&Sigma;}}}{(x_{i,k}-x_{j,k})}^2}$

Wherein: β ₀Be greatest attraction forces, can get β ₀=1.γ is the absorption coefficient of light, and absorption coefficient of light γ has indicated the variation of attractive force, and its value has very big influence to firefly convergence of algorithm speed and optimization effect, can get γ ∈ [0.01,100].r _IjBe the distance of firefly i to firefly j, n is a firefly position vector

Dimension, x _{I, k}Be firefly i position vector

K dimension component, x _{J, k}Be firefly j position vector

K dimension component.

State in realization in the process of iteration, preserving the firefly that does not receive any other firefly domination is the noninferior solution collection of this iteration.

Step 4: upgrade outside files.

Because path planning problem is a multi-objective optimization question; Can not find all optimum path of a paths length, path smoothness and path security; Can only to three mutually the performance index of conflict carry out tradeoffs, confirm some all properties all preferably the path supply the decision maker to select.

The all properties that adopts outside files to be used for being kept to obtain in the iterative process is path, i.e. noninferior solution preferably all.Initial outside files are empty, along with the carrying out of iteration, with each upgrades outside files for the noninferior solution collection that produces in the step 3.The update strategy of files is: for each noninferior solution that noninferior solution is concentrated, the member arranges if noninferior solution receives archives, then refuses noninferior solution and adds in the archives; If noninferior solution has been arranged part archives member, then remove those ridden members, simultaneously noninferior solution is added in the archives; If all members insubjection each other in noninferior solution and the archives then directly adds noninferior solution in the archives.

If the size to files does not limit, all noninferior solutions that satisfy above-mentioned condition can get into files so.But because files all will upgrade in the iteration each time, if do not limit the size of files, calculation cost is excessive.Therefore, consider, should limit the size of files from the angle of actual computation.In case the scale of files reaches the upper limit of prior setting, need the criterion of part noninferior solution in the file that deletes files.The maximum-norm N that surpasses setting when the size of files _aThe time, the method for the noninferior solution that has more in the file that deletes files is: calculate the density of the individual neighborhood of all files member, and ordering from small to large, keep the wherein minimum N of neighborhood density _aIndividual archives member, other members delete from files.

For the definition of individual neighborhood density, a variety of forms have been arranged in the multi-target evolution algorithm.PAES algorithm use adaptive mesh method defines the density of individual neighborhood; This method is simple; Its operand is significantly less than the density Estimation strategy of the S of popular SPEA2 apart from density Estimation and NEAG2; But not inferior on the performance, therefore, we adopt this method to define individual neighborhood density in the outside files.Concrete method of estimation is: the search volume is divided into plurality of grids, and individual neighborhood density is defined as with it and is in all the individual numbers in the same grid.The division of grid is adaptively modifying along with outside files member's variation, and the individuality in being inserted into archives is positioned at outside the existing boundary of grid, then repartitions grid

Step 5: judge whether to reach predefined maximum iteration time.

If reached the maximum iteration time of setting in the step 2, then forward step 6 to; Otherwise, forward step 3 to.

Step 6: confirm the Pareto optimal path, path planning finishes.

Export the noninferior solution collection in the outside files, then obtain the set of one group of Pareto optimal path.Need based on practical problem, therefrom select the result of a Pareto optimal path as path planning.

Claims (4)

1. paths planning method based on multiple goal firefly algorithm is characterized in that: specifically comprise following step:

Step 1: path planning problem is carried out mathematical modeling:

(1) environment to path planning carries out mathematical modeling:

In two dimensional surface, carry out path planning, S is the starting point of robot, and G is a terminal point, in the path planning scope, sets up global coordinate system O-XY, if n path point formed a path, then path representation is P={S, p ₁, p ₂..., p _n, G}, wherein (p ₁, p ₂..., p _n) be the sequence of the path point in the global map, be the target of path planning;

