CN107368075A - Mobile robot global path planning algorithm based on hybrid particle swarm - Google Patents
Mobile robot global path planning algorithm based on hybrid particle swarm Download PDFInfo
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Abstract
The invention discloses a kind of mobile robot global path planning algorithm based on hybrid particle swarm, the sports rule that particle is first according to defined in NTVPSO updates their speed and position.Then, evolved using RBSADE after iteration each time terminates the personal best particle of each particle, stagnated so as to avoid algorithm from being absorbed in iteration too early, improve its optimization efficiency in robot path planning.
Description
Technical field
The invention belongs to robotic technology field, is related to robot global path planning algorithm, more particularly to one kind is based on
The mobile robot global path planning algorithm of hybrid particle swarm.
Background technology
In the past few decades, mobile robot be successfully used to industry and military field go to perform some it is crucial and
Important unmanned task, such as ground location, monitoring and search and rescue.When robot performs these unmanned tasks, advised for robot
It is an important research topic in robotic technology field to mark a feasible secure path.The mesh of robot path planning
Be in a relative complex working environment for robot cooked up from original position to final position one it is optimal or
The avoidance path of suboptimum.Because it is in the extensive use of different field, since middle 1960s, path planning problem
Cause the research interest of numerous scholars.
Being suggested since path planning this problem, there are many paths planning methods.However, under complex environment
Path planning problem, be a uncertainty time polynomial difficulty (non-deterministic polynomial-time
Hard, NP-hard) problem.The NP-hard characteristics of the problem bring huge challenge to its solution.So in order to effectively
Such issues that solution, propose that various forms of high-effect optimized algorithms are very necessary.
Due to characteristic of the evolution algorithm (Evolutionary Algorithms, EAS) based on population and to complicated NP-
The good search capability of hard problems, in recent years, many scholars propose different evolution algorithms such as genetic algorithm
(Genetic Algorithm, GA), simulated annealing (Simulation Annealing Algorithm SAA) and artificial
Neural network algorithm (Artificial Neural Network, ANN) etc. carrys out solution path planning problem.
As one of most powerful evolution algorithm, due to its simplicity and the characteristic of Fast Convergent, particle group optimizing
(Particle Swarm Optimization, PSO) algorithm has been widely used in solving path planning problem.It is however, basic
The performance of particle cluster algorithm has the defects of two typical cases：First defect be can not the global and local of equilibrium particle well search
Suo Nengli, so as to easily cause algorithm to be absorbed in local optimum too early；In addition, after certain iterations, when particle can not be searched
When rope is to than history optimal location better position, basic particle group algorithm will be absorbed in iteration stagnation.In order to improve PSO property
Can be, it is necessary to which the two exemplary shortcomings be made up or overcome.
The evolution algorithm very popular as another, differential evolution (Differential Evolution, DE) are calculated
Method the advantages that searching for reliability and be easily achieved, has been widely used in solving various types due to its simplicity
Optimization problem.However, for different optimization problems, the evolution strategy of DE algorithms and the setting of control parameter are not quite similar,
This needs user to be set according to the priori of oneself.So the evolution strategy of DE algorithms and the setting of control parameter are
Closely bound up with particular problem, this brings challenge to effective DE algorithms are designed.
Global path planning is carried out to mobile robot, during using elementary particle Algorithm for Solving, faces following two problems：
Can not the global and local search capability of equilibrium particle well cause algorithm to be absorbed in local optimum too early, it is impossible to reach it is global most
It is excellent；Iteration stagnation will be absorbed in after certain iterations, so as to influence performance of the particle cluster algorithm in path planning.
Because PSO and DE belong to the evolution algorithm of processing colony EVOLUTIONARY COMPUTATION, naturally, it may be considered that to by this two
Kind algorithm mixes, so as to ensure the advantages of algorithm after mixing can borrow two algorithms to improve the optimization of algorithm
Performance.
Bibliography
[1][1]Zexuan Zhu,Fangxiao Wang,Shan He,and Yiwen Sun.Global Path
Planning of Mobile Robots Using a Memetic Algorithm.International Journal of
Systems Science,46(11):1982–1993,2015.
[2][2]Yong Zhang,Dun‐wei Gong,and Jian‐hua Zhang.Robot path planning
in uncertain environment using multi‐objective particle swarm
optimization.Neurocomputing,103:172–185,2013.
[3][3]N Geng,D W Gong,and Y Zhang.PSO‐Based Robot Path Planning for
Multisurvivor Rescue in Limited Survival Time.Mathematical Problems in
Engineering,2014(2014),2014.
[4][4]Reza Akbari and Koorush Ziarati.A rank based particle swarm
optimization algorithm with dynamic adaptation.Journal of Computational and
Applied Mathematics,235(8):2694–2714,2011.
[5][5]Hui Liu,Zixing Cai,and Yong Wang.Hybridizing particle swarm
optimization with differential evolution for constrained numerical and
engineering optimization.Applied Soft Computing Journal,10(2):629–640,2010.