In global coordinate system O-XY; The coordinate of path point sequence is a two dimension; Be the length that reduces to encode; Setting up a coordinate system S-X ' Y ', is coordinate origin with starting point S, makes X ' axle with

; The ray of order perpendicular to X ' and through S is as Y ' axle, and the coordinate transform of correspondence is:

$\left[\begin{array}{c}{x}^{\&prime;}\\ y^{\&prime;}\end{array}\right]=\left[\begin{array}{cc}\cos \mathrm{\&theta;}& -\sin \mathrm{\&theta;}\\ \sin \mathrm{\&theta;}& \cos \mathrm{\&theta;}\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}{x}_s\\ y_s\end{array}\right]$

Wherein (x, y), (x ', y ') be respectively certain some coordinate under coordinate system O-XY and S-X ' Y ' in the map, θ is the angle between coordinate axis X and the coordinate axis X ', (x _s, y _s) be the coordinate of S point under coordinate system O-XY;

Line segment SG is carried out the n+1 five equilibrium, make vertical line, obtain (the l of parallel lines family at each Along ent ₁, l ₂..., l _n), the intersection point of parallel lines family and path P to be determined is destination path point sequence (p ₁, p ₂..., p _n); Definition S is initial path point p ₀, G is for stopping path point p _N+1, a path candidate is expressed as the set P=(p of a series of available path points ₀, p ₁, p ₂..., p _n, p _N+1), the purpose of path planning is n the path point (p that finds outside starting point and the terminating point ₁, p ₂..., p _n);

Because (the l of parallel lines family ₁, l ₂..., l _n) distance between the adjacent straight line is identical; So confirm its horizontal ordinate in S-X ' Y ' coordinate system according to the sequence number of these path points in the set of path point; Ordinate is initialized as the random number in the perform region, is part to be optimized, therefore; For a certain sequence number is the path point of i, its horizontal ordinate in S-X ' Y ' coordinate system

And ordinate

Be expressed as respectively:

$\left\{\begin{array}{c}{x}_i^{\&prime;}=i\&CenterDot;\frac{L_{\mathrm{SG}}}{n+1}\\ y_i^{\&prime;}=\mathrm{rand}(Y_{\min }^{\&prime;},Y_{\max }^{\&prime;})\end{array}\right.$

Wherein,

Represent that respectively sequence number is abscissa value and the ordinate value of path point in S-X ' Y ' coordinate system of i; L _SGBe the distance between starting point S and the impact point G,

With

Be respectively the minimum value and the maximal value of ordinate,

Be illustrated in

The equally distributed random number of interior obedience;

(2) confirm three evaluation functions in path, weigh path, path smoothness and path security respectively:

If any feasible path is P=(p ₀, p ₁, p ₂..., p _n, p _N+1), then 3 of the multiple goal path planning problem performance index definitions are following:

(1) path f ₁(P)

For a paths P=(p ₀, p ₁, p ₂..., p _n, p _N+1), to form by the n+1 path segments, the length in this path is the length sum of n+1 path segments;

$f_1\left(P\right)=\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}\left|p_i{p}_{i+1}\right|=\left|p_0{p}_1\right|+\underset{i=1}{\overset{n-1}{\mathrm{\&Sigma;}}}\sqrt{{(y_{i+1}^{\&prime;}-y_i^{\&prime;})}^2+{(x_{i+1}^{\&prime;}-x_i^{\&prime;})}^2}+\left|p_n{p}_{n+1}\right|$

Wherein i representes the subscript of path point; | p _ip _I+1| the length of the route segment between expression path point i and the path point i+1;

With

Represent that respectively sequence number is abscissa value and the ordinate value of path point in S-X ' Y ' coordinate system of i+1;

With

Represent that respectively sequence number is abscissa value and the ordinate value of path point in S-X ' Y ' coordinate system of i,