[6]Pei Li and HaiBin Duan.Path planning of unmanned aerial vehicle
based on improved gravitational search algorithm.Science China Technological
Sciences, 55(10):2712–2719,2012.
The content of the invention
In order to solve the problems, such as existing mobile robot global path planning, the present invention proposes one kind and is based on hybrid particle swarm
Mobile robot global path planning algorithm, basic particle group algorithm and differential evolution algorithm are mixed, for solving moving machine
Device people's global path planning problem.
The technical solution adopted for the present invention to solve the technical problems is：
Mobile robot global path planning algorithm based on hybrid particle swarm, comprises the following steps：
Step 1: establish Hybrid Particle Swarm
Particle is updated to its speed and position in such a way：
Wherein, ω represents the inertia weight coefficient of particle；c1Represent the cognition acceleration factor of particle；c2Represent particle
Social recognition acceleration factor；WithRepresent particle m in kth and the speed at k+1 iteration moment respectively；WithPoint
Not Biao Shi particle m in kth and the position at k+1 iteration moment；WithIt is illustrated respectively in kth time iteration moment grain
Sub- m personal best particle and the global optimum position of colony；
ω、c1、c2It is three main control parameters for influenceing algorithm performance, these three is joined using following adaptive rule
Number is updated：
Wherein：
ωmaxAnd ωminRepresent ω upper and lower bound；c1sAnd c1fRepresent c1Initial value and end value；c2sAnd c2fRepresent
c2Initial value and end value；kmaxRepresent maximum iteration；WithRepresent in kth -1 and kth iteration moment grain
Two norms of sub- m velocities；Δ is to prevent βmA denominator minimum normal number being changed into 0 and introduce, Δ=1e-
25；ω is required to meet in algorithmmax>ωmin, c1s>c1fAnd c2s<c2f；
Step 2: establish Differential Evolution Algorithm
Mutation operation produces new according to the following formula by changing the gene information of three different former generation at random
Body：
Wherein,WithIt is from i ≠ i1≠i2≠i3Current population in randomly selected individual, Fi
Represent the zoom factor of differential evolution algorithm；
Scale vectors F is updated using the new adaptive approach based on population diversityiSo that each element is according to such as
Under formula be updated：
Fi,j=1-divi,j(k) (11)
Wherein, Fi,jIt is scale vectors FiJ-th of element；SP represents the size of population；divi,j(k) represent in kth time
The normalized diversity of i-th of former generation's individual of iteration moment；Represent all former generation in kth time iteration moment population
The jth of body ties up multifarious average value；E represents that i-th of former generation's individual is average relative to all individuals in colony in current group
Multifarious difference；
After the mutation operation provided in Differential Evolution Algorithm according to formula (10)-(14) produces new individual, using standard
Intersection and mutation operation in differential evolution algorithm are updated to new individual；
Step 3: mobile work robot environmental modeling
Assuming that robot is operated in two-dimensional space, and the obstacle of certain amount is included in the environment of its work
Thing, a global coordinate system o-xy is initially set up, wherein st and ta represent starting and the final position of robot respectively, in o-xy
In, line segment st-ta is divided into n+1 sections by n navigation spots, and wherein n is a predefined constant parameter；Pass through these navigation spots
N bar vertical lines are drawn, obtain a series of line l1,l2,...,ln；Any one path ph=[st, p to be selected of robot1,...,pn,
Ta] randomly generated on these vertical lines；
Step 4: mobile robot path planning problem mathematical modeling
Assuming that use p0And pn+1The starting point st and final position ta of the robot described in step 4 are represented respectively, then road
Footpath planning problem is created as following mathematical modeling：
Wherein, JLAnd JsPath length and path security are represented respectively；w1And w2It is two weighting parameters, represents JLAnd Js
Relative importance；
Step 5: the mobile robot path planning framework based on hybrid particle swarm and Differential Evolution Algorithm
WithTo represent navigation spots p1,...pnY-coordinate, drilled as using hybrid particle swarm and difference
Change algorithm and treat the particle coding that optimized amount optimizes；
Each variable to be optimized is adjusted by following saturation mechanism
Wherein, width represents the width of working space；
Assuming that Nob barrier is given in the planning space of robot, particle m in HNTVPSO-RBSADE1Violation
Constraint degree can be calculated by formula below：
Assuming that representing the total number of particles that colony is included in HNTVPSO-RBSADE with NP, and useThe individual optimal solution location sets of the particle in kth time iterative process are represented, foundation is based on
The framework of HNTVPSO-RBSADE Algorithm for Solving robot path planning's problems is as follows：
Step1. an initial population is randomly generated in the planning space of robot, and obtains initial time algorithm
Global optimum position gbest and BP；
Step2. judge whether current iteration number reaches maximum iteration, if it is, jumping to Step14；It is no
Then, Step3 is jumped into；
Step3. row variation, intersection and selection operation are entered for each individual in BP；
Step4. the zoom factor of each individual in BP is updated according to formula (11)-(14)；
Step5. the speed of each particle in HNTVPSO-RBSADE is updated according to formula (1)；
Step6. the position of each particle in HNTVPSO-RBSADE is updated according to formula (2)；
Step7. the position of each particle in the saturation mechanism amendment HNTVPSO-RBSADE provided according to formula (18)；
Step8. the target function value of each particle in HNTVPSO-RBSADE is calculated according to formula (15)-(17)；
Step9. the violation constraint degree of each particle in HNTVPSO-RBSADE is calculated according to formula (19)；
Step10. three control parameters of each particle in HNTVPSO-RBSADE are updated according to formula (3)-(9)；
Step11. according to the personal best particle of each particle in the feasibility principle renewal HNTVPSO-RBSADE of solution
pbest；
Step12. the pbest of each particle is stored in BP according to the feasibility principle of solution；
Step13. according to the global optimum position of colony in the feasibility principle renewal HNTVPSO-RBSADE of solution
gbest；
Step14. end loop, and export gbest.