With Represent that respectively sequence number is abscissa value and the ordinate value of path point in S-X ' Y ' coordinate system of i+1; | p ₀p ₁| expression path point p ₀With path point p ₁Between the length of route segment;

(2) path smoothness f ₂(P)

Regard each route segment as a vector, calculate the angle of itself and X ' axle, be the path direction angle, calculate the difference of the deflection of adjacent two sections route segments, obtain deflection angle α according to slope meter _i, the level and smooth degree in path is described with deflection angle size:

$f_2\left(P\right)=\frac{\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_i+k\&times;\mathrm{\&pi;}/2}{n}$

f ₂(P) be the average corner value of path P, α _i(i=1,2 ... n) expression two vectorial p _I-1p _iAnd p _ip _I+1Angle (0≤α _i＜π); N is the number of angle between the adjacent path vector paragraph in the n+1 path segments; K is α _iIn more than or equal to the number of pi/2, when a certain angle during, desired value is punished more than or equal to pi/2;

(3) path safe distance f ₃(P)

Degree of safety is meant the distance size between robot and the barrier, if mobile robot's size is bigger, then can not be regarded as a particle, bumps in order to prevent its a certain position and barrier, makes itself and barrier keep certain path safe distance f ₃(P):

$f_3\left(P\right)=\frac{1}{d}$

Wherein, d representes the bee-line of path P apart from all barriers;

(4) infeasible path is punished

Whether can collision obstacle based on the path, the path is divided into feasible path and infeasible path, judge whether a paths is feasible path; A given paths is judged the crossing information of it and environment; Barrier is set at polygon, is described out by one group of apex coordinate, therefore; Calculate the crossing information on every path segments and every limit of barrier, draw the crossing information of whole piece path and barrier;

For the target function value that guarantees every infeasible path is bigger than the fitness value of all feasible paths, when calculating the infeasible path target function value, add a penalty value, for infeasible path, more than the calculating of three target function values following:

f _i(P)＝W _i+m×C _i,i＝1,2,3

Wherein, W _iFor all feasible paths at objective function f _iOn worst-case value; M is the number of infeasible path section among the infeasible path P; C _iBe penalty factor;

Step 2: initialization multiple goal firefly algorithm:

The parameter of at first, initialization multiple goal firefly algorithm: population size N, outside files size N _aWith maximum iteration time T _MaxThe position of initialization firefly; Every firefly is represented an alternative path; Through the simplification of step 1 to path code, each dimension component of firefly position vector is represented the ordinate of each path point on the alternative path, the initial position of random initializtion firefly in the search volume successively; If confirm n path point, then the position vector of firefly is n-dimensional vector;

Step 3: upgrade the position of firefly, and definite noninferior solution collection:

In FA, firefly adopts brightness to distinguish the quality of separating of firefly representative through luminous realization information sharing; The firefly that brightness is big attracts the little firefly of brightness to move to it; Thereby make whole population move towards better zone, be defined as its brightness to the target function value of firefly position, the notion that adopts Pareto to arrange is distinguished the brightness size of the firefly among the FA; The information sharing mechanism that combines FA simultaneously, firefly is constantly moved in guiding;

The concrete grammar that upgrades firefly position and definite noninferior solution collection is:

At first, be updated to these three objective functions of path, path smoothness and path security to the position vector of each firefly successively, and judge whether the path is feasible, and infeasible path is punished, obtain the corresponding objective function vector of every firefly;

To any two fireflies in the population,, judge the Pareto dominance relation between the firefly based on the notion of Pareto domination; If certain firefly i Pareto domination firefly j; The path of then representing the i representative is more excellent, and j can be attracted and the position of renewal oneself by i, and its position renewal formula is:

${\stackrel{\&RightArrow;}{x}}_j(t+1)={\stackrel{\&RightArrow;}{x}}_j\left(t\right)+{\mathrm{\&beta;}}_{\mathrm{ij}}\left(r_{\mathrm{ij}}\right)({\stackrel{\&RightArrow;}{x}}_i\left(t\right)-{\stackrel{\&RightArrow;}{x}}_j\left(t\right))+\mathrm{\&alpha;}{\stackrel{\&RightArrow;}{\mathrm{\&epsiv;}}}_i$