Further, in step 4, in order to cook up a point-to-point path, path length JLPass through formula below meter
Obtain：
Wherein, dis (pi,pi+1) represent navigation spots piAnd pi+1Between Euclidean distance；
The security performance J in pathsBeeline of the path relative to nearest threat source is represented, passes through formula below meter
Obtain：
Further, two weighting parameters w in mathematical modeling in step 41=w2=1.
Further, in step 5, particle m in HNTVPSO-RBSADE is calculated1Violation constraint degree after, using based on
The feasibility rule of solution, a more preferable path is selected in any two paths to be selected.
The present invention is in order to overcome the two of basic particle group algorithm typical defects, so as to improve it in robot path planning
In optimization efficiency, the present invention proposes this method, and the sports rule that particle is first according to defined in NTVPSO updates theirs
Speed and position.Then, evolved using RBSADE after iteration each time terminates the personal best particle of each particle, so as to
Algorithm is avoided to be absorbed in iteration stagnation too early.
In order to preferably adjust the global and local search capability of particle in HNTVPSO-RBSADE, the present invention exists
Three main control parameters that a kind of new adaptive rule removes more new particle are proposed in HNTVPSO.In addition, in order to improve
RBSADE search performance, the present invention proposes a kind of mutation operation operator based on sequence in RBSADE, while also exists
A kind of adaptive rule based on population diversity is proposed in RBSADE to adjust the zoom factor of algorithm.
Using the HNTVPSO-RBSADE algorithms of proposition, the present invention devises the calculation of a solution robot path planning
Method.In order to verify the validity on the problem is solved of put forward algorithm, by four simulation examples, the present invention compared for being carried
The method gone out and other four kinds of typical path planning algorithms.Obtained simulation result shows proposed algorithm in the superior of solution
Above and there is very strong competitiveness, while the calculating time of carried algorithm and the calculating time of other algorithms have comparativity.
Brief description of the drawings
Fig. 1 is robot two-dimensional working environmental modeling
Fig. 2 is the mobile robot global path planning algorithm flow chart based on hybrid particle swarm and Differential Evolution Algorithm.
Fig. 3 is the optimal trajectory that all algorithms search in four simulation examples
(a)-(d) optimal trajectory of the first to the 4th simulation example of difference
Fig. 4 is all algorithms iterativecurve corresponding to optimal path in four simulation examples
(a)-(d) iterativecurve corresponding to optimal path in the first to the 4th simulation example of difference
Embodiment
The invention will be further described with reference to the accompanying drawings and examples.
The mobile robot global path planning algorithm based on hybrid particle swarm of the present invention, be based on hybrid particle swarm and
The mobile robot global path planning algorithm of Differential Evolution Algorithm (HNTVPSO-RBSADE), its basic thought are as follows：
First, for the full algorithm of particle can not the global and local search capability of equilibrium particle well cause algorithm too early
The problem of being absorbed in local optimum, propose three control ginsengs of each particle in parameter adaptive rule de-regulation particle cluster algorithm
Number, form hybrid particle swarm (NTVPSO) algorithm.
On this basis, a kind of differential evolution (RBSADE) algorithm is proposed, after each iteration, uses differential evolution
(RBSADE) algorithm goes to evolve the personal best particle of each particle, HNTVPSO-RBSADE algorithms is formed, so as to avoid particle
Particle is absorbed in the defects of iteration is stagnated too early in algorithm.
Finally, for mobile robot path planning problem, establish optimal using the progress of HNTVPSO-RBSADE algorithms
The framework of solution.
Mobile robot global path planning algorithm based on hybrid particle swarm and Differential Evolution Algorithm, is specifically included following
Step：
Step 1: establish hybrid particle swarm (NTVPSO) algorithm
Particle is updated to its speed and position in such a way：
Wherein, ω represents the inertia weight coefficient of particle；c1Represent the cognition acceleration factor of particle；c2Represent particle
Social recognition acceleration factor；WithRepresent particle m in kth and the speed at k+1 iteration moment respectively；WithPoint
Not Biao Shi particle m in kth and the position at k+1 iteration moment；WithIt is illustrated respectively in kth time iteration moment grain
Sub- m personal best particle and the global optimum position of colony；
ω、c1、c2It is three main control parameters for influenceing algorithm performance, these three is joined using following adaptive rule
Number is updated：
Wherein：
Here, ωmaxAnd ωminRepresent ω upper and lower bound；c1sAnd c1fRepresent c1Initial value and end value；c2sWith
c2fRepresent c2Initial value and end value；kmaxRepresent maximum iteration；WithRepresent in kth -1 and kth iteration
Two norms of moment particle m velocity；Δ is to prevent βmA denominator minimum normal number being changed into 0 and introduce,
Δ=1e-25 in the present invention；ω is required to meet in algorithmmax>ωmin, c1s>c1fAnd c2s<c2f。
Step 2: establish differential evolution (RBSADE) algorithm
Mutation operation, crossover operation and selection operation are all evolutional operations, new to producing in differential evolution (DE) algorithm
Filial generation play key effect.
Mutation operation produces new according to the following formula by changing the gene information of three different former generation at random
Body：
Wherein,WithIt is from i ≠ i1≠i2≠i3Current population in randomly selected individual, Fi
The zoom factor of DE algorithms is represented, is the key for influenceing individual renewing speciality.
The present invention proposes a kind of new adaptive approach based on population diversity to update scale vectors FiSo that it is each
Element is updated according to formula below：
Fi,j=1-divi,j(k) (11)
Wherein, Fi,jIt is scale vectors FiJ-th of element；SP represents the size of population；divi,j(k) represent in kth time
The normalized diversity of i-th of former generation's individual of iteration moment；Represent all former generation's individuals in kth time iteration moment population
Jth tie up multifarious average value；E represents that i-th of former generation's individual is average more relative to all individuals in colony in current group
The difference of sample.
After the mutation operation provided in RBSADE according to formula (10)-(14) produces new individual, using standard DE algorithms
In intersection and mutation operation this new individual is updated.
Step 3: mobile work robot environmental modeling
Assuming that robot is operated in two-dimensional space, and the obstacle of certain amount is included in the environment of its work
Thing, as shown in Figure 1.A global coordinate system o-xy is initially set up, wherein st and ta represent starting and the terminal of robot respectively
Position.In o-xy, line segment st-ta is divided into n+1 sections by n navigation spots, and wherein n is a predefined constant parameter.It is logical
Cross these navigation spots and draw n bar vertical lines, obtain a series of line l1,l2,...,ln.Any one path ph=to be selected of robot
[st,p1,...,pn, ta] randomly generated on these vertical lines.
Step 4: mobile robot path planning problem mathematical modeling
Two performance indications to be optimized of security of the main length and path for considering path.Assuming that use p0And pn+1Point
The starting point st and final position ta of robot that Biao Shi be described in step 4, the then path planning studied in the present invention are asked
Topic can be created as following mathematical modeling：
Wherein, JLAnd JsPath length and path security are represented respectively；w1And w2It is two weighting parameters, represents JLAnd Js
Relative importance, w in the present invention1=w2=1.
In order to cook up a point-to-point path, path length JLIt can be calculated by formula below：
Wherein, dis (pi,pi+1) represent navigation spots piAnd pi+1Between Euclidean distance.
The security performance J in pathsBeeline of the path relative to nearest threat source is represented, passes through formula below meter
Obtain：
Step 5: the mobile robot path planning framework based on HNTVPSO-RBSADE algorithms
As carried in step 3, when drawing l in the planning space of robot according to the method for the step introduction1,
l2,...,lnAfter these vertical lines, it may be determined that the x coordinate of these vertical lines.Therefore, what is randomly generated on these vertical lines treats routing
The navigation spots p in footpath1,...pnX coordinate just determine therewith.So going to solve the road of robot using HNTVPSO-RBSADE
During the planning problem of footpath, useTo represent navigation spots p1,...pnY-coordinate, as using HNTVPSO-RBSADE
Algorithm treats the particle coding that optimized amount optimizes.
In order to ensure each particle removal search path in the planning space of robot in HNTVPSO-RBSADE, pass through
Following saturation mechanism adjusts each variable to be optimized
Wherein, width represents the width of working space；
Assuming that Nob barrier is given in the planning space of robot, grain in hybrid particle swarm and Differential Evolution Algorithm
Sub- m1Violation constraint degree be calculated by formula below：
After the violation constraint degree of each particle has been calculated, using the feasibility rule based on solution, at any two
A more preferable path is selected in path to be selected；For minimum optimization problem, the feasibility rule of solution is：(1) for appointing
Meaning two has the solution to be selected of identical violation constraint degree, selects the less solution of cost function；(2) have not for any two
With the solution to be selected for violating constraint degree, the small solution of constraint degree is violated in selection；
Assuming that representing the total number of particles that colony is included in hybrid particle swarm and Differential Evolution Algorithm with NP, and useThe individual optimal solution location sets of the particle in kth time iterative process are represented, are established based on mixed
The framework for closing population and Differential Evolution Algorithm solution robot path planning's problem is as follows：
Step1. an initial population is randomly generated in the planning space of robot, and obtains initial time algorithm
Gbest and BP；
Step2. judge whether current iteration number reaches maximum iteration, if it is, jumping to Step14；It is no
Then, Step3 is jumped into；
Step3. row variation, intersection and selection operation are entered for each individual in BP；
Step4. the zoom factor of each individual in BP is updated according to formula (11)-(14)；
Step5. the speed of each particle in HNTVPSO-RBSADE is updated according to formula (1)；
Step6. the position of each particle in HNTVPSO-RBSADE is updated according to formula (2)；
Step7. the position of each particle in the saturation mechanism amendment HNTVPSO-RBSADE provided according to formula (18)；
Step8. the target function value of each particle in HNTVPSO-RBSADE is calculated according to formula (15)-(17)；
Step9. the violation constraint degree of each particle in HNTVPSO-RBSADE is calculated according to formula (19)；
Step10. three control parameters of each particle in HNTVPSO-RBSADE are updated according to formula (3)-(9)；
Step11. according to the personal best particle of each particle in the feasibility principle renewal HNTVPSO-RBSADE of solution
pbest；
Step12. the pbest of each particle is stored in BP according to the feasibility principle of solution；
Step13. according to the global optimum position of colony in the feasibility principle renewal HNTVPSO-RBSADE of solution
gbest；
Step14. end loop, and export gbest.