Wherein t is an iterations;

is firefly i and j location in space; α is a constant; Get α ∈ [0; 1],

is the random number vector;

β _Ij(r _Ij) be the attractive force of firefly i to firefly j, be defined as:

${\mathrm{\&beta;}}_{\mathrm{ij}}\left(r_{\mathrm{ij}}\right)={\mathrm{\&beta;}}_0{e}^{-\mathrm{\&gamma;}{r_{\mathrm{ij}}}^2},$ $r_{\mathrm{ij}}=\left|\right|{\stackrel{\&RightArrow;}{x}}_i-{\stackrel{\&RightArrow;}{x}}_j\left|\right|=\sqrt{\underset{k=1}{\overset{n}{\mathrm{\&Sigma;}}}{(x_{i,k}-x_{j,k})}^2}$

β wherein ₀Be greatest attraction forces, γ is the absorption coefficient of light, r _IjBe the distance of firefly i to firefly j; N is a firefly position vector Dimension, x _{I, k}Be firefly i position vector

K dimension component, x _{J, k}Be firefly j position vector K dimension component, state in realization in the process of iteration, preserve the firefly that do not receive any other firefly domination noninferior solution collection for this iteration;

Step 4: upgrade outside files:

The all properties that adopts outside files to be used for being kept to obtain in the iterative process is path preferably all; Initial outside files are empty; Along with the carrying out of iteration, with each upgrades outside files for the noninferior solution collection that produces in the step 3, the update strategy of files is: for each concentrated noninferior solution of noninferior solution; The member arranges if noninferior solution receives archives, then refuses noninferior solution and adds in the archives; If noninferior solution has been arranged part archives member, then remove those ridden members, simultaneously noninferior solution is added in the archives; If all members insubjection each other in noninferior solution and the archives then directly adds noninferior solution in the archives;

Limit the size of files,, have the criterion of part noninferior solution in the file that deletes files, surpass the maximum-norm N of setting when the size of files when the upper limit that the scale of files is set in advance _aThe time, the method for the noninferior solution that has more in the file that deletes files is: calculate the density of the individual neighborhood of all files member, and ordering from small to large, keep the wherein minimum N of neighborhood density _aIndividual archives member, other members delete from files;

Definition for individual neighborhood density; PAES algorithm use adaptive mesh method defines the density of individual neighborhood; Concrete is: the search volume is divided into plurality of grids, and individual neighborhood density is defined as with it and is in all the individual numbers in the same grid, and the division of grid is adaptively modifying along with outside files member's variation; Individuality in being inserted into archives is positioned at outside the existing boundary of grid, then repartitions grid;

Step 5: judge whether to reach predefined maximum iteration time:

If reached the maximum iteration time of setting in the step 2, then forward step 6 to; Otherwise, forward step 3 to; Step 6: confirm the Pareto optimal path, path planning finishes:

Export the noninferior solution collection in the outside files, then obtain the set of one group of Pareto optimal path, need, therefrom select the result of a Pareto optimal path as path planning based on practical problem.

2. according to a kind of paths planning method based on multiple goal firefly algorithm described in the claim 1, it is characterized in that: described population size N gets 20～40; Described outside files size N _aConfirm the size of the final set of paths that obtains, N _aValue big more, then final selective number of path is many more; Described maximum iteration time gets 300～600.

3. according to a kind of paths planning method described in the claim 1, it is characterized in that: described greatest attraction forces β based on multiple goal firefly algorithm ₀Value is β ₀=1.

4. according to a kind of paths planning method based on multiple goal firefly algorithm described in the claim 1, it is characterized in that: described absorption coefficient of light γ value is γ ∈ [0.01,100].

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