Embodiment
In order to verify proposed method, by four simulation examples, by itself and JADE, TVPSO, GS and mGA algorithms enter
Row compares.In each simulation example, the simulation parameter of every kind of algorithm is as shown in table 1.Meanwhile in each simulation example, often
The group size and maximum iteration of kind algorithm are respectively set to 40 and 200 generations.In order to strictly be compared, the present invention will
The superiority in the path searched of every kind of algorithm and calculating time are contrasted.It is imitative at this four that Fig. 2 illustrates all algorithms
The optimal trajectory searched in true example.Fig. 3 shows every kind of method iteration corresponding to optimal path in each simulation example
Curve.Table 2-5 have recorded the Numerical Simulation Results obtained by every kind of algorithm in this four numerical simulations respectively.
Table 1
Table 2
Table 3
Table 4
Table 5
Above content is further description made for the present invention, it is impossible to assert the embodiment of the present invention only
It is limited to this, for general technical staff of the technical field of the invention, without departing from the inventive concept of the premise, also
Some simple deduction or replace can be made, the present invention should be all considered as belonging to and determine invention by the claims submitted
Protection domain.
Claims (4)
1. the mobile robot global path planning algorithm based on hybrid particle swarm, it is characterised in that comprise the following steps：
Step 1: establish Hybrid Particle Swarm
Particle is updated to its speed and position in such a way：
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</mrow>
</msub>
<mo>-</mo>
<msub>
<mi>c</mi>
<mrow>
<mn>1</mn>
<mi>f</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi>exp</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>&delta;</mi>
<mrow>
<mi>c</mi>
<mn>1</mn>
</mrow>
</msub>
<mo>&CenterDot;</mo>
<msub>
<mi>&beta;</mi>
<mi>m</mi>
</msub>
<mo>&CenterDot;</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<msub>
<mi>c</mi>
<mrow>
<mn>1</mn>
<mi>f</mi>
</mrow>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msub>
<mi>c</mi>
<mn>2</mn>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<msub>
<mi>c</mi>
<mrow>
<mn>2</mn>
<mi>s</mi>
</mrow>
</msub>
<mo>-</mo>
<msub>
<mi>c</mi>
<mrow>
<mn>2</mn>
<mi>f</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi>exp</mi>
<mrow>
<mo>(</mo>
<mo>-</mo>
<msub>
<mi>&delta;</mi>
<mrow>
<mi>c</mi>
<mn>2</mn>
</mrow>
</msub>
<mo>&CenterDot;</mo>
<msub>
<mi>&beta;</mi>
<mi>m</mi>
</msub>
<mo>&CenterDot;</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<msub>
<mi>c</mi>
<mrow>
<mn>2</mn>
<mi>f</mi>
</mrow>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein：
<mrow>
<msub>
<mi>&delta;</mi>
<mi>m</mi>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<msub>
<mi>&omega;</mi>
<mrow>
<mi>m</mi>
<mi>a</mi>
<mi>x</mi>
</mrow>
</msub>
<mo>-</mo>
<msub>
<mi>&omega;</mi>
<mrow>
<mi>m</mi>
<mi>i</mi>
<mi>n</mi>
</mrow>
</msub>
</mrow>
<msub>
<mi>k</mi>
<mrow>
<mi>m</mi>
<mi>a</mi>
<mi>x</mi>
</mrow>
</msub>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msub>
<mi>&delta;</mi>
<mrow>
<mi>c</mi>
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<msub>
<mi>c</mi>
<mrow>
<mn>1</mn>
<mi>s</mi>
</mrow>
</msub>
<mo>-</mo>
<msub>
<mi>c</mi>
<mrow>
<mn>1</mn>
<mi>f</mi>
</mrow>
</msub>
</mrow>
<msub>
<mi>k</mi>
<mrow>
<mi>m</mi>
<mi>a</mi>
<mi>x</mi>
</mrow>
</msub>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msub>
<mi>&delta;</mi>
<mrow>
<mi>c</mi>
<mn>2</mn>
</mrow>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<msub>
<mi>c</mi>
<mrow>
<mn>2</mn>
<mi>s</mi>
</mrow>
</msub>
<mo>-</mo>
<msub>
<mi>c</mi>
<mrow>
<mn>2</mn>
<mi>f</mi>
</mrow>
</msub>
</mrow>
<msub>
<mi>k</mi>
<mrow>
<mi>m</mi>
<mi>a</mi>
<mi>x</mi>
</mrow>
</msub>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>8</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msub>
<mi>&beta;</mi>
<mi>m</mi>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<mo>|</mo>
<mo>|</mo>
<msubsup>
<mi>V</mi>
<mi>m</mi>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>|</mo>
<mo>|</mo>
<mo>+</mo>
<mo>|</mo>
<mo>|</mo>
<msubsup>
<mi>V</mi>
<mi>m</mi>
<mi>k</mi>
</msubsup>
<mo>|</mo>
<mo>|</mo>
</mrow>
<mrow>
<mn>2</mn>
<mo>|</mo>
<mo>|</mo>
<msubsup>
<mi>V</mi>
<mi>m</mi>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>|</mo>
<mo>|</mo>
<mo>+</mo>
<mi>&Delta;</mi>
</mrow>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>9</mn>
<mo>)</mo>
</mrow>
</mrow>
ωmaxAnd ωminRepresent ω upper and lower bound；c1sAnd c1fRepresent c1Initial value and end value；c2sAnd c2fRepresent c2's
Initial value and end value；kmaxRepresent maximum iteration；WithRepresent in kth -1 and kth iteration moment particle m speed
Spend two norms of vector；Δ is to prevent βmA denominator minimum normal number being changed into 0 and introduce, Δ=1e-25；Calculate
ω is required to meet in methodmax>ωmin, c1s>c1fAnd c2s<c2f；
Step 2: establish Differential Evolution Algorithm
Mutation operation produces a new individual according to the following formula by changing the gene information of three different former generation at random：
<mrow>
<msub>
<mi>v</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>x</mi>
<msub>
<mi>i</mi>
<mn>3</mn>
</msub>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>F</mi>
<mi>i</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>x</mi>
<msub>
<mi>i</mi>
<mn>1</mn>
</msub>
</msub>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
<mo>-</mo>
<msub>
<mi>x</mi>
<msub>
<mi>i</mi>
<mn>2</mn>
</msub>
</msub>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>10</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, WithIt is from i ≠ i1≠i2≠i3Current population in randomly selected individual, FiIt is poor to represent
Divide the zoom factor of evolution algorithm；
Scale vectors F is updated using the new adaptive approach based on population diversityiSo that each element is according to following
Formula is updated：
Fi,j=1-divi,j(k) (11)
<mrow>
<msub>
<mi>div</mi>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>S</mi>
<mi>P</mi>
</mrow>
</munderover>
<msqrt>
<mi>E</mi>
</msqrt>
</mrow>
<mrow>
<mi>S</mi>
<mi>P</mi>
<mo>&CenterDot;</mo>
<mi>m</mi>
<mi>a</mi>
<mi>x</mi>
<mrow>
<mo>(</mo>
<msqrt>
<mi>E</mi>
</msqrt>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>12</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>E</mi>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>D</mi>
</munderover>
<msup>
<mrow>
<mo>&lsqb;</mo>
<msub>
<mi>x</mi>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msub>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>13</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msub>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>S</mi>
<mi>P</mi>
</mrow>
</mfrac>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>S</mi>
<mi>P</mi>
</mrow>
</munderover>
<msub>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>k</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>14</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, Fi,jIt is scale vectors FiJ-th of element；SP represents the size of population；divi,j(k) represent in kth time iteration
Carve the normalized diversity of i-th of former generation's individual；Represent the jth of all former generation's individuals in kth time iteration moment population
Tie up multifarious average value；E represents that i-th of former generation's individual is average multifarious relative to all individuals in colony in current group
Difference；
After the mutation operation provided in Differential Evolution Algorithm according to formula (10)-(14) produces new individual, using standard difference
Intersection and mutation operation in evolution algorithm are updated to new individual；
Step 3: mobile work robot environmental modeling
Assuming that robot is operated in two-dimensional space, and includes the barrier of certain amount in the environment of its work, it is first
A global coordinate system o-xy is first established, wherein st and ta represent starting and the final position of robot, in o-xy, line respectively
Section st-ta is divided into n+1 sections by n navigation spots, and wherein n is a predefined constant parameter；N is drawn by these navigation spots
Bar vertical line, obtain a series of line l1,l2,...,ln；Any one path ph=[st, p to be selected of robot1,...,pn, ta]
Randomly generated on these vertical lines；
Step 4: mobile robot path planning problem mathematical modeling
Assuming that use p0And pn+1The starting point st and final position ta of the robot described in step 4 are represented respectively, then path planning
Problem is created as following mathematical modeling：
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>f</mi>
<mi>i</mi>
<mi>n</mi>
<mi>d</mi>
<mi>i</mi>
<mi>n</mi>
<mi>g</mi>
<mo>:</mo>
<mi>p</mi>
<mi>h</mi>
<mo>=</mo>
<mo>&lsqb;</mo>
<mi>s</mi>
<mi>t</mi>
<mo>=</mo>
<msub>
<mi>p</mi>
<mn>0</mn>
</msub>
<mo>,</mo>
<msub>
<mi>p</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>p</mi>
<mn>2</mn>
</msub>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<msub>
<mi>p</mi>
<mi>n</mi>
</msub>
<mo>,</mo>
<mi>t</mi>
<mi>a</mi>
<mo>=</mo>
<msub>
<mi>p</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>&rsqb;</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>min</mi>
<mi>i</mi>
<mi>m</mi>
<mi>i</mi>
<mi>z</mi>
<mi>e</mi>
<mo>:</mo>
<msub>
<mi>J</mi>
<mrow>
<mi>cos</mi>
<mi>t</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>w</mi>
<mn>1</mn>
</msub>
<mo>&CenterDot;</mo>
<msub>
<mi>J</mi>
<mi>L</mi>
</msub>
<mo>+</mo>
<msub>
<mi>w</mi>
<mn>2</mn>
</msub>
<mo>&CenterDot;</mo>
<mfrac>
<mn>1</mn>
<msub>
<mi>J</mi>
<mi>s</mi>
</msub>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>s</mi>
<mo>.</mo>
<mi>t</mi>
<mo>.</mo>
<msub>
<mi>p</mi>
<mi>i</mi>
</msub>
<msub>
<mi>p</mi>
<mrow>
<mi>i</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>&Element;</mo>
<mi>s</mi>
<mi>e</mi>
<mi>m</mi>
<mi>i</mi>
<mo>-</mo>
<mi>f</mi>
<mi>r</mi>
<mi>e</mi>
<mi>e</mi>
<mi> </mi>
<mi>w</mi>
<mi>o</mi>
<mi>r</mi>
<mi>k</mi>
<mi>s</mi>
<mi>p</mi>
<mi>a</mi>
<mi>c</mi>
<mi>e</mi>
<mo>,</mo>
<mn>0</mn>
<mo>&le;</mo>
<mi>i</mi>
<mo>&le;</mo>
<mi>n</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>15</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, JLAnd JsPath length and path security are represented respectively；w1And w2It is two weighting parameters, represents JLAnd JsPhase
To importance；
Step 5: the mobile robot path planning framework based on hybrid particle swarm and Differential Evolution Algorithm
WithTo represent navigation spots p1,...pnY-coordinate, calculated as using hybrid particle swarm and differential evolution
Method treats the particle coding that optimized amount optimizes；
Each variable to be optimized is adjusted by following saturation mechanism
<mrow>
<msub>
<mi>y</mi>
<msub>
<mi>p</mi>
<mi>i</mi>
</msub>
</msub>
<mo>=</mo>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mi>w</mi>
<mi>i</mi>
<mi>d</mi>
<mi>t</mi>
<mi>h</mi>
</mrow>
<mn>2</mn>
</mfrac>
<mo>,</mo>
<mi>i</mi>
<mi>f</mi>
<mi> </mi>
<msub>
<mi>y</mi>
<msub>
<mi>p</mi>
<mi>i</mi>
</msub>
</msub>
<mo>></mo>
<mfrac>
<mrow>
<mi>w</mi>
<mi>i</mi>
<mi>d</mi>
<mi>t</mi>
<mi>h</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>-</mo>
<mfrac>
<mrow>
<mi>w</mi>
<mi>i</mi>
<mi>d</mi>
<mi>t</mi>
<mi>h</mi>
</mrow>
<mn>2</mn>
</mfrac>
<mo>,</mo>
<mi>i</mi>
<mi>f</mi>
<mi> </mi>
<msub>
<mi>y</mi>
<msub>
<mi>p</mi>
<mi>i</mi>
</msub>
</msub>
<mo><</mo>
<mfrac>
<mrow>
<mi>w</mi>
<mi>i</mi>
<mi>d</mi>
<mi>t</mi>
<mi>h</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>y</mi>
<msub>
<mi>p</mi>
<mi>i</mi>
</msub>
</msub>
<mo>,</mo>
<mi>o</mi>
<mi>t</mi>
<mi>h</mi>
<mi>e</mi>
<mi>r</mi>
<mi>w</mi>
<mi>i</mi>
<mi>s</mi>
<mi>e</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>18</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, width represents the width of working space；
Assuming that Nob barrier is given in the planning space of robot, particle m in HNTVPSO-RBSADE1Violation constraint degree
It can be calculated by formula below：
<mrow>
<msub>
<mi>violation</mi>
<msub>
<mi>m</mi>
<mn>1</mn>
</msub>
</msub>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>N</mi>
<mi>o</mi>
<mi>b</mi>
</mrow>
</mfrac>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>N</mi>
<mi>o</mi>
<mi>b</mi>
</mrow>
</munderover>
<msub>
<mi>viol</mi>
<mrow>
<msub>
<mi>m</mi>
<mn>1</mn>
</msub>
<mi>j</mi>
</mrow>
</msub>
</mrow>
<mrow>
<msub>
<mi>viol</mi>
<mrow>
<msub>
<mi>m</mi>
<mn>1</mn>
</msub>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mi>i</mi>
<mi>f</mi>
<mi> </mi>
<msub>
<mi>m</mi>
<mn>1</mn>
</msub>
<mi>c</mi>
<mi>o</mi>
<mi>l</mi>
<mi>l</mi>
<mi>i</mi>
<mi>d</mi>
<mi>e</mi>
<mi>s</mi>
<mi> </mi>
<mi>w</mi>
<mi>i</mi>
<mi>t</mi>
<mi>h</mi>
<mi> </mi>
<mi>o</mi>
<mi>b</mi>
<mi>s</mi>
<mi>t</mi>
<mi>a</mi>
<mi>c</mi>
<mi>l</mi>
<mi>e</mi>
<mi> </mi>
<mi>j</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mi>o</mi>
<mi>t</mi>
<mi>h</mi>
<mi>e</mi>
<mi>r</mi>
<mi>w</mi>
<mi>i</mi>
<mi>s</mi>
<mi>e</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>19</mn>
<mo>)</mo>
</mrow>
</mrow>
Assuming that representing the total number of particles that colony is included in HNTVPSO-RBSADE with NP, and useThe individual optimal solution location sets of the particle in kth time iterative process are represented, foundation is based on
The framework of HNTVPSO-RBSADE Algorithm for Solving robot path planning's problems is as follows：
Step1. an initial population is randomly generated in the planning space of robot, and obtains the overall situation of initial time algorithm
Optimal location gbest and BP；
Step2. judge whether current iteration number reaches maximum iteration, if it is, jumping to Step14；Otherwise, jump
Enter Step3；
Step3. row variation, intersection and selection operation are entered for each individual in BP；
Step4. the zoom factor of each individual in BP is updated according to formula (11)-(14)；
Step5. the speed of each particle in HNTVPSO-RBSADE is updated according to formula (1)；
Step6. the position of each particle in HNTVPSO-RBSADE is updated according to formula (2)；
Step7. the position of each particle in the saturation mechanism amendment HNTVPSO-RBSADE provided according to formula (18)；
Step8. the target function value of each particle in HNTVPSO-RBSADE is calculated according to formula (15)-(17)；
Step9. the violation constraint degree of each particle in HNTVPSO-RBSADE is calculated according to formula (19)；
Step10. three control parameters of each particle in HNTVPSO-RBSADE are updated according to formula (3)-(9)；
Step11. according to the personal best particle pbest of each particle in the feasibility principle renewal HNTVPSO-RBSADE of solution；
Step12. the pbest of each particle is stored in BP according to the feasibility principle of solution；
Step13. according to the global optimum position gbest of colony in the feasibility principle renewal HNTVPSO-RBSADE of solution；
Step14. end loop, and export gbest.
2. the mobile robot global path planning algorithm according to claim 1 based on hybrid particle swarm, its feature exist
In：In step 4, in order to cook up a point-to-point path, path length JLIt is calculated by formula below：
<mrow>
<msub>
<mi>J</mi>
<mi>L</mi>
</msub>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mi>n</mi>
</munderover>
<mi>d</mi>
<mi>i</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>p</mi>
<mi>i</mi>
</msub>
<mo>,</mo>
<msub>
<mi>p</mi>
<mrow>
<mi>i</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>16</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, dis (pi,pi+1) represent navigation spots piAnd pi+1Between Euclidean distance；
The security performance J in pathsBeeline of the path relative to nearest threat source is represented, is calculated by formula below
Arrive：
<mrow>
<msub>
<mi>J</mi>
<mi>s</mi>
</msub>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mi>n</mi>
</munderover>
<munder>
<mrow>
<mi>m</mi>
<mi>i</mi>
<mi>n</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>&le;</mo>
<mi>j</mi>
<mo>&le;</mo>
<mi>N</mi>
<mi>o</mi>
<mi>b</mi>
</mrow>
</munder>
<mo>{</mo>
<mi>M</mi>
<mi>i</mi>
<mi>n</mi>
<mi>d</mi>
<mi>i</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>p</mi>
<mi>i</mi>
</msub>
<msub>
<mi>p</mi>
<mrow>
<mi>i</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>O</mi>
<mi>j</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>}</mo>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>17</mn>
<mo>)</mo>
</mrow>
<mo>.</mo>
</mrow>
3. the mobile robot global path planning algorithm according to claim 1 based on hybrid particle swarm, its feature exist
In：Two weighting parameters w in mathematical modeling in step 41=w2=1.
4. the mobile robot global path planning algorithm according to claim 1 based on hybrid particle swarm, its feature exist
In：In step 5, particle m in HNTVPSO-RBSADE is calculated1Violation constraint degree after, using the feasibility method based on solution
Then, a more preferable path is selected in any two paths to be selected.
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